\(\int \frac {F^{c (a+b x)}}{(d-e x^2)^2} \, dx\) [18]

Optimal result

Integrand size = 20, antiderivative size = 304 \[ \int \frac {F^{c (a+b x)}}{\left (d-e x^2\right )^2} \, dx=\frac {F^{c (a+b x)}}{4 d \sqrt {e} \left (\sqrt {d}-\sqrt {e} x\right )}-\frac {F^{c (a+b x)}}{4 d \sqrt {e} \left (\sqrt {d}+\sqrt {e} x\right )}-\frac {F^{c \left (a+\frac {b \sqrt {d}}{\sqrt {e}}\right )} \operatorname {ExpIntegralEi}\left (-\frac {b c \left (\sqrt {d}-\sqrt {e} x\right ) \log (F)}{\sqrt {e}}\right )}{4 d^{3/2} \sqrt {e}}+\frac {F^{c \left (a-\frac {b \sqrt {d}}{\sqrt {e}}\right )} \operatorname {ExpIntegralEi}\left (\frac {b c \left (\sqrt {d}+\sqrt {e} x\right ) \log (F)}{\sqrt {e}}\right )}{4 d^{3/2} \sqrt {e}}+\frac {b c F^{c \left (a+\frac {b \sqrt {d}}{\sqrt {e}}\right )} \operatorname {ExpIntegralEi}\left (-\frac {b c \left (\sqrt {d}-\sqrt {e} x\right ) \log (F)}{\sqrt {e}}\right ) \log (F)}{4 d e}+\frac {b c F^{c \left (a-\frac {b \sqrt {d}}{\sqrt {e}}\right )} \operatorname {ExpIntegralEi}\left (\frac {b c \left (\sqrt {d}+\sqrt {e} x\right ) \log (F)}{\sqrt {e}}\right ) \log (F)}{4 d e} \] Output:

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

Mathematica [A] (verified)

Time = 0.67 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.58 \[ \int \frac {F^{c (a+b x)}}{\left (d-e x^2\right )^2} \, dx=\frac {F^{c \left (a-\frac {b \sqrt {d}}{\sqrt {e}}\right )} \left (2 \sqrt {d} e F^{b c \left (\frac {\sqrt {d}}{\sqrt {e}}+x\right )} x+F^{\frac {2 b c \sqrt {d}}{\sqrt {e}}} \left (d-e x^2\right ) \operatorname {ExpIntegralEi}\left (b c \left (-\frac {\sqrt {d}}{\sqrt {e}}+x\right ) \log (F)\right ) \left (-\sqrt {e}+b c \sqrt {d} \log (F)\right )+\left (d-e x^2\right ) \operatorname {ExpIntegralEi}\left (b c \left (\frac {\sqrt {d}}{\sqrt {e}}+x\right ) \log (F)\right ) \left (\sqrt {e}+b c \sqrt {d} \log (F)\right )\right )}{4 d^{3/2} e \left (d-e x^2\right )} \] Input:

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

Output:

(F^(c*(a - (b*Sqrt[d])/Sqrt[e]))*(2*Sqrt[d]*e*F^(b*c*(Sqrt[d]/Sqrt[e] + x) 
)*x + F^((2*b*c*Sqrt[d])/Sqrt[e])*(d - e*x^2)*ExpIntegralEi[b*c*(-(Sqrt[d] 
/Sqrt[e]) + x)*Log[F]]*(-Sqrt[e] + b*c*Sqrt[d]*Log[F]) + (d - e*x^2)*ExpIn 
tegralEi[b*c*(Sqrt[d]/Sqrt[e] + x)*Log[F]]*(Sqrt[e] + b*c*Sqrt[d]*Log[F])) 
)/(4*d^(3/2)*e*(d - e*x^2))
 

Rubi [A] (verified)

Time = 1.17 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.01, 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.

