3.629 \(\int \frac{(a+b x^4)^2}{x^3} \, dx\)

Optimal. Leaf size=27 \[ -\frac{a^2}{2 x^2}+a b x^2+\frac{b^2 x^6}{6} \]

[Out]

-a^2/(2*x^2) + a*b*x^2 + (b^2*x^6)/6

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Rubi [A]  time = 0.0093361, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {270} \[ -\frac{a^2}{2 x^2}+a b x^2+\frac{b^2 x^6}{6} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^4)^2/x^3,x]

[Out]

-a^2/(2*x^2) + a*b*x^2 + (b^2*x^6)/6

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \frac{\left (a+b x^4\right )^2}{x^3} \, dx &=\int \left (\frac{a^2}{x^3}+2 a b x+b^2 x^5\right ) \, dx\\ &=-\frac{a^2}{2 x^2}+a b x^2+\frac{b^2 x^6}{6}\\ \end{align*}

Mathematica [A]  time = 0.0007915, size = 27, normalized size = 1. \[ -\frac{a^2}{2 x^2}+a b x^2+\frac{b^2 x^6}{6} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^4)^2/x^3,x]

[Out]

-a^2/(2*x^2) + a*b*x^2 + (b^2*x^6)/6

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Maple [A]  time = 0.002, size = 24, normalized size = 0.9 \begin{align*} -{\frac{{a}^{2}}{2\,{x}^{2}}}+ab{x}^{2}+{\frac{{b}^{2}{x}^{6}}{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^4+a)^2/x^3,x)

[Out]

-1/2/x^2*a^2+a*b*x^2+1/6*b^2*x^6

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Maxima [A]  time = 0.990109, size = 31, normalized size = 1.15 \begin{align*} \frac{1}{6} \, b^{2} x^{6} + a b x^{2} - \frac{a^{2}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^2/x^3,x, algorithm="maxima")

[Out]

1/6*b^2*x^6 + a*b*x^2 - 1/2*a^2/x^2

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Fricas [A]  time = 1.3818, size = 53, normalized size = 1.96 \begin{align*} \frac{b^{2} x^{8} + 6 \, a b x^{4} - 3 \, a^{2}}{6 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^2/x^3,x, algorithm="fricas")

[Out]

1/6*(b^2*x^8 + 6*a*b*x^4 - 3*a^2)/x^2

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Sympy [A]  time = 0.334676, size = 22, normalized size = 0.81 \begin{align*} - \frac{a^{2}}{2 x^{2}} + a b x^{2} + \frac{b^{2} x^{6}}{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**4+a)**2/x**3,x)

[Out]

-a**2/(2*x**2) + a*b*x**2 + b**2*x**6/6

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Giac [A]  time = 1.11119, size = 31, normalized size = 1.15 \begin{align*} \frac{1}{6} \, b^{2} x^{6} + a b x^{2} - \frac{a^{2}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^2/x^3,x, algorithm="giac")

[Out]

1/6*b^2*x^6 + a*b*x^2 - 1/2*a^2/x^2