3.2.43 \(\int \frac {1}{x^8 (a+b x^2)} \, dx\) [143]

Optimal. Leaf size=69 \[ -\frac {1}{7 a x^7}+\frac {b}{5 a^2 x^5}-\frac {b^2}{3 a^3 x^3}+\frac {b^3}{a^4 x}+\frac {b^{7/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{9/2}} \]

[Out]

-1/7/a/x^7+1/5*b/a^2/x^5-1/3*b^2/a^3/x^3+b^3/a^4/x+b^(7/2)*arctan(x*b^(1/2)/a^(1/2))/a^(9/2)

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Rubi [A]
time = 0.02, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {331, 211} \begin {gather*} \frac {b^{7/2} \text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{9/2}}+\frac {b^3}{a^4 x}-\frac {b^2}{3 a^3 x^3}+\frac {b}{5 a^2 x^5}-\frac {1}{7 a x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^8*(a + b*x^2)),x]

[Out]

-1/7*1/(a*x^7) + b/(5*a^2*x^5) - b^2/(3*a^3*x^3) + b^3/(a^4*x) + (b^(7/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/a^(9/2)

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 331

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c
*(m + 1))), x] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rubi steps

\begin {align*} \int \frac {1}{x^8 \left (a+b x^2\right )} \, dx &=-\frac {1}{7 a x^7}-\frac {b \int \frac {1}{x^6 \left (a+b x^2\right )} \, dx}{a}\\ &=-\frac {1}{7 a x^7}+\frac {b}{5 a^2 x^5}+\frac {b^2 \int \frac {1}{x^4 \left (a+b x^2\right )} \, dx}{a^2}\\ &=-\frac {1}{7 a x^7}+\frac {b}{5 a^2 x^5}-\frac {b^2}{3 a^3 x^3}-\frac {b^3 \int \frac {1}{x^2 \left (a+b x^2\right )} \, dx}{a^3}\\ &=-\frac {1}{7 a x^7}+\frac {b}{5 a^2 x^5}-\frac {b^2}{3 a^3 x^3}+\frac {b^3}{a^4 x}+\frac {b^4 \int \frac {1}{a+b x^2} \, dx}{a^4}\\ &=-\frac {1}{7 a x^7}+\frac {b}{5 a^2 x^5}-\frac {b^2}{3 a^3 x^3}+\frac {b^3}{a^4 x}+\frac {b^{7/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{9/2}}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 69, normalized size = 1.00 \begin {gather*} -\frac {1}{7 a x^7}+\frac {b}{5 a^2 x^5}-\frac {b^2}{3 a^3 x^3}+\frac {b^3}{a^4 x}+\frac {b^{7/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^8*(a + b*x^2)),x]

[Out]

-1/7*1/(a*x^7) + b/(5*a^2*x^5) - b^2/(3*a^3*x^3) + b^3/(a^4*x) + (b^(7/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/a^(9/2)

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Maple [A]
time = 0.04, size = 61, normalized size = 0.88

method	result	size
default	\(\frac {b^{4} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{a^{4} \sqrt {a b}}-\frac {1}{7 a \,x^{7}}-\frac {b^{2}}{3 a^{3} x^{3}}+\frac {b}{5 a^{2} x^{5}}+\frac {b^{3}}{a^{4} x}\)	\(61\)
risch	\(\frac {\frac {b^{3} x^{6}}{a^{4}}-\frac {b^{2} x^{4}}{3 a^{3}}+\frac {b \,x^{2}}{5 a^{2}}-\frac {1}{7 a}}{x^{7}}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (a^{9} \textit {\_Z}^{2}+b^{7}\right )}{\sum }\textit {\_R} \ln \left (\left (3 \textit {\_R}^{2} a^{9}+2 b^{7}\right ) x -a^{5} b^{3} \textit {\_R} \right )\right )}{2}\)	\(88\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^8/(b*x^2+a),x,method=_RETURNVERBOSE)

[Out]

b^4/a^4/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))-1/7/a/x^7-1/3*b^2/a^3/x^3+1/5*b/a^2/x^5+b^3/a^4/x

