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3.41 \(\int (a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} (A+B x+C x^2) \, dx\)

Optimal. Leaf size=1348 \[ \frac {C (c+d x)^{3/2} (e+f x)^{3/2} (a+b x)^3}{6 b d f}-\frac {(2 a C d f-b (4 B d f-3 C (d e+c f))) (c+d x)^{3/2} (e+f x)^{3/2} (a+b x)^2}{20 b d^2 f^2}-\frac {(c+d x)^{3/2} (e+f x)^{3/2} \left (\left (7 C \left (15 d^3 e^3+17 c d^2 f e^2+17 c^2 d f^2 e+15 c^3 f^3\right )+4 d f \left (50 A d f (d e+c f)-B \left (35 d^2 e^2+38 c d f e+35 c^2 f^2\right )\right )\right ) b^3-8 a d f \left (C \left (35 d^2 e^2+38 c d f e+35 c^2 f^2\right )+10 d f (8 A d f-5 B (d e+c f))\right ) b^2-8 a^2 d^2 f^2 (16 B d f-7 C (d e+c f)) b+6 d f (10 b d f (2 b c C e+a C d e+a c C f-4 A b d f)+(4 a d f-7 b (d e+c f)) (2 a C d f-b (4 B d f-3 C (d e+c f)))) x b+64 a^3 C d^3 f^3\right )}{960 b d^4 f^4}-\frac {(d e-c f)^2 \left (\left (C \left (21 d^4 e^4+28 c d^3 f e^3+30 c^2 d^2 f^2 e^2+28 c^3 d f^3 e+21 c^4 f^4\right )+4 d f \left (2 A d f \left (5 d^2 e^2+6 c d f e+5 c^2 f^2\right )-B \left (7 d^3 e^3+9 c d^2 f e^2+9 c^2 d f^2 e+7 c^3 f^3\right )\right )\right ) b^2-8 a d f \left (C \left (7 d^3 e^3+9 c d^2 f e^2+9 c^2 d f^2 e+7 c^3 f^3\right )+2 d f \left (8 A d f (d e+c f)-B \left (5 d^2 e^2+6 c d f e+5 c^2 f^2\right )\right )\right ) b+8 a^2 d^2 f^2 \left (C \left (5 d^2 e^2+6 c d f e+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d} \sqrt {e+f x}}\right )}{512 d^{11/2} f^{11/2}}+\frac {\left (\left (C \left (21 d^4 e^4+28 c d^3 f e^3+30 c^2 d^2 f^2 e^2+28 c^3 d f^3 e+21 c^4 f^4\right )+4 d f \left (2 A d f \left (5 d^2 e^2+6 c d f e+5 c^2 f^2\right )-B \left (7 d^3 e^3+9 c d^2 f e^2+9 c^2 d f^2 e+7 c^3 f^3\right )\right )\right ) b^2-8 a d f \left (C \left (7 d^3 e^3+9 c d^2 f e^2+9 c^2 d f^2 e+7 c^3 f^3\right )+2 d f \left (8 A d f (d e+c f)-B \left (5 d^2 e^2+6 c d f e+5 c^2 f^2\right )\right )\right ) b+8 a^2 d^2 f^2 \left (C \left (5 d^2 e^2+6 c d f e+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right )\right ) (c+d x)^{3/2} \sqrt {e+f x}}{256 d^5 f^4}+\frac {(d e-c f) \left (\left (C \left (21 d^4 e^4+28 c d^3 f e^3+30 c^2 d^2 f^2 e^2+28 c^3 d f^3 e+21 c^4 f^4\right )+4 d f \left (2 A d f \left (5 d^2 e^2+6 c d f e+5 c^2 f^2\right )-B \left (7 d^3 e^3+9 c d^2 f e^2+9 c^2 d f^2 e+7 c^3 f^3\right )\right )\right ) b^2-8 a d f \left (C \left (7 d^3 e^3+9 c d^2 f e^2+9 c^2 d f^2 e+7 c^3 f^3\right )+2 d f \left (8 A d f (d e+c f)-B \left (5 d^2 e^2+6 c d f e+5 c^2 f^2\right )\right )\right ) b+8 a^2 d^2 f^2 \left (C \left (5 d^2 e^2+6 c d f e+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{512 d^5 f^5} \]

[Out]

-1/20*(2*a*C*d*f-b*(4*B*d*f-3*C*(c*f+d*e)))*(b*x+a)^2*(d*x+c)^(3/2)*(f*x+e)^(3/2)/b/d^2/f^2+1/6*C*(b*x+a)^3*(d
*x+c)^(3/2)*(f*x+e)^(3/2)/b/d/f-1/960*(d*x+c)^(3/2)*(f*x+e)^(3/2)*(64*a^3*C*d^3*f^3-8*a^2*b*d^2*f^2*(16*B*d*f-
7*C*(c*f+d*e))-8*a*b^2*d*f*(C*(35*c^2*f^2+38*c*d*e*f+35*d^2*e^2)+10*d*f*(8*A*d*f-5*B*(c*f+d*e)))+b^3*(7*C*(15*
c^3*f^3+17*c^2*d*e*f^2+17*c*d^2*e^2*f+15*d^3*e^3)+4*d*f*(50*A*d*f*(c*f+d*e)-B*(35*c^2*f^2+38*c*d*e*f+35*d^2*e^
2)))+6*b*d*f*(10*b*d*f*(-4*A*b*d*f+C*a*c*f+C*a*d*e+2*C*b*c*e)+(4*a*d*f-7*b*(c*f+d*e))*(2*a*C*d*f-b*(4*B*d*f-3*
C*(c*f+d*e))))*x)/b/d^4/f^4-1/512*(-c*f+d*e)^2*(8*a^2*d^2*f^2*(C*(5*c^2*f^2+6*c*d*e*f+5*d^2*e^2)+8*d*f*(2*A*d*
f-B*(c*f+d*e)))-8*a*b*d*f*(C*(7*c^3*f^3+9*c^2*d*e*f^2+9*c*d^2*e^2*f+7*d^3*e^3)+2*d*f*(8*A*d*f*(c*f+d*e)-B*(5*c
^2*f^2+6*c*d*e*f+5*d^2*e^2)))+b^2*(C*(21*c^4*f^4+28*c^3*d*e*f^3+30*c^2*d^2*e^2*f^2+28*c*d^3*e^3*f+21*d^4*e^4)+
4*d*f*(2*A*d*f*(5*c^2*f^2+6*c*d*e*f+5*d^2*e^2)-B*(7*c^3*f^3+9*c^2*d*e*f^2+9*c*d^2*e^2*f+7*d^3*e^3))))*arctanh(
f^(1/2)*(d*x+c)^(1/2)/d^(1/2)/(f*x+e)^(1/2))/d^(11/2)/f^(11/2)+1/256*(8*a^2*d^2*f^2*(C*(5*c^2*f^2+6*c*d*e*f+5*
d^2*e^2)+8*d*f*(2*A*d*f-B*(c*f+d*e)))-8*a*b*d*f*(C*(7*c^3*f^3+9*c^2*d*e*f^2+9*c*d^2*e^2*f+7*d^3*e^3)+2*d*f*(8*
A*d*f*(c*f+d*e)-B*(5*c^2*f^2+6*c*d*e*f+5*d^2*e^2)))+b^2*(C*(21*c^4*f^4+28*c^3*d*e*f^3+30*c^2*d^2*e^2*f^2+28*c*
d^3*e^3*f+21*d^4*e^4)+4*d*f*(2*A*d*f*(5*c^2*f^2+6*c*d*e*f+5*d^2*e^2)-B*(7*c^3*f^3+9*c^2*d*e*f^2+9*c*d^2*e^2*f+
7*d^3*e^3))))*(d*x+c)^(3/2)*(f*x+e)^(1/2)/d^5/f^4+1/512*(-c*f+d*e)*(8*a^2*d^2*f^2*(C*(5*c^2*f^2+6*c*d*e*f+5*d^
2*e^2)+8*d*f*(2*A*d*f-B*(c*f+d*e)))-8*a*b*d*f*(C*(7*c^3*f^3+9*c^2*d*e*f^2+9*c*d^2*e^2*f+7*d^3*e^3)+2*d*f*(8*A*
d*f*(c*f+d*e)-B*(5*c^2*f^2+6*c*d*e*f+5*d^2*e^2)))+b^2*(C*(21*c^4*f^4+28*c^3*d*e*f^3+30*c^2*d^2*e^2*f^2+28*c*d^
3*e^3*f+21*d^4*e^4)+4*d*f*(2*A*d*f*(5*c^2*f^2+6*c*d*e*f+5*d^2*e^2)-B*(7*c^3*f^3+9*c^2*d*e*f^2+9*c*d^2*e^2*f+7*
d^3*e^3))))*(d*x+c)^(1/2)*(f*x+e)^(1/2)/d^5/f^5

