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

Optimal. Leaf size=1032 \[ \frac{C (c+d x)^{3/2} \sqrt{e+f x} (a+b x)^3}{5 b d f}-\frac{(4 a C d f+b (9 C d e+7 c C f-10 B d f)) (c+d x)^{3/2} \sqrt{e+f x} (a+b x)^2}{40 b d^2 f^2}-\frac{(c+d x)^{3/2} \sqrt{e+f x} \left (\left (C \left (315 d^3 e^3+203 c d^2 f e^2+145 c^2 d f^2 e+105 c^3 f^3\right )+10 d f \left (8 A d f (5 d e+3 c f)-B \left (35 d^2 e^2+22 c d f e+15 c^2 f^2\right )\right )\right ) b^3+20 a d f \left (8 d f (5 B d e+3 B c f-6 A d f)-C \left (35 d^2 e^2+22 c d f e+15 c^2 f^2\right )\right ) b^2+8 a^2 d^2 f^2 (23 C d e+9 c C f-30 B d f) b+4 d f (8 b d f (6 b c C e+3 a C d e+a c C f-10 A b d f)-(7 b d e+5 b c f-4 a d f) (4 a C d f+b (9 C d e+7 c C f-10 B d f))) x b+96 a^3 C d^3 f^3\right )}{960 b d^4 f^4}+\frac{(d e-c f) \left (-\left (C \left (63 d^4 e^4+28 c d^3 f e^3+18 c^2 d^2 f^2 e^2+12 c^3 d f^3 e+7 c^4 f^4\right )+2 d f \left (8 A d f \left (5 d^2 e^2+2 c d f e+c^2 f^2\right )-B \left (35 d^3 e^3+15 c d^2 f e^2+9 c^2 d f^2 e+5 c^3 f^3\right )\right )\right ) b^2+4 a d f \left (C \left (35 d^3 e^3+15 c d^2 f e^2+9 c^2 d f^2 e+5 c^3 f^3\right )+8 d f \left (2 A d f (3 d e+c f)-B \left (5 d^2 e^2+2 c d f e+c^2 f^2\right )\right )\right ) b+16 a^2 d^2 f^2 \left (2 d f (3 B d e+B c f-4 A d f)-C \left (5 d^2 e^2+2 c d f e+c^2 f^2\right )\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d} \sqrt{e+f x}}\right )}{128 d^{9/2} f^{11/2}}-\frac{\left (-\left (C \left (63 d^4 e^4+28 c d^3 f e^3+18 c^2 d^2 f^2 e^2+12 c^3 d f^3 e+7 c^4 f^4\right )+2 d f \left (8 A d f \left (5 d^2 e^2+2 c d f e+c^2 f^2\right )-B \left (35 d^3 e^3+15 c d^2 f e^2+9 c^2 d f^2 e+5 c^3 f^3\right )\right )\right ) b^2+4 a d f \left (C \left (35 d^3 e^3+15 c d^2 f e^2+9 c^2 d f^2 e+5 c^3 f^3\right )+8 d f \left (2 A d f (3 d e+c f)-B \left (5 d^2 e^2+2 c d f e+c^2 f^2\right )\right )\right ) b+16 a^2 d^2 f^2 \left (2 d f (3 B d e+B c f-4 A d f)-C \left (5 d^2 e^2+2 c d f e+c^2 f^2\right )\right )\right ) \sqrt{c+d x} \sqrt{e+f x}}{128 d^4 f^5} \]

[Out]

-((16*a^2*d^2*f^2*(2*d*f*(3*B*d*e + B*c*f - 4*A*d*f) - C*(5*d^2*e^2 + 2*c*d*e*f + c^2*f^2)) + 4*a*b*d*f*(C*(35
*d^3*e^3 + 15*c*d^2*e^2*f + 9*c^2*d*e*f^2 + 5*c^3*f^3) + 8*d*f*(2*A*d*f*(3*d*e + c*f) - B*(5*d^2*e^2 + 2*c*d*e
*f + c^2*f^2))) - b^2*(C*(63*d^4*e^4 + 28*c*d^3*e^3*f + 18*c^2*d^2*e^2*f^2 + 12*c^3*d*e*f^3 + 7*c^4*f^4) + 2*d
*f*(8*A*d*f*(5*d^2*e^2 + 2*c*d*e*f + c^2*f^2) - B*(35*d^3*e^3 + 15*c*d^2*e^2*f + 9*c^2*d*e*f^2 + 5*c^3*f^3))))
*Sqrt[c + d*x]*Sqrt[e + f*x])/(128*d^4*f^5) - ((4*a*C*d*f + b*(9*C*d*e + 7*c*C*f - 10*B*d*f))*(a + b*x)^2*(c +
d*x)^(3/2)*Sqrt[e + f*x])/(40*b*d^2*f^2) + (C*(a + b*x)^3*(c + d*x)^(3/2)*Sqrt[e + f*x])/(5*b*d*f) - ((c + d*
x)^(3/2)*Sqrt[e + f*x]*(96*a^3*C*d^3*f^3 + 8*a^2*b*d^2*f^2*(23*C*d*e + 9*c*C*f - 30*B*d*f) + 20*a*b^2*d*f*(8*d
*f*(5*B*d*e + 3*B*c*f - 6*A*d*f) - C*(35*d^2*e^2 + 22*c*d*e*f + 15*c^2*f^2)) + b^3*(C*(315*d^3*e^3 + 203*c*d^2
*e^2*f + 145*c^2*d*e*f^2 + 105*c^3*f^3) + 10*d*f*(8*A*d*f*(5*d*e + 3*c*f) - B*(35*d^2*e^2 + 22*c*d*e*f + 15*c^
2*f^2))) + 4*b*d*f*(8*b*d*f*(6*b*c*C*e + 3*a*C*d*e + a*c*C*f - 10*A*b*d*f) - (7*b*d*e + 5*b*c*f - 4*a*d*f)*(4*
a*C*d*f + b*(9*C*d*e + 7*c*C*f - 10*B*d*f)))*x))/(960*b*d^4*f^4) + ((d*e - c*f)*(16*a^2*d^2*f^2*(2*d*f*(3*B*d*
e + B*c*f - 4*A*d*f) - C*(5*d^2*e^2 + 2*c*d*e*f + c^2*f^2)) + 4*a*b*d*f*(C*(35*d^3*e^3 + 15*c*d^2*e^2*f + 9*c^
2*d*e*f^2 + 5*c^3*f^3) + 8*d*f*(2*A*d*f*(3*d*e + c*f) - B*(5*d^2*e^2 + 2*c*d*e*f + c^2*f^2))) - b^2*(C*(63*d^4
*e^4 + 28*c*d^3*e^3*f + 18*c^2*d^2*e^2*f^2 + 12*c^3*d*e*f^3 + 7*c^4*f^4) + 2*d*f*(8*A*d*f*(5*d^2*e^2 + 2*c*d*e
*f + c^2*f^2) - B*(35*d^3*e^3 + 15*c*d^2*e^2*f + 9*c^2*d*e*f^2 + 5*c^3*f^3))))*ArcTanh[(Sqrt[f]*Sqrt[c + d*x])
/(Sqrt[d]*Sqrt[e + f*x])])/(128*d^(9/2)*f^(11/2))

