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

Optimal. Leaf size=202 \[ -\frac{a^2}{d (c+d x)}+\frac{2 a b f g n \log (F) \left (F^{e g+f g x}\right )^n F^{g n \left (e-\frac{c f}{d}\right )-g n (e+f x)} \text{ExpIntegralEi}\left (\frac{f g n \log (F) (c+d x)}{d}\right )}{d^2}-\frac{2 a b \left (F^{e g+f g x}\right )^n}{d (c+d x)}+\frac{2 b^2 f g n \log (F) \left (F^{e g+f g x}\right )^{2 n} F^{2 g n \left (e-\frac{c f}{d}\right )-2 g n (e+f x)} \text{ExpIntegralEi}\left (\frac{2 f g n \log (F) (c+d x)}{d}\right )}{d^2}-\frac{b^2 \left (F^{e g+f g x}\right )^{2 n}}{d (c+d x)} \]

[Out]

-(a^2/(d*(c + d*x))) - (2*a*b*(F^(e*g + f*g*x))^n)/(d*(c + d*x)) - (b^2*(F^(e*g
+ f*g*x))^(2*n))/(d*(c + d*x)) + (2*a*b*f*F^((e - (c*f)/d)*g*n - g*n*(e + f*x))*
(F^(e*g + f*g*x))^n*g*n*ExpIntegralEi[(f*g*n*(c + d*x)*Log[F])/d]*Log[F])/d^2 +
(2*b^2*f*F^(2*(e - (c*f)/d)*g*n - 2*g*n*(e + f*x))*(F^(e*g + f*g*x))^(2*n)*g*n*E
xpIntegralEi[(2*f*g*n*(c + d*x)*Log[F])/d]*Log[F])/d^2

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Rubi [A]  time = 0.588462, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16 \[ -\frac{a^2}{d (c+d x)}+\frac{2 a b f g n \log (F) \left (F^{e g+f g x}\right )^n F^{g n \left (e-\frac{c f}{d}\right )-g n (e+f x)} \text{ExpIntegralEi}\left (\frac{f g n \log (F) (c+d x)}{d}\right )}{d^2}-\frac{2 a b \left (F^{e g+f g x}\right )^n}{d (c+d x)}+\frac{2 b^2 f g n \log (F) \left (F^{e g+f g x}\right )^{2 n} F^{2 g n \left (e-\frac{c f}{d}\right )-2 g n (e+f x)} \text{ExpIntegralEi}\left (\frac{2 f g n \log (F) (c+d x)}{d}\right )}{d^2}-\frac{b^2 \left (F^{e g+f g x}\right )^{2 n}}{d (c+d x)} \]

Antiderivative was successfully verified.

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

[Out]

-(a^2/(d*(c + d*x))) - (2*a*b*(F^(e*g + f*g*x))^n)/(d*(c + d*x)) - (b^2*(F^(e*g
+ f*g*x))^(2*n))/(d*(c + d*x)) + (2*a*b*f*F^((e - (c*f)/d)*g*n - g*n*(e + f*x))*
(F^(e*g + f*g*x))^n*g*n*ExpIntegralEi[(f*g*n*(c + d*x)*Log[F])/d]*Log[F])/d^2 +
(2*b^2*f*F^(2*(e - (c*f)/d)*g*n - 2*g*n*(e + f*x))*(F^(e*g + f*g*x))^(2*n)*g*n*E
xpIntegralEi[(2*f*g*n*(c + d*x)*Log[F])/d]*Log[F])/d^2

