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

Optimal. Leaf size=718 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d} \sqrt {e+f x}}\right ) \left (16 a^2 d^2 f^2 \left (4 d f (2 A d f-B (c f+d e))+C \left (3 c^2 f^2+2 c d e f+3 d^2 e^2\right )\right )-16 a b d f \left (2 d f \left (4 A d f (c f+d e)-B \left (3 c^2 f^2+2 c d e f+3 d^2 e^2\right )\right )+C \left (5 c^3 f^3+3 c^2 d e f^2+3 c d^2 e^2 f+5 d^3 e^3\right )\right )+b^2 \left (8 d f \left (2 A d f \left (3 c^2 f^2+2 c d e f+3 d^2 e^2\right )-B \left (5 c^3 f^3+3 c^2 d e f^2+3 c d^2 e^2 f+5 d^3 e^3\right )\right )+C \left (35 c^4 f^4+20 c^3 d e f^3+18 c^2 d^2 e^2 f^2+20 c d^3 e^3 f+35 d^4 e^4\right )\right )\right )}{64 d^{9/2} f^{9/2}}-\frac {\sqrt {c+d x} \sqrt {e+f x} \left (32 a^3 C d^3 f^3-8 a^2 b d^2 f^2 (16 B d f-11 C (c f+d e))-16 a b^2 d f \left (6 d f (4 A d f-3 B (c f+d e))+C \left (15 c^2 f^2+14 c d e f+15 d^2 e^2\right )\right )+2 b d f x (6 b d f (a c C f+a C d e-8 A b d f+6 b c C e)+(4 a d f-5 b (c f+d e)) (2 a C d f-b (8 B d f-7 C (c f+d e))))+b^3 \left (8 d f \left (18 A d f (c f+d e)-B \left (15 c^2 f^2+14 c d e f+15 d^2 e^2\right )\right )+5 C \left (21 c^3 f^3+19 c^2 d e f^2+19 c d^2 e^2 f+21 d^3 e^3\right )\right )\right )}{192 b d^4 f^4}-\frac {(a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} (2 a C d f-b (8 B d f-7 C (c f+d e)))}{24 b d^2 f^2}+\frac {C (a+b x)^3 \sqrt {c+d x} \sqrt {e+f x}}{4 b d f} \]

[Out]

1/64*(16*a^2*d^2*f^2*(C*(3*c^2*f^2+2*c*d*e*f+3*d^2*e^2)+4*d*f*(2*A*d*f-B*(c*f+d*e)))-16*a*b*d*f*(C*(5*c^3*f^3+
3*c^2*d*e*f^2+3*c*d^2*e^2*f+5*d^3*e^3)+2*d*f*(4*A*d*f*(c*f+d*e)-B*(3*c^2*f^2+2*c*d*e*f+3*d^2*e^2)))+b^2*(C*(35
*c^4*f^4+20*c^3*d*e*f^3+18*c^2*d^2*e^2*f^2+20*c*d^3*e^3*f+35*d^4*e^4)+8*d*f*(2*A*d*f*(3*c^2*f^2+2*c*d*e*f+3*d^
2*e^2)-B*(5*c^3*f^3+3*c^2*d*e*f^2+3*c*d^2*e^2*f+5*d^3*e^3))))*arctanh(f^(1/2)*(d*x+c)^(1/2)/d^(1/2)/(f*x+e)^(1
/2))/d^(9/2)/f^(9/2)-1/24*(2*a*C*d*f-b*(8*B*d*f-7*C*(c*f+d*e)))*(b*x+a)^2*(d*x+c)^(1/2)*(f*x+e)^(1/2)/b/d^2/f^
2+1/4*C*(b*x+a)^3*(d*x+c)^(1/2)*(f*x+e)^(1/2)/b/d/f-1/192*(32*a^3*C*d^3*f^3-8*a^2*b*d^2*f^2*(16*B*d*f-11*C*(c*
f+d*e))-16*a*b^2*d*f*(C*(15*c^2*f^2+14*c*d*e*f+15*d^2*e^2)+6*d*f*(4*A*d*f-3*B*(c*f+d*e)))+b^3*(5*C*(21*c^3*f^3
+19*c^2*d*e*f^2+19*c*d^2*e^2*f+21*d^3*e^3)+8*d*f*(18*A*d*f*(c*f+d*e)-B*(15*c^2*f^2+14*c*d*e*f+15*d^2*e^2)))+2*
b*d*f*(6*b*d*f*(-8*A*b*d*f+C*a*c*f+C*a*d*e+6*C*b*c*e)+(4*a*d*f-5*b*(c*f+d*e))*(2*a*C*d*f-b*(8*B*d*f-7*C*(c*f+d
*e))))*x)*(d*x+c)^(1/2)*(f*x+e)^(1/2)/b/d^4/f^4

