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

Optimal. Leaf size=87 \[ \frac{3 a^2 x^2}{b^5}-\frac{5 a^4}{2 b^6 \left (a+b x^2\right )}+\frac{a^5}{4 b^6 \left (a+b x^2\right )^2}-\frac{5 a^3 \log \left (a+b x^2\right )}{b^6}-\frac{3 a x^4}{4 b^4}+\frac{x^6}{6 b^3} \]

[Out]

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

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Rubi [A]  time = 0.0715145, antiderivative size = 87, 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} \[ \frac{3 a^2 x^2}{b^5}-\frac{5 a^4}{2 b^6 \left (a+b x^2\right )}+\frac{a^5}{4 b^6 \left (a+b x^2\right )^2}-\frac{5 a^3 \log \left (a+b x^2\right )}{b^6}-\frac{3 a x^4}{4 b^4}+\frac{x^6}{6 b^3} \]

Antiderivative was successfully verified.

[In]

Int[x^11/(a + b*x^2)^3,x]

[Out]

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

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^{11}}{\left (a+b x^2\right )^3} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^5}{(a+b x)^3} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{6 a^2}{b^5}-\frac{3 a x}{b^4}+\frac{x^2}{b^3}-\frac{a^5}{b^5 (a+b x)^3}+\frac{5 a^4}{b^5 (a+b x)^2}-\frac{10 a^3}{b^5 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=\frac{3 a^2 x^2}{b^5}-\frac{3 a x^4}{4 b^4}+\frac{x^6}{6 b^3}+\frac{a^5}{4 b^6 \left (a+b x^2\right )^2}-\frac{5 a^4}{2 b^6 \left (a+b x^2\right )}-\frac{5 a^3 \log \left (a+b x^2\right )}{b^6}\\ \end{align*}

Mathematica [A]  time = 0.0232649, size = 75, normalized size = 0.86 \[ \frac{36 a^2 b x^2-\frac{30 a^4}{a+b x^2}+\frac{3 a^5}{\left (a+b x^2\right )^2}-60 a^3 \log \left (a+b x^2\right )-9 a b^2 x^4+2 b^3 x^6}{12 b^6} \]

Antiderivative was successfully verified.

[In]

Integrate[x^11/(a + b*x^2)^3,x]

[Out]

(36*a^2*b*x^2 - 9*a*b^2*x^4 + 2*b^3*x^6 + (3*a^5)/(a + b*x^2)^2 - (30*a^4)/(a + b*x^2) - 60*a^3*Log[a + b*x^2]
)/(12*b^6)

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Maple [A]  time = 0.01, size = 80, normalized size = 0.9 \begin{align*} 3\,{\frac{{a}^{2}{x}^{2}}{{b}^{5}}}-{\frac{3\,a{x}^{4}}{4\,{b}^{4}}}+{\frac{{x}^{6}}{6\,{b}^{3}}}+{\frac{{a}^{5}}{4\,{b}^{6} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{5\,{a}^{4}}{2\,{b}^{6} \left ( b{x}^{2}+a \right ) }}-5\,{\frac{{a}^{3}\ln \left ( b{x}^{2}+a \right ) }{{b}^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^11/(b*x^2+a)^3,x)

[Out]

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

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Maxima [A]  time = 1.25385, size = 120, normalized size = 1.38 \begin{align*} -\frac{10 \, a^{4} b x^{2} + 9 \, a^{5}}{4 \,{\left (b^{8} x^{4} + 2 \, a b^{7} x^{2} + a^{2} b^{6}\right )}} - \frac{5 \, a^{3} \log \left (b x^{2} + a\right )}{b^{6}} + \frac{2 \, b^{2} x^{6} - 9 \, a b x^{4} + 36 \, a^{2} x^{2}}{12 \, b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11/(b*x^2+a)^3,x, algorithm="maxima")

[Out]

-1/4*(10*a^4*b*x^2 + 9*a^5)/(b^8*x^4 + 2*a*b^7*x^2 + a^2*b^6) - 5*a^3*log(b*x^2 + a)/b^6 + 1/12*(2*b^2*x^6 - 9
*a*b*x^4 + 36*a^2*x^2)/b^5

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Fricas [A]  time = 1.17652, size = 240, normalized size = 2.76 \begin{align*} \frac{2 \, b^{5} x^{10} - 5 \, a b^{4} x^{8} + 20 \, a^{2} b^{3} x^{6} + 63 \, a^{3} b^{2} x^{4} + 6 \, a^{4} b x^{2} - 27 \, a^{5} - 60 \,{\left (a^{3} b^{2} x^{4} + 2 \, a^{4} b x^{2} + a^{5}\right )} \log \left (b x^{2} + a\right )}{12 \,{\left (b^{8} x^{4} + 2 \, a b^{7} x^{2} + a^{2} b^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11/(b*x^2+a)^3,x, algorithm="fricas")

[Out]

1/12*(2*b^5*x^10 - 5*a*b^4*x^8 + 20*a^2*b^3*x^6 + 63*a^3*b^2*x^4 + 6*a^4*b*x^2 - 27*a^5 - 60*(a^3*b^2*x^4 + 2*
a^4*b*x^2 + a^5)*log(b*x^2 + a))/(b^8*x^4 + 2*a*b^7*x^2 + a^2*b^6)

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Sympy [A]  time = 0.579237, size = 90, normalized size = 1.03 \begin{align*} - \frac{5 a^{3} \log{\left (a + b x^{2} \right )}}{b^{6}} + \frac{3 a^{2} x^{2}}{b^{5}} - \frac{3 a x^{4}}{4 b^{4}} - \frac{9 a^{5} + 10 a^{4} b x^{2}}{4 a^{2} b^{6} + 8 a b^{7} x^{2} + 4 b^{8} x^{4}} + \frac{x^{6}}{6 b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**11/(b*x**2+a)**3,x)

[Out]

-5*a**3*log(a + b*x**2)/b**6 + 3*a**2*x**2/b**5 - 3*a*x**4/(4*b**4) - (9*a**5 + 10*a**4*b*x**2)/(4*a**2*b**6 +
8*a*b**7*x**2 + 4*b**8*x**4) + x**6/(6*b**3)

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Giac [A]  time = 1.80929, size = 124, normalized size = 1.43 \begin{align*} -\frac{5 \, a^{3} \log \left ({\left | b x^{2} + a \right |}\right )}{b^{6}} + \frac{30 \, a^{3} b^{2} x^{4} + 50 \, a^{4} b x^{2} + 21 \, a^{5}}{4 \,{\left (b x^{2} + a\right )}^{2} b^{6}} + \frac{2 \, b^{6} x^{6} - 9 \, a b^{5} x^{4} + 36 \, a^{2} b^{4} x^{2}}{12 \, b^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11/(b*x^2+a)^3,x, algorithm="giac")

[Out]

-5*a^3*log(abs(b*x^2 + a))/b^6 + 1/4*(30*a^3*b^2*x^4 + 50*a^4*b*x^2 + 21*a^5)/((b*x^2 + a)^2*b^6) + 1/12*(2*b^
6*x^6 - 9*a*b^5*x^4 + 36*a^2*b^4*x^2)/b^9