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

Optimal. Leaf size=39 \[ a^3 \log (x)+\frac{3}{2} a^2 b x^2+\frac{3}{4} a b^2 x^4+\frac{b^3 x^6}{6} \]

[Out]

(3*a^2*b*x^2)/2 + (3*a*b^2*x^4)/4 + (b^3*x^6)/6 + a^3*Log[x]

_______________________________________________________________________________________

Rubi [A]  time = 0.0519704, antiderivative size = 39, 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 \[ a^3 \log (x)+\frac{3}{2} a^2 b x^2+\frac{3}{4} a b^2 x^4+\frac{b^3 x^6}{6} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^3/x,x]

[Out]

(3*a^2*b*x^2)/2 + (3*a*b^2*x^4)/4 + (b^3*x^6)/6 + a^3*Log[x]

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{a^{3} \log{\left (x^{2} \right )}}{2} + \frac{3 a^{2} b x^{2}}{2} + \frac{3 a b^{2} \int ^{x^{2}} x\, dx}{2} + \frac{b^{3} x^{6}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**3*log(x**2)/2 + 3*a**2*b*x**2/2 + 3*a*b**2*Integral(x, (x, x**2))/2 + b**3*x*
*6/6

_______________________________________________________________________________________

Mathematica [A]  time = 0.00654973, size = 39, normalized size = 1. \[ a^3 \log (x)+\frac{3}{2} a^2 b x^2+\frac{3}{4} a b^2 x^4+\frac{b^3 x^6}{6} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^3/x,x]

[Out]

(3*a^2*b*x^2)/2 + (3*a*b^2*x^4)/4 + (b^3*x^6)/6 + a^3*Log[x]

_______________________________________________________________________________________

Maple [A]  time = 0.003, size = 34, normalized size = 0.9 \[{\frac{3\,{a}^{2}b{x}^{2}}{2}}+{\frac{3\,a{b}^{2}{x}^{4}}{4}}+{\frac{{b}^{3}{x}^{6}}{6}}+{a}^{3}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^3/x,x)

[Out]

3/2*a^2*b*x^2+3/4*a*b^2*x^4+1/6*b^3*x^6+a^3*ln(x)

_______________________________________________________________________________________

Maxima [A]  time = 1.34637, size = 49, normalized size = 1.26 \[ \frac{1}{6} \, b^{3} x^{6} + \frac{3}{4} \, a b^{2} x^{4} + \frac{3}{2} \, a^{2} b x^{2} + \frac{1}{2} \, a^{3} \log \left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*b^3*x^6 + 3/4*a*b^2*x^4 + 3/2*a^2*b*x^2 + 1/2*a^3*log(x^2)

_______________________________________________________________________________________

Fricas [A]  time = 0.20157, size = 45, normalized size = 1.15 \[ \frac{1}{6} \, b^{3} x^{6} + \frac{3}{4} \, a b^{2} x^{4} + \frac{3}{2} \, a^{2} b x^{2} + a^{3} \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*b^3*x^6 + 3/4*a*b^2*x^4 + 3/2*a^2*b*x^2 + a^3*log(x)

_______________________________________________________________________________________

Sympy [A]  time = 1.05655, size = 37, normalized size = 0.95 \[ a^{3} \log{\left (x \right )} + \frac{3 a^{2} b x^{2}}{2} + \frac{3 a b^{2} x^{4}}{4} + \frac{b^{3} x^{6}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**3/x,x)

[Out]

a**3*log(x) + 3*a**2*b*x**2/2 + 3*a*b**2*x**4/4 + b**3*x**6/6

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.209217, size = 49, normalized size = 1.26 \[ \frac{1}{6} \, b^{3} x^{6} + \frac{3}{4} \, a b^{2} x^{4} + \frac{3}{2} \, a^{2} b x^{2} + \frac{1}{2} \, a^{3}{\rm ln}\left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*b^3*x^6 + 3/4*a*b^2*x^4 + 3/2*a^2*b*x^2 + 1/2*a^3*ln(x^2)