\(\int \frac {F^{c (a+b x)}}{(d-e x^3)^2} \, dx\) [22]

Optimal result

Integrand size = 20, antiderivative size = 617 \[ \int \frac {F^{c (a+b x)}}{\left (d-e x^3\right )^2} \, dx=\frac {F^{c (a+b x)}}{9 d^{4/3} \sqrt [3]{e} \left (\sqrt [3]{d}-\sqrt [3]{e} x\right )}-\frac {(-1)^{2/3} F^{c (a+b x)}}{9 d^{4/3} \sqrt [3]{e} \left (\sqrt [3]{-1} \sqrt [3]{d}+\sqrt [3]{e} x\right )}-\frac {\sqrt [3]{-1} F^{c (a+b x)}}{\left (1+\sqrt [3]{-1}\right )^4 d^{4/3} \sqrt [3]{e} \left (\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x\right )}-\frac {2 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 )}{9 d^{5/3} \sqrt [3]{e}}+\frac {2 i \sqrt {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 )}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \sqrt [3]{e}}+\frac {4 F^{c \left (a-\frac {2 i b \sqrt [3]{d}}{\left (i+\sqrt {3}\right ) \sqrt [3]{e}}\right )} \operatorname {ExpIntegralEi}\left (\frac {b c \left (2 \sqrt [3]{d}+\left (1-i \sqrt {3}\right ) \sqrt [3]{e} x\right ) \log (F)}{\left (1-i \sqrt {3}\right ) \sqrt [3]{e}}\right )}{9 \left (1-i \sqrt {3}\right ) d^{5/3} \sqrt [3]{e}}+\frac {b c 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 ) \log (F)}{9 d^{4/3} e^{2/3}}+\frac {b c 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 ) \log (F)}{\left (1+\sqrt [3]{-1}\right )^4 d^{4/3} e^{2/3}}+\frac {(-1)^{2/3} b c 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 ) \log (F)}{9 d^{4/3} e^{2/3}} \] Output:

1/9*F^(c*(b*x+a))/d^(4/3)/e^(1/3)/(d^(1/3)-e^(1/3)*x)-1/9*(-1)^(2/3)*F^(c* 
(b*x+a))/d^(4/3)/e^(1/3)/((-1)^(1/3)*d^(1/3)+e^(1/3)*x)-(-1)^(1/3)*F^(c*(b 
*x+a))/(1+(-1)^(1/3))^4/d^(4/3)/e^(1/3)/(d^(1/3)+(-1)^(1/3)*e^(1/3)*x)-2/9 
*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^ 
(5/3)/e^(1/3)+2*I*3^(1/2)*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))/(1+(-1)^(1/3))^5/d^(5/3)/e^(1 
/3)+4/9*F^(c*(a-2*I*b*d^(1/3)/(3^(1/2)+I)/e^(1/3)))*Ei(b*c*(2*d^(1/3)+(1-I 
*3^(1/2))*e^(1/3)*x)*ln(F)/(1-I*3^(1/2))/e^(1/3))/(1-I*3^(1/2))/d^(5/3)/e^ 
(1/3)+1/9*b*c*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))*ln(F)/d^(4/3)/e^(2/3)+b*c*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))*ln(F)/(1+(-1)^(1/3 
))^4/d^(4/3)/e^(2/3)+1/9*(-1)^(2/3)*b*c*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))*ln(F) 
/d^(4/3)/e^(2/3)
 

Mathematica [C] (verified)

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

Time = 0.21 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.20 \[ \int \frac {F^{c (a+b x)}}{\left (d-e x^3\right )^2} \, dx=\frac {\frac {3 F^{c (a+b x)} x}{d-e x^3}-\frac {2 \text {RootSum}\left [d-e \text {$\#$1}^3\&,\frac {F^{c (a+b \text {$\#$1})} \operatorname {ExpIntegralEi}(b c \log (F) (x-\text {$\#$1}))}{\text {$\#$1}^2}\&\right ]}{e}+\frac {b c \log (F) \text {RootSum}\left [d-e \text {$\#$1}^3\&,\frac {F^{c (a+b \text {$\#$1})} \operatorname {ExpIntegralEi}(b c \log (F) (x-\text {$\#$1}))}{\text {$\#$1}}\&\right ]}{e}}{9 d} \] Input:

Integrate[F^(c*(a + b*x))/(d - e*x^3)^2,x]
 

Output:

((3*F^(c*(a + b*x))*x)/(d - e*x^3) - (2*RootSum[d - e*#1^3 & , (F^(c*(a + 
b*#1))*ExpIntegralEi[b*c*Log[F]*(x - #1)])/#1^2 & ])/e + (b*c*Log[F]*RootS 
um[d - e*#1^3 & , (F^(c*(a + b*#1))*ExpIntegralEi[b*c*Log[F]*(x - #1)])/#1 
 & ])/e)/(9*d)
 

Rubi [A] (verified)

Time = 2.24 (sec) , antiderivative size = 618, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {7292, 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.

\(\displaystyle \int \frac {F^{c (a+b x)}}{\left (d-e x^3\right )^2} \, dx\)

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {2 F^{a c+b c x}}{9 d^{5/3} \left (\sqrt [3]{d}-\sqrt [3]{e} x\right )}+\frac {2 (-1)^{5/6} \sqrt {3} F^{a c+b c x}}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \left (\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x\right )}+\frac {4 i F^{a c+b c x}}{9 d^{5/3} \left (2 i \sqrt [3]{d}+\left (\sqrt {3}+i\right ) \sqrt [3]{e} x\right )}+\frac {F^{a c+b c x}}{9 d^{4/3} \left (\sqrt [3]{d}-\sqrt [3]{e} x\right )^2}+\frac {(-1)^{2/3} F^{a c+b c x}}{\left (1+\sqrt [3]{-1}\right )^4 d^{4/3} \left (\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x\right )^2}+\frac {F^{a c+b c x}}{\left (\sqrt [3]{-1}-1\right )^2 \left (1+\sqrt [3]{-1}\right )^4 d^{4/3} \left ((-1)^{2/3} \sqrt [3]{e} x-\sqrt [3]{d}\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {b c \log (F) F^{c \left (a+\frac {b \sqrt [3]{d}}{\sqrt [3]{e}}\right )} \operatorname {ExpIntegralEi}\left (-\frac {b c \left (\sqrt [3]{d}-\sqrt [3]{e} x\right ) \log (F)}{\sqrt [3]{e}}\right )}{9 d^{4/3} e^{2/3}}+\frac {(-1)^{2/3} b c \log (F) F^{c \left (a-\frac {\sqrt [3]{-1} b \sqrt [3]{d}}{\sqrt [3]{e}}\right )} \operatorname {ExpIntegralEi}\left (\frac {\sqrt [3]{-1} b c \left (\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{e} x\right ) \log (F)}{\sqrt [3]{e}}\right )}{9 d^{4/3} e^{2/3}}+\frac {b c \log (F) F^{c \left (a+\frac {(-1)^{2/3} b \sqrt [3]{d}}{\sqrt [3]{e}}\right )} \operatorname {ExpIntegralEi}\left (-\frac {b c \left ((-1)^{2/3} \sqrt [3]{d}-\sqrt [3]{e} x\right ) \log (F)}{\sqrt [3]{e}}\right )}{\left (1+\sqrt [3]{-1}\right )^4 d^{4/3} e^{2/3}}-\frac {2 F^{c \left (a+\frac {b \sqrt [3]{d}}{\sqrt [3]{e}}\right )} \operatorname {ExpIntegralEi}\left (-\frac {b c \left (\sqrt [3]{d}-\sqrt [3]{e} x\right ) \log (F)}{\sqrt [3]{e}}\right )}{9 d^{5/3} \sqrt [3]{e}}+\frac {2 i \sqrt {3} F^{c \left (a+\frac {(-1)^{2/3} b \sqrt [3]{d}}{\sqrt [3]{e}}\right )} \operatorname {ExpIntegralEi}\left (-\frac {b c \left ((-1)^{2/3} \sqrt [3]{d}-\sqrt [3]{e} x\right ) \log (F)}{\sqrt [3]{e}}\right )}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \sqrt [3]{e}}+\frac {4 F^{c \left (a-\frac {2 i b \sqrt [3]{d}}{\left (\sqrt {3}+i\right ) \sqrt [3]{e}}\right )} \operatorname {ExpIntegralEi}\left (\frac {b c \left (\left (i+\sqrt {3}\right ) \sqrt [3]{e} x+2 i \sqrt [3]{d}\right ) \log (F)}{\left (i+\sqrt {3}\right ) \sqrt [3]{e}}\right )}{9 \left (1-i \sqrt {3}\right ) d^{5/3} \sqrt [3]{e}}+\frac {F^{a c+b c x}}{9 d^{4/3} \sqrt [3]{e} \left (\sqrt [3]{d}-\sqrt [3]{e} x\right )}-\frac {(-1)^{2/3} F^{a c+b c x}}{9 d^{4/3} \sqrt [3]{e} \left (\sqrt [3]{-1} \sqrt [3]{d}+\sqrt [3]{e} x\right )}-\frac {\sqrt [3]{-1} F^{a c+b c x}}{\left (1+\sqrt [3]{-1}\right )^4 d^{4/3} \sqrt [3]{e} \left (\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x\right )}\)

