Integrand size = 15, antiderivative size = 291 \[ \int f^{\frac {c}{(a+b x)^2}} x^3 \, dx=-\frac {a^3 f^{\frac {c}{(a+b x)^2}} (a+b x)}{b^4}+\frac {3 a^2 f^{\frac {c}{(a+b x)^2}} (a+b x)^2}{2 b^4}-\frac {a f^{\frac {c}{(a+b x)^2}} (a+b x)^3}{b^4}+\frac {f^{\frac {c}{(a+b x)^2}} (a+b x)^4}{4 b^4}+\frac {a^3 \sqrt {c} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {c} \sqrt {\log (f)}}{a+b x}\right ) \sqrt {\log (f)}}{b^4}-\frac {2 a c f^{\frac {c}{(a+b x)^2}} (a+b x) \log (f)}{b^4}+\frac {c f^{\frac {c}{(a+b x)^2}} (a+b x)^2 \log (f)}{4 b^4}-\frac {3 a^2 c \operatorname {ExpIntegralEi}\left (\frac {c \log (f)}{(a+b x)^2}\right ) \log (f)}{2 b^4}+\frac {2 a c^{3/2} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {c} \sqrt {\log (f)}}{a+b x}\right ) \log ^{\frac {3}{2}}(f)}{b^4}-\frac {c^2 \operatorname {ExpIntegralEi}\left (\frac {c \log (f)}{(a+b x)^2}\right ) \log ^2(f)}{4 b^4} \] Output:
-a^3*f^(c/(b*x+a)^2)*(b*x+a)/b^4+3/2*a^2*f^(c/(b*x+a)^2)*(b*x+a)^2/b^4-a*f ^(c/(b*x+a)^2)*(b*x+a)^3/b^4+1/4*f^(c/(b*x+a)^2)*(b*x+a)^4/b^4+a^3*c^(1/2) *Pi^(1/2)*erfi(c^(1/2)*ln(f)^(1/2)/(b*x+a))*ln(f)^(1/2)/b^4-2*a*c*f^(c/(b* x+a)^2)*(b*x+a)*ln(f)/b^4+1/4*c*f^(c/(b*x+a)^2)*(b*x+a)^2*ln(f)/b^4-3/2*a^ 2*c*Ei(c*ln(f)/(b*x+a)^2)*ln(f)/b^4+2*a*c^(3/2)*Pi^(1/2)*erfi(c^(1/2)*ln(f )^(1/2)/(b*x+a))*ln(f)^(3/2)/b^4-1/4*c^2*Ei(c*ln(f)/(b*x+a)^2)*ln(f)^2/b^4
Time = 0.15 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.51 \[ \int f^{\frac {c}{(a+b x)^2}} x^3 \, dx=-\frac {a^2 f^{\frac {c}{(a+b x)^2}} \left (a^2+7 c \log (f)\right )}{4 b^4}+\frac {-c \operatorname {ExpIntegralEi}\left (\frac {c \log (f)}{(a+b x)^2}\right ) \log (f) \left (6 a^2+c \log (f)\right )+4 a \sqrt {c} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {c} \sqrt {\log (f)}}{a+b x}\right ) \sqrt {\log (f)} \left (a^2+2 c \log (f)\right )+b f^{\frac {c}{(a+b x)^2}} x \left (b^3 x^3-6 a c \log (f)+b c x \log (f)\right )}{4 b^4} \] Input:
Integrate[f^(c/(a + b*x)^2)*x^3,x]
Output:
-1/4*(a^2*f^(c/(a + b*x)^2)*(a^2 + 7*c*Log[f]))/b^4 + (-(c*ExpIntegralEi[( c*Log[f])/(a + b*x)^2]*Log[f]*(6*a^2 + c*Log[f])) + 4*a*Sqrt[c]*Sqrt[Pi]*E rfi[(Sqrt[c]*Sqrt[Log[f]])/(a + b*x)]*Sqrt[Log[f]]*(a^2 + 2*c*Log[f]) + b* f^(c/(a + b*x)^2)*x*(b^3*x^3 - 6*a*c*Log[f] + b*c*x*Log[f]))/(4*b^4)
Time = 0.91 (sec) , antiderivative size = 291, 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.
