3.232 \(\int f^{\frac{c}{(a+b x)^3}} x^4 \, dx\)

Optimal. Leaf size=239 \[ -\frac{4 a^3 (a+b x)^2 \left (-\frac{c \log (f)}{(a+b x)^3}\right )^{2/3} \text{Gamma}\left (-\frac{2}{3},-\frac{c \log (f)}{(a+b x)^3}\right )}{3 b^5}+\frac{a^4 (a+b x) \sqrt [3]{-\frac{c \log (f)}{(a+b x)^3}} \text{Gamma}\left (-\frac{1}{3},-\frac{c \log (f)}{(a+b x)^3}\right )}{3 b^5}+\frac{(a+b x)^5 \left (-\frac{c \log (f)}{(a+b x)^3}\right )^{5/3} \text{Gamma}\left (-\frac{5}{3},-\frac{c \log (f)}{(a+b x)^3}\right )}{3 b^5}-\frac{4 a (a+b x)^4 \left (-\frac{c \log (f)}{(a+b x)^3}\right )^{4/3} \text{Gamma}\left (-\frac{4}{3},-\frac{c \log (f)}{(a+b x)^3}\right )}{3 b^5}-\frac{2 a^2 c \log (f) \text{Ei}\left (\frac{c \log (f)}{(a+b x)^3}\right )}{b^5}+\frac{2 a^2 (a+b x)^3 f^{\frac{c}{(a+b x)^3}}}{b^5} \]

[Out]

(2*a^2*f^(c/(a + b*x)^3)*(a + b*x)^3)/b^5 - (2*a^2*c*ExpIntegralEi[(c*Log[f])/(a + b*x)^3]*Log[f])/b^5 + (a^4*
(a + b*x)*Gamma[-1/3, -((c*Log[f])/(a + b*x)^3)]*(-((c*Log[f])/(a + b*x)^3))^(1/3))/(3*b^5) - (4*a^3*(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^5) - (4*a*(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^5) + ((a + b*x)^5*Gamma[-5/3, -((c*Lo
g[f])/(a + b*x)^3)]*(-((c*Log[f])/(a + b*x)^3))^(5/3))/(3*b^5)

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Rubi [A]  time = 0.200715, antiderivative size = 239, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ -\frac{4 a^3 (a+b x)^2 \left (-\frac{c \log (f)}{(a+b x)^3}\right )^{2/3} \text{Gamma}\left (-\frac{2}{3},-\frac{c \log (f)}{(a+b x)^3}\right )}{3 b^5}+\frac{a^4 (a+b x) \sqrt [3]{-\frac{c \log (f)}{(a+b x)^3}} \text{Gamma}\left (-\frac{1}{3},-\frac{c \log (f)}{(a+b x)^3}\right )}{3 b^5}+\frac{(a+b x)^5 \left (-\frac{c \log (f)}{(a+b x)^3}\right )^{5/3} \text{Gamma}\left (-\frac{5}{3},-\frac{c \log (f)}{(a+b x)^3}\right )}{3 b^5}-\frac{4 a (a+b x)^4 \left (-\frac{c \log (f)}{(a+b x)^3}\right )^{4/3} \text{Gamma}\left (-\frac{4}{3},-\frac{c \log (f)}{(a+b x)^3}\right )}{3 b^5}-\frac{2 a^2 c \log (f) \text{Ei}\left (\frac{c \log (f)}{(a+b x)^3}\right )}{b^5}+\frac{2 a^2 (a+b x)^3 f^{\frac{c}{(a+b x)^3}}}{b^5} \]

Antiderivative was successfully verified.