________________________________________________________________________________________

Rubi [A]  time = 2.37, antiderivative size = 1345, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {1615, 153, 147, 50, 63, 217, 206} \[ \frac {C (c+d x)^{3/2} (e+f x)^{3/2} (a+b x)^3}{6 b d f}+\frac {(4 b B d f-2 a C d f-3 b C (d e+c f)) (c+d x)^{3/2} (e+f x)^{3/2} (a+b x)^2}{20 b d^2 f^2}-\frac {(c+d x)^{3/2} (e+f x)^{3/2} \left (\left (7 C \left (15 d^3 e^3+17 c d^2 f e^2+17 c^2 d f^2 e+15 c^3 f^3\right )+4 d f \left (50 A d f (d e+c f)-B \left (35 d^2 e^2+38 c d f e+35 c^2 f^2\right )\right )\right ) b^3-8 a d f \left (C \left (35 d^2 e^2+38 c d f e+35 c^2 f^2\right )+10 d f (8 A d f-5 B (d e+c f))\right ) b^2-8 a^2 d^2 f^2 (16 B d f-7 C (d e+c f)) b+6 d f (10 b d f (2 b c C e+a C d e+a c C f-4 A b d f)-(4 a d f-7 b (d e+c f)) (4 b B d f-2 a C d f-3 b C (d e+c f))) x b+64 a^3 C d^3 f^3\right )}{960 b d^4 f^4}-\frac {(d e-c f)^2 \left (\left (C \left (21 d^4 e^4+28 c d^3 f e^3+30 c^2 d^2 f^2 e^2+28 c^3 d f^3 e+21 c^4 f^4\right )+4 d f \left (2 A d f \left (5 d^2 e^2+6 c d f e+5 c^2 f^2\right )-B \left (7 d^3 e^3+9 c d^2 f e^2+9 c^2 d f^2 e+7 c^3 f^3\right )\right )\right ) b^2-8 a d f \left (C \left (7 d^3 e^3+9 c d^2 f e^2+9 c^2 d f^2 e+7 c^3 f^3\right )+2 d f \left (8 A d f (d e+c f)-B \left (5 d^2 e^2+6 c d f e+5 c^2 f^2\right )\right )\right ) b+8 a^2 d^2 f^2 \left (C \left (5 d^2 e^2+6 c d f e+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d} \sqrt {e+f x}}\right )}{512 d^{11/2} f^{11/2}}+\frac {\left (\left (C \left (21 d^4 e^4+28 c d^3 f e^3+30 c^2 d^2 f^2 e^2+28 c^3 d f^3 e+21 c^4 f^4\right )+4 d f \left (2 A d f \left (5 d^2 e^2+6 c d f e+5 c^2 f^2\right )-B \left (7 d^3 e^3+9 c d^2 f e^2+9 c^2 d f^2 e+7 c^3 f^3\right )\right )\right ) b^2-8 a d f \left (C \left (7 d^3 e^3+9 c d^2 f e^2+9 c^2 d f^2 e+7 c^3 f^3\right )+2 d f \left (8 A d f (d e+c f)-B \left (5 d^2 e^2+6 c d f e+5 c^2 f^2\right )\right )\right ) b+8 a^2 d^2 f^2 \left (C \left (5 d^2 e^2+6 c d f e+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right )\right ) (c+d x)^{3/2} \sqrt {e+f x}}{256 d^5 f^4}+\frac {(d e-c f) \left (\left (C \left (21 d^4 e^4+28 c d^3 f e^3+30 c^2 d^2 f^2 e^2+28 c^3 d f^3 e+21 c^4 f^4\right )+4 d f \left (2 A d f \left (5 d^2 e^2+6 c d f e+5 c^2 f^2\right )-B \left (7 d^3 e^3+9 c d^2 f e^2+9 c^2 d f^2 e+7 c^3 f^3\right )\right )\right ) b^2-8 a d f \left (C \left (7 d^3 e^3+9 c d^2 f e^2+9 c^2 d f^2 e+7 c^3 f^3\right )+2 d f \left (8 A d f (d e+c f)-B \left (5 d^2 e^2+6 c d f e+5 c^2 f^2\right )\right )\right ) b+8 a^2 d^2 f^2 \left (C \left (5 d^2 e^2+6 c d f e+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{512 d^5 f^5} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^2*Sqrt[c + d*x]*Sqrt[e + f*x]*(A + B*x + C*x^2),x]

[Out]

((d*e - c*f)*(8*a^2*d^2*f^2*(C*(5*d^2*e^2 + 6*c*d*e*f + 5*c^2*f^2) + 8*d*f*(2*A*d*f - B*(d*e + c*f))) - 8*a*b*
d*f*(C*(7*d^3*e^3 + 9*c*d^2*e^2*f + 9*c^2*d*e*f^2 + 7*c^3*f^3) + 2*d*f*(8*A*d*f*(d*e + c*f) - B*(5*d^2*e^2 + 6
*c*d*e*f + 5*c^2*f^2))) + b^2*(C*(21*d^4*e^4 + 28*c*d^3*e^3*f + 30*c^2*d^2*e^2*f^2 + 28*c^3*d*e*f^3 + 21*c^4*f
^4) + 4*d*f*(2*A*d*f*(5*d^2*e^2 + 6*c*d*e*f + 5*c^2*f^2) - B*(7*d^3*e^3 + 9*c*d^2*e^2*f + 9*c^2*d*e*f^2 + 7*c^
3*f^3))))*Sqrt[c + d*x]*Sqrt[e + f*x])/(512*d^5*f^5) + ((8*a^2*d^2*f^2*(C*(5*d^2*e^2 + 6*c*d*e*f + 5*c^2*f^2)
+ 8*d*f*(2*A*d*f - B*(d*e + c*f))) - 8*a*b*d*f*(C*(7*d^3*e^3 + 9*c*d^2*e^2*f + 9*c^2*d*e*f^2 + 7*c^3*f^3) + 2*
d*f*(8*A*d*f*(d*e + c*f) - B*(5*d^2*e^2 + 6*c*d*e*f + 5*c^2*f^2))) + b^2*(C*(21*d^4*e^4 + 28*c*d^3*e^3*f + 30*
c^2*d^2*e^2*f^2 + 28*c^3*d*e*f^3 + 21*c^4*f^4) + 4*d*f*(2*A*d*f*(5*d^2*e^2 + 6*c*d*e*f + 5*c^2*f^2) - B*(7*d^3
*e^3 + 9*c*d^2*e^2*f + 9*c^2*d*e*f^2 + 7*c^3*f^3))))*(c + d*x)^(3/2)*Sqrt[e + f*x])/(256*d^5*f^4) + ((4*b*B*d*
f - 2*a*C*d*f - 3*b*C*(d*e + c*f))*(a + b*x)^2*(c + d*x)^(3/2)*(e + f*x)^(3/2))/(20*b*d^2*f^2) + (C*(a + b*x)^
3*(c + d*x)^(3/2)*(e + f*x)^(3/2))/(6*b*d*f) - ((c + d*x)^(3/2)*(e + f*x)^(3/2)*(64*a^3*C*d^3*f^3 - 8*a^2*b*d^
2*f^2*(16*B*d*f - 7*C*(d*e + c*f)) - 8*a*b^2*d*f*(C*(35*d^2*e^2 + 38*c*d*e*f + 35*c^2*f^2) + 10*d*f*(8*A*d*f -
 5*B*(d*e + c*f))) + b^3*(7*C*(15*d^3*e^3 + 17*c*d^2*e^2*f + 17*c^2*d*e*f^2 + 15*c^3*f^3) + 4*d*f*(50*A*d*f*(d
*e + c*f) - B*(35*d^2*e^2 + 38*c*d*e*f + 35*c^2*f^2))) + 6*b*d*f*(10*b*d*f*(2*b*c*C*e + a*C*d*e + a*c*C*f - 4*
A*b*d*f) - (4*a*d*f - 7*b*(d*e + c*f))*(4*b*B*d*f - 2*a*C*d*f - 3*b*C*(d*e + c*f)))*x))/(960*b*d^4*f^4) - ((d*
e - c*f)^2*(8*a^2*d^2*f^2*(C*(5*d^2*e^2 + 6*c*d*e*f + 5*c^2*f^2) + 8*d*f*(2*A*d*f - B*(d*e + c*f))) - 8*a*b*d*
f*(C*(7*d^3*e^3 + 9*c*d^2*e^2*f + 9*c^2*d*e*f^2 + 7*c^3*f^3) + 2*d*f*(8*A*d*f*(d*e + c*f) - B*(5*d^2*e^2 + 6*c
*d*e*f + 5*c^2*f^2))) + b^2*(C*(21*d^4*e^4 + 28*c*d^3*e^3*f + 30*c^2*d^2*e^2*f^2 + 28*c^3*d*e*f^3 + 21*c^4*f^4
) + 4*d*f*(2*A*d*f*(5*d^2*e^2 + 6*c*d*e*f + 5*c^2*f^2) - B*(7*d^3*e^3 + 9*c*d^2*e^2*f + 9*c^2*d*e*f^2 + 7*c^3*
f^3))))*ArcTanh[(Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[e + f*x])])/(512*d^(11/2)*f^(11/2))

