3.460 \(\int \frac{(a^2+2 a b x^2+b^2 x^4)^3}{x^8} \, dx\)

Optimal. Leaf size=72 \[ -\frac{5 a^4 b^2}{x^3}-\frac{20 a^3 b^3}{x}+15 a^2 b^4 x-\frac{6 a^5 b}{5 x^5}-\frac{a^6}{7 x^7}+2 a b^5 x^3+\frac{b^6 x^5}{5} \]

[Out]

-a^6/(7*x^7) - (6*a^5*b)/(5*x^5) - (5*a^4*b^2)/x^3 - (20*a^3*b^3)/x + 15*a^2*b^4*x + 2*a*b^5*x^3 + (b^6*x^5)/5

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Rubi [A]  time = 0.0369146, antiderivative size = 72, 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} \[ -\frac{5 a^4 b^2}{x^3}-\frac{20 a^3 b^3}{x}+15 a^2 b^4 x-\frac{6 a^5 b}{5 x^5}-\frac{a^6}{7 x^7}+2 a b^5 x^3+\frac{b^6 x^5}{5} \]

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^8,x]

[Out]

-a^6/(7*x^7) - (6*a^5*b)/(5*x^5) - (5*a^4*b^2)/x^3 - (20*a^3*b^3)/x + 15*a^2*b^4*x + 2*a*b^5*x^3 + (b^6*x^5)/5

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 )^3}{x^8} \, dx &=\frac{\int \frac{\left (a b+b^2 x^2\right )^6}{x^8} \, dx}{b^6}\\ &=\frac{\int \left (15 a^2 b^{10}+\frac{a^6 b^6}{x^8}+\frac{6 a^5 b^7}{x^6}+\frac{15 a^4 b^8}{x^4}+\frac{20 a^3 b^9}{x^2}+6 a b^{11} x^2+b^{12} x^4\right ) \, dx}{b^6}\\ &=-\frac{a^6}{7 x^7}-\frac{6 a^5 b}{5 x^5}-\frac{5 a^4 b^2}{x^3}-\frac{20 a^3 b^3}{x}+15 a^2 b^4 x+2 a b^5 x^3+\frac{b^6 x^5}{5}\\ \end{align*}

Mathematica [A]  time = 0.0092379, size = 72, normalized size = 1. \[ -\frac{5 a^4 b^2}{x^3}-\frac{20 a^3 b^3}{x}+15 a^2 b^4 x-\frac{6 a^5 b}{5 x^5}-\frac{a^6}{7 x^7}+2 a b^5 x^3+\frac{b^6 x^5}{5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^8,x]

[Out]

-a^6/(7*x^7) - (6*a^5*b)/(5*x^5) - (5*a^4*b^2)/x^3 - (20*a^3*b^3)/x + 15*a^2*b^4*x + 2*a*b^5*x^3 + (b^6*x^5)/5

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Maple [A]  time = 0.049, size = 67, normalized size = 0.9 \begin{align*} -{\frac{{a}^{6}}{7\,{x}^{7}}}-{\frac{6\,{a}^{5}b}{5\,{x}^{5}}}-5\,{\frac{{a}^{4}{b}^{2}}{{x}^{3}}}-20\,{\frac{{a}^{3}{b}^{3}}{x}}+15\,{a}^{2}{b}^{4}x+2\,a{b}^{5}{x}^{3}+{\frac{{b}^{6}{x}^{5}}{5}} \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)^3/x^8,x)

[Out]

-1/7*a^6/x^7-6/5*a^5*b/x^5-5*a^4*b^2/x^3-20*a^3*b^3/x+15*a^2*b^4*x+2*a*b^5*x^3+1/5*b^6*x^5

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Maxima [A]  time = 0.97182, size = 93, normalized size = 1.29 \begin{align*} \frac{1}{5} \, b^{6} x^{5} + 2 \, a b^{5} x^{3} + 15 \, a^{2} b^{4} x - \frac{700 \, a^{3} b^{3} x^{6} + 175 \, a^{4} b^{2} x^{4} + 42 \, a^{5} b x^{2} + 5 \, a^{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)^3/x^8,x, algorithm="maxima")

[Out]

1/5*b^6*x^5 + 2*a*b^5*x^3 + 15*a^2*b^4*x - 1/35*(700*a^3*b^3*x^6 + 175*a^4*b^2*x^4 + 42*a^5*b*x^2 + 5*a^6)/x^7

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Fricas [A]  time = 1.7017, size = 157, normalized size = 2.18 \begin{align*} \frac{7 \, b^{6} x^{12} + 70 \, a b^{5} x^{10} + 525 \, a^{2} b^{4} x^{8} - 700 \, a^{3} b^{3} x^{6} - 175 \, a^{4} b^{2} x^{4} - 42 \, a^{5} b x^{2} - 5 \, a^{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)^3/x^8,x, algorithm="fricas")

[Out]

1/35*(7*b^6*x^12 + 70*a*b^5*x^10 + 525*a^2*b^4*x^8 - 700*a^3*b^3*x^6 - 175*a^4*b^2*x^4 - 42*a^5*b*x^2 - 5*a^6)
/x^7

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Sympy [A]  time = 0.470031, size = 71, normalized size = 0.99 \begin{align*} 15 a^{2} b^{4} x + 2 a b^{5} x^{3} + \frac{b^{6} x^{5}}{5} - \frac{5 a^{6} + 42 a^{5} b x^{2} + 175 a^{4} b^{2} x^{4} + 700 a^{3} 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)**3/x**8,x)

[Out]

15*a**2*b**4*x + 2*a*b**5*x**3 + b**6*x**5/5 - (5*a**6 + 42*a**5*b*x**2 + 175*a**4*b**2*x**4 + 700*a**3*b**3*x
**6)/(35*x**7)

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Giac [A]  time = 1.1558, size = 93, normalized size = 1.29 \begin{align*} \frac{1}{5} \, b^{6} x^{5} + 2 \, a b^{5} x^{3} + 15 \, a^{2} b^{4} x - \frac{700 \, a^{3} b^{3} x^{6} + 175 \, a^{4} b^{2} x^{4} + 42 \, a^{5} b x^{2} + 5 \, a^{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)^3/x^8,x, algorithm="giac")

[Out]

1/5*b^6*x^5 + 2*a*b^5*x^3 + 15*a^2*b^4*x - 1/35*(700*a^3*b^3*x^6 + 175*a^4*b^2*x^4 + 42*a^5*b*x^2 + 5*a^6)/x^7