3.298 \(\int \frac{x^3 (d+e x)}{a^2-c^2 x^2} \, dx\)

Optimal. Leaf size=81 \[ -\frac{a^2 (a e+c d) \log (a-c x)}{2 c^5}-\frac{a^2 (c d-a e) \log (a+c x)}{2 c^5}-\frac{a^2 e x}{c^4}-\frac{d x^2}{2 c^2}-\frac{e x^3}{3 c^2} \]

[Out]

-((a^2*e*x)/c^4) - (d*x^2)/(2*c^2) - (e*x^3)/(3*c^2) - (a^2*(c*d + a*e)*Log[a -
c*x])/(2*c^5) - (a^2*(c*d - a*e)*Log[a + c*x])/(2*c^5)

_______________________________________________________________________________________

Rubi [A]  time = 0.152679, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13 \[ -\frac{a^2 (a e+c d) \log (a-c x)}{2 c^5}-\frac{a^2 (c d-a e) \log (a+c x)}{2 c^5}-\frac{a^2 e x}{c^4}-\frac{d x^2}{2 c^2}-\frac{e x^3}{3 c^2} \]

Antiderivative was successfully verified.

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

[Out]

-((a^2*e*x)/c^4) - (d*x^2)/(2*c^2) - (e*x^3)/(3*c^2) - (a^2*(c*d + a*e)*Log[a -
c*x])/(2*c^5) - (a^2*(c*d - a*e)*Log[a + c*x])/(2*c^5)

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{2} \int e\, dx}{c^{4}} + \frac{a^{2} \left (a e - c d\right ) \log{\left (a + c x \right )}}{2 c^{5}} - \frac{a^{2} \left (a e + c d\right ) \log{\left (a - c x \right )}}{2 c^{5}} - \frac{d \int x\, dx}{c^{2}} - \frac{e x^{3}}{3 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(e*x+d)/(-c**2*x**2+a**2),x)

[Out]

-a**2*Integral(e, x)/c**4 + a**2*(a*e - c*d)*log(a + c*x)/(2*c**5) - a**2*(a*e +
c*d)*log(a - c*x)/(2*c**5) - d*Integral(x, x)/c**2 - e*x**3/(3*c**2)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0243619, size = 72, normalized size = 0.89 \[ \frac{a^3 e \tanh ^{-1}\left (\frac{c x}{a}\right )}{c^5}-\frac{a^2 e x}{c^4}-\frac{a^2 d \log \left (a^2-c^2 x^2\right )}{2 c^4}-\frac{d x^2}{2 c^2}-\frac{e x^3}{3 c^2} \]

Antiderivative was successfully verified.

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

[Out]

-((a^2*e*x)/c^4) - (d*x^2)/(2*c^2) - (e*x^3)/(3*c^2) + (a^3*e*ArcTanh[(c*x)/a])/
c^5 - (a^2*d*Log[a^2 - c^2*x^2])/(2*c^4)

_______________________________________________________________________________________

Maple [A]  time = 0.01, size = 94, normalized size = 1.2 \[ -{\frac{e{x}^{3}}{3\,{c}^{2}}}-{\frac{d{x}^{2}}{2\,{c}^{2}}}-{\frac{{a}^{2}ex}{{c}^{4}}}+{\frac{{a}^{3}\ln \left ( cx+a \right ) e}{2\,{c}^{5}}}-{\frac{{a}^{2}\ln \left ( cx+a \right ) d}{2\,{c}^{4}}}-{\frac{{a}^{3}\ln \left ( cx-a \right ) e}{2\,{c}^{5}}}-{\frac{{a}^{2}\ln \left ( cx-a \right ) d}{2\,{c}^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*e*x^3/c^2-1/2*d*x^2/c^2-a^2*e*x/c^4+1/2/c^5*a^3*ln(c*x+a)*e-1/2/c^4*a^2*ln(
c*x+a)*d-1/2/c^5*a^3*ln(c*x-a)*e-1/2/c^4*a^2*ln(c*x-a)*d