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 147

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol]
:> -Simp[((a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x)*(a + b*x)^
(m + 1)*(c + d*x)^(n + 1))/(b^2*d^2*(m + n + 2)*(m + n + 3)), x] + Dist[(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d*(
n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*
(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)), Int[(a + b*x)^m*(c + d*x)^n
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]

Rule 153

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(h*(a + b*x)^m*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 2)), x] + Dist[1/(d*f*(m + n
 + p + 2)), Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(
n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x]
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegerQ[m]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
 /; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 1615

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> With[
{q = Expon[Px, x], k = Coeff[Px, x, Expon[Px, x]]}, Simp[(k*(a + b*x)^(m + q - 1)*(c + d*x)^(n + 1)*(e + f*x)^
(p + 1))/(d*f*b^(q - 1)*(m + n + p + q + 1)), x] + Dist[1/(d*f*b^q*(m + n + p + q + 1)), Int[(a + b*x)^m*(c +
d*x)^n*(e + f*x)^p*ExpandToSum[d*f*b^q*(m + n + p + q + 1)*Px - d*f*k*(m + n + p + q + 1)*(a + b*x)^q + k*(a +
 b*x)^(q - 2)*(a^2*d*f*(m + n + p + q + 1) - b*(b*c*e*(m + q - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*
(2*(m + q) + n + p) - b*(d*e*(m + q + n) + c*f*(m + q + p)))*x), x], x], x] /; NeQ[m + n + p + q + 1, 0]] /; F
reeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x] && IntegersQ[2*m, 2*n, 2*p]