________________________________________________________________________________________

Rubi [A]  time = 1.34, antiderivative size = 715, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1615, 153, 147, 63, 217, 206} \[ -\frac {\sqrt {c+d x} \sqrt {e+f x} \left (-8 a^2 b d^2 f^2 (16 B d f-11 C (c f+d e))+32 a^3 C d^3 f^3-16 a b^2 d f \left (6 d f (4 A d f-3 B (c f+d e))+C \left (15 c^2 f^2+14 c d e f+15 d^2 e^2\right )\right )+2 b d f x (6 b d f (a c C f+a C d e-8 A b d f+6 b c C e)-(4 a d f-5 b (c f+d e)) (-2 a C d f+8 b B d f-7 b C (c f+d e)))+b^3 \left (8 d f \left (18 A d f (c f+d e)-B \left (15 c^2 f^2+14 c d e f+15 d^2 e^2\right )\right )+5 C \left (19 c^2 d e f^2+21 c^3 f^3+19 c d^2 e^2 f+21 d^3 e^3\right )\right )\right )}{192 b d^4 f^4}+\frac {\tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d} \sqrt {e+f x}}\right ) \left (16 a^2 d^2 f^2 \left (4 d f (2 A d f-B (c f+d e))+C \left (3 c^2 f^2+2 c d e f+3 d^2 e^2\right )\right )-16 a b d f \left (2 d f \left (4 A d f (c f+d e)-B \left (3 c^2 f^2+2 c d e f+3 d^2 e^2\right )\right )+C \left (3 c^2 d e f^2+5 c^3 f^3+3 c d^2 e^2 f+5 d^3 e^3\right )\right )+b^2 \left (8 d f \left (2 A d f \left (3 c^2 f^2+2 c d e f+3 d^2 e^2\right )-B \left (3 c^2 d e f^2+5 c^3 f^3+3 c d^2 e^2 f+5 d^3 e^3\right )\right )+C \left (18 c^2 d^2 e^2 f^2+20 c^3 d e f^3+35 c^4 f^4+20 c d^3 e^3 f+35 d^4 e^4\right )\right )\right )}{64 d^{9/2} f^{9/2}}+\frac {(a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} (-2 a C d f+8 b B d f-7 b C (c f+d e))}{24 b d^2 f^2}+\frac {C (a+b x)^3 \sqrt {c+d x} \sqrt {e+f x}}{4 b d f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((8*b*B*d*f - 2*a*C*d*f - 7*b*C*(d*e + c*f))*(a + b*x)^2*Sqrt[c + d*x]*Sqrt[e + f*x])/(24*b*d^2*f^2) + (C*(a +
b*x)^3*Sqrt[c + d*x]*Sqrt[e + f*x])/(4*b*d*f) - (Sqrt[c + d*x]*Sqrt[e + f*x]*(32*a^3*C*d^3*f^3 - 8*a^2*b*d^2*
f^2*(16*B*d*f - 11*C*(d*e + c*f)) - 16*a*b^2*d*f*(C*(15*d^2*e^2 + 14*c*d*e*f + 15*c^2*f^2) + 6*d*f*(4*A*d*f -
3*B*(d*e + c*f))) + b^3*(5*C*(21*d^3*e^3 + 19*c*d^2*e^2*f + 19*c^2*d*e*f^2 + 21*c^3*f^3) + 8*d*f*(18*A*d*f*(d*
e + c*f) - B*(15*d^2*e^2 + 14*c*d*e*f + 15*c^2*f^2))) + 2*b*d*f*(6*b*d*f*(6*b*c*C*e + a*C*d*e + a*c*C*f - 8*A*
b*d*f) - (4*a*d*f - 5*b*(d*e + c*f))*(8*b*B*d*f - 2*a*C*d*f - 7*b*C*(d*e + c*f)))*x))/(192*b*d^4*f^4) + ((16*a
^2*d^2*f^2*(C*(3*d^2*e^2 + 2*c*d*e*f + 3*c^2*f^2) + 4*d*f*(2*A*d*f - B*(d*e + c*f))) - 16*a*b*d*f*(C*(5*d^3*e^
3 + 3*c*d^2*e^2*f + 3*c^2*d*e*f^2 + 5*c^3*f^3) + 2*d*f*(4*A*d*f*(d*e + c*f) - B*(3*d^2*e^2 + 2*c*d*e*f + 3*c^2
*f^2))) + b^2*(C*(35*d^4*e^4 + 20*c*d^3*e^3*f + 18*c^2*d^2*e^2*f^2 + 20*c^3*d*e*f^3 + 35*c^4*f^4) + 8*d*f*(2*A
*d*f*(3*d^2*e^2 + 2*c*d*e*f + 3*c^2*f^2) - B*(5*d^3*e^3 + 3*c*d^2*e^2*f + 3*c^2*d*e*f^2 + 5*c^3*f^3))))*ArcTan
h[(Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[e + f*x])])/(64*d^(9/2)*f^(9/2))