________________________________________________________________________________________

Rubi [A]  time = 1.78768, antiderivative size = 1032, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ \frac{C (c+d x)^{3/2} \sqrt{e+f x} (a+b x)^3}{5 b d f}-\frac{(4 a C d f+b (9 C d e+7 c C f-10 B d f)) (c+d x)^{3/2} \sqrt{e+f x} (a+b x)^2}{40 b d^2 f^2}-\frac{(c+d x)^{3/2} \sqrt{e+f x} \left (\left (C \left (315 d^3 e^3+203 c d^2 f e^2+145 c^2 d f^2 e+105 c^3 f^3\right )+10 d f \left (8 A d f (5 d e+3 c f)-B \left (35 d^2 e^2+22 c d f e+15 c^2 f^2\right )\right )\right ) b^3+20 a d f \left (8 d f (5 B d e+3 B c f-6 A d f)-C \left (35 d^2 e^2+22 c d f e+15 c^2 f^2\right )\right ) b^2+8 a^2 d^2 f^2 (23 C d e+9 c C f-30 B d f) b+4 d f (8 b d f (6 b c C e+3 a C d e+a c C f-10 A b d f)-(7 b d e+5 b c f-4 a d f) (4 a C d f+b (9 C d e+7 c C f-10 B d f))) x b+96 a^3 C d^3 f^3\right )}{960 b d^4 f^4}+\frac{(d e-c f) \left (-\left (C \left (63 d^4 e^4+28 c d^3 f e^3+18 c^2 d^2 f^2 e^2+12 c^3 d f^3 e+7 c^4 f^4\right )+2 d f \left (8 A d f \left (5 d^2 e^2+2 c d f e+c^2 f^2\right )-B \left (35 d^3 e^3+15 c d^2 f e^2+9 c^2 d f^2 e+5 c^3 f^3\right )\right )\right ) b^2+4 a d f \left (C \left (35 d^3 e^3+15 c d^2 f e^2+9 c^2 d f^2 e+5 c^3 f^3\right )+8 d f \left (2 A d f (3 d e+c f)-B \left (5 d^2 e^2+2 c d f e+c^2 f^2\right )\right )\right ) b+16 a^2 d^2 f^2 \left (2 d f (3 B d e+B c f-4 A d f)-C \left (5 d^2 e^2+2 c d f e+c^2 f^2\right )\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d} \sqrt{e+f x}}\right )}{128 d^{9/2} f^{11/2}}-\frac{\left (-\left (C \left (63 d^4 e^4+28 c d^3 f e^3+18 c^2 d^2 f^2 e^2+12 c^3 d f^3 e+7 c^4 f^4\right )+2 d f \left (8 A d f \left (5 d^2 e^2+2 c d f e+c^2 f^2\right )-B \left (35 d^3 e^3+15 c d^2 f e^2+9 c^2 d f^2 e+5 c^3 f^3\right )\right )\right ) b^2+4 a d f \left (C \left (35 d^3 e^3+15 c d^2 f e^2+9 c^2 d f^2 e+5 c^3 f^3\right )+8 d f \left (2 A d f (3 d e+c f)-B \left (5 d^2 e^2+2 c d f e+c^2 f^2\right )\right )\right ) b+16 a^2 d^2 f^2 \left (2 d f (3 B d e+B c f-4 A d f)-C \left (5 d^2 e^2+2 c d f e+c^2 f^2\right )\right )\right ) \sqrt{c+d x} \sqrt{e+f x}}{128 d^4 f^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((16*a^2*d^2*f^2*(2*d*f*(3*B*d*e + B*c*f - 4*A*d*f) - C*(5*d^2*e^2 + 2*c*d*e*f + c^2*f^2)) + 4*a*b*d*f*(C*(35
*d^3*e^3 + 15*c*d^2*e^2*f + 9*c^2*d*e*f^2 + 5*c^3*f^3) + 8*d*f*(2*A*d*f*(3*d*e + c*f) - B*(5*d^2*e^2 + 2*c*d*e
*f + c^2*f^2))) - b^2*(C*(63*d^4*e^4 + 28*c*d^3*e^3*f + 18*c^2*d^2*e^2*f^2 + 12*c^3*d*e*f^3 + 7*c^4*f^4) + 2*d
*f*(8*A*d*f*(5*d^2*e^2 + 2*c*d*e*f + c^2*f^2) - B*(35*d^3*e^3 + 15*c*d^2*e^2*f + 9*c^2*d*e*f^2 + 5*c^3*f^3))))
*Sqrt[c + d*x]*Sqrt[e + f*x])/(128*d^4*f^5) - ((4*a*C*d*f + b*(9*C*d*e + 7*c*C*f - 10*B*d*f))*(a + b*x)^2*(c +
d*x)^(3/2)*Sqrt[e + f*x])/(40*b*d^2*f^2) + (C*(a + b*x)^3*(c + d*x)^(3/2)*Sqrt[e + f*x])/(5*b*d*f) - ((c + d*
x)^(3/2)*Sqrt[e + f*x]*(96*a^3*C*d^3*f^3 + 8*a^2*b*d^2*f^2*(23*C*d*e + 9*c*C*f - 30*B*d*f) + 20*a*b^2*d*f*(8*d
*f*(5*B*d*e + 3*B*c*f - 6*A*d*f) - C*(35*d^2*e^2 + 22*c*d*e*f + 15*c^2*f^2)) + b^3*(C*(315*d^3*e^3 + 203*c*d^2
*e^2*f + 145*c^2*d*e*f^2 + 105*c^3*f^3) + 10*d*f*(8*A*d*f*(5*d*e + 3*c*f) - B*(35*d^2*e^2 + 22*c*d*e*f + 15*c^
2*f^2))) + 4*b*d*f*(8*b*d*f*(6*b*c*C*e + 3*a*C*d*e + a*c*C*f - 10*A*b*d*f) - (7*b*d*e + 5*b*c*f - 4*a*d*f)*(4*
a*C*d*f + b*(9*C*d*e + 7*c*C*f - 10*B*d*f)))*x))/(960*b*d^4*f^4) + ((d*e - c*f)*(16*a^2*d^2*f^2*(2*d*f*(3*B*d*
e + B*c*f - 4*A*d*f) - C*(5*d^2*e^2 + 2*c*d*e*f + c^2*f^2)) + 4*a*b*d*f*(C*(35*d^3*e^3 + 15*c*d^2*e^2*f + 9*c^
2*d*e*f^2 + 5*c^3*f^3) + 8*d*f*(2*A*d*f*(3*d*e + c*f) - B*(5*d^2*e^2 + 2*c*d*e*f + c^2*f^2))) - b^2*(C*(63*d^4
*e^4 + 28*c*d^3*e^3*f + 18*c^2*d^2*e^2*f^2 + 12*c^3*d*e*f^3 + 7*c^4*f^4) + 2*d*f*(8*A*d*f*(5*d^2*e^2 + 2*c*d*e
*f + c^2*f^2) - B*(35*d^3*e^3 + 15*c*d^2*e^2*f + 9*c^2*d*e*f^2 + 5*c^3*f^3))))*ArcTanh[(Sqrt[f]*Sqrt[c + d*x])
/(Sqrt[d]*Sqrt[e + f*x])])/(128*d^(9/2)*f^(11/2))