Rubi steps

\begin {align*} \int (a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right ) \, dx &=\frac {C (a+b x)^3 (c+d x)^{3/2} (e+f x)^{3/2}}{6 b d f}+\frac {\int (a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} \left (-\frac {3}{2} b (2 b c C e+a C d e+a c C f-4 A b d f)+\frac {3}{2} b (4 b B d f-2 a C d f-3 b C (d e+c f)) x\right ) \, dx}{6 b^2 d f}\\ &=\frac {(4 b B d f-2 a C d f-3 b C (d e+c f)) (a+b x)^2 (c+d x)^{3/2} (e+f x)^{3/2}}{20 b d^2 f^2}+\frac {C (a+b x)^3 (c+d x)^{3/2} (e+f x)^{3/2}}{6 b d f}+\frac {\int (a+b x) \sqrt {c+d x} \sqrt {e+f x} \left (-\frac {3}{4} b (10 a d f (2 b c C e+a C d e+a c C f-4 A b d f)+(4 b c e+3 a (d e+c f)) (4 b B d f-2 a C d f-3 b C (d e+c f)))-\frac {3}{4} b (10 b d f (2 b c C e+a C d e+a c C f-4 A b d f)-(4 a d f-7 b (d e+c f)) (4 b B d f-2 a C d f-3 b C (d e+c f))) x\right ) \, dx}{30 b^2 d^2 f^2}\\ &=\frac {(4 b B d f-2 a C d f-3 b C (d e+c f)) (a+b x)^2 (c+d x)^{3/2} (e+f x)^{3/2}}{20 b d^2 f^2}+\frac {C (a+b x)^3 (c+d x)^{3/2} (e+f x)^{3/2}}{6 b d f}-\frac {(c+d x)^{3/2} (e+f x)^{3/2} \left (64 a^3 C d^3 f^3-8 a^2 b d^2 f^2 (16 B d f-7 C (d e+c f))-8 a b^2 d f \left (C \left (35 d^2 e^2+38 c d e f+35 c^2 f^2\right )+10 d f (8 A d f-5 B (d e+c f))\right )+b^3 \left (7 C \left (15 d^3 e^3+17 c d^2 e^2 f+17 c^2 d e f^2+15 c^3 f^3\right )+4 d f \left (50 A d f (d e+c f)-B \left (35 d^2 e^2+38 c d e f+35 c^2 f^2\right )\right )\right )+6 b d f (10 b d f (2 b c C e+a C d e+a c C f-4 A b d f)-(4 a d f-7 b (d e+c f)) (4 b B d f-2 a C d f-3 b C (d e+c f))) x\right )}{960 b d^4 f^4}+\frac {\left (8 a^2 d^2 f^2 \left (C \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right )-8 a b d f \left (C \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )+2 d f \left (8 A d f (d e+c f)-B \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )\right )\right )+b^2 \left (C \left (21 d^4 e^4+28 c d^3 e^3 f+30 c^2 d^2 e^2 f^2+28 c^3 d e f^3+21 c^4 f^4\right )+4 d f \left (2 A d f \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )-B \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )\right )\right )\right ) \int \sqrt {c+d x} \sqrt {e+f x} \, dx}{128 d^4 f^4}\\ &=\frac {\left (8 a^2 d^2 f^2 \left (C \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right )-8 a b d f \left (C \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )+2 d f \left (8 A d f (d e+c f)-B \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )\right )\right )+b^2 \left (C \left (21 d^4 e^4+28 c d^3 e^3 f+30 c^2 d^2 e^2 f^2+28 c^3 d e f^3+21 c^4 f^4\right )+4 d f \left (2 A d f \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )-B \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )\right )\right )\right ) (c+d x)^{3/2} \sqrt {e+f x}}{256 d^5 f^4}+\frac {(4 b B d f-2 a C d f-3 b C (d e+c f)) (a+b x)^2 (c+d x)^{3/2} (e+f x)^{3/2}}{20 b d^2 f^2}+\frac {C (a+b x)^3 (c+d x)^{3/2} (e+f x)^{3/2}}{6 b d f}-\frac {(c+d x)^{3/2} (e+f x)^{3/2} \left (64 a^3 C d^3 f^3-8 a^2 b d^2 f^2 (16 B d f-7 C (d e+c f))-8 a b^2 d f \left (C \left (35 d^2 e^2+38 c d e f+35 c^2 f^2\right )+10 d f (8 A d f-5 B (d e+c f))\right )+b^3 \left (7 C \left (15 d^3 e^3+17 c d^2 e^2 f+17 c^2 d e f^2+15 c^3 f^3\right )+4 d f \left (50 A d f (d e+c f)-B \left (35 d^2 e^2+38 c d e f+35 c^2 f^2\right )\right )\right )+6 b d f (10 b d f (2 b c C e+a C d e+a c C f-4 A b d f)-(4 a d f-7 b (d e+c f)) (4 b B d f-2 a C d f-3 b C (d e+c f))) x\right )}{960 b d^4 f^4}+\frac {\left ((d e-c f) \left (8 a^2 d^2 f^2 \left (C \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right )-8 a b d f \left (C \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )+2 d f \left (8 A d f (d e+c f)-B \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )\right )\right )+b^2 \left (C \left (21 d^4 e^4+28 c d^3 e^3 f+30 c^2 d^2 e^2 f^2+28 c^3 d e f^3+21 c^4 f^4\right )+4 d f \left (2 A d f \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )-B \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )\right )\right )\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {e+f x}} \, dx}{512 d^5 f^4}\\ &=\frac {(d e-c f) \left (8 a^2 d^2 f^2 \left (C \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right )-8 a b d f \left (C \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )+2 d f \left (8 A d f (d e+c f)-B \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )\right )\right )+b^2 \left (C \left (21 d^4 e^4+28 c d^3 e^3 f+30 c^2 d^2 e^2 f^2+28 c^3 d e f^3+21 c^4 f^4\right )+4 d f \left (2 A d f \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )-B \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )\right )\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{512 d^5 f^5}+\frac {\left (8 a^2 d^2 f^2 \left (C \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right )-8 a b d f \left (C \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )+2 d f \left (8 A d f (d e+c f)-B \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )\right )\right )+b^2 \left (C \left (21 d^4 e^4+28 c d^3 e^3 f+30 c^2 d^2 e^2 f^2+28 c^3 d e f^3+21 c^4 f^4\right )+4 d f \left (2 A d f \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )-B \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )\right )\right )\right ) (c+d x)^{3/2} \sqrt {e+f x}}{256 d^5 f^4}+\frac {(4 b B d f-2 a C d f-3 b C (d e+c f)) (a+b x)^2 (c+d x)^{3/2} (e+f x)^{3/2}}{20 b d^2 f^2}+\frac {C (a+b x)^3 (c+d x)^{3/2} (e+f x)^{3/2}}{6 b d f}-\frac {(c+d x)^{3/2} (e+f x)^{3/2} \left (64 a^3 C d^3 f^3-8 a^2 b d^2 f^2 (16 B d f-7 C (d e+c f))-8 a b^2 d f \left (C \left (35 d^2 e^2+38 c d e f+35 c^2 f^2\right )+10 d f (8 A d f-5 B (d e+c f))\right )+b^3 \left (7 C \left (15 d^3 e^3+17 c d^2 e^2 f+17 c^2 d e f^2+15 c^3 f^3\right )+4 d f \left (50 A d f (d e+c f)-B \left (35 d^2 e^2+38 c d e f+35 c^2 f^2\right )\right )\right )+6 b d f (10 b d f (2 b c C e+a C d e+a c C f-4 A b d f)-(4 a d f-7 b (d e+c f)) (4 b B d f-2 a C d f-3 b C (d e+c f))) x\right )}{960 b d^4 f^4}-\frac {\left ((d e-c f)^2 \left (8 a^2 d^2 f^2 \left (C \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right )-8 a b d f \left (C \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )+2 d f \left (8 A d f (d e+c f)-B \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )\right )\right )+b^2 \left (C \left (21 d^4 e^4+28 c d^3 e^3 f+30 c^2 d^2 e^2 f^2+28 c^3 d e f^3+21 c^4 f^4\right )+4 d f \left (2 A d f \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )-B \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )\right )\right )\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {e+f x}} \, dx}{1024 d^5 f^5}\\ &=\frac {(d e-c f) \left (8 a^2 d^2 f^2 \left (C \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right )-8 a b d f \left (C \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )+2 d f \left (8 A d f (d e+c f)-B \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )\right )\right )+b^2 \left (C \left (21 d^4 e^4+28 c d^3 e^3 f+30 c^2 d^2 e^2 f^2+28 c^3 d e f^3+21 c^4 f^4\right )+4 d f \left (2 A d f \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )-B \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )\right )\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{512 d^5 f^5}+\frac {\left (8 a^2 d^2 f^2 \left (C \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right )-8 a b d f \left (C \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )+2 d f \left (8 A d f (d e+c f)-B \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )\right )\right )+b^2 \left (C \left (21 d^4 e^4+28 c d^3 e^3 f+30 c^2 d^2 e^2 f^2+28 c^3 d e f^3+21 c^4 f^4\right )+4 d f \left (2 A d f \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )-B \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )\right )\right )\right ) (c+d x)^{3/2} \sqrt {e+f x}}{256 d^5 f^4}+\frac {(4 b B d f-2 a C d f-3 b C (d e+c f)) (a+b x)^2 (c+d x)^{3/2} (e+f x)^{3/2}}{20 b d^2 f^2}+\frac {C (a+b x)^3 (c+d x)^{3/2} (e+f x)^{3/2}}{6 b d f}-\frac {(c+d x)^{3/2} (e+f x)^{3/2} \left (64 a^3 C d^3 f^3-8 a^2 b d^2 f^2 (16 B d f-7 C (d e+c f))-8 a b^2 d f \left (C \left (35 d^2 e^2+38 c d e f+35 c^2 f^2\right )+10 d f (8 A d f-5 B (d e+c f))\right )+b^3 \left (7 C \left (15 d^3 e^3+17 c d^2 e^2 f+17 c^2 d e f^2+15 c^3 f^3\right )+4 d f \left (50 A d f (d e+c f)-B \left (35 d^2 e^2+38 c d e f+35 c^2 f^2\right )\right )\right )+6 b d f (10 b d f (2 b c C e+a C d e+a c C f-4 A b d f)-(4 a d f-7 b (d e+c f)) (4 b B d f-2 a C d f-3 b C (d e+c f))) x\right )}{960 b d^4 f^4}-\frac {\left ((d e-c f)^2 \left (8 a^2 d^2 f^2 \left (C \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right )-8 a b d f \left (C \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )+2 d f \left (8 A d f (d e+c f)-B \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )\right )\right )+b^2 \left (C \left (21 d^4 e^4+28 c d^3 e^3 f+30 c^2 d^2 e^2 f^2+28 c^3 d e f^3+21 c^4 f^4\right )+4 d f \left (2 A d f \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )-B \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )\right )\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {e-\frac {c f}{d}+\frac {f x^2}{d}}} \, dx,x,\sqrt {c+d x}\right )}{512 d^6 f^5}\\ &=\frac {(d e-c f) \left (8 a^2 d^2 f^2 \left (C \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right )-8 a b d f \left (C \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )+2 d f \left (8 A d f (d e+c f)-B \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )\right )\right )+b^2 \left (C \left (21 d^4 e^4+28 c d^3 e^3 f+30 c^2 d^2 e^2 f^2+28 c^3 d e f^3+21 c^4 f^4\right )+4 d f \left (2 A d f \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )-B \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )\right )\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{512 d^5 f^5}+\frac {\left (8 a^2 d^2 f^2 \left (C \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right )-8 a b d f \left (C \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )+2 d f \left (8 A d f (d e+c f)-B \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )\right )\right )+b^2 \left (C \left (21 d^4 e^4+28 c d^3 e^3 f+30 c^2 d^2 e^2 f^2+28 c^3 d e f^3+21 c^4 f^4\right )+4 d f \left (2 A d f \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )-B \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )\right )\right )\right ) (c+d x)^{3/2} \sqrt {e+f x}}{256 d^5 f^4}+\frac {(4 b B d f-2 a C d f-3 b C (d e+c f)) (a+b x)^2 (c+d x)^{3/2} (e+f x)^{3/2}}{20 b d^2 f^2}+\frac {C (a+b x)^3 (c+d x)^{3/2} (e+f x)^{3/2}}{6 b d f}-\frac {(c+d x)^{3/2} (e+f x)^{3/2} \left (64 a^3 C d^3 f^3-8 a^2 b d^2 f^2 (16 B d f-7 C (d e+c f))-8 a b^2 d f \left (C \left (35 d^2 e^2+38 c d e f+35 c^2 f^2\right )+10 d f (8 A d f-5 B (d e+c f))\right )+b^3 \left (7 C \left (15 d^3 e^3+17 c d^2 e^2 f+17 c^2 d e f^2+15 c^3 f^3\right )+4 d f \left (50 A d f (d e+c f)-B \left (35 d^2 e^2+38 c d e f+35 c^2 f^2\right )\right )\right )+6 b d f (10 b d f (2 b c C e+a C d e+a c C f-4 A b d f)-(4 a d f-7 b (d e+c f)) (4 b B d f-2 a C d f-3 b C (d e+c f))) x\right )}{960 b d^4 f^4}-\frac {\left ((d e-c f)^2 \left (8 a^2 d^2 f^2 \left (C \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right )-8 a b d f \left (C \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )+2 d f \left (8 A d f (d e+c f)-B \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )\right )\right )+b^2 \left (C \left (21 d^4 e^4+28 c d^3 e^3 f+30 c^2 d^2 e^2 f^2+28 c^3 d e f^3+21 c^4 f^4\right )+4 d f \left (2 A d f \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )-B \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )\right )\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-\frac {f x^2}{d}} \, dx,x,\frac {\sqrt {c+d x}}{\sqrt {e+f x}}\right )}{512 d^6 f^5}\\ &=\frac {(d e-c f) \left (8 a^2 d^2 f^2 \left (C \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right )-8 a b d f \left (C \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )+2 d f \left (8 A d f (d e+c f)-B \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )\right )\right )+b^2 \left (C \left (21 d^4 e^4+28 c d^3 e^3 f+30 c^2 d^2 e^2 f^2+28 c^3 d e f^3+21 c^4 f^4\right )+4 d f \left (2 A d f \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )-B \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )\right )\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{512 d^5 f^5}+\frac {\left (8 a^2 d^2 f^2 \left (C \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right )-8 a b d f \left (C \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )+2 d f \left (8 A d f (d e+c f)-B \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )\right )\right )+b^2 \left (C \left (21 d^4 e^4+28 c d^3 e^3 f+30 c^2 d^2 e^2 f^2+28 c^3 d e f^3+21 c^4 f^4\right )+4 d f \left (2 A d f \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )-B \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )\right )\right )\right ) (c+d x)^{3/2} \sqrt {e+f x}}{256 d^5 f^4}+\frac {(4 b B d f-2 a C d f-3 b C (d e+c f)) (a+b x)^2 (c+d x)^{3/2} (e+f x)^{3/2}}{20 b d^2 f^2}+\frac {C (a+b x)^3 (c+d x)^{3/2} (e+f x)^{3/2}}{6 b d f}-\frac {(c+d x)^{3/2} (e+f x)^{3/2} \left (64 a^3 C d^3 f^3-8 a^2 b d^2 f^2 (16 B d f-7 C (d e+c f))-8 a b^2 d f \left (C \left (35 d^2 e^2+38 c d e f+35 c^2 f^2\right )+10 d f (8 A d f-5 B (d e+c f))\right )+b^3 \left (7 C \left (15 d^3 e^3+17 c d^2 e^2 f+17 c^2 d e f^2+15 c^3 f^3\right )+4 d f \left (50 A d f (d e+c f)-B \left (35 d^2 e^2+38 c d e f+35 c^2 f^2\right )\right )\right )+6 b d f (10 b d f (2 b c C e+a C d e+a c C f-4 A b d f)-(4 a d f-7 b (d e+c f)) (4 b B d f-2 a C d f-3 b C (d e+c f))) x\right )}{960 b d^4 f^4}-\frac {(d e-c f)^2 \left (8 a^2 d^2 f^2 \left (C \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right )-8 a b d f \left (C \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )+2 d f \left (8 A d f (d e+c f)-B \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )\right )\right )+b^2 \left (C \left (21 d^4 e^4+28 c d^3 e^3 f+30 c^2 d^2 e^2 f^2+28 c^3 d e f^3+21 c^4 f^4\right )+4 d f \left (2 A d f \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )-B \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )\right )\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d} \sqrt {e+f x}}\right )}{512 d^{11/2} f^{11/2}}\\ \end {align*}

