Integrand size = 47, antiderivative size = 110 \[ \int \frac {\left (a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {d f-e f x}}}\right )^2}{d^2-e^2 x^2} \, dx=\frac {2 a b \operatorname {ExpIntegralEi}\left (\frac {c \sqrt {d+e x} \log (F)}{\sqrt {d f-e f x}}\right )}{d e}+\frac {b^2 \operatorname {ExpIntegralEi}\left (\frac {2 c \sqrt {d+e x} \log (F)}{\sqrt {d f-e f x}}\right )}{d e}+\frac {a^2 \log \left (\frac {\sqrt {d+e x}}{\sqrt {d f-e f x}}\right )}{d e} \]
2*a*b*Ei(c*ln(F)*(e*x+d)^(1/2)/(-e*f*x+d*f)^(1/2))/d/e+b^2*Ei(2*c*ln(F)*(e *x+d)^(1/2)/(-e*f*x+d*f)^(1/2))/d/e+a^2*ln((e*x+d)^(1/2)/(-e*f*x+d*f)^(1/2 ))/d/e
\[ \int \frac {\left (a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {d f-e f x}}}\right )^2}{d^2-e^2 x^2} \, dx=\int \frac {\left (a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {d f-e f x}}}\right )^2}{d^2-e^2 x^2} \, dx \]
Time = 0.45 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.90, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.064, Rules used = {2729, 2614, 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.
\(\displaystyle \int \frac {\left (a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {d f-e f x}}}\right )^2}{d^2-e^2 x^2} \, dx\) |
\(\Big \downarrow \) 2729 |
\(\displaystyle \frac {\int \frac {\left (b F^{\frac {c \sqrt {d+e x}}{\sqrt {d f-e f x}}}+a\right )^2 \sqrt {d f-e f x}}{\sqrt {d+e x}}d\frac {\sqrt {d+e x}}{\sqrt {d f-e f x}}}{d e}\) |
\(\Big \downarrow \) 2614 |
\(\displaystyle \frac {\int \left (\frac {2 a b \sqrt {d f-e f x} F^{\frac {c \sqrt {d+e x}}{\sqrt {d f-e f x}}}}{\sqrt {d+e x}}+\frac {b^2 \sqrt {d f-e f x} F^{\frac {2 c \sqrt {d+e x}}{\sqrt {d f-e f x}}}}{\sqrt {d+e x}}+\frac {a^2 \sqrt {d f-e f x}}{\sqrt {d+e x}}\right )d\frac {\sqrt {d+e x}}{\sqrt {d f-e f x}}}{d e}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a^2 \log \left (\frac {\sqrt {d+e x}}{\sqrt {d f-e f x}}\right )+2 a b \operatorname {ExpIntegralEi}\left (\frac {c \sqrt {d+e x} \log (F)}{\sqrt {d f-e f x}}\right )+b^2 \operatorname {ExpIntegralEi}\left (\frac {2 c \sqrt {d+e x} \log (F)}{\sqrt {d f-e f x}}\right )}{d e}\) |
(2*a*b*ExpIntegralEi[(c*Sqrt[d + e*x]*Log[F])/Sqrt[d*f - e*f*x]] + b^2*Exp IntegralEi[(2*c*Sqrt[d + e*x]*Log[F])/Sqrt[d*f - e*f*x]] + a^2*Log[Sqrt[d + e*x]/Sqrt[d*f - e*f*x]])/(d*e)
3.6.52.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.))^(p_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c + d*x)^m, (a + b*(F ^(g*(e + f*x)))^n)^p, x], x] /; FreeQ[{F, a, b, c, d, e, f, g, m, n}, x] && IGtQ[p, 0]
