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

Optimal. Leaf size=98 \[ -\frac{d \text{PolyLog}\left (2,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a f^2 g^2 n^2 \log ^2(F)}-\frac{(c+d x) \log \left (\frac{b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{a f g n \log (F)}+\frac{(c+d x)^2}{2 a d} \]

[Out]

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

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Rubi [A]  time = 0.256859, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174 \[ -\frac{d \text{PolyLog}\left (2,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a f^2 g^2 n^2 \log ^2(F)}-\frac{(c+d x) \log \left (\frac{b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{a f g n \log (F)}+\frac{(c+d x)^2}{2 a d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 26.6092, size = 66, normalized size = 0.67 \[ \frac{d \operatorname{Li}_{2}\left (- \frac{a \left (F^{g \left (e + f x\right )}\right )^{- n}}{b}\right )}{a f^{2} g^{2} n^{2} \log{\left (F \right )}^{2}} - \frac{\left (c + d x\right ) \log{\left (\frac{a \left (F^{g \left (e + f x\right )}\right )^{- n}}{b} + 1 \right )}}{a f g n \log{\left (F \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 89.7332, size = 0, normalized size = 0. \[ \int \frac{c+d x}{a+b \left (F^{g (e+f x)}\right )^n} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [B]  time = 0.06, size = 526, normalized size = 5.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.266359, size = 198, normalized size = 2.02 \[ \frac{2 \,{\left (d e - c f\right )} g n \log \left (F^{f g n x + e g n} b + a\right ) \log \left (F\right ) +{\left (d f^{2} g^{2} n^{2} x^{2} + 2 \, c f^{2} g^{2} n^{2} x\right )} \log \left (F\right )^{2} - 2 \,{\left (d f g n x + d e g n\right )} \log \left (F\right ) \log \left (\frac{F^{f g n x + e g n} b + a}{a}\right ) - 2 \, d{\rm Li}_2\left (-\frac{F^{f g n x + e g n} b + a}{a} + 1\right )}{2 \, a f^{2} g^{2} n^{2} \log \left (F\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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