3.1737 \(\int \frac {(A+B x) (a^2+2 a b x+b^2 x^2)^{3/2}}{(d+e x)^{12}} \, dx\)

Optimal. Leaf size=298 \[ \frac {b^2 \sqrt {a^2+2 a b x+b^2 x^2} (-3 a B e-A b e+4 b B d)}{8 e^5 (a+b x) (d+e x)^8}-\frac {b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) (-a B e-A b e+2 b B d)}{3 e^5 (a+b x) (d+e x)^9}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{10 e^5 (a+b x) (d+e x)^{10}}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3 (B d-A e)}{11 e^5 (a+b x) (d+e x)^{11}}-\frac {b^3 B \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^5 (a+b x) (d+e x)^7} \]

[Out]

-1/11*(-a*e+b*d)^3*(-A*e+B*d)*((b*x+a)^2)^(1/2)/e^5/(b*x+a)/(e*x+d)^11+1/10*(-a*e+b*d)^2*(-3*A*b*e-B*a*e+4*B*b
*d)*((b*x+a)^2)^(1/2)/e^5/(b*x+a)/(e*x+d)^10-1/3*b*(-a*e+b*d)*(-A*b*e-B*a*e+2*B*b*d)*((b*x+a)^2)^(1/2)/e^5/(b*
x+a)/(e*x+d)^9+1/8*b^2*(-A*b*e-3*B*a*e+4*B*b*d)*((b*x+a)^2)^(1/2)/e^5/(b*x+a)/(e*x+d)^8-1/7*b^3*B*((b*x+a)^2)^
(1/2)/e^5/(b*x+a)/(e*x+d)^7

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Rubi [A]  time = 0.18, antiderivative size = 298, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {770, 77} \[ \frac {b^2 \sqrt {a^2+2 a b x+b^2 x^2} (-3 a B e-A b e+4 b B d)}{8 e^5 (a+b x) (d+e x)^8}-\frac {b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) (-a B e-A b e+2 b B d)}{3 e^5 (a+b x) (d+e x)^9}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{10 e^5 (a+b x) (d+e x)^{10}}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3 (B d-A e)}{11 e^5 (a+b x) (d+e x)^{11}}-\frac {b^3 B \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^5 (a+b x) (d+e x)^7} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/(d + e*x)^12,x]

[Out]

-((b*d - a*e)^3*(B*d - A*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^5*(a + b*x)*(d + e*x)^11) + ((b*d - a*e)^2*(4
*b*B*d - 3*A*b*e - a*B*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(10*e^5*(a + b*x)*(d + e*x)^10) - (b*(b*d - a*e)*(2*b
*B*d - A*b*e - a*B*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^5*(a + b*x)*(d + e*x)^9) + (b^2*(4*b*B*d - A*b*e - 3
*a*B*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(8*e^5*(a + b*x)*(d + e*x)^8) - (b^3*B*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(
7*e^5*(a + b*x)*(d + e*x)^7)

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 770

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dis
t[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(f + g*x)*(b/2 + c
*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && EqQ[b^2 - 4*a*c, 0]

Rubi steps

\begin {align*} \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^{12}} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right )^3 (A+B x)}{(d+e x)^{12}} \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (-\frac {b^3 (b d-a e)^3 (-B d+A e)}{e^4 (d+e x)^{12}}+\frac {b^3 (b d-a e)^2 (-4 b B d+3 A b e+a B e)}{e^4 (d+e x)^{11}}-\frac {3 b^4 (b d-a e) (-2 b B d+A b e+a B e)}{e^4 (d+e x)^{10}}+\frac {b^5 (-4 b B d+A b e+3 a B e)}{e^4 (d+e x)^9}+\frac {b^6 B}{e^4 (d+e x)^8}\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=-\frac {(b d-a e)^3 (B d-A e) \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^5 (a+b x) (d+e x)^{11}}+\frac {(b d-a e)^2 (4 b B d-3 A b e-a B e) \sqrt {a^2+2 a b x+b^2 x^2}}{10 e^5 (a+b x) (d+e x)^{10}}-\frac {b (b d-a e) (2 b B d-A b e-a B e) \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^5 (a+b x) (d+e x)^9}+\frac {b^2 (4 b B d-A b e-3 a B e) \sqrt {a^2+2 a b x+b^2 x^2}}{8 e^5 (a+b x) (d+e x)^8}-\frac {b^3 B \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^5 (a+b x) (d+e x)^7}\\ \end {align*}

