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

Optimal. Leaf size=439 \[ -\frac{2 d (c+d x) \text{PolyLog}\left (2,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^2 g^2 n^2 \log ^2(F)}+\frac{3 d^2 \text{PolyLog}\left (2,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^3 g^3 n^3 \log ^3(F)}+\frac{2 d^2 \text{PolyLog}\left (3,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^3 g^3 n^3 \log ^3(F)}+\frac{3 d (c+d x) \log \left (\frac{b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{a^3 f^2 g^2 n^2 \log ^2(F)}-\frac{(c+d x)^2 \log \left (\frac{b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{a^3 f g n \log (F)}-\frac{d^2 \log \left (a+b \left (F^{g (e+f x)}\right )^n\right )}{a^3 f^3 g^3 n^3 \log ^3(F)}-\frac{3 (c+d x)^2}{2 a^3 f g n \log (F)}+\frac{(c+d x)^3}{3 a^3 d}+\frac{d^2 x}{a^3 f^2 g^2 n^2 \log ^2(F)}-\frac{d (c+d x)}{a^2 f^2 g^2 n^2 \log ^2(F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}+\frac{(c+d x)^2}{a^2 f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}+\frac{(c+d x)^2}{2 a f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2} \]

[Out]

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

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Rubi [A]  time = 2.18456, antiderivative size = 439, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 13, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.52 \[ -\frac{2 d (c+d x) \text{PolyLog}\left (2,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^2 g^2 n^2 \log ^2(F)}+\frac{3 d^2 \text{PolyLog}\left (2,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^3 g^3 n^3 \log ^3(F)}+\frac{2 d^2 \text{PolyLog}\left (3,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^3 g^3 n^3 \log ^3(F)}+\frac{3 d (c+d x) \log \left (\frac{b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{a^3 f^2 g^2 n^2 \log ^2(F)}-\frac{(c+d x)^2 \log \left (\frac{b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{a^3 f g n \log (F)}-\frac{d^2 \log \left (a+b \left (F^{g (e+f x)}\right )^n\right )}{a^3 f^3 g^3 n^3 \log ^3(F)}-\frac{3 (c+d x)^2}{2 a^3 f g n \log (F)}+\frac{(c+d x)^3}{3 a^3 d}+\frac{d^2 x}{a^3 f^2 g^2 n^2 \log ^2(F)}-\frac{d (c+d x)}{a^2 f^2 g^2 n^2 \log ^2(F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}+\frac{(c+d x)^2}{a^2 f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}+\frac{(c+d x)^2}{2 a f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [A]  time = 3.53597, size = 0, normalized size = 0. \[ \int \frac{(c+d x)^2}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^3} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [B]  time = 0.053, size = 1457, normalized size = 3.