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 \frac {(a+b x)^2 \left (A+B x+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x}} \, dx &=\frac {C (a+b x)^3 \sqrt {c+d x} \sqrt {e+f x}}{4 b d f}+\frac {\int \frac {(a+b x)^2 \left (-\frac {1}{2} b (6 b c C e+a C d e+a c C f-8 A b d f)+\frac {1}{2} b (8 b B d f-2 a C d f-7 b C (d e+c f)) x\right )}{\sqrt {c+d x} \sqrt {e+f x}} \, dx}{4 b^2 d f}\\ &=\frac {(8 b B d f-2 a C d f-7 b C (d e+c f)) (a+b x)^2 \sqrt {c+d x} \sqrt {e+f x}}{24 b d^2 f^2}+\frac {C (a+b x)^3 \sqrt {c+d x} \sqrt {e+f x}}{4 b d f}+\frac {\int \frac {(a+b x) \left (-\frac {1}{4} b \left (4 a^2 C d f (d e+c f)+4 b^2 c e (8 B d f-7 C (d e+c f))-a b \left (7 C (d e-c f)^2+8 d f (6 A d f-B (d e+c f))\right )\right )-\frac {1}{4} b (6 b d f (6 b c C e+a C d e+a c C f-8 A b d f)-(4 a d f-5 b (d e+c f)) (8 b B d f-2 a C d f-7 b C (d e+c f))) x\right )}{\sqrt {c+d x} \sqrt {e+f x}} \, dx}{12 b^2 d^2 f^2}\\ &=\frac {(8 b B d f-2 a C d f-7 b C (d e+c f)) (a+b x)^2 \sqrt {c+d x} \sqrt {e+f x}}{24 b d^2 f^2}+\frac {C (a+b x)^3 \sqrt {c+d x} \sqrt {e+f x}}{4 b d f}-\frac {\sqrt {c+d x} \sqrt {e+f x} \left (32 a^3 C d^3 f^3-8 a^2 b d^2 f^2 (16 B d f-11 C (d e+c f))-16 a b^2 d f \left (C \left (15 d^2 e^2+14 c d e f+15 c^2 f^2\right )+6 d f (4 A d f-3 B (d e+c f))\right )+b^3 \left (5 C \left (21 d^3 e^3+19 c d^2 e^2 f+19 c^2 d e f^2+21 c^3 f^3\right )+8 d f \left (18 A d f (d e+c f)-B \left (15 d^2 e^2+14 c d e f+15 c^2 f^2\right )\right )\right )+2 b d f (6 b d f (6 b c C e+a C d e+a c C f-8 A b d f)-(4 a d f-5 b (d e+c f)) (8 b B d f-2 a C d f-7 b C (d e+c f))) x\right )}{192 b d^4 f^4}+\frac {\left (16 a^2 d^2 f^2 \left (C \left (3 d^2 e^2+2 c d e f+3 c^2 f^2\right )+4 d f (2 A d f-B (d e+c f))\right )-16 a b d f \left (C \left (5 d^3 e^3+3 c d^2 e^2 f+3 c^2 d e f^2+5 c^3 f^3\right )+2 d f \left (4 A d f (d e+c f)-B \left (3 d^2 e^2+2 c d e f+3 c^2 f^2\right )\right )\right )+b^2 \left (C \left (35 d^4 e^4+20 c d^3 e^3 f+18 c^2 d^2 e^2 f^2+20 c^3 d e f^3+35 c^4 f^4\right )+8 d f \left (2 A d f \left (3 d^2 e^2+2 c d e f+3 c^2 f^2\right )-B \left (5 d^3 e^3+3 c d^2 e^2 f+3 c^2 d e f^2+5 c^3 f^3\right )\right )\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {e+f x}} \, dx}{128 d^4 f^4}\\ &=\frac {(8 b B d f-2 a C d f-7 b C (d e+c f)) (a+b x)^2 \sqrt {c+d x} \sqrt {e+f x}}{24 b d^2 f^2}+\frac {C (a+b x)^3 \sqrt {c+d x} \sqrt {e+f x}}{4 b d f}-\frac {\sqrt {c+d x} \sqrt {e+f x} \left (32 a^3 C d^3 f^3-8 a^2 b d^2 f^2 (16 B d f-11 C (d e+c f))-16 a b^2 d f \left (C \left (15 d^2 e^2+14 c d e f+15 c^2 f^2\right )+6 d f (4 A d f-3 B (d e+c f))\right )+b^3 \left (5 C \left (21 d^3 e^3+19 c d^2 e^2 f+19 c^2 d e f^2+21 c^3 f^3\right )+8 d f \left (18 A d f (d e+c f)-B \left (15 d^2 e^2+14 c d e f+15 c^2 f^2\right )\right )\right )+2 b d f (6 b d f (6 b c C e+a C d e+a c C f-8 A b d f)-(4 a d f-5 b (d e+c f)) (8 b B d f-2 a C d f-7 b C (d e+c f))) x\right )}{192 b d^4 f^4}+\frac {\left (16 a^2 d^2 f^2 \left (C \left (3 d^2 e^2+2 c d e f+3 c^2 f^2\right )+4 d f (2 A d f-B (d e+c f))\right )-16 a b d f \left (C \left (5 d^3 e^3+3 c d^2 e^2 f+3 c^2 d e f^2+5 c^3 f^3\right )+2 d f \left (4 A d f (d e+c f)-B \left (3 d^2 e^2+2 c d e f+3 c^2 f^2\right )\right )\right )+b^2 \left (C \left (35 d^4 e^4+20 c d^3 e^3 f+18 c^2 d^2 e^2 f^2+20 c^3 d e f^3+35 c^4 f^4\right )+8 d f \left (2 A d f \left (3 d^2 e^2+2 c d e f+3 c^2 f^2\right )-B \left (5 d^3 e^3+3 c d^2 e^2 f+3 c^2 d e f^2+5 c^3 f^3\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 )}{64 d^5 f^4}\\ &=\frac {(8 b B d f-2 a C d f-7 b C (d e+c f)) (a+b x)^2 \sqrt {c+d x} \sqrt {e+f x}}{24 b d^2 f^2}+\frac {C (a+b x)^3 \sqrt {c+d x} \sqrt {e+f x}}{4 b d f}-\frac {\sqrt {c+d x} \sqrt {e+f x} \left (32 a^3 C d^3 f^3-8 a^2 b d^2 f^2 (16 B d f-11 C (d e+c f))-16 a b^2 d f \left (C \left (15 d^2 e^2+14 c d e f+15 c^2 f^2\right )+6 d f (4 A d f-3 B (d e+c f))\right )+b^3 \left (5 C \left (21 d^3 e^3+19 c d^2 e^2 f+19 c^2 d e f^2+21 c^3 f^3\right )+8 d f \left (18 A d f (d e+c f)-B \left (15 d^2 e^2+14 c d e f+15 c^2 f^2\right )\right )\right )+2 b d f (6 b d f (6 b c C e+a C d e+a c C f-8 A b d f)-(4 a d f-5 b (d e+c f)) (8 b B d f-2 a C d f-7 b C (d e+c f))) x\right )}{192 b d^4 f^4}+\frac {\left (16 a^2 d^2 f^2 \left (C \left (3 d^2 e^2+2 c d e f+3 c^2 f^2\right )+4 d f (2 A d f-B (d e+c f))\right )-16 a b d f \left (C \left (5 d^3 e^3+3 c d^2 e^2 f+3 c^2 d e f^2+5 c^3 f^3\right )+2 d f \left (4 A d f (d e+c f)-B \left (3 d^2 e^2+2 c d e f+3 c^2 f^2\right )\right )\right )+b^2 \left (C \left (35 d^4 e^4+20 c d^3 e^3 f+18 c^2 d^2 e^2 f^2+20 c^3 d e f^3+35 c^4 f^4\right )+8 d f \left (2 A d f \left (3 d^2 e^2+2 c d e f+3 c^2 f^2\right )-B \left (5 d^3 e^3+3 c d^2 e^2 f+3 c^2 d e f^2+5 c^3 f^3\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 )}{64 d^5 f^4}\\ &=\frac {(8 b B d f-2 a C d f-7 b C (d e+c f)) (a+b x)^2 \sqrt {c+d x} \sqrt {e+f x}}{24 b d^2 f^2}+\frac {C (a+b x)^3 \sqrt {c+d x} \sqrt {e+f x}}{4 b d f}-\frac {\sqrt {c+d x} \sqrt {e+f x} \left (32 a^3 C d^3 f^3-8 a^2 b d^2 f^2 (16 B d f-11 C (d e+c f))-16 a b^2 d f \left (C \left (15 d^2 e^2+14 c d e f+15 c^2 f^2\right )+6 d f (4 A d f-3 B (d e+c f))\right )+b^3 \left (5 C \left (21 d^3 e^3+19 c d^2 e^2 f+19 c^2 d e f^2+21 c^3 f^3\right )+8 d f \left (18 A d f (d e+c f)-B \left (15 d^2 e^2+14 c d e f+15 c^2 f^2\right )\right )\right )+2 b d f (6 b d f (6 b c C e+a C d e+a c C f-8 A b d f)-(4 a d f-5 b (d e+c f)) (8 b B d f-2 a C d f-7 b C (d e+c f))) x\right )}{192 b d^4 f^4}+\frac {\left (16 a^2 d^2 f^2 \left (C \left (3 d^2 e^2+2 c d e f+3 c^2 f^2\right )+4 d f (2 A d f-B (d e+c f))\right )-16 a b d f \left (C \left (5 d^3 e^3+3 c d^2 e^2 f+3 c^2 d e f^2+5 c^3 f^3\right )+2 d f \left (4 A d f (d e+c f)-B \left (3 d^2 e^2+2 c d e f+3 c^2 f^2\right )\right )\right )+b^2 \left (C \left (35 d^4 e^4+20 c d^3 e^3 f+18 c^2 d^2 e^2 f^2+20 c^3 d e f^3+35 c^4 f^4\right )+8 d f \left (2 A d f \left (3 d^2 e^2+2 c d e f+3 c^2 f^2\right )-B \left (5 d^3 e^3+3 c d^2 e^2 f+3 c^2 d e f^2+5 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 )}{64 d^{9/2} f^{9/2}}\\ \end {align*}

