3.300 \(\int \frac{x (d+e x)}{a^2-c^2 x^2} \, dx\)

Optimal. Leaf size=50 \[ -\frac{(a e+c d) \log (a-c x)}{2 c^3}-\frac{(c d-a e) \log (a+c x)}{2 c^3}-\frac{e x}{c^2} \]

[Out]

-((e*x)/c^2) - ((c*d + a*e)*Log[a - c*x])/(2*c^3) - ((c*d - a*e)*Log[a + c*x])/(
2*c^3)

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Rubi [A]  time = 0.0995554, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ -\frac{(a e+c d) \log (a-c x)}{2 c^3}-\frac{(c d-a e) \log (a+c x)}{2 c^3}-\frac{e x}{c^2} \]

Antiderivative was successfully verified.

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

[Out]

-((e*x)/c^2) - ((c*d + a*e)*Log[a - c*x])/(2*c^3) - ((c*d - a*e)*Log[a + c*x])/(
2*c^3)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{\int e\, dx}{c^{2}} + \frac{\left (a e - c d\right ) \log{\left (a + c x \right )}}{2 c^{3}} - \frac{\left (a e + c d\right ) \log{\left (a - c x \right )}}{2 c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(e, x)/c**2 + (a*e - c*d)*log(a + c*x)/(2*c**3) - (a*e + c*d)*log(a - c
*x)/(2*c**3)

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Mathematica [A]  time = 0.0159777, size = 42, normalized size = 0.84 \[ -\frac{d \log \left (a^2-c^2 x^2\right )}{2 c^2}+\frac{a e \tanh ^{-1}\left (\frac{c x}{a}\right )}{c^3}-\frac{e x}{c^2} \]

Antiderivative was successfully verified.

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

[Out]

-((e*x)/c^2) + (a*e*ArcTanh[(c*x)/a])/c^3 - (d*Log[a^2 - c^2*x^2])/(2*c^2)

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Maple [A]  time = 0.009, size = 63, normalized size = 1.3 \[ -{\frac{ex}{{c}^{2}}}+{\frac{\ln \left ( cx+a \right ) ae}{2\,{c}^{3}}}-{\frac{\ln \left ( cx+a \right ) d}{2\,{c}^{2}}}-{\frac{\ln \left ( cx-a \right ) ae}{2\,{c}^{3}}}-{\frac{\ln \left ( cx-a \right ) d}{2\,{c}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(e*x+d)/(-c^2*x^2+a^2),x)

[Out]

-e*x/c^2+1/2/c^3*ln(c*x+a)*a*e-1/2/c^2*ln(c*x+a)*d-1/2/c^3*ln(c*x-a)*a*e-1/2/c^2
*ln(c*x-a)*d

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Maxima [A]  time = 0.689189, size = 63, normalized size = 1.26 \[ -\frac{e x}{c^{2}} - \frac{{\left (c d - a e\right )} \log \left (c x + a\right )}{2 \, c^{3}} - \frac{{\left (c d + a e\right )} \log \left (c x - a\right )}{2 \, c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-e*x/c^2 - 1/2*(c*d - a*e)*log(c*x + a)/c^3 - 1/2*(c*d + a*e)*log(c*x - a)/c^3

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Fricas [A]  time = 0.288444, size = 57, normalized size = 1.14 \[ -\frac{2 \, c e x +{\left (c d - a e\right )} \log \left (c x + a\right ) +{\left (c d + a e\right )} \log \left (c x - a\right )}{2 \, c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(2*c*e*x + (c*d - a*e)*log(c*x + a) + (c*d + a*e)*log(c*x - a))/c^3

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Sympy [A]  time = 1.77534, size = 60, normalized size = 1.2 \[ - \frac{e x}{c^{2}} + \frac{\left (a e - c d\right ) \log{\left (x + \frac{d + \frac{a e - c d}{c}}{e} \right )}}{2 c^{3}} - \frac{\left (a e + c d\right ) \log{\left (x + \frac{d - \frac{a e + c d}{c}}{e} \right )}}{2 c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-e*x/c**2 + (a*e - c*d)*log(x + (d + (a*e - c*d)/c)/e)/(2*c**3) - (a*e + c*d)*lo
g(x + (d - (a*e + c*d)/c)/e)/(2*c**3)

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GIAC/XCAS [A]  time = 0.269304, size = 70, normalized size = 1.4 \[ -\frac{x e}{c^{2}} - \frac{{\left (c d - a e\right )}{\rm ln}\left ({\left | c x + a \right |}\right )}{2 \, c^{3}} - \frac{{\left (c d + a e\right )}{\rm ln}\left ({\left | c x - a \right |}\right )}{2 \, c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x*e/c^2 - 1/2*(c*d - a*e)*ln(abs(c*x + a))/c^3 - 1/2*(c*d + a*e)*ln(abs(c*x - a
))/c^3