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Mathematica [B]  time = 7.13, size = 3599, normalized size = 2.67 \[ \text {Result too large to show} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^2*Sqrt[c + d*x]*Sqrt[e + f*x]*(A + B*x + C*x^2),x]

[Out]

(2*b^2*C*(d*e - c*f)^4*(c + d*x)^(3/2)*Sqrt[e + f*x]*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) -
(c*d*f)/(d*e - c*f))))^(11/2)*((63/(128*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e
- c*f))))^5) + 21/(32*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^4) + 63/
(80*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^3) + 9/(10*(1 + (d*f*(c +
d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^2) + (1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*
e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^(-1))/4 + (63*(d*e - c*f)^2*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)
)^2*((2*d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))) - (2*Sqrt[d]*Sqrt[f]*Sqrt[c +
 d*x]*ArcSinh[(Sqrt[d]*Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)]
)])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)]*Sqrt[1 + (d*f*(c + d*x))/((d*e - c*f)*((d
^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)))])))/(2048*d^2*f^2*(c + d*x)^2*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^
2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^5)))/(3*d^5*f^4*(d/((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)))^(9/2
)*Sqrt[(d*(e + f*x))/(d*e - c*f)]) + (2*b*(d*e - c*f)^3*(-4*b*C*e + b*B*f + 2*a*C*f)*(c + d*x)^(3/2)*Sqrt[e +
f*x]*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^(9/2)*((3*(35/(64*(1 + (d
*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^4) + 35/(48*(1 + (d*f*(c + d*x))/((d*
e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^3) + 7/(8*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d
*e - c*f) - (c*d*f)/(d*e - c*f))))^2) + (1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e
- c*f))))^(-1)))/10 + (21*(d*e - c*f)^2*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))^2*((2*d*f*(c + d*x))/((d*e
 - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))) - (2*Sqrt[d]*Sqrt[f]*Sqrt[c + d*x]*ArcSinh[(Sqrt[d]*Sqrt[
f]*Sqrt[c + d*x])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)])])/(Sqrt[d*e - c*f]*Sqrt[(d
^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)]*Sqrt[1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/
(d*e - c*f)))])))/(512*d^2*f^2*(c + d*x)^2*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d
*e - c*f))))^4)))/(3*d^4*f^4*(d/((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)))^(7/2)*Sqrt[(d*(e + f*x))/(d*e - c
*f)]) + (2*(d*e - c*f)^2*(6*b^2*C*e^2 - 3*b^2*B*e*f - 6*a*b*C*e*f + A*b^2*f^2 + 2*a*b*B*f^2 + a^2*C*f^2)*(c +
d*x)^(3/2)*Sqrt[e + f*x]*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^(7/2)
*((3*(5/(8*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^3) + 5/(6*(1 + (d*f
*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^2) + (1 + (d*f*(c + d*x))/((d*e - c*f)*
((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^(-1)))/8 + (15*(d*e - c*f)^2*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e
 - c*f))^2*((2*d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))) - (2*Sqrt[d]*Sqrt[f]*S
qrt[c + d*x]*ArcSinh[(Sqrt[d]*Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e
- c*f)])])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)]*Sqrt[1 + (d*f*(c + d*x))/((d*e - c
*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)))])))/(256*d^2*f^2*(c + d*x)^2*(1 + (d*f*(c + d*x))/((d*e - c*f
)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^3)))/(3*d^3*f^4*(d/((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))
)^(5/2)*Sqrt[(d*(e + f*x))/(d*e - c*f)]) + (2*(-(b*e) + a*f)*(d*e - c*f)*(4*b*C*e^2 - 3*b*B*e*f - 2*a*C*e*f +
2*A*b*f^2 + a*B*f^2)*(c + d*x)^(3/2)*Sqrt[e + f*x]*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c
*d*f)/(d*e - c*f))))^(5/2)*((3/(4*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)
)))^2) + (1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^(-1))/2 + (3*(d*e - c
*f)^2*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))^2*((2*d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*
d*f)/(d*e - c*f))) - (2*Sqrt[d]*Sqrt[f]*Sqrt[c + d*x]*ArcSinh[(Sqrt[d]*Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d*e - c*f]
*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)])])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e -
 c*f)]*Sqrt[1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)))])))/(32*d^2*f^2*(c +
 d*x)^2*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^2)))/(3*d^2*f^4*(d/((d
^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)))^(3/2)*Sqrt[(d*(e + f*x))/(d*e - c*f)]) + (2*(-(b*e) + a*f)^2*(C*e^2
- B*e*f + A*f^2)*(c + d*x)^(3/2)*Sqrt[e + f*x]*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f
)/(d*e - c*f))))^(3/2)*(3/(4*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)))))
+ (3*(d*e - c*f)^2*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))^2*((2*d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e
 - c*f) - (c*d*f)/(d*e - c*f))) - (2*Sqrt[d]*Sqrt[f]*Sqrt[c + d*x]*ArcSinh[(Sqrt[d]*Sqrt[f]*Sqrt[c + d*x])/(Sq
rt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)])])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (
c*d*f)/(d*e - c*f)]*Sqrt[1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)))])))/(16
*d^2*f^2*(c + d*x)^2*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)))))))/(3*d*f
^4*Sqrt[d/((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))]*Sqrt[(d*(e + f*x))/(d*e - c*f)])

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fricas [A]  time = 6.80, size = 3096, normalized size = 2.30 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*(C*x^2+B*x+A)*(d*x+c)^(1/2)*(f*x+e)^(1/2),x, algorithm="fricas")

[Out]

