3.2.44 \(\int \frac {1}{x^9 (a+b x^2)} \, dx\) [144]

Optimal. Leaf size=75 \[ -\frac {1}{8 a x^8}+\frac {b}{6 a^2 x^6}-\frac {b^2}{4 a^3 x^4}+\frac {b^3}{2 a^4 x^2}+\frac {b^4 \log (x)}{a^5}-\frac {b^4 \log \left (a+b x^2\right )}{2 a^5} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 75, 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 = {272, 46} \begin {gather*} -\frac {b^4 \log \left (a+b x^2\right )}{2 a^5}+\frac {b^4 \log (x)}{a^5}+\frac {b^3}{2 a^4 x^2}-\frac {b^2}{4 a^3 x^4}+\frac {b}{6 a^2 x^6}-\frac {1}{8 a x^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^9*(a + b*x^2)),x]

[Out]

-1/8*1/(a*x^8) + b/(6*a^2*x^6) - b^2/(4*a^3*x^4) + b^3/(2*a^4*x^2) + (b^4*Log[x])/a^5 - (b^4*Log[a + b*x^2])/(
2*a^5)

Rule 46

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] && Lt
Q[m + n + 2, 0])

Rule 272

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^9 \left (a+b x^2\right )} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{x^5 (a+b x)} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {1}{a x^5}-\frac {b}{a^2 x^4}+\frac {b^2}{a^3 x^3}-\frac {b^3}{a^4 x^2}+\frac {b^4}{a^5 x}-\frac {b^5}{a^5 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {1}{8 a x^8}+\frac {b}{6 a^2 x^6}-\frac {b^2}{4 a^3 x^4}+\frac {b^3}{2 a^4 x^2}+\frac {b^4 \log (x)}{a^5}-\frac {b^4 \log \left (a+b x^2\right )}{2 a^5}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 75, normalized size = 1.00 \begin {gather*} -\frac {1}{8 a x^8}+\frac {b}{6 a^2 x^6}-\frac {b^2}{4 a^3 x^4}+\frac {b^3}{2 a^4 x^2}+\frac {b^4 \log (x)}{a^5}-\frac {b^4 \log \left (a+b x^2\right )}{2 a^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^9*(a + b*x^2)),x]

[Out]

-1/8*1/(a*x^8) + b/(6*a^2*x^6) - b^2/(4*a^3*x^4) + b^3/(2*a^4*x^2) + (b^4*Log[x])/a^5 - (b^4*Log[a + b*x^2])/(
2*a^5)

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Maple [A]
time = 0.04, size = 66, normalized size = 0.88

method	result	size
default	\(-\frac {1}{8 a \,x^{8}}+\frac {b}{6 a^{2} x^{6}}-\frac {b^{2}}{4 a^{3} x^{4}}+\frac {b^{3}}{2 a^{4} x^{2}}+\frac {b^{4} \ln \left (x \right )}{a^{5}}-\frac {b^{4} \ln \left (b \,x^{2}+a \right )}{2 a^{5}}\)	\(66\)
norman	\(\frac {-\frac {1}{8 a}+\frac {b \,x^{2}}{6 a^{2}}-\frac {b^{2} x^{4}}{4 a^{3}}+\frac {b^{3} x^{6}}{2 a^{4}}}{x^{8}}+\frac {b^{4} \ln \left (x \right )}{a^{5}}-\frac {b^{4} \ln \left (b \,x^{2}+a \right )}{2 a^{5}}\)	\(68\)
risch	\(\frac {-\frac {1}{8 a}+\frac {b \,x^{2}}{6 a^{2}}-\frac {b^{2} x^{4}}{4 a^{3}}+\frac {b^{3} x^{6}}{2 a^{4}}}{x^{8}}+\frac {b^{4} \ln \left (x \right )}{a^{5}}-\frac {b^{4} \ln \left (b \,x^{2}+a \right )}{2 a^{5}}\)	\(68\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^9/(b*x^2+a),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.35, size = 69, normalized size = 0.92 \begin {gather*} -\frac {b^{4} \log \left (b x^{2} + a\right )}{2 \, a^{5}} + \frac {b^{4} \log \left (x^{2}\right )}{2 \, a^{5}} + \frac {12 \, b^{3} x^{6} - 6 \, a b^{2} x^{4} + 4 \, a^{2} b x^{2} - 3 \, a^{3}}{24 \, a^{4} x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^9/(b*x^2+a),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 1.23, size = 69, normalized size = 0.92 \begin {gather*} -\frac {12 \, b^{4} x^{8} \log \left (b x^{2} + a\right ) - 24 \, b^{4} x^{8} \log \left (x\right ) - 12 \, a b^{3} x^{6} + 6 \, a^{2} b^{2} x^{4} - 4 \, a^{3} b x^{2} + 3 \, a^{4}}{24 \, a^{5} x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^9/(b*x^2+a),x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.16, size = 68, normalized size = 0.91 \begin {gather*} \frac {- 3 a^{3} + 4 a^{2} b x^{2} - 6 a b^{2} x^{4} + 12 b^{3} x^{6}}{24 a^{4} x^{8}} + \frac {b^{4} \log {\left (x \right )}}{a^{5}} - \frac {b^{4} \log {\left (\frac {a}{b} + x^{2} \right )}}{2 a^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**9/(b*x**2+a),x)

[Out]

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

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Giac [A]
time = 0.47, size = 81, normalized size = 1.08 \begin {gather*} \frac {b^{4} \log \left (x^{2}\right )}{2 \, a^{5}} - \frac {b^{4} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{5}} - \frac {25 \, b^{4} x^{8} - 12 \, a b^{3} x^{6} + 6 \, a^{2} b^{2} x^{4} - 4 \, a^{3} b x^{2} + 3 \, a^{4}}{24 \, a^{5} x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^9/(b*x^2+a),x, algorithm="giac")

[Out]

1/2*b^4*log(x^2)/a^5 - 1/2*b^4*log(abs(b*x^2 + a))/a^5 - 1/24*(25*b^4*x^8 - 12*a*b^3*x^6 + 6*a^2*b^2*x^4 - 4*a
^3*b*x^2 + 3*a^4)/(a^5*x^8)

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Mupad [B]
time = 4.67, size = 68, normalized size = 0.91 \begin {gather*} \frac {b^4\,\ln \left (x\right )}{a^5}-\frac {b^4\,\ln \left (b\,x^2+a\right )}{2\,a^5}-\frac {\frac {1}{8\,a}-\frac {b\,x^2}{6\,a^2}+\frac {b^2\,x^4}{4\,a^3}-\frac {b^3\,x^6}{2\,a^4}}{x^8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^9*(a + b*x^2)),x)

[Out]

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

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