3.342 \(\int F^{a+\frac{b}{(c+d x)^3}} (c+d x)^{11} \, dx\)

Optimal. Leaf size=31 \[ \frac{b^4 F^a \log ^4(F) \text{Gamma}\left (-4,-\frac{b \log (F)}{(c+d x)^3}\right )}{3 d} \]

[Out]

(b^4*F^a*Gamma[-4, -((b*Log[F])/(c + d*x)^3)]*Log[F]^4)/(3*d)

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Rubi [A]  time = 0.0775168, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048 \[ \frac{b^4 F^a \log ^4(F) \text{Gamma}\left (-4,-\frac{b \log (F)}{(c+d x)^3}\right )}{3 d} \]

Antiderivative was successfully verified.

[In]  Int[F^(a + b/(c + d*x)^3)*(c + d*x)^11,x]

[Out]

(b^4*F^a*Gamma[-4, -((b*Log[F])/(c + d*x)^3)]*Log[F]^4)/(3*d)

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Rubi in Sympy [A]  time = 7.75368, size = 31, normalized size = 1. \[ \frac{F^{a} b^{4} \Gamma{\left (-4,- \frac{b \log{\left (F \right )}}{\left (c + d x\right )^{3}} \right )} \log{\left (F \right )}^{4}}{3 d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(F**(a+b/(d*x+c)**3)*(d*x+c)**11,x)

[Out]

F**a*b**4*Gamma(-4, -b*log(F)/(c + d*x)**3)*log(F)**4/(3*d)

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Mathematica [B]  time = 0.119253, size = 96, normalized size = 3.1 \[ \frac{F^a \left ((c+d x)^3 F^{\frac{b}{(c+d x)^3}} \left (b^3 \log ^3(F)+b^2 \log ^2(F) (c+d x)^3+2 b \log (F) (c+d x)^6+6 (c+d x)^9\right )-b^4 \log ^4(F) \text{ExpIntegralEi}\left (\frac{b \log (F)}{(c+d x)^3}\right )\right )}{72 d} \]

Antiderivative was successfully verified.

[In]  Integrate[F^(a + b/(c + d*x)^3)*(c + d*x)^11,x]

[Out]

(F^a*(-(b^4*ExpIntegralEi[(b*Log[F])/(c + d*x)^3]*Log[F]^4) + F^(b/(c + d*x)^3)*
(c + d*x)^3*(6*(c + d*x)^9 + 2*b*(c + d*x)^6*Log[F] + b^2*(c + d*x)^3*Log[F]^2 +
b^3*Log[F]^3)))/(72*d)

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Maple [F]  time = 0.106, size = 0, normalized size = 0. \[ \int{F}^{a+{\frac{b}{ \left ( dx+c \right ) ^{3}}}} \left ( dx+c \right ) ^{11}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(F^(a+b/(d*x+c)^3)*(d*x+c)^11,x)

[Out]