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]

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 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 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 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 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])

Rubi steps

\begin{align*} \int \frac{(a+b x)^2 \sqrt{c+d x} \left (A+B x+C x^2\right )}{\sqrt{e+f x}} \, dx &=\frac{C (a+b x)^3 (c+d x)^{3/2} \sqrt{e+f x}}{5 b d f}+\frac{\int \frac{(a+b x)^2 \sqrt{c+d x} \left (-\frac{1}{2} b (6 b c C e+3 a C d e+a c C f-10 A b d f)-\frac{1}{2} b (4 a C d f+b (9 C d e+7 c C f-10 B d f)) x\right )}{\sqrt{e+f x}} \, dx}{5 b^2 d f}\\ &=-\frac{(4 a C d f+b (9 C d e+7 c C f-10 B d f)) (a+b x)^2 (c+d x)^{3/2} \sqrt{e+f x}}{40 b d^2 f^2}+\frac{C (a+b x)^3 (c+d x)^{3/2} \sqrt{e+f x}}{5 b d f}+\frac{\int \frac{(a+b x) \sqrt{c+d x} \left (-\frac{1}{4} b (8 a d f (6 b c C e+3 a C d e+a c C f-10 A b d f)-(4 b c e+3 a d e+a c f) (4 a C d f+b (9 C d e+7 c C f-10 B d f)))-\frac{1}{4} b (8 b d f (6 b c C e+3 a C d e+a c C f-10 A b d f)-(7 b d e+5 b c f-4 a d f) (4 a C d f+b (9 C d e+7 c C f-10 B d f))) x\right )}{\sqrt{e+f x}} \, dx}{20 b^2 d^2 f^2}\\ &=-\frac{(4 a C d f+b (9 C d e+7 c C f-10 B d f)) (a+b x)^2 (c+d x)^{3/2} \sqrt{e+f x}}{40 b d^2 f^2}+\frac{C (a+b x)^3 (c+d x)^{3/2} \sqrt{e+f x}}{5 b d f}-\frac{(c+d x)^{3/2} \sqrt{e+f x} \left (96 a^3 C d^3 f^3+8 a^2 b d^2 f^2 (23 C d e+9 c C f-30 B d f)+20 a b^2 d f \left (8 d f (5 B d e+3 B c f-6 A d f)-C \left (35 d^2 e^2+22 c d e f+15 c^2 f^2\right )\right )+b^3 \left (C \left (315 d^3 e^3+203 c d^2 e^2 f+145 c^2 d e f^2+105 c^3 f^3\right )+10 d f \left (8 A d f (5 d e+3 c f)-B \left (35 d^2 e^2+22 c d e f+15 c^2 f^2\right )\right )\right )+4 b d f (8 b d f (6 b c C e+3 a C d e+a c C f-10 A b d f)-(7 b d e+5 b c f-4 a d f) (4 a C d f+b (9 C d e+7 c C f-10 B d f))) x\right )}{960 b d^4 f^4}-\frac{\left (16 a^2 d^2 f^2 \left (2 d f (3 B d e+B c f-4 A d f)-C \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right )+4 a b d f \left (C \left (35 d^3 e^3+15 c d^2 e^2 f+9 c^2 d e f^2+5 c^3 f^3\right )+8 d f \left (2 A d f (3 d e+c f)-B \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right )\right )-b^2 \left (C \left (63 d^4 e^4+28 c d^3 e^3 f+18 c^2 d^2 e^2 f^2+12 c^3 d e f^3+7 c^4 f^4\right )+2 d f \left (8 A d f \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )-B \left (35 d^3 e^3+15 c d^2 e^2 f+9 c^2 d e f^2+5 c^3 f^3\right )\right )\right )\right ) \int \frac{\sqrt{c+d x}}{\sqrt{e+f x}} \, dx}{128 d^4 f^4}\\ &=-\frac{\left (16 a^2 d^2 f^2 \left (2 d f (3 B d e+B c f-4 A d f)-C \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right )+4 a b d f \left (C \left (35 d^3 e^3+15 c d^2 e^2 f+9 c^2 d e f^2+5 c^3 f^3\right )+8 d f \left (2 A d f (3 d e+c f)-B \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right )\right )-b^2 \left (C \left (63 d^4 e^4+28 c d^3 e^3 f+18 c^2 d^2 e^2 f^2+12 c^3 d e f^3+7 c^4 f^4\right )+2 d f \left (8 A d f \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )-B \left (35 d^3 e^3+15 c d^2 e^2 f+9 c^2 d e f^2+5 c^3 f^3\right )\right )\right )\right ) \sqrt{c+d x} \sqrt{e+f x}}{128 d^4 f^5}-\frac{(4 a C d f+b (9 C d e+7 c C f-10 B d f)) (a+b x)^2 (c+d x)^{3/2} \sqrt{e+f x}}{40 b d^2 f^2}+\frac{C (a+b x)^3 (c+d x)^{3/2} \sqrt{e+f x}}{5 b d f}-\frac{(c+d x)^{3/2} \sqrt{e+f x} \left (96 a^3 C d^3 f^3+8 a^2 b d^2 f^2 (23 C d e+9 c C f-30 B d f)+20 a b^2 d f \left (8 d f (5 B d e+3 B c f-6 A d f)-C \left (35 d^2 e^2+22 c d e f+15 c^2 f^2\right )\right )+b^3 \left (C \left (315 d^3 e^3+203 c d^2 e^2 f+145 c^2 d e f^2+105 c^3 f^3\right )+10 d f \left (8 A d f (5 d e+3 c f)-B \left (35 d^2 e^2+22 c d e f+15 c^2 f^2\right )\right )\right )+4 b d f (8 b d f (6 b c C e+3 a C d e+a c C f-10 A b d f)-(7 b d e+5 b c f-4 a d f) (4 a C d f+b (9 C d e+7 c C f-10 B d f))) x\right )}{960 b d^4 f^4}+\frac{\left ((d e-c f) \left (16 a^2 d^2 f^2 \left (2 d f (3 B d e+B c f-4 A d f)-C \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right )+4 a b d f \left (C \left (35 d^3 e^3+15 c d^2 e^2 f+9 c^2 d e f^2+5 c^3 f^3\right )+8 d f \left (2 A d f (3 d e+c f)-B \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right )\right )-b^2 \left (C \left (63 d^4 e^4+28 c d^3 e^3 f+18 c^2 d^2 e^2 f^2+12 c^3 d e f^3+7 c^4 f^4\right )+2 d f \left (8 A d f \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )-B \left (35 d^3 e^3+15 c d^2 e^2 f+9 c^2 d e f^2+5 c^3 f^3\right )\right )\right )\right )\right ) \int \frac{1}{\sqrt{c+d x} \sqrt{e+f x}} \, dx}{256 d^4 f^5}\\ &=-\frac{\left (16 a^2 d^2 f^2 \left (2 d f (3 B d e+B c f-4 A d f)-C \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right )+4 a b d f \left (C \left (35 d^3 e^3+15 c d^2 e^2 f+9 c^2 d e f^2+5 c^3 f^3\right )+8 d f \left (2 A d f (3 d e+c f)-B \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right )\right )-b^2 \left (C \left (63 d^4 e^4+28 c d^3 e^3 f+18 c^2 d^2 e^2 f^2+12 c^3 d e f^3+7 c^4 f^4\right )+2 d f \left (8 A d f \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )-B \left (35 d^3 e^3+15 c d^2 e^2 f+9 c^2 d e f^2+5 c^3 f^3\right )\right )\right )\right ) \sqrt{c+d x} \sqrt{e+f x}}{128 d^4 f^5}-\frac{(4 a C d f+b (9 C d e+7 c C