3.578 \(\int \frac{\left (b e-a e e^{c+d x}\right ) x}{b e-2 a e e^{c+d x}-b e e^{2 (c+d x)}} \, dx\)

Optimal. Leaf size=150 \[ -\frac{\text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{2 d^2}-\frac{\text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{2 d^2}-\frac{x \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{2 d}-\frac{x \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{2 d}+\frac{x^2}{2} \]

[Out]

x^2/2 - (x*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(2*d) - (x*Log[1 + (b
*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(2*d) - PolyLog[2, -((b*E^(c + d*x))/(a -
Sqrt[a^2 + b^2]))]/(2*d^2) - PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))
]/(2*d^2)

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Rubi [A]  time = 0.986193, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 47, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.106 \[ -\frac{\text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{2 d^2}-\frac{\text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{2 d^2}-\frac{x \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{2 d}-\frac{x \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{2 d}+\frac{x^2}{2} \]

Antiderivative was successfully verified.

[In]  Int[((b*e - a*e*E^(c + d*x))*x)/(b*e - 2*a*e*E^(c + d*x) - b*e*E^(2*(c + d*x))),x]

[Out]

x^2/2 - (x*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(2*d) - (x*Log[1 + (b
*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(2*d) - PolyLog[2, -((b*E^(c + d*x))/(a -
Sqrt[a^2 + b^2]))]/(2*d^2) - PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))
]/(2*d^2)

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Rubi in Sympy [A]  time = 68.3132, size = 121, normalized size = 0.81 \[ - \frac{x \log{\left (1 - \frac{\left (- a + \sqrt{a^{2} + b^{2}}\right ) e^{- c - d x}}{b} \right )}}{2 d} - \frac{x \log{\left (1 + \frac{\left (a + \sqrt{a^{2} + b^{2}}\right ) e^{- c - d x}}{b} \right )}}{2 d} + \frac{\operatorname{Li}_{2}\left (- \frac{\left (a - \sqrt{a^{2} + b^{2}}\right ) e^{- c - d x}}{b}\right )}{2 d^{2}} + \frac{\operatorname{Li}_{2}\left (- \frac{\left (a + \sqrt{a^{2} + b^{2}}\right ) e^{- c - d x}}{b}\right )}{2 d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*e-a*e*exp(d*x+c))*x/(b*e-2*a*e*exp(d*x+c)-b*e*exp(2*d*x+2*c)),x)

[Out]

-x*log(1 - (-a + sqrt(a**2 + b**2))*exp(-c - d*x)/b)/(2*d) - x*log(1 + (a + sqrt
(a**2 + b**2))*exp(-c - d*x)/b)/(2*d) + polylog(2, -(a - sqrt(a**2 + b**2))*exp(
-c - d*x)/b)/(2*d**2) + polylog(2, -(a + sqrt(a**2 + b**2))*exp(-c - d*x)/b)/(2*
d**2)

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Mathematica [A]  time = 0.391094, size = 176, normalized size = 1.17 \[ -\frac{\text{PolyLog}\left (2,-\frac{b e^{2 c+d x}}{a e^c-\sqrt{e^{2 c} \left (a^2+b^2\right )}}\right )+\text{PolyLog}\left (2,-\frac{b e^{2 c+d x}}{\sqrt{e^{2 c} \left (a^2+b^2\right )}+a e^c}\right )+d x \left (\log \left (\frac{b e^{2 c+d x}}{a e^c-\sqrt{e^{2 c} \left (a^2+b^2\right )}}+1\right )+\log \left (\frac{b e^{2 c+d x}}{\sqrt{e^{2 c} \left (a^2+b^2\right )}+a e^c}+1\right )-d x\right )}{2 d^2} \]

Antiderivative was successfully verified.

[In]  Integrate[((b*e - a*e*E^(c + d*x))*x)/(b*e - 2*a*e*E^(c + d*x) - b*e*E^(2*(c + d*x))),x]

[Out]

-(d*x*(-(d*x) + Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])] +
Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])]) + PolyLog[2, -(
(b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))] + PolyLog[2, -((b*E^(2*c
+ d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))])/(2*d^2)

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Maple [C]  time = 0.054, size = 285, normalized size = 1.9 \[ -{\frac{x}{2\,d}\ln \left ( -{1 \left ({{\rm e}^{2\,c}}{{\rm e}^{dx}}b+{{\rm e}^{c}}a-\sqrt{ \left ({{\rm e}^{c}} \right ) ^{2}{a}^{2}+{{\rm e}^{2\,c}}{b}^{2}} \right ) \left ( -{{\rm e}^{c}}a+\sqrt{ \left ({{\rm e}^{c}} \right ) ^{2}{a}^{2}+{{\rm e}^{2\,c}}{b}^{2}} \right ) ^{-1}} \right ) }-{\frac{x}{2\,d}\ln \left ({1 \left ({{\rm e}^{2\,c}}{{\rm e}^{dx}}b+{{\rm e}^{c}}a+\sqrt{ \left ({{\rm e}^{c}} \right ) ^{2}{a}^{2}+{{\rm e}^{2\,c}}{b}^{2}} \right ) \left ({{\rm e}^{c}}a+\sqrt{ \left ({{\rm e}^{c}} \right ) ^{2}{a}^{2}+{{\rm e}^{2\,c}}{b}^{2}} \right ) ^{-1}} \right ) }-{\frac{1}{2\,{d}^{2}}{\it dilog} \left ( -{1 \left ({{\rm e}^{2\,c}}{{\rm e}^{dx}}b+{{\rm e}^{c}}a-\sqrt{ \left ({{\rm e}^{c}} \right ) ^{2}{a}^{2}+{{\rm e}^{2\,c}}{b}^{2}} \right ) \left ( -{{\rm e}^{c}}a+\sqrt{ \left ({{\rm e}^{c}} \right ) ^{2}{a}^{2}+{{\rm e}^{2\,c}}{b}^{2}} \right ) ^{-1}} \right ) }-{\frac{1}{2\,{d}^{2}}{\it dilog} \left ({1 \left ({{\rm e}^{2\,c}}{{\rm e}^{dx}}b+{{\rm e}^{c}}a+\sqrt{ \left ({{\rm e}^{c}} \right ) ^{2}{a}^{2}+{{\rm e}^{2\,c}}{b}^{2}} \right ) \left ({{\rm e}^{c}}a+\sqrt{ \left ({{\rm e}^{c}} \right ) ^{2}{a}^{2}+{{\rm e}^{2\,c}}{b}^{2}} \right ) ^{-1}} \right ) }+{\frac{{x}^{2}}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*e-a*e*exp(d*x+c))*x/(b*e-2*a*e*exp(d*x+c)-b*e*exp(2*d*x+2*c)),x)

