Integrand size = 20, antiderivative size = 116 \[ \int \frac {F^{c (a+b x)}}{d-e x^2} \, dx=-\frac {F^{c \left (a+\frac {b \sqrt {d}}{\sqrt {e}}\right )} \operatorname {ExpIntegralEi}\left (-\frac {b c \left (\sqrt {d}-\sqrt {e} x\right ) \log (F)}{\sqrt {e}}\right )}{2 \sqrt {d} \sqrt {e}}+\frac {F^{c \left (a-\frac {b \sqrt {d}}{\sqrt {e}}\right )} \operatorname {ExpIntegralEi}\left (\frac {b c \left (\sqrt {d}+\sqrt {e} x\right ) \log (F)}{\sqrt {e}}\right )}{2 \sqrt {d} \sqrt {e}} \] Output:
-1/2*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^(1/2)/e^(1/2)+1/2*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^(1/2)/e^(1/2)
Time = 0.26 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.78 \[ \int \frac {F^{c (a+b x)}}{d-e x^2} \, dx=\frac {F^{c \left (a-\frac {b \sqrt {d}}{\sqrt {e}}\right )} \left (-F^{\frac {2 b c \sqrt {d}}{\sqrt {e}}} \operatorname {ExpIntegralEi}\left (b c \left (-\frac {\sqrt {d}}{\sqrt {e}}+x\right ) \log (F)\right )+\operatorname {ExpIntegralEi}\left (b c \left (\frac {\sqrt {d}}{\sqrt {e}}+x\right ) \log (F)\right )\right )}{2 \sqrt {d} \sqrt {e}} \] Input:
Integrate[F^(c*(a + b*x))/(d - e*x^2),x]
Output:
(F^(c*(a - (b*Sqrt[d])/Sqrt[e]))*(-(F^((2*b*c*Sqrt[d])/Sqrt[e])*ExpIntegra lEi[b*c*(-(Sqrt[d]/Sqrt[e]) + x)*Log[F]]) + ExpIntegralEi[b*c*(Sqrt[d]/Sqr t[e] + x)*Log[F]]))/(2*Sqrt[d]*Sqrt[e])
Time = 0.49 (sec) , antiderivative size = 116, 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 = {2699, 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.
\(\displaystyle \int \frac {F^{c (a+b x)}}{d-e x^2} \, dx\) |
\(\Big \downarrow \) 2699 |
\(\displaystyle \int \left (\frac {F^{c (a+b x)}}{2 \sqrt {d} \left (\sqrt {d}-\sqrt {e} x\right )}+\frac {F^{c (a+b x)}}{2 \sqrt {d} \left (\sqrt {d}+\sqrt {e} x\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {F^{c \left (a-\frac {b \sqrt {d}}{\sqrt {e}}\right )} \operatorname {ExpIntegralEi}\left (\frac {b c \left (\sqrt {e} x+\sqrt {d}\right ) \log (F)}{\sqrt {e}}\right )}{2 \sqrt {d} \sqrt {e}}-\frac {F^{c \left (a+\frac {b \sqrt {d}}{\sqrt {e}}\right )} \operatorname {ExpIntegralEi}\left (-\frac {b c \left (\sqrt {d}-\sqrt {e} x\right ) \log (F)}{\sqrt {e}}\right )}{2 \sqrt {d} \sqrt {e}}\) |
Input:
Int[F^(c*(a + b*x))/(d - e*x^2),x]
Output:
-1/2*(F^(c*(a + (b*Sqrt[d])/Sqrt[e]))*ExpIntegralEi[-((b*c*(Sqrt[d] - Sqrt [e]*x)*Log[F])/Sqrt[e])])/(Sqrt[d]*Sqrt[e]) + (F^(c*(a - (b*Sqrt[d])/Sqrt[ e]))*ExpIntegralEi[(b*c*(Sqrt[d] + Sqrt[e]*x)*Log[F])/Sqrt[e]])/(2*Sqrt[d] *Sqrt[e])
Int[(F_)^((g_.)*((d_.) + (e_.)*(x_))^(n_.))/((a_) + (c_.)*(x_)^2), x_Symbol ] :> Int[ExpandIntegrand[F^(g*(d + e*x)^n), 1/(a + c*x^2), x], x] /; FreeQ[ {F, a, c, d, e, g, n}, x]
Time = 0.10 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.12
method | result | size |