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/2/n/g^3/f^3/ln(F)^3/a^3*d^2*ln(F^(g*(f*x+e)))^2-1/n/g/f/ln(F)/a^3*c^2*ln(a+b*
(F^(g*(f*x+e)))^n)+1/n^3/g^3/f^3/ln(F)^3/a^3*d^2*ln((F^(g*(f*x+e)))^n)+1/n/g/f/l
n(F)/a^3*c^2*ln((F^(g*(f*x+e)))^n)+1/g^2/f^2/ln(F)^2/a^3*d*c*ln(F^(g*(f*x+e)))^2
+1/g^2/f^2/ln(F)^2/a^3*d^2*ln(F^(g*(f*x+e)))^2*x-d^2*ln(a+b*(F^(g*(f*x+e)))^n)/a
^3/f^3/g^3/n^3/ln(F)^3+3*d^2*polylog(2,-b*(F^(g*(f*x+e)))^n/a)/a^3/f^3/g^3/n^3/l
n(F)^3+2*d^2*polylog(3,-b*(F^(g*(f*x+e)))^n/a)/a^3/f^3/g^3/n^3/ln(F)^3+1/2*(2*ln
(F)*b*d^2*f*g*n*x^2*(F^(g*(f*x+e)))^n+3*ln(F)*a*d^2*f*g*n*x^2+4*ln(F)*b*c*d*f*g*
n*x*(F^(g*(f*x+e)))^n+6*ln(F)*a*c*d*f*g*n*x+2*ln(F)*b*c^2*f*g*n*(F^(g*(f*x+e)))^
n+3*ln(F)*a*c^2*f*g*n-2*b*d^2*x*(F^(g*(f*x+e)))^n-2*a*d^2*x-2*b*c*d*(F^(g*(f*x+e
)))^n-2*a*c*d)/n^2/g^2/f^2/ln(F)^2/a^2/(a+b*(F^(g*(f*x+e)))^n)^2+1/n/g^3/f^3/ln(
F)^3/a^3*d^2*ln((F^(g*(f*x+e)))^n)*ln(F^(g*(f*x+e)))^2-3/n^2/g^3/f^3/ln(F)^3/a^3
*d^2*ln(a+b*(F^(g*(f*x+e)))^n)*ln(F^(g*(f*x+e)))+3/n^2/g^3/f^3/ln(F)^3/a^3*d^2*l
n(1+b*(F^(g*(f*x+e)))^n/a)*ln(F^(g*(f*x+e)))-3/n^2/g^2/f^2/ln(F)^2/a^3*d^2*ln((F
^(g*(f*x+e)))^n)*x+3/n^2/g^3/f^3/ln(F)^3/a^3*d^2*ln((F^(g*(f*x+e)))^n)*ln(F^(g*(
f*x+e)))+3/n^2/g^2/f^2/ln(F)^2/a^3*d^2*ln(a+b*(F^(g*(f*x+e)))^n)*x-2/n^2/g^2/f^2
/ln(F)^2/a^3*c*d*polylog(2,-b*(F^(g*(f*x+e)))^n/a)-3/n^2/g^2/f^2/ln(F)^2/a^3*c*d
*ln((F^(g*(f*x+e)))^n)-2/n^2/g^2/f^2/ln(F)^2/a^3*d^2*polylog(2,-b*(F^(g*(f*x+e))
)^n/a)*x+3/n^2/g^2/f^2/ln(F)^2/a^3*c*d*ln(a+b*(F^(g*(f*x+e)))^n)+1/n/g^3/f^3/ln(
F)^3/a^3*d^2*ln(1+b*(F^(g*(f*x+e)))^n/a)*ln(F^(g*(f*x+e)))^2-1/n/g/f/ln(F)/a^3*d
^2*ln(a+b*(F^(g*(f*x+e)))^n)*x^2-1/n/g^3/f^3/ln(F)^3/a^3*d^2*ln(a+b*(F^(g*(f*x+e
)))^n)*ln(F^(g*(f*x+e)))^2+1/n/g/f/ln(F)/a^3*d^2*ln((F^(g*(f*x+e)))^n)*x^2-2/3/g
^3/f^3/ln(F)^3/a^3*d^2*ln(F^(g*(f*x+e)))^3+2/n/g/f/ln(F)/a^3*c*d*ln((F^(g*(f*x+e
)))^n)*x-2/n/g/f/ln(F)/a^3*c*d*ln(a+b*(F^(g*(f*x+e)))^n)*x-2/n/g^2/f^2/ln(F)^2/a
^3*d^2*ln(1+b*(F^(g*(f*x+e)))^n/a)*ln(F^(g*(f*x+e)))*x+2/n/g^2/f^2/ln(F)^2/a^3*d
^2*ln(a+b*(F^(g*(f*x+e)))^n)*ln(F^(g*(f*x+e)))*x-2/n/g^2/f^2/ln(F)^2/a^3*d^2*ln(
