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

Optimal. Leaf size=184 \[ \frac{a^2 (a+b x)^2 \left (-\frac{c \log (f)}{(a+b x)^3}\right )^{2/3} \text{Gamma}\left (-\frac{2}{3},-\frac{c \log (f)}{(a+b x)^3}\right )}{b^4}-\frac{a^3 (a+b x) \sqrt [3]{-\frac{c \log (f)}{(a+b x)^3}} \text{Gamma}\left (-\frac{1}{3},-\frac{c \log (f)}{(a+b x)^3}\right )}{3 b^4}+\frac{(a+b x)^4 \left (-\frac{c \log (f)}{(a+b x)^3}\right )^{4/3} \text{Gamma}\left (-\frac{4}{3},-\frac{c \log (f)}{(a+b x)^3}\right )}{3 b^4}+\frac{a c \log (f) \text{Ei}\left (\frac{c \log (f)}{(a+b x)^3}\right )}{b^4}-\frac{a (a+b x)^3 f^{\frac{c}{(a+b x)^3}}}{b^4} \]

[Out]

-((a*f^(c/(a + b*x)^3)*(a + b*x)^3)/b^4) + (a*c*ExpIntegralEi[(c*Log[f])/(a + b*x)^3]*Log[f])/b^4 - (a^3*(a +
b*x)*Gamma[-1/3, -((c*Log[f])/(a + b*x)^3)]*(-((c*Log[f])/(a + b*x)^3))^(1/3))/(3*b^4) + (a^2*(a + b*x)^2*Gamm
a[-2/3, -((c*Log[f])/(a + b*x)^3)]*(-((c*Log[f])/(a + b*x)^3))^(2/3))/b^4 + ((a + b*x)^4*Gamma[-4/3, -((c*Log[
f])/(a + b*x)^3)]*(-((c*Log[f])/(a + b*x)^3))^(4/3))/(3*b^4)

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Rubi [A]  time = 0.149328, antiderivative size = 184, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2226, 2208, 2218, 2214, 2210} \[ \frac{a^2 (a+b x)^2 \left (-\frac{c \log (f)}{(a+b x)^3}\right )^{2/3} \text{Gamma}\left (-\frac{2}{3},-\frac{c \log (f)}{(a+b x)^3}\right )}{b^4}-\frac{a^3 (a+b x) \sqrt [3]{-\frac{c \log (f)}{(a+b x)^3}} \text{Gamma}\left (-\frac{1}{3},-\frac{c \log (f)}{(a+b x)^3}\right )}{3 b^4}+\frac{(a+b x)^4 \left (-\frac{c \log (f)}{(a+b x)^3}\right )^{4/3} \text{Gamma}\left (-\frac{4}{3},-\frac{c \log (f)}{(a+b x)^3}\right )}{3 b^4}+\frac{a c \log (f) \text{Ei}\left (\frac{c \log (f)}{(a+b x)^3}\right )}{b^4}-\frac{a (a+b x)^3 f^{\frac{c}{(a+b x)^3}}}{b^4} \]

Antiderivative was successfully verified.

[In]

Int[f^(c/(a + b*x)^3)*x^3,x]

[Out]

-((a*f^(c/(a + b*x)^3)*(a + b*x)^3)/b^4) + (a*c*ExpIntegralEi[(c*Log[f])/(a + b*x)^3]*Log[f])/b^4 - (a^3*(a +
b*x)*Gamma[-1/3, -((c*Log[f])/(a + b*x)^3)]*(-((c*Log[f])/(a + b*x)^3))^(1/3))/(3*b^4) + (a^2*(a + b*x)^2*Gamm
a[-2/3, -((c*Log[f])/(a + b*x)^3)]*(-((c*Log[f])/(a + b*x)^3))^(2/3))/b^4 + ((a + b*x)^4*Gamma[-4/3, -((c*Log[
f])/(a + b*x)^3)]*(-((c*Log[f])/(a + b*x)^3))^(4/3))/(3*b^4)

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]

Rule 2214

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m
+ 1)*F^(a + b*(c + d*x)^n))/(d*(m + 1)), x] - Dist[(b*n*Log[F])/(m + 1), Int[(c + d*x)^(m + n)*F^(a + b*(c +
d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[(2*(m + 1))/n] && LtQ[-4, (m + 1)/n, 5] && IntegerQ[n
] && ((GtQ[n, 0] && LtQ[m, -1]) || (GtQ[-n, 0] && LeQ[-n, m + 1]))

