\(\int \frac {F^{c (a+b x)}}{d-e x^3} \, dx\) [21]

Optimal result

Integrand size = 20, antiderivative size = 211 \[ \int \frac {F^{c (a+b x)}}{d-e x^3} \, dx=-\frac {F^{c \left (a+\frac {b \sqrt [3]{d}}{\sqrt [3]{e}}\right )} \operatorname {ExpIntegralEi}\left (-\frac {b c \left (\sqrt [3]{d}-\sqrt [3]{e} x\right ) \log (F)}{\sqrt [3]{e}}\right )}{3 d^{2/3} \sqrt [3]{e}}-\frac {(-1)^{2/3} F^{c \left (a+\frac {(-1)^{2/3} b \sqrt [3]{d}}{\sqrt [3]{e}}\right )} \operatorname {ExpIntegralEi}\left (-\frac {b c \left ((-1)^{2/3} \sqrt [3]{d}-\sqrt [3]{e} x\right ) \log (F)}{\sqrt [3]{e}}\right )}{3 d^{2/3} \sqrt [3]{e}}+\frac {\sqrt [3]{-1} F^{c \left (a-\frac {\sqrt [3]{-1} b \sqrt [3]{d}}{\sqrt [3]{e}}\right )} \operatorname {ExpIntegralEi}\left (\frac {\sqrt [3]{-1} b c \left (\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{e} x\right ) \log (F)}{\sqrt [3]{e}}\right )}{3 d^{2/3} \sqrt [3]{e}} \] Output:

-1/3*F^(c*(a+b*d^(1/3)/e^(1/3)))*Ei(-b*c*(d^(1/3)-e^(1/3)*x)*ln(F)/e^(1/3) 
)/d^(2/3)/e^(1/3)-1/3*(-1)^(2/3)*F^(c*(a+(-1)^(2/3)*b*d^(1/3)/e^(1/3)))*Ei 
(-b*c*((-1)^(2/3)*d^(1/3)-e^(1/3)*x)*ln(F)/e^(1/3))/d^(2/3)/e^(1/3)+1/3*(- 
1)^(1/3)*F^(c*(a-(-1)^(1/3)*b*d^(1/3)/e^(1/3)))*Ei((-1)^(1/3)*b*c*(d^(1/3) 
-(-1)^(2/3)*e^(1/3)*x)*ln(F)/e^(1/3))/d^(2/3)/e^(1/3)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 4 in optimal.

Time = 5.10 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.22 \[ \int \frac {F^{c (a+b x)}}{d-e x^3} \, dx=-\frac {\text {RootSum}\left [d-e \text {$\#$1}^3\&,\frac {F^{c (a+b \text {$\#$1})} \operatorname {ExpIntegralEi}(b c \log (F) (x-\text {$\#$1}))}{\text {$\#$1}^2}\&\right ]}{3 e} \] Input:

Integrate[F^(c*(a + b*x))/(d - e*x^3),x]
 

Output:

-1/3*RootSum[d - e*#1^3 & , (F^(c*(a + b*#1))*ExpIntegralEi[b*c*Log[F]*(x 
- #1)])/#1^2 & ]/e
 

Rubi [A] (verified)

Time = 0.89 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {7276, 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.

\(\displaystyle \int \frac {F^{c (a+b x)}}{d-e x^3} \, dx\)

\(\Big \downarrow \) 7276

\(\displaystyle \int \left (\frac {F^{a c+b c x}}{3 d^{2/3} \left (\sqrt [3]{d}-\sqrt [3]{e} x\right )}+\frac {F^{a c+b c x}}{3 d^{2/3} \left (\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x\right )}+\frac {F^{a c+b c x}}{3 d^{2/3} \left (\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{e} x\right )}\right )dx\)

\(\Big \downarrow \) 2009

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

Input:

Int[F^(c*(a + b*x))/(d - e*x^3),x]
 

Output:

-1/3*(F^(c*(a + (b*d^(1/3))/e^(1/3)))*ExpIntegralEi[-((b*c*(d^(1/3) - e^(1 
/3)*x)*Log[F])/e^(1/3))])/(d^(2/3)*e^(1/3)) - ((-1)^(2/3)*F^(c*(a + ((-1)^ 
(2/3)*b*d^(1/3))/e^(1/3)))*ExpIntegralEi[-((b*c*((-1)^(2/3)*d^(1/3) - e^(1 
/3)*x)*Log[F])/e^(1/3))])/(3*d^(2/3)*e^(1/3)) + ((-1)^(1/3)*F^(c*(a - ((-1 
)^(1/3)*b*d^(1/3))/e^(1/3)))*ExpIntegralEi[((-1)^(1/3)*b*c*(d^(1/3) - (-1) 
^(2/3)*e^(1/3)*x)*Log[F])/e^(1/3)])/(3*d^(2/3)*e^(1/3))
 

