Integrand size = 15, antiderivative size = 206 \[ \int f^{\frac {c}{(a+b x)^2}} x^2 \, dx=\frac {a^2 f^{\frac {c}{(a+b x)^2}} (a+b x)}{b^3}-\frac {a f^{\frac {c}{(a+b x)^2}} (a+b x)^2}{b^3}+\frac {f^{\frac {c}{(a+b x)^2}} (a+b x)^3}{3 b^3}-\frac {a^2 \sqrt {c} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {c} \sqrt {\log (f)}}{a+b x}\right ) \sqrt {\log (f)}}{b^3}+\frac {2 c f^{\frac {c}{(a+b x)^2}} (a+b x) \log (f)}{3 b^3}+\frac {a c \operatorname {ExpIntegralEi}\left (\frac {c \log (f)}{(a+b x)^2}\right ) \log (f)}{b^3}-\frac {2 c^{3/2} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {c} \sqrt {\log (f)}}{a+b x}\right ) \log ^{\frac {3}{2}}(f)}{3 b^3} \] Output:
a^2*f^(c/(b*x+a)^2)*(b*x+a)/b^3-a*f^(c/(b*x+a)^2)*(b*x+a)^2/b^3+1/3*f^(c/( b*x+a)^2)*(b*x+a)^3/b^3-a^2*c^(1/2)*Pi^(1/2)*erfi(c^(1/2)*ln(f)^(1/2)/(b*x +a))*ln(f)^(1/2)/b^3+2/3*c*f^(c/(b*x+a)^2)*(b*x+a)*ln(f)/b^3+a*c*Ei(c*ln(f )/(b*x+a)^2)*ln(f)/b^3-2/3*c^(3/2)*Pi^(1/2)*erfi(c^(1/2)*ln(f)^(1/2)/(b*x+ a))*ln(f)^(3/2)/b^3
Time = 0.12 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.64 \[ \int f^{\frac {c}{(a+b x)^2}} x^2 \, dx=\frac {a f^{\frac {c}{(a+b x)^2}} \left (a^2+2 c \log (f)\right )}{3 b^3}+\frac {3 a c \operatorname {ExpIntegralEi}\left (\frac {c \log (f)}{(a+b x)^2}\right ) \log (f)-\sqrt {c} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {c} \sqrt {\log (f)}}{a+b x}\right ) \sqrt {\log (f)} \left (3 a^2+2 c \log (f)\right )+b f^{\frac {c}{(a+b x)^2}} x \left (b^2 x^2+2 c \log (f)\right )}{3 b^3} \] Input:
Integrate[f^(c/(a + b*x)^2)*x^2,x]
Output:
(a*f^(c/(a + b*x)^2)*(a^2 + 2*c*Log[f]))/(3*b^3) + (3*a*c*ExpIntegralEi[(c *Log[f])/(a + b*x)^2]*Log[f] - Sqrt[c]*Sqrt[Pi]*Erfi[(Sqrt[c]*Sqrt[Log[f]] )/(a + b*x)]*Sqrt[Log[f]]*(3*a^2 + 2*c*Log[f]) + b*f^(c/(a + b*x)^2)*x*(b^ 2*x^2 + 2*c*Log[f]))/(3*b^3)
Time = 0.71 (sec) , antiderivative size = 206, 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.
| \(\displaystyle \int x^2 f^{\frac {c}{(a+b x)^2}} \, dx\) |
| \(\Big \downarrow \) 2656 |
| \(\displaystyle \int \left (\frac {a^2 f^{\frac {c}{(a+b x)^2}}}{b^2}+\frac {(a+b x)^2 f^{\frac {c}{(a+b x)^2}}}{b^2}-\frac {2 a (a+b x) f^{\frac {c}{(a+b x)^2}}}{b^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\sqrt {\pi } a^2 \sqrt {c} \sqrt {\log (f)} \text {erfi}\left (\frac {\sqrt {c} \sqrt {\log (f)}}{a+b x}\right )}{b^3}+\frac {a^2 (a+b x) f^{\frac {c}{(a+b x)^2}}}{b^3}-\frac {2 \sqrt {\pi } c^{3/2} \log ^{\frac {3}{2}}(f) \text {erfi}\left (\frac {\sqrt {c} \sqrt {\log (f)}}{a+b x}\right )}{3 b^3}+\frac {a c \log (f) \operatorname {ExpIntegralEi}\left (\frac {c \log (f)}{(a+b x)^2}\right )}{b^3}+\frac {(a+b x)^3 f^{\frac {c}{(a+b x)^2}}}{3 b^3}-\frac {a (a+b x)^2 f^{\frac {c}{(a+b x)^2}}}{b^3}+\frac {2 c \log (f) (a+b x) f^{\frac {c}{(a+b x)^2}}}{3 b^3}\) |