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Mathematica [B]  time = 6.49, size = 2195, normalized size = 3.06 \[ \text {Result too large to show} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(b*e - a*f)^2*Sqrt[d*e - c*f]*(C*e^2 - f*(B*e - A*f))*Sqrt[(d*(e + f*x))/(d*e - c*f)]*ArcSinh[(Sqrt[f]*Sqrt
[c + d*x])/Sqrt[d*e - c*f]])/(d*f^(9/2)*Sqrt[e + f*x]) + (2*b^2*C*(d*e - c*f)^3*Sqrt[c + d*x]*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)*((35/(16*(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/(24*(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/(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 + (35*Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)]*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)])])/(128*Sqrt[d]*Sqrt[f]*Sqrt[c + d*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))))/(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*b*(d*e - c*f)^2*(-4*b*C*e + b
*B*f + 2*a*C*f)*Sqrt[c + d*x]*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)*((15/(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/(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))/6 + (5*Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e
- c*f) - (c*d*f)/(d*e - c*f)]*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)])])/(16*Sqrt[d]*Sqrt[f]*Sqrt[c + d*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))))/(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*(d*e - c*f)*(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)*Sqrt[c + d*x]*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/(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) +
(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 + (3*Sqrt[d*e - c*f]*S
qrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)]*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)])])/(8*Sqrt[d]*Sqrt[f]*Sqrt[c + d*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))))/(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)*(4*b*C*e^2 - 3*b*B*e*f - 2*a*C*e*f + 2*A*b*f^2
+ a*B*f^2)*Sqrt[c + d*x]*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)*(1/(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))))) + (Sqrt[
d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)]*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)])])/(2*Sqrt[d]*Sqrt[f]*Sqrt[c + d*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))))/(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 = 6.27, size = 1436, normalized size = 2.00 \[ \text {result too large to display} \]