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

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {e F^{a c+b c x}}{2 d \left (d e-e^2 x^2\right )}+\frac {e F^{a c+b c x}}{4 d \left (\sqrt {d} \sqrt {e}-e x\right )^2}+\frac {e F^{a c+b c x}}{4 d \left (\sqrt {d} \sqrt {e}+e x\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

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

Input:

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

Output:

F^(a*c + b*c*x)/(4*d*Sqrt[e]*(Sqrt[d] - Sqrt[e]*x)) - F^(a*c + b*c*x)/(4*d 
*Sqrt[e]*(Sqrt[d] + Sqrt[e]*x)) - (F^(c*(a + (b*Sqrt[d])/Sqrt[e]))*ExpInte 
gralEi[-((b*c*(Sqrt[d] - Sqrt[e]*x)*Log[F])/Sqrt[e])])/(4*d^(3/2)*Sqrt[e]) 
 + (F^(c*(a - (b*Sqrt[d])/Sqrt[e]))*ExpIntegralEi[(b*c*(Sqrt[d] + Sqrt[e]* 
x)*Log[F])/Sqrt[e]])/(4*d^(3/2)*Sqrt[e]) + (b*c*F^(c*(a + (b*Sqrt[d])/Sqrt 
[e]))*ExpIntegralEi[-((b*c*(Sqrt[d] - Sqrt[e]*x)*Log[F])/Sqrt[e])]*Log[F]) 
/(4*d*e) + (b*c*F^(c*(a - (b*Sqrt[d])/Sqrt[e]))*ExpIntegralEi[(b*c*(Sqrt[d 
] + Sqrt[e]*x)*Log[F])/Sqrt[e]]*Log[F])/(4*d*e)
 

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]

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 276, normalized size of antiderivative = 0.91 \[ \int \frac {F^{c (a+b x)}}{\left (d-e x^2\right )^2} \, dx=-\frac {2 \, F^{b c x + a c} b c d e x \log \left (F\right ) - {\left ({\left (b^{2} c^{2} d e x^{2} - b^{2} c^{2} d^{2}\right )} \log \left (F\right )^{2} - \sqrt {\frac {b^{2} c^{2} d \log \left (F\right )^{2}}{e}} {\left (e^{2} x^{2} - d e\right )}\right )} {\rm Ei}\left (b c x \log \left (F\right ) - \sqrt {\frac {b^{2} c^{2} d \log \left (F\right )^{2}}{e}}\right ) e^{\left (a c \log \left (F\right ) + \sqrt {\frac {b^{2} c^{2} d \log \left (F\right )^{2}}{e}}\right )} - {\left ({\left (b^{2} c^{2} d e x^{2} - b^{2} c^{2} d^{2}\right )} \log \left (F\right )^{2} + \sqrt {\frac {b^{2} c^{2} d \log \left (F\right )^{2}}{e}} {\left (e^{2} x^{2} - d e\right )}\right )} {\rm Ei}\left (b c x \log \left (F\right ) + \sqrt {\frac {b^{2} c^{2} d \log \left (F\right )^{2}}{e}}\right ) e^{\left (a c \log \left (F\right ) - \sqrt {\frac {b^{2} c^{2} d \log \left (F\right )^{2}}{e}}\right )}}{4 \, {\left (b c d^{2} e^{2} x^{2} - b c d^{3} e\right )} \log \left (F\right )} \] Input:

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

Output:

-1/4*(2*F^(b*c*x + a*c)*b*c*d*e*x*log(F) - ((b^2*c^2*d*e*x^2 - b^2*c^2*d^2 
)*log(F)^2 - sqrt(b^2*c^2*d*log(F)^2/e)*(e^2*x^2 - d*e))*Ei(b*c*x*log(F) - 
 sqrt(b^2*c^2*d*log(F)^2/e))*e^(a*c*log(F) + sqrt(b^2*c^2*d*log(F)^2/e)) - 
 ((b^2*c^2*d*e*x^2 - b^2*c^2*d^2)*log(F)^2 + sqrt(b^2*c^2*d*log(F)^2/e)*(e 
^2*x^2 - d*e))*Ei(b*c*x*log(F) + sqrt(b^2*c^2*d*log(F)^2/e))*e^(a*c*log(F) 
 - sqrt(b^2*c^2*d*log(F)^2/e)))/((b*c*d^2*e^2*x^2 - b*c*d^3*e)*log(F))
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {F^{c (a+b x)}}{\left (d-e x^2\right )^2} \, dx=f^{a c} \left (\int \frac {f^{b c x}}{e^{2} x^{4}-2 d e \,x^{2}+d^{2}}d x \right ) \] Input:

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

Output:

f**(a*c)*int(f**(b*c*x)/(d**2 - 2*d*e*x**2 + e**2*x**4),x)