Integrand size = 31, antiderivative size = 126 \[ \int \frac {F^{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2}}{(d g+e g x)^3} \, dx=-\frac {e^{-\frac {1-2 a b f n \log (F)}{b^2 f n^2 \log (F)}} \sqrt {\pi } \left (c (d+e x)^n\right )^{2/n} \text {erfi}\left (\frac {\frac {1}{n}-a b f \log (F)-b^2 f \log (F) \log \left (c (d+e x)^n\right )}{b \sqrt {f} \sqrt {\log (F)}}\right )}{2 b e \sqrt {f} g^3 n (d+e x)^2 \sqrt {\log (F)}} \]
1/2*(c*(e*x+d)^n)^(2/n)*erfi((-1/n+a*b*f*ln(F)+b^2*f*ln(F)*ln(c*(e*x+d)^n) )/b/f^(1/2)/ln(F)^(1/2))*Pi^(1/2)/b/e/exp((1-2*a*b*f*n*ln(F))/b^2/f/n^2/ln (F))/g^3/n/(e*x+d)^2/f^(1/2)/ln(F)^(1/2)
Time = 0.22 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.96 \[ \int \frac {F^{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2}}{(d g+e g x)^3} \, dx=\frac {e^{\frac {-1+2 a b f n \log (F)}{b^2 f n^2 \log (F)}} \sqrt {\pi } \left (c (d+e x)^n\right )^{2/n} \text {erfi}\left (\frac {-1+b f n \log (F) \left (a+b \log \left (c (d+e x)^n\right )\right )}{b \sqrt {f} n \sqrt {\log (F)}}\right )}{2 b e \sqrt {f} g^3 n (d+e x)^2 \sqrt {\log (F)}} \]
(E^((-1 + 2*a*b*f*n*Log[F])/(b^2*f*n^2*Log[F]))*Sqrt[Pi]*(c*(d + e*x)^n)^( 2/n)*Erfi[(-1 + b*f*n*Log[F]*(a + b*Log[c*(d + e*x)^n]))/(b*Sqrt[f]*n*Sqrt [Log[F]])])/(2*b*e*Sqrt[f]*g^3*n*(d + e*x)^2*Sqrt[Log[F]])
Time = 0.54 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.13, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {2712, 2706, 2725, 2664, 2633}
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.
| \(\displaystyle \int \frac {F^{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2}}{(d g+e g x)^3} \, dx\) |
| \(\Big \downarrow \) 2712 |
| \(\displaystyle \frac {(d+e x)^{-2 a b f n \log (F)} \left (c (d+e x)^n\right )^{2 a b f \log (F)} \int F^{f a^2+b^2 f \log ^2\left (c (d+e x)^n\right )} (d+e x)^{2 a b f n \log (F)-3}dx}{g^3}\) |
| \(\Big \downarrow \) 2706 |
| \(\displaystyle \frac {\left (c (d+e x)^n\right )^{2 \left (\frac {1}{n}-a b f \log (F)\right )+2 a b f \log (F)} \int F^{f a^2+2 b f \log \left (c (d+e x)^n\right ) a+b^2 f \log ^2\left (c (d+e x)^n\right )} \left (c (d+e x)^n\right )^{-2/n}d\log \left (c (d+e x)^n\right )}{e g^3 n (d+e x)^2}\) |
| \(\Big \downarrow \) 2725 |
| \(\displaystyle \frac {\left (c (d+e x)^n\right )^{2 \left (\frac {1}{n}-a b f \log (F)\right )+2 a b f \log (F)} \int \exp \left (f \log (F) a^2+b^2 f \log (F) \log ^2\left (c (d+e x)^n\right )-\frac {2 (1-a b f n \log (F)) \log \left (c (d+e x)^n\right )}{n}\right )d\log \left (c (d+e x)^n\right )}{e g^3 n (d+e x)^2}\) |
| \(\Big \downarrow \) 2664 |
| \(\displaystyle \frac {e^{-\frac {1-2 a b f n \log (F)}{b^2 f n^2 \log (F)}} \left (c (d+e x)^n\right )^{2 \left (\frac {1}{n}-a b f \log (F)\right )+2 a b f \log (F)} \int \exp \left (\frac {\left (-f \log (F) \log \left (c (d+e x)^n\right ) b^2-a f \log (F) b+\frac {1}{n}\right )^2}{b^2 f \log (F)}\right )d\log \left (c (d+e x)^n\right )}{e g^3 n (d+e x)^2}\) |
| \(\Big \downarrow \) 2633 |
| \(\displaystyle -\frac {\sqrt {\pi } e^{-\frac {1-2 a b f n \log (F)}{b^2 f n^2 \log (F)}} \left (c (d+e x)^n\right )^{2 \left (\frac {1}{n}-a b f \log (F)\right )+2 a b f \log (F)} \text {erfi}\left (\frac {-a b f \log (F)+b^2 (-f) \log (F) \log \left (c (d+e x)^n\right )+\frac {1}{n}}{b \sqrt {f} \sqrt {\log (F)}}\right )}{2 b e \sqrt {f} g^3 n \sqrt {\log (F)} (d+e x)^2}\) |