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]

[1/768*(3*(35*C*b^2*d^4*e^4 + 20*(C*b^2*c*d^3 - 2*(2*C*a*b + B*b^2)*d^4)*e^3*f + 6*(3*C*b^2*c^2*d^2 - 4*(2*C*a
*b + B*b^2)*c*d^3 + 8*(C*a^2 + 2*B*a*b + A*b^2)*d^4)*e^2*f^2 + 4*(5*C*b^2*c^3*d - 6*(2*C*a*b + B*b^2)*c^2*d^2
+ 8*(C*a^2 + 2*B*a*b + A*b^2)*c*d^3 - 16*(B*a^2 + 2*A*a*b)*d^4)*e*f^3 + (35*C*b^2*c^4 + 128*A*a^2*d^4 - 40*(2*
C*a*b + B*b^2)*c^3*d + 48*(C*a^2 + 2*B*a*b + A*b^2)*c^2*d^2 - 64*(B*a^2 + 2*A*a*b)*c*d^3)*f^4)*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*(48*C*b^2*d^4*f^4*x^3 - 105*C*b^2*d^4*e^3*f - 5*(19*C*b^2*c*d^3 - 24*(2*C*a*b +
B*b^2)*d^4)*e^2*f^2 - (95*C*b^2*c^2*d^2 - 112*(2*C*a*b + B*b^2)*c*d^3 + 144*(C*a^2 + 2*B*a*b + A*b^2)*d^4)*e*f
^3 - 3*(35*C*b^2*c^3*d - 40*(2*C*a*b + B*b^2)*c^2*d^2 + 48*(C*a^2 + 2*B*a*b + A*b^2)*c*d^3 - 64*(B*a^2 + 2*A*a
*b)*d^4)*f^4 - 8*(7*C*b^2*d^4*e*f^3 + (7*C*b^2*c*d^3 - 8*(2*C*a*b + B*b^2)*d^4)*f^4)*x^2 + 2*(35*C*b^2*d^4*e^2
*f^2 + 2*(17*C*b^2*c*d^3 - 20*(2*C*a*b + B*b^2)*d^4)*e*f^3 + (35*C*b^2*c^2*d^2 - 40*(2*C*a*b + B*b^2)*c*d^3 +
48*(C*a^2 + 2*B*a*b + A*b^2)*d^4)*f^4)*x)*sqrt(d*x + c)*sqrt(f*x + e))/(d^5*f^5), -1/384*(3*(35*C*b^2*d^4*e^4
+ 20*(C*b^2*c*d^3 - 2*(2*C*a*b + B*b^2)*d^4)*e^3*f + 6*(3*C*b^2*c^2*d^2 - 4*(2*C*a*b + B*b^2)*c*d^3 + 8*(C*a^2
+ 2*B*a*b + A*b^2)*d^4)*e^2*f^2 + 4*(5*C*b^2*c^3*d - 6*(2*C*a*b + B*b^2)*c^2*d^2 + 8*(C*a^2 + 2*B*a*b + A*b^2
)*c*d^3 - 16*(B*a^2 + 2*A*a*b)*d^4)*e*f^3 + (35*C*b^2*c^4 + 128*A*a^2*d^4 - 40*(2*C*a*b + B*b^2)*c^3*d + 48*(C
*a^2 + 2*B*a*b + A*b^2)*c^2*d^2 - 64*(B*a^2 + 2*A*a*b)*c*d^3)*f^4)*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*(48*C*b^2*d^4*f^4
*x^3 - 105*C*b^2*d^4*e^3*f - 5*(19*C*b^2*c*d^3 - 24*(2*C*a*b + B*b^2)*d^4)*e^2*f^2 - (95*C*b^2*c^2*d^2 - 112*(
2*C*a*b + B*b^2)*c*d^3 + 144*(C*a^2 + 2*B*a*b + A*b^2)*d^4)*e*f^3 - 3*(35*C*b^2*c^3*d - 40*(2*C*a*b + B*b^2)*c
^2*d^2 + 48*(C*a^2 + 2*B*a*b + A*b^2)*c*d^3 - 64*(B*a^2 + 2*A*a*b)*d^4)*f^4 - 8*(7*C*b^2*d^4*e*f^3 + (7*C*b^2*
c*d^3 - 8*(2*C*a*b + B*b^2)*d^4)*f^4)*x^2 + 2*(35*C*b^2*d^4*e^2*f^2 + 2*(17*C*b^2*c*d^3 - 20*(2*C*a*b + B*b^2)
*d^4)*e*f^3 + (35*C*b^2*c^2*d^2 - 40*(2*C*a*b + B*b^2)*c*d^3 + 48*(C*a^2 + 2*B*a*b + A*b^2)*d^4)*f^4)*x)*sqrt(
d*x + c)*sqrt(f*x + e))/(d^5*f^5)]