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Mathematica [A]  time = 0.11, size = 233, normalized size = 0.78 \[ -\frac {\sqrt {(a+b x)^2} \left (84 a^3 e^3 (10 A e+B (d+11 e x))+28 a^2 b e^2 \left (9 A e (d+11 e x)+2 B \left (d^2+11 d e x+55 e^2 x^2\right )\right )+7 a b^2 e \left (8 A e \left (d^2+11 d e x+55 e^2 x^2\right )+3 B \left (d^3+11 d^2 e x+55 d e^2 x^2+165 e^3 x^3\right )\right )+b^3 \left (7 A e \left (d^3+11 d^2 e x+55 d e^2 x^2+165 e^3 x^3\right )+4 B \left (d^4+11 d^3 e x+55 d^2 e^2 x^2+165 d e^3 x^3+330 e^4 x^4\right )\right )\right )}{9240 e^5 (a+b x) (d+e x)^{11}} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/(d + e*x)^12,x]

[Out]

-1/9240*(Sqrt[(a + b*x)^2]*(84*a^3*e^3*(10*A*e + B*(d + 11*e*x)) + 28*a^2*b*e^2*(9*A*e*(d + 11*e*x) + 2*B*(d^2
+ 11*d*e*x + 55*e^2*x^2)) + 7*a*b^2*e*(8*A*e*(d^2 + 11*d*e*x + 55*e^2*x^2) + 3*B*(d^3 + 11*d^2*e*x + 55*d*e^2
*x^2 + 165*e^3*x^3)) + b^3*(7*A*e*(d^3 + 11*d^2*e*x + 55*d*e^2*x^2 + 165*e^3*x^3) + 4*B*(d^4 + 11*d^3*e*x + 55
*d^2*e^2*x^2 + 165*d*e^3*x^3 + 330*e^4*x^4))))/(e^5*(a + b*x)*(d + e*x)^11)

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fricas [A]  time = 0.94, size = 376, normalized size = 1.26 \[ -\frac {1320 \, B b^{3} e^{4} x^{4} + 4 \, B b^{3} d^{4} + 840 \, A a^{3} e^{4} + 7 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 56 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} + 84 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} + 165 \, {\left (4 \, B b^{3} d e^{3} + 7 \, {\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + 55 \, {\left (4 \, B b^{3} d^{2} e^{2} + 7 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 56 \, {\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} + 11 \, {\left (4 \, B b^{3} d^{3} e + 7 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 56 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{3} + 84 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x}{9240 \, {\left (e^{16} x^{11} + 11 \, d e^{15} x^{10} + 55 \, d^{2} e^{14} x^{9} + 165 \, d^{3} e^{13} x^{8} + 330 \, d^{4} e^{12} x^{7} + 462 \, d^{5} e^{11} x^{6} + 462 \, d^{6} e^{10} x^{5} + 330 \, d^{7} e^{9} x^{4} + 165 \, d^{8} e^{8} x^{3} + 55 \, d^{9} e^{7} x^{2} + 11 \, d^{10} e^{6} x + d^{11} e^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(3/2)/(e*x+d)^12,x, algorithm="fricas")

[Out]