_______________________________________________________________________________________

Maxima [A]  time = 0.690319, size = 109, normalized size = 1.35 \[ -\frac{2 \, c^{2} e x^{3} + 3 \, c^{2} d x^{2} + 6 \, a^{2} e x}{6 \, c^{4}} - \frac{{\left (a^{2} c d - a^{3} e\right )} \log \left (c x + a\right )}{2 \, c^{5}} - \frac{{\left (a^{2} c d + a^{3} e\right )} \log \left (c x - a\right )}{2 \, c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(e*x + d)*x^3/(c^2*x^2 - a^2),x, algorithm="maxima")

[Out]

-1/6*(2*c^2*e*x^3 + 3*c^2*d*x^2 + 6*a^2*e*x)/c^4 - 1/2*(a^2*c*d - a^3*e)*log(c*x
+ a)/c^5 - 1/2*(a^2*c*d + a^3*e)*log(c*x - a)/c^5

_______________________________________________________________________________________

Fricas [A]  time = 0.28811, size = 101, normalized size = 1.25 \[ -\frac{2 \, c^{3} e x^{3} + 3 \, c^{3} d x^{2} + 6 \, a^{2} c e x + 3 \,{\left (a^{2} c d - a^{3} e\right )} \log \left (c x + a\right ) + 3 \,{\left (a^{2} c d + a^{3} e\right )} \log \left (c x - a\right )}{6 \, c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(e*x + d)*x^3/(c^2*x^2 - a^2),x, algorithm="fricas")

[Out]

-1/6*(2*c^3*e*x^3 + 3*c^3*d*x^2 + 6*a^2*c*e*x + 3*(a^2*c*d - a^3*e)*log(c*x + a)
+ 3*(a^2*c*d + a^3*e)*log(c*x - a))/c^5

_______________________________________________________________________________________

Sympy [A]  time = 2.04243, size = 110, normalized size = 1.36 \[ - \frac{a^{2} e x}{c^{4}} + \frac{a^{2} \left (a e - c d\right ) \log{\left (x + \frac{a^{2} d + \frac{a^{2} \left (a e - c d\right )}{c}}{a^{2} e} \right )}}{2 c^{5}} - \frac{a^{2} \left (a e + c d\right ) \log{\left (x + \frac{a^{2} d - \frac{a^{2} \left (a e + c d\right )}{c}}{a^{2} e} \right )}}{2 c^{5}} - \frac{d x^{2}}{2 c^{2}} - \frac{e x^{3}}{3 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(e*x+d)/(-c**2*x**2+a**2),x)

[Out]

-a**2*e*x/c**4 + a**2*(a*e - c*d)*log(x + (a**2*d + a**2*(a*e - c*d)/c)/(a**2*e)
)/(2*c**5) - a**2*(a*e + c*d)*log(x + (a**2*d - a**2*(a*e + c*d)/c)/(a**2*e))/(2
*c**5) - d*x**2/(2*c**2) - e*x**3/(3*c**2)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.272138, size = 122, normalized size = 1.51 \[ -\frac{{\left (a^{2} c d - a^{3} e\right )}{\rm ln}\left ({\left | c x + a \right |}\right )}{2 \, c^{5}} - \frac{{\left (a^{2} c d + a^{3} e\right )}{\rm ln}\left ({\left | c x - a \right |}\right )}{2 \, c^{5}} - \frac{2 \, c^{4} x^{3} e + 3 \, c^{4} d x^{2} + 6 \, a^{2} c^{2} x e}{6 \, c^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(e*x + d)*x^3/(c^2*x^2 - a^2),x, algorithm="giac")

[Out]

-1/2*(a^2*c*d - a^3*e)*ln(abs(c*x + a))/c^5 - 1/2*(a^2*c*d + a^3*e)*ln(abs(c*x -
a))/c^5 - 1/6*(2*c^4*x^3*e + 3*c^4*d*x^2 + 6*a^2*c^2*x*e)/c^6