\(\int f^{\frac {c}{(a+b x)^3}} x \, dx\) [168]

Optimal result

Integrand size = 13, antiderivative size = 92 \[ \int f^{\frac {c}{(a+b x)^3}} x \, dx=-\frac {a (a+b x) \Gamma \left (-\frac {1}{3},-\frac {c \log (f)}{(a+b x)^3}\right ) \sqrt [3]{-\frac {c \log (f)}{(a+b x)^3}}}{3 b^2}+\frac {(a+b x)^2 \Gamma \left (-\frac {2}{3},-\frac {c \log (f)}{(a+b x)^3}\right ) \left (-\frac {c \log (f)}{(a+b x)^3}\right )^{2/3}}{3 b^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.93 \[ \int f^{\frac {c}{(a+b x)^3}} x \, dx=\frac {(a+b x) \left (-a \Gamma \left (-\frac {1}{3},-\frac {c \log (f)}{(a+b x)^3}\right ) \sqrt [3]{-\frac {c \log (f)}{(a+b x)^3}}+(a+b x) \Gamma \left (-\frac {2}{3},-\frac {c \log (f)}{(a+b x)^3}\right ) \left (-\frac {c \log (f)}{(a+b x)^3}\right )^{2/3}\right )}{3 b^2} \] Input:

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

Output:

((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*L 
og[f])/(a + b*x)^3))^(2/3)))/(3*b^2)
 

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, 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.

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

\(\Big \downarrow \) 2656

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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]

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.67 \[ \int f^{\frac {c}{(a+b x)^3}} x \, dx=-\frac {b^{2} \left (-\frac {c \log \left (f\right )}{b^{3}}\right )^{\frac {2}{3}} \Gamma \left (\frac {1}{3}, -\frac {c \log \left (f\right )}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}\right ) - 2 \, a b \left (-\frac {c \log \left (f\right )}{b^{3}}\right )^{\frac {1}{3}} \Gamma \left (\frac {2}{3}, -\frac {c \log \left (f\right )}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}\right ) - {\left (b^{2} x^{2} - a^{2}\right )} f^{\frac {c}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}}}{2 \, b^{2}} \] Input:

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

Output:

-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*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^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
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(9*f**(c/(a**3 + 3*a**2*b*x + 3*a*b**2*x**2 + b**3*x**3))*log(f)**2*a*c**2 
 + 45*f**(c/(a**3 + 3*a**2*b*x + 3*a*b**2*x**2 + b**3*x**3))*log(f)**2*b*c 
**2*x - 60*f**(c/(a**3 + 3*a**2*b*x + 3*a*b**2*x**2 + b**3*x**3))*log(f)*a 
**3*b*c*x - 180*f**(c/(a**3 + 3*a**2*b*x + 3*a*b**2*x**2 + b**3*x**3))*log 
(f)*a**2*b**2*c*x**2 - 180*f**(c/(a**3 + 3*a**2*b*x + 3*a*b**2*x**2 + b**3 
*x**3))*log(f)*a*b**3*c*x**3 - 60*f**(c/(a**3 + 3*a**2*b*x + 3*a*b**2*x**2 
 + b**3*x**3))*log(f)*b**4*c*x**4 - 20*f**(c/(a**3 + 3*a**2*b*x + 3*a*b**2 
*x**2 + b**3*x**3))*a**7 - 100*f**(c/(a**3 + 3*a**2*b*x + 3*a*b**2*x**2 + 
b**3*x**3))*a**6*b*x - 180*f**(c/(a**3 + 3*a**2*b*x + 3*a*b**2*x**2 + b**3 
*x**3))*a**5*b**2*x**2 - 100*f**(c/(a**3 + 3*a**2*b*x + 3*a*b**2*x**2 + b* 
*3*x**3))*a**4*b**3*x**3 + 100*f**(c/(a**3 + 3*a**2*b*x + 3*a*b**2*x**2 + 
b**3*x**3))*a**3*b**4*x**4 + 180*f**(c/(a**3 + 3*a**2*b*x + 3*a*b**2*x**2 
+ b**3*x**3))*a**2*b**5*x**5 + 100*f**(c/(a**3 + 3*a**2*b*x + 3*a*b**2*x** 
2 + b**3*x**3))*a*b**6*x**6 + 20*f**(c/(a**3 + 3*a**2*b*x + 3*a*b**2*x**2 
+ b**3*x**3))*b**7*x**7 + 27*int(f**(c/(a**3 + 3*a**2*b*x + 3*a*b**2*x**2 
+ b**3*x**3))/(a**9 + 9*a**8*b*x + 36*a**7*b**2*x**2 + 84*a**6*b**3*x**3 + 
 126*a**5*b**4*x**4 + 126*a**4*b**5*x**5 + 84*a**3*b**6*x**6 + 36*a**2*b** 
7*x**7 + 9*a*b**8*x**8 + b**9*x**9),x)*log(f)**3*a**6*b*c**3 + 135*int(f** 
(c/(a**3 + 3*a**2*b*x + 3*a*b**2*x**2 + b**3*x**3))/(a**9 + 9*a**8*b*x + 3 
6*a**7*b**2*x**2 + 84*a**6*b**3*x**3 + 126*a**5*b**4*x**4 + 126*a**4*b*...