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

Optimal. Leaf size=63 \[ -\frac{b^2}{2 a^3 x^2}+\frac{b^3 \log \left (a+b x^2\right )}{2 a^4}-\frac{b^3 \log (x)}{a^4}+\frac{b}{4 a^2 x^4}-\frac{1}{6 a x^6} \]

[Out]

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

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Rubi [A]  time = 0.0346818, antiderivative size = 63, 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} \[ -\frac{b^2}{2 a^3 x^2}+\frac{b^3 \log \left (a+b x^2\right )}{2 a^4}-\frac{b^3 \log (x)}{a^4}+\frac{b}{4 a^2 x^4}-\frac{1}{6 a x^6} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^7*(a + b*x^2)),x]

[Out]

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

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

Mathematica [A]  time = 0.0068349, size = 63, normalized size = 1. \[ -\frac{b^2}{2 a^3 x^2}+\frac{b^3 \log \left (a+b x^2\right )}{2 a^4}-\frac{b^3 \log (x)}{a^4}+\frac{b}{4 a^2 x^4}-\frac{1}{6 a x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^7*(a + b*x^2)),x]

[Out]

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

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Maple [A]  time = 0.006, size = 56, normalized size = 0.9 \begin{align*} -{\frac{1}{6\,a{x}^{6}}}+{\frac{b}{4\,{a}^{2}{x}^{4}}}-{\frac{{b}^{2}}{2\,{a}^{3}{x}^{2}}}-{\frac{{b}^{3}\ln \left ( x \right ) }{{a}^{4}}}+{\frac{{b}^{3}\ln \left ( b{x}^{2}+a \right ) }{2\,{a}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 2.08265, size = 78, normalized size = 1.24 \begin{align*} \frac{b^{3} \log \left (b x^{2} + a\right )}{2 \, a^{4}} - \frac{b^{3} \log \left (x^{2}\right )}{2 \, a^{4}} - \frac{6 \, b^{2} x^{4} - 3 \, a b x^{2} + 2 \, a^{2}}{12 \, a^{3} x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.27312, size = 134, normalized size = 2.13 \begin{align*} \frac{6 \, b^{3} x^{6} \log \left (b x^{2} + a\right ) - 12 \, b^{3} x^{6} \log \left (x\right ) - 6 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} - 2 \, a^{3}}{12 \, a^{4} x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.538223, size = 56, normalized size = 0.89 \begin{align*} - \frac{2 a^{2} - 3 a b x^{2} + 6 b^{2} x^{4}}{12 a^{3} x^{6}} - \frac{b^{3} \log{\left (x \right )}}{a^{4}} + \frac{b^{3} \log{\left (\frac{a}{b} + x^{2} \right )}}{2 a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 2.87401, size = 95, normalized size = 1.51 \begin{align*} -\frac{b^{3} \log \left (x^{2}\right )}{2 \, a^{4}} + \frac{b^{3} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{4}} + \frac{11 \, b^{3} x^{6} - 6 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} - 2 \, a^{3}}{12 \, a^{4} x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*b^3*log(x^2)/a^4 + 1/2*b^3*log(abs(b*x^2 + a))/a^4 + 1/12*(11*b^3*x^6 - 6*a*b^2*x^4 + 3*a^2*b*x^2 - 2*a^3
)/(a^4*x^6)