Input:
Int[f^(c/(a + b*x)^2)*x^2,x]
Output:
(a^2*f^(c/(a + b*x)^2)*(a + b*x))/b^3 - (a*f^(c/(a + b*x)^2)*(a + b*x)^2)/ b^3 + (f^(c/(a + b*x)^2)*(a + b*x)^3)/(3*b^3) - (a^2*Sqrt[c]*Sqrt[Pi]*Erfi [(Sqrt[c]*Sqrt[Log[f]])/(a + b*x)]*Sqrt[Log[f]])/b^3 + (2*c*f^(c/(a + b*x) ^2)*(a + b*x)*Log[f])/(3*b^3) + (a*c*ExpIntegralEi[(c*Log[f])/(a + b*x)^2] *Log[f])/b^3 - (2*c^(3/2)*Sqrt[Pi]*Erfi[(Sqrt[c]*Sqrt[Log[f]])/(a + b*x)]* Log[f]^(3/2))/(3*b^3)
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]
Time = 0.16 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.85
| method | result | size |
| risch | \(\frac {f^{\frac {c}{\left (b x +a \right )^{2}}} x^{3}}{3}+\frac {a^{3} f^{\frac {c}{\left (b x +a \right )^{2}}}}{3 b^{3}}+\frac {2 f^{\frac {c}{\left (b x +a \right )^{2}}} \ln \left (f \right ) c x}{3 b^{2}}+\frac {2 f^{\frac {c}{\left (b x +a \right )^{2}}} \ln \left (f \right ) a c}{3 b^{3}}-\frac {2 \ln \left (f \right )^{2} c^{2} \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-c \ln \left (f \right )}}{b x +a}\right )}{3 b^{3} \sqrt {-c \ln \left (f \right )}}-\frac {a^{2} \ln \left (f \right ) c \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-c \ln \left (f \right )}}{b x +a}\right )}{b^{3} \sqrt {-c \ln \left (f \right )}}-\frac {a \ln \left (f \right ) c \,\operatorname {expIntegral}_{1}\left (-\frac {c \ln \left (f \right )}{\left (b x +a \right )^{2}}\right )}{b^{3}}\) | \(175\) |
Input:
int(f^(c/(b*x+a)^2)*x^2,x,method=_RETURNVERBOSE)
Output:
1/3*f^(c/(b*x+a)^2)*x^3+1/3/b^3*a^3*f^(c/(b*x+a)^2)+2/3/b^2*f^(c/(b*x+a)^2 )*ln(f)*c*x+2/3/b^3*f^(c/(b*x+a)^2)*ln(f)*a*c-2/3/b^3*ln(f)^2*c^2*Pi^(1/2) /(-c*ln(f))^(1/2)*erf((-c*ln(f))^(1/2)/(b*x+a))-1/b^3*a^2*ln(f)*c*Pi^(1/2) /(-c*ln(f))^(1/2)*erf((-c*ln(f))^(1/2)/(b*x+a))-1/b^3*a*ln(f)*c*Ei(1,-c*ln (f)/(b*x+a)^2)
Time = 0.08 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.62 \[ \int f^{\frac {c}{(a+b x)^2}} x^2 \, dx=\frac {3 \, a c {\rm Ei}\left (\frac {c \log \left (f\right )}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right ) \log \left (f\right ) + \sqrt {\pi } {\left (3 \, a^{2} b + 2 \, b c \log \left (f\right )\right )} \sqrt {-\frac {c \log \left (f\right )}{b^{2}}} \operatorname {erf}\left (\frac {b \sqrt {-\frac {c \log \left (f\right )}{b^{2}}}}{b x + a}\right ) + {\left (b^{3} x^{3} + a^{3} + 2 \, {\left (b c x + a c\right )} \log \left (f\right )\right )} f^{\frac {c}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{3 \, b^{3}} \] Input:
integrate(f^(c/(b*x+a)^2)*x^2,x, algorithm="fricas")
Output:
1/3*(3*a*c*Ei(c*log(f)/(b^2*x^2 + 2*a*b*x + a^2))*log(f) + sqrt(pi)*(3*a^2 *b + 2*b*c*log(f))*sqrt(-c*log(f)/b^2)*erf(b*sqrt(-c*log(f)/b^2)/(b*x + a) ) + (b^3*x^3 + a^3 + 2*(b*c*x + a*c)*log(f))*f^(c/(b^2*x^2 + 2*a*b*x + a^2 )))/b^3