-1/2*(Sqrt[Pi]*(c*(d + e*x)^n)^(2*a*b*f*Log[F] + 2*(n^(-1) - a*b*f*Log[F]) )*Erfi[(n^(-1) - a*b*f*Log[F] - b^2*f*Log[F]*Log[c*(d + e*x)^n])/(b*Sqrt[f ]*Sqrt[Log[F]])])/(b*e*E^((1 - 2*a*b*f*n*Log[F])/(b^2*f*n^2*Log[F]))*Sqrt[ f]*g^3*n*(d + e*x)^2*Sqrt[Log[F]])
3.7.8.3.1 Defintions of rubi rules used
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt [Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{ F, a, b, c, d}, x] && PosQ[b]
Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[F^(a - b^2/ (4*c)) Int[F^((b + 2*c*x)^2/(4*c)), x], x] /; FreeQ[{F, a, b, c}, x]
Int[(F_)^(((a_.) + Log[(c_.)*((d_.) + (e_.)*(x_))^(n_.)]^2*(b_.))*(f_.))*(( g_.) + (h_.)*(x_))^(m_.), x_Symbol] :> Simp[(g + h*x)^(m + 1)/(h*n*(c*(d + e*x)^n)^((m + 1)/n)) Subst[Int[E^(a*f*Log[F] + ((m + 1)*x)/n + b*f*Log[F] *x^2), x], x, Log[c*(d + e*x)^n]], x] /; FreeQ[{F, a, b, c, d, e, f, g, h, m, n}, x] && EqQ[e*g - d*h, 0]
Int[(F_)^(((a_.) + Log[(c_.)*((d_.) + (e_.)*(x_))^(n_.)]*(b_.))^2*(f_.))*(( g_.) + (h_.)*(x_))^(m_.), x_Symbol] :> Simp[(g + h*x)^m*((c*(d + e*x)^n)^(2 *a*b*f*Log[F])/(d + e*x)^(m + 2*a*b*f*n*Log[F]))*Int[(d + e*x)^(m + 2*a*b*f *n*Log[F])*F^(a^2*f + b^2*f*Log[c*(d + e*x)^n]^2), x], x] /; FreeQ[{F, a, b , c, d, e, f, g, h, m, n}, x] && EqQ[e*g - d*h, 0]
Int[(u_.)*(F_)^(v_)*(G_)^(w_), x_Symbol] :> With[{z = v*Log[F] + w*Log[G]}, Int[u*NormalizeIntegrand[E^z, x], x] /; BinomialQ[z, x] || (PolynomialQ[z, x] && LeQ[Exponent[z, x], 2])] /; FreeQ[{F, G}, x]
\[\int \frac {F^{f {\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )}^{2}}}{\left (e g x +d g \right )^{3}}d x\]
Time = 0.30 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.02 \[ \int \frac {F^{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2}}{(d g+e g x)^3} \, dx=-\frac {\sqrt {\pi } \sqrt {-b^{2} f n^{2} \log \left (F\right )} \operatorname {erf}\left (\frac {{\left (b^{2} f n^{2} \log \left (e x + d\right ) \log \left (F\right ) + b^{2} f n \log \left (F\right ) \log \left (c\right ) + a b f n \log \left (F\right ) - 1\right )} \sqrt {-b^{2} f n^{2} \log \left (F\right )}}{b^{2} f n^{2} \log \left (F\right )}\right ) e^{\left (\frac {2 \, b^{2} f n \log \left (F\right ) \log \left (c\right ) + 2 \, a b f n \log \left (F\right ) - 1}{b^{2} f n^{2} \log \left (F\right )}\right )}}{2 \, b e g^{3} n} \]
-1/2*sqrt(pi)*sqrt(-b^2*f*n^2*log(F))*erf((b^2*f*n^2*log(e*x + d)*log(F) + b^2*f*n*log(F)*log(c) + a*b*f*n*log(F) - 1)*sqrt(-b^2*f*n^2*log(F))/(b^2* f*n^2*log(F)))*e^((2*b^2*f*n*log(F)*log(c) + 2*a*b*f*n*log(F) - 1)/(b^2*f* n^2*log(F)))/(b*e*g^3*n)
Timed out. \[ \int \frac {F^{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2}}{(d g+e g x)^3} \, dx=\text {Timed out} \]
\[ \int \frac {F^{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2}}{(d g+e g x)^3} \, dx=\int { \frac {F^{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} f}}{{\left (e g x + d g\right )}^{3}} \,d x } \]
\[ \int \frac {F^{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2}}{(d g+e g x)^3} \, dx=\int { \frac {F^{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} f}}{{\left (e g x + d g\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {F^{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2}}{(d g+e g x)^3} \, dx=\int \frac {{\mathrm {e}}^{f\,\ln \left (F\right )\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2}}{{\left (d\,g+e\,g\,x\right )}^3} \,d x \]