3.38 \(\int \frac{(a+b (F^{g (e+f x)})^n)^2}{(c+d x)^3} \, dx\)

Optimal. Leaf size=286 \[ -\frac{a^2}{2 d (c+d x)^2}+\frac{a b f^2 g^2 n^2 \log ^2(F) \left (F^{e g+f g x}\right )^n F^{g n \left (e-\frac{c f}{d}\right )-g n (e+f x)} \text{Ei}\left (\frac{f g n (c+d x) \log (F)}{d}\right )}{d^3}-\frac{a b f g n \log (F) \left (F^{e g+f g x}\right )^n}{d^2 (c+d x)}-\frac{a b \left (F^{e g+f g x}\right )^n}{d (c+d x)^2}+\frac{2 b^2 f^2 g^2 n^2 \log ^2(F) \left (F^{e g+f g x}\right )^{2 n} F^{2 g n \left (e-\frac{c f}{d}\right )-2 g n (e+f x)} \text{Ei}\left (\frac{2 f g n (c+d x) \log (F)}{d}\right )}{d^3}-\frac{b^2 f g n \log (F) \left (F^{e g+f g x}\right )^{2 n}}{d^2 (c+d x)}-\frac{b^2 \left (F^{e g+f g x}\right )^{2 n}}{2 d (c+d x)^2} \]

[Out]

-a^2/(2*d*(c + d*x)^2) - (a*b*(F^(e*g + f*g*x))^n)/(d*(c + d*x)^2) - (b^2*(F^(e*g + f*g*x))^(2*n))/(2*d*(c + d
*x)^2) - (a*b*f*(F^(e*g + f*g*x))^n*g*n*Log[F])/(d^2*(c + d*x)) - (b^2*f*(F^(e*g + f*g*x))^(2*n)*g*n*Log[F])/(
d^2*(c + d*x)) + (a*b*f^2*F^((e - (c*f)/d)*g*n - g*n*(e + f*x))*(F^(e*g + f*g*x))^n*g^2*n^2*ExpIntegralEi[(f*g
*n*(c + d*x)*Log[F])/d]*Log[F]^2)/d^3 + (2*b^2*f^2*F^(2*(e - (c*f)/d)*g*n - 2*g*n*(e + f*x))*(F^(e*g + f*g*x))
^(2*n)*g^2*n^2*ExpIntegralEi[(2*f*g*n*(c + d*x)*Log[F])/d]*Log[F]^2)/d^3

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Rubi [A]  time = 0.474569, antiderivative size = 286, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {2183, 2177, 2182, 2178} \[ -\frac{a^2}{2 d (c+d x)^2}+\frac{a b f^2 g^2 n^2 \log ^2(F) \left (F^{e g+f g x}\right )^n F^{g n \left (e-\frac{c f}{d}\right )-g n (e+f x)} \text{Ei}\left (\frac{f g n (c+d x) \log (F)}{d}\right )}{d^3}-\frac{a b f g n \log (F) \left (F^{e g+f g x}\right )^n}{d^2 (c+d x)}-\frac{a b \left (F^{e g+f g x}\right )^n}{d (c+d x)^2}+\frac{2 b^2 f^2 g^2 n^2 \log ^2(F) \left (F^{e g+f g x}\right )^{2 n} F^{2 g n \left (e-\frac{c f}{d}\right )-2 g n (e+f x)} \text{Ei}\left (\frac{2 f g n (c+d x) \log (F)}{d}\right )}{d^3}-\frac{b^2 f g n \log (F) \left (F^{e g+f g x}\right )^{2 n}}{d^2 (c+d x)}-\frac{b^2 \left (F^{e g+f g x}\right )^{2 n}}{2 d (c+d x)^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*(F^(g*(e + f*x)))^n)^2/(c + d*x)^3,x]

[Out]

-a^2/(2*d*(c + d*x)^2) - (a*b*(F^(e*g + f*g*x))^n)/(d*(c + d*x)^2) - (b^2*(F^(e*g + f*g*x))^(2*n))/(2*d*(c + d
*x)^2) - (a*b*f*(F^(e*g + f*g*x))^n*g*n*Log[F])/(d^2*(c + d*x)) - (b^2*f*(F^(e*g + f*g*x))^(2*n)*g*n*Log[F])/(
d^2*(c + d*x)) + (a*b*f^2*F^((e - (c*f)/d)*g*n - g*n*(e + f*x))*(F^(e*g + f*g*x))^n*g^2*n^2*ExpIntegralEi[(f*g
*n*(c + d*x)*Log[F])/d]*Log[F]^2)/d^3 + (2*b^2*f^2*F^(2*(e - (c*f)/d)*g*n - 2*g*n*(e + f*x))*(F^(e*g + f*g*x))
^(2*n)*g^2*n^2*ExpIntegralEi[(2*f*g*n*(c + d*x)*Log[F])/d]*Log[F]^2)/d^3

Rule 2183

Int[((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.))^(p_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> In
t[ExpandIntegrand[(c + d*x)^m, (a + b*(F^(g*(e + f*x)))^n)^p, x], x] /; FreeQ[{F, a, b, c, d, e, f, g, m, n},
x] && IGtQ[p, 0]

Rule 2177

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[((c + d*x)^(m
+ 1)*(b*F^(g*(e + f*x)))^n)/(d*(m + 1)), x] - Dist[(f*g*n*Log[F])/(d*(m + 1)), Int[(c + d*x)^(m + 1)*(b*F^(g*
(e + f*x)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && LtQ[m, -1] && IntegerQ[2*m] &&  !$UseGamma ===
True

