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

Optimal. Leaf size=192 \[ \frac{6 d^2 (c+d x) \text{PolyLog}\left (3,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a f^3 g^3 n^3 \log ^3(F)}-\frac{3 d (c+d x)^2 \text{PolyLog}\left (2,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a f^2 g^2 n^2 \log ^2(F)}-\frac{6 d^3 \text{PolyLog}\left (4,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a f^4 g^4 n^4 \log ^4(F)}-\frac{(c+d x)^3 \log \left (\frac{b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{a f g n \log (F)}+\frac{(c+d x)^4}{4 a d} \]

[Out]

(c + d*x)^4/(4*a*d) - ((c + d*x)^3*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(a*f*g*n*
Log[F]) - (3*d*(c + d*x)^2*PolyLog[2, -((b*(F^(g*(e + f*x)))^n)/a)])/(a*f^2*g^2*
n^2*Log[F]^2) + (6*d^2*(c + d*x)*PolyLog[3, -((b*(F^(g*(e + f*x)))^n)/a)])/(a*f^
3*g^3*n^3*Log[F]^3) - (6*d^3*PolyLog[4, -((b*(F^(g*(e + f*x)))^n)/a)])/(a*f^4*g^
4*n^4*Log[F]^4)

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Rubi [A]  time = 0.547824, antiderivative size = 192, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24 \[ \frac{6 d^2 (c+d x) \text{PolyLog}\left (3,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a f^3 g^3 n^3 \log ^3(F)}-\frac{3 d (c+d x)^2 \text{PolyLog}\left (2,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a f^2 g^2 n^2 \log ^2(F)}-\frac{6 d^3 \text{PolyLog}\left (4,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a f^4 g^4 n^4 \log ^4(F)}-\frac{(c+d x)^3 \log \left (\frac{b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{a f g n \log (F)}+\frac{(c+d x)^4}{4 a d} \]

Antiderivative was successfully verified.

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

[Out]

(c + d*x)^4/(4*a*d) - ((c + d*x)^3*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(a*f*g*n*
Log[F]) - (3*d*(c + d*x)^2*PolyLog[2, -((b*(F^(g*(e + f*x)))^n)/a)])/(a*f^2*g^2*
n^2*Log[F]^2) + (6*d^2*(c + d*x)*PolyLog[3, -((b*(F^(g*(e + f*x)))^n)/a)])/(a*f^
3*g^3*n^3*Log[F]^3) - (6*d^3*PolyLog[4, -((b*(F^(g*(e + f*x)))^n)/a)])/(a*f^4*g^
4*n^4*Log[F]^4)

