3.179 \(\int \frac{1}{x^7 (a+b x^2)^3} \, dx\)

Optimal. Leaf size=95 \[ -\frac{2 b^3}{a^5 \left (a+b x^2\right )}-\frac{b^3}{4 a^4 \left (a+b x^2\right )^2}-\frac{3 b^2}{a^5 x^2}+\frac{5 b^3 \log \left (a+b x^2\right )}{a^6}-\frac{10 b^3 \log (x)}{a^6}+\frac{3 b}{4 a^4 x^4}-\frac{1}{6 a^3 x^6} \]

[Out]

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

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Rubi [A]  time = 0.067537, antiderivative size = 95, 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} \[ -\frac{2 b^3}{a^5 \left (a+b x^2\right )}-\frac{b^3}{4 a^4 \left (a+b x^2\right )^2}-\frac{3 b^2}{a^5 x^2}+\frac{5 b^3 \log \left (a+b x^2\right )}{a^6}-\frac{10 b^3 \log (x)}{a^6}+\frac{3 b}{4 a^4 x^4}-\frac{1}{6 a^3 x^6} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^7*(a + b*x^2)^3),x]

[Out]

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

Mathematica [A]  time = 0.0668509, size = 85, normalized size = 0.89 \[ -\frac{\frac{a \left (20 a^2 b^2 x^4-5 a^3 b x^2+2 a^4+90 a b^3 x^6+60 b^4 x^8\right )}{x^6 \left (a+b x^2\right )^2}-60 b^3 \log \left (a+b x^2\right )+120 b^3 \log (x)}{12 a^6} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^7*(a + b*x^2)^3),x]

[Out]

-((a*(2*a^4 - 5*a^3*b*x^2 + 20*a^2*b^2*x^4 + 90*a*b^3*x^6 + 60*b^4*x^8))/(x^6*(a + b*x^2)^2) + 120*b^3*Log[x]
- 60*b^3*Log[a + b*x^2])/(12*a^6)

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Maple [A]  time = 0.012, size = 90, normalized size = 1. \begin{align*} -{\frac{1}{6\,{a}^{3}{x}^{6}}}+{\frac{3\,b}{4\,{a}^{4}{x}^{4}}}-3\,{\frac{{b}^{2}}{{a}^{5}{x}^{2}}}-{\frac{{b}^{3}}{4\,{a}^{4} \left ( b{x}^{2}+a \right ) ^{2}}}-2\,{\frac{{b}^{3}}{{a}^{5} \left ( b{x}^{2}+a \right ) }}-10\,{\frac{{b}^{3}\ln \left ( x \right ) }{{a}^{6}}}+5\,{\frac{{b}^{3}\ln \left ( b{x}^{2}+a \right ) }{{a}^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^7/(b*x^2+a)^3,x)

[Out]

-1/6/a^3/x^6+3/4*b/a^4/x^4-3*b^2/a^5/x^2-1/4*b^3/a^4/(b*x^2+a)^2-2*b^3/a^5/(b*x^2+a)-10*b^3*ln(x)/a^6+5*b^3*ln
(b*x^2+a)/a^6

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Maxima [A]  time = 1.80771, size = 139, normalized size = 1.46 \begin{align*} -\frac{60 \, b^{4} x^{8} + 90 \, a b^{3} x^{6} + 20 \, a^{2} b^{2} x^{4} - 5 \, a^{3} b x^{2} + 2 \, a^{4}}{12 \,{\left (a^{5} b^{2} x^{10} + 2 \, a^{6} b x^{8} + a^{7} x^{6}\right )}} + \frac{5 \, b^{3} \log \left (b x^{2} + a\right )}{a^{6}} - \frac{5 \, b^{3} \log \left (x^{2}\right )}{a^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(b*x^2+a)^3,x, algorithm="maxima")

[Out]