Rule 2182

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Dist[(b*F^(g*(e +
f*x)))^n/F^(g*n*(e + f*x)), Int[(c + d*x)^m*F^(g*n*(e + f*x)), x], x] /; FreeQ[{F, b, c, d, e, f, g, m, n}, x]

Rule 2178

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - (c*f)/d))*ExpIntegral
Ei[(f*g*(c + d*x)*Log[F])/d])/d, x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

Rubi steps

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

Mathematica [A]  time = 0.628889, size = 217, normalized size = 0.76 \[ -\frac{a^2 d^2-2 a b f^2 g^2 n^2 \log ^2(F) (c+d x)^2 \left (F^{g (e+f x)}\right )^n F^{-\frac{f g n (c+d x)}{d}} \text{Ei}\left (\frac{f g n (c+d x) \log (F)}{d}\right )+2 a b d \left (F^{g (e+f x)}\right )^n (f g n \log (F) (c+d x)+d)-4 b^2 f^2 g^2 n^2 \log ^2(F) (c+d x)^2 \left (F^{g (e+f x)}\right )^{2 n} F^{-\frac{2 f g n (c+d x)}{d}} \text{Ei}\left (\frac{2 f g n (c+d x) \log (F)}{d}\right )+b^2 d \left (F^{g (e+f x)}\right )^{2 n} (2 f g n \log (F) (c+d x)+d)}{2 d^3 (c+d x)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*(F^(g*(e + f*x)))^n)^2/(c + d*x)^3,x]

[Out]

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

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Maple [F]  time = 0.026, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b \left ({F}^{g \left ( fx+e \right ) } \right ) ^{n} \right ) ^{2}}{ \left ( dx+c \right ) ^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*(F^(g*(f*x+e)))^n)^2/(d*x+c)^3,x)

[Out]

int((a+b*(F^(g*(f*x+e)))^n)^2/(d*x+c)^3,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\left (F^{e g}\right )}^{2 \, n} b^{2} \int \frac{{\left (F^{f g x}\right )}^{2 \, n}}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\,{d x} + 2 \,{\left (F^{e g}\right )}^{n} a b \int \frac{{\left (F^{f g x}\right )}^{n}}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\,{d x} - \frac{a^{2}}{2 \,{\left (d^{3} x^{2} + 2 \, c d^{2} x + c^{2} d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*(F^(g*(f*x+e)))^n)^2/(d*x+c)^3,x, algorithm="maxima")

[Out]

(F^(e*g))^(2*n)*b^2*integrate((F^(f*g*x))^(2*n)/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3), x) + 2*(F^(e*g))^n*
a*b*integrate((F^(f*g*x))^n/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3), x) - 1/2*a^2/(d^3*x^2 + 2*c*d^2*x + c^2
*d)

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Fricas [A]  time = 1.59511, size = 671, normalized size = 2.35 \begin{align*} \frac{4 \,{\left (b^{2} d^{2} f^{2} g^{2} n^{2} x^{2} + 2 \, b^{2} c d f^{2} g^{2} n^{2} x + b^{2} c^{2} f^{2} g^{2} n^{2}\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} + 2 \,{\left (a b d^{2} f^{2} g^{2} n^{2} x^{2} + 2 \, a b c d f^{2} g^{2} n^{2} x + a b c^{2} f^{2} g^{2} n^{2}\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 )^{2} - a^{2} d^{2} -{\left (b^{2} d^{2} + 2 \,{\left (b^{2} d^{2} f g n x + b^{2} c d f g n\right )} \log \left (F\right )\right )} F^{2 \, f g n x + 2 \, e g n} - 2 \,{\left (a b d^{2} +{\left (a b d^{2} f g n x + a b c d f g n\right )} \log \left (F\right )\right )} F^{f g n x + e g n}}{2 \,{\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*(F^(g*(f*x+e)))^n)^2/(d*x+c)^3,x, algorithm="fricas")

[Out]

1/2*(4*(b^2*d^2*f^2*g^2*n^2*x^2 + 2*b^2*c*d*f^2*g^2*n^2*x + b^2*c^2*f^2*g^2*n^2)*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 + 2*(a*b*d^2*f^2*g^2*n^2*x^2 + 2*a*b*c*d*f^2*g^2*n^2*x + a*b*c^2*f^2*
g^2*n^2)*F^((d*e - c*f)*g*n/d)*Ei((d*f*g*n*x + c*f*g*n)*log(F)/d)*log(F)^2 - a^2*d^2 - (b^2*d^2 + 2*(b^2*d^2*f
*g*n*x + b^2*c*d*f*g*n)*log(F))*F^(2*f*g*n*x + 2*e*g*n) - 2*(a*b*d^2 + (a*b*d^2*f*g*n*x + a*b*c*d*f*g*n)*log(F
))*F^(f*g*n*x + e*g*n))/(d^5*x^2 + 2*c*d^4*x + c^2*d^3)

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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((a+b*(F**(g*(f*x+e)))**n)**2/(d*x+c)**3,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left ({\left (F^{{\left (f x + e\right )} g}\right )}^{n} b + a\right )}^{2}}{{\left (d x + c\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*(F^(g*(f*x+e)))^n)^2/(d*x+c)^3,x, algorithm="giac")

[Out]

integrate(((F^((f*x + e)*g))^n*b + a)^2/(d*x + c)^3, x)