(F^(g*(f*x+e)))^n)*ln(F^(g*(f*x+e)))*x-2/n/g^2/f^2/ln(F)^2/a^3*c*d*ln(1+b*(F^(g*
(f*x+e)))^n/a)*ln(F^(g*(f*x+e)))-2/n/g^2/f^2/ln(F)^2/a^3*c*d*ln((F^(g*(f*x+e)))^
n)*ln(F^(g*(f*x+e)))+2/n/g^2/f^2/ln(F)^2/a^3*c*d*ln(a+b*(F^(g*(f*x+e)))^n)*ln(F^
(g*(f*x+e)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \frac{1}{2} \, c^{2}{\left (\frac{2 \,{\left (F^{f g x + e g}\right )}^{n} b + 3 \, a}{{\left (2 \,{\left (F^{f g x + e g}\right )}^{n} a^{3} b n +{\left (F^{f g x + e g}\right )}^{2 \, n} a^{2} b^{2} n + a^{4} n\right )} f g \log \left (F\right )} + \frac{2 \, \log \left (F^{f g x + e g}\right )}{a^{3} f g \log \left (F\right )} - \frac{2 \, \log \left (\frac{{\left (F^{f g x + e g}\right )}^{n} b + a}{b}\right )}{a^{3} f g n \log \left (F\right )}\right )} + \frac{3 \, a d^{2} f g n x^{2} \log \left (F\right ) - 2 \, a c d + 2 \,{\left ({\left (F^{e g}\right )}^{n} b d^{2} f g n x^{2} \log \left (F\right ) -{\left (F^{e g}\right )}^{n} b c d +{\left (2 \,{\left (F^{e g}\right )}^{n} b c d f g n \log \left (F\right ) -{\left (F^{e g}\right )}^{n} b d^{2}\right )} x\right )}{\left (F^{f g x}\right )}^{n} + 2 \,{\left (3 \, a c d f g n \log \left (F\right ) - a d^{2}\right )} x}{2 \,{\left (2 \,{\left (F^{f g x}\right )}^{n}{\left (F^{e g}\right )}^{n} a^{3} b f^{2} g^{2} n^{2} \log \left (F\right )^{2} +{\left (F^{f g x}\right )}^{2 \, n}{\left (F^{e g}\right )}^{2 \, n} a^{2} b^{2} f^{2} g^{2} n^{2} \log \left (F\right )^{2} + a^{4} f^{2} g^{2} n^{2} \log \left (F\right )^{2}\right )}} + \int \frac{d^{2} f^{2} g^{2} n^{2} x^{2} \log \left (F\right )^{2} - 3 \, c d f g n \log \left (F\right ) + d^{2} +{\left (2 \, c d f^{2} g^{2} n^{2} \log \left (F\right )^{2} - 3 \, d^{2} f g n \log \left (F\right )\right )} x}{{\left (F^{f g x}\right )}^{n}{\left (F^{e g}\right )}^{n} a^{2} b f^{2} g^{2} n^{2} \log \left (F\right )^{2} + a^{3} f^{2} g^{2} n^{2} \log \left (F\right )^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.269916, size = 2049, normalized size = 4.67 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(9*(a^2*d^2*e^2 - 2*a^2*c*d*e*f + a^2*c^2*f^2)*g^2*n^2*log(F)^2 + 6*(a^2*d^2
*e - a^2*c*d*f)*g*n*log(F) + 2*(a^2*d^2*f^3*g^3*n^3*x^3 + 3*a^2*c*d*f^3*g^3*n^3*
x^2 + 3*a^2*c^2*f^3*g^3*n^3*x + (a^2*d^2*e^3 - 3*a^2*c*d*e^2*f + 3*a^2*c^2*e*f^2
)*g^3*n^3)*log(F)^3 + (2*(b^2*d^2*f^3*g^3*n^3*x^3 + 3*b^2*c*d*f^3*g^3*n^3*x^2 +
