\(\int \frac {f^{\frac {c}{a+b x}}}{x^3} \, dx\) [156]

Optimal result

Integrand size = 15, antiderivative size = 166 \[ \int \frac {f^{\frac {c}{a+b x}}}{x^3} \, dx=\frac {b^2 f^{\frac {c}{a+b x}}}{2 a^2}-\frac {f^{\frac {c}{a+b x}}}{2 x^2}+\frac {b^2 c f^{\frac {c}{a+b x}} \log (f)}{2 a^3}+\frac {b c f^{\frac {c}{a+b x}} \log (f)}{2 a^2 x}+\frac {b^2 c f^{\frac {c}{a}} \operatorname {ExpIntegralEi}\left (-\frac {b c x \log (f)}{a (a+b x)}\right ) \log (f)}{a^3}+\frac {b^2 c^2 f^{\frac {c}{a}} \operatorname {ExpIntegralEi}\left (-\frac {b c x \log (f)}{a (a+b x)}\right ) \log ^2(f)}{2 a^4} \] Output:

1/2*b^2*f^(c/(b*x+a))/a^2-1/2*f^(c/(b*x+a))/x^2+1/2*b^2*c*f^(c/(b*x+a))*ln 
(f)/a^3+1/2*b*c*f^(c/(b*x+a))*ln(f)/a^2/x+b^2*c*f^(c/a)*Ei(-b*c*x*ln(f)/a/ 
(b*x+a))*ln(f)/a^3+1/2*b^2*c^2*f^(c/a)*Ei(-b*c*x*ln(f)/a/(b*x+a))*ln(f)^2/ 
a^4
 

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.69 \[ \int \frac {f^{\frac {c}{a+b x}}}{x^3} \, dx=\frac {b^2 f^{\frac {c}{a+b x}} (2 a+c \log (f))}{2 a^3}+\frac {b^2 c f^{\frac {c}{a}} \operatorname {ExpIntegralEi}\left (-\frac {b c x \log (f)}{a^2+a b x}\right ) \log (f) (2 a+c \log (f))-\frac {a^2 f^{\frac {c}{a+b x}} \left (a^2+b^2 x^2-b c x \log (f)\right )}{x^2}}{2 a^4} \] Input:

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

Output:

(b^2*f^(c/(a + b*x))*(2*a + c*Log[f]))/(2*a^3) + (b^2*c*f^(c/a)*ExpIntegra 
lEi[-((b*c*x*Log[f])/(a^2 + a*b*x))]*Log[f]*(2*a + c*Log[f]) - (a^2*f^(c/( 
a + b*x))*(a^2 + b^2*x^2 - b*c*x*Log[f]))/x^2)/(2*a^4)
 

Rubi [A] (verified)

Time = 1.42 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.92, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2653, 7293, 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.

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

\(\Big \downarrow \) 2653

\(\displaystyle -\frac {1}{2} b c \log (f) \int \frac {f^{\frac {c}{a+b x}}}{x^2 (a+b x)^2}dx-\frac {f^{\frac {c}{a+b x}}}{2 x^2}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {1}{2} b c \log (f) \int \left (-\frac {2 b f^{\frac {c}{a+b x}}}{a^3 x}+\frac {2 b^2 f^{\frac {c}{a+b x}}}{a^3 (a+b x)}+\frac {f^{\frac {c}{a+b x}}}{a^2 x^2}+\frac {b^2 f^{\frac {c}{a+b x}}}{a^2 (a+b x)^2}\right )dx-\frac {f^{\frac {c}{a+b x}}}{2 x^2}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {1}{2} b c \log (f) \left (-\frac {b c \log (f) f^{\frac {c}{a}} \operatorname {ExpIntegralEi}\left (-\frac {b c x \log (f)}{a (a+b x)}\right )}{a^4}-\frac {2 b f^{\frac {c}{a}} \operatorname {ExpIntegralEi}\left (-\frac {b c x \log (f)}{a (a+b x)}\right )}{a^3}-\frac {b f^{\frac {c}{a+b x}}}{a^3}-\frac {f^{\frac {c}{a+b x}}}{a^2 x}-\frac {b f^{\frac {c}{a+b x}}}{a^2 c \log (f)}\right )-\frac {f^{\frac {c}{a+b x}}}{2 x^2}\)

Input:

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

Output:

-1/2*f^(c/(a + b*x))/x^2 - (b*c*Log[f]*(-((b*f^(c/(a + b*x)))/a^3) - f^(c/ 
(a + b*x))/(a^2*x) - (2*b*f^(c/a)*ExpIntegralEi[-((b*c*x*Log[f])/(a*(a + b 
*x)))])/a^3 - (b*f^(c/(a + b*x)))/(a^2*c*Log[f]) - (b*c*f^(c/a)*ExpIntegra 
lEi[-((b*c*x*Log[f])/(a*(a + b*x)))]*Log[f])/a^4))/2
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2653

Int[(F_)^((a_.) + (b_.)/((c_.) + (d_.)*(x_)))*((e_.) + (f_.)*(x_))^(m_), x_ 
Symbol] :> Simp[(e + f*x)^(m + 1)*(F^(a + b/(c + d*x))/(f*(m + 1))), x] + S 
imp[b*d*(Log[F]/(f*(m + 1)))   Int[(e + f*x)^(m + 1)*(F^(a + b/(c + d*x))/( 
c + d*x)^2), x], x] /; FreeQ[{F, a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] 
 && ILtQ[m, -1]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

Maple [A] (verified)

