3.435 \(\int \frac{(a^2+2 a b x^2+b^2 x^4)^2}{x^8} \, dx\)

Optimal. Leaf size=47 \[ -\frac{2 a^2 b^2}{x^3}-\frac{4 a^3 b}{5 x^5}-\frac{a^4}{7 x^7}-\frac{4 a b^3}{x}+b^4 x \]

[Out]

-a^4/(7*x^7) - (4*a^3*b)/(5*x^5) - (2*a^2*b^2)/x^3 - (4*a*b^3)/x + b^4*x

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Rubi [A]  time = 0.02538, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {28, 270} \[ -\frac{2 a^2 b^2}{x^3}-\frac{4 a^3 b}{5 x^5}-\frac{a^4}{7 x^7}-\frac{4 a b^3}{x}+b^4 x \]

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^2/x^8,x]

[Out]

-a^4/(7*x^7) - (4*a^3*b)/(5*x^5) - (2*a^2*b^2)/x^3 - (4*a*b^3)/x + b^4*x

Rule 28

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*
p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

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^2+2 a b x^2+b^2 x^4\right )^2}{x^8} \, dx &=\frac{\int \frac{\left (a b+b^2 x^2\right )^4}{x^8} \, dx}{b^4}\\ &=\frac{\int \left (b^8+\frac{a^4 b^4}{x^8}+\frac{4 a^3 b^5}{x^6}+\frac{6 a^2 b^6}{x^4}+\frac{4 a b^7}{x^2}\right ) \, dx}{b^4}\\ &=-\frac{a^4}{7 x^7}-\frac{4 a^3 b}{5 x^5}-\frac{2 a^2 b^2}{x^3}-\frac{4 a b^3}{x}+b^4 x\\ \end{align*}

Mathematica [A]  time = 0.0052106, size = 47, normalized size = 1. \[ -\frac{2 a^2 b^2}{x^3}-\frac{4 a^3 b}{5 x^5}-\frac{a^4}{7 x^7}-\frac{4 a b^3}{x}+b^4 x \]

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^2/x^8,x]

[Out]

-a^4/(7*x^7) - (4*a^3*b)/(5*x^5) - (2*a^2*b^2)/x^3 - (4*a*b^3)/x + b^4*x

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Maple [A]  time = 0.049, size = 44, normalized size = 0.9 \begin{align*} -{\frac{{a}^{4}}{7\,{x}^{7}}}-{\frac{4\,{a}^{3}b}{5\,{x}^{5}}}-2\,{\frac{{b}^{2}{a}^{2}}{{x}^{3}}}-4\,{\frac{a{b}^{3}}{x}}+{b}^{4}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b^2*x^4+2*a*b*x^2+a^2)^2/x^8,x)

[Out]

-1/7*a^4/x^7-4/5*a^3*b/x^5-2*a^2*b^2/x^3-4*a*b^3/x+b^4*x

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Maxima [A]  time = 0.994078, size = 62, normalized size = 1.32 \begin{align*} b^{4} x - \frac{140 \, a b^{3} x^{6} + 70 \, a^{2} b^{2} x^{4} + 28 \, a^{3} b x^{2} + 5 \, a^{4}}{35 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^4+2*a*b*x^2+a^2)^2/x^8,x, algorithm="maxima")

[Out]

b^4*x - 1/35*(140*a*b^3*x^6 + 70*a^2*b^2*x^4 + 28*a^3*b*x^2 + 5*a^4)/x^7

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Fricas [A]  time = 1.59812, size = 107, normalized size = 2.28 \begin{align*} \frac{35 \, b^{4} x^{8} - 140 \, a b^{3} x^{6} - 70 \, a^{2} b^{2} x^{4} - 28 \, a^{3} b x^{2} - 5 \, a^{4}}{35 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^4+2*a*b*x^2+a^2)^2/x^8,x, algorithm="fricas")

[Out]

1/35*(35*b^4*x^8 - 140*a*b^3*x^6 - 70*a^2*b^2*x^4 - 28*a^3*b*x^2 - 5*a^4)/x^7

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Sympy [A]  time = 0.42316, size = 46, normalized size = 0.98 \begin{align*} b^{4} x - \frac{5 a^{4} + 28 a^{3} b x^{2} + 70 a^{2} b^{2} x^{4} + 140 a b^{3} x^{6}}{35 x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b**2*x**4+2*a*b*x**2+a**2)**2/x**8,x)

[Out]

b**4*x - (5*a**4 + 28*a**3*b*x**2 + 70*a**2*b**2*x**4 + 140*a*b**3*x**6)/(35*x**7)

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Giac [A]  time = 1.10015, size = 62, normalized size = 1.32 \begin{align*} b^{4} x - \frac{140 \, a b^{3} x^{6} + 70 \, a^{2} b^{2} x^{4} + 28 \, a^{3} b x^{2} + 5 \, a^{4}}{35 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^4+2*a*b*x^2+a^2)^2/x^8,x, algorithm="giac")

[Out]

b^4*x - 1/35*(140*a*b^3*x^6 + 70*a^2*b^2*x^4 + 28*a^3*b*x^2 + 5*a^4)/x^7