Integrand size = 28, antiderivative size = 502 \[ \int F^{f \left (a+b \log ^2\left (c (d+e x)^n\right )\right )} (g+h x)^3 \, dx=\frac {3 e^{-\frac {1}{b f n^2 \log (F)}} F^{a f} h (e g-d h)^2 \sqrt {\pi } (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text {erfi}\left (\frac {1+b f n \log (F) \log \left (c (d+e x)^n\right )}{\sqrt {b} \sqrt {f} n \sqrt {\log (F)}}\right )}{2 \sqrt {b} e^4 \sqrt {f} n \sqrt {\log (F)}}+\frac {e^{-\frac {4}{b f n^2 \log (F)}} F^{a f} h^3 \sqrt {\pi } (d+e x)^4 \left (c (d+e x)^n\right )^{-4/n} \text {erfi}\left (\frac {2+b f n \log (F) \log \left (c (d+e x)^n\right )}{\sqrt {b} \sqrt {f} n \sqrt {\log (F)}}\right )}{2 \sqrt {b} e^4 \sqrt {f} n \sqrt {\log (F)}}+\frac {e^{-\frac {1}{4 b f n^2 \log (F)}} F^{a f} (e g-d h)^3 \sqrt {\pi } (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {1+2 b f n \log (F) \log \left (c (d+e x)^n\right )}{2 \sqrt {b} \sqrt {f} n \sqrt {\log (F)}}\right )}{2 \sqrt {b} e^4 \sqrt {f} n \sqrt {\log (F)}}+\frac {3 e^{-\frac {9}{4 b f n^2 \log (F)}} F^{a f} h^2 (e g-d h) \sqrt {\pi } (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \text {erfi}\left (\frac {3+2 b f n \log (F) \log \left (c (d+e x)^n\right )}{2 \sqrt {b} \sqrt {f} n \sqrt {\log (F)}}\right )}{2 \sqrt {b} e^4 \sqrt {f} n \sqrt {\log (F)}} \]
3/2*F^(a*f)*h*(-d*h+e*g)^2*(e*x+d)^2*erfi((1+b*f*n*ln(F)*ln(c*(e*x+d)^n))/ n/b^(1/2)/f^(1/2)/ln(F)^(1/2))*Pi^(1/2)/e^4/exp(1/b/f/n^2/ln(F))/n/((c*(e* x+d)^n)^(2/n))/b^(1/2)/f^(1/2)/ln(F)^(1/2)+1/2*F^(a*f)*h^3*(e*x+d)^4*erfi( (2+b*f*n*ln(F)*ln(c*(e*x+d)^n))/n/b^(1/2)/f^(1/2)/ln(F)^(1/2))*Pi^(1/2)/e^ 4/exp(4/b/f/n^2/ln(F))/n/((c*(e*x+d)^n)^(4/n))/b^(1/2)/f^(1/2)/ln(F)^(1/2) +1/2*F^(a*f)*(-d*h+e*g)^3*(e*x+d)*erfi(1/2*(1+2*b*f*n*ln(F)*ln(c*(e*x+d)^n ))/n/b^(1/2)/f^(1/2)/ln(F)^(1/2))*Pi^(1/2)/e^4/exp(1/4/b/f/n^2/ln(F))/n/(( c*(e*x+d)^n)^(1/n))/b^(1/2)/f^(1/2)/ln(F)^(1/2)+3/2*F^(a*f)*h^2*(-d*h+e*g) *(e*x+d)^3*erfi(1/2*(3+2*b*f*n*ln(F)*ln(c*(e*x+d)^n))/n/b^(1/2)/f^(1/2)/ln (F)^(1/2))*Pi^(1/2)/e^4/exp(9/4/b/f/n^2/ln(F))/n/((c*(e*x+d)^n)^(3/n))/b^( 1/2)/f^(1/2)/ln(F)^(1/2)
Time = 1.53 (sec) , antiderivative size = 396, normalized size of antiderivative = 0.79 \[ \int F^{f \left (a+b \log ^2\left (c (d+e x)^n\right )\right )} (g+h x)^3 \, dx=\frac {e^{-\frac {4}{b f n^2 \log (F)}} F^{a f} \sqrt {\pi } (d+e x) \left (c (d+e x)^n\right )^{-4/n} \left (3 e^{\frac {3}{b f n^2 \log (F)}} h (e g-d h)^2 (d+e x) \left (c (d+e x)^n\right )^{2/n} \text {erfi}\left (\frac {1+b f n \log (F) \log \left (c (d+e x)^n\right )}{\sqrt {b} \sqrt {f} n \sqrt {\log (F)}}\right )+h^3 (d+e x)^3 \text {erfi}\left (\frac {2+b f n \log (F) \log \left (c (d+e x)^n\right )}{\sqrt {b} \sqrt {f} n \sqrt {\log (F)}}\right )+e^{\frac {7}{4 b f n^2 \log (F)}} (e g-d h) \left (c (d+e x)^n\right )^{\frac {1}{n}} \left (e^{\frac {2}{b f n^2 \log (F)}} (e g-d h)^2 \left (c (d+e x)^n\right )^{2/n} \text {erfi}\left (\frac {1+2 b f n \log (F) \log \left (c (d+e x)^n\right )}{2 \sqrt {b} \sqrt {f} n \sqrt {\log (F)}}\right )+3 h^2 (d+e x)^2 \text {erfi}\left (\frac {3+2 b f n \log (F) \log \left (c (d+e x)^n\right )}{2 \sqrt {b} \sqrt {f} n \sqrt {\log (F)}}\right )\right )\right )}{2 \sqrt {b} e^4 \sqrt {f} n \sqrt {\log (F)}} \]
(F^(a*f)*Sqrt[Pi]*(d + e*x)*(3*E^(3/(b*f*n^2*Log[F]))*h*(e*g - d*h)^2*(d + e*x)*(c*(d + e*x)^n)^(2/n)*Erfi[(1 + b*f*n*Log[F]*Log[c*(d + e*x)^n])/(Sq rt[b]*Sqrt[f]*n*Sqrt[Log[F]])] + h^3*(d + e*x)^3*Erfi[(2 + b*f*n*Log[F]*Lo g[c*(d + e*x)^n])/(Sqrt[b]*Sqrt[f]*n*Sqrt[Log[F]])] + E^(7/(4*b*f*n^2*Log[ F]))*(e*g - d*h)*(c*(d + e*x)^n)^n^(-1)*(E^(2/(b*f*n^2*Log[F]))*(e*g - d*h )^2*(c*(d + e*x)^n)^(2/n)*Erfi[(1 + 2*b*f*n*Log[F]*Log[c*(d + e*x)^n])/(2* Sqrt[b]*Sqrt[f]*n*Sqrt[Log[F]])] + 3*h^2*(d + e*x)^2*Erfi[(3 + 2*b*f*n*Log [F]*Log[c*(d + e*x)^n])/(2*Sqrt[b]*Sqrt[f]*n*Sqrt[Log[F]])])))/(2*Sqrt[b]* e^4*E^(4/(b*f*n^2*Log[F]))*Sqrt[f]*n*(c*(d + e*x)^n)^(4/n)*Sqrt[Log[F]])
Time = 0.97 (sec) , antiderivative size = 494, normalized size of antiderivative = 0.98, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2707, 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.
| \(\displaystyle \int (g+h x)^3 F^{f \left (a+b \log ^2\left (c (d+e x)^n\right )\right )} \, dx\) |
| \(\Big \downarrow \) 2707 |
| \(\displaystyle \frac {\int \left ((e g-d h)^3 F^{f \left (b \log ^2\left (c (d+e x)^n\right )+a\right )}+h^3 (d+e x)^3 F^{f \left (b \log ^2\left (c (d+e x)^n\right )+a\right )}+3 h^2 (e g-d h) (d+e x)^2 F^{f \left (b \log ^2\left (c (d+e x)^n\right )+a\right )}+3 h (e g-d h)^2 (d+e x) F^{f \left (b \log ^2\left (c (d+e x)^n\right )+a\right )}\right )d(d+e x)}{e^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {3 \sqrt {\pi } h^2 F^{a f} (d+e x)^3 (e g-d h) e^{-\frac {9}{4 b f n^2 \log (F)}} \left (c (d+e x)^n\right )^{-3/n} \text {erfi}\left (\frac {2 b f n \log (F) \log \left (c (d+e x)^n\right )+3}{2 \sqrt {b} \sqrt {f} n \sqrt {\log (F)}}\right )}{2 \sqrt {b} \sqrt {f} n \sqrt {\log (F)}}+\frac {3 \sqrt {\pi } h F^{a f} (d+e x)^2 (e g-d h)^2 e^{-\frac {1}{b f n^2 \log (F)}} \left (c (d+e x)^n\right )^{-2/n} \text {erfi}\left (\frac {b f n \log (F) \log \left (c (d+e x)^n\right )+1}{\sqrt {b} \sqrt {f} n \sqrt {\log (F)}}\right )}{2 \sqrt {b} \sqrt {f} n \sqrt {\log (F)}}+\frac {\sqrt {\pi } F^{a f} (d+e x) (e g-d h)^3 e^{-\frac {1}{4 b f n^2 \log (F)}} \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {2 b f n \log (F) \log \left (c (d+e x)^n\right )+1}{2 \sqrt {b} \sqrt {f} n \sqrt {\log (F)}}\right )}{2 \sqrt {b} \sqrt {f} n \sqrt {\log (F)}}+\frac {\sqrt {\pi } h^3 F^{a f} (d+e x)^4 e^{-\frac {4}{b f n^2 \log (F)}} \left (c (d+e x)^n\right )^{-4/n} \text {erfi}\left (\frac {b f n \log (F) \log \left (c (d+e x)^n\right )+2}{\sqrt {b} \sqrt {f} n \sqrt {\log (F)}}\right )}{2 \sqrt {b} \sqrt {f} n \sqrt {\log (F)}}}{e^4}\) |