Time = 0.15 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.05

method	result	size
risch	\(\frac {-f^{\frac {c}{a}} \operatorname {expIntegral}_{1}\left (-\frac {c \ln \left (f \right )}{b x +a}+\frac {c \ln \left (f \right )}{a}\right ) c^{2} b^{2} \ln \left (f \right )^{2} x^{2}-2 \ln \left (f \right ) f^{\frac {c}{a}} \operatorname {expIntegral}_{1}\left (-\frac {c \ln \left (f \right )}{b x +a}+\frac {c \ln \left (f \right )}{a}\right ) a \,b^{2} c \,x^{2}+f^{\frac {c}{b x +a}} \ln \left (f \right ) a \,b^{2} c \,x^{2}+f^{\frac {c}{b x +a}} \ln \left (f \right ) a^{2} b c x +f^{\frac {c}{b x +a}} a^{2} b^{2} x^{2}-f^{\frac {c}{b x +a}} a^{4}}{2 a^{4} x^{2}}\)	\(174\)

Input:

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

Output:

1/2*(-f^(1/a*c)*Ei(1,-c*ln(f)/(b*x+a)+c*ln(f)/a)*c^2*b^2*ln(f)^2*x^2-2*ln( 
f)*f^(1/a*c)*Ei(1,-c*ln(f)/(b*x+a)+c*ln(f)/a)*a*b^2*c*x^2+f^(c/(b*x+a))*ln 
(f)*a*b^2*c*x^2+f^(c/(b*x+a))*ln(f)*a^2*b*c*x+f^(c/(b*x+a))*a^2*b^2*x^2-f^ 
(c/(b*x+a))*a^4)/a^4/x^2
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.66 \[ \int \frac {f^{\frac {c}{a+b x}}}{x^3} \, dx=\frac {{\left (b^{2} c^{2} x^{2} \log \left (f\right )^{2} + 2 \, a b^{2} c x^{2} \log \left (f\right )\right )} f^{\frac {c}{a}} {\rm Ei}\left (-\frac {b c x \log \left (f\right )}{a b x + a^{2}}\right ) + {\left (a^{2} b^{2} x^{2} - a^{4} + {\left (a b^{2} c x^{2} + a^{2} b c x\right )} \log \left (f\right )\right )} f^{\frac {c}{b x + a}}}{2 \, a^{4} x^{2}} \] Input:

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

Output:

1/2*((b^2*c^2*x^2*log(f)^2 + 2*a*b^2*c*x^2*log(f))*f^(c/a)*Ei(-b*c*x*log(f 
)/(a*b*x + a^2)) + (a^2*b^2*x^2 - a^4 + (a*b^2*c*x^2 + a^2*b*c*x)*log(f))* 
f^(c/(b*x + a)))/(a^4*x^2)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 4*f**(c/(a + b*x))*log(f)**3*a**3*c**3 - 6*f**(c/(a + b*x))*log(f)**3* 
a**2*b*c**3*x - 4*f**(c/(a + b*x))*log(f)**3*a*b**2*c**3*x**2 - f**(c/(a + 
 b*x))*log(f)**3*b**3*c**3*x**3 - 10*f**(c/(a + b*x))*log(f)**2*a**4*c**2 
- 20*f**(c/(a + b*x))*log(f)**2*a**3*b*c**2*x - 15*f**(c/(a + b*x))*log(f) 
**2*a**2*b**2*c**2*x**2 - 2*f**(c/(a + b*x))*log(f)**2*a*b**3*c**2*x**3 + 
f**(c/(a + b*x))*log(f)**2*b**4*c**2*x**4 + 8*f**(c/(a + b*x))*log(f)*a**3 
*b**2*c*x**2 + 12*f**(c/(a + b*x))*log(f)*a**2*b**3*c*x**3 + 4*f**(c/(a + 
b*x))*log(f)*a*b**4*c*x**4 - 4*f**(c/(a + b*x))*a**4*b**2*x**2 - 8*f**(c/( 
a + b*x))*a**3*b**3*x**3 - 4*f**(c/(a + b*x))*a**2*b**4*x**4 + int(f**(c/( 
a + b*x))/(log(f)**2*a**4*c**2*x**3 + 4*log(f)**2*a**3*b*c**2*x**4 + 6*log 
(f)**2*a**2*b**2*c**2*x**5 + 4*log(f)**2*a*b**3*c**2*x**6 + log(f)**2*b**4 
*c**2*x**7 + 10*log(f)*a**5*c*x**3 + 40*log(f)*a**4*b*c*x**4 + 60*log(f)*a 
**3*b**2*c*x**5 + 40*log(f)*a**2*b**3*c*x**6 + 10*log(f)*a*b**4*c*x**7 + 2 
0*a**6*x**3 + 80*a**5*b*x**4 + 120*a**4*b**2*x**5 + 80*a**3*b**3*x**6 + 20 
*a**2*b**4*x**7),x)*log(f)**6*a**6*c**6*x**2 + 2*int(f**(c/(a + b*x))/(log 
(f)**2*a**4*c**2*x**3 + 4*log(f)**2*a**3*b*c**2*x**4 + 6*log(f)**2*a**2*b* 
*2*c**2*x**5 + 4*log(f)**2*a*b**3*c**2*x**6 + log(f)**2*b**4*c**2*x**7 + 1 
0*log(f)*a**5*c*x**3 + 40*log(f)*a**4*b*c*x**4 + 60*log(f)*a**3*b**2*c*x** 
5 + 40*log(f)*a**2*b**3*c*x**6 + 10*log(f)*a*b**4*c*x**7 + 20*a**6*x**3 + 
80*a**5*b*x**4 + 120*a**4*b**2*x**5 + 80*a**3*b**3*x**6 + 20*a**2*b**4*...