\(\int f^{\frac {c}{(a+b x)^3}} x^2 \, dx\) [167]

Optimal result

Integrand size = 15, antiderivative size = 142 \[ \int f^{\frac {c}{(a+b x)^3}} x^2 \, dx=\frac {f^{\frac {c}{(a+b x)^3}} (a+b x)^3}{3 b^3}-\frac {c \operatorname {ExpIntegralEi}\left (\frac {c \log (f)}{(a+b x)^3}\right ) \log (f)}{3 b^3}+\frac {a^2 (a+b x) \Gamma \left (-\frac {1}{3},-\frac {c \log (f)}{(a+b x)^3}\right ) \sqrt [3]{-\frac {c \log (f)}{(a+b x)^3}}}{3 b^3}-\frac {2 a (a+b x)^2 \Gamma \left (-\frac {2}{3},-\frac {c \log (f)}{(a+b x)^3}\right ) \left (-\frac {c \log (f)}{(a+b x)^3}\right )^{2/3}}{3 b^3} \] Output:

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

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.89 \[ \int f^{\frac {c}{(a+b x)^3}} x^2 \, dx=\frac {f^{\frac {c}{(a+b x)^3}} (a+b x)^3-c \operatorname {ExpIntegralEi}\left (\frac {c \log (f)}{(a+b x)^3}\right ) \log (f)+a^2 (a+b x) \Gamma \left (-\frac {1}{3},-\frac {c \log (f)}{(a+b x)^3}\right ) \sqrt [3]{-\frac {c \log (f)}{(a+b x)^3}}-2 a (a+b x)^2 \Gamma \left (-\frac {2}{3},-\frac {c \log (f)}{(a+b x)^3}\right ) \left (-\frac {c \log (f)}{(a+b x)^3}\right )^{2/3}}{3 b^3} \] Input:

Integrate[f^(c/(a + b*x)^3)*x^2,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 142, 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.

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

\(\Big \downarrow \) 2656

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

\(\Big \downarrow \) 2009

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

Input:

Int[f^(c/(a + b*x)^3)*x^2,x]
 

Output:

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

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]

\[\int f^{\frac {c}{\left (b x +a \right )^{3}}} x^{2}d x\]

Input:

int(f^(c/(b*x+a)^3)*x^2,x)
 

Output:

int(f^(c/(b*x+a)^3)*x^2,x)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.37 \[ \int f^{\frac {c}{(a+b x)^3}} x^2 \, dx=\frac {3 \, a b^{2} \left (-\frac {c \log \left (f\right )}{b^{3}}\right )^{\frac {2}{3}} \Gamma \left (\frac {1}{3}, -\frac {c \log \left (f\right )}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}\right ) - 3 \, a^{2} b \left (-\frac {c \log \left (f\right )}{b^{3}}\right )^{\frac {1}{3}} \Gamma \left (\frac {2}{3}, -\frac {c \log \left (f\right )}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}\right ) - c {\rm Ei}\left (\frac {c \log \left (f\right )}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}\right ) \log \left (f\right ) + {\left (b^{3} x^{3} + a^{3}\right )} f^{\frac {c}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}}}{3 \, b^{3}} \] Input:

integrate(f^(c/(b*x+a)^3)*x^2,x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int f^{\frac {c}{(a+b x)^3}} x^2 \, dx=\int f^{\frac {c}{\left (a + b x\right )^{3}}} x^{2}\, dx \] Input:

integrate(f**(c/(b*x+a)**3)*x**2,x)
 

Output:

Integral(f**(c/(a + b*x)**3)*x**2, x)
 

Maxima [F]

\[ \int f^{\frac {c}{(a+b x)^3}} x^2 \, dx=\int { f^{\frac {c}{{\left (b x + a\right )}^{3}}} x^{2} \,d x } \] Input:

integrate(f^(c/(b*x+a)^3)*x^2,x, algorithm="maxima")
 

Output:

1/3*f^(c/(b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x + a^3))*x^3 + b*c*integrate(f^ 
(c/(b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x + a^3))*x^3/(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)
 

Giac [F]

\[ \int f^{\frac {c}{(a+b x)^3}} x^2 \, dx=\int { f^{\frac {c}{{\left (b x + a\right )}^{3}}} x^{2} \,d x } \] Input:

integrate(f^(c/(b*x+a)^3)*x^2,x, algorithm="giac")
 

Output:

integrate(f^(c/(b*x + a)^3)*x^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int f^{\frac {c}{(a+b x)^3}} x^2 \, dx=\int f^{\frac {c}{{\left (a+b\,x\right )}^3}}\,x^2 \,d x \] Input:

int(f^(c/(a + b*x)^3)*x^2,x)
 

Output:

int(f^(c/(a + b*x)^3)*x^2, x)
 

Reduce [F]

\[ \int f^{\frac {c}{(a+b x)^3}} x^2 \, dx=\text {too large to display} \] Input:

int(f^(c/(b*x+a)^3)*x^2,x)
 

Output:

(54*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*c**3*x + 135*f**(c/(a**3 + 3*a**2*b*x + 3*a*b**2*x**2 + b**3*x**3))*log( 
f)**3*a*b**2*c**3*x**2 + 18*f**(c/(a**3 + 3*a**2*b*x + 3*a*b**2*x**2 + b** 
3*x**3))*log(f)**2*a**6*c**2 + 432*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*c**2*x + 99*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**2*c**2*x**2 - 522*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**3*c**2*x**3 - 
 540*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**4*c**2*x**4 - 180*f**(c/(a**3 + 3*a**2*b*x + 3*a*b**2*x**2 + b**3*x**3 
))*log(f)**2*a*b**5*c**2*x**5 + 108*f**(c/(a**3 + 3*a**2*b*x + 3*a*b**2*x* 
*2 + b**3*x**3))*log(f)*a**9*c - 330*f**(c/(a**3 + 3*a**2*b*x + 3*a*b**2*x 
**2 + b**3*x**3))*log(f)*a**8*b*c*x - 2292*f**(c/(a**3 + 3*a**2*b*x + 3*a* 
b**2*x**2 + b**3*x**3))*log(f)*a**7*b**2*c*x**2 - 4014*f**(c/(a**3 + 3*a** 
2*b*x + 3*a*b**2*x**2 + b**3*x**3))*log(f)*a**6*b**3*c*x**3 - 3150*f**(c/( 
a**3 + 3*a**2*b*x + 3*a*b**2*x**2 + b**3*x**3))*log(f)*a**5*b**4*c*x**4 - 
1008*f**(c/(a**3 + 3*a**2*b*x + 3*a*b**2*x**2 + b**3*x**3))*log(f)*a**4*b* 
*5*c*x**5 + 182*f**(c/(a**3 + 3*a**2*b*x + 3*a*b**2*x**2 + b**3*x**3))*log 
(f)*a**3*b**6*c*x**6 + 300*f**(c/(a**3 + 3*a**2*b*x + 3*a*b**2*x**2 + b**3 
*x**3))*log(f)*a**2*b**7*c*x**7 + 120*f**(c/(a**3 + 3*a**2*b*x + 3*a*b**2* 
x**2 + b**3*x**3))*log(f)*a*b**8*c*x**8 + 20*f**(c/(a**3 + 3*a**2*b*x +...