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

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

[Out]

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

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Rubi [A]  time = 0.0518568, 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, Rules used = {2226, 2208, 2218} \[ \frac{(a+b x)^2 \left (-\frac{c \log (f)}{(a+b x)^3}\right )^{2/3} \text{Gamma}\left (-\frac{2}{3},-\frac{c \log (f)}{(a+b x)^3}\right )}{3 b^2}-\frac{a (a+b x) \sqrt [3]{-\frac{c \log (f)}{(a+b x)^3}} \text{Gamma}\left (-\frac{1}{3},-\frac{c \log (f)}{(a+b x)^3}\right )}{3 b^2} \]

Antiderivative was successfully verified.

[In]

Int[f^(c/(a + b*x)^3)*x,x]

[Out]

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

Rule 2226

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*(u_), x_Symbol] :> Int[ExpandLinearProduct[F^(a + b*(c + d*
x)^n), u, c, d, x], x] /; FreeQ[{F, a, b, c, d, n}, x] && PolynomialQ[u, x]

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]

Rule 2218

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> -Simp[(F^a*(e + f*
x)^(m + 1)*Gamma[(m + 1)/n, -(b*(c + d*x)^n*Log[F])])/(f*n*(-(b*(c + d*x)^n*Log[F]))^((m + 1)/n)), x] /; FreeQ
[{F, a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0]

Rubi steps

\begin{align*} \int f^{\frac{c}{(a+b x)^3}} x \, dx &=\int \left (-\frac{a f^{\frac{c}{(a+b x)^3}}}{b}+\frac{f^{\frac{c}{(a+b x)^3}} (a+b x)}{b}\right ) \, dx\\ &=\frac{\int f^{\frac{c}{(a+b x)^3}} (a+b x) \, dx}{b}-\frac{a \int f^{\frac{c}{(a+b x)^3}} \, dx}{b}\\ &=-\frac{a (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^2}+\frac{(a+b x)^2 \Gamma \left (-\frac{2}{3},-\frac{c \log (f)}{(a+b x)^3}\right ) \left (-\frac{c \log (f)}{(a+b x)^3}\right )^{2/3}}{3 b^2}\\ \end{align*}

Mathematica [A]  time = 0.067011, size = 86, normalized size = 0.93 \[ \frac{(a+b x) \left ((a+b x) \left (-\frac{c \log (f)}{(a+b x)^3}\right )^{2/3} \text{Gamma}\left (-\frac{2}{3},-\frac{c \log (f)}{(a+b x)^3}\right )-a \sqrt [3]{-\frac{c \log (f)}{(a+b x)^3}} \text{Gamma}\left (-\frac{1}{3},-\frac{c \log (f)}{(a+b x)^3}\right )\right )}{3 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(c/(a + b*x)^3)*x,x]

[Out]

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

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Maple [F]  time = 0.033, size = 0, normalized size = 0. \begin{align*} \int{f}^{{\frac{c}{ \left ( bx+a \right ) ^{3}}}}x\, dx \end{align*}

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. \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^{2}}{2 \,{\left (b^{4} x^{4} + 4 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} + 4 \, a^{3} b x + a^{4}\right )}}\,{d x} \log \left (f\right ) + \frac{1}{2} \, f^{\frac{c}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*b*c*integrate(1/2*f^(c/(b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x + a^3))*x^2/(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) + 1/2*f^(c/(b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x + a^3))*x^2

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Fricas [A]  time = 1.5793, size = 354, normalized size = 3.85 \begin{align*} -\frac{b^{2} \left (-\frac{c \log \left (f\right )}{b^{3}}\right )^{\frac{2}{3}} \Gamma \left (\frac{1}{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 ) - 2 \, a 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^{2} x^{2} - a^{2}\right )} f^{\frac{c}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}}}{2 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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,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}}} x\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(f^(c/(b*x + a)^3)*x, x)