| \(\displaystyle \int x^3 f^{\frac {c}{(a+b x)^2}} \, dx\) |
| \(\Big \downarrow \) 2656 |
| \(\displaystyle \int \left (-\frac {a^3 f^{\frac {c}{(a+b x)^2}}}{b^3}+\frac {3 a^2 (a+b x) f^{\frac {c}{(a+b x)^2}}}{b^3}+\frac {(a+b x)^3 f^{\frac {c}{(a+b x)^2}}}{b^3}-\frac {3 a (a+b x)^2 f^{\frac {c}{(a+b x)^2}}}{b^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {\pi } a^3 \sqrt {c} \sqrt {\log (f)} \text {erfi}\left (\frac {\sqrt {c} \sqrt {\log (f)}}{a+b x}\right )}{b^4}-\frac {a^3 (a+b x) f^{\frac {c}{(a+b x)^2}}}{b^4}-\frac {3 a^2 c \log (f) \operatorname {ExpIntegralEi}\left (\frac {c \log (f)}{(a+b x)^2}\right )}{2 b^4}+\frac {3 a^2 (a+b x)^2 f^{\frac {c}{(a+b x)^2}}}{2 b^4}+\frac {2 \sqrt {\pi } a c^{3/2} \log ^{\frac {3}{2}}(f) \text {erfi}\left (\frac {\sqrt {c} \sqrt {\log (f)}}{a+b x}\right )}{b^4}-\frac {c^2 \log ^2(f) \operatorname {ExpIntegralEi}\left (\frac {c \log (f)}{(a+b x)^2}\right )}{4 b^4}+\frac {(a+b x)^4 f^{\frac {c}{(a+b x)^2}}}{4 b^4}-\frac {a (a+b x)^3 f^{\frac {c}{(a+b x)^2}}}{b^4}+\frac {c \log (f) (a+b x)^2 f^{\frac {c}{(a+b x)^2}}}{4 b^4}-\frac {2 a c \log (f) (a+b x) f^{\frac {c}{(a+b x)^2}}}{b^4}\) |
Input:
Int[f^(c/(a + b*x)^2)*x^3,x]
Output:
-((a^3*f^(c/(a + b*x)^2)*(a + b*x))/b^4) + (3*a^2*f^(c/(a + b*x)^2)*(a + b *x)^2)/(2*b^4) - (a*f^(c/(a + b*x)^2)*(a + b*x)^3)/b^4 + (f^(c/(a + b*x)^2 )*(a + b*x)^4)/(4*b^4) + (a^3*Sqrt[c]*Sqrt[Pi]*Erfi[(Sqrt[c]*Sqrt[Log[f]]) /(a + b*x)]*Sqrt[Log[f]])/b^4 - (2*a*c*f^(c/(a + b*x)^2)*(a + b*x)*Log[f]) /b^4 + (c*f^(c/(a + b*x)^2)*(a + b*x)^2*Log[f])/(4*b^4) - (3*a^2*c*ExpInte gralEi[(c*Log[f])/(a + b*x)^2]*Log[f])/(2*b^4) + (2*a*c^(3/2)*Sqrt[Pi]*Erf i[(Sqrt[c]*Sqrt[Log[f]])/(a + b*x)]*Log[f]^(3/2))/b^4 - (c^2*ExpIntegralEi [(c*Log[f])/(a + b*x)^2]*Log[f]^2)/(4*b^4)
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 = 0.20 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.78
| method | result | size |
| risch | \(\frac {f^{\frac {c}{\left (b x +a \right )^{2}}} x^{4}}{4}+\frac {f^{\frac {c}{\left (b x +a \right )^{2}}} \ln \left (f \right ) c \,x^{2}}{4 b^{2}}-\frac {3 f^{\frac {c}{\left (b x +a \right )^{2}}} \ln \left (f \right ) a c x}{2 b^{3}}+\frac {2 \ln \left (f \right )^{2} \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-c \ln \left (f \right )}}{b x +a}\right ) a \,c^{2}}{b^{4} \sqrt {-c \ln \left (f \right )}}+\frac {\ln \left (f \right ) \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-c \ln \left (f \right )}}{b x +a}\right ) a^{3} c}{b^{4} \sqrt {-c \ln \left (f \right )}}-\frac {7 f^{\frac {c}{\left (b x +a \right )^{2}}} \ln \left (f \right ) a^{2} c}{4 b^{4}}-\frac {f^{\frac {c}{\left (b x +a \right )^{2}}} a^{4}}{4 b^{4}}+\frac {\ln \left (f \right )^{2} \operatorname {expIntegral}_{1}\left (-\frac {c \ln \left (f \right )}{\left (b x +a \right )^{2}}\right ) c^{2}}{4 b^{4}}+\frac {3 \ln \left (f \right ) \operatorname {expIntegral}_{1}\left (-\frac {c \ln \left (f \right )}{\left (b x +a \right )^{2}}\right ) a^{2} c}{2 b^{4}}\) | \(228\) |
Input:
int(f^(c/(b*x+a)^2)*x^3,x,method=_RETURNVERBOSE)
Output:
1/4*f^(c/(b*x+a)^2)*x^4+1/4/b^2*f^(c/(b*x+a)^2)*ln(f)*c*x^2-3/2/b^3*f^(c/( b*x+a)^2)*ln(f)*a*c*x+2/b^4/(-c*ln(f))^(1/2)*ln(f)^2*Pi^(1/2)*erf((-c*ln(f ))^(1/2)/(b*x+a))*a*c^2+1/b^4/(-c*ln(f))^(1/2)*ln(f)*Pi^(1/2)*erf((-c*ln(f ))^(1/2)/(b*x+a))*a^3*c-7/4/b^4*f^(c/(b*x+a)^2)*ln(f)*a^2*c-1/4/b^4*f^(c/( b*x+a)^2)*a^4+1/4/b^4*ln(f)^2*Ei(1,-c*ln(f)/(b*x+a)^2)*c^2+3/2/b^4*ln(f)*E i(1,-c*ln(f)/(b*x+a)^2)*a^2*c
Time = 0.08 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.54 \[ \int f^{\frac {c}{(a+b x)^2}} x^3 \, dx=-\frac {4 \, \sqrt {\pi } {\left (a^{3} b + 2 \, a b c \log \left (f\right )\right )} \sqrt {-\frac {c \log \left (f\right )}{b^{2}}} \operatorname {erf}\left (\frac {b \sqrt {-\frac {c \log \left (f\right )}{b^{2}}}}{b x + a}\right ) - {\left (b^{4} x^{4} - a^{4} + {\left (b^{2} c x^{2} - 6 \, a b c x - 7 \, a^{2} c\right )} \log \left (f\right )\right )} f^{\frac {c}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + {\left (6 \, a^{2} c \log \left (f\right ) + c^{2} \log \left (f\right )^{2}\right )} {\rm Ei}\left (\frac {c \log \left (f\right )}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right )}{4 \, b^{4}} \] Input:
integrate(f^(c/(b*x+a)^2)*x^3,x, algorithm="fricas")
Output:
-1/4*(4*sqrt(pi)*(a^3*b + 2*a*b*c*log(f))*sqrt(-c*log(f)/b^2)*erf(b*sqrt(- c*log(f)/b^2)/(b*x + a)) - (b^4*x^4 - a^4 + (b^2*c*x^2 - 6*a*b*c*x - 7*a^2 *c)*log(f))*f^(c/(b^2*x^2 + 2*a*b*x + a^2)) + (6*a^2*c*log(f) + c^2*log(f) ^2)*Ei(c*log(f)/(b^2*x^2 + 2*a*b*x + a^2)))/b^4
\[ \int f^{\frac {c}{(a+b x)^2}} x^3 \, dx=\int f^{\frac {c}{\left (a + b x\right )^{2}}} x^{3}\, dx \] Input:
integrate(f**(c/(b*x+a)**2)*x**3,x)
Output:
Integral(f**(c/(a + b*x)**2)*x**3, x)
\[ \int f^{\frac {c}{(a+b x)^2}} x^3 \, dx=\int { f^{\frac {c}{{\left (b x + a\right )}^{2}}} x^{3} \,d x } \] Input:
integrate(f^(c/(b*x+a)^2)*x^3,x, algorithm="maxima")
Output:
1/4*(b^3*x^4 + b*c*x^2*log(f) - 6*a*c*x*log(f))*f^(c/(b^2*x^2 + 2*a*b*x + a^2))/b^3 + integrate(1/2*(3*a^4*c*log(f) + (6*a^2*b^2*c*log(f) + b^2*c^2* log(f)^2)*x^2 + 2*(4*a^3*b*c*log(f) - 3*a*b*c^2*log(f)^2)*x)*f^(c/(b^2*x^2 + 2*a*b*x + a^2))/(b^6*x^3 + 3*a*b^5*x^2 + 3*a^2*b^4*x + a^3*b^3), x)
\[ \int f^{\frac {c}{(a+b x)^2}} x^3 \, dx=\int { f^{\frac {c}{{\left (b x + a\right )}^{2}}} x^{3} \,d x } \] Input:
integrate(f^(c/(b*x+a)^2)*x^3,x, algorithm="giac")
Output:
integrate(f^(c/(b*x + a)^2)*x^3, x)
Timed out. \[ \int f^{\frac {c}{(a+b x)^2}} x^3 \, dx=\int f^{\frac {c}{{\left (a+b\,x\right )}^2}}\,x^3 \,d x \] Input:
int(f^(c/(a + b*x)^2)*x^3,x)
Output:
int(f^(c/(a + b*x)^2)*x^3, x)
\[ \int f^{\frac {c}{(a+b x)^2}} x^3 \, dx=\text {too large to display} \] Input:
int(f^(c/(b*x+a)^2)*x^3,x)
Output:
( - 16*f**(c/(a**2 + 2*a*b*x + b**2*x**2))*log(f)**4*a*b*c**4*x - 8*f**(c/ (a**2 + 2*a*b*x + b**2*x**2))*log(f)**3*a**4*c**3 + 80*f**(c/(a**2 + 2*a*b *x + b**2*x**2))*log(f)**3*a**3*b*c**3*x + 40*f**(c/(a**2 + 2*a*b*x + b**2 *x**2))*log(f)**3*a**2*b**2*c**3*x**2 + 24*f**(c/(a**2 + 2*a*b*x + b**2*x* *2))*log(f)**3*a*b**3*c**3*x**3 + 56*f**(c/(a**2 + 2*a*b*x + b**2*x**2))*l og(f)**2*a**6*c**2 - 16*f**(c/(a**2 + 2*a*b*x + b**2*x**2))*log(f)**2*a**5 *b*c**2*x - 108*f**(c/(a**2 + 2*a*b*x + b**2*x**2))*log(f)**2*a**4*b**2*c* *2*x**2 - 100*f**(c/(a**2 + 2*a*b*x + b**2*x**2))*log(f)**2*a**3*b**3*c**2 *x**3 - 25*f**(c/(a**2 + 2*a*b*x + b**2*x**2))*log(f)**2*a**2*b**4*c**2*x* *4 - 6*f**(c/(a**2 + 2*a*b*x + b**2*x**2))*log(f)**2*a*b**5*c**2*x**5 + 3* f**(c/(a**2 + 2*a*b*x + b**2*x**2))*log(f)**2*b**6*c**2*x**6 - 120*f**(c/( a**2 + 2*a*b*x + b**2*x**2))*log(f)*a**8*c - 324*f**(c/(a**2 + 2*a*b*x + b **2*x**2))*log(f)*a**7*b*c*x - 330*f**(c/(a**2 + 2*a*b*x + b**2*x**2))*log (f)*a**6*b**2*c*x**2 - 168*f**(c/(a**2 + 2*a*b*x + b**2*x**2))*log(f)*a**5 *b**3*c*x**3 - 39*f**(c/(a**2 + 2*a*b*x + b**2*x**2))*log(f)*a**4*b**4*c*x **4 + 12*f**(c/(a**2 + 2*a*b*x + b**2*x**2))*log(f)*a**3*b**5*c*x**5 + 18* f**(c/(a**2 + 2*a*b*x + b**2*x**2))*log(f)*a**2*b**6*c*x**6 + 12*f**(c/(a* *2 + 2*a*b*x + b**2*x**2))*log(f)*a*b**7*c*x**7 + 3*f**(c/(a**2 + 2*a*b*x + b**2*x**2))*log(f)*b**8*c*x**8 + 78*f**(c/(a**2 + 2*a*b*x + b**2*x**2))* a**10 + 312*f**(c/(a**2 + 2*a*b*x + b**2*x**2))*a**9*b*x + 468*f**(c/(a...