Input:

Int[F^(c*(a + b*x))/(d - e*x^3)^2,x]
 

Output:

F^(a*c + b*c*x)/(9*d^(4/3)*e^(1/3)*(d^(1/3) - e^(1/3)*x)) - ((-1)^(2/3)*F^ 
(a*c + b*c*x))/(9*d^(4/3)*e^(1/3)*((-1)^(1/3)*d^(1/3) + e^(1/3)*x)) - ((-1 
)^(1/3)*F^(a*c + b*c*x))/((1 + (-1)^(1/3))^4*d^(4/3)*e^(1/3)*(d^(1/3) + (- 
1)^(1/3)*e^(1/3)*x)) - (2*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))])/(9*d^(5/3)*e^(1/3)) + ((2*I) 
*Sqrt[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))])/((1 + (-1)^(1/3))^5*d^( 
5/3)*e^(1/3)) + (4*F^(c*(a - ((2*I)*b*d^(1/3))/((I + Sqrt[3])*e^(1/3))))*E 
xpIntegralEi[(b*c*((2*I)*d^(1/3) + (I + Sqrt[3])*e^(1/3)*x)*Log[F])/((I + 
Sqrt[3])*e^(1/3))])/(9*(1 - I*Sqrt[3])*d^(5/3)*e^(1/3)) + (b*c*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))]*Log[F])/(9*d^(4/3)*e^(2/3)) + (b*c*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))]*Log[F])/((1 + (-1)^(1/3))^4*d^(4/3)*e^(2/3)) + ((-1)^(2/3)*b*c*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)]*Log[F])/(9*d^(4/3)*e^(2/3) 
)
 

Defintions of rubi rules used

rule 2009

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

rule 7292

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

rule 7293

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

Maple [F]

\[\int \frac {F^{c \left (b x +a \right )}}{\left (-e \,x^{3}+d \right )^{2}}d x\]

Input:

int(F^(c*(b*x+a))/(-e*x^3+d)^2,x)
 

Output:

int(F^(c*(b*x+a))/(-e*x^3+d)^2,x)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 460, normalized size of antiderivative = 0.75 \[ \int \frac {F^{c (a+b x)}}{\left (d-e x^3\right )^2} \, dx=-\frac {6 \, F^{b c x + a c} b c d x \log \left (F\right ) + {\left (\left (-\frac {b^{3} c^{3} d \log \left (F\right )^{3}}{e}\right )^{\frac {2}{3}} {\left (e x^{3} - \sqrt {-3} {\left (e x^{3} - d\right )} - d\right )} + 2 \, \left (-\frac {b^{3} c^{3} d \log \left (F\right )^{3}}{e}\right )^{\frac {1}{3}} {\left (e x^{3} + \sqrt {-3} {\left (e x^{3} - d\right )} - d\right )}\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 (\left (-\frac {b^{3} c^{3} d \log \left (F\right )^{3}}{e}\right )^{\frac {2}{3}} {\left (e x^{3} + \sqrt {-3} {\left (e x^{3} - d\right )} - d\right )} + 2 \, \left (-\frac {b^{3} c^{3} d \log \left (F\right )^{3}}{e}\right )^{\frac {1}{3}} {\left (e x^{3} - \sqrt {-3} {\left (e x^{3} - d\right )} - d\right )}\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 (\left (-\frac {b^{3} c^{3} d \log \left (F\right )^{3}}{e}\right )^{\frac {2}{3}} {\left (e x^{3} - d\right )} + 2 \, \left (-\frac {b^{3} c^{3} d \log \left (F\right )^{3}}{e}\right )^{\frac {1}{3}} {\left (e x^{3} - d\right )}\right )} {\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 )}}{18 \, {\left (b c d^{2} e x^{3} - b c d^{3}\right )} \log \left (F\right )} \] Input:

integrate(F^((b*x+a)*c)/(-e*x^3+d)^2,x, algorithm="fricas")
 

Output:

-1/18*(6*F^(b*c*x + a*c)*b*c*d*x*log(F) + ((-b^3*c^3*d*log(F)^3/e)^(2/3)*( 
e*x^3 - sqrt(-3)*(e*x^3 - d) - d) + 2*(-b^3*c^3*d*log(F)^3/e)^(1/3)*(e*x^3 
 + sqrt(-3)*(e*x^3 - d) - d))*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)^(2/3)*(e*x^3 + sqrt(-3)*(e*x^3 
- d) - d) + 2*(-b^3*c^3*d*log(F)^3/e)^(1/3)*(e*x^3 - sqrt(-3)*(e*x^3 - d) 
- d))*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)^(2/3)*(e*x^3 - d) + 2*(-b^3*c^3*d*log(F)^3/e)^(1/3)*( 
e*x^3 - d))*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^2*e*x^3 - b*c*d^3)*log(F))
 

Sympy [F]

\[ \int \frac {F^{c (a+b x)}}{\left (d-e x^3\right )^2} \, dx=\int \frac {F^{c \left (a + b x\right )}}{\left (- d + e x^{3}\right )^{2}}\, dx \] Input:

integrate(F**((b*x+a)*c)/(-e*x**3+d)**2,x)
 

Output:

Integral(F**(c*(a + b*x))/(-d + e*x**3)**2, x)
 

Maxima [F]

\[ \int \frac {F^{c (a+b x)}}{\left (d-e x^3\right )^2} \, dx=\int { \frac {F^{{\left (b x + a\right )} c}}{{\left (e x^{3} - d\right )}^{2}} \,d x } \] Input:

integrate(F^((b*x+a)*c)/(-e*x^3+d)^2,x, algorithm="maxima")
 

Output:

integrate(F^((b*x + a)*c)/(e*x^3 - d)^2, x)
 

Giac [F]

\[ \int \frac {F^{c (a+b x)}}{\left (d-e x^3\right )^2} \, dx=\int { \frac {F^{{\left (b x + a\right )} c}}{{\left (e x^{3} - d\right )}^{2}} \,d x } \] Input:

integrate(F^((b*x+a)*c)/(-e*x^3+d)^2,x, algorithm="giac")
 

Output:

integrate(F^((b*x + a)*c)/(e*x^3 - d)^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {F^{c (a+b x)}}{\left (d-e x^3\right )^2} \, dx=\int \frac {F^{c\,\left (a+b\,x\right )}}{{\left (d-e\,x^3\right )}^2} \,d x \] Input:

int(F^(c*(a + b*x))/(d - e*x^3)^2,x)
 

Output:

int(F^(c*(a + b*x))/(d - e*x^3)^2, x)
 

Reduce [F]

\[ \int \frac {F^{c (a+b x)}}{\left (d-e x^3\right )^2} \, dx=f^{a c} \left (\int \frac {f^{b c x}}{e^{2} x^{6}-2 d e \,x^{3}+d^{2}}d x \right ) \] Input:

int(F^((b*x+a)*c)/(-e*x^3+d)^2,x)
 

Output:

f**(a*c)*int(f**(b*c*x)/(d**2 - 2*d*e*x**3 + e**2*x**6),x)