risch | \(\frac {F^{\frac {\left (b \sqrt {d e}+e a \right ) c}{e}} \operatorname {expIntegral}_{1}\left (\frac {\sqrt {d e}\, \ln \left (F \right ) b c +e a \ln \left (F \right ) c -e \left (b c x \ln \left (F \right )+a c \ln \left (F \right )\right )}{e}\right )}{2 \sqrt {d e}}-\frac {F^{\frac {\left (-b \sqrt {d e}+e a \right ) c}{e}} \operatorname {expIntegral}_{1}\left (-\frac {\sqrt {d e}\, \ln \left (F \right ) b c -e a \ln \left (F \right ) c +e \left (b c x \ln \left (F \right )+a c \ln \left (F \right )\right )}{e}\right )}{2 \sqrt {d e}}\) | \(130\) |
Input:
int(F^(c*(b*x+a))/(-e*x^2+d),x,method=_RETURNVERBOSE)
Output:
1/2/(d*e)^(1/2)*F^((b*(d*e)^(1/2)+e*a)/e*c)*Ei(1,((d*e)^(1/2)*ln(F)*b*c+e* a*ln(F)*c-e*(b*c*x*ln(F)+a*c*ln(F)))/e)-1/2/(d*e)^(1/2)*F^((-b*(d*e)^(1/2) +e*a)/e*c)*Ei(1,-((d*e)^(1/2)*ln(F)*b*c-e*a*ln(F)*c+e*(b*c*x*ln(F)+a*c*ln( F)))/e)
Time = 0.09 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.34 \[ \int \frac {F^{c (a+b x)}}{d-e x^2} \, dx=-\frac {\sqrt {\frac {b^{2} c^{2} d \log \left (F\right )^{2}}{e}} {\rm Ei}\left (b c x \log \left (F\right ) - \sqrt {\frac {b^{2} c^{2} d \log \left (F\right )^{2}}{e}}\right ) e^{\left (a c \log \left (F\right ) + \sqrt {\frac {b^{2} c^{2} d \log \left (F\right )^{2}}{e}}\right )} - \sqrt {\frac {b^{2} c^{2} d \log \left (F\right )^{2}}{e}} {\rm Ei}\left (b c x \log \left (F\right ) + \sqrt {\frac {b^{2} c^{2} d \log \left (F\right )^{2}}{e}}\right ) e^{\left (a c \log \left (F\right ) - \sqrt {\frac {b^{2} c^{2} d \log \left (F\right )^{2}}{e}}\right )}}{2 \, b c d \log \left (F\right )} \] Input:
integrate(F^((b*x+a)*c)/(-e*x^2+d),x, algorithm="fricas")
Output:
-1/2*(sqrt(b^2*c^2*d*log(F)^2/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)) - sqrt(b^2*c^2*d*log(F)^2 /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*log(F))
\[ \int \frac {F^{c (a+b x)}}{d-e x^2} \, dx=- \int \frac {F^{a c + b c x}}{- d + e x^{2}}\, dx \] Input:
integrate(F**((b*x+a)*c)/(-e*x**2+d),x)
Output:
-Integral(F**(a*c + b*c*x)/(-d + e*x**2), x)
\[ \int \frac {F^{c (a+b x)}}{d-e x^2} \, dx=\int { -\frac {F^{{\left (b x + a\right )} c}}{e x^{2} - d} \,d x } \] Input:
integrate(F^((b*x+a)*c)/(-e*x^2+d),x, algorithm="maxima")
Output:
-integrate(F^((b*x + a)*c)/(e*x^2 - d), x)
\[ \int \frac {F^{c (a+b x)}}{d-e x^2} \, dx=\int { -\frac {F^{{\left (b x + a\right )} c}}{e x^{2} - d} \,d x } \] Input:
integrate(F^((b*x+a)*c)/(-e*x^2+d),x, algorithm="giac")
Output:
integrate(-F^((b*x + a)*c)/(e*x^2 - d), x)
Timed out. \[ \int \frac {F^{c (a+b x)}}{d-e x^2} \, dx=\int \frac {F^{c\,\left (a+b\,x\right )}}{d-e\,x^2} \,d x \] Input:
int(F^(c*(a + b*x))/(d - e*x^2),x)
Output:
int(F^(c*(a + b*x))/(d - e*x^2), x)
\[ \int \frac {F^{c (a+b x)}}{d-e x^2} \, dx=f^{a c} \left (\int \frac {f^{b c x}}{-e \,x^{2}+d}d x \right ) \] Input:
int(F^((b*x+a)*c)/(-e*x^2+d),x)
Output:
f**(a*c)*int(f**(b*c*x)/(d - e*x**2),x)