\(\int F^{f (a+b \log ^2(c (d+e x)^n))} (g+h x)^2 \, dx\) [519]

Optimal result

Integrand size = 28, antiderivative size = 372 \[ \int F^{f \left (a+b \log ^2\left (c (d+e x)^n\right )\right )} (g+h x)^2 \, dx=\frac {e^{-\frac {1}{b f n^2 \log (F)}} F^{a f} h (e g-d h) \sqrt {\pi } (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text {erfi}\left (\frac {1+b f n \log (F) \log \left (c (d+e x)^n\right )}{\sqrt {b} \sqrt {f} n \sqrt {\log (F)}}\right )}{\sqrt {b} e^3 \sqrt {f} n \sqrt {\log (F)}}+\frac {e^{-\frac {1}{4 b f n^2 \log (F)}} F^{a f} (e g-d h)^2 \sqrt {\pi } (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {1+2 b f n \log (F) \log \left (c (d+e x)^n\right )}{2 \sqrt {b} \sqrt {f} n \sqrt {\log (F)}}\right )}{2 \sqrt {b} e^3 \sqrt {f} n \sqrt {\log (F)}}+\frac {e^{-\frac {9}{4 b f n^2 \log (F)}} F^{a f} h^2 \sqrt {\pi } (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \text {erfi}\left (\frac {3+2 b f n \log (F) \log \left (c (d+e x)^n\right )}{2 \sqrt {b} \sqrt {f} n \sqrt {\log (F)}}\right )}{2 \sqrt {b} e^3 \sqrt {f} n \sqrt {\log (F)}} \] Output:

F^(a*f)*h*(-d*h+e*g)*Pi^(1/2)*(e*x+d)^2*erfi((1+b*f*n*ln(F)*ln(c*(e*x+d)^n 
))/b^(1/2)/f^(1/2)/n/ln(F)^(1/2))/b^(1/2)/e^3/exp(1/b/f/n^2/ln(F))/f^(1/2) 
/n/((c*(e*x+d)^n)^(2/n))/ln(F)^(1/2)+1/2*F^(a*f)*(-d*h+e*g)^2*Pi^(1/2)*(e* 
x+d)*erfi(1/2*(1+2*b*f*n*ln(F)*ln(c*(e*x+d)^n))/b^(1/2)/f^(1/2)/n/ln(F)^(1 
/2))/b^(1/2)/e^3/exp(1/4/b/f/n^2/ln(F))/f^(1/2)/n/((c*(e*x+d)^n)^(1/n))/ln 
(F)^(1/2)+1/2*F^(a*f)*h^2*Pi^(1/2)*(e*x+d)^3*erfi(1/2*(3+2*b*f*n*ln(F)*ln( 
c*(e*x+d)^n))/b^(1/2)/f^(1/2)/n/ln(F)^(1/2))/b^(1/2)/e^3/exp(9/4/b/f/n^2/l 
n(F))/f^(1/2)/n/((c*(e*x+d)^n)^(3/n))/ln(F)^(1/2)
 

Mathematica [A] (verified)

Time = 0.63 (sec) , antiderivative size = 303, normalized size of antiderivative = 0.81 \[ \int F^{f \left (a+b \log ^2\left (c (d+e x)^n\right )\right )} (g+h x)^2 \, dx=\frac {e^{-\frac {9}{4 b f n^2 \log (F)}} F^{a f} \sqrt {\pi } (d+e x) \left (c (d+e x)^n\right )^{-3/n} \left (-2 e^{\frac {5}{4 b f n^2 \log (F)}} h (-e g+d h) (d+e x) \left (c (d+e x)^n\right )^{\frac {1}{n}} \text {erfi}\left (\frac {1+b f n \log (F) \log \left (c (d+e x)^n\right )}{\sqrt {b} \sqrt {f} n \sqrt {\log (F)}}\right )+e^{\frac {2}{b f n^2 \log (F)}} (e g-d h)^2 \left (c (d+e x)^n\right )^{2/n} \text {erfi}\left (\frac {1+2 b f n \log (F) \log \left (c (d+e x)^n\right )}{2 \sqrt {b} \sqrt {f} n \sqrt {\log (F)}}\right )+h^2 (d+e x)^2 \text {erfi}\left (\frac {3+2 b f n \log (F) \log \left (c (d+e x)^n\right )}{2 \sqrt {b} \sqrt {f} n \sqrt {\log (F)}}\right )\right )}{2 \sqrt {b} e^3 \sqrt {f} n \sqrt {\log (F)}} \] Input:

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

Output:

(F^(a*f)*Sqrt[Pi]*(d + e*x)*(-2*E^(5/(4*b*f*n^2*Log[F]))*h*(-(e*g) + d*h)* 
(d + e*x)*(c*(d + e*x)^n)^n^(-1)*Erfi[(1 + b*f*n*Log[F]*Log[c*(d + e*x)^n] 
)/(Sqrt[b]*Sqrt[f]*n*Sqrt[Log[F]])] + 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*Sqr 
t[b]*Sqrt[f]*n*Sqrt[Log[F]])] + h^2*(d + e*x)^2*Erfi[(3 + 2*b*f*n*Log[F]*L 
og[c*(d + e*x)^n])/(2*Sqrt[b]*Sqrt[f]*n*Sqrt[Log[F]])]))/(2*Sqrt[b]*e^3*E^ 
(9/(4*b*f*n^2*Log[F]))*Sqrt[f]*n*(c*(d + e*x)^n)^(3/n)*Sqrt[Log[F]])
 

Rubi [A] (verified)

Time = 1.07 (sec) , antiderivative size = 367, normalized size of antiderivative = 0.99, 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.

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

\(\Big \downarrow \) 2707

\(\displaystyle \frac {\int \left ((e g-d h)^2 F^{f \left (b \log ^2\left (c (d+e x)^n\right )+a\right )}+h^2 (d+e x)^2 F^{f \left (b \log ^2\left (c (d+e x)^n\right )+a\right )}+2 h (e g-d h) (d+e x) F^{f \left (b \log ^2\left (c (d+e x)^n\right )+a\right )}\right )d(d+e x)}{e^3}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {\sqrt {\pi } h F^{a f} (d+e x)^2 (e g-d h) e^{-\frac {1}{b f n^2 \log (F)}} \left (c (d+e x)^n\right )^{-2/n} \text {erfi}\left (\frac {b f n \log (F) \log \left (c (d+e x)^n\right )+1}{\sqrt {b} \sqrt {f} n \sqrt {\log (F)}}\right )}{\sqrt {b} \sqrt {f} n \sqrt {\log (F)}}+\frac {\sqrt {\pi } F^{a f} (d+e x) (e g-d h)^2 e^{-\frac {1}{4 b f n^2 \log (F)}} \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {2 b f n \log (F) \log \left (c (d+e x)^n\right )+1}{2 \sqrt {b} \sqrt {f} n \sqrt {\log (F)}}\right )}{2 \sqrt {b} \sqrt {f} n \sqrt {\log (F)}}+\frac {\sqrt {\pi } h^2 F^{a f} (d+e x)^3 e^{-\frac {9}{4 b f n^2 \log (F)}} \left (c (d+e x)^n\right )^{-3/n} \text {erfi}\left (\frac {2 b f n \log (F) \log \left (c (d+e x)^n\right )+3}{2 \sqrt {b} \sqrt {f} n \sqrt {\log (F)}}\right )}{2 \sqrt {b} \sqrt {f} n \sqrt {\log (F)}}}{e^3}\)

Input:

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

Output:

((F^(a*f)*h*(e*g - d*h)*Sqrt[Pi]*(d + e*x)^2*Erfi[(1 + b*f*n*Log[F]*Log[c* 
(d + e*x)^n])/(Sqrt[b]*Sqrt[f]*n*Sqrt[Log[F]])])/(Sqrt[b]*E^(1/(b*f*n^2*Lo 
g[F]))*Sqrt[f]*n*(c*(d + e*x)^n)^(2/n)*Sqrt[Log[F]]) + (F^(a*f)*(e*g - d*h 
)^2*Sqrt[Pi]*(d + e*x)*Erfi[(1 + 2*b*f*n*Log[F]*Log[c*(d + e*x)^n])/(2*Sqr 
t[b]*Sqrt[f]*n*Sqrt[Log[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]]) + (F^(a*f)*h^2*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[L 
og[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^3
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2707

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]
 

Maple [F(-1)]

Timed out.