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Rubi in Sympy [A]  time = 94.0252, size = 160, normalized size = 0.83 \[ \frac{6 d^{3} \operatorname{Li}_{4}\left (- \frac{a \left (F^{g \left (e + f x\right )}\right )^{- n}}{b}\right )}{a f^{4} g^{4} n^{4} \log{\left (F \right )}^{4}} + \frac{6 d^{2} \left (c + d x\right ) \operatorname{Li}_{3}\left (- \frac{a \left (F^{g \left (e + f x\right )}\right )^{- n}}{b}\right )}{a f^{3} g^{3} n^{3} \log{\left (F \right )}^{3}} + \frac{3 d \left (c + d x\right )^{2} \operatorname{Li}_{2}\left (- \frac{a \left (F^{g \left (e + f x\right )}\right )^{- n}}{b}\right )}{a f^{2} g^{2} n^{2} \log{\left (F \right )}^{2}} - \frac{\left (c + d x\right )^{3} \log{\left (\frac{a \left (F^{g \left (e + f x\right )}\right )^{- n}}{b} + 1 \right )}}{a f g n \log{\left (F \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 2.84445, size = 0, normalized size = 0. \[ \int \frac{(c+d x)^3}{a+b \left (F^{g (e+f x)}\right )^n} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [B]  time = 0.083, size = 2495, normalized size = 13. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/n/g^2/f^2/ln(F)^2*c^2*d*(ln(F^(g*(f*x+e)))-g*(f*x+e)*ln(F))/a*ln((F^(g*(f*x+e
)))^n)+3/n/g^2/f^2/ln(F)^2*c^2*d*(ln(F^(g*(f*x+e)))-g*(f*x+e)*ln(F))/a*ln(a+b*(F
^(g*(f*x+e)))^n)-3/n/g/f/ln(F)*c*d^2/a*ln(1+b*(F^(g*(f*x+e)))^n/a)*x^2+3/n/g/f^3
/ln(F)*c*d^2/a*ln(1+b*(F^(g*(f*x+e)))^n/a)*e^2+3/n/g^3/f^3/ln(F)^3*c*d^2/a*ln(1+
b*(F^(g*(f*x+e)))^n/a)*(ln(F^(g*(f*x+e)))-g*(f*x+e)*ln(F))^2-6/n^2/g^2/f^2/ln(F)
^2*c*d^2/a*polylog(2,-b*(F^(g*(f*x+e)))^n/a)*x-3/n/g^2/f^4/ln(F)^2*d^3*(ln(F^(g*
(f*x+e)))-g*(f*x+e)*ln(F))/a*ln(1+b*(F^(g*(f*x+e)))^n/a)*e^2-3/n/g^3/f^4/ln(F)^3
*d^3*(ln(F^(g*(f*x+e)))-g*(f*x+e)*ln(F))^2/a*ln(1+b*(F^(g*(f*x+e)))^n/a)*e-3/n/g
/f/ln(F)*c^2*d/a*ln(1+b*(F^(g*(f*x+e)))^n/a)*x-3/n/g/f^2/ln(F)*c^2*d/a*ln(1+b*(F
^(g*(f*x+e)))^n/a)*e-3/n/g^2/f^2/ln(F)^2*c^2*d/a*ln(1+b*(F^(g*(f*x+e)))^n/a)*(ln
(F^(g*(f*x+e)))-g*(f*x+e)*ln(F))-3/n/g^3/f^4/ln(F)^3*d^3*e*(ln(F^(g*(f*x+e)))-g*
(f*x+e)*ln(F))^2/a*ln((F^(g*(f*x+e)))^n)-6*d^3*polylog(4,-b*(F^(g*(f*x+e)))^n/a)
/a/f^4/g^4/n^4/ln(F)^4+3*c*d^2/a*x^3-6/g/f^2/ln(F)*c*d^2*(ln(F^(g*(f*x+e)))-g*(f
*x+e)*ln(F))/a*x*e+3/n/g/f^3/ln(F)*c*d^2*e^2/a*ln((F^(g*(f*x+e)))^n)-3/n/g/f^3/l
n(F)*c*d^2*e^2/a*ln(a+b*(F^(g*(f*x+e)))^n)+3/n/g^3/f^4/ln(F)^3*d^3*e*(ln(F^(g*(f
*x+e)))-g*(f*x+e)*ln(F))^2/a*ln(a+b*(F^(g*(f*x+e)))^n)+3/n/g^3/f^3/ln(F)^3*c*d^2