f-10 B d f)) (a+b x)^2 (c+d x)^{3/2} \sqrt{e+f x}}{40 b d^2 f^2}+\frac{C (a+b x)^3 (c+d x)^{3/2} \sqrt{e+f x}}{5 b d f}-\frac{(c+d x)^{3/2} \sqrt{e+f x} \left (96 a^3 C d^3 f^3+8 a^2 b d^2 f^2 (23 C d e+9 c C f-30 B d f)+20 a b^2 d f \left (8 d f (5 B d e+3 B c f-6 A d f)-C \left (35 d^2 e^2+22 c d e f+15 c^2 f^2\right )\right )+b^3 \left (C \left (315 d^3 e^3+203 c d^2 e^2 f+145 c^2 d e f^2+105 c^3 f^3\right )+10 d f \left (8 A d f (5 d e+3 c f)-B \left (35 d^2 e^2+22 c d e f+15 c^2 f^2\right )\right )\right )+4 b d f (8 b d f (6 b c C e+3 a C d e+a c C f-10 A b d f)-(7 b d e+5 b c f-4 a d f) (4 a C d f+b (9 C d e+7 c C f-10 B d f))) x\right )}{960 b d^4 f^4}+\frac{\left ((d e-c f) \left (16 a^2 d^2 f^2 \left (2 d f (3 B d e+B c f-4 A d f)-C \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right )+4 a b d f \left (C \left (35 d^3 e^3+15 c d^2 e^2 f+9 c^2 d e f^2+5 c^3 f^3\right )+8 d f \left (2 A d f (3 d e+c f)-B \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right )\right )-b^2 \left (C \left (63 d^4 e^4+28 c d^3 e^3 f+18 c^2 d^2 e^2 f^2+12 c^3 d e f^3+7 c^4 f^4\right )+2 d f \left (8 A d f \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )-B \left (35 d^3 e^3+15 c d^2 e^2 f+9 c^2 d e f^2+5 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 )}{128 d^5 f^5}\\ &=-\frac{\left (16 a^2 d^2 f^2 \left (2 d f (3 B d e+B c f-4 A d f)-C \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right )+4 a b d f \left (C \left (35 d^3 e^3+15 c d^2 e^2 f+9 c^2 d e f^2+5 c^3 f^3\right )+8 d f \left (2 A d f (3 d e+c f)-B \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right )\right )-b^2 \left (C \left (63 d^4 e^4+28 c d^3 e^3 f+18 c^2 d^2 e^2 f^2+12 c^3 d e f^3+7 c^4 f^4\right )+2 d f \left (8 A d f \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )-B \left (35 d^3 e^3+15 c d^2 e^2 f+9 c^2 d e f^2+5 c^3 f^3\right )\right )\right )\right ) \sqrt{c+d x} \sqrt{e+f x}}{128 d^4 f^5}-\frac{(4 a C d f+b (9 C d e+7 c C f-10 B d f)) (a+b x)^2 (c+d x)^{3/2} \sqrt{e+f x}}{40 b d^2 f^2}+\frac{C (a+b x)^3 (c+d x)^{3/2} \sqrt{e+f x}}{5 b d f}-\frac{(c+d x)^{3/2} \sqrt{e+f x} \left (96 a^3 C d^3 f^3+8 a^2 b d^2 f^2 (23 C d e+9 c C f-30 B d f)+20 a b^2 d f \left (8 d f (5 B d e+3 B c f-6 A d f)-C \left (35 d^2 e^2+22 c d e f+15 c^2 f^2\right )\right )+b^3 \left (C \left (315 d^3 e^3+203 c d^2 e^2 f+145 c^2 d e f^2+105 c^3 f^3\right )+10 d f \left (8 A d f (5 d e+3 c f)-B \left (35 d^2 e^2+22 c d e f+15 c^2 f^2\right )\right )\right )+4 b d f (8 b d f (6 b c C e+3 a C d e+a c C f-10 A b d f)-(7 b d e+5 b c f-4 a d f) (4 a C d f+b (9 C d e+7 c C f-10 B d f))) x\right )}{960 b d^4 f^4}+\frac{\left ((d e-c f) \left (16 a^2 d^2 f^2 \left (2 d f (3 B d e+B c f-4 A d f)-C \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right )+4 a b d f \left (C \left (35 d^3 e^3+15 c d^2 e^2 f+9 c^2 d e f^2+5 c^3 f^3\right )+8 d f \left (2 A d f (3 d e+c f)-B \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right )\right )-b^2 \left (C \left (63 d^4 e^4+28 c d^3 e^3 f+18 c^2 d^2 e^2 f^2+12 c^3 d e f^3+7 c^4 f^4\right )+2 d f \left (8 A d f \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )-B \left (35 d^3 e^3+15 c d^2 e^2 f+9 c^2 d e f^2+5 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 )}{128 d^5 f^5}\\ &=-\frac{\left (16 a^2 d^2 f^2 \left (2 d f (3 B d e+B c f-4 A d f)-C \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right )+4 a b d f \left (C \left (35 d^3 e^3+15 c d^2 e^2 f+9 c^2 d e f^2+5 c^3 f^3\right )+8 d f \left (2 A d f (3 d e+c f)-B \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right )\right )-b^2 \left (C \left (63 d^4 e^4+28 c d^3 e^3 f+18 c^2 d^2 e^2 f^2+12 c^3 d e f^3+7 c^4 f^4\right )+2 d f \left (8 A d f \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )-B \left (35 d^3 e^3+15 c d^2 e^2 f+9 c^2 d e f^2+5 c^3 f^3\right )\right )\right )\right ) \sqrt{c+d x} \sqrt{e+f x}}{128 d^4 f^5}-\frac{(4 a C d f+b (9 C d e+7 c C f-10 B d f)) (a+b x)^2 (c+d x)^{3/2} \sqrt{e+f x}}{40 b d^2 f^2}+\frac{C (a+b x)^3 (c+d x)^{3/2} \sqrt{e+f x}}{5 b d f}-\frac{(c+d x)^{3/2} \sqrt{e+f x} \left (96 a^3 C d^3 f^3+8 a^2 b d^2 f^2 (23 C d e+9 c C f-30 B d f)+20 a b^2 d f \left (8 d f (5 B d e+3 B c f-6 A d f)-C \left (35 d^2 e^2+22 c d e f+15 c^2 f^2\right )\right )+b^3 \left (C \left (315 d^3 e^3+203 c d^2 e^2 f+145 c^2 d e f^2+105 c^3 f^3\right )+10 d f \left (8 A d f (5 d e+3 c f)-B \left (35 d^2 e^2+22 c d e f+15 c^2 f^2\right )\right )\right )+4 b d f (8 b d f (6 b c C e+3 a C d e+a c C f-10 A b d f)-(7 b d e+5 b c f-4 a d f) (4 a C d f+b (9 C d e+7 c C f-10 B d f))) x\right )}{960 b d^4 f^4}+\frac{(d e-c f) \left (16 a^2 d^2 f^2 \left (2 d f (3 B d e+B c f-4 A d f)-C \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right )+4 a b d f \left (C \left (35 d^3 e^3+15 c d^2 e^2 f+9 c^2 d e f^2+5 c^3 f^3\right )+8 d f \left (2 A d f (3 d e+c f)-B \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right )\right )-b^2 \left (C \left (63 d^4 e^4+28 c d^3 e^3 f+18 c^2 d^2 e^2 f^2+12 c^3 d e f^3+7 c^4 f^4\right )+2 d f \left (8 A d f \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )-B \left (35 d^3 e^3+15 c d^2 e^2 f+9 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 )}{128 d^{9/2} f^{11/2}}\\ \end{align*}