[In]

Int[f^(c/(a + b*x)^3)*x^4,x]

[Out]

(2*a^2*f^(c/(a + b*x)^3)*(a + b*x)^3)/b^5 - (2*a^2*c*ExpIntegralEi[(c*Log[f])/(a + b*x)^3]*Log[f])/b^5 + (a^4*
(a + b*x)*Gamma[-1/3, -((c*Log[f])/(a + b*x)^3)]*(-((c*Log[f])/(a + b*x)^3))^(1/3))/(3*b^5) - (4*a^3*(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^5) - (4*a*(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^5) + ((a + b*x)^5*Gamma[-5/3, -((c*Lo
g[f])/(a + b*x)^3)]*(-((c*Log[f])/(a + b*x)^3))^(5/3))/(3*b^5)

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^4 \, dx &=\int \left (\frac{a^4 f^{\frac{c}{(a+b x)^3}}}{b^4}-\frac{4 a^3 f^{\frac{c}{(a+b x)^3}} (a+b x)}{b^4}+\frac{6 a^2 f^{\frac{c}{(a+b x)^3}} (a+b x)^2}{b^4}-\frac{4 a f^{\frac{c}{(a+b x)^3}} (a+b x)^3}{b^4}+\frac{f^{\frac{c}{(a+b x)^3}} (a+b x)^4}{b^4}\right ) \, dx\\ &=\frac{\int f^{\frac{c}{(a+b x)^3}} (a+b x)^4 \, dx}{b^4}-\frac{(4 a) \int f^{\frac{c}{(a+b x)^3}} (a+b x)^3 \, dx}{b^4}+\frac{\left (6 a^2\right ) \int f^{\frac{c}{(a+b x)^3}} (a+b x)^2 \, dx}{b^4}-\frac{\left (4 a^3\right ) \int f^{\frac{c}{(a+b x)^3}} (a+b x) \, dx}{b^4}+\frac{a^4 \int f^{\frac{c}{(a+b x)^3}} \, dx}{b^4}\\ &=\frac{2 a^2 f^{\frac{c}{(a+b x)^3}} (a+b x)^3}{b^5}+\frac{a^4 (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^5}-\frac{4 a^3 (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^5}-\frac{4 a (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^5}+\frac{(a+b x)^5 \Gamma \left (-\frac{5}{3},-\frac{c \log (f)}{(a+b x)^3}\right ) \left (-\frac{c \log (f)}{(a+b x)^3}\right )^{5/3}}{3 b^5}+\frac{\left (6 a^2 c \log (f)\right ) \int \frac{f^{\frac{c}{(a+b x)^3}}}{a+b x} \, dx}{b^4}\\ &=\frac{2 a^2 f^{\frac{c}{(a+b x)^3}} (a+b x)^3}{b^5}-\frac{2 a^2 c \text{Ei}\left (\frac{c \log (f)}{(a+b x)^3}\right ) \log (f)}{b^5}+\frac{a^4 (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^5}-\frac{4 a^3 (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^5}-\frac{4 a (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^5}+\frac{(a+b x)^5 \Gamma \left (-\frac{5}{3},-\frac{c \log (f)}{(a+b x)^3}\right ) \left (-\frac{c \log (f)}{(a+b x)^3}\right )^{5/3}}{3 b^5}\\ \end{align*}

Mathematica [A]  time = 0.193196, size = 219, normalized size = 0.92 \[ \frac{-4 a^3 (a+b x)^2 \left (-\frac{c \log (f)}{(a+b x)^3}\right )^{2/3} \text{Gamma}\left (-\frac{2}{3},-\frac{c \log (f)}{(a+b x)^3}\right )+a^4 (a+b x) \sqrt [3]{-\frac{c \log (f)}{(a+b x)^3}} \text{Gamma}\left (-\frac{1}{3},-\frac{c \log (f)}{(a+b x)^3}\right )+(a+b x)^5 \left (-\frac{c \log (f)}{(a+b x)^3}\right )^{5/3} \text{Gamma}\left (-\frac{5}{3},-\frac{c \log (f)}{(a+b x)^3}\right )+4 a c \log (f) (a+b x) \sqrt [3]{-\frac{c \log (f)}{(a+b x)^3}} \text{Gamma}\left (-\frac{4}{3},-\frac{c \log (f)}{(a+b x)^3}\right )-6 a^2 c \log (f) \text{Ei}\left (\frac{c \log (f)}{(a+b x)^3}\right )+6 a^2 (a+b x)^3 f^{\frac{c}{(a+b x)^3}}}{3 b^5} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(c/(a + b*x)^3)*x^4,x]

