Integrand size = 13, antiderivative size = 120 \[ \int f^{\frac {c}{a+b x}} x \, dx=-\frac {a f^{\frac {c}{a+b x}} (a+b x)}{b^2}+\frac {f^{\frac {c}{a+b x}} (a+b x)^2}{2 b^2}+\frac {c f^{\frac {c}{a+b x}} (a+b x) \log (f)}{2 b^2}+\frac {a c \operatorname {ExpIntegralEi}\left (\frac {c \log (f)}{a+b x}\right ) \log (f)}{b^2}-\frac {c^2 \operatorname {ExpIntegralEi}\left (\frac {c \log (f)}{a+b x}\right ) \log ^2(f)}{2 b^2} \] Output:
-a*f^(c/(b*x+a))*(b*x+a)/b^2+1/2*f^(c/(b*x+a))*(b*x+a)^2/b^2+1/2*c*f^(c/(b *x+a))*(b*x+a)*ln(f)/b^2+a*c*Ei(c*ln(f)/(b*x+a))*ln(f)/b^2-1/2*c^2*Ei(c*ln (f)/(b*x+a))*ln(f)^2/b^2
Time = 0.07 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.68 \[ \int f^{\frac {c}{a+b x}} x \, dx=-\frac {a f^{\frac {c}{a+b x}} (a-c \log (f))}{2 b^2}+\frac {c \operatorname {ExpIntegralEi}\left (\frac {c \log (f)}{a+b x}\right ) \log (f) (2 a-c \log (f))+b f^{\frac {c}{a+b x}} x (b x+c \log (f))}{2 b^2} \] Input:
Integrate[f^(c/(a + b*x))*x,x]
Output:
-1/2*(a*f^(c/(a + b*x))*(a - c*Log[f]))/b^2 + (c*ExpIntegralEi[(c*Log[f])/ (a + b*x)]*Log[f]*(2*a - c*Log[f]) + b*f^(c/(a + b*x))*x*(b*x + c*Log[f])) /(2*b^2)
Time = 0.49 (sec) , antiderivative size = 120, 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.
| \(\displaystyle \int x f^{\frac {c}{a+b x}} \, dx\) |
| \(\Big \downarrow \) 2656 |
| \(\displaystyle \int \left (\frac {(a+b x) f^{\frac {c}{a+b x}}}{b}-\frac {a f^{\frac {c}{a+b x}}}{b}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {c^2 \log ^2(f) \operatorname {ExpIntegralEi}\left (\frac {c \log (f)}{a+b x}\right )}{2 b^2}+\frac {a c \log (f) \operatorname {ExpIntegralEi}\left (\frac {c \log (f)}{a+b x}\right )}{b^2}+\frac {(a+b x)^2 f^{\frac {c}{a+b x}}}{2 b^2}-\frac {a (a+b x) f^{\frac {c}{a+b x}}}{b^2}+\frac {c \log (f) (a+b x) f^{\frac {c}{a+b x}}}{2 b^2}\) |
Input:
Int[f^(c/(a + b*x))*x,x]
Output:
-((a*f^(c/(a + b*x))*(a + b*x))/b^2) + (f^(c/(a + b*x))*(a + b*x)^2)/(2*b^ 2) + (c*f^(c/(a + b*x))*(a + b*x)*Log[f])/(2*b^2) + (a*c*ExpIntegralEi[(c* Log[f])/(a + b*x)]*Log[f])/b^2 - (c^2*ExpIntegralEi[(c*Log[f])/(a + b*x)]* Log[f]^2)/(2*b^2)
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 = 0.11 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.05
| method | result | size |