3*b^2*c^2*f^3*g^3*n^3*x + (b^2*d^2*e^3 - 3*b^2*c*d*e^2*f + 3*b^2*c^2*e*f^2)*g^3*
n^3)*log(F)^3 - 9*(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*d^2*
e^2 - 2*b^2*c*d*e*f)*g^2*n^2)*log(F)^2 + 6*(b^2*d^2*f*g*n*x + b^2*d^2*e*g*n)*log
(F))*F^(2*f*g*n*x + 2*e*g*n) + 2*(2*(a*b*d^2*f^3*g^3*n^3*x^3 + 3*a*b*c*d*f^3*g^3
*n^3*x^2 + 3*a*b*c^2*f^3*g^3*n^3*x + (a*b*d^2*e^3 - 3*a*b*c*d*e^2*f + 3*a*b*c^2*
e*f^2)*g^3*n^3)*log(F)^3 - 3*(2*a*b*d^2*f^2*g^2*n^2*x^2 + 4*a*b*c*d*f^2*g^2*n^2*
x - (3*a*b*d^2*e^2 - 6*a*b*c*d*e*f + a*b*c^2*f^2)*g^2*n^2)*log(F)^2 + 3*(a*b*d^2
*f*g*n*x + (2*a*b*d^2*e - a*b*c*d*f)*g*n)*log(F))*F^(f*g*n*x + e*g*n) + 6*(3*a^2
*d^2 + (3*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*(3*a*b*d^2 - 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) - 2*(a^2*d^2*f*g*n*x + a^2*c*d*f*g*n)*log(F))*dilog(-(F^(f*g*n*x + e*g*
n)*b + a)/a + 1) - 6*((a^2*d^2*e^2 - 2*a^2*c*d*e*f + a^2*c^2*f^2)*g^2*n^2*log(F)
^2 + a^2*d^2 + 3*(a^2*d^2*e - a^2*c*d*f)*g*n*log(F) + ((b^2*d^2*e^2 - 2*b^2*c*d*
e*f + b^2*c^2*f^2)*g^2*n^2*log(F)^2 + b^2*d^2 + 3*(b^2*d^2*e - b^2*c*d*f)*g*n*lo
g(F))*F^(2*f*g*n*x + 2*e*g*n) + 2*((a*b*d^2*e^2 - 2*a*b*c*d*e*f + a*b*c^2*f^2)*g
^2*n^2*log(F)^2 + a*b*d^2 + 3*(a*b*d^2*e - a*b*c*d*f)*g*n*log(F))*F^(f*g*n*x + e
*g*n))*log(F^(f*g*n*x + e*g*n)*b + a) - 6*((a^2*d^2*f^2*g^2*n^2*x^2 + 2*a^2*c*d*
f^2*g^2*n^2*x - (a^2*d^2*e^2 - 2*a^2*c*d*e*f)*g^2*n^2)*log(F)^2 + ((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*d^2*e^2 - 2*b^2*c*d*e*f)*g^2*n^2)*l
og(F)^2 - 3*(b^2*d^2*f*g*n*x + b^2*d^2*e*g*n)*log(F))*F^(2*f*g*n*x + 2*e*g*n) +
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*d^2*e^2 - 2*a*b*c*d
*e*f)*g^2*n^2)*log(F)^2 - 3*(a*b*d^2*f*g*n*x + a*b*d^2*e*g*n)*log(F))*F^(f*g*n*x
+ e*g*n) - 3*(a^2*d^2*f*g*n*x + a^2*d^2*e*g*n)*log(F))*log((F^(f*g*n*x + e*g*n)
*b + a)/a) + 12*(2*F^(f*g*n*x + e*g*n)*a*b*d^2 + F^(2*f*g*n*x + 2*e*g*n)*b^2*d^2
+ a^2*d^2)*polylog(3, -F^(f*g*n*x + e*g*n)*b/a))/(2*F^(f*g*n*x + e*g*n)*a^4*b*f
^3*g^3*n^3*log(F)^3 + F^(2*f*g*n*x + 2*e*g*n)*a^3*b^2*f^3*g^3*n^3*log(F)^3 + a^5
