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

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

[Out]

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

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Rubi [A]  time = 0.105393, 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 \[ -\frac{F^a (c+d x)^4 \text{Gamma}\left (\frac{4}{3},-b \log (F) (c+d x)^3\right )}{3 d \left (-b \log (F) (c+d x)^3\right )^{4/3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 5.49816, size = 48, normalized size = 0.98 \[ - \frac{F^{a} \left (c + d x\right )^{4} \Gamma{\left (\frac{4}{3},- b \left (c + d x\right )^{3} \log{\left (F \right )} \right )}}{3 d \left (- b \left (c + d x\right )^{3} \log{\left (F \right )}\right )^{\frac{4}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.107883, size = 79, normalized size = 1.61 \[ -\frac{F^a (c+d x)^4 \left (\text{Gamma}\left (\frac{1}{3},-b \log (F) (c+d x)^3\right )+3 F^{b (c+d x)^3} \sqrt [3]{-b \log (F) (c+d x)^3}\right )}{9 d \left (-b \log (F) (c+d x)^3\right )^{4/3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.05, size = 0, normalized size = 0. \[ \int{F}^{a+b \left ( dx+c \right ) ^{3}} \left ( dx+c \right ) ^{3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.296052, size = 159, normalized size = 3.24 \[ \frac{F^{a} d \Gamma \left (\frac{1}{3}, -{\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \log \left (F\right )\right ) + 3 \, \left (-b d^{3} \log \left (F\right )\right )^{\frac{1}{3}}{\left (d x + c\right )} F^{b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a}}{9 \, \left (-b d^{3} \log \left (F\right )\right )^{\frac{1}{3}} b d \log \left (F\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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)**3,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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