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

Optimal. Leaf size=147 \[ -\frac{a}{2 d (c+d x)^2}+\frac{b f^2 g^2 n^2 \log ^2(F) \left (F^{e g+f g x}\right )^n F^{g n \left (e-\frac{c f}{d}\right )-g n (e+f x)} \text{Ei}\left (\frac{f g n (c+d x) \log (F)}{d}\right )}{2 d^3}-\frac{b f g n \log (F) \left (F^{e g+f g x}\right )^n}{2 d^2 (c+d x)}-\frac{b \left (F^{e g+f g x}\right )^n}{2 d (c+d x)^2} \]

[Out]

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

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Rubi [A]  time = 0.239608, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {2183, 2177, 2182, 2178} \[ -\frac{a}{2 d (c+d x)^2}+\frac{b f^2 g^2 n^2 \log ^2(F) \left (F^{e g+f g x}\right )^n F^{g n \left (e-\frac{c f}{d}\right )-g n (e+f x)} \text{Ei}\left (\frac{f g n (c+d x) \log (F)}{d}\right )}{2 d^3}-\frac{b f g n \log (F) \left (F^{e g+f g x}\right )^n}{2 d^2 (c+d x)}-\frac{b \left (F^{e g+f g x}\right )^n}{2 d (c+d x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.327488, size = 111, normalized size = 0.76 \[ -\frac{a d^2-b f^2 g^2 n^2 \log ^2(F) (c+d x)^2 \left (F^{g (e+f x)}\right )^n F^{-\frac{f g n (c+d x)}{d}} \text{Ei}\left (\frac{f g n (c+d x) \log (F)}{d}\right )+b d \left (F^{g (e+f x)}\right )^n (f g n \log (F) (c+d x)+d)}{2 d^3 (c+d x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a*d^2 - (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) + b*d*(F^(g*(e + f*x)))^n*(d + 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{a+b \left ({F}^{g \left ( fx+e \right ) } \right ) ^{n}}{ \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)/(d*x+c)^3,x)

[Out]

int((a+b*(F^(g*(f*x+e)))^n)/(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 )}^{n} 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 \,{\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)/(d*x+c)^3,x, algorithm="maxima")

[Out]

(F^(e*g))^n*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/(d^3*x^2 + 2*c*d^2
*x + c^2*d)

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Fricas [A]  time = 1.58503, size = 338, normalized size = 2.3 \begin{align*} \frac{{\left (b d^{2} f^{2} g^{2} n^{2} x^{2} + 2 \, b c d f^{2} g^{2} n^{2} x + 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 d^{2} -{\left (b d^{2} +{\left (b d^{2} f g n x + 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)/(d*x+c)^3,x, algorithm="fricas")

[Out]

1/2*((b*d^2*f^2*g^2*n^2*x^2 + 2*b*c*d*f^2*g^2*n^2*x + 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*d^2 - (b*d^2 + (b*d^2*f*g*n*x + 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)/(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 (F^{{\left (f x + e\right )} g}\right )}^{n} b + a}{{\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)/(d*x+c)^3,x, algorithm="giac")

[Out]

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