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giac [A]  time = 2.51, size = 951, normalized size = 1.32 \[ \frac {{\left (\sqrt {{\left (d x + c\right )} d f - c d f + d^{2} e} {\left (2 \, {\left (d x + c\right )} {\left (4 \, {\left (d x + c\right )} {\left (\frac {6 \, {\left (d x + c\right )} C b^{2}}{d^{5} f} - \frac {25 \, C b^{2} c d^{19} f^{6} - 16 \, C a b d^{20} f^{6} - 8 \, B b^{2} d^{20} f^{6} + 7 \, C b^{2} d^{20} f^{5} e}{d^{24} f^{7}}\right )} + \frac {163 \, C b^{2} c^{2} d^{19} f^{6} - 208 \, C a b c d^{20} f^{6} - 104 \, B b^{2} c d^{20} f^{6} + 48 \, C a^{2} d^{21} f^{6} + 96 \, B a b d^{21} f^{6} + 48 \, A b^{2} d^{21} f^{6} + 90 \, C b^{2} c d^{20} f^{5} e - 80 \, C a b d^{21} f^{5} e - 40 \, B b^{2} d^{21} f^{5} e + 35 \, C b^{2} d^{21} f^{4} e^{2}}{d^{24} f^{7}}\right )} - \frac {3 \, {\left (93 \, C b^{2} c^{3} d^{19} f^{6} - 176 \, C a b c^{2} d^{20} f^{6} - 88 \, B b^{2} c^{2} d^{20} f^{6} + 80 \, C a^{2} c d^{21} f^{6} + 160 \, B a b c d^{21} f^{6} + 80 \, A b^{2} c d^{21} f^{6} - 64 \, B a^{2} d^{22} f^{6} - 128 \, A a b d^{22} f^{6} + 73 \, C b^{2} c^{2} d^{20} f^{5} e - 128 \, C a b c d^{21} f^{5} e - 64 \, B b^{2} c d^{21} f^{5} e + 48 \, C a^{2} d^{22} f^{5} e + 96 \, B a b d^{22} f^{5} e + 48 \, A b^{2} d^{22} f^{5} e + 55 \, C b^{2} c d^{21} f^{4} e^{2} - 80 \, C a b d^{22} f^{4} e^{2} - 40 \, B b^{2} d^{22} f^{4} e^{2} + 35 \, C b^{2} d^{22} f^{3} e^{3}\right )}}{d^{24} f^{7}}\right )} \sqrt {d x + c} - \frac {3 \, {\left (35 \, C b^{2} c^{4} f^{4} - 80 \, C a b c^{3} d f^{4} - 40 \, B b^{2} c^{3} d f^{4} + 48 \, C a^{2} c^{2} d^{2} f^{4} + 96 \, B a b c^{2} d^{2} f^{4} + 48 \, A b^{2} c^{2} d^{2} f^{4} - 64 \, B a^{2} c d^{3} f^{4} - 128 \, A a b c d^{3} f^{4} + 128 \, A a^{2} d^{4} f^{4} + 20 \, C b^{2} c^{3} d f^{3} e - 48 \, C a b c^{2} d^{2} f^{3} e - 24 \, B b^{2} c^{2} d^{2} f^{3} e + 32 \, C a^{2} c d^{3} f^{3} e + 64 \, B a b c d^{3} f^{3} e + 32 \, A b^{2} c d^{3} f^{3} e - 64 \, B a^{2} d^{4} f^{3} e - 128 \, A a b d^{4} f^{3} e + 18 \, C b^{2} c^{2} d^{2} f^{2} e^{2} - 48 \, C a b c d^{3} f^{2} e^{2} - 24 \, B b^{2} c d^{3} f^{2} e^{2} + 48 \, C a^{2} d^{4} f^{2} e^{2} + 96 \, B a b d^{4} f^{2} e^{2} + 48 \, A b^{2} d^{4} f^{2} e^{2} + 20 \, C b^{2} c d^{3} f e^{3} - 80 \, C a b d^{4} f e^{3} - 40 \, B b^{2} d^{4} f e^{3} + 35 \, C b^{2} d^{4} e^{4}\right )} \log \left ({\left | -\sqrt {d f} \sqrt {d x + c} + \sqrt {{\left (d x + c\right )} d f - c d f + d^{2} e} \right |}\right )}{\sqrt {d f} d^{4} f^{4}}\right )} d}{192 \, {\left | d \right |}} \]

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]

