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

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

[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*Gamma[-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 = 0.268448, 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 \[ -\frac{a^3 (a+b x) \sqrt [3]{-\frac{c \log (f)}{(a+b x)^3}} \text{Gamma}\left (-\frac{1}{3},-\frac{c \log (f)}{(a+b x)^3}\right )}{3 b^4}+\frac{a^2 (a+b x)^2 \left (-\frac{c \log (f)}{(a+b x)^3}\right )^{2/3} \text{Gamma}\left (-\frac{2}{3},-\frac{c \log (f)}{(a+b x)^3}\right )}{b^4}+\frac{(a+b x)^4 \left (-\frac{c \log (f)}{(a+b x)^3}\right )^{4/3} \text{Gamma}\left (-\frac{4}{3},-\frac{c \log (f)}{(a+b x)^3}\right )}{3 b^4}+\frac{a c \log (f) \text{ExpIntegralEi}\left (\frac{c \log (f)}{(a+b x)^3}\right )}{b^4}-\frac{a (a+b x)^3 f^{\frac{c}{(a+b x)^3}}}{b^4} \]

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*Gamma[-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 in Sympy [A]  time = 28.4839, size = 190, normalized size = 1.03 \[ - \frac{a^{3} \sqrt [3]{- \frac{c \log{\left (f \right )}}{\left (a + b x\right )^{3}}} \left (a + b x\right ) \Gamma{\left (- \frac{1}{3},- \frac{c \log{\left (f \right )}}{\left (a + b x\right )^{3}} \right )}}{3 b^{4}} + \frac{a^{2} \left (- \frac{c \log{\left (f \right )}}{\left (a + b x\right )^{3}}\right )^{\frac{2}{3}} \left (a + b x\right )^{2} \Gamma{\left (- \frac{2}{3},- \frac{c \log{\left (f \right )}}{\left (a + b x\right )^{3}} \right )}}{b^{4}} + \frac{a c \log{\left (f \right )} \operatorname{Ei}{\left (\frac{c \log{\left (f \right )}}{\left (a + b x\right )^{3}} \right )}}{b^{4}} - \frac{a f^{\frac{c}{\left (a + b x\right )^{3}}} \left (a + b x\right )^{3}}{b^{4}} + \frac{\left (- \frac{c \log{\left (f \right )}}{\left (a + b x\right )^{3}}\right )^{\frac{4}{3}} \left (a + b x\right )^{4} \Gamma{\left (- \frac{4}{3},- \frac{c \log{\left (f \right )}}{\left (a + b x\right )^{3}} \right )}}{3 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(f**(c/(b*x+a)**3)*x**3,x)

[Out]

-a**3*(-c*log(f)/(a + b*x)**3)**(1/3)*(a + b*x)*Gamma(-1/3, -c*log(f)/(a + b*x)*
*3)/(3*b**4) + a**2*(-c*log(f)/(a + b*x)**3)**(2/3)*(a + b*x)**2*Gamma(-2/3, -c*
log(f)/(a + b*x)**3)/b**4 + a*c*log(f)*Ei(c*log(f)/(a + b*x)**3)/b**4 - a*f**(c/
(a + b*x)**3)*(a + b*x)**3/b**4 + (-c*log(f)/(a + b*x)**3)**(4/3)*(a + b*x)**4*G
amma(-4/3, -c*log(f)/(a + b*x)**3)/(3*b**4)

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Mathematica [A]  time = 0.553319, size = 201, normalized size = 1.09 \[ \frac{a (a+b x) \sqrt [3]{-\frac{c \log (f)}{(a+b x)^3}} \left (6 a c \log (f) \text{Gamma}\left (\frac{1}{3},-\frac{c \log (f)}{(a+b x)^3}\right )+(a+b x) \left (3 c \log (f)-a^3\right ) f^{\frac{c}{(a+b x)^3}} \sqrt [3]{-\frac{c \log (f)}{(a+b x)^3}}\right )+c \log (f) \left (3 c \log (f)-4 a^3\right ) \text{Gamma}\left (\frac{2}{3},-\frac{c \log (f)}{(a+b x)^3}\right )}{4 b^4 (a+b x)^2 \left (-\frac{c \log (f)}{(a+b x)^3}\right )^{2/3}}+\frac{a c \log (f) \text{ExpIntegralEi}\left (\frac{c \log (f)}{(a+b x)^3}\right )}{b^4}+\frac{1}{4} x f^{\frac{c}{(a+b x)^3}} \left (\frac{3 c \log (f)}{b^3}+x^3\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[f^(c/(a + b*x)^3)*x^3,x]

[Out]

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

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Maple [F]  time = 0.043, size = 0, normalized size = 0. \[ \int{f}^{{\frac{c}{ \left ( bx+a \right ) ^{3}}}}{x}^{3}\, dx \]

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. \[ \frac{{\left (b^{3} x^{4} + 3 \, c x \log \left (f\right )\right )} f^{\frac{c}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}}}{4 \, b^{3}} - \int \frac{3 \,{\left (4 \, a b^{3} c x^{3} \log \left (f\right ) + 6 \, a^{2} b^{2} c x^{2} \log \left (f\right ) + a^{4} c \log \left (f\right ) +{\left (4 \, a^{3} b c \log \left (f\right ) - 3 \, b c^{2} \log \left (f\right )^{2}\right )} x\right )} f^{\frac{c}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}}}{4 \,{\left (b^{7} x^{4} + 4 \, a b^{6} x^{3} + 6 \, a^{2} b^{5} x^{2} + 4 \, a^{3} b^{4} x + a^{4} b^{3}\right )}}\,{d x} \]

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 = 0.288293, size = 336, normalized size = 1.83 \[ \frac{6 \, a^{2} b c \left (-\frac{c \log \left (f\right )}{b^{3}}\right )^{\frac{1}{3}} \Gamma \left (\frac{1}{3}, -\frac{c \log \left (f\right )}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}\right ) \log \left (f\right ) -{\left (4 \, a^{3} c \log \left (f\right ) - 3 \, c^{2} \log \left (f\right )^{2}\right )} \Gamma \left (\frac{2}{3}, -\frac{c \log \left (f\right )}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}\right ) +{\left (4 \, a b^{2} c{\rm Ei}\left (\frac{c \log \left (f\right )}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}\right ) \log \left (f\right ) +{\left (b^{6} x^{4} - a^{4} b^{2} + 3 \,{\left (b^{3} c x + a b^{2} c\right )} \log \left (f\right )\right )} f^{\frac{c}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}}\right )} \left (-\frac{c \log \left (f\right )}{b^{3}}\right )^{\frac{2}{3}}}{4 \, b^{6} \left (-\frac{c \log \left (f\right )}{b^{3}}\right )^{\frac{2}{3}}} \]

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*c*(-c*log(f)/b^3)^(1/3)*gamma(1/3, -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*c*log(f) - 3*c^2*log(f)^2)*gamma(2/3, -c*lo
g(f)/(b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x + a^3)) + (4*a*b^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) + (b^6*x^4 - a^4*b^2 + 3*(b^3*c*x + a
*b^2*c)*log(f))*f^(c/(b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x + a^3)))*(-c*log(f)/b^3)
^(2/3))/(b^6*(-c*log(f)/b^3)^(2/3))

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

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

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)