Mathematica [B]  time = 6.67163, size = 3220, normalized size = 3.12 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-(b*e) + a*f)^2*(d*e - c*f)^2*(C*e^2 - B*e*f + A*f^2)*Sqrt[d/((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))]*((
d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))^2*Sqrt[(d*(e + f*x))/(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)))]*((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]*S
qrt[(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)))])))/(2*d^3*f^6*Sqrt[c
+ d*x]*Sqrt[e + f*x]) + (2*b^2*C*(d*e - c*f)^3*(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]*Sqr
t[(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*b*(d*e - c*f)^2*(-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))))^(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]*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)))])))/(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*(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)*(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]*S
qrt[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)*(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))))^(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]*Sqr
t[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)))])))/(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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Maple [B]  time = 0.042, size = 3958, normalized size = 3.8 \begin{align*} \text{output too large to display} \end{align*}

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]

1/3840*(d*x+c)^(1/2)*(f*x+e)^(1/2)*(768*C*x^4*b^2*d^4*f^4*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)+1280*C*x^2*a^2*d
^4*f^4*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)+1280*A*x^2*b^2*d^4*f^4*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)+1920*B*(
d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*x*a^2*d^4*f^4+680*C*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*a*b*c^2*d^2*e*f^3+1
000*C*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*a*b*c*d^3*e^2*f^2+156*C*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*x*b^2*c^
2*d^2*e*f^3-2240*C*x^2*a*b*d^4*e*f^3*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)-128*C*x^2*b^2*c*d^3*e*f^3*((d*x+c)*(f
*x+e))^(1/2)*(d*f)^(1/2)+320*C*x^2*a*b*c*d^3*f^4*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)+196*C*(d*f)^(1/2)*((d*x+c
)*(f*x+e))^(1/2)*x*b^2*c*d^3*e^2*f^2-1280*B*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*a*b*c*d^3*e*f^3-400*C*(d*f)^(1
/2)*((d*x+c)*(f*x+e))^(1/2)*x*a*b*c^2*d^2*f^4+2800*C*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*x*a*b*d^4*e^2*f^2-240
*B*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*x*b^2*c*d^3*e*f^3-3200*B*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*x*a*b*d^4*
e*f^3+105*C*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*b^2*c^5*f^5+340*B*(d*f
)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*b^2*c^2*d^2*e*f^3+500*B*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*b^2*c*d^3*e^2*f^2-
640*C*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*a^2*c*d^3*e*f^3+600*C*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*a*b*c^3*d*
f^4-220*C*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*b^2*c^3*d*e*f^3-272*C*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*b^2*c^
2*d^2*e^2*f^2-420*C*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*b^2*c*d^3*e^3*f+140*C*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1
/2)*x*b^2*c^3*d*f^4-1260*C*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*x*b^2*d^4*e^3*f+1920*A*(d*f)^(1/2)*((d*x+c)*(f*
x+e))^(1/2)*a*b*c*d^3*f^4-640*A*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*b^2*c*d^3*e*f^3+3840*A*(d*f)^(1/2)*((d*x+c
)*(f*x+e))^(1/2)*x*a*b*d^4*f^4+320*A*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*x*b^2*c*d^3*f^4+480*B*ln(1/2*(2*d*f*x
+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*a*b*c^2*d^3*e*f^4-240*C*ln(1/2*(2*d*f*x+2*((d*x+c
)*(f*x+e))^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*a*b*c^3*d^2*e*f^4-360*C*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))
^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*a*b*c^2*d^3*e^2*f^3+96*C*x^3*b^2*c*d^3*f^4*((d*x+c)*(f*x+e))^(1/2)*(d
*f)^(1/2)-864*C*x^3*b^2*d^4*e*f^3*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)+2560*B*x^2*a*b*d^4*f^4*((d*x+c)*(f*x+e))
^(1/2)*(d*f)^(1/2)+160*B*x^2*b^2*c*d^3*f^4*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)-1120*B*x^2*b^2*d^4*e*f^3*((d*x+
c)*(f*x+e))^(1/2)*(d*f)^(1/2)-112*C*x^2*b^2*c^2*d^2*f^4*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)+1008*C*x^2*b^2*d^4
*e^2*f^2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)+1920*C*x^3*a*b*d^4*f^4*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)-1600*A
*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*x*b^2*d^4*e*f^3-200*B*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*x*b^2*c^2*d^2*f
^4+1400*B*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*x*b^2*d^4*e^2*f^2+320*C*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*x*a^
