\(\int f^{\frac {c}{(a+b x)^3}} x^3 \, dx\) [166]

Optimal result

Integrand size = 15, antiderivative size = 184 \[ \int f^{\frac {c}{(a+b x)^3}} x^3 \, dx=-\frac {a f^{\frac {c}{(a+b x)^3}} (a+b x)^3}{b^4}+\frac {a c \operatorname {ExpIntegralEi}\left (\frac {c \log (f)}{(a+b x)^3}\right ) \log (f)}{b^4}-\frac {a^3 (a+b x) \Gamma \left (-\frac {1}{3},-\frac {c \log (f)}{(a+b x)^3}\right ) \sqrt [3]{-\frac {c \log (f)}{(a+b x)^3}}}{3 b^4}+\frac {a^2 (a+b x)^2 \Gamma \left (-\frac {2}{3},-\frac {c \log (f)}{(a+b x)^3}\right ) \left (-\frac {c \log (f)}{(a+b x)^3}\right )^{2/3}}{b^4}+\frac {(a+b x)^4 \Gamma \left (-\frac {4}{3},-\frac {c \log (f)}{(a+b x)^3}\right ) \left (-\frac {c \log (f)}{(a+b x)^3}\right )^{4/3}}{3 b^4} \] Output:

-a*f^(c/(b*x+a)^3)*(b*x+a)^3/b^4+a*c*Ei(c*ln(f)/(b*x+a)^3)*ln(f)/b^4-1/3*a 
^3*(b*x+a)*GAMMA(-1/3,-c*ln(f)/(b*x+a)^3)*(-c*ln(f)/(b*x+a)^3)^(1/3)/b^4+a 
^2*(b*x+a)^2*GAMMA(-2/3,-c*ln(f)/(b*x+a)^3)*(-c*ln(f)/(b*x+a)^3)^(2/3)/b^4 
+1/3*(b*x+a)^4*GAMMA(-4/3,-c*ln(f)/(b*x+a)^3)*(-c*ln(f)/(b*x+a)^3)^(4/3)/b 
^4
 

Mathematica [A] (verified)

Time = 0.40 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.91 \[ \int f^{\frac {c}{(a+b x)^3}} x^3 \, dx=\frac {3 a c \operatorname {ExpIntegralEi}\left (\frac {c \log (f)}{(a+b x)^3}\right ) \log (f)-(a+b x) \left (a^3 \Gamma \left (-\frac {1}{3},-\frac {c \log (f)}{(a+b x)^3}\right ) \sqrt [3]{-\frac {c \log (f)}{(a+b x)^3}}+c \Gamma \left (-\frac {4}{3},-\frac {c \log (f)}{(a+b x)^3}\right ) \log (f) \sqrt [3]{-\frac {c \log (f)}{(a+b x)^3}}+3 a (a+b x) \left (f^{\frac {c}{(a+b x)^3}} (a+b x)-a \Gamma \left (-\frac {2}{3},-\frac {c \log (f)}{(a+b x)^3}\right ) \left (-\frac {c \log (f)}{(a+b x)^3}\right )^{2/3}\right )\right )}{3 b^4} \] Input:

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

Output:

(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*Ga 
mma[-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)
 

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2656, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 2656

\(\displaystyle \int \left (-\frac {a^3 f^{\frac {c}{(a+b x)^3}}}{b^3}+\frac {3 a^2 (a+b x) f^{\frac {c}{(a+b x)^3}}}{b^3}+\frac {(a+b x)^3 f^{\frac {c}{(a+b x)^3}}}{b^3}-\frac {3 a (a+b x)^2 f^{\frac {c}{(a+b x)^3}}}{b^3}\right )dx\)

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-((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)
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2656

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

Maple [F]

\[\int f^{\frac {c}{\left (b x +a \right )^{3}}} x^{3}d x\]

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.20 \[ \int f^{\frac {c}{(a+b x)^3}} x^3 \, dx=-\frac {6 \, a^{2} b^{2} \left (-\frac {c \log \left (f\right )}{b^{3}}\right )^{\frac {2}{3}} \Gamma \left (\frac {1}{3}, -\frac {c \log \left (f\right )}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}\right ) - 4 \, a c {\rm Ei}\left (\frac {c \log \left (f\right )}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}\right ) \log \left (f\right ) - {\left (4 \, a^{3} b - 3 \, b c \log \left (f\right )\right )} \left (-\frac {c \log \left (f\right )}{b^{3}}\right )^{\frac {1}{3}} \Gamma \left (\frac {2}{3}, -\frac {c \log \left (f\right )}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}\right ) - {\left (b^{4} x^{4} - a^{4} + 3 \, {\left (b c x + a c\right )} \log \left (f\right )\right )} f^{\frac {c}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}}}{4 \, b^{4}} \] Input:

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

Output:

-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)*g 
amma(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
 

Sympy [F]

\[ \int f^{\frac {c}{(a+b x)^3}} x^3 \, dx=\int f^{\frac {c}{\left (a + b x\right )^{3}}} x^{3}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int f^{\frac {c}{(a+b x)^3}} x^3 \, dx=\int { f^{\frac {c}{{\left (b x + a\right )}^{3}}} x^{3} \,d x } \] Input:

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

Output:

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)
 

Giac [F]

\[ \int f^{\frac {c}{(a+b x)^3}} x^3 \, dx=\int { f^{\frac {c}{{\left (b x + a\right )}^{3}}} x^{3} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int f^{\frac {c}{(a+b x)^3}} x^3 \, dx=\int f^{\frac {c}{{\left (a+b\,x\right )}^3}}\,x^3 \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int f^{\frac {c}{(a+b x)^3}} x^3 \, dx=\text {too large to display} \] Input:

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

Output:

(27*f**(c/(a**3 + 3*a**2*b*x + 3*a*b**2*x**2 + b**3*x**3))*log(f)**4*b*c** 
4*x + 9*f**(c/(a**3 + 3*a**2*b*x + 3*a*b**2*x**2 + b**3*x**3))*log(f)**3*a 
**4*c**3 - 324*f**(c/(a**3 + 3*a**2*b*x + 3*a*b**2*x**2 + b**3*x**3))*log( 
f)**3*a**3*b*c**3*x - 513*f**(c/(a**3 + 3*a**2*b*x + 3*a*b**2*x**2 + b**3* 
x**3))*log(f)**3*a**2*b**2*c**3*x**2 - 126*f**(c/(a**3 + 3*a**2*b*x + 3*a* 
b**2*x**2 + b**3*x**3))*log(f)**3*a*b**3*c**3*x**3 - 45*f**(c/(a**3 + 3*a* 
*2*b*x + 3*a*b**2*x**2 + b**3*x**3))*log(f)**3*b**4*c**3*x**4 - 126*f**(c/ 
(a**3 + 3*a**2*b*x + 3*a*b**2*x**2 + b**3*x**3))*log(f)**2*a**7*c**2 - 936 
*f**(c/(a**3 + 3*a**2*b*x + 3*a*b**2*x**2 + b**3*x**3))*log(f)**2*a**6*b*c 
**2*x + 324*f**(c/(a**3 + 3*a**2*b*x + 3*a*b**2*x**2 + b**3*x**3))*log(f)* 
*2*a**5*b**2*c**2*x**2 + 2208*f**(c/(a**3 + 3*a**2*b*x + 3*a*b**2*x**2 + b 
**3*x**3))*log(f)**2*a**4*b**3*c**2*x**3 + 2100*f**(c/(a**3 + 3*a**2*b*x + 
 3*a*b**2*x**2 + b**3*x**3))*log(f)**2*a**3*b**4*c**2*x**4 + 846*f**(c/(a* 
*3 + 3*a**2*b*x + 3*a*b**2*x**2 + b**3*x**3))*log(f)**2*a**2*b**5*c**2*x** 
5 + 156*f**(c/(a**3 + 3*a**2*b*x + 3*a*b**2*x**2 + b**3*x**3))*log(f)**2*a 
*b**6*c**2*x**6 + 30*f**(c/(a**3 + 3*a**2*b*x + 3*a*b**2*x**2 + b**3*x**3) 
)*log(f)**2*b**7*c**2*x**7 - 60*f**(c/(a**3 + 3*a**2*b*x + 3*a*b**2*x**2 + 
 b**3*x**3))*log(f)*a**10*c + 1596*f**(c/(a**3 + 3*a**2*b*x + 3*a*b**2*x** 
2 + b**3*x**3))*log(f)*a**9*b*c*x + 6924*f**(c/(a**3 + 3*a**2*b*x + 3*a*b* 
*2*x**2 + b**3*x**3))*log(f)*a**8*b**2*c*x**2 + 11188*f**(c/(a**3 + 3*a...