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

Optimal. Leaf size=118 \[ -\frac{a^6}{10 b^7 \left (a+b x^2\right )^5}+\frac{3 a^5}{4 b^7 \left (a+b x^2\right )^4}-\frac{5 a^4}{2 b^7 \left (a+b x^2\right )^3}+\frac{5 a^3}{b^7 \left (a+b x^2\right )^2}-\frac{15 a^2}{2 b^7 \left (a+b x^2\right )}-\frac{3 a \log \left (a+b x^2\right )}{b^7}+\frac{x^2}{2 b^6} \]

[Out]

x^2/(2*b^6) - a^6/(10*b^7*(a + b*x^2)^5) + (3*a^5)/(4*b^7*(a + b*x^2)^4) - (5*a^4)/(2*b^7*(a + b*x^2)^3) + (5*
a^3)/(b^7*(a + b*x^2)^2) - (15*a^2)/(2*b^7*(a + b*x^2)) - (3*a*Log[a + b*x^2])/b^7

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Rubi [A]  time = 0.114487, antiderivative size = 118, 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} \[ -\frac{a^6}{10 b^7 \left (a+b x^2\right )^5}+\frac{3 a^5}{4 b^7 \left (a+b x^2\right )^4}-\frac{5 a^4}{2 b^7 \left (a+b x^2\right )^3}+\frac{5 a^3}{b^7 \left (a+b x^2\right )^2}-\frac{15 a^2}{2 b^7 \left (a+b x^2\right )}-\frac{3 a \log \left (a+b x^2\right )}{b^7}+\frac{x^2}{2 b^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x^2/(2*b^6) - a^6/(10*b^7*(a + b*x^2)^5) + (3*a^5)/(4*b^7*(a + b*x^2)^4) - (5*a^4)/(2*b^7*(a + b*x^2)^3) + (5*
a^3)/(b^7*(a + b*x^2)^2) - (15*a^2)/(2*b^7*(a + b*x^2)) - (3*a*Log[a + b*x^2])/b^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 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{x^{13}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx &=b^6 \int \frac{x^{13}}{\left (a b+b^2 x^2\right )^6} \, dx\\ &=\frac{1}{2} b^6 \operatorname{Subst}\left (\int \frac{x^6}{\left (a b+b^2 x\right )^6} \, dx,x,x^2\right )\\ &=\frac{1}{2} b^6 \operatorname{Subst}\left (\int \left (\frac{1}{b^{12}}+\frac{a^6}{b^{12} (a+b x)^6}-\frac{6 a^5}{b^{12} (a+b x)^5}+\frac{15 a^4}{b^{12} (a+b x)^4}-\frac{20 a^3}{b^{12} (a+b x)^3}+\frac{15 a^2}{b^{12} (a+b x)^2}-\frac{6 a}{b^{12} (a+b x)}\right ) \, dx,x,x^2\right )\\ &=\frac{x^2}{2 b^6}-\frac{a^6}{10 b^7 \left (a+b x^2\right )^5}+\frac{3 a^5}{4 b^7 \left (a+b x^2\right )^4}-\frac{5 a^4}{2 b^7 \left (a+b x^2\right )^3}+\frac{5 a^3}{b^7 \left (a+b x^2\right )^2}-\frac{15 a^2}{2 b^7 \left (a+b x^2\right )}-\frac{3 a \log \left (a+b x^2\right )}{b^7}\\ \end{align*}

Mathematica [A]  time = 0.0268393, size = 101, normalized size = 0.86 \[ -\frac{50 a^2 b^4 x^8+400 a^3 b^3 x^6+600 a^4 b^2 x^4+375 a^5 b x^2+87 a^6-50 a b^5 x^{10}+60 a \left (a+b x^2\right )^5 \log \left (a+b x^2\right )-10 b^6 x^{12}}{20 b^7 \left (a+b x^2\right )^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(87*a^6 + 375*a^5*b*x^2 + 600*a^4*b^2*x^4 + 400*a^3*b^3*x^6 + 50*a^2*b^4*x^8 - 50*a*b^5*x^10 - 10*b^6*x^12 +
60*a*(a + b*x^2)^5*Log[a + b*x^2])/(20*b^7*(a + b*x^2)^5)

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Maple [A]  time = 0.056, size = 109, normalized size = 0.9 \begin{align*}{\frac{{x}^{2}}{2\,{b}^{6}}}-{\frac{{a}^{6}}{10\,{b}^{7} \left ( b{x}^{2}+a \right ) ^{5}}}+{\frac{3\,{a}^{5}}{4\,{b}^{7} \left ( b{x}^{2}+a \right ) ^{4}}}-{\frac{5\,{a}^{4}}{2\,{b}^{7} \left ( b{x}^{2}+a \right ) ^{3}}}+5\,{\frac{{a}^{3}}{{b}^{7} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{15\,{a}^{2}}{2\,{b}^{7} \left ( b{x}^{2}+a \right ) }}-3\,{\frac{a\ln \left ( b{x}^{2}+a \right ) }{{b}^{7}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^13/(b^2*x^4+2*a*b*x^2+a^2)^3,x)

[Out]

1/2*x^2/b^6-1/10*a^6/b^7/(b*x^2+a)^5+3/4*a^5/b^7/(b*x^2+a)^4-5/2*a^4/b^7/(b*x^2+a)^3+5*a^3/b^7/(b*x^2+a)^2-15/
2*a^2/b^7/(b*x^2+a)-3*a*ln(b*x^2+a)/b^7

