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

Optimal. Leaf size=44 \[ -\frac{(a+b x) \text{Gamma}\left (\frac{1}{3},-c \log (f) (a+b x)^3\right )}{3 b \sqrt [3]{-c \log (f) (a+b x)^3}} \]

[Out]

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

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Rubi [A]  time = 0.0055393, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {2208} \[ -\frac{(a+b x) \text{Gamma}\left (\frac{1}{3},-c \log (f) (a+b x)^3\right )}{3 b \sqrt [3]{-c \log (f) (a+b x)^3}} \]

Antiderivative was successfully verified.

[In]

Int[f^(c*(a + b*x)^3),x]

[Out]

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

Rule 2208

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_)), x_Symbol] :> -Simp[(F^a*(c + d*x)*Gamma[1/n, -(b*(c + d*x)
^n*Log[F])])/(d*n*(-(b*(c + d*x)^n*Log[F]))^(1/n)), x] /; FreeQ[{F, a, b, c, d, n}, x] &&  !IntegerQ[2/n]

Rubi steps

\begin{align*} \int f^{c (a+b x)^3} \, dx &=-\frac{(a+b x) \Gamma \left (\frac{1}{3},-c (a+b x)^3 \log (f)\right )}{3 b \sqrt [3]{-c (a+b x)^3 \log (f)}}\\ \end{align*}

Mathematica [A]  time = 0.0090916, size = 44, normalized size = 1. \[ -\frac{(a+b x) \text{Gamma}\left (\frac{1}{3},-c \log (f) (a+b x)^3\right )}{3 b \sqrt [3]{-c \log (f) (a+b x)^3}} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(c*(a + b*x)^3),x]

[Out]

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

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Maple [F]  time = 0.016, size = 0, normalized size = 0. \begin{align*} \int{f}^{c \left ( bx+a \right ) ^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(f^((b*x + a)^3*c), x)

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Fricas [A]  time = 1.52329, size = 151, normalized size = 3.43 \begin{align*} \frac{\left (-b^{3} c \log \left (f\right )\right )^{\frac{2}{3}} \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 )}{3 \, b^{3} c \log \left (f\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(-b^3*c*log(f))^(2/3)*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^3*c*log(f))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(f**(c*(a + b*x)**3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(f^((b*x + a)^3*c), x)