hanged

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 367, normalized size of antiderivative = 0.99 \[ \int F^{f \left (a+b \log ^2\left (c (d+e x)^n\right )\right )} (g+h x)^2 \, dx=-\frac {\sqrt {\pi } \sqrt {-b f n^{2} \log \left (F\right )} h^{2} \operatorname {erf}\left (\frac {{\left (2 \, b f n^{2} \log \left (e x + d\right ) \log \left (F\right ) + 2 \, b f n \log \left (F\right ) \log \left (c\right ) + 3\right )} \sqrt {-b f n^{2} \log \left (F\right )}}{2 \, b f n^{2} \log \left (F\right )}\right ) e^{\left (\frac {4 \, a b f^{2} n^{2} \log \left (F\right )^{2} - 12 \, b f n \log \left (F\right ) \log \left (c\right ) - 9}{4 \, b f n^{2} \log \left (F\right )}\right )} + \sqrt {\pi } \sqrt {-b f n^{2} \log \left (F\right )} {\left (e^{2} g^{2} - 2 \, d e g h + d^{2} h^{2}\right )} \operatorname {erf}\left (\frac {{\left (2 \, b f n^{2} \log \left (e x + d\right ) \log \left (F\right ) + 2 \, b f n \log \left (F\right ) \log \left (c\right ) + 1\right )} \sqrt {-b f n^{2} \log \left (F\right )}}{2 \, b f n^{2} \log \left (F\right )}\right ) e^{\left (\frac {4 \, a b f^{2} n^{2} \log \left (F\right )^{2} - 4 \, b f n \log \left (F\right ) \log \left (c\right ) - 1}{4 \, b f n^{2} \log \left (F\right )}\right )} + 2 \, \sqrt {\pi } \sqrt {-b f n^{2} \log \left (F\right )} {\left (e g h - d h^{2}\right )} \operatorname {erf}\left (\frac {{\left (b f n^{2} \log \left (e x + d\right ) \log \left (F\right ) + b f n \log \left (F\right ) \log \left (c\right ) + 1\right )} \sqrt {-b f n^{2} \log \left (F\right )}}{b f n^{2} \log \left (F\right )}\right ) e^{\left (\frac {a b f^{2} n^{2} \log \left (F\right )^{2} - 2 \, b f n \log \left (F\right ) \log \left (c\right ) - 1}{b f n^{2} \log \left (F\right )}\right )}}{2 \, e^{3} n} \] Input:

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

Output:

-1/2*(sqrt(pi)*sqrt(-b*f*n^2*log(F))*h^2*erf(1/2*(2*b*f*n^2*log(e*x + d)*l 
og(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))) + sqrt(pi)*sqrt(-b*f*n^2*log(F))*(e^2*g^2 - 2*d*e*g*h + d^2*h^2)*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*lo 
g(F)*log(c) - 1)/(b*f*n^2*log(F))) + 2*sqrt(pi)*sqrt(-b*f*n^2*log(F))*(e*g 
*h - d*h^2)*erf((b*f*n^2*log(e*x + d)*log(F) + b*f*n*log(F)*log(c) + 1)*sq 
rt(-b*f*n^2*log(F))/(b*f*n^2*log(F)))*e^((a*b*f^2*n^2*log(F)^2 - 2*b*f*n*l 
og(F)*log(c) - 1)/(b*f*n^2*log(F))))/(e^3*n)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1027 vs. \(2 (343) = 686\).

Time = 105.67 (sec) , antiderivative size = 1027, normalized size of antiderivative = 2.76 \[ \int F^{f \left (a+b \log ^2\left (c (d+e x)^n\right )\right )} (g+h x)^2 \, dx=\text {Too large to display} \] Input:

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

Output:

Piecewise((-11*F**(a*f + b*f*log(c*(d + e*x)**n)**2)*b*d**3*f*h**2*n**2*lo 
g(F)/(9*e**3) - 11*F**(a*f + b*f*log(c*(d + e*x)**n)**2)*b*d**3*f*h**2*n*l 
og(F)*log(c*(d + e*x)**n)/(9*e**3) + 3*F**(a*f + b*f*log(c*(d + e*x)**n)** 
2)*b*d**2*f*g*h*n**2*log(F)/e**2 + 3*F**(a*f + b*f*log(c*(d + e*x)**n)**2) 
*b*d**2*f*g*h*n*log(F)*log(c*(d + e*x)**n)/e**2 + 11*F**(a*f + b*f*log(c*( 
d + e*x)**n)**2)*b*d**2*f*h**2*n**2*x*log(F)/(9*e**2) - 2*F**(a*f + b*f*lo 
g(c*(d + e*x)**n)**2)*b*d**2*f*h**2*n*x*log(F)*log(c*(d + e*x)**n)/(3*e**2 
) - 2*F**(a*f + b*f*log(c*(d + e*x)**n)**2)*b*d*f*g**2*n**2*log(F)/e - 2*F 
**(a*f + b*f*log(c*(d + e*x)**n)**2)*b*d*f*g**2*n*log(F)*log(c*(d + e*x)** 
n)/e - 3*F**(a*f + b*f*log(c*(d + e*x)**n)**2)*b*d*f*g*h*n**2*x*log(F)/e + 
 2*F**(a*f + b*f*log(c*(d + e*x)**n)**2)*b*d*f*g*h*n*x*log(F)*log(c*(d + e 
*x)**n)/e - 5*F**(a*f + b*f*log(c*(d + e*x)**n)**2)*b*d*f*h**2*n**2*x**2*l 
og(F)/(18*e) + F**(a*f + b*f*log(c*(d + e*x)**n)**2)*b*d*f*h**2*n*x**2*log 
(F)*log(c*(d + e*x)**n)/(3*e) + 2*F**(a*f + b*f*log(c*(d + e*x)**n)**2)*b* 
f*g**2*n**2*x*log(F) - 2*F**(a*f + b*f*log(c*(d + e*x)**n)**2)*b*f*g**2*n* 
x*log(F)*log(c*(d + e*x)**n) + F**(a*f + b*f*log(c*(d + e*x)**n)**2)*b*f*g 
*h*n**2*x**2*log(F)/2 - F**(a*f + b*f*log(c*(d + e*x)**n)**2)*b*f*g*h*n*x* 
*2*log(F)*log(c*(d + e*x)**n) + 2*F**(a*f + b*f*log(c*(d + e*x)**n)**2)*b* 
f*h**2*n**2*x**3*log(F)/27 - 2*F**(a*f + b*f*log(c*(d + e*x)**n)**2)*b*f*h 
**2*n*x**3*log(F)*log(c*(d + e*x)**n)/9 + F**(a*f + b*f*log(c*(d + e*x)...
                                                                                    
                                                                                    
 

Maxima [F]

\[ \int F^{f \left (a+b \log ^2\left (c (d+e x)^n\right )\right )} (g+h x)^2 \, dx=\int { {\left (h x + g\right )}^{2} F^{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + a\right )} f} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int F^{f \left (a+b \log ^2\left (c (d+e x)^n\right )\right )} (g+h x)^2 \, dx=\int { {\left (h x + g\right )}^{2} F^{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + a\right )} f} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int F^{f \left (a+b \log ^2\left (c (d+e x)^n\right )\right )} (g+h x)^2 \, dx=\int {\mathrm {e}}^{f\,\ln \left (F\right )\,\left (b\,{\ln \left (c\,{\left (d+e\,x\right )}^n\right )}^2+a\right )}\,{\left (g+h\,x\right )}^2 \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int F^{f \left (a+b \log ^2\left (c (d+e x)^n\right )\right )} (g+h x)^2 \, dx=f^{a f} \left (\left (\int f^{\mathrm {log}\left (\left (e x +d \right )^{n} c \right )^{2} b f}d x \right ) g^{2}+\left (\int f^{\mathrm {log}\left (\left (e x +d \right )^{n} c \right )^{2} b f} x^{2}d x \right ) h^{2}+2 \left (\int f^{\mathrm {log}\left (\left (e x +d \right )^{n} c \right )^{2} b f} x d x \right ) g h \right ) \] Input:

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

Output:

f**(a*f)*(int(f**(log((d + e*x)**n*c)**2*b*f),x)*g**2 + int(f**(log((d + e 
*x)**n*c)**2*b*f)*x**2,x)*h**2 + 2*int(f**(log((d + e*x)**n*c)**2*b*f)*x,x 
)*g*h)