3.461 \(\int \frac{(a^2+2 a b x^2+b^2 x^4)^3}{x^9} \, dx\)

Optimal. Leaf size=73 \[ -\frac{15 a^4 b^2}{4 x^4}-\frac{10 a^3 b^3}{x^2}+15 a^2 b^4 \log (x)-\frac{a^5 b}{x^6}-\frac{a^6}{8 x^8}+3 a b^5 x^2+\frac{b^6 x^4}{4} \]

[Out]

-a^6/(8*x^8) - (a^5*b)/x^6 - (15*a^4*b^2)/(4*x^4) - (10*a^3*b^3)/x^2 + 3*a*b^5*x^2 + (b^6*x^4)/4 + 15*a^2*b^4*
Log[x]

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Rubi [A]  time = 0.0520787, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {28, 266, 43} \[ -\frac{15 a^4 b^2}{4 x^4}-\frac{10 a^3 b^3}{x^2}+15 a^2 b^4 \log (x)-\frac{a^5 b}{x^6}-\frac{a^6}{8 x^8}+3 a b^5 x^2+\frac{b^6 x^4}{4} \]

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^9,x]

[Out]

-a^6/(8*x^8) - (a^5*b)/x^6 - (15*a^4*b^2)/(4*x^4) - (10*a^3*b^3)/x^2 + 3*a*b^5*x^2 + (b^6*x^4)/4 + 15*a^2*b^4*
Log[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 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0080131, size = 73, normalized size = 1. \[ -\frac{15 a^4 b^2}{4 x^4}-\frac{10 a^3 b^3}{x^2}+15 a^2 b^4 \log (x)-\frac{a^5 b}{x^6}-\frac{a^6}{8 x^8}+3 a b^5 x^2+\frac{b^6 x^4}{4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^9,x]

[Out]

-a^6/(8*x^8) - (a^5*b)/x^6 - (15*a^4*b^2)/(4*x^4) - (10*a^3*b^3)/x^2 + 3*a*b^5*x^2 + (b^6*x^4)/4 + 15*a^2*b^4*
Log[x]

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Maple [A]  time = 0.049, size = 68, normalized size = 0.9 \begin{align*} -{\frac{{a}^{6}}{8\,{x}^{8}}}-{\frac{{a}^{5}b}{{x}^{6}}}-{\frac{15\,{a}^{4}{b}^{2}}{4\,{x}^{4}}}-10\,{\frac{{a}^{3}{b}^{3}}{{x}^{2}}}+3\,a{b}^{5}{x}^{2}+{\frac{{b}^{6}{x}^{4}}{4}}+15\,{a}^{2}{b}^{4}\ln \left ( x \right ) \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^9,x)

[Out]

-1/8*a^6/x^8-a^5*b/x^6-15/4*a^4*b^2/x^4-10*a^3*b^3/x^2+3*a*b^5*x^2+1/4*b^6*x^4+15*a^2*b^4*ln(x)

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Maxima [A]  time = 0.983707, size = 95, normalized size = 1.3 \begin{align*} \frac{1}{4} \, b^{6} x^{4} + 3 \, a b^{5} x^{2} + \frac{15}{2} \, a^{2} b^{4} \log \left (x^{2}\right ) - \frac{80 \, a^{3} b^{3} x^{6} + 30 \, a^{4} b^{2} x^{4} + 8 \, a^{5} b x^{2} + a^{6}}{8 \, x^{8}} \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^9,x, algorithm="maxima")

[Out]

1/4*b^6*x^4 + 3*a*b^5*x^2 + 15/2*a^2*b^4*log(x^2) - 1/8*(80*a^3*b^3*x^6 + 30*a^4*b^2*x^4 + 8*a^5*b*x^2 + a^6)/
x^8

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Fricas [A]  time = 1.69756, size = 158, normalized size = 2.16 \begin{align*} \frac{2 \, b^{6} x^{12} + 24 \, a b^{5} x^{10} + 120 \, a^{2} b^{4} x^{8} \log \left (x\right ) - 80 \, a^{3} b^{3} x^{6} - 30 \, a^{4} b^{2} x^{4} - 8 \, a^{5} b x^{2} - a^{6}}{8 \, x^{8}} \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^9,x, algorithm="fricas")

[Out]

1/8*(2*b^6*x^12 + 24*a*b^5*x^10 + 120*a^2*b^4*x^8*log(x) - 80*a^3*b^3*x^6 - 30*a^4*b^2*x^4 - 8*a^5*b*x^2 - a^6
)/x^8

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Sympy [A]  time = 0.527583, size = 71, normalized size = 0.97 \begin{align*} 15 a^{2} b^{4} \log{\left (x \right )} + 3 a b^{5} x^{2} + \frac{b^{6} x^{4}}{4} - \frac{a^{6} + 8 a^{5} b x^{2} + 30 a^{4} b^{2} x^{4} + 80 a^{3} b^{3} x^{6}}{8 x^{8}} \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**9,x)

[Out]

15*a**2*b**4*log(x) + 3*a*b**5*x**2 + b**6*x**4/4 - (a**6 + 8*a**5*b*x**2 + 30*a**4*b**2*x**4 + 80*a**3*b**3*x
**6)/(8*x**8)

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Giac [A]  time = 1.16452, size = 109, normalized size = 1.49 \begin{align*} \frac{1}{4} \, b^{6} x^{4} + 3 \, a b^{5} x^{2} + \frac{15}{2} \, a^{2} b^{4} \log \left (x^{2}\right ) - \frac{125 \, a^{2} b^{4} x^{8} + 80 \, a^{3} b^{3} x^{6} + 30 \, a^{4} b^{2} x^{4} + 8 \, a^{5} b x^{2} + a^{6}}{8 \, x^{8}} \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^9,x, algorithm="giac")

[Out]

1/4*b^6*x^4 + 3*a*b^5*x^2 + 15/2*a^2*b^4*log(x^2) - 1/8*(125*a^2*b^4*x^8 + 80*a^3*b^3*x^6 + 30*a^4*b^2*x^4 + 8
*a^5*b*x^2 + a^6)/x^8