3.163 \(\int \frac{a^x x}{(1+b x)^2} \, dx\)

Optimal. Leaf size=64 \[ -\frac{a^{-1/b} \log (a) \text{ExpIntegralEi}\left (\frac{\log (a) (b x+1)}{b}\right )}{b^3}+\frac{a^{-1/b} \text{ExpIntegralEi}\left (\frac{\log (a) (b x+1)}{b}\right )}{b^2}+\frac{a^x}{b^2 (b x+1)} \]

[Out]

a^x/(b^2*(1 + b*x)) + ExpIntegralEi[((1 + b*x)*Log[a])/b]/(a^b^(-1)*b^2) - (ExpI
ntegralEi[((1 + b*x)*Log[a])/b]*Log[a])/(a^b^(-1)*b^3)

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Rubi [A]  time = 0.153365, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{a^{-1/b} \log (a) \text{ExpIntegralEi}\left (\frac{\log (a) (b x+1)}{b}\right )}{b^3}+\frac{a^{-1/b} \text{ExpIntegralEi}\left (\frac{\log (a) (b x+1)}{b}\right )}{b^2}+\frac{a^x}{b^2 (b x+1)} \]

Antiderivative was successfully verified.

[In]  Int[(a^x*x)/(1 + b*x)^2,x]

[Out]

a^x/(b^2*(1 + b*x)) + ExpIntegralEi[((1 + b*x)*Log[a])/b]/(a^b^(-1)*b^2) - (ExpI
ntegralEi[((1 + b*x)*Log[a])/b]*Log[a])/(a^b^(-1)*b^3)

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Rubi in Sympy [A]  time = 7.41885, size = 54, normalized size = 0.84 \[ \frac{a^{x}}{b^{2} \left (b x + 1\right )} + \frac{a^{- \frac{1}{b}} \operatorname{Ei}{\left (\frac{\left (b x + 1\right ) \log{\left (a \right )}}{b} \right )}}{b^{2}} - \frac{a^{- \frac{1}{b}} \log{\left (a \right )} \operatorname{Ei}{\left (\frac{\left (b x + 1\right ) \log{\left (a \right )}}{b} \right )}}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(a**x*x/(b*x+1)**2,x)

[Out]

a**x/(b**2*(b*x + 1)) + a**(-1/b)*Ei((b*x + 1)*log(a)/b)/b**2 - a**(-1/b)*log(a)
*Ei((b*x + 1)*log(a)/b)/b**3

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Mathematica [A]  time = 0.0507419, size = 43, normalized size = 0.67 \[ \frac{a^{-1/b} (b-\log (a)) \text{ExpIntegralEi}\left (\frac{\log (a) (b x+1)}{b}\right )+\frac{b a^x}{b x+1}}{b^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^x*x)/(1 + b*x)^2,x]

[Out]

((a^x*b)/(1 + b*x) + (ExpIntegralEi[((1 + b*x)*Log[a])/b]*(b - Log[a]))/a^b^(-1)
)/b^3

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Maple [A]  time = 0.02, size = 79, normalized size = 1.2 \[ -{\frac{1}{{b}^{2}}{a}^{-{b}^{-1}}{\it Ei} \left ( 1,-x\ln \left ( a \right ) -{\frac{\ln \left ( a \right ) }{b}} \right ) }+{\frac{{a}^{x}\ln \left ( a \right ) }{{b}^{3}} \left ( x\ln \left ( a \right ) +{\frac{\ln \left ( a \right ) }{b}} \right ) ^{-1}}+{\frac{\ln \left ( a \right ) }{{b}^{3}}{a}^{-{b}^{-1}}{\it Ei} \left ( 1,-x\ln \left ( a \right ) -{\frac{\ln \left ( a \right ) }{b}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(a^x*x/(b*x+1)^2,x)

[Out]

-1/b^2*a^(-1/b)*Ei(1,-x*ln(a)-ln(a)/b)+ln(a)/b^3*a^x/(x*ln(a)+ln(a)/b)+ln(a)/b^3
*a^(-1/b)*Ei(1,-x*ln(a)-ln(a)/b)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(a^x*x/(b*x + 1)^2,x, algorithm="maxima")

[Out]

a^x*x/(b^2*x^2*log(a) + 2*b*x*log(a) + log(a)) + integrate((b*x - 1)*a^x/(b^3*x^
3*log(a) + 3*b^2*x^2*log(a) + 3*b*x*log(a) + log(a)), x)

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Fricas [A]  time = 0.25096, size = 73, normalized size = 1.14 \[ \frac{a^{x} b + \frac{{\left (b^{2} x -{\left (b x + 1\right )} \log \left (a\right ) + b\right )}{\rm Ei}\left (\frac{{\left (b x + 1\right )} \log \left (a\right )}{b}\right )}{a^{\left (\frac{1}{b}\right )}}}{b^{4} x + b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(a^x*x/(b*x + 1)^2,x, algorithm="fricas")

[Out]

(a^x*b + (b^2*x - (b*x + 1)*log(a) + b)*Ei((b*x + 1)*log(a)/b)/a^(1/b))/(b^4*x +
b^3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{a^{x} x}{\left (b x + 1\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(a**x*x/(b*x+1)**2,x)

[Out]

Integral(a**x*x/(b*x + 1)**2, x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{a^{x} x}{{\left (b x + 1\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(a^x*x/(b*x + 1)^2,x, algorithm="giac")

[Out]

integrate(a^x*x/(b*x + 1)^2, x)