1/192*(sqrt((d*x + c)*d*f - c*d*f + d^2*e)*(2*(d*x + c)*(4*(d*x + c)*(6*(d*x + c)*C*b^2/(d^5*f) - (25*C*b^2*c*
d^19*f^6 - 16*C*a*b*d^20*f^6 - 8*B*b^2*d^20*f^6 + 7*C*b^2*d^20*f^5*e)/(d^24*f^7)) + (163*C*b^2*c^2*d^19*f^6 -
208*C*a*b*c*d^20*f^6 - 104*B*b^2*c*d^20*f^6 + 48*C*a^2*d^21*f^6 + 96*B*a*b*d^21*f^6 + 48*A*b^2*d^21*f^6 + 90*C
*b^2*c*d^20*f^5*e - 80*C*a*b*d^21*f^5*e - 40*B*b^2*d^21*f^5*e + 35*C*b^2*d^21*f^4*e^2)/(d^24*f^7)) - 3*(93*C*b
^2*c^3*d^19*f^6 - 176*C*a*b*c^2*d^20*f^6 - 88*B*b^2*c^2*d^20*f^6 + 80*C*a^2*c*d^21*f^6 + 160*B*a*b*c*d^21*f^6
+ 80*A*b^2*c*d^21*f^6 - 64*B*a^2*d^22*f^6 - 128*A*a*b*d^22*f^6 + 73*C*b^2*c^2*d^20*f^5*e - 128*C*a*b*c*d^21*f^
5*e - 64*B*b^2*c*d^21*f^5*e + 48*C*a^2*d^22*f^5*e + 96*B*a*b*d^22*f^5*e + 48*A*b^2*d^22*f^5*e + 55*C*b^2*c*d^2
1*f^4*e^2 - 80*C*a*b*d^22*f^4*e^2 - 40*B*b^2*d^22*f^4*e^2 + 35*C*b^2*d^22*f^3*e^3)/(d^24*f^7))*sqrt(d*x + c) -
3*(35*C*b^2*c^4*f^4 - 80*C*a*b*c^3*d*f^4 - 40*B*b^2*c^3*d*f^4 + 48*C*a^2*c^2*d^2*f^4 + 96*B*a*b*c^2*d^2*f^4 +
48*A*b^2*c^2*d^2*f^4 - 64*B*a^2*c*d^3*f^4 - 128*A*a*b*c*d^3*f^4 + 128*A*a^2*d^4*f^4 + 20*C*b^2*c^3*d*f^3*e -
48*C*a*b*c^2*d^2*f^3*e - 24*B*b^2*c^2*d^2*f^3*e + 32*C*a^2*c*d^3*f^3*e + 64*B*a*b*c*d^3*f^3*e + 32*A*b^2*c*d^3
*f^3*e - 64*B*a^2*d^4*f^3*e - 128*A*a*b*d^4*f^3*e + 18*C*b^2*c^2*d^2*f^2*e^2 - 48*C*a*b*c*d^3*f^2*e^2 - 24*B*b
^2*c*d^3*f^2*e^2 + 48*C*a^2*d^4*f^2*e^2 + 96*B*a*b*d^4*f^2*e^2 + 48*A*b^2*d^4*f^2*e^2 + 20*C*b^2*c*d^3*f*e^3 -
80*C*a*b*d^4*f*e^3 - 40*B*b^2*d^4*f*e^3 + 35*C*b^2*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^4*f^4))*d/abs(d)

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maple [B]  time = 0.05, size = 2528, normalized size = 3.52 \[ \text {result too large to display} \]

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]