int(F^(a+b/(d*x+c)^3)*(d*x+c)^11,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \frac{1}{72} \,{\left (6 \, F^{a} d^{11} x^{12} + 72 \, F^{a} c d^{10} x^{11} + 396 \, F^{a} c^{2} d^{9} x^{10} + 2 \,{\left (660 \, F^{a} c^{3} d^{8} + F^{a} b d^{8} \log \left (F\right )\right )} x^{9} + 18 \,{\left (165 \, F^{a} c^{4} d^{7} + F^{a} b c d^{7} \log \left (F\right )\right )} x^{8} + 72 \,{\left (66 \, F^{a} c^{5} d^{6} + F^{a} b c^{2} d^{6} \log \left (F\right )\right )} x^{7} +{\left (5544 \, F^{a} c^{6} d^{5} + 168 \, F^{a} b c^{3} d^{5} \log \left (F\right ) + F^{a} b^{2} d^{5} \log \left (F\right )^{2}\right )} x^{6} + 6 \,{\left (792 \, F^{a} c^{7} d^{4} + 42 \, F^{a} b c^{4} d^{4} \log \left (F\right ) + F^{a} b^{2} c d^{4} \log \left (F\right )^{2}\right )} x^{5} + 3 \,{\left (990 \, F^{a} c^{8} d^{3} + 84 \, F^{a} b c^{5} d^{3} \log \left (F\right ) + 5 \, F^{a} b^{2} c^{2} d^{3} \log \left (F\right )^{2}\right )} x^{4} +{\left (1320 \, F^{a} c^{9} d^{2} + 168 \, F^{a} b c^{6} d^{2} \log \left (F\right ) + 20 \, F^{a} b^{2} c^{3} d^{2} \log \left (F\right )^{2} + F^{a} b^{3} d^{2} \log \left (F\right )^{3}\right )} x^{3} + 3 \,{\left (132 \, F^{a} c^{10} d + 24 \, F^{a} b c^{7} d \log \left (F\right ) + 5 \, F^{a} b^{2} c^{4} d \log \left (F\right )^{2} + F^{a} b^{3} c d \log \left (F\right )^{3}\right )} x^{2} + 3 \,{\left (24 \, F^{a} c^{11} + 6 \, F^{a} b c^{8} \log \left (F\right ) + 2 \, F^{a} b^{2} c^{5} \log \left (F\right )^{2} + F^{a} b^{3} c^{2} \log \left (F\right )^{3}\right )} x\right )} F^{\frac{b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}} + \int -\frac{{\left (6 \, F^{a} b c^{12} \log \left (F\right ) - F^{a} b^{4} d^{3} x^{3} \log \left (F\right )^{4} + 2 \, F^{a} b^{2} c^{9} \log \left (F\right )^{2} - 3 \, F^{a} b^{4} c d^{2} x^{2} \log \left (F\right )^{4} + F^{a} b^{3} c^{6} \log \left (F\right )^{3} - 3 \, F^{a} b^{4} c^{2} d x \log \left (F\right )^{4}\right )} F^{\frac{b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}}}{24 \,{\left (d^{4} x^{4} + 4 \, c d^{3} x^{3} + 6 \, c^{2} d^{2} x^{2} + 4 \, c^{3} d x + c^{4}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^11*F^(a + b/(d*x + c)^3),x, algorithm="maxima")

[Out]

1/72*(6*F^a*d^11*x^12 + 72*F^a*c*d^10*x^11 + 396*F^a*c^2*d^9*x^10 + 2*(660*F^a*c
^3*d^8 + F^a*b*d^8*log(F))*x^9 + 18*(165*F^a*c^4*d^7 + F^a*b*c*d^7*log(F))*x^8 +
72*(66*F^a*c^5*d^6 + F^a*b*c^2*d^6*log(F))*x^7 + (5544*F^a*c^6*d^5 + 168*F^a*b*
c^3*d^5*log(F) + F^a*b^2*d^5*log(F)^2)*x^6 + 6*(792*F^a*c^7*d^4 + 42*F^a*b*c^4*d
^4*log(F) + F^a*b^2*c*d^4*log(F)^2)*x^5 + 3*(990*F^a*c^8*d^3 + 84*F^a*b*c^5*d^3*
log(F) + 5*F^a*b^2*c^2*d^3*log(F)^2)*x^4 + (1320*F^a*c^9*d^2 + 168*F^a*b*c^6*d^2
*log(F) + 20*F^a*b^2*c^3*d^2*log(F)^2 + F^a*b^3*d^2*log(F)^3)*x^3 + 3*(132*F^a*c
^10*d + 24*F^a*b*c^7*d*log(F) + 5*F^a*b^2*c^4*d*log(F)^2 + F^a*b^3*c*d*log(F)^3)
*x^2 + 3*(24*F^a*c^11 + 6*F^a*b*c^8*log(F) + 2*F^a*b^2*c^5*log(F)^2 + F^a*b^3*c^
2*log(F)^3)*x)*F^(b/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3)) + integrate(-1/24
*(6*F^a*b*c^12*log(F) - F^a*b^4*d^3*x^3*log(F)^4 + 2*F^a*b^2*c^9*log(F)^2 - 3*F^
a*b^4*c*d^2*x^2*log(F)^4 + F^a*b^3*c^6*log(F)^3 - 3*F^a*b^4*c^2*d*x*log(F)^4)*F^
(b/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3))/(d^4*x^4 + 4*c*d^3*x^3 + 6*c^2*d^2
*x^2 + 4*c^3*d*x + c^4), x)

