3.132 \(\int \frac{x^3}{a+b x^2} \, dx\)

Optimal. Leaf size=27 \[ \frac{x^2}{2 b}-\frac{a \log \left (a+b x^2\right )}{2 b^2} \]

[Out]

x^2/(2*b) - (a*Log[a + b*x^2])/(2*b^2)

_______________________________________________________________________________________

Rubi [A]  time = 0.04987, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{x^2}{2 b}-\frac{a \log \left (a+b x^2\right )}{2 b^2} \]

Antiderivative was successfully verified.

[In]  Int[x^3/(a + b*x^2),x]

[Out]

x^2/(2*b) - (a*Log[a + b*x^2])/(2*b^2)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3/(b*x**2+a),x)

[Out]

-a*log(a + b*x**2)/(2*b**2) + Integral(1/b, (x, x**2))/2

_______________________________________________________________________________________

Mathematica [A]  time = 0.00759544, size = 27, normalized size = 1. \[ \frac{x^2}{2 b}-\frac{a \log \left (a+b x^2\right )}{2 b^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/(a + b*x^2),x]

[Out]

x^2/(2*b) - (a*Log[a + b*x^2])/(2*b^2)

_______________________________________________________________________________________

Maple [A]  time = 0.003, size = 24, normalized size = 0.9 \[{\frac{{x}^{2}}{2\,b}}-{\frac{a\ln \left ( b{x}^{2}+a \right ) }{2\,{b}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(b*x^2+a),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.34588, size = 31, normalized size = 1.15 \[ \frac{x^{2}}{2 \, b} - \frac{a \log \left (b x^{2} + a\right )}{2 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(b*x^2 + a),x, algorithm="maxima")

[Out]

1/2*x^2/b - 1/2*a*log(b*x^2 + a)/b^2

_______________________________________________________________________________________

Fricas [A]  time = 0.198994, size = 30, normalized size = 1.11 \[ \frac{b x^{2} - a \log \left (b x^{2} + a\right )}{2 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(b*x^2 + a),x, algorithm="fricas")

[Out]

1/2*(b*x^2 - a*log(b*x^2 + a))/b^2

_______________________________________________________________________________________

Sympy [A]  time = 1.16468, size = 20, normalized size = 0.74 \[ - \frac{a \log{\left (a + b x^{2} \right )}}{2 b^{2}} + \frac{x^{2}}{2 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3/(b*x**2+a),x)

[Out]

-a*log(a + b*x**2)/(2*b**2) + x**2/(2*b)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.224063, size = 32, normalized size = 1.19 \[ \frac{x^{2}}{2 \, b} - \frac{a{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(b*x^2 + a),x, algorithm="giac")

[Out]

1/2*x^2/b - 1/2*a*ln(abs(b*x^2 + a))/b^2