2*c*d^3*f^4-1600*C*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*x*a^2*d^4*e*f^3+4800*B*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1
/2)*a*b*d^4*e^2*f^2-4200*C*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*a*b*d^4*e^3*f-5760*A*(d*f)^(1/2)*((d*x+c)*(f*x+
e))^(1/2)*a*b*d^4*e*f^3+1440*B*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*a*b
*c*d^4*e^2*f^3-1200*C*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*a*b*c*d^4*e^
3*f^2+1440*B*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*a^2*d^5*e^2*f^3-480*C
*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*x*a*b*c*d^3*e*f^3+2880*A*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^
(1/2)+c*f+d*e)/(d*f)^(1/2))*a*b*d^5*e^2*f^3+720*A*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)+c*f+d*
e)/(d*f)^(1/2))*b^2*c*d^4*e^2*f^3-480*C*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*a^2*c^2*d^2*f^4+240*A*ln(1/2*(2*d*
f*x+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*b^2*c^2*d^3*e*f^4+480*B*ln(1/2*(2*d*f*x+2*((d*
x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*a*b*c^3*d^2*f^5-120*B*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e)
)^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*b^2*c^3*d^2*e*f^4-180*B*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(d
*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*b^2*c^2*d^3*e^2*f^3+240*C*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2
)+c*f+d*e)/(d*f)^(1/2))*a^2*c^2*d^3*e*f^4+960*B*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*a^2*c*d^3*f^4+300*B*(d*f)^
(1/2)*((d*x+c)*(f*x+e))^(1/2)*b^2*c^3*d*f^4-480*A*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*b^2*c^2*d^2*f^4-960*A*ln
(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*a*b*c^2*d^3*f^5+720*C*ln(1/2*(2*d*f*
x+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*a^2*c*d^4*e^2*f^3+2100*C*ln(1/2*(2*d*f*x+2*((d*x
+c)*(f*x+e))^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*a*b*d^5*e^4*f+525*C*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(
1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*b^2*c*d^4*e^4*f+2400*A*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*b^2*d^4*e^2*
f^2-2880*B*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*a^2*d^4*e*f^3-2100*B*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*b^2*d^
4*e^3*f+2400*C*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*a^2*d^4*e^2*f^2-300*C*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(
1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*a*b*c^4*d*f^5+75*C*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2
)+c*f+d*e)/(d*f)^(1/2))*b^2*c^4*d*e*f^4+90*C*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d
*f)^(1/2))*b^2*c^3*d^2*e^2*f^3+150*C*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2
))*b^2*c^2*d^3*e^3*f^2+960*B*x^3*b^2*d^4*f^4*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)-960*B*ln(1/2*(2*d*f*x+2*((d*x
+c)*(f*x+e))^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*a^2*c*d^4*e*f^4-2400*B*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e)
)^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*a*b*d^5*e^3*f^2-600*B*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(d*f
)^(1/2)+c*f+d*e)/(d*f)^(1/2))*b^2*c*d^4*e^3*f^2-945*C*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)+c*
f+d*e)/(d*f)^(1/2))*b^2*d^5*e^5+640*B*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*x*a*b*c*d^3*f^4-1920*A*ln(1/2*(2*d*f
*x+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*a*b*c*d^4*e*f^4-960*B*(d*f)^(1/2)*((d*x+c)*(f*x
+e))^(1/2)*a*b*c^2*d^2*f^4+1050*B*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*
b^2*d^5*e^4*f-1200*C*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*a^2*d^5*e^3*f
^2+3840*A*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*a^2*d^4*f^4+1890*C*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*b^2*d^4*e
^4-1200*A*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*b^2*d^5*e^3*f^2+1920*A*l
n(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*a^2*c*d^4*f^5-1920*A*ln(1/2*(2*d*f*
x+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*a^2*d^5*e*f^4-150*B*ln(1/2*(2*d*f*x+2*((d*x+c)*(
f*x+e))^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*b^2*c^4*d*f^5+240*C*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*
(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*a^2*c^3*d^2*f^5-480*B*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)+
c*f+d*e)/(d*f)^(1/2))*a^2*c^2*d^3*f^5+240*A*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*
f)^(1/2))*b^2*c^3*d^2*f^5-210*C*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*b^2*c^4*f^4)/((d*x+c)*(f*x+e))^(1/2)/f^5/d
^4/(d*f)^(1/2)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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

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Fricas [A]  time = 43.625, size = 4766, normalized size = 4.62 \begin{align*} \text{result too large to display} \end{align*}

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]

