3.165 \(\int \frac{1}{x^7 (a+b x^2)^2} \, dx\)

Optimal. Leaf size=80 \[ -\frac{b^3}{2 a^4 \left (a+b x^2\right )}-\frac{3 b^2}{2 a^4 x^2}+\frac{2 b^3 \log \left (a+b x^2\right )}{a^5}-\frac{4 b^3 \log (x)}{a^5}+\frac{b}{2 a^3 x^4}-\frac{1}{6 a^2 x^6} \]

[Out]

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

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Rubi [A]  time = 0.0539037, antiderivative size = 80, 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, 44} \[ -\frac{b^3}{2 a^4 \left (a+b x^2\right )}-\frac{3 b^2}{2 a^4 x^2}+\frac{2 b^3 \log \left (a+b x^2\right )}{a^5}-\frac{4 b^3 \log (x)}{a^5}+\frac{b}{2 a^3 x^4}-\frac{1}{6 a^2 x^6} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^7*(a + b*x^2)^2),x]

[Out]

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

Rubi steps

\begin{align*} \int \frac{1}{x^7 \left (a+b x^2\right )^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^4 (a+b x)^2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{1}{a^2 x^4}-\frac{2 b}{a^3 x^3}+\frac{3 b^2}{a^4 x^2}-\frac{4 b^3}{a^5 x}+\frac{b^4}{a^4 (a+b x)^2}+\frac{4 b^4}{a^5 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{1}{6 a^2 x^6}+\frac{b}{2 a^3 x^4}-\frac{3 b^2}{2 a^4 x^2}-\frac{b^3}{2 a^4 \left (a+b x^2\right )}-\frac{4 b^3 \log (x)}{a^5}+\frac{2 b^3 \log \left (a+b x^2\right )}{a^5}\\ \end{align*}

Mathematica [A]  time = 0.0536143, size = 68, normalized size = 0.85 \[ \frac{a \left (-\frac{a^2}{x^6}-\frac{3 b^3}{a+b x^2}+\frac{3 a b}{x^4}-\frac{9 b^2}{x^2}\right )+12 b^3 \log \left (a+b x^2\right )-24 b^3 \log (x)}{6 a^5} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^7*(a + b*x^2)^2),x]

[Out]

(a*(-(a^2/x^6) + (3*a*b)/x^4 - (9*b^2)/x^2 - (3*b^3)/(a + b*x^2)) - 24*b^3*Log[x] + 12*b^3*Log[a + b*x^2])/(6*
a^5)

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Maple [A]  time = 0.012, size = 73, normalized size = 0.9 \begin{align*} -{\frac{1}{6\,{a}^{2}{x}^{6}}}+{\frac{b}{2\,{a}^{3}{x}^{4}}}-{\frac{3\,{b}^{2}}{2\,{a}^{4}{x}^{2}}}-{\frac{{b}^{3}}{2\,{a}^{4} \left ( b{x}^{2}+a \right ) }}-4\,{\frac{{b}^{3}\ln \left ( x \right ) }{{a}^{5}}}+2\,{\frac{{b}^{3}\ln \left ( b{x}^{2}+a \right ) }{{a}^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^7/(b*x^2+a)^2,x)

[Out]

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

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Maxima [A]  time = 2.20893, size = 107, normalized size = 1.34 \begin{align*} -\frac{12 \, b^{3} x^{6} + 6 \, a b^{2} x^{4} - 2 \, a^{2} b x^{2} + a^{3}}{6 \,{\left (a^{4} b x^{8} + a^{5} x^{6}\right )}} + \frac{2 \, b^{3} \log \left (b x^{2} + a\right )}{a^{5}} - \frac{2 \, b^{3} \log \left (x^{2}\right )}{a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(b*x^2+a)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.36778, size = 209, normalized size = 2.61 \begin{align*} -\frac{12 \, a b^{3} x^{6} + 6 \, a^{2} b^{2} x^{4} - 2 \, a^{3} b x^{2} + a^{4} - 12 \,{\left (b^{4} x^{8} + a b^{3} x^{6}\right )} \log \left (b x^{2} + a\right ) + 24 \,{\left (b^{4} x^{8} + a b^{3} x^{6}\right )} \log \left (x\right )}{6 \,{\left (a^{5} b x^{8} + a^{6} x^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(b*x^2+a)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.833802, size = 78, normalized size = 0.98 \begin{align*} - \frac{a^{3} - 2 a^{2} b x^{2} + 6 a b^{2} x^{4} + 12 b^{3} x^{6}}{6 a^{5} x^{6} + 6 a^{4} b x^{8}} - \frac{4 b^{3} \log{\left (x \right )}}{a^{5}} + \frac{2 b^{3} \log{\left (\frac{a}{b} + x^{2} \right )}}{a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**7/(b*x**2+a)**2,x)

[Out]

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

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Giac [A]  time = 2.76559, size = 134, normalized size = 1.68 \begin{align*} -\frac{2 \, b^{3} \log \left (x^{2}\right )}{a^{5}} + \frac{2 \, b^{3} \log \left ({\left | b x^{2} + a \right |}\right )}{a^{5}} - \frac{4 \, b^{4} x^{2} + 5 \, a b^{3}}{2 \,{\left (b x^{2} + a\right )} a^{5}} + \frac{22 \, b^{3} x^{6} - 9 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} - a^{3}}{6 \, a^{5} x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(b*x^2+a)^2,x, algorithm="giac")

[Out]

-2*b^3*log(x^2)/a^5 + 2*b^3*log(abs(b*x^2 + a))/a^5 - 1/2*(4*b^4*x^2 + 5*a*b^3)/((b*x^2 + a)*a^5) + 1/6*(22*b^
3*x^6 - 9*a*b^2*x^4 + 3*a^2*b*x^2 - a^3)/(a^5*x^6)