((3*F^(a*f)*h*(e*g - d*h)^2*Sqrt[Pi]*(d + e*x)^2*Erfi[(1 + b*f*n*Log[F]*Lo g[c*(d + e*x)^n])/(Sqrt[b]*Sqrt[f]*n*Sqrt[Log[F]])])/(2*Sqrt[b]*E^(1/(b*f* n^2*Log[F]))*Sqrt[f]*n*(c*(d + e*x)^n)^(2/n)*Sqrt[Log[F]]) + (F^(a*f)*h^3* Sqrt[Pi]*(d + e*x)^4*Erfi[(2 + b*f*n*Log[F]*Log[c*(d + e*x)^n])/(Sqrt[b]*S qrt[f]*n*Sqrt[Log[F]])])/(2*Sqrt[b]*E^(4/(b*f*n^2*Log[F]))*Sqrt[f]*n*(c*(d + e*x)^n)^(4/n)*Sqrt[Log[F]]) + (F^(a*f)*(e*g - d*h)^3*Sqrt[Pi]*(d + e*x) *Erfi[(1 + 2*b*f*n*Log[F]*Log[c*(d + e*x)^n])/(2*Sqrt[b]*Sqrt[f]*n*Sqrt[Lo g[F]])])/(2*Sqrt[b]*E^(1/(4*b*f*n^2*Log[F]))*Sqrt[f]*n*(c*(d + e*x)^n)^n^( -1)*Sqrt[Log[F]]) + (3*F^(a*f)*h^2*(e*g - d*h)*Sqrt[Pi]*(d + e*x)^3*Erfi[( 3 + 2*b*f*n*Log[F]*Log[c*(d + e*x)^n])/(2*Sqrt[b]*Sqrt[f]*n*Sqrt[Log[F]])] )/(2*Sqrt[b]*E^(9/(4*b*f*n^2*Log[F]))*Sqrt[f]*n*(c*(d + e*x)^n)^(3/n)*Sqrt [Log[F]]))/e^4
3.6.95.3.1 Defintions of rubi rules used
Int[(F_)^(((a_.) + Log[(c_.)*((d_.) + (e_.)*(x_))^(n_.)]^2*(b_.))*(f_.))*(( g_.) + (h_.)*(x_))^(m_.), x_Symbol] :> Simp[1/e^(m + 1) Subst[Int[ExpandI ntegrand[F^(f*(a + b*Log[c*x^n]^2)), (e*g - d*h + h*x)^m, x], x], x, d + e* x], x] /; FreeQ[{F, a, b, c, d, e, f, g, h, n}, x] && IGtQ[m, 0]
\[\int F^{f \left (a +b \ln \left (c \left (e x +d \right )^{n}\right )^{2}\right )} \left (h x +g \right )^{3}d x\]
Time = 0.29 (sec) , antiderivative size = 513, normalized size of antiderivative = 1.02 \[ \int F^{f \left (a+b \log ^2\left (c (d+e x)^n\right )\right )} (g+h x)^3 \, dx=-\frac {\sqrt {\pi } \sqrt {-b f n^{2} \log \left (F\right )} h^{3} \operatorname {erf}\left (\frac {{\left (b f n^{2} \log \left (e x + d\right ) \log \left (F\right ) + b f n \log \left (F\right ) \log \left (c\right ) + 2\right )} \sqrt {-b f n^{2} \log \left (F\right )}}{b f n^{2} \log \left (F\right )}\right ) e^{\left (\frac {a b f^{2} n^{2} \log \left (F\right )^{2} - 4 \, b f n \log \left (F\right ) \log \left (c\right ) - 4}{b f n^{2} \log \left (F\right )}\right )} + \sqrt {\pi } {\left (e^{3} g^{3} - 3 \, d e^{2} g^{2} h + 3 \, d^{2} e g h^{2} - d^{3} h^{3}\right )} \sqrt {-b f n^{2} \log \left (F\right )} \operatorname {erf}\left (\frac {{\left (2 \, b f n^{2} \log \left (e x + d\right ) \log \left (F\right ) + 2 \, b f n \log \left (F\right ) \log \left (c\right ) + 1\right )} \sqrt {-b f n^{2} \log \left (F\right )}}{2 \, b f n^{2} \log \left (F\right )}\right ) e^{\left (\frac {4 \, a b f^{2} n^{2} \log \left (F\right )^{2} - 4 \, b f n \log \left (F\right ) \log \left (c\right ) - 1}{4 \, b f n^{2} \log \left (F\right )}\right )} + 3 \, \sqrt {\pi } \sqrt {-b f n^{2} \log \left (F\right )} {\left (e g h^{2} - d h^{3}\right )} \operatorname {erf}\left (\frac {{\left (2 \, b f n^{2} \log \left (e x + d\right ) \log \left (F\right ) + 2 \, b f n \log \left (F\right ) \log \left (c\right ) + 3\right )} \sqrt {-b f n^{2} \log \left (F\right )}}{2 \, b f n^{2} \log \left (F\right )}\right ) e^{\left (\frac {4 \, a b f^{2} n^{2} \log \left (F\right )^{2} - 12 \, b f n \log \left (F\right ) \log \left (c\right ) - 9}{4 \, b f n^{2} \log \left (F\right )}\right )} + 3 \, \sqrt {\pi } {\left (e^{2} g^{2} h - 2 \, d e g h^{2} + d^{2} h^{3}\right )} \sqrt {-b f n^{2} \log \left (F\right )} \operatorname {erf}\left (\frac {{\left (b f n^{2} \log \left (e x + d\right ) \log \left (F\right ) + b f n \log \left (F\right ) \log \left (c\right ) + 1\right )} \sqrt {-b f n^{2} \log \left (F\right )}}{b f n^{2} \log \left (F\right )}\right ) e^{\left (\frac {a b f^{2} n^{2} \log \left (F\right )^{2} - 2 \, b f n \log \left (F\right ) \log \left (c\right ) - 1}{b f n^{2} \log \left (F\right )}\right )}}{2 \, e^{4} n} \]
-1/2*(sqrt(pi)*sqrt(-b*f*n^2*log(F))*h^3*erf((b*f*n^2*log(e*x + d)*log(F) + b*f*n*log(F)*log(c) + 2)*sqrt(-b*f*n^2*log(F))/(b*f*n^2*log(F)))*e^((a*b *f^2*n^2*log(F)^2 - 4*b*f*n*log(F)*log(c) - 4)/(b*f*n^2*log(F))) + sqrt(pi )*(e^3*g^3 - 3*d*e^2*g^2*h + 3*d^2*e*g*h^2 - d^3*h^3)*sqrt(-b*f*n^2*log(F) )*erf(1/2*(2*b*f*n^2*log(e*x + d)*log(F) + 2*b*f*n*log(F)*log(c) + 1)*sqrt (-b*f*n^2*log(F))/(b*f*n^2*log(F)))*e^(1/4*(4*a*b*f^2*n^2*log(F)^2 - 4*b*f *n*log(F)*log(c) - 1)/(b*f*n^2*log(F))) + 3*sqrt(pi)*sqrt(-b*f*n^2*log(F)) *(e*g*h^2 - d*h^3)*erf(1/2*(2*b*f*n^2*log(e*x + d)*log(F) + 2*b*f*n*log(F) *log(c) + 3)*sqrt(-b*f*n^2*log(F))/(b*f*n^2*log(F)))*e^(1/4*(4*a*b*f^2*n^2 *log(F)^2 - 12*b*f*n*log(F)*log(c) - 9)/(b*f*n^2*log(F))) + 3*sqrt(pi)*(e^ 2*g^2*h - 2*d*e*g*h^2 + d^2*h^3)*sqrt(-b*f*n^2*log(F))*erf((b*f*n^2*log(e* x + d)*log(F) + b*f*n*log(F)*log(c) + 1)*sqrt(-b*f*n^2*log(F))/(b*f*n^2*lo g(F)))*e^((a*b*f^2*n^2*log(F)^2 - 2*b*f*n*log(F)*log(c) - 1)/(b*f*n^2*log( F))))/(e^4*n)
Timed out. \[ \int F^{f \left (a+b \log ^2\left (c (d+e x)^n\right )\right )} (g+h x)^3 \, dx=\text {Timed out} \]
\[ \int F^{f \left (a+b \log ^2\left (c (d+e x)^n\right )\right )} (g+h x)^3 \, dx=\int { {\left (h x + g\right )}^{3} F^{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + a\right )} f} \,d x } \]
\[ \int F^{f \left (a+b \log ^2\left (c (d+e x)^n\right )\right )} (g+h x)^3 \, dx=\int { {\left (h x + g\right )}^{3} F^{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + a\right )} f} \,d x } \]
Timed out. \[ \int F^{f \left (a+b \log ^2\left (c (d+e x)^n\right )\right )} (g+h x)^3 \, dx=\int {\mathrm {e}}^{f\,\ln \left (F\right )\,\left (b\,{\ln \left (c\,{\left (d+e\,x\right )}^n\right )}^2+a\right )}\,{\left (g+h\,x\right )}^3 \,d x \]