[1/30720*(15*(21*C*b^2*d^6*e^6 - 14*(C*b^2*c*d^5 + 2*(2*C*a*b + B*b^2)*d^6)*e^5*f - 5*(C*b^2*c^2*d^4 - 4*(2*C*
a*b + B*b^2)*c*d^5 - 8*(C*a^2 + 2*B*a*b + A*b^2)*d^6)*e^4*f^2 - 4*(C*b^2*c^3*d^3 - 2*(2*C*a*b + B*b^2)*c^2*d^4
 + 8*(C*a^2 + 2*B*a*b + A*b^2)*c*d^5 + 16*(B*a^2 + 2*A*a*b)*d^6)*e^3*f^3 - (5*C*b^2*c^4*d^2 - 128*A*a^2*d^6 -
8*(2*C*a*b + B*b^2)*c^3*d^3 + 16*(C*a^2 + 2*B*a*b + A*b^2)*c^2*d^4 - 64*(B*a^2 + 2*A*a*b)*c*d^5)*e^2*f^4 - 2*(
7*C*b^2*c^5*d + 128*A*a^2*c*d^5 - 10*(2*C*a*b + B*b^2)*c^4*d^2 + 16*(C*a^2 + 2*B*a*b + A*b^2)*c^3*d^3 - 32*(B*
a^2 + 2*A*a*b)*c^2*d^4)*e*f^5 + (21*C*b^2*c^6 + 128*A*a^2*c^2*d^4 - 28*(2*C*a*b + B*b^2)*c^5*d + 40*(C*a^2 + 2
*B*a*b + A*b^2)*c^4*d^2 - 64*(B*a^2 + 2*A*a*b)*c^3*d^3)*f^6)*sqrt(d*f)*log(8*d^2*f^2*x^2 + d^2*e^2 + 6*c*d*e*f
 + c^2*f^2 - 4*(2*d*f*x + d*e + c*f)*sqrt(d*f)*sqrt(d*x + c)*sqrt(f*x + e) + 8*(d^2*e*f + c*d*f^2)*x) + 4*(128
0*C*b^2*d^6*f^6*x^5 + 315*C*b^2*d^6*e^5*f - 105*(C*b^2*c*d^5 + 4*(2*C*a*b + B*b^2)*d^6)*e^4*f^2 - 2*(41*C*b^2*
c^2*d^4 - 80*(2*C*a*b + B*b^2)*c*d^5 - 300*(C*a^2 + 2*B*a*b + A*b^2)*d^6)*e^3*f^3 - 2*(41*C*b^2*c^3*d^3 - 68*(
2*C*a*b + B*b^2)*c^2*d^4 + 140*(C*a^2 + 2*B*a*b + A*b^2)*c*d^5 + 480*(B*a^2 + 2*A*a*b)*d^6)*e^2*f^4 - 5*(21*C*
b^2*c^4*d^2 - 384*A*a^2*d^6 - 32*(2*C*a*b + B*b^2)*c^3*d^3 + 56*(C*a^2 + 2*B*a*b + A*b^2)*c^2*d^4 - 128*(B*a^2
 + 2*A*a*b)*c*d^5)*e*f^5 + 15*(21*C*b^2*c^5*d + 128*A*a^2*c*d^5 - 28*(2*C*a*b + B*b^2)*c^4*d^2 + 40*(C*a^2 + 2
*B*a*b + A*b^2)*c^3*d^3 - 64*(B*a^2 + 2*A*a*b)*c^2*d^4)*f^6 + 128*(C*b^2*d^6*e*f^5 + (C*b^2*c*d^5 + 12*(2*C*a*
b + B*b^2)*d^6)*f^6)*x^4 - 16*(9*C*b^2*d^6*e^2*f^4 - 2*(C*b^2*c*d^5 + 6*(2*C*a*b + B*b^2)*d^6)*e*f^5 + 3*(3*C*
b^2*c^2*d^4 - 4*(2*C*a*b + B*b^2)*c*d^5 - 40*(C*a^2 + 2*B*a*b + A*b^2)*d^6)*f^6)*x^3 + 8*(21*C*b^2*d^6*e^3*f^3
 - (5*C*b^2*c*d^5 + 28*(2*C*a*b + B*b^2)*d^6)*e^2*f^4 - (5*C*b^2*c^2*d^4 - 8*(2*C*a*b + B*b^2)*c*d^5 - 40*(C*a
^2 + 2*B*a*b + A*b^2)*d^6)*e*f^5 + (21*C*b^2*c^3*d^3 - 28*(2*C*a*b + B*b^2)*c^2*d^4 + 40*(C*a^2 + 2*B*a*b + A*
b^2)*c*d^5 + 320*(B*a^2 + 2*A*a*b)*d^6)*f^6)*x^2 - 2*(105*C*b^2*d^6*e^4*f^2 - 28*(C*b^2*c*d^5 + 5*(2*C*a*b + B
*b^2)*d^6)*e^3*f^3 - 2*(13*C*b^2*c^2*d^4 - 22*(2*C*a*b + B*b^2)*c*d^5 - 100*(C*a^2 + 2*B*a*b + A*b^2)*d^6)*e^2
*f^4 - 4*(7*C*b^2*c^3*d^3 - 11*(2*C*a*b + B*b^2)*c^2*d^4 + 20*(C*a^2 + 2*B*a*b + A*b^2)*c*d^5 + 80*(B*a^2 + 2*
A*a*b)*d^6)*e*f^5 + 5*(21*C*b^2*c^4*d^2 - 384*A*a^2*d^6 - 28*(2*C*a*b + B*b^2)*c^3*d^3 + 40*(C*a^2 + 2*B*a*b +
 A*b^2)*c^2*d^4 - 64*(B*a^2 + 2*A*a*b)*c*d^5)*f^6)*x)*sqrt(d*x + c)*sqrt(f*x + e))/(d^6*f^6), 1/15360*(15*(21*
C*b^2*d^6*e^6 - 14*(C*b^2*c*d^5 + 2*(2*C*a*b + B*b^2)*d^6)*e^5*f - 5*(C*b^2*c^2*d^4 - 4*(2*C*a*b + B*b^2)*c*d^
5 - 8*(C*a^2 + 2*B*a*b + A*b^2)*d^6)*e^4*f^2 - 4*(C*b^2*c^3*d^3 - 2*(2*C*a*b + B*b^2)*c^2*d^4 + 8*(C*a^2 + 2*B
*a*b + A*b^2)*c*d^5 + 16*(B*a^2 + 2*A*a*b)*d^6)*e^3*f^3 - (5*C*b^2*c^4*d^2 - 128*A*a^2*d^6 - 8*(2*C*a*b + B*b^
2)*c^3*d^3 + 16*(C*a^2 + 2*B*a*b + A*b^2)*c^2*d^4 - 64*(B*a^2 + 2*A*a*b)*c*d^5)*e^2*f^4 - 2*(7*C*b^2*c^5*d + 1
28*A*a^2*c*d^5 - 10*(2*C*a*b + B*b^2)*c^4*d^2 + 16*(C*a^2 + 2*B*a*b + A*b^2)*c^3*d^3 - 32*(B*a^2 + 2*A*a*b)*c^
2*d^4)*e*f^5 + (21*C*b^2*c^6 + 128*A*a^2*c^2*d^4 - 28*(2*C*a*b + B*b^2)*c^5*d + 40*(C*a^2 + 2*B*a*b + A*b^2)*c
^4*d^2 - 64*(B*a^2 + 2*A*a*b)*c^3*d^3)*f^6)*sqrt(-d*f)*arctan(1/2*(2*d*f*x + d*e + c*f)*sqrt(-d*f)*sqrt(d*x +
c)*sqrt(f*x + e)/(d^2*f^2*x^2 + c*d*e*f + (d^2*e*f + c*d*f^2)*x)) + 2*(1280*C*b^2*d^6*f^6*x^5 + 315*C*b^2*d^6*
e^5*f - 105*(C*b^2*c*d^5 + 4*(2*C*a*b + B*b^2)*d^6)*e^4*f^2 - 2*(41*C*b^2*c^2*d^4 - 80*(2*C*a*b + B*b^2)*c*d^5
 - 300*(C*a^2 + 2*B*a*b + A*b^2)*d^6)*e^3*f^3 - 2*(41*C*b^2*c^3*d^3 - 68*(2*C*a*b + B*b^2)*c^2*d^4 + 140*(C*a^
2 + 2*B*a*b + A*b^2)*c*d^5 + 480*(B*a^2 + 2*A*a*b)*d^6)*e^2*f^4 - 5*(21*C*b^2*c^4*d^2 - 384*A*a^2*d^6 - 32*(2*
C*a*b + B*b^2)*c^3*d^3 + 56*(C*a^2 + 2*B*a*b + A*b^2)*c^2*d^4 - 128*(B*a^2 + 2*A*a*b)*c*d^5)*e*f^5 + 15*(21*C*
b^2*c^5*d + 128*A*a^2*c*d^5 - 28*(2*C*a*b + B*b^2)*c^4*d^2 + 40*(C*a^2 + 2*B*a*b + A*b^2)*c^3*d^3 - 64*(B*a^2
+ 2*A*a*b)*c^2*d^4)*f^6 + 128*(C*b^2*d^6*e*f^5 + (C*b^2*c*d^5 + 12*(2*C*a*b + B*b^2)*d^6)*f^6)*x^4 - 16*(9*C*b
^2*d^6*e^2*f^4 - 2*(C*b^2*c*d^5 + 6*(2*C*a*b + B*b^2)*d^6)*e*f^5 + 3*(3*C*b^2*c^2*d^4 - 4*(2*C*a*b + B*b^2)*c*
d^5 - 40*(C*a^2 + 2*B*a*b + A*b^2)*d^6)*f^6)*x^3 + 8*(21*C*b^2*d^6*e^3*f^3 - (5*C*b^2*c*d^5 + 28*(2*C*a*b + B*
b^2)*d^6)*e^2*f^4 - (5*C*b^2*c^2*d^4 - 8*(2*C*a*b + B*b^2)*c*d^5 - 40*(C*a^2 + 2*B*a*b + A*b^2)*d^6)*e*f^5 + (
21*C*b^2*c^3*d^3 - 28*(2*C*a*b + B*b^2)*c^2*d^4 + 40*(C*a^2 + 2*B*a*b + A*b^2)*c*d^5 + 320*(B*a^2 + 2*A*a*b)*d
^6)*f^6)*x^2 - 2*(105*C*b^2*d^6*e^4*f^2 - 28*(C*b^2*c*d^5 + 5*(2*C*a*b + B*b^2)*d^6)*e^3*f^3 - 2*(13*C*b^2*c^2
*d^4 - 22*(2*C*a*b + B*b^2)*c*d^5 - 100*(C*a^2 + 2*B*a*b + A*b^2)*d^6)*e^2*f^4 - 4*(7*C*b^2*c^3*d^3 - 11*(2*C*
a*b + B*b^2)*c^2*d^4 + 20*(C*a^2 + 2*B*a*b + A*b^2)*c*d^5 + 80*(B*a^2 + 2*A*a*b)*d^6)*e*f^5 + 5*(21*C*b^2*c^4*
d^2 - 384*A*a^2*d^6 - 28*(2*C*a*b + B*b^2)*c^3*d^3 + 40*(C*a^2 + 2*B*a*b + A*b^2)*c^2*d^4 - 64*(B*a^2 + 2*A*a*
b)*c*d^5)*f^6)*x)*sqrt(d*x + c)*sqrt(f*x + e))/(d^6*f^6)]

________________________________________________________________________________________

giac [B]  time = 6.33, size = 4708, normalized size = 3.49 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*(C*x^2+B*x+A)*(d*x+c)^(1/2)*(f*x+e)^(1/2),x, algorithm="giac")

[Out]