Defintions of rubi rules used

rule 2009

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

rule 7276

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE 
xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ 
[n, 0]
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.09 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.57

method	result	size
risch	\(\frac {\ln \left (F \right )^{2} b^{2} c^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (-\ln \left (F \right )^{3} a^{3} c^{3} e -\ln \left (F \right )^{3} b^{3} c^{3} d +3 \ln \left (F \right )^{2} a^{2} c^{2} e \textit {\_Z} -3 \ln \left (F \right ) a c e \,\textit {\_Z}^{2}+e \,\textit {\_Z}^{3}\right )}{\sum }\frac {{\mathrm e}^{\textit {\_R1}} \operatorname {expIntegral}_{1}\left (-b c x \ln \left (F \right )-a c \ln \left (F \right )+\textit {\_R1} \right )}{\ln \left (F \right )^{2} a^{2} c^{2}-2 \ln \left (F \right ) \textit {\_R1} a c +\textit {\_R1}^{2}}\right )}{3 e}\)	\(120\)

Input:

int(F^(c*(b*x+a))/(-e*x^3+d),x,method=_RETURNVERBOSE)
 

Output:

1/3*ln(F)^2*b^2*c^2/e*sum(1/(ln(F)^2*a^2*c^2-2*ln(F)*_R1*a*c+_R1^2)*exp(_R 
1)*Ei(1,-b*c*x*ln(F)-a*c*ln(F)+_R1),_R1=RootOf(-ln(F)^3*a^3*c^3*e-ln(F)^3* 
b^3*c^3*d+3*ln(F)^2*a^2*c^2*e*_Z-3*ln(F)*a*c*e*_Z^2+e*_Z^3))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.27 \[ \int \frac {F^{c (a+b x)}}{d-e x^3} \, dx=-\frac {\left (-\frac {b^{3} c^{3} d \log \left (F\right )^{3}}{e}\right )^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )} {\rm Ei}\left (b c x \log \left (F\right ) - \frac {1}{2} \, \left (-\frac {b^{3} c^{3} d \log \left (F\right )^{3}}{e}\right )^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )}\right ) e^{\left (a c \log \left (F\right ) + \frac {1}{2} \, \left (-\frac {b^{3} c^{3} d \log \left (F\right )^{3}}{e}\right )^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )}\right )} - \left (-\frac {b^{3} c^{3} d \log \left (F\right )^{3}}{e}\right )^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )} {\rm Ei}\left (b c x \log \left (F\right ) + \frac {1}{2} \, \left (-\frac {b^{3} c^{3} d \log \left (F\right )^{3}}{e}\right )^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )}\right ) e^{\left (a c \log \left (F\right ) - \frac {1}{2} \, \left (-\frac {b^{3} c^{3} d \log \left (F\right )^{3}}{e}\right )^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )}\right )} - 2 \, \left (-\frac {b^{3} c^{3} d \log \left (F\right )^{3}}{e}\right )^{\frac {1}{3}} {\rm Ei}\left (b c x \log \left (F\right ) + \left (-\frac {b^{3} c^{3} d \log \left (F\right )^{3}}{e}\right )^{\frac {1}{3}}\right ) e^{\left (a c \log \left (F\right ) - \left (-\frac {b^{3} c^{3} d \log \left (F\right )^{3}}{e}\right )^{\frac {1}{3}}\right )}}{6 \, b c d \log \left (F\right )} \] Input:

integrate(F^((b*x+a)*c)/(-e*x^3+d),x, algorithm="fricas")
 

Output:

-1/6*((-b^3*c^3*d*log(F)^3/e)^(1/3)*(sqrt(-3) + 1)*Ei(b*c*x*log(F) - 1/2*( 
-b^3*c^3*d*log(F)^3/e)^(1/3)*(sqrt(-3) + 1))*e^(a*c*log(F) + 1/2*(-b^3*c^3 
*d*log(F)^3/e)^(1/3)*(sqrt(-3) + 1)) - (-b^3*c^3*d*log(F)^3/e)^(1/3)*(sqrt 
(-3) - 1)*Ei(b*c*x*log(F) + 1/2*(-b^3*c^3*d*log(F)^3/e)^(1/3)*(sqrt(-3) - 
1))*e^(a*c*log(F) - 1/2*(-b^3*c^3*d*log(F)^3/e)^(1/3)*(sqrt(-3) - 1)) - 2* 
(-b^3*c^3*d*log(F)^3/e)^(1/3)*Ei(b*c*x*log(F) + (-b^3*c^3*d*log(F)^3/e)^(1 
/3))*e^(a*c*log(F) - (-b^3*c^3*d*log(F)^3/e)^(1/3)))/(b*c*d*log(F))
 

Sympy [F]

\[ \int \frac {F^{c (a+b x)}}{d-e x^3} \, dx=- \int \frac {F^{a c + b c x}}{- d + e x^{3}}\, dx \] Input:

integrate(F**((b*x+a)*c)/(-e*x**3+d),x)
 

Output:

-Integral(F**(a*c + b*c*x)/(-d + e*x**3), x)
 

Maxima [F]

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

integrate(F^((b*x+a)*c)/(-e*x^3+d),x, algorithm="maxima")
 

Output:

-integrate(F^((b*x + a)*c)/(e*x^3 - d), x)
 

Giac [F]

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

integrate(F^((b*x+a)*c)/(-e*x^3+d),x, algorithm="giac")
 

Output:

integrate(-F^((b*x + a)*c)/(e*x^3 - d), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

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

int(F^(c*(a + b*x))/(d - e*x^3),x)
 

Output:

int(F^(c*(a + b*x))/(d - e*x^3), x)
 

Reduce [F]

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

int(F^((b*x+a)*c)/(-e*x^3+d),x)
 

Output:

f**(a*c)*int(f**(b*c*x)/(d - e*x**3),x)