*(ln(F^(g*(f*x+e)))-g*(f*x+e)*ln(F))^2/a*ln((F^(g*(f*x+e)))^n)-3/n/g/f^2/ln(F)*c
^2*d*e/a*ln((F^(g*(f*x+e)))^n)+3/n/g/f^2/ln(F)*c^2*d*e/a*ln(a+b*(F^(g*(f*x+e)))^
n)-3/n/g^3/f^3/ln(F)^3*c*d^2*(ln(F^(g*(f*x+e)))-g*(f*x+e)*ln(F))^2/a*ln(a+b*(F^(
g*(f*x+e)))^n)-3/n/g^2/f^4/ln(F)^2*d^3*e^2*(ln(F^(g*(f*x+e)))-g*(f*x+e)*ln(F))/a
*ln((F^(g*(f*x+e)))^n)+3/n/g^2/f^4/ln(F)^2*d^3*e^2*(ln(F^(g*(f*x+e)))-g*(f*x+e)*
ln(F))/a*ln(a+b*(F^(g*(f*x+e)))^n)+1/n/g^4/f^4/ln(F)^4*d^3*(ln(F^(g*(f*x+e)))-g*
(f*x+e)*ln(F))^3/a*ln(a+b*(F^(g*(f*x+e)))^n)-3/n^2/g^2/f^2/ln(F)^2*c^2*d/a*polyl
og(2,-b*(F^(g*(f*x+e)))^n/a)+6/n^3/g^3/f^3/ln(F)^3*c*d^2/a*polylog(3,-b*(F^(g*(f
*x+e)))^n/a)-1/n/g^4/f^4/ln(F)^4*d^3*(ln(F^(g*(f*x+e)))-g*(f*x+e)*ln(F))^3/a*ln(
1+b*(F^(g*(f*x+e)))^n/a)-1/n/g/f^4/ln(F)*d^3*e^3/a*ln(1+b*(F^(g*(f*x+e)))^n/a)-3
/n^2/g^2/f^2/ln(F)^2*d^3/a*polylog(2,-b*(F^(g*(f*x+e)))^n/a)*x^2+3/g/f/ln(F)*c^2
*d/a*x*(ln(F^(g*(f*x+e)))-g*(f*x+e)*ln(F))-1/n/g^4/f^4/ln(F)^4*d^3*(ln(F^(g*(f*x
+e)))-g*(f*x+e)*ln(F))^3/a*ln((F^(g*(f*x+e)))^n)-3/g/f/ln(F)*c^2*d/a*ln(F^(g*(f*
x+e)))*x-1/n/g/f^4/ln(F)*d^3*e^3/a*ln((F^(g*(f*x+e)))^n)+1/n/g/f^4/ln(F)*d^3*e^3
/a*ln(a+b*(F^(g*(f*x+e)))^n)+3/g/f^3/ln(F)*d^3/a*x*e^2*(ln(F^(g*(f*x+e)))-g*(f*x
+e)*ln(F))+3/g^2/f^3/ln(F)^2*d^3/a*x*e*(ln(F^(g*(f*x+e)))-g*(f*x+e)*ln(F))^2-3/g
^2/f^2/ln(F)^2*c*d^2*(ln(F^(g*(f*x+e)))-g*(f*x+e)*ln(F))^2/a*x-1/n/g/f/ln(F)*d^3
/a*ln(1+b*(F^(g*(f*x+e)))^n/a)*x^3+6/n^3/g^3/f^3/ln(F)^3*d^3/a*polylog(3,-b*(F^(
g*(f*x+e)))^n/a)*x-6/g/f/ln(F)*c*d^2/a*ln(F^(g*(f*x+e)))*x^2+6/g^2/f^2/ln(F)^2*c
*d^2/a*ln(F^(g*(f*x+e)))^2*x-3/g/f/ln(F)*d^3/a*ln(F^(g*(f*x+e)))*x^3+9/2/g^2/f^2
/ln(F)^2*d^3/a*ln(F^(g*(f*x+e)))^2*x^2-3/g^3/f^3/ln(F)^3*d^3/a*ln(F^(g*(f*x+e)))
^3*x-2/g^3/f^3/ln(F)^3*c*d^2/a*ln(F^(g*(f*x+e)))^3+3/2/g^2/f^2/ln(F)^2*c^2*d/a*l
n(F^(g*(f*x+e)))^2+1/g^3/f^3/ln(F)^3*d^3/a*x*(ln(F^(g*(f*x+e)))-g*(f*x+e)*ln(F))
^3+1/n/g/f/ln(F)*c^3/a*ln((F^(g*(f*x+e)))^n)-1/n/g/f/ln(F)*c^3/a*ln(a+b*(F^(g*(f
*x+e)))^n)+3*c^2*d/a*x^2+6/n/g^2/f^3/ln(F)^2*c*d^2*e*(ln(F^(g*(f*x+e)))-g*(f*x+e
)*ln(F))/a*ln((F^(g*(f*x+e)))^n)+6/n/g^2/f^3/ln(F)^2*c*d^2/a*ln(1+b*(F^(g*(f*x+e
)))^n/a)*e*(ln(F^(g*(f*x+e)))-g*(f*x+e)*ln(F))-6/n/g^2/f^3/ln(F)^2*c*d^2*e*(ln(F
^(g*(f*x+e)))-g*(f*x+e)*ln(F))/a*ln(a+b*(F^(g*(f*x+e)))^n)+d^3/a*x^4+3/f*c^2*d/a
*x*e-3/f^2*c*d^2*e^2/a*x+1/f^3*d^3/a*x*e^3+3/4/g^4/f^4/ln(F)^4*d^3/a*ln(F^(g*(f*
x+e)))^4