-1/9240*(1320*B*b^3*e^4*x^4 + 4*B*b^3*d^4 + 840*A*a^3*e^4 + 7*(3*B*a*b^2 + A*b^3)*d^3*e + 56*(B*a^2*b + A*a*b^
2)*d^2*e^2 + 84*(B*a^3 + 3*A*a^2*b)*d*e^3 + 165*(4*B*b^3*d*e^3 + 7*(3*B*a*b^2 + A*b^3)*e^4)*x^3 + 55*(4*B*b^3*
d^2*e^2 + 7*(3*B*a*b^2 + A*b^3)*d*e^3 + 56*(B*a^2*b + A*a*b^2)*e^4)*x^2 + 11*(4*B*b^3*d^3*e + 7*(3*B*a*b^2 + A
*b^3)*d^2*e^2 + 56*(B*a^2*b + A*a*b^2)*d*e^3 + 84*(B*a^3 + 3*A*a^2*b)*e^4)*x)/(e^16*x^11 + 11*d*e^15*x^10 + 55
*d^2*e^14*x^9 + 165*d^3*e^13*x^8 + 330*d^4*e^12*x^7 + 462*d^5*e^11*x^6 + 462*d^6*e^10*x^5 + 330*d^7*e^9*x^4 +
165*d^8*e^8*x^3 + 55*d^9*e^7*x^2 + 11*d^10*e^6*x + d^11*e^5)

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giac [A]  time = 0.20, size = 427, normalized size = 1.43 \[ -\frac {{\left (1320 \, B b^{3} x^{4} e^{4} \mathrm {sgn}\left (b x + a\right ) + 660 \, B b^{3} d x^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 220 \, B b^{3} d^{2} x^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 44 \, B b^{3} d^{3} x e \mathrm {sgn}\left (b x + a\right ) + 4 \, B b^{3} d^{4} \mathrm {sgn}\left (b x + a\right ) + 3465 \, B a b^{2} x^{3} e^{4} \mathrm {sgn}\left (b x + a\right ) + 1155 \, A b^{3} x^{3} e^{4} \mathrm {sgn}\left (b x + a\right ) + 1155 \, B a b^{2} d x^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 385 \, A b^{3} d x^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 231 \, B a b^{2} d^{2} x e^{2} \mathrm {sgn}\left (b x + a\right ) + 77 \, A b^{3} d^{2} x e^{2} \mathrm {sgn}\left (b x + a\right ) + 21 \, B a b^{2} d^{3} e \mathrm {sgn}\left (b x + a\right ) + 7 \, A b^{3} d^{3} e \mathrm {sgn}\left (b x + a\right ) + 3080 \, B a^{2} b x^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 3080 \, A a b^{2} x^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 616 \, B a^{2} b d x e^{3} \mathrm {sgn}\left (b x + a\right ) + 616 \, A a b^{2} d x e^{3} \mathrm {sgn}\left (b x + a\right ) + 56 \, B a^{2} b d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 56 \, A a b^{2} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 924 \, B a^{3} x e^{4} \mathrm {sgn}\left (b x + a\right ) + 2772 \, A a^{2} b x e^{4} \mathrm {sgn}\left (b x + a\right ) + 84 \, B a^{3} d e^{3} \mathrm {sgn}\left (b x + a\right ) + 252 \, A a^{2} b d e^{3} \mathrm {sgn}\left (b x + a\right ) + 840 \, A a^{3} e^{4} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-5\right )}}{9240 \, {\left (x e + d\right )}^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(3/2)/(e*x+d)^12,x, algorithm="giac")

[Out]