1/384*(144*A*ln(1/2*(2*d*f*x+c*f+d*e+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2))/(d*f)^(1/2))*b^2*d^4*e^2*f^2+192*A
*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*x*b^2*d^3*f^3-384*A*ln(1/2*(2*d*f*x+c*f+d*e+2*((d*x+c)*(f*x+e))^(1/2)*(d*
f)^(1/2))/(d*f)^(1/2))*a*b*c*d^3*f^4-384*A*ln(1/2*(2*d*f*x+c*f+d*e+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2))/(d*f
)^(1/2))*a*b*d^4*e*f^3+96*C*x^3*b^2*d^3*f^3*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)+128*B*x^2*b^2*d^3*f^3*(d*f)^(1
/2)*((d*x+c)*(f*x+e))^(1/2)+96*A*ln(1/2*(2*d*f*x+c*f+d*e+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2))/(d*f)^(1/2))*b
^2*c*d^3*e*f^3+60*C*ln(1/2*(2*d*f*x+c*f+d*e+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2))/(d*f)^(1/2))*b^2*c*d^3*e^3*
f-72*B*ln(1/2*(2*d*f*x+c*f+d*e+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2))/(d*f)^(1/2))*b^2*c^2*d^2*e*f^3-72*B*ln(1
/2*(2*d*f*x+c*f+d*e+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2))/(d*f)^(1/2))*b^2*c*d^3*e^2*f^2+96*C*ln(1/2*(2*d*f*x
+c*f+d*e+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2))/(d*f)^(1/2))*a^2*c*d^3*e*f^3+60*C*ln(1/2*(2*d*f*x+c*f+d*e+2*((
d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2))/(d*f)^(1/2))*b^2*c^3*d*e*f^3+54*C*ln(1/2*(2*d*f*x+c*f+d*e+2*((d*x+c)*(f*x+e
))^(1/2)*(d*f)^(1/2))/(d*f)^(1/2))*b^2*c^2*d^2*e^2*f^2+192*C*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*x*a^2*d^3*f^3
-240*C*ln(1/2*(2*d*f*x+c*f+d*e+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2))/(d*f)^(1/2))*a*b*c^3*d*f^4-240*C*ln(1/2*
(2*d*f*x+c*f+d*e+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2))/(d*f)^(1/2))*a*b*d^4*e^3*f+768*A*(d*f)^(1/2)*((d*x+c)*
(f*x+e))^(1/2)*a*b*d^3*f^3-288*A*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*b^2*c*d^2*f^3-288*A*(d*f)^(1/2)*((d*x+c)*
(f*x+e))^(1/2)*b^2*d^3*e*f^2+288*B*ln(1/2*(2*d*f*x+c*f+d*e+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2))/(d*f)^(1/2))
*a*b*c^2*d^2*f^4+288*B*ln(1/2*(2*d*f*x+c*f+d*e+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2))/(d*f)^(1/2))*a*b*d^4*e^2
*f^2+240*B*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*b^2*c^2*d*f^3+240*B*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*b^2*d^3
*e^2*f+105*C*ln(1/2*(2*d*f*x+c*f+d*e+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2))/(d*f)^(1/2))*b^2*c^4*f^4+105*C*ln(
1/2*(2*d*f*x+c*f+d*e+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2))/(d*f)^(1/2))*b^2*d^4*e^4-192*B*ln(1/2*(2*d*f*x+c*f
+d*e+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2))/(d*f)^(1/2))*a^2*c*d^3*f^4-192*B*ln(1/2*(2*d*f*x+c*f+d*e+2*((d*x+c
)*(f*x+e))^(1/2)*(d*f)^(1/2))/(d*f)^(1/2))*a^2*d^4*e*f^3+144*A*ln(1/2*(2*d*f*x+c*f+d*e+2*((d*x+c)*(f*x+e))^(1/
2)*(d*f)^(1/2))/(d*f)^(1/2))*b^2*c^2*d^2*f^4-120*B*ln(1/2*(2*d*f*x+c*f+d*e+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/
2))/(d*f)^(1/2))*b^2*c^3*d*f^4-120*B*ln(1/2*(2*d*f*x+c*f+d*e+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2))/(d*f)^(1/2
))*b^2*d^4*e^3*f+144*C*ln(1/2*(2*d*f*x+c*f+d*e+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2))/(d*f)^(1/2))*a^2*c^2*d^2
*f^4+144*C*ln(1/2*(2*d*f*x+c*f+d*e+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2))/(d*f)^(1/2))*a^2*d^4*e^2*f^2+384*B*(
d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*a^2*d^3*f^3-210*C*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*b^2*c^3*f^3-210*C*(d*
f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*b^2*d^3*e^3+384*A*ln(1/2*(2*d*f*x+c*f+d*e+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/
2))/(d*f)^(1/2))*a^2*d^4*f^4-288*C*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*a^2*c*d^2*f^3-288*C*(d*f)^(1/2)*((d*x+c
)*(f*x+e))^(1/2)*a^2*d^3*e*f^2-190*C*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*b^2*c^2*d*e*f^2-190*C*(d*f)^(1/2)*((d
*x+c)*(f*x+e))^(1/2)*b^2*c*d^2*e^2*f-160*B*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*x*b^2*d^3*e*f^2+140*C*(d*f)^(1/
2)*((d*x+c)*(f*x+e))^(1/2)*x*b^2*c^2*d*f^3-144*C*ln(1/2*(2*d*f*x+c*f+d*e+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)
)/(d*f)^(1/2))*a*b*c*d^3*e^2*f^2+192*B*ln(1/2*(2*d*f*x+c*f+d*e+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2))/(d*f)^(1
/2))*a*b*c*d^3*e*f^3-144*C*ln(1/2*(2*d*f*x+c*f+d*e+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2))/(d*f)^(1/2))*a*b*c^2
*d^2*e*f^3-576*B*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*a*b*c*d^2*f^3-576*B*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*a
*b*d^3*e*f^2+224*B*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*b^2*c*d^2*e*f^2+480*C*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/
2)*a*b*c^2*d*f^3+140*C*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*x*b^2*d^3*e^2*f+256*C*x^2*a*b*d^3*f^3*(d*f)^(1/2)*(
(d*x+c)*(f*x+e))^(1/2)-112*C*x^2*b^2*c*d^2*f^3*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)-112*C*x^2*b^2*d^3*e*f^2*(d*
f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)+384*B*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*x*a*b*d^3*f^3-160*B*(d*f)^(1/2)*((d
*x+c)*(f*x+e))^(1/2)*x*b^2*c*d^2*f^3+480*C*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*a*b*d^3*e^2*f+448*C*(d*f)^(1/2)
*((d*x+c)*(f*x+e))^(1/2)*a*b*c*d^2*e*f^2-320*C*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*x*a*b*c*d^2*f^3-320*C*(d*f)
^(1/2)*((d*x+c)*(f*x+e))^(1/2)*x*a*b*d^3*e*f^2+136*C*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*x*b^2*c*d^2*e*f^2)*(d
*x+c)^(1/2)*(f*x+e)^(1/2)/(d*f)^(1/2)/f^4/d^4/((d*x+c)*(f*x+e))^(1/2)

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