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

Optimal. Leaf size=26 \[ -\frac{a^2}{3 x^3}+2 a b x+\frac{b^2 x^5}{5} \]

[Out]

-a^2/(3*x^3) + 2*a*b*x + (b^2*x^5)/5

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Rubi [A]  time = 0.0087323, antiderivative size = 26, 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} \[ -\frac{a^2}{3 x^3}+2 a b x+\frac{b^2 x^5}{5} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^4)^2/x^4,x]

[Out]

-a^2/(3*x^3) + 2*a*b*x + (b^2*x^5)/5

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

Mathematica [A]  time = 0.0006119, size = 26, normalized size = 1. \[ -\frac{a^2}{3 x^3}+2 a b x+\frac{b^2 x^5}{5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^4)^2/x^4,x]

[Out]

-a^2/(3*x^3) + 2*a*b*x + (b^2*x^5)/5

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Maple [A]  time = 0.003, size = 23, normalized size = 0.9 \begin{align*} -{\frac{{a}^{2}}{3\,{x}^{3}}}+2\,xab+{\frac{{b}^{2}{x}^{5}}{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*a^2/x^3+2*x*a*b+1/5*b^2*x^5

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Maxima [A]  time = 0.975209, size = 30, normalized size = 1.15 \begin{align*} \frac{1}{5} \, b^{2} x^{5} + 2 \, a b x - \frac{a^{2}}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*b^2*x^5 + 2*a*b*x - 1/3*a^2/x^3

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Fricas [A]  time = 1.36813, size = 58, normalized size = 2.23 \begin{align*} \frac{3 \, b^{2} x^{8} + 30 \, a b x^{4} - 5 \, a^{2}}{15 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*(3*b^2*x^8 + 30*a*b*x^4 - 5*a^2)/x^3

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Sympy [A]  time = 0.363274, size = 22, normalized size = 0.85 \begin{align*} - \frac{a^{2}}{3 x^{3}} + 2 a b x + \frac{b^{2} x^{5}}{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a**2/(3*x**3) + 2*a*b*x + b**2*x**5/5

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Giac [A]  time = 1.09158, size = 30, normalized size = 1.15 \begin{align*} \frac{1}{5} \, b^{2} x^{5} + 2 \, a b x - \frac{a^{2}}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*b^2*x^5 + 2*a*b*x - 1/3*a^2/x^3