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Maxima [A]
time = 0.53, size = 62, normalized size = 0.90 \begin {gather*} \frac {b^{4} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a^{4}} + \frac {105 \, b^{3} x^{6} - 35 \, a b^{2} x^{4} + 21 \, a^{2} b x^{2} - 15 \, a^{3}}{105 \, a^{4} x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^8/(b*x^2+a),x, algorithm="maxima")

[Out]

b^4*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^4) + 1/105*(105*b^3*x^6 - 35*a*b^2*x^4 + 21*a^2*b*x^2 - 15*a^3)/(a^4*x^
7)

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Fricas [A]
time = 1.59, size = 154, normalized size = 2.23 \begin {gather*} \left [\frac {105 \, b^{3} x^{7} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} + 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right ) + 210 \, b^{3} x^{6} - 70 \, a b^{2} x^{4} + 42 \, a^{2} b x^{2} - 30 \, a^{3}}{210 \, a^{4} x^{7}}, \frac {105 \, b^{3} x^{7} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right ) + 105 \, b^{3} x^{6} - 35 \, a b^{2} x^{4} + 21 \, a^{2} b x^{2} - 15 \, a^{3}}{105 \, a^{4} x^{7}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^8/(b*x^2+a),x, algorithm="fricas")

[Out]

[1/210*(105*b^3*x^7*sqrt(-b/a)*log((b*x^2 + 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)) + 210*b^3*x^6 - 70*a*b^2*x^4 +
42*a^2*b*x^2 - 30*a^3)/(a^4*x^7), 1/105*(105*b^3*x^7*sqrt(b/a)*arctan(x*sqrt(b/a)) + 105*b^3*x^6 - 35*a*b^2*x^
4 + 21*a^2*b*x^2 - 15*a^3)/(a^4*x^7)]

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Sympy [A]
time = 0.12, size = 112, normalized size = 1.62 \begin {gather*} - \frac {\sqrt {- \frac {b^{7}}{a^{9}}} \log {\left (- \frac {a^{5} \sqrt {- \frac {b^{7}}{a^{9}}}}{b^{4}} + x \right )}}{2} + \frac {\sqrt {- \frac {b^{7}}{a^{9}}} \log {\left (\frac {a^{5} \sqrt {- \frac {b^{7}}{a^{9}}}}{b^{4}} + x \right )}}{2} + \frac {- 15 a^{3} + 21 a^{2} b x^{2} - 35 a b^{2} x^{4} + 105 b^{3} x^{6}}{105 a^{4} x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**8/(b*x**2+a),x)

[Out]

-sqrt(-b**7/a**9)*log(-a**5*sqrt(-b**7/a**9)/b**4 + x)/2 + sqrt(-b**7/a**9)*log(a**5*sqrt(-b**7/a**9)/b**4 + x
)/2 + (-15*a**3 + 21*a**2*b*x**2 - 35*a*b**2*x**4 + 105*b**3*x**6)/(105*a**4*x**7)

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Giac [A]
time = 0.49, size = 62, normalized size = 0.90 \begin {gather*} \frac {b^{4} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a^{4}} + \frac {105 \, b^{3} x^{6} - 35 \, a b^{2} x^{4} + 21 \, a^{2} b x^{2} - 15 \, a^{3}}{105 \, a^{4} x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^8/(b*x^2+a),x, algorithm="giac")

[Out]

b^4*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^4) + 1/105*(105*b^3*x^6 - 35*a*b^2*x^4 + 21*a^2*b*x^2 - 15*a^3)/(a^4*x^
7)

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Mupad [B]
time = 0.06, size = 59, normalized size = 0.86 \begin {gather*} \frac {b^{7/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{a^{9/2}}-\frac {\frac {1}{7\,a}-\frac {b\,x^2}{5\,a^2}+\frac {b^2\,x^4}{3\,a^3}-\frac {b^3\,x^6}{a^4}}{x^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^8*(a + b*x^2)),x)

[Out]

(b^(7/2)*atan((b^(1/2)*x)/a^(1/2)))/a^(9/2) - (1/(7*a) - (b*x^2)/(5*a^2) + (b^2*x^4)/(3*a^3) - (b^3*x^6)/a^4)/
x^7

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