[-1/7680*(15*(63*C*b^2*d^5*e^5 - 35*(C*b^2*c*d^4 + 2*(2*C*a*b + B*b^2)*d^5)*e^4*f - 10*(C*b^2*c^2*d^3 - 4*(2*C
*a*b + B*b^2)*c*d^4 - 8*(C*a^2 + 2*B*a*b + A*b^2)*d^5)*e^3*f^2 - 6*(C*b^2*c^3*d^2 - 2*(2*C*a*b + B*b^2)*c^2*d^
3 + 8*(C*a^2 + 2*B*a*b + A*b^2)*c*d^4 + 16*(B*a^2 + 2*A*a*b)*d^5)*e^2*f^3 - (5*C*b^2*c^4*d - 128*A*a^2*d^5 - 8
*(2*C*a*b + B*b^2)*c^3*d^2 + 16*(C*a^2 + 2*B*a*b + A*b^2)*c^2*d^3 - 64*(B*a^2 + 2*A*a*b)*c*d^4)*e*f^4 - (7*C*b
^2*c^5 + 128*A*a^2*c*d^4 - 10*(2*C*a*b + B*b^2)*c^4*d + 16*(C*a^2 + 2*B*a*b + A*b^2)*c^3*d^2 - 32*(B*a^2 + 2*A
*a*b)*c^2*d^3)*f^5)*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*(384*C*b^2*d^5*f^5*x^4 + 945*C*b^2*d^5*e^4*f
- 210*(C*b^2*c*d^4 + 5*(2*C*a*b + B*b^2)*d^5)*e^3*f^2 - 2*(68*C*b^2*c^2*d^3 - 125*(2*C*a*b + B*b^2)*c*d^4 - 60
0*(C*a^2 + 2*B*a*b + A*b^2)*d^5)*e^2*f^3 - 10*(11*C*b^2*c^3*d^2 - 17*(2*C*a*b + B*b^2)*c^2*d^3 + 32*(C*a^2 + 2
*B*a*b + A*b^2)*c*d^4 + 144*(B*a^2 + 2*A*a*b)*d^5)*e*f^4 - 15*(7*C*b^2*c^4*d - 128*A*a^2*d^5 - 10*(2*C*a*b + B
*b^2)*c^3*d^2 + 16*(C*a^2 + 2*B*a*b + A*b^2)*c^2*d^3 - 32*(B*a^2 + 2*A*a*b)*c*d^4)*f^5 - 48*(9*C*b^2*d^5*e*f^4
- (C*b^2*c*d^4 + 10*(2*C*a*b + B*b^2)*d^5)*f^5)*x^3 + 8*(63*C*b^2*d^5*e^2*f^3 - 2*(4*C*b^2*c*d^4 + 35*(2*C*a*
b + B*b^2)*d^5)*e*f^4 - (7*C*b^2*c^2*d^3 - 10*(2*C*a*b + B*b^2)*c*d^4 - 80*(C*a^2 + 2*B*a*b + A*b^2)*d^5)*f^5)
*x^2 - 2*(315*C*b^2*d^5*e^3*f^2 - 7*(7*C*b^2*c*d^4 + 50*(2*C*a*b + B*b^2)*d^5)*e^2*f^3 - (39*C*b^2*c^2*d^3 - 6
0*(2*C*a*b + B*b^2)*c*d^4 - 400*(C*a^2 + 2*B*a*b + A*b^2)*d^5)*e*f^4 - 5*(7*C*b^2*c^3*d^2 - 10*(2*C*a*b + B*b^
2)*c^2*d^3 + 16*(C*a^2 + 2*B*a*b + A*b^2)*c*d^4 + 96*(B*a^2 + 2*A*a*b)*d^5)*f^5)*x)*sqrt(d*x + c)*sqrt(f*x + e
))/(d^5*f^6), 1/3840*(15*(63*C*b^2*d^5*e^5 - 35*(C*b^2*c*d^4 + 2*(2*C*a*b + B*b^2)*d^5)*e^4*f - 10*(C*b^2*c^2*
d^3 - 4*(2*C*a*b + B*b^2)*c*d^4 - 8*(C*a^2 + 2*B*a*b + A*b^2)*d^5)*e^3*f^2 - 6*(C*b^2*c^3*d^2 - 2*(2*C*a*b + B
*b^2)*c^2*d^3 + 8*(C*a^2 + 2*B*a*b + A*b^2)*c*d^4 + 16*(B*a^2 + 2*A*a*b)*d^5)*e^2*f^3 - (5*C*b^2*c^4*d - 128*A
*a^2*d^5 - 8*(2*C*a*b + B*b^2)*c^3*d^2 + 16*(C*a^2 + 2*B*a*b + A*b^2)*c^2*d^3 - 64*(B*a^2 + 2*A*a*b)*c*d^4)*e*
f^4 - (7*C*b^2*c^5 + 128*A*a^2*c*d^4 - 10*(2*C*a*b + B*b^2)*c^4*d + 16*(C*a^2 + 2*B*a*b + A*b^2)*c^3*d^2 - 32*
(B*a^2 + 2*A*a*b)*c^2*d^3)*f^5)*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*(384*C*b^2*d^5*f^5*x^4 + 945*C*b^2*d^5*e^4*f - 210*(
C*b^2*c*d^4 + 5*(2*C*a*b + B*b^2)*d^5)*e^3*f^2 - 2*(68*C*b^2*c^2*d^3 - 125*(2*C*a*b + B*b^2)*c*d^4 - 600*(C*a^
2 + 2*B*a*b + A*b^2)*d^5)*e^2*f^3 - 10*(11*C*b^2*c^3*d^2 - 17*(2*C*a*b + B*b^2)*c^2*d^3 + 32*(C*a^2 + 2*B*a*b
+ A*b^2)*c*d^4 + 144*(B*a^2 + 2*A*a*b)*d^5)*e*f^4 - 15*(7*C*b^2*c^4*d - 128*A*a^2*d^5 - 10*(2*C*a*b + B*b^2)*c
^3*d^2 + 16*(C*a^2 + 2*B*a*b + A*b^2)*c^2*d^3 - 32*(B*a^2 + 2*A*a*b)*c*d^4)*f^5 - 48*(9*C*b^2*d^5*e*f^4 - (C*b
^2*c*d^4 + 10*(2*C*a*b + B*b^2)*d^5)*f^5)*x^3 + 8*(63*C*b^2*d^5*e^2*f^3 - 2*(4*C*b^2*c*d^4 + 35*(2*C*a*b + B*b
^2)*d^5)*e*f^4 - (7*C*b^2*c^2*d^3 - 10*(2*C*a*b + B*b^2)*c*d^4 - 80*(C*a^2 + 2*B*a*b + A*b^2)*d^5)*f^5)*x^2 -
2*(315*C*b^2*d^5*e^3*f^2 - 7*(7*C*b^2*c*d^4 + 50*(2*C*a*b + B*b^2)*d^5)*e^2*f^3 - (39*C*b^2*c^2*d^3 - 60*(2*C*
a*b + B*b^2)*c*d^4 - 400*(C*a^2 + 2*B*a*b + A*b^2)*d^5)*e*f^4 - 5*(7*C*b^2*c^3*d^2 - 10*(2*C*a*b + B*b^2)*c^2*
d^3 + 16*(C*a^2 + 2*B*a*b + A*b^2)*c*d^4 + 96*(B*a^2 + 2*A*a*b)*d^5)*f^5)*x)*sqrt(d*x + c)*sqrt(f*x + e))/(d^5
*f^6)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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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Giac [A]  time = 1.97709, size = 2032, normalized size = 1.97 \begin{align*} \text{result too large to display} \end{align*}

