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

Optimal. Leaf size=191 \[ -\frac{d \text{PolyLog}\left (2,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^2 g^2 n^2 \log ^2(F)}-\frac{(c+d x) \log \left (\frac{b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{a^2 f g n \log (F)}+\frac{d \log \left (a+b \left (F^{g (e+f x)}\right )^n\right )}{a^2 f^2 g^2 n^2 \log ^2(F)}+\frac{(c+d x)^2}{2 a^2 d}-\frac{d x}{a^2 f g n \log (F)}+\frac{c+d x}{a f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )} \]

[Out]

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

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Rubi [A]  time = 0.56891, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 11, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.478 \[ -\frac{d \text{PolyLog}\left (2,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^2 g^2 n^2 \log ^2(F)}-\frac{(c+d x) \log \left (\frac{b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{a^2 f g n \log (F)}+\frac{d \log \left (a+b \left (F^{g (e+f x)}\right )^n\right )}{a^2 f^2 g^2 n^2 \log ^2(F)}+\frac{(c+d x)^2}{2 a^2 d}-\frac{d x}{a^2 f g n \log (F)}+\frac{c+d x}{a f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{c + d x}{a f g n \left (a + b \left (F^{g \left (e + f x\right )}\right )^{n}\right ) \log{\left (F \right )}} + \frac{d \int x\, dx}{a^{2}} - \frac{d x \log{\left (1 + \frac{b \left (F^{g \left (e + f x\right )}\right )^{n}}{a} \right )}}{a^{2} f g n \log{\left (F \right )}} + \frac{d x \log{\left (a + b \left (F^{g \left (e + f x\right )}\right )^{n} \right )}}{a^{2} f g n \log{\left (F \right )}} - \frac{d x \log{\left (\left (F^{g \left (e + f x\right )}\right )^{n} \right )}}{a^{2} f g n \log{\left (F \right )}} + \frac{d \log{\left (a + b \left (F^{g \left (e + f x\right )}\right )^{n} \right )}}{a^{2} f^{2} g^{2} n^{2} \log{\left (F \right )}^{2}} - \frac{d \log{\left (\left (F^{g \left (e + f x\right )}\right )^{n} \right )}}{a^{2} f^{2} g^{2} n^{2} \log{\left (F \right )}^{2}} - \frac{d \operatorname{Li}_{2}\left (- \frac{b \left (F^{g \left (e + f x\right )}\right )^{n}}{a}\right )}{a^{2} f^{2} g^{2} n^{2} \log{\left (F \right )}^{2}} - \frac{\left (c + d x\right ) \log{\left (a + b \left (F^{g \left (e + f x\right )}\right )^{n} \right )}}{a^{2} f g n \log{\left (F \right )}} + \frac{\left (c + d x\right ) \log{\left (\left (F^{g \left (e + f x\right )}\right )^{n} \right )}}{a^{2} f g n \log{\left (F \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 90.7211, size = 0, normalized size = 0. \[ \int \frac{c+d x}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^2} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [B]  time = 0.085, size = 437, normalized size = 2.3 \[{\frac{dx+c}{af \left ( a+b \left ({F}^{g \left ( fx+e \right ) } \right ) ^{n} \right ) gn\ln \left ( F \right ) }}+{\frac{c\ln \left ( \left ({F}^{g \left ( fx+e \right ) } \right ) ^{n} \right ) }{\ln \left ( F \right ){a}^{2}fgn}}-{\frac{c\ln \left ( a+b \left ({F}^{g \left ( fx+e \right ) } \right ) ^{n} \right ) }{\ln \left ( F \right ){a}^{2}fgn}}-{\frac{d}{{a}^{2}{f}^{2}{g}^{2}{n}^{2} \left ( \ln \left ( F \right ) \right ) ^{2}}{\it polylog} \left ( 2,-{\frac{b \left ({F}^{g \left ( fx+e \right ) } \right ) ^{n}}{a}} \right ) }-{\frac{d\ln \left ( \left ({F}^{g \left ( fx+e \right ) } \right ) ^{n} \right ) }{{a}^{2}{f}^{2}{g}^{2}{n}^{2} \left ( \ln \left ( F \right ) \right ) ^{2}}}+{\frac{d\ln \left ( a+b \left ({F}^{g \left ( fx+e \right ) } \right ) ^{n} \right ) }{{a}^{2}{f}^{2}{g}^{2}{n}^{2} \left ( \ln \left ( F \right ) \right ) ^{2}}}-{\frac{d\ln \left ({F}^{g \left ( fx+e \right ) } \right ) }{ \left ( \ln \left ( F \right ) \right ) ^{2}{a}^{2}{f}^{2}{g}^{2}n}\ln \left ( 1+{\frac{b \left ({F}^{g \left ( fx+e \right ) } \right ) ^{n}}{a}} \right ) }+{\frac{d\ln \left ( \left ({F}^{g \left ( fx+e \right ) } \right ) ^{n} \right ) x}{\ln \left ( F \right ){a}^{2}fgn}}-{\frac{d\ln \left ( \left ({F}^{g \left ( fx+e \right ) } \right ) ^{n} \right ) \ln \left ({F}^{g \left ( fx+e \right ) } \right ) }{ \left ( \ln \left ( F \right ) \right ) ^{2}{a}^{2}{f}^{2}{g}^{2}n}}-{\frac{d\ln \left ( a+b \left ({F}^{g \left ( fx+e \right ) } \right ) ^{n} \right ) x}{\ln \left ( F \right ){a}^{2}fgn}}+{\frac{d\ln \left ( a+b \left ({F}^{g \left ( fx+e \right ) } \right ) ^{n} \right ) \ln \left ({F}^{g \left ( fx+e \right ) } \right ) }{ \left ( \ln \left ( F \right ) \right ) ^{2}{a}^{2}{f}^{2}{g}^{2}n}}+{\frac{d \left ( \ln \left ({F}^{g \left ( fx+e \right ) } \right ) \right ) ^{2}}{2\, \left ( \ln \left ( F \right ) \right ) ^{2}{a}^{2}{f}^{2}{g}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.267044, size = 540, normalized size = 2.83 \[ -\frac{2 \,{\left (a d e - a c f\right )} g n \log \left (F\right ) -{\left (a d f^{2} g^{2} n^{2} x^{2} + 2 \, a c f^{2} g^{2} n^{2} x -{\left (a d e^{2} - 2 \, a c e f\right )} g^{2} n^{2}\right )} \log \left (F\right )^{2} -{\left ({\left (b d f^{2} g^{2} n^{2} x^{2} + 2 \, b c f^{2} g^{2} n^{2} x -{\left (b d e^{2} - 2 \, b c e f\right )} g^{2} n^{2}\right )} \log \left (F\right )^{2} - 2 \,{\left (b d f g n x + b d e g n\right )} \log \left (F\right )\right )} F^{f g n x + e g n} + 2 \,{\left (F^{f g n x + e g n} b d + a d\right )}{\rm Li}_2\left (-\frac{F^{f g n x + e g n} b + a}{a} + 1\right ) - 2 \,{\left ({\left (a d e - a c f\right )} g n \log \left (F\right ) +{\left ({\left (b d e - b c f\right )} g n \log \left (F\right ) + b d\right )} F^{f g n x + e g n} + a d\right )} \log \left (F^{f g n x + e g n} b + a\right ) + 2 \,{\left ({\left (b d f g n x + b d e g n\right )} F^{f g n x + e g n} \log \left (F\right ) +{\left (a d f g n x + a d e g n\right )} \log \left (F\right )\right )} \log \left (\frac{F^{f g n x + e g n} b + a}{a}\right )}{2 \,{\left (F^{f g n x + e g n} a^{2} b f^{2} g^{2} n^{2} \log \left (F\right )^{2} + a^{3} f^{2} g^{2} n^{2} \log \left (F\right )^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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