3.356 \(\int \frac{F^{a+\frac{b}{(c+d x)^3}}}{(c+d x)^2} \, dx\)

Optimal. Leaf size=49 \[ \frac{F^a \text{Gamma}\left (\frac{1}{3},-\frac{b \log (F)}{(c+d x)^3}\right )}{3 d (c+d x) \sqrt [3]{-\frac{b \log (F)}{(c+d x)^3}}} \]

[Out]

(F^a*Gamma[1/3, -((b*Log[F])/(c + d*x)^3)])/(3*d*(c + d*x)*(-((b*Log[F])/(c + d*
x)^3))^(1/3))

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Rubi [A]  time = 0.0778039, antiderivative size = 49, 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 \[ \frac{F^a \text{Gamma}\left (\frac{1}{3},-\frac{b \log (F)}{(c+d x)^3}\right )}{3 d (c+d x) \sqrt [3]{-\frac{b \log (F)}{(c+d x)^3}}} \]

Antiderivative was successfully verified.

[In]  Int[F^(a + b/(c + d*x)^3)/(c + d*x)^2,x]

[Out]

(F^a*Gamma[1/3, -((b*Log[F])/(c + d*x)^3)])/(3*d*(c + d*x)*(-((b*Log[F])/(c + d*
x)^3))^(1/3))

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Rubi in Sympy [A]  time = 5.9025, size = 44, normalized size = 0.9 \[ \frac{F^{a} \Gamma{\left (\frac{1}{3},- \frac{b \log{\left (F \right )}}{\left (c + d x\right )^{3}} \right )}}{3 d \sqrt [3]{- \frac{b \log{\left (F \right )}}{\left (c + d x\right )^{3}}} \left (c + d x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(F**(a+b/(d*x+c)**3)/(d*x+c)**2,x)

[Out]

F**a*Gamma(1/3, -b*log(F)/(c + d*x)**3)/(3*d*(-b*log(F)/(c + d*x)**3)**(1/3)*(c
+ d*x))

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Mathematica [A]  time = 0.0367174, size = 49, normalized size = 1. \[ \frac{F^a \text{Gamma}\left (\frac{1}{3},-\frac{b \log (F)}{(c+d x)^3}\right )}{3 d (c+d x) \sqrt [3]{-\frac{b \log (F)}{(c+d x)^3}}} \]

Antiderivative was successfully verified.

[In]  Integrate[F^(a + b/(c + d*x)^3)/(c + d*x)^2,x]

[Out]

(F^a*Gamma[1/3, -((b*Log[F])/(c + d*x)^3)])/(3*d*(c + d*x)*(-((b*Log[F])/(c + d*
x)^3))^(1/3))

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Maple [F]  time = 0.051, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( dx+c \right ) ^{2}}{F}^{a+{\frac{b}{ \left ( dx+c \right ) ^{3}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(F^(a+b/(d*x+c)^3)/(d*x+c)^2,x)

[Out]

int(F^(a+b/(d*x+c)^3)/(d*x+c)^2,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{F^{a + \frac{b}{{\left (d x + c\right )}^{3}}}}{{\left (d x + c\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^(a + b/(d*x + c)^3)/(d*x + c)^2,x, algorithm="maxima")

[Out]

integrate(F^(a + b/(d*x + c)^3)/(d*x + c)^2, x)

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Fricas [A]  time = 0.265686, size = 73, normalized size = 1.49 \[ \frac{F^{a} \Gamma \left (\frac{1}{3}, -\frac{b \log \left (F\right )}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right )}{3 \, d^{2} \left (-\frac{b \log \left (F\right )}{d^{3}}\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^(a + b/(d*x + c)^3)/(d*x + c)^2,x, algorithm="fricas")

[Out]

1/3*F^a*gamma(1/3, -b*log(F)/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3))/(d^2*(-b
*log(F)/d^3)^(1/3))

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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)**2,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{F^{a + \frac{b}{{\left (d x + c\right )}^{3}}}}{{\left (d x + c\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^(a + b/(d*x + c)^3)/(d*x + c)^2,x, algorithm="giac")

[Out]

integrate(F^(a + b/(d*x + c)^3)/(d*x + c)^2, x)