[Out]

-1/2/d*x*ln(-(exp(2*c)*exp(d*x)*b+exp(c)*a-(exp(c)^2*a^2+exp(2*c)*b^2)^(1/2))/(-
exp(c)*a+(exp(c)^2*a^2+exp(2*c)*b^2)^(1/2)))-1/2/d*x*ln((exp(2*c)*exp(d*x)*b+exp
(c)*a+(exp(c)^2*a^2+exp(2*c)*b^2)^(1/2))/(exp(c)*a+(exp(c)^2*a^2+exp(2*c)*b^2)^(
1/2)))-1/2/d^2*dilog(-(exp(2*c)*exp(d*x)*b+exp(c)*a-(exp(c)^2*a^2+exp(2*c)*b^2)^
(1/2))/(-exp(c)*a+(exp(c)^2*a^2+exp(2*c)*b^2)^(1/2)))-1/2/d^2*dilog((exp(2*c)*ex
p(d*x)*b+exp(c)*a+(exp(c)^2*a^2+exp(2*c)*b^2)^(1/2))/(exp(c)*a+(exp(c)^2*a^2+exp
(2*c)*b^2)^(1/2)))+1/2*x^2

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((a*e*e^(d*x + c) - b*e)*x/(b*e*e^(2*d*x + 2*c) + 2*a*e*e^(d*x + c) - b
*e), x)

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Fricas [A]  time = 0.324853, size = 393, normalized size = 2.62 \[ \frac{d^{2} x^{2} + c \log \left (2 \, b e^{\left (d x + c\right )} + 2 \, b \sqrt{\frac{a^{2} + b^{2}}{b^{2}}} + 2 \, a\right ) + c \log \left (2 \, b e^{\left (d x + c\right )} - 2 \, b \sqrt{\frac{a^{2} + b^{2}}{b^{2}}} + 2 \, a\right ) -{\left (d x + c\right )} \log \left (\frac{b e^{\left (d x + c\right )} + b \sqrt{\frac{a^{2} + b^{2}}{b^{2}}} + a}{b \sqrt{\frac{a^{2} + b^{2}}{b^{2}}} + a}\right ) -{\left (d x + c\right )} \log \left (-\frac{b e^{\left (d x + c\right )} - b \sqrt{\frac{a^{2} + b^{2}}{b^{2}}} + a}{b \sqrt{\frac{a^{2} + b^{2}}{b^{2}}} - a}\right ) -{\rm Li}_2\left (-\frac{b e^{\left (d x + c\right )} + b \sqrt{\frac{a^{2} + b^{2}}{b^{2}}} + a}{b \sqrt{\frac{a^{2} + b^{2}}{b^{2}}} + a} + 1\right ) -{\rm Li}_2\left (\frac{b e^{\left (d x + c\right )} - b \sqrt{\frac{a^{2} + b^{2}}{b^{2}}} + a}{b \sqrt{\frac{a^{2} + b^{2}}{b^{2}}} - a} + 1\right )}{2 \, d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(d^2*x^2 + c*log(2*b*e^(d*x + c) + 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) + c*log(
2*b*e^(d*x + c) - 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) - (d*x + c)*log((b*e^(d*x + c
) + b*sqrt((a^2 + b^2)/b^2) + a)/(b*sqrt((a^2 + b^2)/b^2) + a)) - (d*x + c)*log(
-(b*e^(d*x + c) - b*sqrt((a^2 + b^2)/b^2) + a)/(b*sqrt((a^2 + b^2)/b^2) - a)) -
dilog(-(b*e^(d*x + c) + b*sqrt((a^2 + b^2)/b^2) + a)/(b*sqrt((a^2 + b^2)/b^2) +
a) + 1) - dilog((b*e^(d*x + c) - b*sqrt((a^2 + b^2)/b^2) + a)/(b*sqrt((a^2 + b^2
)/b^2) - a) + 1))/d^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*e-a*e*exp(d*x+c))*x/(b*e-2*a*e*exp(d*x+c)-b*e*exp(2*d*x+2*c)),x)

[Out]

Integral(x*(a*exp(c)*exp(d*x) - b)/(2*a*exp(c)*exp(d*x) + b*exp(2*c)*exp(2*d*x)
- b), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((a*e*e^(d*x + c) - b*e)*x/(b*e*e^(2*d*x + 2*c) + 2*a*e*e^(d*x + c) - b
*e), x)