3.49 \(\int F^{c (a+b x)} (d+e x)^{4/3} \, dx\)

Optimal. Leaf size=71 \[ -\frac{e \sqrt [3]{d+e x} F^{c \left (a-\frac{b d}{e}\right )} \text{Gamma}\left (\frac{7}{3},-\frac{b c \log (F) (d+e x)}{e}\right )}{b^2 c^2 \log ^2(F) \sqrt [3]{-\frac{b c \log (F) (d+e x)}{e}}} \]

[Out]

-((e*F^(c*(a - (b*d)/e))*(d + e*x)^(1/3)*Gamma[7/3, -((b*c*(d + e*x)*Log[F])/e)]
)/(b^2*c^2*Log[F]^2*(-((b*c*(d + e*x)*Log[F])/e))^(1/3)))

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Rubi [A]  time = 0.057213, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053 \[ -\frac{e \sqrt [3]{d+e x} F^{c \left (a-\frac{b d}{e}\right )} \text{Gamma}\left (\frac{7}{3},-\frac{b c \log (F) (d+e x)}{e}\right )}{b^2 c^2 \log ^2(F) \sqrt [3]{-\frac{b c \log (F) (d+e x)}{e}}} \]

Antiderivative was successfully verified.

[In]  Int[F^(c*(a + b*x))*(d + e*x)^(4/3),x]

[Out]

-((e*F^(c*(a - (b*d)/e))*(d + e*x)^(1/3)*Gamma[7/3, -((b*c*(d + e*x)*Log[F])/e)]
)/(b^2*c^2*Log[F]^2*(-((b*c*(d + e*x)*Log[F])/e))^(1/3)))

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Rubi in Sympy [A]  time = 6.84765, size = 71, normalized size = 1. \[ - \frac{F^{\frac{c \left (a e - b d\right )}{e}} e \sqrt [3]{d + e x} \Gamma{\left (\frac{7}{3},\frac{b c \left (- d - e x\right ) \log{\left (F \right )}}{e} \right )}}{b^{2} c^{2} \sqrt [3]{\frac{b c \left (- d - e x\right ) \log{\left (F \right )}}{e}} \log{\left (F \right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(F**(c*(b*x+a))*(e*x+d)**(4/3),x)

[Out]

-F**(c*(a*e - b*d)/e)*e*(d + e*x)**(1/3)*Gamma(7/3, b*c*(-d - e*x)*log(F)/e)/(b*
*2*c**2*(b*c*(-d - e*x)*log(F)/e)**(1/3)*log(F)**2)

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Mathematica [A]  time = 0.203369, size = 63, normalized size = 0.89 \[ -\frac{(d+e x)^{7/3} F^{c \left (a-\frac{b d}{e}\right )} \text{Gamma}\left (\frac{7}{3},-\frac{b c \log (F) (d+e x)}{e}\right )}{e \left (-\frac{b c \log (F) (d+e x)}{e}\right )^{7/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[F^(c*(a + b*x))*(d + e*x)^(4/3),x]

[Out]

-((F^(c*(a - (b*d)/e))*(d + e*x)^(7/3)*Gamma[7/3, -((b*c*(d + e*x)*Log[F])/e)])/
(e*(-((b*c*(d + e*x)*Log[F])/e))^(7/3)))

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Maple [F]  time = 0.026, size = 0, normalized size = 0. \[ \int{F}^{c \left ( bx+a \right ) } \left ( ex+d \right ) ^{{\frac{4}{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(F^(c*(b*x+a))*(e*x+d)^(4/3),x)

[Out]

int(F^(c*(b*x+a))*(e*x+d)^(4/3),x)

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Maxima [A]  time = 0.904104, size = 81, normalized size = 1.14 \[ -\frac{{\left (e x + d\right )}^{\frac{7}{3}} F^{a c} \Gamma \left (\frac{7}{3}, -\frac{{\left (e x + d\right )} b c \log \left (F\right )}{e}\right )}{\left (-\frac{{\left (e x + d\right )} b c \log \left (F\right )}{e}\right )^{\frac{7}{3}} F^{\frac{b c d}{e}} e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(4/3)*F^((b*x + a)*c),x, algorithm="maxima")

[Out]

-(e*x + d)^(7/3)*F^(a*c)*gamma(7/3, -(e*x + d)*b*c*log(F)/e)/((-(e*x + d)*b*c*lo
g(F)/e)^(7/3)*F^(b*c*d/e)*e)

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Fricas [A]  time = 0.283813, size = 151, normalized size = 2.13 \[ \frac{3 \,{\left (e x + d\right )}^{\frac{1}{3}}{\left (3 \,{\left (b c e x + b c d\right )} \log \left (F\right ) - 4 \, e\right )} \left (-\frac{b c \log \left (F\right )}{e}\right )^{\frac{1}{3}} F^{b c x + a c} - \frac{4 \, e \Gamma \left (\frac{1}{3}, -\frac{{\left (b c e x + b c d\right )} \log \left (F\right )}{e}\right )}{F^{\frac{b c d - a c e}{e}}}}{9 \, \left (-\frac{b c \log \left (F\right )}{e}\right )^{\frac{1}{3}} b^{2} c^{2} \log \left (F\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(4/3)*F^((b*x + a)*c),x, algorithm="fricas")

[Out]

1/9*(3*(e*x + d)^(1/3)*(3*(b*c*e*x + b*c*d)*log(F) - 4*e)*(-b*c*log(F)/e)^(1/3)*
F^(b*c*x + a*c) - 4*e*gamma(1/3, -(b*c*e*x + b*c*d)*log(F)/e)/F^((b*c*d - a*c*e)
/e))/((-b*c*log(F)/e)^(1/3)*b^2*c^2*log(F)^2)

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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**(c*(b*x+a))*(e*x+d)**(4/3),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (e x + d\right )}^{\frac{4}{3}} F^{{\left (b x + a\right )} c}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(4/3)*F^((b*x + a)*c),x, algorithm="giac")

[Out]

integrate((e*x + d)^(4/3)*F^((b*x + a)*c), x)