*f^3*g^3*n^3*log(F)^3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{3 a c^{2} f g n \log{\left (F \right )} + 6 a c d f g n x \log{\left (F \right )} - 2 a c d + 3 a d^{2} f g n x^{2} \log{\left (F \right )} - 2 a d^{2} x + \left (2 b c^{2} f g n \log{\left (F \right )} + 4 b c d f g n x \log{\left (F \right )} - 2 b c d + 2 b d^{2} f g n x^{2} \log{\left (F \right )} - 2 b d^{2} x\right ) \left (F^{g \left (e + f x\right )}\right )^{n}}{2 a^{4} f^{2} g^{2} n^{2} \log{\left (F \right )}^{2} + 4 a^{3} b f^{2} g^{2} n^{2} \left (F^{g \left (e + f x\right )}\right )^{n} \log{\left (F \right )}^{2} + 2 a^{2} b^{2} f^{2} g^{2} n^{2} \left (F^{g \left (e + f x\right )}\right )^{2 n} \log{\left (F \right )}^{2}} + \frac{\int \frac{d^{2}}{a + b e^{e g n \log{\left (F \right )}} e^{f g n x \log{\left (F \right )}}}\, dx + \int \frac{c^{2} f^{2} g^{2} n^{2} \log{\left (F \right )}^{2}}{a + b e^{e g n \log{\left (F \right )}} e^{f g n x \log{\left (F \right )}}}\, dx + \int \left (- \frac{3 c d f g n \log{\left (F \right )}}{a + b e^{e g n \log{\left (F \right )}} e^{f g n x \log{\left (F \right )}}}\right )\, dx + \int \left (- \frac{3 d^{2} f g n x \log{\left (F \right )}}{a + b e^{e g n \log{\left (F \right )}} e^{f g n x \log{\left (F \right )}}}\right )\, dx + \int \frac{d^{2} f^{2} g^{2} n^{2} x^{2} \log{\left (F \right )}^{2}}{a + b e^{e g n \log{\left (F \right )}} e^{f g n x \log{\left (F \right )}}}\, dx + \int \frac{2 c d f^{2} g^{2} n^{2} x \log{\left (F \right )}^{2}}{a + b e^{e g n \log{\left (F \right )}} e^{f g n x \log{\left (F \right )}}}\, dx}{a^{2} f^{2} g^{2} n^{2} \log{\left (F \right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(3*a*c**2*f*g*n*log(F) + 6*a*c*d*f*g*n*x*log(F) - 2*a*c*d + 3*a*d**2*f*g*n*x**2*
log(F) - 2*a*d**2*x + (2*b*c**2*f*g*n*log(F) + 4*b*c*d*f*g*n*x*log(F) - 2*b*c*d
+ 2*b*d**2*f*g*n*x**2*log(F) - 2*b*d**2*x)*(F**(g*(e + f*x)))**n)/(2*a**4*f**2*g
**2*n**2*log(F)**2 + 4*a**3*b*f**2*g**2*n**2*(F**(g*(e + f*x)))**n*log(F)**2 + 2
*a**2*b**2*f**2*g**2*n**2*(F**(g*(e + f*x)))**(2*n)*log(F)**2) + (Integral(d**2/
(a + b*exp(e*g*n*log(F))*exp(f*g*n*x*log(F))), x) + Integral(c**2*f**2*g**2*n**2
*log(F)**2/(a + b*exp(e*g*n*log(F))*exp(f*g*n*x*log(F))), x) + Integral(-3*c*d*f
*g*n*log(F)/(a + b*exp(e*g*n*log(F))*exp(f*g*n*x*log(F))), x) + Integral(-3*d**2
*f*g*n*x*log(F)/(a + b*exp(e*g*n*log(F))*exp(f*g*n*x*log(F))), x) + Integral(d**
2*f**2*g**2*n**2*x**2*log(F)**2/(a + b*exp(e*g*n*log(F))*exp(f*g*n*x*log(F))), x
) + Integral(2*c*d*f**2*g**2*n**2*x*log(F)**2/(a + b*exp(e*g*n*log(F))*exp(f*g*n
*x*log(F))), x))/(a**2*f**2*g**2*n**2*log(F)**2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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