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

Optimal. Leaf size=98 \[ -\frac{e \sqrt [3]{d+e x} \left (F^{c (a+b x)}\right )^n F^{c n \left (a-\frac{b d}{e}\right )-c n (a+b x)} \text{Gamma}\left (\frac{7}{3},-\frac{b c n \log (F) (d+e x)}{e}\right )}{b^2 c^2 n^2 \log ^2(F) \sqrt [3]{-\frac{b c n \log (F) (d+e x)}{e}}} \]

[Out]

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

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Rubi [A]  time = 0.169855, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ -\frac{e \sqrt [3]{d+e x} \left (F^{c (a+b x)}\right )^n F^{c n \left (a-\frac{b d}{e}\right )-c n (a+b x)} \text{Gamma}\left (\frac{7}{3},-\frac{b c n \log (F) (d+e x)}{e}\right )}{b^2 c^2 n^2 \log ^2(F) \sqrt [3]{-\frac{b c n \log (F) (d+e x)}{e}}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 14.8097, size = 102, normalized size = 1.04 \[ - \frac{F^{c n \left (- a - b x\right )} F^{\frac{c n \left (a e - b d\right )}{e}} e \sqrt [3]{d + e x} \left (F^{c \left (a + b x\right )}\right )^{n} \Gamma{\left (\frac{7}{3},\frac{b c n \left (- d - e x\right ) \log{\left (F \right )}}{e} \right )}}{b^{2} c^{2} n^{2} \sqrt [3]{\frac{b c n \left (- d - e x\right ) \log{\left (F \right )}}{e}} \log{\left (F \right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.30187, size = 78, normalized size = 0.8 \[ -\frac{(d+e x)^{7/3} \left (F^{c (a+b x)}\right )^n F^{-\frac{b c n (d+e x)}{e}} \text{Gamma}\left (\frac{7}{3},-\frac{b c n \log (F) (d+e x)}{e}\right )}{e \left (-\frac{b c n \log (F) (d+e x)}{e}\right )^{7/3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.093, size = 0, normalized size = 0. \[ \int \left ({F}^{c \left ( bx+a \right ) } \right ) ^{n} \left ( ex+d \right ) ^{{\frac{4}{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.271391, size = 167, normalized size = 1.7 \[ \frac{3 \, \left (-\frac{b c n \log \left (F\right )}{e}\right )^{\frac{1}{3}}{\left (e x + d\right )}^{\frac{1}{3}}{\left (3 \,{\left (b c e n x + b c d n\right )} \log \left (F\right ) - 4 \, e\right )} F^{b c n x + a c n} - \frac{4 \, e \Gamma \left (\frac{1}{3}, -\frac{{\left (b c e n x + b c d n\right )} \log \left (F\right )}{e}\right )}{F^{\frac{{\left (b c d - a c e\right )} n}{e}}}}{9 \, \left (-\frac{b c n \log \left (F\right )}{e}\right )^{\frac{1}{3}} b^{2} c^{2} n^{2} \log \left (F\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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