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.266186, size = 558, normalized size = 2.91 \[ \frac{4 \,{\left (d^{3} e^{3} - 3 \, c d^{2} e^{2} f + 3 \, c^{2} d e f^{2} - c^{3} f^{3}\right )} g^{3} n^{3} \log \left (F^{f g n x + e g n} b + a\right ) \log \left (F\right )^{3} +{\left (d^{3} f^{4} g^{4} n^{4} x^{4} + 4 \, c d^{2} f^{4} g^{4} n^{4} x^{3} + 6 \, c^{2} d f^{4} g^{4} n^{4} x^{2} + 4 \, c^{3} f^{4} g^{4} n^{4} x\right )} \log \left (F\right )^{4} - 4 \,{\left (d^{3} f^{3} g^{3} n^{3} x^{3} + 3 \, c d^{2} f^{3} g^{3} n^{3} x^{2} + 3 \, c^{2} d f^{3} g^{3} n^{3} x +{\left (d^{3} e^{3} - 3 \, c d^{2} e^{2} f + 3 \, c^{2} d e f^{2}\right )} g^{3} n^{3}\right )} \log \left (F\right )^{3} \log \left (\frac{F^{f g n x + e g n} b + a}{a}\right ) - 12 \,{\left (d^{3} f^{2} g^{2} n^{2} x^{2} + 2 \, c d^{2} f^{2} g^{2} n^{2} x + c^{2} d f^{2} g^{2} n^{2}\right )}{\rm Li}_2\left (-\frac{F^{f g n x + e g n} b + a}{a} + 1\right ) \log \left (F\right )^{2} - 24 \, d^{3}{\rm Li}_{4}(-\frac{F^{f g n x + e g n} b}{a}) + 24 \,{\left (d^{3} f g n x + c d^{2} f g n\right )} \log \left (F\right ){\rm Li}_{3}(-\frac{F^{f g n x + e g n} b}{a})}{4 \, a f^{4} g^{4} n^{4} \log \left (F\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(4*(d^3*e^3 - 3*c*d^2*e^2*f + 3*c^2*d*e*f^2 - c^3*f^3)*g^3*n^3*log(F^(f*g*n*
x + e*g*n)*b + a)*log(F)^3 + (d^3*f^4*g^4*n^4*x^4 + 4*c*d^2*f^4*g^4*n^4*x^3 + 6*
c^2*d*f^4*g^4*n^4*x^2 + 4*c^3*f^4*g^4*n^4*x)*log(F)^4 - 4*(d^3*f^3*g^3*n^3*x^3 +
3*c*d^2*f^3*g^3*n^3*x^2 + 3*c^2*d*f^3*g^3*n^3*x + (d^3*e^3 - 3*c*d^2*e^2*f + 3*
c^2*d*e*f^2)*g^3*n^3)*log(F)^3*log((F^(f*g*n*x + e*g*n)*b + a)/a) - 12*(d^3*f^2*
g^2*n^2*x^2 + 2*c*d^2*f^2*g^2*n^2*x + c^2*d*f^2*g^2*n^2)*dilog(-(F^(f*g*n*x + e*
g*n)*b + a)/a + 1)*log(F)^2 - 24*d^3*polylog(4, -F^(f*g*n*x + e*g*n)*b/a) + 24*(
d^3*f*g*n*x + c*d^2*f*g*n)*log(F)*polylog(3, -F^(f*g*n*x + e*g*n)*b/a))/(a*f^4*g
^4*n^4*log(F)^4)

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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