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Rubi in Sympy [A]  time = 48.1219, size = 202, normalized size = 1. \[ \frac{2 F^{g n \left (- 2 e - 2 f x\right )} F^{- \frac{2 g n \left (c f - d e\right )}{d}} b^{2} f g n \left (F^{g \left (e + f x\right )}\right )^{2 n} \log{\left (F \right )} \operatorname{Ei}{\left (\frac{f g n \left (2 c + 2 d x\right ) \log{\left (F \right )}}{d} \right )}}{d^{2}} + \frac{2 F^{g n \left (- e - f x\right )} F^{- \frac{g n \left (c f - d e\right )}{d}} a b f g n \left (F^{g \left (e + f x\right )}\right )^{n} \log{\left (F \right )} \operatorname{Ei}{\left (\frac{f g n \left (c + d x\right ) \log{\left (F \right )}}{d} \right )}}{d^{2}} - \frac{a^{2}}{d \left (c + d x\right )} - \frac{2 a b \left (F^{g \left (e + f x\right )}\right )^{n}}{d \left (c + d x\right )} - \frac{b^{2} \left (F^{g \left (e + f x\right )}\right )^{2 n}}{d \left (c + d x\right )} \]

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)**2,x)

[Out]

2*F**(g*n*(-2*e - 2*f*x))*F**(-2*g*n*(c*f - d*e)/d)*b**2*f*g*n*(F**(g*(e + f*x))
)**(2*n)*log(F)*Ei(f*g*n*(2*c + 2*d*x)*log(F)/d)/d**2 + 2*F**(g*n*(-e - f*x))*F*
*(-g*n*(c*f - d*e)/d)*a*b*f*g*n*(F**(g*(e + f*x)))**n*log(F)*Ei(f*g*n*(c + d*x)*
log(F)/d)/d**2 - a**2/(d*(c + d*x)) - 2*a*b*(F**(g*(e + f*x)))**n/(d*(c + d*x))
- b**2*(F**(g*(e + f*x)))**(2*n)/(d*(c + d*x))

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Mathematica [A]  time = 0.812669, size = 136, normalized size = 0.67 \[ \frac{2 a b f g n \log (F) \left (F^{g (e+f x)}\right )^n F^{-\frac{f g n (c+d x)}{d}} \text{ExpIntegralEi}\left (\frac{f g n \log (F) (c+d x)}{d}\right )-\frac{d \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2}{c+d x}+2 b^2 f g n \log (F) \left (F^{g (e+f x)}\right )^{2 n} F^{-\frac{2 f g n (c+d x)}{d}} \text{ExpIntegralEi}\left (\frac{2 f g n \log (F) (c+d x)}{d}\right )}{d^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.032, size = 0, normalized size = 0. \[ \int{\frac{ \left ( a+b \left ({F}^{g \left ( fx+e \right ) } \right ) ^{n} \right ) ^{2}}{ \left ( dx+c \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.279562, size = 231, normalized size = 1.14 \[ -\frac{2 \, F^{f g n x + e g n} a b d + F^{2 \, f g n x + 2 \, e g n} b^{2} d - 2 \,{\left (b^{2} d f g n x + b^{2} c f g n\right )} F^{\frac{2 \,{\left (d e - c f\right )} g n}{d}}{\rm Ei}\left (\frac{2 \,{\left (d f g n x + c f g n\right )} \log \left (F\right )}{d}\right ) \log \left (F\right ) - 2 \,{\left (a b d f g n x + a b c f g n\right )} F^{\frac{{\left (d e - c f\right )} g n}{d}}{\rm Ei}\left (\frac{{\left (d f g n x + c f g n\right )} \log \left (F\right )}{d}\right ) \log \left (F\right ) + a^{2} d}{d^{3} x + c d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(2*F^(f*g*n*x + e*g*n)*a*b*d + F^(2*f*g*n*x + 2*e*g*n)*b^2*d - 2*(b^2*d*f*g*n*x
+ b^2*c*f*g*n)*F^(2*(d*e - c*f)*g*n/d)*Ei(2*(d*f*g*n*x + c*f*g*n)*log(F)/d)*log
(F) - 2*(a*b*d*f*g*n*x + a*b*c*f*g*n)*F^((d*e - c*f)*g*n/d)*Ei((d*f*g*n*x + c*f*
g*n)*log(F)/d)*log(F) + a^2*d)/(d^3*x + c*d^2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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