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

Optimal. Leaf size=228 \[ \frac{2 a b (c+d x)^m \left (F^{e g+f g x}\right )^n F^{g n \left (e-\frac{c f}{d}\right )-g n (e+f x)} \left (-\frac{f g n \log (F) (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{f g n \log (F) (c+d x)}{d}\right )}{f g n \log (F)}+\frac{b^2 2^{-m-1} (c+d x)^m \left (F^{e g+f g x}\right )^{2 n} F^{2 g n \left (e-\frac{c f}{d}\right )-2 g n (e+f x)} \left (-\frac{f g n \log (F) (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{2 f g n \log (F) (c+d x)}{d}\right )}{f g n \log (F)}+\frac{a^2 (c+d x)^{m+1}}{d (m+1)} \]

[Out]

(a^2*(c + d*x)^(1 + m))/(d*(1 + m)) + (2^(-1 - m)*b^2*F^(2*(e - (c*f)/d)*g*n - 2
*g*n*(e + f*x))*(F^(e*g + f*g*x))^(2*n)*(c + d*x)^m*Gamma[1 + m, (-2*f*g*n*(c +
d*x)*Log[F])/d])/(f*g*n*Log[F]*(-((f*g*n*(c + d*x)*Log[F])/d))^m) + (2*a*b*F^((e
- (c*f)/d)*g*n - g*n*(e + f*x))*(F^(e*g + f*g*x))^n*(c + d*x)^m*Gamma[1 + m, -(
(f*g*n*(c + d*x)*Log[F])/d)])/(f*g*n*Log[F]*(-((f*g*n*(c + d*x)*Log[F])/d))^m)

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Rubi [A]  time = 0.498875, antiderivative size = 228, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12 \[ \frac{2 a b (c+d x)^m \left (F^{e g+f g x}\right )^n F^{g n \left (e-\frac{c f}{d}\right )-g n (e+f x)} \left (-\frac{f g n \log (F) (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{f g n \log (F) (c+d x)}{d}\right )}{f g n \log (F)}+\frac{b^2 2^{-m-1} (c+d x)^m \left (F^{e g+f g x}\right )^{2 n} F^{2 g n \left (e-\frac{c f}{d}\right )-2 g n (e+f x)} \left (-\frac{f g n \log (F) (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{2 f g n \log (F) (c+d x)}{d}\right )}{f g n \log (F)}+\frac{a^2 (c+d x)^{m+1}}{d (m+1)} \]

Antiderivative was successfully verified.

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

[Out]

(a^2*(c + d*x)^(1 + m))/(d*(1 + m)) + (2^(-1 - m)*b^2*F^(2*(e - (c*f)/d)*g*n - 2
*g*n*(e + f*x))*(F^(e*g + f*g*x))^(2*n)*(c + d*x)^m*Gamma[1 + m, (-2*f*g*n*(c +
d*x)*Log[F])/d])/(f*g*n*Log[F]*(-((f*g*n*(c + d*x)*Log[F])/d))^m) + (2*a*b*F^((e
- (c*f)/d)*g*n - g*n*(e + f*x))*(F^(e*g + f*g*x))^n*(c + d*x)^m*Gamma[1 + m, -(
(f*g*n*(c + d*x)*Log[F])/d)])/(f*g*n*Log[F]*(-((f*g*n*(c + d*x)*Log[F])/d))^m)