[Out]

(6*a^2*f^(c/(a + b*x)^3)*(a + b*x)^3 - 6*a^2*c*ExpIntegralEi[(c*Log[f])/(a + b*x)^3]*Log[f] + a^4*(a + b*x)*Ga
mma[-1/3, -((c*Log[f])/(a + b*x)^3)]*(-((c*Log[f])/(a + b*x)^3))^(1/3) + 4*a*c*(a + b*x)*Gamma[-4/3, -((c*Log[
f])/(a + b*x)^3)]*Log[f]*(-((c*Log[f])/(a + b*x)^3))^(1/3) - 4*a^3*(a + b*x)^2*Gamma[-2/3, -((c*Log[f])/(a + b
*x)^3)]*(-((c*Log[f])/(a + b*x)^3))^(2/3) + (a + b*x)^5*Gamma[-5/3, -((c*Log[f])/(a + b*x)^3)]*(-((c*Log[f])/(
a + b*x)^3))^(5/3))/(3*b^5)

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Maple [F]  time = 0.046, size = 0, normalized size = 0. \begin{align*} \int{f}^{{\frac{c}{ \left ( bx+a \right ) ^{3}}}}{x}^{4}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(c/(b*x+a)^3)*x^4,x)

[Out]

int(f^(c/(b*x+a)^3)*x^4,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left (2 \, b^{4} x^{5} + 3 \, b c x^{2} \log \left (f\right ) - 24 \, a 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}}}}{10 \, b^{4}} + \int \frac{3 \,{\left (20 \, a^{2} b^{3} c x^{3} \log \left (f\right ) + 8 \, a^{5} c \log \left (f\right ) +{\left (40 \, a^{3} b^{2} c \log \left (f\right ) + 3 \, b^{2} c^{2} \log \left (f\right )^{2}\right )} x^{2} + 6 \,{\left (5 \, a^{4} b c \log \left (f\right ) - 4 \, a 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}}}}{10 \,{\left (b^{8} x^{4} + 4 \, a b^{7} x^{3} + 6 \, a^{2} b^{6} x^{2} + 4 \, a^{3} b^{5} x + a^{4} b^{4}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(c/(b*x+a)^3)*x^4,x, algorithm="maxima")

[Out]

1/10*(2*b^4*x^5 + 3*b*c*x^2*log(f) - 24*a*c*x*log(f))*f^(c/(b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x + a^3))/b^4 + in
tegrate(3/10*(20*a^2*b^3*c*x^3*log(f) + 8*a^5*c*log(f) + (40*a^3*b^2*c*log(f) + 3*b^2*c^2*log(f)^2)*x^2 + 6*(5
*a^4*b*c*log(f) - 4*a*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^8*x^4 + 4*a*b^7*x^
3 + 6*a^2*b^6*x^2 + 4*a^3*b^5*x + a^4*b^4), x)

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Fricas [A]  time = 1.60878, size = 585, normalized size = 2.45 \begin{align*} -\frac{20 \, a^{2} 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 (20 \, a^{3} b^{2} - 3 \, b^{2} c \log \left (f\right )\right )} \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 ) + 10 \,{\left (a^{4} b - 3 \, a 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 (2 \, b^{5} x^{5} + 2 \, a^{5} + 3 \,{\left (b^{2} c x^{2} - 8 \, a b c x - 9 \, a^{2} 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}}}}{10 \, b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(c/(b*x+a)^3)*x^4,x, algorithm="fricas")

[Out]

-1/10*(20*a^2*c*Ei(c*log(f)/(b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x + a^3))*log(f) - (20*a^3*b^2 - 3*b^2*c*log(f))*
(-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)) + 10*(a^4*b - 3*a*b*c*lo
g(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)) - (2*b^5*x^5 + 2*a
^5 + 3*(b^2*c*x^2 - 8*a*b*c*x - 9*a^2*c)*log(f))*f^(c/(b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x + a^3)))/b^5

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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**4,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^{4}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(c/(b*x+a)^3)*x^4,x, algorithm="giac")

[Out]

integrate(f^(c/(b*x + a)^3)*x^4, x)