3.1859 \(\int \frac {x^5}{(a+\frac {b}{x^2})^2} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.05, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {263, 266, 43} \[ \frac {3 b^2 x^2}{2 a^4}-\frac {b^4}{2 a^5 \left (a x^2+b\right )}-\frac {2 b^3 \log \left (a x^2+b\right )}{a^5}-\frac {b x^4}{2 a^3}+\frac {x^6}{6 a^2} \]

Antiderivative was successfully verified.

[In]

Int[x^5/(a + b/x^2)^2,x]

[Out]

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

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])

Rule 263

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[n]

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]]

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 60, normalized size = 0.86 \[ \frac {a^3 x^6-3 a^2 b x^4-\frac {3 b^4}{a x^2+b}-12 b^3 \log \left (a x^2+b\right )+9 a b^2 x^2}{6 a^5} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/(a + b/x^2)^2,x]

[Out]

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

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fricas [A]  time = 0.94, size = 81, normalized size = 1.16 \[ \frac {a^{4} x^{8} - 2 \, a^{3} b x^{6} + 6 \, a^{2} b^{2} x^{4} + 9 \, a b^{3} x^{2} - 3 \, b^{4} - 12 \, {\left (a b^{3} x^{2} + b^{4}\right )} \log \left (a x^{2} + b\right )}{6 \, {\left (a^{6} x^{2} + a^{5} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(a+b/x^2)^2,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.16, size = 80, normalized size = 1.14 \[ -\frac {2 \, b^{3} \log \left ({\left | a x^{2} + b \right |}\right )}{a^{5}} + \frac {a^{4} x^{6} - 3 \, a^{3} b x^{4} + 9 \, a^{2} b^{2} x^{2}}{6 \, a^{6}} + \frac {4 \, a b^{3} x^{2} + 3 \, b^{4}}{2 \, {\left (a x^{2} + b\right )} a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(a+b/x^2)^2,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.01, size = 63, normalized size = 0.90 \[ \frac {x^{6}}{6 a^{2}}-\frac {b \,x^{4}}{2 a^{3}}+\frac {3 b^{2} x^{2}}{2 a^{4}}-\frac {b^{4}}{2 \left (a \,x^{2}+b \right ) a^{5}}-\frac {2 b^{3} \ln \left (a \,x^{2}+b \right )}{a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(a+b/x^2)^2,x)

[Out]

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

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maxima [A]  time = 0.88, size = 65, normalized size = 0.93 \[ -\frac {b^{4}}{2 \, {\left (a^{6} x^{2} + a^{5} b\right )}} - \frac {2 \, b^{3} \log \left (a x^{2} + b\right )}{a^{5}} + \frac {a^{2} x^{6} - 3 \, a b x^{4} + 9 \, b^{2} x^{2}}{6 \, a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(a+b/x^2)^2,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.04, size = 68, normalized size = 0.97 \[ \frac {x^6}{6\,a^2}-\frac {b^4}{2\,a\,\left (a^5\,x^2+b\,a^4\right )}-\frac {b\,x^4}{2\,a^3}-\frac {2\,b^3\,\ln \left (a\,x^2+b\right )}{a^5}+\frac {3\,b^2\,x^2}{2\,a^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(a + b/x^2)^2,x)

[Out]

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

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sympy [A]  time = 0.29, size = 66, normalized size = 0.94 \[ - \frac {b^{4}}{2 a^{6} x^{2} + 2 a^{5} b} + \frac {x^{6}}{6 a^{2}} - \frac {b x^{4}}{2 a^{3}} + \frac {3 b^{2} x^{2}}{2 a^{4}} - \frac {2 b^{3} \log {\left (a x^{2} + b \right )}}{a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(a+b/x**2)**2,x)

[Out]

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

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