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Fricas [A]  time = 0.278766, size = 657, normalized size = 21.19 \[ -\frac{F^{a} b^{4}{\rm Ei}\left (\frac{b \log \left (F\right )}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) \log \left (F\right )^{4} -{\left (6 \, d^{12} x^{12} + 72 \, c d^{11} x^{11} + 396 \, c^{2} d^{10} x^{10} + 1320 \, c^{3} d^{9} x^{9} + 2970 \, c^{4} d^{8} x^{8} + 4752 \, c^{5} d^{7} x^{7} + 5544 \, c^{6} d^{6} x^{6} + 4752 \, c^{7} d^{5} x^{5} + 2970 \, c^{8} d^{4} x^{4} + 1320 \, c^{9} d^{3} x^{3} + 396 \, c^{10} d^{2} x^{2} + 72 \, c^{11} d x + 6 \, c^{12} +{\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3}\right )} \log \left (F\right )^{3} +{\left (b^{2} d^{6} x^{6} + 6 \, b^{2} c d^{5} x^{5} + 15 \, b^{2} c^{2} d^{4} x^{4} + 20 \, b^{2} c^{3} d^{3} x^{3} + 15 \, b^{2} c^{4} d^{2} x^{2} + 6 \, b^{2} c^{5} d x + b^{2} c^{6}\right )} \log \left (F\right )^{2} + 2 \,{\left (b d^{9} x^{9} + 9 \, b c d^{8} x^{8} + 36 \, b c^{2} d^{7} x^{7} + 84 \, b c^{3} d^{6} x^{6} + 126 \, b c^{4} d^{5} x^{5} + 126 \, b c^{5} d^{4} x^{4} + 84 \, b c^{6} d^{3} x^{3} + 36 \, b c^{7} d^{2} x^{2} + 9 \, b c^{8} d x + b c^{9}\right )} \log \left (F\right )\right )} F^{\frac{a d^{3} x^{3} + 3 \, a c d^{2} x^{2} + 3 \, a c^{2} d x + a c^{3} + b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}}}{72 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^11*F^(a + b/(d*x + c)^3),x, algorithm="fricas")

[Out]

-1/72*(F^a*b^4*Ei(b*log(F)/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3))*log(F)^4 -
(6*d^12*x^12 + 72*c*d^11*x^11 + 396*c^2*d^10*x^10 + 1320*c^3*d^9*x^9 + 2970*c^4
*d^8*x^8 + 4752*c^5*d^7*x^7 + 5544*c^6*d^6*x^6 + 4752*c^7*d^5*x^5 + 2970*c^8*d^4
*x^4 + 1320*c^9*d^3*x^3 + 396*c^10*d^2*x^2 + 72*c^11*d*x + 6*c^12 + (b^3*d^3*x^3
+ 3*b^3*c*d^2*x^2 + 3*b^3*c^2*d*x + b^3*c^3)*log(F)^3 + (b^2*d^6*x^6 + 6*b^2*c*
d^5*x^5 + 15*b^2*c^2*d^4*x^4 + 20*b^2*c^3*d^3*x^3 + 15*b^2*c^4*d^2*x^2 + 6*b^2*c
^5*d*x + b^2*c^6)*log(F)^2 + 2*(b*d^9*x^9 + 9*b*c*d^8*x^8 + 36*b*c^2*d^7*x^7 + 8
4*b*c^3*d^6*x^6 + 126*b*c^4*d^5*x^5 + 126*b*c^5*d^4*x^4 + 84*b*c^6*d^3*x^3 + 36*
b*c^7*d^2*x^2 + 9*b*c^8*d*x + b*c^9)*log(F))*F^((a*d^3*x^3 + 3*a*c*d^2*x^2 + 3*a
*c^2*d*x + a*c^3 + b)/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3)))/d

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F**(a+b/(d*x+c)**3)*(d*x+c)**11,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (d x + c\right )}^{11} F^{a + \frac{b}{{\left (d x + c\right )}^{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^11*F^(a + b/(d*x + c)^3),x, algorithm="giac")

[Out]

integrate((d*x + c)^11*F^(a + b/(d*x + c)^3), x)