3.298 \(\int \frac{F^{a+b (c+d x)^3}}{(c+d x)^5} \, dx\)

Optimal. Leaf size=49 \[ -\frac{F^a \left (-b \log (F) (c+d x)^3\right )^{4/3} \text{Gamma}\left (-\frac{4}{3},-b \log (F) (c+d x)^3\right )}{3 d (c+d x)^4} \]

[Out]

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

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Rubi [A]  time = 0.0995934, 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 \[ -\frac{F^a \left (-b \log (F) (c+d x)^3\right )^{4/3} \text{Gamma}\left (-\frac{4}{3},-b \log (F) (c+d x)^3\right )}{3 d (c+d x)^4} \]

Antiderivative was successfully verified.

[In]  Int[F^(a + b*(c + d*x)^3)/(c + d*x)^5,x]

[Out]

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

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Rubi in Sympy [A]  time = 5.31346, size = 49, normalized size = 1. \[ - \frac{F^{a} \left (- b \left (c + d x\right )^{3} \log{\left (F \right )}\right )^{\frac{4}{3}} \Gamma{\left (- \frac{4}{3},- b \left (c + d x\right )^{3} \log{\left (F \right )} \right )}}{3 d \left (c + d x\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(F**(a+b*(d*x+c)**3)/(d*x+c)**5,x)

[Out]

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

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Mathematica [A]  time = 0.248423, size = 93, normalized size = 1.9 \[ \frac{F^a \left (\frac{3 b^3 \log ^3(F) (c+d x)^9 \text{Gamma}\left (\frac{2}{3},-b \log (F) (c+d x)^3\right )}{\left (-b \log (F) (c+d x)^3\right )^{5/3}}-F^{b (c+d x)^3} \left (3 b \log (F) (c+d x)^3+1\right )\right )}{4 d (c+d x)^4} \]

Antiderivative was successfully verified.

[In]  Integrate[F^(a + b*(c + d*x)^3)/(c + d*x)^5,x]

[Out]

(F^a*((3*b^3*(c + d*x)^9*Gamma[2/3, -(b*(c + d*x)^3*Log[F])]*Log[F]^3)/(-(b*(c +
d*x)^3*Log[F]))^(5/3) - F^(b*(c + d*x)^3)*(1 + 3*b*(c + d*x)^3*Log[F])))/(4*d*(
c + d*x)^4)

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Maple [F]  time = 0.071, size = 0, normalized size = 0. \[ \int{\frac{{F}^{a+b \left ( dx+c \right ) ^{3}}}{ \left ( dx+c \right ) ^{5}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(F^(a+b*(d*x+c)^3)/(d*x+c)^5,x)

[Out]

int(F^(a+b*(d*x+c)^3)/(d*x+c)^5,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{F^{{\left (d x + c\right )}^{3} b + a}}{{\left (d x + c\right )}^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^((d*x + c)^3*b + a)/(d*x + c)^5,x, algorithm="maxima")

[Out]

integrate(F^((d*x + c)^3*b + a)/(d*x + c)^5, x)

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Fricas [A]  time = 0.278192, size = 333, normalized size = 6.8 \[ -\frac{3 \,{\left (b^{2} d^{6} x^{4} + 4 \, b^{2} c d^{5} x^{3} + 6 \, b^{2} c^{2} d^{4} x^{2} + 4 \, b^{2} c^{3} d^{3} x + b^{2} c^{4} d^{2}\right )} F^{a} \Gamma \left (\frac{2}{3}, -{\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \log \left (F\right )\right ) \log \left (F\right )^{2} + \left (-b d^{3} \log \left (F\right )\right )^{\frac{2}{3}}{\left (3 \,{\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \log \left (F\right ) + 1\right )} F^{b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a}}{4 \,{\left (d^{5} x^{4} + 4 \, c d^{4} x^{3} + 6 \, c^{2} d^{3} x^{2} + 4 \, c^{3} d^{2} x + c^{4} d\right )} \left (-b d^{3} \log \left (F\right )\right )^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^((d*x + c)^3*b + a)/(d*x + c)^5,x, algorithm="fricas")

[Out]

-1/4*(3*(b^2*d^6*x^4 + 4*b^2*c*d^5*x^3 + 6*b^2*c^2*d^4*x^2 + 4*b^2*c^3*d^3*x + b
^2*c^4*d^2)*F^a*gamma(2/3, -(b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3)*lo
g(F))*log(F)^2 + (-b*d^3*log(F))^(2/3)*(3*(b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d
*x + b*c^3)*log(F) + 1)*F^(b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3 + a))
/((d^5*x^4 + 4*c*d^4*x^3 + 6*c^2*d^3*x^2 + 4*c^3*d^2*x + c^4*d)*(-b*d^3*log(F))^
(2/3))

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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)**5,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{F^{{\left (d x + c\right )}^{3} b + a}}{{\left (d x + c\right )}^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^((d*x + c)^3*b + a)/(d*x + c)^5,x, algorithm="giac")

[Out]

integrate(F^((d*x + c)^3*b + a)/(d*x + c)^5, x)