1/7680*(7680*((c*d*f - d^2*e)*log(abs(-sqrt(d*f)*sqrt(d*x + c) + sqrt((d*x + c)*d*f - c*d*f + d^2*e)))/sqrt(d*
f) + sqrt((d*x + c)*d*f - c*d*f + d^2*e)*sqrt(d*x + c))*A*a^2*c*abs(d)/d^2 + 320*(sqrt((d*x + c)*d*f - c*d*f +
 d^2*e)*sqrt(d*x + c)*(2*(d*x + c)*(4*(d*x + c)/d^2 - (13*c*d^5*f^4 - d^6*f^3*e)/(d^7*f^4)) + 3*(11*c^2*d^5*f^
4 - 2*c*d^6*f^3*e - d^7*f^2*e^2)/(d^7*f^4)) + 3*(5*c^3*f^3 - 3*c^2*d*f^2*e - c*d^2*f*e^2 - d^3*e^3)*log(abs(-s
qrt(d*f)*sqrt(d*x + c) + sqrt((d*x + c)*d*f - c*d*f + d^2*e)))/(sqrt(d*f)*d*f^2))*C*a^2*c*abs(d)/d^2 + 640*(sq
rt((d*x + c)*d*f - c*d*f + d^2*e)*sqrt(d*x + c)*(2*(d*x + c)*(4*(d*x + c)/d^2 - (13*c*d^5*f^4 - d^6*f^3*e)/(d^
7*f^4)) + 3*(11*c^2*d^5*f^4 - 2*c*d^6*f^3*e - d^7*f^2*e^2)/(d^7*f^4)) + 3*(5*c^3*f^3 - 3*c^2*d*f^2*e - c*d^2*f
*e^2 - d^3*e^3)*log(abs(-sqrt(d*f)*sqrt(d*x + c) + sqrt((d*x + c)*d*f - c*d*f + d^2*e)))/(sqrt(d*f)*d*f^2))*B*
a*b*c*abs(d)/d^2 + 80*(sqrt((d*x + c)*d*f - c*d*f + d^2*e)*(2*(d*x + c)*(4*(d*x + c)*(6*(d*x + c)/d^3 - (25*c*
d^11*f^6 - d^12*f^5*e)/(d^14*f^6)) + (163*c^2*d^11*f^6 - 14*c*d^12*f^5*e - 5*d^13*f^4*e^2)/(d^14*f^6)) - 3*(93
*c^3*d^11*f^6 - 15*c^2*d^12*f^5*e - 9*c*d^13*f^4*e^2 - 5*d^14*f^3*e^3)/(d^14*f^6))*sqrt(d*x + c) - 3*(35*c^4*f
^4 - 20*c^3*d*f^3*e - 6*c^2*d^2*f^2*e^2 - 4*c*d^3*f*e^3 - 5*d^4*e^4)*log(abs(-sqrt(d*f)*sqrt(d*x + c) + sqrt((
d*x + c)*d*f - c*d*f + d^2*e)))/(sqrt(d*f)*d^2*f^3))*C*a*b*c*abs(d)/d^2 + 320*(sqrt((d*x + c)*d*f - c*d*f + d^
2*e)*sqrt(d*x + c)*(2*(d*x + c)*(4*(d*x + c)/d^2 - (13*c*d^5*f^4 - d^6*f^3*e)/(d^7*f^4)) + 3*(11*c^2*d^5*f^4 -
 2*c*d^6*f^3*e - d^7*f^2*e^2)/(d^7*f^4)) + 3*(5*c^3*f^3 - 3*c^2*d*f^2*e - c*d^2*f*e^2 - d^3*e^3)*log(abs(-sqrt
(d*f)*sqrt(d*x + c) + sqrt((d*x + c)*d*f - c*d*f + d^2*e)))/(sqrt(d*f)*d*f^2))*A*b^2*c*abs(d)/d^2 + 40*(sqrt((
d*x + c)*d*f - c*d*f + d^2*e)*(2*(d*x + c)*(4*(d*x + c)*(6*(d*x + c)/d^3 - (25*c*d^11*f^6 - d^12*f^5*e)/(d^14*
f^6)) + (163*c^2*d^11*f^6 - 14*c*d^12*f^5*e - 5*d^13*f^4*e^2)/(d^14*f^6)) - 3*(93*c^3*d^11*f^6 - 15*c^2*d^12*f
^5*e - 9*c*d^13*f^4*e^2 - 5*d^14*f^3*e^3)/(d^14*f^6))*sqrt(d*x + c) - 3*(35*c^4*f^4 - 20*c^3*d*f^3*e - 6*c^2*d
^2*f^2*e^2 - 4*c*d^3*f*e^3 - 5*d^4*e^4)*log(abs(-sqrt(d*f)*sqrt(d*x + c) + sqrt((d*x + c)*d*f - c*d*f + d^2*e)
))/(sqrt(d*f)*d^2*f^3))*B*b^2*c*abs(d)/d^2 + 4*(sqrt((d*x + c)*d*f - c*d*f + d^2*e)*(2*(4*(d*x + c)*(6*(d*x +
c)*(8*(d*x + c)/d^4 - (41*c*d^19*f^8 - d^20*f^7*e)/(d^23*f^8)) + (513*c^2*d^19*f^8 - 26*c*d^20*f^7*e - 7*d^21*
f^6*e^2)/(d^23*f^8)) - 5*(447*c^3*d^19*f^8 - 37*c^2*d^20*f^7*e - 19*c*d^21*f^6*e^2 - 7*d^22*f^5*e^3)/(d^23*f^8
))*(d*x + c) + 15*(193*c^4*d^19*f^8 - 28*c^3*d^20*f^7*e - 18*c^2*d^21*f^6*e^2 - 12*c*d^22*f^5*e^3 - 7*d^23*f^4
*e^4)/(d^23*f^8))*sqrt(d*x + c) + 15*(63*c^5*f^5 - 35*c^4*d*f^4*e - 10*c^3*d^2*f^3*e^2 - 6*c^2*d^3*f^2*e^3 - 5
*c*d^4*f*e^4 - 7*d^5*e^5)*log(abs(-sqrt(d*f)*sqrt(d*x + c) + sqrt((d*x + c)*d*f - c*d*f + d^2*e)))/(sqrt(d*f)*
d^3*f^4))*C*b^2*c*abs(d)/d^2 + 320*(sqrt((d*x + c)*d*f - c*d*f + d^2*e)*sqrt(d*x + c)*(2*(d*x + c)*(4*(d*x + c
)/d^2 - (13*c*d^5*f^4 - d^6*f^3*e)/(d^7*f^4)) + 3*(11*c^2*d^5*f^4 - 2*c*d^6*f^3*e - d^7*f^2*e^2)/(d^7*f^4)) +
3*(5*c^3*f^3 - 3*c^2*d*f^2*e - c*d^2*f*e^2 - d^3*e^3)*log(abs(-sqrt(d*f)*sqrt(d*x + c) + sqrt((d*x + c)*d*f -
c*d*f + d^2*e)))/(sqrt(d*f)*d*f^2))*B*a^2*abs(d)/d + 40*(sqrt((d*x + c)*d*f - c*d*f + d^2*e)*(2*(d*x + c)*(4*(
d*x + c)*(6*(d*x + c)/d^3 - (25*c*d^11*f^6 - d^12*f^5*e)/(d^14*f^6)) + (163*c^2*d^11*f^6 - 14*c*d^12*f^5*e - 5
*d^13*f^4*e^2)/(d^14*f^6)) - 3*(93*c^3*d^11*f^6 - 15*c^2*d^12*f^5*e - 9*c*d^13*f^4*e^2 - 5*d^14*f^3*e^3)/(d^14
*f^6))*sqrt(d*x + c) - 3*(35*c^4*f^4 - 20*c^3*d*f^3*e - 6*c^2*d^2*f^2*e^2 - 4*c*d^3*f*e^3 - 5*d^4*e^4)*log(abs
(-sqrt(d*f)*sqrt(d*x + c) + sqrt((d*x + c)*d*f - c*d*f + d^2*e)))/(sqrt(d*f)*d^2*f^3))*C*a^2*abs(d)/d + 640*(s
qrt((d*x + c)*d*f - c*d*f + d^2*e)*sqrt(d*x + c)*(2*(d*x + c)*(4*(d*x + c)/d^2 - (13*c*d^5*f^4 - d^6*f^3*e)/(d
^7*f^4)) + 3*(11*c^2*d^5*f^4 - 2*c*d^6*f^3*e - d^7*f^2*e^2)/(d^7*f^4)) + 3*(5*c^3*f^3 - 3*c^2*d*f^2*e - c*d^2*
f*e^2 - d^3*e^3)*log(abs(-sqrt(d*f)*sqrt(d*x + c) + sqrt((d*x + c)*d*f - c*d*f + d^2*e)))/(sqrt(d*f)*d*f^2))*A
*a*b*abs(d)/d + 80*(sqrt((d*x + c)*d*f - c*d*f + d^2*e)*(2*(d*x + c)*(4*(d*x + c)*(6*(d*x + c)/d^3 - (25*c*d^1
1*f^6 - d^12*f^5*e)/(d^14*f^6)) + (163*c^2*d^11*f^6 - 14*c*d^12*f^5*e - 5*d^13*f^4*e^2)/(d^14*f^6)) - 3*(93*c^
3*d^11*f^6 - 15*c^2*d^12*f^5*e - 9*c*d^13*f^4*e^2 - 5*d^14*f^3*e^3)/(d^14*f^6))*sqrt(d*x + c) - 3*(35*c^4*f^4