\[ \int f^{\frac {c}{(a+b x)^2}} x^2 \, dx=\int f^{\frac {c}{\left (a + b x\right )^{2}}} x^{2}\, dx \] Input:
integrate(f**(c/(b*x+a)**2)*x**2,x)
Output:
Integral(f**(c/(a + b*x)**2)*x**2, x)
\[ \int f^{\frac {c}{(a+b x)^2}} x^2 \, dx=\int { f^{\frac {c}{{\left (b x + a\right )}^{2}}} x^{2} \,d x } \] Input:
integrate(f^(c/(b*x+a)^2)*x^2,x, algorithm="maxima")
Output:
1/3*(b^2*x^3 + 2*c*x*log(f))*f^(c/(b^2*x^2 + 2*a*b*x + a^2))/b^2 - integra te(2/3*(3*a*b^2*c*x^2*log(f) + a^3*c*log(f) + (3*a^2*b*c*log(f) - 2*b*c^2* log(f)^2)*x)*f^(c/(b^2*x^2 + 2*a*b*x + a^2))/(b^5*x^3 + 3*a*b^4*x^2 + 3*a^ 2*b^3*x + a^3*b^2), x)
\[ \int f^{\frac {c}{(a+b x)^2}} x^2 \, dx=\int { f^{\frac {c}{{\left (b x + a\right )}^{2}}} x^{2} \,d x } \] Input:
integrate(f^(c/(b*x+a)^2)*x^2,x, algorithm="giac")
Output:
integrate(f^(c/(b*x + a)^2)*x^2, x)
Timed out. \[ \int f^{\frac {c}{(a+b x)^2}} x^2 \, dx=\int f^{\frac {c}{{\left (a+b\,x\right )}^2}}\,x^2 \,d x \] Input:
int(f^(c/(a + b*x)^2)*x^2,x)
Output:
int(f^(c/(a + b*x)^2)*x^2, x)
\[ \int f^{\frac {c}{(a+b x)^2}} x^2 \, dx=\text {too large to display} \] Input:
int(f^(c/(b*x+a)^2)*x^2,x)
Output:
(8*f**(c/(a**2 + 2*a*b*x + b**2*x**2))*log(f)**4*b*c**4*x + 4*f**(c/(a**2 + 2*a*b*x + b**2*x**2))*log(f)**3*a**3*c**3 - 40*f**(c/(a**2 + 2*a*b*x + b **2*x**2))*log(f)**3*a**2*b*c**3*x - 20*f**(c/(a**2 + 2*a*b*x + b**2*x**2) )*log(f)**3*a*b**2*c**3*x**2 - 12*f**(c/(a**2 + 2*a*b*x + b**2*x**2))*log( f)**3*b**3*c**3*x**3 - 28*f**(c/(a**2 + 2*a*b*x + b**2*x**2))*log(f)**2*a* *5*c**2 + 8*f**(c/(a**2 + 2*a*b*x + b**2*x**2))*log(f)**2*a**4*b*c**2*x + 54*f**(c/(a**2 + 2*a*b*x + b**2*x**2))*log(f)**2*a**3*b**2*c**2*x**2 + 50* f**(c/(a**2 + 2*a*b*x + b**2*x**2))*log(f)**2*a**2*b**3*c**2*x**3 + 14*f** (c/(a**2 + 2*a*b*x + b**2*x**2))*log(f)**2*a*b**4*c**2*x**4 + 6*f**(c/(a** 2 + 2*a*b*x + b**2*x**2))*log(f)**2*b**5*c**2*x**5 + 60*f**(c/(a**2 + 2*a* b*x + b**2*x**2))*log(f)*a**7*c + 162*f**(c/(a**2 + 2*a*b*x + b**2*x**2))* log(f)*a**6*b*c*x + 165*f**(c/(a**2 + 2*a*b*x + b**2*x**2))*log(f)*a**5*b* *2*c*x**2 + 87*f**(c/(a**2 + 2*a*b*x + b**2*x**2))*log(f)*a**4*b**3*c*x**3 + 33*f**(c/(a**2 + 2*a*b*x + b**2*x**2))*log(f)*a**3*b**4*c*x**4 + 18*f** (c/(a**2 + 2*a*b*x + b**2*x**2))*log(f)*a**2*b**5*c*x**5 + 12*f**(c/(a**2 + 2*a*b*x + b**2*x**2))*log(f)*a*b**6*c*x**6 + 3*f**(c/(a**2 + 2*a*b*x + b **2*x**2))*log(f)*b**7*c*x**7 - 39*f**(c/(a**2 + 2*a*b*x + b**2*x**2))*a** 9 - 156*f**(c/(a**2 + 2*a*b*x + b**2*x**2))*a**8*b*x - 234*f**(c/(a**2 + 2 *a*b*x + b**2*x**2))*a**7*b**2*x**2 - 156*f**(c/(a**2 + 2*a*b*x + b**2*x** 2))*a**6*b**3*x**3 - 39*f**(c/(a**2 + 2*a*b*x + b**2*x**2))*a**5*b**4*x...