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Rubi in Sympy [A]  time = 50.2532, size = 219, normalized size = 0.96 \[ \frac{F^{g n \left (- 2 e - 2 f x\right )} F^{- \frac{2 g n \left (c f - d e\right )}{d}} b^{2} \left (\frac{f g n \left (- 2 c - 2 d x\right ) \log{\left (F \right )}}{d}\right )^{- m} \left (c + d x\right )^{m} \left (F^{g \left (e + f x\right )}\right )^{2 n} \Gamma{\left (m + 1,\frac{f g n \left (- 2 c - 2 d x\right ) \log{\left (F \right )}}{d} \right )}}{2 f g n \log{\left (F \right )}} + \frac{2 F^{g n \left (- e - f x\right )} F^{- \frac{g n \left (c f - d e\right )}{d}} a b \left (\frac{f g n \left (- c - d x\right ) \log{\left (F \right )}}{d}\right )^{- m} \left (c + d x\right )^{m} \left (F^{g \left (e + f x\right )}\right )^{n} \Gamma{\left (m + 1,\frac{f g n \left (- c - d x\right ) \log{\left (F \right )}}{d} \right )}}{f g n \log{\left (F \right )}} + \frac{a^{2} \left (c + d x\right )^{m + 1}}{d \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*(F**(g*(f*x+e)))**n)**2*(d*x+c)**m,x)

[Out]

F**(g*n*(-2*e - 2*f*x))*F**(-2*g*n*(c*f - d*e)/d)*b**2*(f*g*n*(-2*c - 2*d*x)*log
(F)/d)**(-m)*(c + d*x)**m*(F**(g*(e + f*x)))**(2*n)*Gamma(m + 1, f*g*n*(-2*c - 2
*d*x)*log(F)/d)/(2*f*g*n*log(F)) + 2*F**(g*n*(-e - f*x))*F**(-g*n*(c*f - d*e)/d)
*a*b*(f*g*n*(-c - d*x)*log(F)/d)**(-m)*(c + d*x)**m*(F**(g*(e + f*x)))**n*Gamma(
m + 1, f*g*n*(-c - d*x)*log(F)/d)/(f*g*n*log(F)) + a**2*(c + d*x)**(m + 1)/(d*(m
+ 1))

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Mathematica [A]  time = 0.200101, size = 0, normalized size = 0. \[ \int \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 (c+d x)^m \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.051, size = 0, normalized size = 0. \[ \int \left ( a+b \left ({F}^{g \left ( fx+e \right ) } \right ) ^{n} \right ) ^{2} \left ( dx+c \right ) ^{m}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*(F^(g*(f*x+e)))^n)^2*(d*x+c)^m,x)

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.275973, size = 258, normalized size = 1.13 \[ \frac{4 \,{\left (a b d m + a b d\right )} e^{\left (\frac{{\left (d e - c f\right )} g n \log \left (F\right ) - d m \log \left (-\frac{f g n \log \left (F\right )}{d}\right )}{d}\right )} \Gamma \left (m + 1, -\frac{{\left (d f g n x + c f g n\right )} \log \left (F\right )}{d}\right ) +{\left (b^{2} d m + b^{2} d\right )} e^{\left (\frac{2 \,{\left (d e - c f\right )} g n \log \left (F\right ) - d m \log \left (-\frac{2 \, f g n \log \left (F\right )}{d}\right )}{d}\right )} \Gamma \left (m + 1, -\frac{2 \,{\left (d f g n x + c f g n\right )} \log \left (F\right )}{d}\right ) + 2 \,{\left (a^{2} d f g n x + a^{2} c f g n\right )}{\left (d x + c\right )}^{m} \log \left (F\right )}{2 \,{\left (d f g m + d f g\right )} n \log \left (F\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(4*(a*b*d*m + a*b*d)*e^(((d*e - c*f)*g*n*log(F) - d*m*log(-f*g*n*log(F)/d))/
d)*gamma(m + 1, -(d*f*g*n*x + c*f*g*n)*log(F)/d) + (b^2*d*m + b^2*d)*e^((2*(d*e
- c*f)*g*n*log(F) - d*m*log(-2*f*g*n*log(F)/d))/d)*gamma(m + 1, -2*(d*f*g*n*x +
c*f*g*n)*log(F)/d) + 2*(a^2*d*f*g*n*x + a^2*c*f*g*n)*(d*x + c)^m*log(F))/((d*f*g
*m + d*f*g)*n*log(F))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*(F**(g*(f*x+e)))**n)**2*(d*x+c)**m,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left ({\left (F^{{\left (f x + e\right )} g}\right )}^{n} b + a\right )}^{2}{\left (d x + c\right )}^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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