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

Optimal. Leaf size=70 \[ \frac{3 a^2 x^2}{2 b^4}-\frac{a^4}{2 b^5 \left (a+b x^2\right )}-\frac{2 a^3 \log \left (a+b x^2\right )}{b^5}-\frac{a x^4}{2 b^3}+\frac{x^6}{6 b^2} \]

[Out]

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

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Rubi [A]  time = 0.0544227, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 43} \[ \frac{3 a^2 x^2}{2 b^4}-\frac{a^4}{2 b^5 \left (a+b x^2\right )}-\frac{2 a^3 \log \left (a+b x^2\right )}{b^5}-\frac{a x^4}{2 b^3}+\frac{x^6}{6 b^2} \]

Antiderivative was successfully verified.

[In]

Int[x^9/(a + b*x^2)^2,x]

[Out]

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

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^9}{\left (a+b x^2\right )^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^4}{(a+b x)^2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{3 a^2}{b^4}-\frac{2 a x}{b^3}+\frac{x^2}{b^2}+\frac{a^4}{b^4 (a+b x)^2}-\frac{4 a^3}{b^4 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=\frac{3 a^2 x^2}{2 b^4}-\frac{a x^4}{2 b^3}+\frac{x^6}{6 b^2}-\frac{a^4}{2 b^5 \left (a+b x^2\right )}-\frac{2 a^3 \log \left (a+b x^2\right )}{b^5}\\ \end{align*}

Mathematica [A]  time = 0.0231995, size = 60, normalized size = 0.86 \[ \frac{9 a^2 b x^2-\frac{3 a^4}{a+b x^2}-12 a^3 \log \left (a+b x^2\right )-3 a b^2 x^4+b^3 x^6}{6 b^5} \]

Antiderivative was successfully verified.

[In]

Integrate[x^9/(a + b*x^2)^2,x]

[Out]

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

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Maple [A]  time = 0.009, size = 63, normalized size = 0.9 \begin{align*}{\frac{3\,{a}^{2}{x}^{2}}{2\,{b}^{4}}}-{\frac{a{x}^{4}}{2\,{b}^{3}}}+{\frac{{x}^{6}}{6\,{b}^{2}}}-{\frac{{a}^{4}}{2\,{b}^{5} \left ( b{x}^{2}+a \right ) }}-2\,{\frac{{a}^{3}\ln \left ( b{x}^{2}+a \right ) }{{b}^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^9/(b*x^2+a)^2,x)

[Out]

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

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Maxima [A]  time = 2.62999, size = 88, normalized size = 1.26 \begin{align*} -\frac{a^{4}}{2 \,{\left (b^{6} x^{2} + a b^{5}\right )}} - \frac{2 \, a^{3} \log \left (b x^{2} + a\right )}{b^{5}} + \frac{b^{2} x^{6} - 3 \, a b x^{4} + 9 \, a^{2} x^{2}}{6 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9/(b*x^2+a)^2,x, algorithm="maxima")

[Out]

-1/2*a^4/(b^6*x^2 + a*b^5) - 2*a^3*log(b*x^2 + a)/b^5 + 1/6*(b^2*x^6 - 3*a*b*x^4 + 9*a^2*x^2)/b^4

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Fricas [A]  time = 1.18547, size = 166, normalized size = 2.37 \begin{align*} \frac{b^{4} x^{8} - 2 \, a b^{3} x^{6} + 6 \, a^{2} b^{2} x^{4} + 9 \, a^{3} b x^{2} - 3 \, a^{4} - 12 \,{\left (a^{3} b x^{2} + a^{4}\right )} \log \left (b x^{2} + a\right )}{6 \,{\left (b^{6} x^{2} + a b^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9/(b*x^2+a)^2,x, algorithm="fricas")

[Out]

1/6*(b^4*x^8 - 2*a*b^3*x^6 + 6*a^2*b^2*x^4 + 9*a^3*b*x^2 - 3*a^4 - 12*(a^3*b*x^2 + a^4)*log(b*x^2 + a))/(b^6*x
^2 + a*b^5)

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Sympy [A]  time = 0.403645, size = 66, normalized size = 0.94 \begin{align*} - \frac{a^{4}}{2 a b^{5} + 2 b^{6} x^{2}} - \frac{2 a^{3} \log{\left (a + b x^{2} \right )}}{b^{5}} + \frac{3 a^{2} x^{2}}{2 b^{4}} - \frac{a x^{4}}{2 b^{3}} + \frac{x^{6}}{6 b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**9/(b*x**2+a)**2,x)

[Out]

-a**4/(2*a*b**5 + 2*b**6*x**2) - 2*a**3*log(a + b*x**2)/b**5 + 3*a**2*x**2/(2*b**4) - a*x**4/(2*b**3) + x**6/(
6*b**2)

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Giac [A]  time = 2.48879, size = 108, normalized size = 1.54 \begin{align*} -\frac{2 \, a^{3} \log \left ({\left | b x^{2} + a \right |}\right )}{b^{5}} + \frac{b^{4} x^{6} - 3 \, a b^{3} x^{4} + 9 \, a^{2} b^{2} x^{2}}{6 \, b^{6}} + \frac{4 \, a^{3} b x^{2} + 3 \, a^{4}}{2 \,{\left (b x^{2} + a\right )} b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9/(b*x^2+a)^2,x, algorithm="giac")

[Out]

-2*a^3*log(abs(b*x^2 + a))/b^5 + 1/6*(b^4*x^6 - 3*a*b^3*x^4 + 9*a^2*b^2*x^2)/b^6 + 1/2*(4*a^3*b*x^2 + 3*a^4)/(
(b*x^2 + a)*b^5)