| risch | \(\frac {f^{\frac {c}{b x +a}} x^{2}}{2}-\frac {f^{\frac {c}{b x +a}} a^{2}}{2 b^{2}}+\frac {\ln \left (f \right ) f^{\frac {c}{b x +a}} c x}{2 b}+\frac {\ln \left (f \right ) f^{\frac {c}{b x +a}} a c}{2 b^{2}}+\frac {c^{2} \ln \left (f \right )^{2} \operatorname {expIntegral}_{1}\left (-\frac {c \ln \left (f \right )}{b x +a}\right )}{2 b^{2}}-\frac {c \ln \left (f \right ) a \,\operatorname {expIntegral}_{1}\left (-\frac {c \ln \left (f \right )}{b x +a}\right )}{b^{2}}\) | \(126\) |
Input:
int(f^(c/(b*x+a))*x,x,method=_RETURNVERBOSE)
Output:
1/2*f^(c/(b*x+a))*x^2-1/2/b^2*f^(c/(b*x+a))*a^2+1/2/b*ln(f)*f^(c/(b*x+a))* c*x+1/2/b^2*ln(f)*f^(c/(b*x+a))*a*c+1/2*c^2*ln(f)^2/b^2*Ei(1,-c*ln(f)/(b*x +a))-c*ln(f)/b^2*a*Ei(1,-c*ln(f)/(b*x+a))
Time = 0.08 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.59 \[ \int f^{\frac {c}{a+b x}} x \, dx=\frac {{\left (b^{2} x^{2} - a^{2} + {\left (b c x + a c\right )} \log \left (f\right )\right )} f^{\frac {c}{b x + a}} - {\left (c^{2} \log \left (f\right )^{2} - 2 \, a c \log \left (f\right )\right )} {\rm Ei}\left (\frac {c \log \left (f\right )}{b x + a}\right )}{2 \, b^{2}} \] Input:
integrate(f^(c/(b*x+a))*x,x, algorithm="fricas")
Output:
1/2*((b^2*x^2 - a^2 + (b*c*x + a*c)*log(f))*f^(c/(b*x + a)) - (c^2*log(f)^ 2 - 2*a*c*log(f))*Ei(c*log(f)/(b*x + a)))/b^2
\[ \int f^{\frac {c}{a+b x}} x \, dx=\int f^{\frac {c}{a + b x}} x\, dx \] Input:
integrate(f**(c/(b*x+a))*x,x)
Output:
Integral(f**(c/(a + b*x))*x, x)
\[ \int f^{\frac {c}{a+b x}} x \, dx=\int { f^{\frac {c}{b x + a}} x \,d x } \] Input:
integrate(f^(c/(b*x+a))*x,x, algorithm="maxima")
Output:
1/2*(b*x^2 + c*x*log(f))*f^(c/(b*x + a))/b - integrate(1/2*(a^2*c*log(f) - (b*c^2*log(f)^2 - 2*a*b*c*log(f))*x)*f^(c/(b*x + a))/(b^3*x^2 + 2*a*b^2*x + a^2*b), x)
\[ \int f^{\frac {c}{a+b x}} x \, dx=\int { f^{\frac {c}{b x + a}} x \,d x } \] Input:
integrate(f^(c/(b*x+a))*x,x, algorithm="giac")
Output:
integrate(f^(c/(b*x + a))*x, x)
Time = 0.18 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.13 \[ \int f^{\frac {c}{a+b x}} x \, dx=\frac {\frac {b\,f^{\frac {c}{a+b\,x}}\,x^3}{2}+f^{\frac {c}{a+b\,x}}\,x^2\,\left (\frac {a}{2}+\frac {c\,\ln \left (f\right )}{2}\right )-\frac {a^2\,f^{\frac {c}{a+b\,x}}\,\left (a-c\,\ln \left (f\right )\right )}{2\,b^2}-\frac {f^{\frac {c}{a+b\,x}}\,x\,\left (a^2-2\,a\,c\,\ln \left (f\right )\right )}{2\,b}}{a+b\,x}-\frac {\mathrm {ei}\left (\frac {c\,\ln \left (f\right )}{a+b\,x}\right )\,\left (c^2\,{\ln \left (f\right )}^2-2\,a\,c\,\ln \left (f\right )\right )}{2\,b^2} \] Input:
int(f^(c/(a + b*x))*x,x)
Output:
((b*f^(c/(a + b*x))*x^3)/2 + f^(c/(a + b*x))*x^2*(a/2 + (c*log(f))/2) - (a ^2*f^(c/(a + b*x))*(a - c*log(f)))/(2*b^2) - (f^(c/(a + b*x))*x*(a^2 - 2*a *c*log(f)))/(2*b))/(a + b*x) - (ei((c*log(f))/(a + b*x))*(c^2*log(f)^2 - 2 *a*c*log(f)))/(2*b^2)
\[ \int f^{\frac {c}{a+b x}} x \, dx=\frac {f^{\frac {c}{b x +a}} \mathrm {log}\left (f \right )^{2} b^{2} c^{2} x^{2}+2 f^{\frac {c}{b x +a}} \mathrm {log}\left (f \right ) a^{2} b c x +f^{\frac {c}{b x +a}} \mathrm {log}\left (f \right ) a \,b^{2} c \,x^{2}+f^{\frac {c}{b x +a}} \mathrm {log}\left (f \right ) b^{3} c \,x^{3}+2 f^{\frac {c}{b x +a}} a^{4}+2 f^{\frac {c}{b x +a}} a^{3} b x +\left (\int \frac {f^{\frac {c}{b x +a}} x^{2}}{b^{3} x^{3}+3 a \,b^{2} x^{2}+3 a^{2} b x +a^{3}}d x \right ) \mathrm {log}\left (f \right )^{3} a \,b^{3} c^{3}+\left (\int \frac {f^{\frac {c}{b x +a}} x^{2}}{b^{3} x^{3}+3 a \,b^{2} x^{2}+3 a^{2} b x +a^{3}}d x \right ) \mathrm {log}\left (f \right )^{3} b^{4} c^{3} x -2 \left (\int \frac {f^{\frac {c}{b x +a}} x^{2}}{b^{3} x^{3}+3 a \,b^{2} x^{2}+3 a^{2} b x +a^{3}}d x \right ) \mathrm {log}\left (f \right )^{2} a^{2} b^{3} c^{2}-2 \left (\int \frac {f^{\frac {c}{b x +a}} x^{2}}{b^{3} x^{3}+3 a \,b^{2} x^{2}+3 a^{2} b x +a^{3}}d x \right ) \mathrm {log}\left (f \right )^{2} a \,b^{4} c^{2} x}{2 \,\mathrm {log}\left (f \right ) b^{2} c \left (b x +a \right )} \] Input:
int(f^(c/(b*x+a))*x,x)
Output:
(f**(c/(a + b*x))*log(f)**2*b**2*c**2*x**2 + 2*f**(c/(a + b*x))*log(f)*a** 2*b*c*x + f**(c/(a + b*x))*log(f)*a*b**2*c*x**2 + f**(c/(a + b*x))*log(f)* b**3*c*x**3 + 2*f**(c/(a + b*x))*a**4 + 2*f**(c/(a + b*x))*a**3*b*x + int( (f**(c/(a + b*x))*x**2)/(a**3 + 3*a**2*b*x + 3*a*b**2*x**2 + b**3*x**3),x) *log(f)**3*a*b**3*c**3 + int((f**(c/(a + b*x))*x**2)/(a**3 + 3*a**2*b*x + 3*a*b**2*x**2 + b**3*x**3),x)*log(f)**3*b**4*c**3*x - 2*int((f**(c/(a + b* x))*x**2)/(a**3 + 3*a**2*b*x + 3*a*b**2*x**2 + b**3*x**3),x)*log(f)**2*a** 2*b**3*c**2 - 2*int((f**(c/(a + b*x))*x**2)/(a**3 + 3*a**2*b*x + 3*a*b**2* x**2 + b**3*x**3),x)*log(f)**2*a*b**4*c**2*x)/(2*log(f)*b**2*c*(a + b*x))