Integrand size = 31, antiderivative size = 133 \[ \int F^{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2} (d g+e g x)^2 \, dx=\frac {e^{-\frac {3 (3+4 a b f n \log (F))}{4 b^2 f n^2 \log (F)}} g^2 \sqrt {\pi } (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \text {erfi}\left (\frac {\frac {3}{n}+2 a b f \log (F)+2 b^2 f \log (F) \log \left (c (d+e x)^n\right )}{2 b \sqrt {f} \sqrt {\log (F)}}\right )}{2 b e \sqrt {f} n \sqrt {\log (F)}} \]
1/2*g^2*(e*x+d)^3*erfi(1/2*(3/n+2*a*b*f*ln(F)+2*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(3/4*(3+4*a*b*f*n*ln(F))/b^2/f/n ^2/ln(F))/n/((c*(e*x+d)^n)^(3/n))/f^(1/2)/ln(F)^(1/2)
Time = 0.24 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.97 \[ \int F^{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2} (d g+e g x)^2 \, dx=\frac {e^{-\frac {3 \left (3+4 b f n \log (F) \left (a+b \left (-n \log (d+e x)+\log \left (c (d+e x)^n\right )\right )\right )\right )}{4 b^2 f n^2 \log (F)}} g^2 \sqrt {\pi } \text {erfi}\left (\frac {3+2 b f n \log (F) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 b \sqrt {f} n \sqrt {\log (F)}}\right )}{2 b e \sqrt {f} n \sqrt {\log (F)}} \]
(g^2*Sqrt[Pi]*Erfi[(3 + 2*b*f*n*Log[F]*(a + b*Log[c*(d + e*x)^n]))/(2*b*Sq rt[f]*n*Sqrt[Log[F]])])/(2*b*e*E^((3*(3 + 4*b*f*n*Log[F]*(a + b*(-(n*Log[d + e*x]) + Log[c*(d + e*x)^n]))))/(4*b^2*f*n^2*Log[F]))*Sqrt[f]*n*Sqrt[Log [F]])
Time = 0.51 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.00, 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.
\(\displaystyle \int (d g+e g x)^2 F^{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx\) |
\(\Big \downarrow \) 2712 |
\(\displaystyle g^2 (d+e x)^{-2 a b f n \log (F)} \left (c (d+e x)^n\right )^{2 a b f \log (F)} \int F^{f a^2+b^2 f \log ^2\left (c (d+e x)^n\right )} (d+e x)^{2 a b f n \log (F)+2}dx\) |
\(\Big \downarrow \) 2706 |
\(\displaystyle \frac {g^2 (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \int F^{f a^2+2 b f \log \left (c (d+e x)^n\right ) a+b^2 f \log ^2\left (c (d+e x)^n\right )} \left (c (d+e x)^n\right )^{3/n}d\log \left (c (d+e x)^n\right )}{e n}\) |
\(\Big \downarrow \) 2725 |
\(\displaystyle \frac {g^2 (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \int \exp \left (f \log (F) a^2+b^2 f \log (F) \log ^2\left (c (d+e x)^n\right )+\frac {(2 a b f n \log (F)+3) \log \left (c (d+e x)^n\right )}{n}\right )d\log \left (c (d+e x)^n\right )}{e n}\) |
\(\Big \downarrow \) 2664 |
\(\displaystyle \frac {g^2 (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \exp \left (-\frac {3 (4 a b f n \log (F)+3)}{4 b^2 f n^2 \log (F)}\right ) \int \exp \left (\frac {\left (2 f \log (F) \log \left (c (d+e x)^n\right ) b^2+2 a f \log (F) b+\frac {3}{n}\right )^2}{4 b^2 f \log (F)}\right )d\log \left (c (d+e x)^n\right )}{e n}\) |
\(\Big \downarrow \) 2633 |
\(\displaystyle \frac {\sqrt {\pi } g^2 (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \exp \left (-\frac {3 (4 a b f n \log (F)+3)}{4 b^2 f n^2 \log (F)}\right ) \text {erfi}\left (\frac {2 a b f \log (F)+2 b^2 f \log (F) \log \left (c (d+e x)^n\right )+\frac {3}{n}}{2 b \sqrt {f} \sqrt {\log (F)}}\right )}{2 b e \sqrt {f} n \sqrt {\log (F)}}\) |
(g^2*Sqrt[Pi]*(d + e*x)^3*Erfi[(3/n + 2*a*b*f*Log[F] + 2*b^2*f*Log[F]*Log[ c*(d + e*x)^n])/(2*b*Sqrt[f]*Sqrt[Log[F]])])/(2*b*e*E^((3*(3 + 4*a*b*f*n*L og[F]))/(4*b^2*f*n^2*Log[F]))*Sqrt[f]*n*(c*(d + e*x)^n)^(3/n)*Sqrt[Log[F]] )
3.7.3.3.1 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]
\[\int F^{f {\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )}^{2}} \left (e g x +d g \right )^{2}d x\]
Time = 0.31 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.01 \[ \int F^{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2} (d g+e g x)^2 \, dx=-\frac {\sqrt {\pi } \sqrt {-b^{2} f n^{2} \log \left (F\right )} g^{2} \operatorname {erf}\left (\frac {{\left (2 \, b^{2} f n^{2} \log \left (e x + d\right ) \log \left (F\right ) + 2 \, b^{2} f n \log \left (F\right ) \log \left (c\right ) + 2 \, a b f n \log \left (F\right ) + 3\right )} \sqrt {-b^{2} f n^{2} \log \left (F\right )}}{2 \, b^{2} f n^{2} \log \left (F\right )}\right ) e^{\left (-\frac {3 \, {\left (4 \, b^{2} f n \log \left (F\right ) \log \left (c\right ) + 4 \, a b f n \log \left (F\right ) + 3\right )}}{4 \, b^{2} f n^{2} \log \left (F\right )}\right )}}{2 \, b e n} \]
-1/2*sqrt(pi)*sqrt(-b^2*f*n^2*log(F))*g^2*erf(1/2*(2*b^2*f*n^2*log(e*x + d )*log(F) + 2*b^2*f*n*log(F)*log(c) + 2*a*b*f*n*log(F) + 3)*sqrt(-b^2*f*n^2 *log(F))/(b^2*f*n^2*log(F)))*e^(-3/4*(4*b^2*f*n*log(F)*log(c) + 4*a*b*f*n* log(F) + 3)/(b^2*f*n^2*log(F)))/(b*e*n)
Timed out. \[ \int F^{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2} (d g+e g x)^2 \, dx=\text {Timed out} \]
\[ \int F^{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2} (d g+e g x)^2 \, dx=\int { {\left (e g x + d g\right )}^{2} F^{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} f} \,d x } \]
\[ \int F^{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2} (d g+e g x)^2 \, dx=\int { {\left (e g x + d g\right )}^{2} F^{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} f} \,d x } \]
Timed out. \[ \int F^{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2} (d g+e g x)^2 \, dx=\int {\mathrm {e}}^{f\,\ln \left (F\right )\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2}\,{\left (d\,g+e\,g\,x\right )}^2 \,d x \]