3.603 \(\int F^{f (a+b \log (c (d+e x)^n))^2} (d g+e g x)^2 \, dx\)

Optimal. Leaf size=133 \[ \frac{\sqrt{\pi } g^2 (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \exp \left (-\frac{3 (4 a b f n \log (F)+3)}{4 b^2 f n^2 \log (F)}\right ) \text{Erfi}\left (\frac{2 a b f \log (F)+2 b^2 f \log (F) \log \left (c (d+e x)^n\right )+\frac{3}{n}}{2 b \sqrt{f} \sqrt{\log (F)}}\right )}{2 b e \sqrt{f} n \sqrt{\log (F)}} \]

[Out]

(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[Lo
g[F]])])/(2*b*e*E^((3*(3 + 4*a*b*f*n*Log[F]))/(4*b^2*f*n^2*Log[F]))*Sqrt[f]*n*(c*(d + e*x)^n)^(3/n)*Sqrt[Log[F
]])

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Rubi [A]  time = 0.410737, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.226, Rules used = {12, 2278, 2274, 15, 2276, 2234, 2204} \[ \frac{\sqrt{\pi } g^2 (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \exp \left (-\frac{3 (4 a b f n \log (F)+3)}{4 b^2 f n^2 \log (F)}\right ) \text{Erfi}\left (\frac{2 a b f \log (F)+2 b^2 f \log (F) \log \left (c (d+e x)^n\right )+\frac{3}{n}}{2 b \sqrt{f} \sqrt{\log (F)}}\right )}{2 b e \sqrt{f} n \sqrt{\log (F)}} \]

Antiderivative was successfully verified.

[In]

Int[F^(f*(a + b*Log[c*(d + e*x)^n])^2)*(d*g + e*g*x)^2,x]

[Out]

(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[Lo
g[F]])])/(2*b*e*E^((3*(3 + 4*a*b*f*n*Log[F]))/(4*b^2*f*n^2*Log[F]))*Sqrt[f]*n*(c*(d + e*x)^n)^(3/n)*Sqrt[Log[F
]])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2278

Int[(F_)^(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^2*(d_.))*((e_.)*(x_))^(m_.), x_Symbol] :> Int[(e*x)^m*F^(a^2*d
+ 2*a*b*d*Log[c*x^n] + b^2*d*Log[c*x^n]^2), x] /; FreeQ[{F, a, b, c, d, e, m, n}, x]

Rule 2274

Int[(u_.)*(F_)^((a_.)*(Log[z_]*(b_.) + (v_.))), x_Symbol] :> Int[u*F^(a*v)*z^(a*b*Log[F]), x] /; FreeQ[{F, a,
b}, x]

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 2276

Int[(F_)^(((a_.) + Log[(c_.)*(x_)^(n_.)]^2*(b_.))*(d_.))*((e_.)*(x_))^(m_.), x_Symbol] :> Dist[(e*x)^(m + 1)/(
e*n*(c*x^n)^((m + 1)/n)), Subst[Int[E^(a*d*Log[F] + ((m + 1)*x)/n + b*d*Log[F]*x^2), x], x, Log[c*x^n]], x] /;
FreeQ[{F, a, b, c, d, e, m, n}, x]

Rule 2234

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[F^(a - b^2/(4*c)), Int[F^((b + 2*c*x)^2/(4*c))
, x], x] /; FreeQ[{F, a, b, c}, x]

Rule 2204

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]