- 20*c^3*d*f^3*e - 6*c^2*d^2*f^2*e^2 - 4*c*d^3*f*e^3 - 5*d^4*e^4)*log(abs(-sqrt(d*f)*sqrt(d*x + c) + sqrt((d*x
 + c)*d*f - c*d*f + d^2*e)))/(sqrt(d*f)*d^2*f^3))*B*a*b*abs(d)/d + 8*(sqrt((d*x + c)*d*f - c*d*f + d^2*e)*(2*(
4*(d*x + c)*(6*(d*x + c)*(8*(d*x + c)/d^4 - (41*c*d^19*f^8 - d^20*f^7*e)/(d^23*f^8)) + (513*c^2*d^19*f^8 - 26*
c*d^20*f^7*e - 7*d^21*f^6*e^2)/(d^23*f^8)) - 5*(447*c^3*d^19*f^8 - 37*c^2*d^20*f^7*e - 19*c*d^21*f^6*e^2 - 7*d
^22*f^5*e^3)/(d^23*f^8))*(d*x + c) + 15*(193*c^4*d^19*f^8 - 28*c^3*d^20*f^7*e - 18*c^2*d^21*f^6*e^2 - 12*c*d^2
2*f^5*e^3 - 7*d^23*f^4*e^4)/(d^23*f^8))*sqrt(d*x + c) + 15*(63*c^5*f^5 - 35*c^4*d*f^4*e - 10*c^3*d^2*f^3*e^2 -
 6*c^2*d^3*f^2*e^3 - 5*c*d^4*f*e^4 - 7*d^5*e^5)*log(abs(-sqrt(d*f)*sqrt(d*x + c) + sqrt((d*x + c)*d*f - c*d*f
+ d^2*e)))/(sqrt(d*f)*d^3*f^4))*C*a*b*abs(d)/d + 40*(sqrt((d*x + c)*d*f - c*d*f + d^2*e)*(2*(d*x + c)*(4*(d*x
+ c)*(6*(d*x + c)/d^3 - (25*c*d^11*f^6 - d^12*f^5*e)/(d^14*f^6)) + (163*c^2*d^11*f^6 - 14*c*d^12*f^5*e - 5*d^1
3*f^4*e^2)/(d^14*f^6)) - 3*(93*c^3*d^11*f^6 - 15*c^2*d^12*f^5*e - 9*c*d^13*f^4*e^2 - 5*d^14*f^3*e^3)/(d^14*f^6
))*sqrt(d*x + c) - 3*(35*c^4*f^4 - 20*c^3*d*f^3*e - 6*c^2*d^2*f^2*e^2 - 4*c*d^3*f*e^3 - 5*d^4*e^4)*log(abs(-sq
rt(d*f)*sqrt(d*x + c) + sqrt((d*x + c)*d*f - c*d*f + d^2*e)))/(sqrt(d*f)*d^2*f^3))*A*b^2*abs(d)/d + 4*(sqrt((d
*x + c)*d*f - c*d*f + d^2*e)*(2*(4*(d*x + c)*(6*(d*x + c)*(8*(d*x + c)/d^4 - (41*c*d^19*f^8 - d^20*f^7*e)/(d^2
3*f^8)) + (513*c^2*d^19*f^8 - 26*c*d^20*f^7*e - 7*d^21*f^6*e^2)/(d^23*f^8)) - 5*(447*c^3*d^19*f^8 - 37*c^2*d^2
0*f^7*e - 19*c*d^21*f^6*e^2 - 7*d^22*f^5*e^3)/(d^23*f^8))*(d*x + c) + 15*(193*c^4*d^19*f^8 - 28*c^3*d^20*f^7*e
 - 18*c^2*d^21*f^6*e^2 - 12*c*d^22*f^5*e^3 - 7*d^23*f^4*e^4)/(d^23*f^8))*sqrt(d*x + c) + 15*(63*c^5*f^5 - 35*c
^4*d*f^4*e - 10*c^3*d^2*f^3*e^2 - 6*c^2*d^3*f^2*e^3 - 5*c*d^4*f*e^4 - 7*d^5*e^5)*log(abs(-sqrt(d*f)*sqrt(d*x +
 c) + sqrt((d*x + c)*d*f - c*d*f + d^2*e)))/(sqrt(d*f)*d^3*f^4))*B*b^2*abs(d)/d + (sqrt((d*x + c)*d*f - c*d*f
+ d^2*e)*(2*(4*(2*(d*x + c)*(8*(d*x + c)*(10*(d*x + c)/d^5 - (61*c*d^29*f^10 - d^30*f^9*e)/(d^34*f^10)) + 3*(4
17*c^2*d^29*f^10 - 14*c*d^30*f^9*e - 3*d^31*f^8*e^2)/(d^34*f^10)) - (3481*c^3*d^29*f^10 - 183*c^2*d^30*f^9*e -
 77*c*d^31*f^8*e^2 - 21*d^32*f^7*e^3)/(d^34*f^10))*(d*x + c) + 5*(2279*c^4*d^29*f^10 - 176*c^3*d^30*f^9*e - 10
6*c^2*d^31*f^8*e^2 - 56*c*d^32*f^7*e^3 - 21*d^33*f^6*e^4)/(d^34*f^10))*(d*x + c) - 15*(793*c^5*d^29*f^10 - 105
*c^4*d^30*f^9*e - 70*c^3*d^31*f^8*e^2 - 50*c^2*d^32*f^7*e^3 - 35*c*d^33*f^6*e^4 - 21*d^34*f^5*e^5)/(d^34*f^10)
)*sqrt(d*x + c) - 15*(231*c^6*f^6 - 126*c^5*d*f^5*e - 35*c^4*d^2*f^4*e^2 - 20*c^3*d^3*f^3*e^3 - 15*c^2*d^4*f^2
*e^4 - 14*c*d^5*f*e^5 - 21*d^6*e^6)*log(abs(-sqrt(d*f)*sqrt(d*x + c) + sqrt((d*x + c)*d*f - c*d*f + d^2*e)))/(
sqrt(d*f)*d^4*f^5))*C*b^2*abs(d)/d + 1920*(sqrt((d*x + c)*d*f - c*d*f + d^2*e)*(2*d*x + 2*c - (5*c*f^2 - d*f*e
)/f^2)*sqrt(d*x + c) - (3*c^2*d*f^2 - 2*c*d^2*f*e - d^3*e^2)*log(abs(-sqrt(d*f)*sqrt(d*x + c) + sqrt((d*x + c)
*d*f - c*d*f + d^2*e)))/(sqrt(d*f)*f))*B*a^2*c*abs(d)/d^3 + 3840*(sqrt((d*x + c)*d*f - c*d*f + d^2*e)*(2*d*x +
 2*c - (5*c*f^2 - d*f*e)/f^2)*sqrt(d*x + c) - (3*c^2*d*f^2 - 2*c*d^2*f*e - d^3*e^2)*log(abs(-sqrt(d*f)*sqrt(d*
x + c) + sqrt((d*x + c)*d*f - c*d*f + d^2*e)))/(sqrt(d*f)*f))*A*a*b*c*abs(d)/d^3 + 1920*(sqrt((d*x + c)*d*f -
c*d*f + d^2*e)*(2*d*x + 2*c - (5*c*f^2 - d*f*e)/f^2)*sqrt(d*x + c) - (3*c^2*d*f^2 - 2*c*d^2*f*e - d^3*e^2)*log
(abs(-sqrt(d*f)*sqrt(d*x + c) + sqrt((d*x + c)*d*f - c*d*f + d^2*e)))/(sqrt(d*f)*f))*A*a^2*abs(d)/d^2)/d

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maple [B]  time = 0.05, size = 6728, normalized size = 4.99 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^2*(C*x^2+B*x+A)*(d*x+c)^(1/2)*(f*x+e)^(1/2),x)

[Out]

result too large to display

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*(C*x^2+B*x+A)*(d*x+c)^(1/2)*(f*x+e)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(c*f-d*e>0)', see `assume?` for
 more details)Is c*f-d*e zero or nonzero?

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mupad [F(-1)]  time = 0.00, size = -1, normalized size = -0.00 \[ \text {Hanged} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e + f*x)^(1/2)*(a + b*x)^2*(c + d*x)^(1/2)*(A + B*x + C*x^2),x)

[Out]

\text{Hanged}

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**2*(C*x**2+B*x+A)*(d*x+c)**(1/2)*(f*x+e)**(1/2),x)

[Out]

Timed out

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