3.458 \(\int \frac{(a^2+2 a b x^2+b^2 x^4)^3}{x^6} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0408074, 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} \[ 5 a^2 b^4 x^3+20 a^3 b^3 x-\frac{15 a^4 b^2}{x}-\frac{2 a^5 b}{x^3}-\frac{a^6}{5 x^5}+\frac{6}{5} a b^5 x^5+\frac{b^6 x^7}{7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0063316, size = 72, normalized size = 1. \[ 5 a^2 b^4 x^3+20 a^3 b^3 x-\frac{15 a^4 b^2}{x}-\frac{2 a^5 b}{x^3}-\frac{a^6}{5 x^5}+\frac{6}{5} a b^5 x^5+\frac{b^6 x^7}{7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.046, size = 67, normalized size = 0.9 \begin{align*} -{\frac{{a}^{6}}{5\,{x}^{5}}}-2\,{\frac{{a}^{5}b}{{x}^{3}}}-15\,{\frac{{a}^{4}{b}^{2}}{x}}+20\,x{a}^{3}{b}^{3}+5\,{a}^{2}{b}^{4}{x}^{3}+{\frac{6\,a{b}^{5}{x}^{5}}{5}}+{\frac{{b}^{6}{x}^{7}}{7}} \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^6,x)

[Out]

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

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Maxima [A]  time = 0.987634, size = 90, normalized size = 1.25 \begin{align*} \frac{1}{7} \, b^{6} x^{7} + \frac{6}{5} \, a b^{5} x^{5} + 5 \, a^{2} b^{4} x^{3} + 20 \, a^{3} b^{3} x - \frac{75 \, a^{4} b^{2} x^{4} + 10 \, a^{5} b x^{2} + a^{6}}{5 \, x^{5}} \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^6,x, algorithm="maxima")

[Out]

1/7*b^6*x^7 + 6/5*a*b^5*x^5 + 5*a^2*b^4*x^3 + 20*a^3*b^3*x - 1/5*(75*a^4*b^2*x^4 + 10*a^5*b*x^2 + a^6)/x^5

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Fricas [A]  time = 1.59192, size = 157, normalized size = 2.18 \begin{align*} \frac{5 \, b^{6} x^{12} + 42 \, a b^{5} x^{10} + 175 \, a^{2} b^{4} x^{8} + 700 \, a^{3} b^{3} x^{6} - 525 \, a^{4} b^{2} x^{4} - 70 \, a^{5} b x^{2} - 7 \, a^{6}}{35 \, x^{5}} \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^6,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.409117, size = 71, normalized size = 0.99 \begin{align*} 20 a^{3} b^{3} x + 5 a^{2} b^{4} x^{3} + \frac{6 a b^{5} x^{5}}{5} + \frac{b^{6} x^{7}}{7} - \frac{a^{6} + 10 a^{5} b x^{2} + 75 a^{4} b^{2} x^{4}}{5 x^{5}} \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**6,x)

[Out]

20*a**3*b**3*x + 5*a**2*b**4*x**3 + 6*a*b**5*x**5/5 + b**6*x**7/7 - (a**6 + 10*a**5*b*x**2 + 75*a**4*b**2*x**4
)/(5*x**5)

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Giac [A]  time = 1.17887, size = 90, normalized size = 1.25 \begin{align*} \frac{1}{7} \, b^{6} x^{7} + \frac{6}{5} \, a b^{5} x^{5} + 5 \, a^{2} b^{4} x^{3} + 20 \, a^{3} b^{3} x - \frac{75 \, a^{4} b^{2} x^{4} + 10 \, a^{5} b x^{2} + a^{6}}{5 \, x^{5}} \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^6,x, algorithm="giac")

[Out]

1/7*b^6*x^7 + 6/5*a*b^5*x^5 + 5*a^2*b^4*x^3 + 20*a^3*b^3*x - 1/5*(75*a^4*b^2*x^4 + 10*a^5*b*x^2 + a^6)/x^5