3.463 \(\int \frac{(a^2+2 a b x^2+b^2 x^4)^3}{x^{11}} \, dx\)

Optimal. Leaf size=77 \[ -\frac{5 a^4 b^2}{2 x^6}-\frac{5 a^3 b^3}{x^4}-\frac{15 a^2 b^4}{2 x^2}-\frac{3 a^5 b}{4 x^8}-\frac{a^6}{10 x^{10}}+6 a b^5 \log (x)+\frac{b^6 x^2}{2} \]

[Out]

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

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Rubi [A]  time = 0.0511226, antiderivative size = 77, 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} \[ -\frac{5 a^4 b^2}{2 x^6}-\frac{5 a^3 b^3}{x^4}-\frac{15 a^2 b^4}{2 x^2}-\frac{3 a^5 b}{4 x^8}-\frac{a^6}{10 x^{10}}+6 a b^5 \log (x)+\frac{b^6 x^2}{2} \]

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^11,x]

[Out]

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

Mathematica [A]  time = 0.0050888, size = 77, normalized size = 1. \[ -\frac{5 a^4 b^2}{2 x^6}-\frac{5 a^3 b^3}{x^4}-\frac{15 a^2 b^4}{2 x^2}-\frac{3 a^5 b}{4 x^8}-\frac{a^6}{10 x^{10}}+6 a b^5 \log (x)+\frac{b^6 x^2}{2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^11,x]

[Out]

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

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Maple [A]  time = 0.049, size = 68, normalized size = 0.9 \begin{align*} -{\frac{{a}^{6}}{10\,{x}^{10}}}-{\frac{3\,{a}^{5}b}{4\,{x}^{8}}}-{\frac{5\,{a}^{4}{b}^{2}}{2\,{x}^{6}}}-5\,{\frac{{a}^{3}{b}^{3}}{{x}^{4}}}-{\frac{15\,{a}^{2}{b}^{4}}{2\,{x}^{2}}}+{\frac{{b}^{6}{x}^{2}}{2}}+6\,a{b}^{5}\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^11,x)

[Out]

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

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Maxima [A]  time = 1.00177, size = 97, normalized size = 1.26 \begin{align*} \frac{1}{2} \, b^{6} x^{2} + 3 \, a b^{5} \log \left (x^{2}\right ) - \frac{150 \, a^{2} b^{4} x^{8} + 100 \, a^{3} b^{3} x^{6} + 50 \, a^{4} b^{2} x^{4} + 15 \, a^{5} b x^{2} + 2 \, a^{6}}{20 \, x^{10}} \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^11,x, algorithm="maxima")

[Out]

1/2*b^6*x^2 + 3*a*b^5*log(x^2) - 1/20*(150*a^2*b^4*x^8 + 100*a^3*b^3*x^6 + 50*a^4*b^2*x^4 + 15*a^5*b*x^2 + 2*a
^6)/x^10

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Fricas [A]  time = 1.63156, size = 169, normalized size = 2.19 \begin{align*} \frac{10 \, b^{6} x^{12} + 120 \, a b^{5} x^{10} \log \left (x\right ) - 150 \, a^{2} b^{4} x^{8} - 100 \, a^{3} b^{3} x^{6} - 50 \, a^{4} b^{2} x^{4} - 15 \, a^{5} b x^{2} - 2 \, a^{6}}{20 \, x^{10}} \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^11,x, algorithm="fricas")

[Out]

1/20*(10*b^6*x^12 + 120*a*b^5*x^10*log(x) - 150*a^2*b^4*x^8 - 100*a^3*b^3*x^6 - 50*a^4*b^2*x^4 - 15*a^5*b*x^2
- 2*a^6)/x^10

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Sympy [A]  time = 0.621393, size = 73, normalized size = 0.95 \begin{align*} 6 a b^{5} \log{\left (x \right )} + \frac{b^{6} x^{2}}{2} - \frac{2 a^{6} + 15 a^{5} b x^{2} + 50 a^{4} b^{2} x^{4} + 100 a^{3} b^{3} x^{6} + 150 a^{2} b^{4} x^{8}}{20 x^{10}} \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**11,x)

[Out]

6*a*b**5*log(x) + b**6*x**2/2 - (2*a**6 + 15*a**5*b*x**2 + 50*a**4*b**2*x**4 + 100*a**3*b**3*x**6 + 150*a**2*b
**4*x**8)/(20*x**10)

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Giac [A]  time = 1.11706, size = 109, normalized size = 1.42 \begin{align*} \frac{1}{2} \, b^{6} x^{2} + 3 \, a b^{5} \log \left (x^{2}\right ) - \frac{137 \, a b^{5} x^{10} + 150 \, a^{2} b^{4} x^{8} + 100 \, a^{3} b^{3} x^{6} + 50 \, a^{4} b^{2} x^{4} + 15 \, a^{5} b x^{2} + 2 \, a^{6}}{20 \, x^{10}} \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^11,x, algorithm="giac")

[Out]

1/2*b^6*x^2 + 3*a*b^5*log(x^2) - 1/20*(137*a*b^5*x^10 + 150*a^2*b^4*x^8 + 100*a^3*b^3*x^6 + 50*a^4*b^2*x^4 + 1
5*a^5*b*x^2 + 2*a^6)/x^10