Rule 2210

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

Rubi steps

\begin{align*} \int f^{\frac{c}{(a+b x)^3}} x^3 \, dx &=\int \left (-\frac{a^3 f^{\frac{c}{(a+b x)^3}}}{b^3}+\frac{3 a^2 f^{\frac{c}{(a+b x)^3}} (a+b x)}{b^3}-\frac{3 a f^{\frac{c}{(a+b x)^3}} (a+b x)^2}{b^3}+\frac{f^{\frac{c}{(a+b x)^3}} (a+b x)^3}{b^3}\right ) \, dx\\ &=\frac{\int f^{\frac{c}{(a+b x)^3}} (a+b x)^3 \, dx}{b^3}-\frac{(3 a) \int f^{\frac{c}{(a+b x)^3}} (a+b x)^2 \, dx}{b^3}+\frac{\left (3 a^2\right ) \int f^{\frac{c}{(a+b x)^3}} (a+b x) \, dx}{b^3}-\frac{a^3 \int f^{\frac{c}{(a+b x)^3}} \, dx}{b^3}\\ &=-\frac{a f^{\frac{c}{(a+b x)^3}} (a+b x)^3}{b^4}-\frac{a^3 (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^4}+\frac{a^2 (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}}{b^4}+\frac{(a+b x)^4 \Gamma \left (-\frac{4}{3},-\frac{c \log (f)}{(a+b x)^3}\right ) \left (-\frac{c \log (f)}{(a+b x)^3}\right )^{4/3}}{3 b^4}-\frac{(3 a c \log (f)) \int \frac{f^{\frac{c}{(a+b x)^3}}}{a+b x} \, dx}{b^3}\\ &=-\frac{a f^{\frac{c}{(a+b x)^3}} (a+b x)^3}{b^4}+\frac{a c \text{Ei}\left (\frac{c \log (f)}{(a+b x)^3}\right ) \log (f)}{b^4}-\frac{a^3 (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^4}+\frac{a^2 (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}}{b^4}+\frac{(a+b x)^4 \Gamma \left (-\frac{4}{3},-\frac{c \log (f)}{(a+b x)^3}\right ) \left (-\frac{c \log (f)}{(a+b x)^3}\right )^{4/3}}{3 b^4}\\ \end{align*}

Mathematica [A]  time = 0.362696, size = 167, normalized size = 0.91 \[ \frac{3 a c \log (f) \text{Ei}\left (\frac{c \log (f)}{(a+b x)^3}\right )-(a+b x) \left (a^3 \sqrt [3]{-\frac{c \log (f)}{(a+b x)^3}} \text{Gamma}\left (-\frac{1}{3},-\frac{c \log (f)}{(a+b x)^3}\right )+3 a (a+b x) \left ((a+b x) f^{\frac{c}{(a+b x)^3}}-a \left (-\frac{c \log (f)}{(a+b x)^3}\right )^{2/3} \text{Gamma}\left (-\frac{2}{3},-\frac{c \log (f)}{(a+b x)^3}\right )\right )+c \log (f) \sqrt [3]{-\frac{c \log (f)}{(a+b x)^3}} \text{Gamma}\left (-\frac{4}{3},-\frac{c \log (f)}{(a+b x)^3}\right )\right )}{3 b^4} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(c/(a + b*x)^3)*x^3,x]

[Out]

(3*a*c*ExpIntegralEi[(c*Log[f])/(a + b*x)^3]*Log[f] - (a + b*x)*(a^3*Gamma[-1/3, -((c*Log[f])/(a + b*x)^3)]*(-
((c*Log[f])/(a + b*x)^3))^(1/3) + c*Gamma[-4/3, -((c*Log[f])/(a + b*x)^3)]*Log[f]*(-((c*Log[f])/(a + b*x)^3))^
(1/3) + 3*a*(a + b*x)*(f^(c/(a + b*x)^3)*(a + b*x) - a*Gamma[-2/3, -((c*Log[f])/(a + b*x)^3)]*(-((c*Log[f])/(a
+ b*x)^3))^(2/3))))/(3*b^4)

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Maple [F]  time = 0.042, size = 0, normalized size = 0. \begin{align*} \int{f}^{{\frac{c}{ \left ( bx+a \right ) ^{3}}}}{x}^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left (b^{3} x^{4} + 3 \, c x \log \left (f\right )\right )} f^{\frac{c}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}}}{4 \, b^{3}} - \int \frac{3 \,{\left (4 \, a b^{3} c x^{3} \log \left (f\right ) + 6 \, a^{2} b^{2} c x^{2} \log \left (f\right ) + a^{4} c \log \left (f\right ) +{\left (4 \, a^{3} b c \log \left (f\right ) - 3 \, b c^{2} \log \left (f\right )^{2}\right )} x\right )} f^{\frac{c}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}}}{4 \,{\left (b^{7} x^{4} + 4 \, a b^{6} x^{3} + 6 \, a^{2} b^{5} x^{2} + 4 \, a^{3} b^{4} x + a^{4} b^{3}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(b^3*x^4 + 3*c*x*log(f))*f^(c/(b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x + a^3))/b^3 - integrate(3/4*(4*a*b^3*c*x^
3*log(f) + 6*a^2*b^2*c*x^2*log(f) + a^4*c*log(f) + (4*a^3*b*c*log(f) - 3*b*c^2*log(f)^2)*x)*f^(c/(b^3*x^3 + 3*
a*b^2*x^2 + 3*a^2*b*x + a^3))/(b^7*x^4 + 4*a*b^6*x^3 + 6*a^2*b^5*x^2 + 4*a^3*b^4*x + a^4*b^3), x)

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Fricas [A]  time = 1.63792, size = 516, normalized size = 2.8 \begin{align*} -\frac{6 \, a^{2} 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 ) - 4 \, a c{\rm Ei}\left (\frac{c \log \left (f\right )}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}\right ) \log \left (f\right ) -{\left (4 \, a^{3} b - 3 \, b c \log \left (f\right )\right )} \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^{4} x^{4} - a^{4} + 3 \,{\left (b c x + a c\right )} \log \left (f\right )\right )} f^{\frac{c}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}}}{4 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(6*a^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)) - 4*a*c*
Ei(c*log(f)/(b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x + a^3))*log(f) - (4*a^3*b - 3*b*c*log(f))*(-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^4*x^4 - a^4 + 3*(b*c*x + a*c)*log(f))*f^
(c/(b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x + a^3)))/b^4

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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**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}}} x^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(f^(c/(b*x + a)^3)*x^3, x)