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

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

[Out]

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

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Rubi [A]  time = 0.0054403, 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} \[ \frac{(a+b x) \sqrt [3]{-\frac{c \log (f)}{(a+b x)^3}} \text{Gamma}\left (-\frac{1}{3},-\frac{c \log (f)}{(a+b x)^3}\right )}{3 b} \]

Antiderivative was successfully verified.

[In]

Int[f^(c/(a + b*x)^3),x]

[Out]

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

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

Mathematica [A]  time = 0.0083877, size = 44, normalized size = 1. \[ \frac{(a+b x) \sqrt [3]{-\frac{c \log (f)}{(a+b x)^3}} \text{Gamma}\left (-\frac{1}{3},-\frac{c \log (f)}{(a+b x)^3}\right )}{3 b} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(c/(a + b*x)^3),x]

[Out]

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

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Maple [F]  time = 0.025, size = 0, normalized size = 0. \begin{align*} \int{f}^{{\frac{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*} 3 \, b c \int \frac{f^{\frac{c}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}} x}{b^{4} x^{4} + 4 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} + 4 \, a^{3} b x + a^{4}}\,{d x} \log \left (f\right ) + f^{\frac{c}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}} 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]

3*b*c*integrate(f^(c/(b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x + a^3))*x/(b^4*x^4 + 4*a*b^3*x^3 + 6*a^2*b^2*x^2 + 4*a
^3*b*x + a^4), x)*log(f) + f^(c/(b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x + a^3))*x

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Fricas [B]  time = 1.5492, size = 208, normalized size = 4.73 \begin{align*} -\frac{b \left (-\frac{c \log \left (f\right )}{b^{3}}\right )^{\frac{1}{3}} \Gamma \left (\frac{2}{3}, -\frac{c \log \left (f\right )}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}\right ) -{\left (b x + a\right )} f^{\frac{c}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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