-1/12*(60*b^4*x^8 + 90*a*b^3*x^6 + 20*a^2*b^2*x^4 - 5*a^3*b*x^2 + 2*a^4)/(a^5*b^2*x^10 + 2*a^6*b*x^8 + a^7*x^6
) + 5*b^3*log(b*x^2 + a)/a^6 - 5*b^3*log(x^2)/a^6

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Fricas [A]  time = 1.26026, size = 308, normalized size = 3.24 \begin{align*} -\frac{60 \, a b^{4} x^{8} + 90 \, a^{2} b^{3} x^{6} + 20 \, a^{3} b^{2} x^{4} - 5 \, a^{4} b x^{2} + 2 \, a^{5} - 60 \,{\left (b^{5} x^{10} + 2 \, a b^{4} x^{8} + a^{2} b^{3} x^{6}\right )} \log \left (b x^{2} + a\right ) + 120 \,{\left (b^{5} x^{10} + 2 \, a b^{4} x^{8} + a^{2} b^{3} x^{6}\right )} \log \left (x\right )}{12 \,{\left (a^{6} b^{2} x^{10} + 2 \, a^{7} b x^{8} + a^{8} x^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(b*x^2+a)^3,x, algorithm="fricas")

[Out]

-1/12*(60*a*b^4*x^8 + 90*a^2*b^3*x^6 + 20*a^3*b^2*x^4 - 5*a^4*b*x^2 + 2*a^5 - 60*(b^5*x^10 + 2*a*b^4*x^8 + a^2
*b^3*x^6)*log(b*x^2 + a) + 120*(b^5*x^10 + 2*a*b^4*x^8 + a^2*b^3*x^6)*log(x))/(a^6*b^2*x^10 + 2*a^7*b*x^8 + a^
8*x^6)

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Sympy [A]  time = 1.71902, size = 104, normalized size = 1.09 \begin{align*} - \frac{2 a^{4} - 5 a^{3} b x^{2} + 20 a^{2} b^{2} x^{4} + 90 a b^{3} x^{6} + 60 b^{4} x^{8}}{12 a^{7} x^{6} + 24 a^{6} b x^{8} + 12 a^{5} b^{2} x^{10}} - \frac{10 b^{3} \log{\left (x \right )}}{a^{6}} + \frac{5 b^{3} \log{\left (\frac{a}{b} + x^{2} \right )}}{a^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**7/(b*x**2+a)**3,x)

[Out]

-(2*a**4 - 5*a**3*b*x**2 + 20*a**2*b**2*x**4 + 90*a*b**3*x**6 + 60*b**4*x**8)/(12*a**7*x**6 + 24*a**6*b*x**8 +
12*a**5*b**2*x**10) - 10*b**3*log(x)/a**6 + 5*b**3*log(a/b + x**2)/a**6

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Giac [A]  time = 2.10982, size = 149, normalized size = 1.57 \begin{align*} -\frac{5 \, b^{3} \log \left (x^{2}\right )}{a^{6}} + \frac{5 \, b^{3} \log \left ({\left | b x^{2} + a \right |}\right )}{a^{6}} - \frac{30 \, b^{5} x^{4} + 68 \, a b^{4} x^{2} + 39 \, a^{2} b^{3}}{4 \,{\left (b x^{2} + a\right )}^{2} a^{6}} + \frac{110 \, b^{3} x^{6} - 36 \, a b^{2} x^{4} + 9 \, a^{2} b x^{2} - 2 \, a^{3}}{12 \, a^{6} x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(b*x^2+a)^3,x, algorithm="giac")

[Out]

-5*b^3*log(x^2)/a^6 + 5*b^3*log(abs(b*x^2 + a))/a^6 - 1/4*(30*b^5*x^4 + 68*a*b^4*x^2 + 39*a^2*b^3)/((b*x^2 + a
)^2*a^6) + 1/12*(110*b^3*x^6 - 36*a*b^2*x^4 + 9*a^2*b*x^2 - 2*a^3)/(a^6*x^6)