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]

1/1920*(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)*C*b^2/(d^5*f) - (31*C*b^
2*c*d^20*f^8 - 20*C*a*b*d^21*f^8 - 10*B*b^2*d^21*f^8 + 9*C*b^2*d^21*f^7*e)/(d^25*f^9)) + (263*C*b^2*c^2*d^20*f
^8 - 340*C*a*b*c*d^21*f^8 - 170*B*b^2*c*d^21*f^8 + 80*C*a^2*d^22*f^8 + 160*B*a*b*d^22*f^8 + 80*A*b^2*d^22*f^8
+ 154*C*b^2*c*d^21*f^7*e - 140*C*a*b*d^22*f^7*e - 70*B*b^2*d^22*f^7*e + 63*C*b^2*d^22*f^6*e^2)/(d^25*f^9)) - 5
*(121*C*b^2*c^3*d^20*f^8 - 236*C*a*b*c^2*d^21*f^8 - 118*B*b^2*c^2*d^21*f^8 + 112*C*a^2*c*d^22*f^8 + 224*B*a*b*
c*d^22*f^8 + 112*A*b^2*c*d^22*f^8 - 96*B*a^2*d^23*f^8 - 192*A*a*b*d^23*f^8 + 109*C*b^2*c^2*d^21*f^7*e - 200*C*
a*b*c*d^22*f^7*e - 100*B*b^2*c*d^22*f^7*e + 80*C*a^2*d^23*f^7*e + 160*B*a*b*d^23*f^7*e + 80*A*b^2*d^23*f^7*e +
91*C*b^2*c*d^22*f^6*e^2 - 140*C*a*b*d^23*f^6*e^2 - 70*B*b^2*d^23*f^6*e^2 + 63*C*b^2*d^23*f^5*e^3)/(d^25*f^9))
*(d*x + c) + 15*(7*C*b^2*c^4*d^20*f^8 - 20*C*a*b*c^3*d^21*f^8 - 10*B*b^2*c^3*d^21*f^8 + 16*C*a^2*c^2*d^22*f^8
+ 32*B*a*b*c^2*d^22*f^8 + 16*A*b^2*c^2*d^22*f^8 - 32*B*a^2*c*d^23*f^8 - 64*A*a*b*c*d^23*f^8 + 128*A*a^2*d^24*f
^8 + 12*C*b^2*c^3*d^21*f^7*e - 36*C*a*b*c^2*d^22*f^7*e - 18*B*b^2*c^2*d^22*f^7*e + 32*C*a^2*c*d^23*f^7*e + 64*
B*a*b*c*d^23*f^7*e + 32*A*b^2*c*d^23*f^7*e - 96*B*a^2*d^24*f^7*e - 192*A*a*b*d^24*f^7*e + 18*C*b^2*c^2*d^22*f^
6*e^2 - 60*C*a*b*c*d^23*f^6*e^2 - 30*B*b^2*c*d^23*f^6*e^2 + 80*C*a^2*d^24*f^6*e^2 + 160*B*a*b*d^24*f^6*e^2 + 8
0*A*b^2*d^24*f^6*e^2 + 28*C*b^2*c*d^23*f^5*e^3 - 140*C*a*b*d^24*f^5*e^3 - 70*B*b^2*d^24*f^5*e^3 + 63*C*b^2*d^2
4*f^4*e^4)/(d^25*f^9))*sqrt(d*x + c) - 15*(7*C*b^2*c^5*f^5 - 20*C*a*b*c^4*d*f^5 - 10*B*b^2*c^4*d*f^5 + 16*C*a^
2*c^3*d^2*f^5 + 32*B*a*b*c^3*d^2*f^5 + 16*A*b^2*c^3*d^2*f^5 - 32*B*a^2*c^2*d^3*f^5 - 64*A*a*b*c^2*d^3*f^5 + 12
8*A*a^2*c*d^4*f^5 + 5*C*b^2*c^4*d*f^4*e - 16*C*a*b*c^3*d^2*f^4*e - 8*B*b^2*c^3*d^2*f^4*e + 16*C*a^2*c^2*d^3*f^
4*e + 32*B*a*b*c^2*d^3*f^4*e + 16*A*b^2*c^2*d^3*f^4*e - 64*B*a^2*c*d^4*f^4*e - 128*A*a*b*c*d^4*f^4*e - 128*A*a
^2*d^5*f^4*e + 6*C*b^2*c^3*d^2*f^3*e^2 - 24*C*a*b*c^2*d^3*f^3*e^2 - 12*B*b^2*c^2*d^3*f^3*e^2 + 48*C*a^2*c*d^4*
f^3*e^2 + 96*B*a*b*c*d^4*f^3*e^2 + 48*A*b^2*c*d^4*f^3*e^2 + 96*B*a^2*d^5*f^3*e^2 + 192*A*a*b*d^5*f^3*e^2 + 10*
C*b^2*c^2*d^3*f^2*e^3 - 80*C*a*b*c*d^4*f^2*e^3 - 40*B*b^2*c*d^4*f^2*e^3 - 80*C*a^2*d^5*f^2*e^3 - 160*B*a*b*d^5
*f^2*e^3 - 80*A*b^2*d^5*f^2*e^3 + 35*C*b^2*c*d^4*f*e^4 + 140*C*a*b*d^5*f*e^4 + 70*B*b^2*d^5*f*e^4 - 63*C*b^2*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^4*f^5))*d/abs(d)