3.465 \(\int \frac{(a^2+2 a b x^2+b^2 x^4)^3}{x^{13}} \, dx\)

Optimal. Leaf size=76 \[ -\frac{15 a^4 b^2}{8 x^8}-\frac{10 a^3 b^3}{3 x^6}-\frac{15 a^2 b^4}{4 x^4}-\frac{3 a^5 b}{5 x^{10}}-\frac{a^6}{12 x^{12}}-\frac{3 a b^5}{x^2}+b^6 \log (x) \]

[Out]

-a^6/(12*x^12) - (3*a^5*b)/(5*x^10) - (15*a^4*b^2)/(8*x^8) - (10*a^3*b^3)/(3*x^6) - (15*a^2*b^4)/(4*x^4) - (3*
a*b^5)/x^2 + b^6*Log[x]

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Rubi [A]  time = 0.0486256, antiderivative size = 76, 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} \[ -\frac{15 a^4 b^2}{8 x^8}-\frac{10 a^3 b^3}{3 x^6}-\frac{15 a^2 b^4}{4 x^4}-\frac{3 a^5 b}{5 x^{10}}-\frac{a^6}{12 x^{12}}-\frac{3 a b^5}{x^2}+b^6 \log (x) \]

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^13,x]

[Out]

-a^6/(12*x^12) - (3*a^5*b)/(5*x^10) - (15*a^4*b^2)/(8*x^8) - (10*a^3*b^3)/(3*x^6) - (15*a^2*b^4)/(4*x^4) - (3*
a*b^5)/x^2 + b^6*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^{13}} \, dx &=\frac{\int \frac{\left (a b+b^2 x^2\right )^6}{x^{13}} \, dx}{b^6}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\left (a b+b^2 x\right )^6}{x^7} \, dx,x,x^2\right )}{2 b^6}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a^6 b^6}{x^7}+\frac{6 a^5 b^7}{x^6}+\frac{15 a^4 b^8}{x^5}+\frac{20 a^3 b^9}{x^4}+\frac{15 a^2 b^{10}}{x^3}+\frac{6 a b^{11}}{x^2}+\frac{b^{12}}{x}\right ) \, dx,x,x^2\right )}{2 b^6}\\ &=-\frac{a^6}{12 x^{12}}-\frac{3 a^5 b}{5 x^{10}}-\frac{15 a^4 b^2}{8 x^8}-\frac{10 a^3 b^3}{3 x^6}-\frac{15 a^2 b^4}{4 x^4}-\frac{3 a b^5}{x^2}+b^6 \log (x)\\ \end{align*}

Mathematica [A]  time = 0.004745, size = 76, normalized size = 1. \[ -\frac{15 a^4 b^2}{8 x^8}-\frac{10 a^3 b^3}{3 x^6}-\frac{15 a^2 b^4}{4 x^4}-\frac{3 a^5 b}{5 x^{10}}-\frac{a^6}{12 x^{12}}-\frac{3 a b^5}{x^2}+b^6 \log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^13,x]

[Out]

-a^6/(12*x^12) - (3*a^5*b)/(5*x^10) - (15*a^4*b^2)/(8*x^8) - (10*a^3*b^3)/(3*x^6) - (15*a^2*b^4)/(4*x^4) - (3*
a*b^5)/x^2 + b^6*Log[x]

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Maple [A]  time = 0.049, size = 67, normalized size = 0.9 \begin{align*} -{\frac{{a}^{6}}{12\,{x}^{12}}}-{\frac{3\,{a}^{5}b}{5\,{x}^{10}}}-{\frac{15\,{a}^{4}{b}^{2}}{8\,{x}^{8}}}-{\frac{10\,{a}^{3}{b}^{3}}{3\,{x}^{6}}}-{\frac{15\,{a}^{2}{b}^{4}}{4\,{x}^{4}}}-3\,{\frac{a{b}^{5}}{{x}^{2}}}+{b}^{6}\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^13,x)

[Out]

-1/12*a^6/x^12-3/5*a^5*b/x^10-15/8*a^4*b^2/x^8-10/3*a^3*b^3/x^6-15/4*a^2*b^4/x^4-3*a*b^5/x^2+b^6*ln(x)

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Maxima [A]  time = 0.982927, size = 97, normalized size = 1.28 \begin{align*} \frac{1}{2} \, b^{6} \log \left (x^{2}\right ) - \frac{360 \, a b^{5} x^{10} + 450 \, a^{2} b^{4} x^{8} + 400 \, a^{3} b^{3} x^{6} + 225 \, a^{4} b^{2} x^{4} + 72 \, a^{5} b x^{2} + 10 \, a^{6}}{120 \, x^{12}} \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^13,x, algorithm="maxima")

[Out]

1/2*b^6*log(x^2) - 1/120*(360*a*b^5*x^10 + 450*a^2*b^4*x^8 + 400*a^3*b^3*x^6 + 225*a^4*b^2*x^4 + 72*a^5*b*x^2
+ 10*a^6)/x^12

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Fricas [A]  time = 1.59287, size = 174, normalized size = 2.29 \begin{align*} \frac{120 \, b^{6} x^{12} \log \left (x\right ) - 360 \, a b^{5} x^{10} - 450 \, a^{2} b^{4} x^{8} - 400 \, a^{3} b^{3} x^{6} - 225 \, a^{4} b^{2} x^{4} - 72 \, a^{5} b x^{2} - 10 \, a^{6}}{120 \, x^{12}} \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^13,x, algorithm="fricas")

[Out]

1/120*(120*b^6*x^12*log(x) - 360*a*b^5*x^10 - 450*a^2*b^4*x^8 - 400*a^3*b^3*x^6 - 225*a^4*b^2*x^4 - 72*a^5*b*x
^2 - 10*a^6)/x^12

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Sympy [A]  time = 0.718432, size = 71, normalized size = 0.93 \begin{align*} b^{6} \log{\left (x \right )} - \frac{10 a^{6} + 72 a^{5} b x^{2} + 225 a^{4} b^{2} x^{4} + 400 a^{3} b^{3} x^{6} + 450 a^{2} b^{4} x^{8} + 360 a b^{5} x^{10}}{120 x^{12}} \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**13,x)

[Out]

b**6*log(x) - (10*a**6 + 72*a**5*b*x**2 + 225*a**4*b**2*x**4 + 400*a**3*b**3*x**6 + 450*a**2*b**4*x**8 + 360*a
*b**5*x**10)/(120*x**12)

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Giac [A]  time = 1.15707, size = 108, normalized size = 1.42 \begin{align*} \frac{1}{2} \, b^{6} \log \left (x^{2}\right ) - \frac{147 \, b^{6} x^{12} + 360 \, a b^{5} x^{10} + 450 \, a^{2} b^{4} x^{8} + 400 \, a^{3} b^{3} x^{6} + 225 \, a^{4} b^{2} x^{4} + 72 \, a^{5} b x^{2} + 10 \, a^{6}}{120 \, x^{12}} \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^13,x, algorithm="giac")

[Out]

1/2*b^6*log(x^2) - 1/120*(147*b^6*x^12 + 360*a*b^5*x^10 + 450*a^2*b^4*x^8 + 400*a^3*b^3*x^6 + 225*a^4*b^2*x^4
+ 72*a^5*b*x^2 + 10*a^6)/x^12