Rubi steps

\begin{align*} \int F^{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2} (d g+e g x)^2 \, dx &=\frac{\operatorname{Subst}\left (\int F^{f \left (a+b \log \left (c x^n\right )\right )^2} g^2 x^2 \, dx,x,d+e x\right )}{e}\\ &=\frac{g^2 \operatorname{Subst}\left (\int F^{f \left (a+b \log \left (c x^n\right )\right )^2} x^2 \, dx,x,d+e x\right )}{e}\\ &=\frac{g^2 \operatorname{Subst}\left (\int F^{a^2 f+2 a b f \log \left (c x^n\right )+b^2 f \log ^2\left (c x^n\right )} x^2 \, dx,x,d+e x\right )}{e}\\ &=\frac{g^2 \operatorname{Subst}\left (\int F^{a^2 f+b^2 f \log ^2\left (c x^n\right )} x^2 \left (c x^n\right )^{2 a b f \log (F)} \, dx,x,d+e x\right )}{e}\\ &=\frac{\left (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)}\right ) \operatorname{Subst}\left (\int F^{a^2 f+b^2 f \log ^2\left (c x^n\right )} x^{2+2 a b f n \log (F)} \, dx,x,d+e x\right )}{e}\\ &=\frac{\left (g^2 (d+e x)^3 \left (c (d+e x)^n\right )^{2 a b f \log (F)-\frac{3+2 a b f n \log (F)}{n}}\right ) \operatorname{Subst}\left (\int \exp \left (a^2 f \log (F)+b^2 f x^2 \log (F)+\frac{x (3+2 a b f n \log (F))}{n}\right ) \, dx,x,\log \left (c (d+e x)^n\right )\right )}{e n}\\ &=\frac{\left (\exp \left (a^2 f \log (F)-\frac{(3+2 a b f n \log (F))^2}{4 b^2 f n^2 \log (F)}\right ) g^2 (d+e x)^3 \left (c (d+e x)^n\right )^{2 a b f \log (F)-\frac{3+2 a b f n \log (F)}{n}}\right ) \operatorname{Subst}\left (\int \exp \left (\frac{\left (2 b^2 f x \log (F)+\frac{3+2 a b f n \log (F)}{n}\right )^2}{4 b^2 f \log (F)}\right ) \, dx,x,\log \left (c (d+e x)^n\right )\right )}{e n}\\ &=\frac{\exp \left (-\frac{3 (3+4 a b f n \log (F))}{4 b^2 f n^2 \log (F)}\right ) 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)}}\\ \end{align*}

Mathematica [A]  time = 0.34262, size = 129, normalized size = 0.97 \[ \frac{\sqrt{\pi } g^2 \text{Erfi}\left (\frac{2 b f n \log (F) \left (a+b \log \left (c (d+e x)^n\right )\right )+3}{2 b \sqrt{f} n \sqrt{\log (F)}}\right ) \exp \left (-\frac{3 \left (4 b f n \log (F) \left (a+b \left (\log \left (c (d+e x)^n\right )-n \log (d+e x)\right )\right )+3\right )}{4 b^2 f n^2 \log (F)}\right )}{2 b e \sqrt{f} n \sqrt{\log (F)}} \]

Antiderivative was successfully verified.

[In]

Integrate[F^(f*(a + b*Log[c*(d + e*x)^n])^2)*(d*g + e*g*x)^2,x]

[Out]

(g^2*Sqrt[Pi]*Erfi[(3 + 2*b*f*n*Log[F]*(a + b*Log[c*(d + e*x)^n]))/(2*b*Sqrt[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[L
og[F]])

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Maple [F]  time = 0.441, size = 0, normalized size = 0. \begin{align*} \int{F}^{f \left ( a+b\ln \left ( c \left ( ex+d \right ) ^{n} \right ) \right ) ^{2}} \left ( egx+dg \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(f*(a+b*ln(c*(e*x+d)^n))^2)*(e*g*x+d*g)^2,x)

[Out]

int(F^(f*(a+b*ln(c*(e*x+d)^n))^2)*(e*g*x+d*g)^2,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(f*(a+b*log(c*(e*x+d)^n))^2)*(e*g*x+d*g)^2,x, algorithm="maxima")

[Out]

integrate((e*g*x + d*g)^2*F^((b*log((e*x + d)^n*c) + a)^2*f), x)

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Fricas [A]  time = 1.0033, size = 355, normalized size = 2.67 \begin{align*} -\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(f*(a+b*log(c*(e*x+d)^n))^2)*(e*g*x+d*g)^2,x, algorithm="fricas")

[Out]

-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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F**(f*(a+b*ln(c*(e*x+d)**n))**2)*(e*g*x+d*g)**2,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(f*(a+b*log(c*(e*x+d)^n))^2)*(e*g*x+d*g)^2,x, algorithm="giac")

[Out]

integrate((e*g*x + d*g)^2*F^((b*log((e*x + d)^n*c) + a)^2*f), x)