Int[((a_.) + (b_.)*(F_)^(((c_.)*Sqrt[(d_.) + (e_.)*(x_)])/Sqrt[(f_.) + (g_. )*(x_)]))^(n_.)/((A_) + (C_.)*(x_)^2), x_Symbol] :> Simp[2*e*(g/(C*(e*f - d *g))) Subst[Int[(a + b*F^(c*x))^n/x, x], x, Sqrt[d + e*x]/Sqrt[f + g*x]], x] /; FreeQ[{a, b, c, d, e, f, g, A, C, F}, x] && EqQ[C*d*f - A*e*g, 0] && EqQ[e*f + d*g, 0] && IGtQ[n, 0]
\[\int \frac {\left (a +b \,F^{\frac {c \sqrt {e x +d}}{\sqrt {-e f x +d f}}}\right )^{2}}{-e^{2} x^{2}+d^{2}}d x\]
\[ \int \frac {\left (a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {d f-e f x}}}\right )^2}{d^2-e^2 x^2} \, dx=\int { -\frac {{\left (F^{\frac {\sqrt {e x + d} c}{\sqrt {-e f x + d f}}} b + a\right )}^{2}}{e^{2} x^{2} - d^{2}} \,d x } \]
integrate((a+b*F^(c*(e*x+d)^(1/2)/(-e*f*x+d*f)^(1/2)))^2/(-e^2*x^2+d^2),x, algorithm="fricas")
integral(-(a^2 + 2*a*b/F^(sqrt(-e*f*x + d*f)*sqrt(e*x + d)*c/(e*f*x - d*f) ) + b^2/F^(2*sqrt(-e*f*x + d*f)*sqrt(e*x + d)*c/(e*f*x - d*f)))/(e^2*x^2 - d^2), x)
\[ \int \frac {\left (a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {d f-e f x}}}\right )^2}{d^2-e^2 x^2} \, dx=- \int \frac {a^{2}}{- d^{2} + e^{2} x^{2}}\, dx - \int \frac {F^{\frac {2 c \sqrt {d + e x}}{\sqrt {d f - e f x}}} b^{2}}{- d^{2} + e^{2} x^{2}}\, dx - \int \frac {2 F^{\frac {c \sqrt {d + e x}}{\sqrt {d f - e f x}}} a b}{- d^{2} + e^{2} x^{2}}\, dx \]
-Integral(a**2/(-d**2 + e**2*x**2), x) - Integral(F**(2*c*sqrt(d + e*x)/sq rt(d*f - e*f*x))*b**2/(-d**2 + e**2*x**2), x) - Integral(2*F**(c*sqrt(d + e*x)/sqrt(d*f - e*f*x))*a*b/(-d**2 + e**2*x**2), x)
\[ \int \frac {\left (a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {d f-e f x}}}\right )^2}{d^2-e^2 x^2} \, dx=\int { -\frac {{\left (F^{\frac {\sqrt {e x + d} c}{\sqrt {-e f x + d f}}} b + a\right )}^{2}}{e^{2} x^{2} - d^{2}} \,d x } \]
integrate((a+b*F^(c*(e*x+d)^(1/2)/(-e*f*x+d*f)^(1/2)))^2/(-e^2*x^2+d^2),x, algorithm="maxima")
1/2*a^2*(log(e*x + d)/(d*e) - log(e*x - d)/(d*e)) - b^2*integrate(F^(2*sqr t(e*x + d)*c/(sqrt(-e*x + d)*sqrt(f)))/(e^2*x^2 - d^2), x) - 2*a*b*integra te(F^(sqrt(e*x + d)*c/(sqrt(-e*x + d)*sqrt(f)))/(e^2*x^2 - d^2), x)
\[ \int \frac {\left (a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {d f-e f x}}}\right )^2}{d^2-e^2 x^2} \, dx=\int { -\frac {{\left (F^{\frac {\sqrt {e x + d} c}{\sqrt {-e f x + d f}}} b + a\right )}^{2}}{e^{2} x^{2} - d^{2}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {d f-e f x}}}\right )^2}{d^2-e^2 x^2} \, dx=\int \frac {{\left (a+b\,{\mathrm {e}}^{\frac {c\,\ln \left (F\right )\,\sqrt {d+e\,x}}{\sqrt {d\,f-e\,f\,x}}}\right )}^2}{d^2-e^2\,x^2} \,d x \]