-1/9240*(1320*B*b^3*x^4*e^4*sgn(b*x + a) + 660*B*b^3*d*x^3*e^3*sgn(b*x + a) + 220*B*b^3*d^2*x^2*e^2*sgn(b*x +
a) + 44*B*b^3*d^3*x*e*sgn(b*x + a) + 4*B*b^3*d^4*sgn(b*x + a) + 3465*B*a*b^2*x^3*e^4*sgn(b*x + a) + 1155*A*b^3
*x^3*e^4*sgn(b*x + a) + 1155*B*a*b^2*d*x^2*e^3*sgn(b*x + a) + 385*A*b^3*d*x^2*e^3*sgn(b*x + a) + 231*B*a*b^2*d
^2*x*e^2*sgn(b*x + a) + 77*A*b^3*d^2*x*e^2*sgn(b*x + a) + 21*B*a*b^2*d^3*e*sgn(b*x + a) + 7*A*b^3*d^3*e*sgn(b*
x + a) + 3080*B*a^2*b*x^2*e^4*sgn(b*x + a) + 3080*A*a*b^2*x^2*e^4*sgn(b*x + a) + 616*B*a^2*b*d*x*e^3*sgn(b*x +
a) + 616*A*a*b^2*d*x*e^3*sgn(b*x + a) + 56*B*a^2*b*d^2*e^2*sgn(b*x + a) + 56*A*a*b^2*d^2*e^2*sgn(b*x + a) + 9
24*B*a^3*x*e^4*sgn(b*x + a) + 2772*A*a^2*b*x*e^4*sgn(b*x + a) + 84*B*a^3*d*e^3*sgn(b*x + a) + 252*A*a^2*b*d*e^
3*sgn(b*x + a) + 840*A*a^3*e^4*sgn(b*x + a))*e^(-5)/(x*e + d)^11

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maple [A]  time = 0.06, size = 317, normalized size = 1.06 \[ -\frac {\left (1320 B \,b^{3} e^{4} x^{4}+1155 A \,b^{3} e^{4} x^{3}+3465 B a \,b^{2} e^{4} x^{3}+660 B \,b^{3} d \,e^{3} x^{3}+3080 A a \,b^{2} e^{4} x^{2}+385 A \,b^{3} d \,e^{3} x^{2}+3080 B \,a^{2} b \,e^{4} x^{2}+1155 B a \,b^{2} d \,e^{3} x^{2}+220 B \,b^{3} d^{2} e^{2} x^{2}+2772 A \,a^{2} b \,e^{4} x +616 A a \,b^{2} d \,e^{3} x +77 A \,b^{3} d^{2} e^{2} x +924 B \,a^{3} e^{4} x +616 B \,a^{2} b d \,e^{3} x +231 B a \,b^{2} d^{2} e^{2} x +44 B \,b^{3} d^{3} e x +840 A \,a^{3} e^{4}+252 A \,a^{2} b d \,e^{3}+56 A a \,b^{2} d^{2} e^{2}+7 A \,b^{3} d^{3} e +84 B \,a^{3} d \,e^{3}+56 B \,a^{2} b \,d^{2} e^{2}+21 B a \,b^{2} d^{3} e +4 B \,b^{3} d^{4}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{9240 \left (e x +d \right )^{11} \left (b x +a \right )^{3} e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(3/2)/(e*x+d)^12,x)

[Out]

-1/9240/e^5*(1320*B*b^3*e^4*x^4+1155*A*b^3*e^4*x^3+3465*B*a*b^2*e^4*x^3+660*B*b^3*d*e^3*x^3+3080*A*a*b^2*e^4*x
^2+385*A*b^3*d*e^3*x^2+3080*B*a^2*b*e^4*x^2+1155*B*a*b^2*d*e^3*x^2+220*B*b^3*d^2*e^2*x^2+2772*A*a^2*b*e^4*x+61
6*A*a*b^2*d*e^3*x+77*A*b^3*d^2*e^2*x+924*B*a^3*e^4*x+616*B*a^2*b*d*e^3*x+231*B*a*b^2*d^2*e^2*x+44*B*b^3*d^3*e*
x+840*A*a^3*e^4+252*A*a^2*b*d*e^3+56*A*a*b^2*d^2*e^2+7*A*b^3*d^3*e+84*B*a^3*d*e^3+56*B*a^2*b*d^2*e^2+21*B*a*b^
2*d^3*e+4*B*b^3*d^4)*((b*x+a)^2)^(3/2)/(e*x+d)^11/(b*x+a)^3

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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)*(b^2*x^2+2*a*b*x+a^2)^(3/2)/(e*x+d)^12,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(a*e-b*d>0)', see `assume?` for
more details)Is a*e-b*d zero or nonzero?