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Maxima [A]  time = 1.23894, size = 178, normalized size = 1.51 \begin{align*} -\frac{150 \, a^{2} b^{4} x^{8} + 500 \, a^{3} b^{3} x^{6} + 650 \, a^{4} b^{2} x^{4} + 385 \, a^{5} b x^{2} + 87 \, a^{6}}{20 \,{\left (b^{12} x^{10} + 5 \, a b^{11} x^{8} + 10 \, a^{2} b^{10} x^{6} + 10 \, a^{3} b^{9} x^{4} + 5 \, a^{4} b^{8} x^{2} + a^{5} b^{7}\right )}} + \frac{x^{2}}{2 \, b^{6}} - \frac{3 \, a \log \left (b x^{2} + a\right )}{b^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^13/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="maxima")

[Out]

-1/20*(150*a^2*b^4*x^8 + 500*a^3*b^3*x^6 + 650*a^4*b^2*x^4 + 385*a^5*b*x^2 + 87*a^6)/(b^12*x^10 + 5*a*b^11*x^8
+ 10*a^2*b^10*x^6 + 10*a^3*b^9*x^4 + 5*a^4*b^8*x^2 + a^5*b^7) + 1/2*x^2/b^6 - 3*a*log(b*x^2 + a)/b^7

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Fricas [A]  time = 1.73678, size = 412, normalized size = 3.49 \begin{align*} \frac{10 \, b^{6} x^{12} + 50 \, a b^{5} x^{10} - 50 \, a^{2} b^{4} x^{8} - 400 \, a^{3} b^{3} x^{6} - 600 \, a^{4} b^{2} x^{4} - 375 \, a^{5} b x^{2} - 87 \, a^{6} - 60 \,{\left (a b^{5} x^{10} + 5 \, a^{2} b^{4} x^{8} + 10 \, a^{3} b^{3} x^{6} + 10 \, a^{4} b^{2} x^{4} + 5 \, a^{5} b x^{2} + a^{6}\right )} \log \left (b x^{2} + a\right )}{20 \,{\left (b^{12} x^{10} + 5 \, a b^{11} x^{8} + 10 \, a^{2} b^{10} x^{6} + 10 \, a^{3} b^{9} x^{4} + 5 \, a^{4} b^{8} x^{2} + a^{5} b^{7}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^13/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="fricas")

[Out]

1/20*(10*b^6*x^12 + 50*a*b^5*x^10 - 50*a^2*b^4*x^8 - 400*a^3*b^3*x^6 - 600*a^4*b^2*x^4 - 375*a^5*b*x^2 - 87*a^
6 - 60*(a*b^5*x^10 + 5*a^2*b^4*x^8 + 10*a^3*b^3*x^6 + 10*a^4*b^2*x^4 + 5*a^5*b*x^2 + a^6)*log(b*x^2 + a))/(b^1
2*x^10 + 5*a*b^11*x^8 + 10*a^2*b^10*x^6 + 10*a^3*b^9*x^4 + 5*a^4*b^8*x^2 + a^5*b^7)

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Sympy [A]  time = 1.4333, size = 136, normalized size = 1.15 \begin{align*} - \frac{3 a \log{\left (a + b x^{2} \right )}}{b^{7}} - \frac{87 a^{6} + 385 a^{5} b x^{2} + 650 a^{4} b^{2} x^{4} + 500 a^{3} b^{3} x^{6} + 150 a^{2} b^{4} x^{8}}{20 a^{5} b^{7} + 100 a^{4} b^{8} x^{2} + 200 a^{3} b^{9} x^{4} + 200 a^{2} b^{10} x^{6} + 100 a b^{11} x^{8} + 20 b^{12} x^{10}} + \frac{x^{2}}{2 b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**13/(b**2*x**4+2*a*b*x**2+a**2)**3,x)

[Out]

-3*a*log(a + b*x**2)/b**7 - (87*a**6 + 385*a**5*b*x**2 + 650*a**4*b**2*x**4 + 500*a**3*b**3*x**6 + 150*a**2*b*
*4*x**8)/(20*a**5*b**7 + 100*a**4*b**8*x**2 + 200*a**3*b**9*x**4 + 200*a**2*b**10*x**6 + 100*a*b**11*x**8 + 20
*b**12*x**10) + x**2/(2*b**6)

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Giac [A]  time = 1.13895, size = 128, normalized size = 1.08 \begin{align*} \frac{x^{2}}{2 \, b^{6}} - \frac{3 \, a \log \left ({\left | b x^{2} + a \right |}\right )}{b^{7}} + \frac{137 \, a b^{5} x^{10} + 535 \, a^{2} b^{4} x^{8} + 870 \, a^{3} b^{3} x^{6} + 720 \, a^{4} b^{2} x^{4} + 300 \, a^{5} b x^{2} + 50 \, a^{6}}{20 \,{\left (b x^{2} + a\right )}^{5} b^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^13/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="giac")

[Out]

1/2*x^2/b^6 - 3*a*log(abs(b*x^2 + a))/b^7 + 1/20*(137*a*b^5*x^10 + 535*a^2*b^4*x^8 + 870*a^3*b^3*x^6 + 720*a^4
*b^2*x^4 + 300*a^5*b*x^2 + 50*a^6)/((b*x^2 + a)^5*b^7)