3.178 \(\int \frac {1}{x^5 (a+b x^2)^3} \, dx\)

Optimal. Leaf size=86 \[ -\frac {3 b^2 \log \left (a+b x^2\right )}{a^5}+\frac {6 b^2 \log (x)}{a^5}+\frac {3 b^2}{2 a^4 \left (a+b x^2\right )}+\frac {3 b}{2 a^4 x^2}+\frac {b^2}{4 a^3 \left (a+b x^2\right )^2}-\frac {1}{4 a^3 x^4} \]

[Out]

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

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Rubi [A]  time = 0.06, antiderivative size = 86, normalized size of antiderivative = 1.00, 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, 44} \[ \frac {3 b^2}{2 a^4 \left (a+b x^2\right )}+\frac {b^2}{4 a^3 \left (a+b x^2\right )^2}-\frac {3 b^2 \log \left (a+b x^2\right )}{a^5}+\frac {6 b^2 \log (x)}{a^5}+\frac {3 b}{2 a^4 x^2}-\frac {1}{4 a^3 x^4} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^5*(a + b*x^2)^3),x]

[Out]

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

Rule 44

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

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

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Mathematica [A]  time = 0.05, size = 74, normalized size = 0.86 \[ \frac {\frac {a \left (-a^3+4 a^2 b x^2+18 a b^2 x^4+12 b^3 x^6\right )}{x^4 \left (a+b x^2\right )^2}-12 b^2 \log \left (a+b x^2\right )+24 b^2 \log (x)}{4 a^5} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^5*(a + b*x^2)^3),x]

[Out]

((a*(-a^3 + 4*a^2*b*x^2 + 18*a*b^2*x^4 + 12*b^3*x^6))/(x^4*(a + b*x^2)^2) + 24*b^2*Log[x] - 12*b^2*Log[a + b*x
^2])/(4*a^5)

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fricas [A]  time = 0.94, size = 134, normalized size = 1.56 \[ \frac {12 \, a b^{3} x^{6} + 18 \, a^{2} b^{2} x^{4} + 4 \, a^{3} b x^{2} - a^{4} - 12 \, {\left (b^{4} x^{8} + 2 \, a b^{3} x^{6} + a^{2} b^{2} x^{4}\right )} \log \left (b x^{2} + a\right ) + 24 \, {\left (b^{4} x^{8} + 2 \, a b^{3} x^{6} + a^{2} b^{2} x^{4}\right )} \log \relax (x)}{4 \, {\left (a^{5} b^{2} x^{8} + 2 \, a^{6} b x^{6} + a^{7} x^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(b*x^2+a)^3,x, algorithm="fricas")

[Out]

1/4*(12*a*b^3*x^6 + 18*a^2*b^2*x^4 + 4*a^3*b*x^2 - a^4 - 12*(b^4*x^8 + 2*a*b^3*x^6 + a^2*b^2*x^4)*log(b*x^2 +
a) + 24*(b^4*x^8 + 2*a*b^3*x^6 + a^2*b^2*x^4)*log(x))/(a^5*b^2*x^8 + 2*a^6*b*x^6 + a^7*x^4)

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giac [A]  time = 0.65, size = 80, normalized size = 0.93 \[ \frac {3 \, b^{2} \log \left (x^{2}\right )}{a^{5}} - \frac {3 \, b^{2} \log \left ({\left | b x^{2} + a \right |}\right )}{a^{5}} + \frac {12 \, b^{3} x^{6} + 18 \, a b^{2} x^{4} + 4 \, a^{2} b x^{2} - a^{3}}{4 \, {\left (b x^{4} + a x^{2}\right )}^{2} a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(b*x^2+a)^3,x, algorithm="giac")

[Out]

3*b^2*log(x^2)/a^5 - 3*b^2*log(abs(b*x^2 + a))/a^5 + 1/4*(12*b^3*x^6 + 18*a*b^2*x^4 + 4*a^2*b*x^2 - a^3)/((b*x
^4 + a*x^2)^2*a^4)

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maple [A]  time = 0.02, size = 79, normalized size = 0.92 \[ \frac {b^{2}}{4 \left (b \,x^{2}+a \right )^{2} a^{3}}+\frac {3 b^{2}}{2 \left (b \,x^{2}+a \right ) a^{4}}+\frac {6 b^{2} \ln \relax (x )}{a^{5}}-\frac {3 b^{2} \ln \left (b \,x^{2}+a \right )}{a^{5}}+\frac {3 b}{2 a^{4} x^{2}}-\frac {1}{4 a^{3} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^5/(b*x^2+a)^3,x)

[Out]

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

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maxima [A]  time = 1.42, size = 92, normalized size = 1.07 \[ \frac {12 \, b^{3} x^{6} + 18 \, a b^{2} x^{4} + 4 \, a^{2} b x^{2} - a^{3}}{4 \, {\left (a^{4} b^{2} x^{8} + 2 \, a^{5} b x^{6} + a^{6} x^{4}\right )}} - \frac {3 \, b^{2} \log \left (b x^{2} + a\right )}{a^{5}} + \frac {3 \, b^{2} \log \left (x^{2}\right )}{a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(b*x^2+a)^3,x, algorithm="maxima")

[Out]

1/4*(12*b^3*x^6 + 18*a*b^2*x^4 + 4*a^2*b*x^2 - a^3)/(a^4*b^2*x^8 + 2*a^5*b*x^6 + a^6*x^4) - 3*b^2*log(b*x^2 +
a)/a^5 + 3*b^2*log(x^2)/a^5

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mupad [B]  time = 4.67, size = 88, normalized size = 1.02 \[ \frac {\frac {b\,x^2}{a^2}-\frac {1}{4\,a}+\frac {9\,b^2\,x^4}{2\,a^3}+\frac {3\,b^3\,x^6}{a^4}}{a^2\,x^4+2\,a\,b\,x^6+b^2\,x^8}-\frac {3\,b^2\,\ln \left (b\,x^2+a\right )}{a^5}+\frac {6\,b^2\,\ln \relax (x)}{a^5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^5*(a + b*x^2)^3),x)

[Out]

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

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sympy [A]  time = 0.55, size = 90, normalized size = 1.05 \[ \frac {- a^{3} + 4 a^{2} b x^{2} + 18 a b^{2} x^{4} + 12 b^{3} x^{6}}{4 a^{6} x^{4} + 8 a^{5} b x^{6} + 4 a^{4} b^{2} x^{8}} + \frac {6 b^{2} \log {\relax (x )}}{a^{5}} - \frac {3 b^{2} \log {\left (\frac {a}{b} + x^{2} \right )}}{a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**5/(b*x**2+a)**3,x)

[Out]

(-a**3 + 4*a**2*b*x**2 + 18*a*b**2*x**4 + 12*b**3*x**6)/(4*a**6*x**4 + 8*a**5*b*x**6 + 4*a**4*b**2*x**8) + 6*b
**2*log(x)/a**5 - 3*b**2*log(a/b + x**2)/a**5

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