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mupad [B]  time = 2.36, size = 577, normalized size = 1.94 \[ -\frac {\left (\frac {A\,b^3\,e-3\,B\,b^3\,d+3\,B\,a\,b^2\,e}{8\,e^5}-\frac {B\,b^3\,d}{8\,e^5}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^8}-\frac {\left (\frac {A\,a^3}{11\,e}-\frac {d\,\left (\frac {B\,a^3+3\,A\,b\,a^2}{11\,e}+\frac {d\,\left (\frac {d\,\left (\frac {A\,b^3+3\,B\,a\,b^2}{11\,e}-\frac {B\,b^3\,d}{11\,e^2}\right )}{e}-\frac {3\,a\,b\,\left (A\,b+B\,a\right )}{11\,e}\right )}{e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^{11}}-\frac {\left (\frac {B\,a^3\,e^3-3\,B\,a^2\,b\,d\,e^2+3\,A\,a^2\,b\,e^3+3\,B\,a\,b^2\,d^2\,e-3\,A\,a\,b^2\,d\,e^2-B\,b^3\,d^3+A\,b^3\,d^2\,e}{10\,e^5}-\frac {d\,\left (\frac {3\,B\,a^2\,b\,e^3-3\,B\,a\,b^2\,d\,e^2+3\,A\,a\,b^2\,e^3+B\,b^3\,d^2\,e-A\,b^3\,d\,e^2}{10\,e^5}-\frac {d\,\left (\frac {b^2\,\left (A\,b\,e+3\,B\,a\,e-B\,b\,d\right )}{10\,e^3}-\frac {B\,b^3\,d}{10\,e^3}\right )}{e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^{10}}-\frac {\left (\frac {3\,B\,a^2\,b\,e^2-6\,B\,a\,b^2\,d\,e+3\,A\,a\,b^2\,e^2+3\,B\,b^3\,d^2-2\,A\,b^3\,d\,e}{9\,e^5}-\frac {d\,\left (\frac {b^2\,\left (A\,b\,e+3\,B\,a\,e-2\,B\,b\,d\right )}{9\,e^4}-\frac {B\,b^3\,d}{9\,e^4}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^9}-\frac {B\,b^3\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{7\,e^5\,\left (a+b\,x\right )\,{\left (d+e\,x\right )}^7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2))/(d + e*x)^12,x)

[Out]

- (((A*b^3*e - 3*B*b^3*d + 3*B*a*b^2*e)/(8*e^5) - (B*b^3*d)/(8*e^5))*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*
x)*(d + e*x)^8) - (((A*a^3)/(11*e) - (d*((B*a^3 + 3*A*a^2*b)/(11*e) + (d*((d*((A*b^3 + 3*B*a*b^2)/(11*e) - (B*
b^3*d)/(11*e^2)))/e - (3*a*b*(A*b + B*a))/(11*e)))/e))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x
)^11) - (((B*a^3*e^3 - B*b^3*d^3 + 3*A*a^2*b*e^3 + A*b^3*d^2*e - 3*A*a*b^2*d*e^2 + 3*B*a*b^2*d^2*e - 3*B*a^2*b
*d*e^2)/(10*e^5) - (d*((3*A*a*b^2*e^3 + 3*B*a^2*b*e^3 - A*b^3*d*e^2 + B*b^3*d^2*e - 3*B*a*b^2*d*e^2)/(10*e^5)
- (d*((b^2*(A*b*e + 3*B*a*e - B*b*d))/(10*e^3) - (B*b^3*d)/(10*e^3)))/e))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/
((a + b*x)*(d + e*x)^10) - (((3*B*b^3*d^2 - 2*A*b^3*d*e + 3*A*a*b^2*e^2 + 3*B*a^2*b*e^2 - 6*B*a*b^2*d*e)/(9*e^
5) - (d*((b^2*(A*b*e + 3*B*a*e - 2*B*b*d))/(9*e^4) - (B*b^3*d)/(9*e^4)))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(
(a + b*x)*(d + e*x)^9) - (B*b^3*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(7*e^5*(a + b*x)*(d + e*x)^7)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**(3/2)/(e*x+d)**12,x)

[Out]

Exception raised: HeuristicGCDFailed

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