3.203 \(\int f^{c (a+b x)^3} x \, dx\)

Optimal. Leaf size=92 \[ \frac{a (a+b x) \text{Gamma}\left (\frac{1}{3},-c \log (f) (a+b x)^3\right )}{3 b^2 \sqrt [3]{-c \log (f) (a+b x)^3}}-\frac{(a+b x)^2 \text{Gamma}\left (\frac{2}{3},-c \log (f) (a+b x)^3\right )}{3 b^2 \left (-c \log (f) (a+b x)^3\right )^{2/3}} \]

[Out]

-((a + b*x)^2*Gamma[2/3, -(c*(a + b*x)^3*Log[f])])/(3*b^2*(-(c*(a + b*x)^3*Log[f
]))^(2/3)) + (a*(a + b*x)*Gamma[1/3, -(c*(a + b*x)^3*Log[f])])/(3*b^2*(-(c*(a +
b*x)^3*Log[f]))^(1/3))

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Rubi [A]  time = 0.0930898, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{a (a+b x) \text{Gamma}\left (\frac{1}{3},-c \log (f) (a+b x)^3\right )}{3 b^2 \sqrt [3]{-c \log (f) (a+b x)^3}}-\frac{(a+b x)^2 \text{Gamma}\left (\frac{2}{3},-c \log (f) (a+b x)^3\right )}{3 b^2 \left (-c \log (f) (a+b x)^3\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]  Int[f^(c*(a + b*x)^3)*x,x]

[Out]

-((a + b*x)^2*Gamma[2/3, -(c*(a + b*x)^3*Log[f])])/(3*b^2*(-(c*(a + b*x)^3*Log[f
]))^(2/3)) + (a*(a + b*x)*Gamma[1/3, -(c*(a + b*x)^3*Log[f])])/(3*b^2*(-(c*(a +
b*x)^3*Log[f]))^(1/3))

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Rubi in Sympy [A]  time = 8.62724, size = 90, normalized size = 0.98 \[ \frac{a \left (a + b x\right ) \Gamma{\left (\frac{1}{3},- c \left (a + b x\right )^{3} \log{\left (f \right )} \right )}}{3 b^{2} \sqrt [3]{- c \left (a + b x\right )^{3} \log{\left (f \right )}}} - \frac{\left (a + b x\right )^{2} \Gamma{\left (\frac{2}{3},- c \left (a + b x\right )^{3} \log{\left (f \right )} \right )}}{3 b^{2} \left (- c \left (a + b x\right )^{3} \log{\left (f \right )}\right )^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*(a + b*x)*Gamma(1/3, -c*(a + b*x)**3*log(f))/(3*b**2*(-c*(a + b*x)**3*log(f))*
*(1/3)) - (a + b*x)**2*Gamma(2/3, -c*(a + b*x)**3*log(f))/(3*b**2*(-c*(a + b*x)*
*3*log(f))**(2/3))

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Mathematica [A]  time = 0.555301, size = 0, normalized size = 0. \[ \int f^{c (a+b x)^3} x \, dx \]

Verification is Not applicable to the result.

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

[Out]

Integrate[f^(c*(a + b*x)^3)*x, x]

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Maple [F]  time = 0.02, size = 0, normalized size = 0. \[ \int{f}^{c \left ( bx+a \right ) ^{3}}x\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(f^(c*(b*x+a)^3)*x,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(f^((b*x + a)^3*c)*x, x)

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Fricas [A]  time = 0.2687, size = 144, normalized size = 1.57 \[ \frac{\left (-b^{3} c \log \left (f\right )\right )^{\frac{1}{3}} a \Gamma \left (\frac{1}{3}, -{\left (b^{3} c x^{3} + 3 \, a b^{2} c x^{2} + 3 \, a^{2} b c x + a^{3} c\right )} \log \left (f\right )\right ) - b \Gamma \left (\frac{2}{3}, -{\left (b^{3} c x^{3} + 3 \, a b^{2} c x^{2} + 3 \, a^{2} b c x + a^{3} c\right )} \log \left (f\right )\right )}{3 \, \left (-b^{3} c \log \left (f\right )\right )^{\frac{2}{3}} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f**(c*(b*x+a)**3)*x,x)

[Out]

Integral(f**(c*(a + b*x)**3)*x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(f^((b*x + a)^3*c)*x, x)