Optimal. Leaf size=69 \[ \frac {b^{7/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{9/2}}+\frac {b^3}{a^4 x}-\frac {b^2}{3 a^3 x^3}+\frac {b}{5 a^2 x^5}-\frac {1}{7 a x^7} \]
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Rubi [A] time = 0.03, 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 = {325, 205} \[ -\frac {b^2}{3 a^3 x^3}+\frac {b^3}{a^4 x}+\frac {b^{7/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{9/2}}+\frac {b}{5 a^2 x^5}-\frac {1}{7 a x^7} \]
Antiderivative was successfully verified.
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Rule 205
Rule 325
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 = 0.03, size = 69, normalized size = 1.00 \[ \frac {b^{7/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{9/2}}+\frac {b^3}{a^4 x}-\frac {b^2}{3 a^3 x^3}+\frac {b}{5 a^2 x^5}-\frac {1}{7 a x^7} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.95, size = 154, normalized size = 2.23 \[ \left [\frac {105 \, b^{3} x^{7} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} + 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right ) + 210 \, b^{3} x^{6} - 70 \, a b^{2} x^{4} + 42 \, a^{2} b x^{2} - 30 \, a^{3}}{210 \, a^{4} x^{7}}, \frac {105 \, b^{3} x^{7} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right ) + 105 \, b^{3} x^{6} - 35 \, a b^{2} x^{4} + 21 \, a^{2} b x^{2} - 15 \, a^{3}}{105 \, a^{4} x^{7}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.64, size = 62, normalized size = 0.90 \[ \frac {b^{4} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a^{4}} + \frac {105 \, b^{3} x^{6} - 35 \, a b^{2} x^{4} + 21 \, a^{2} b x^{2} - 15 \, a^{3}}{105 \, a^{4} x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 61, normalized size = 0.88 \[ \frac {b^{4} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, a^{4}}+\frac {b^{3}}{a^{4} x}-\frac {b^{2}}{3 a^{3} x^{3}}+\frac {b}{5 a^{2} x^{5}}-\frac {1}{7 a \,x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.93, size = 62, normalized size = 0.90 \[ \frac {b^{4} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a^{4}} + \frac {105 \, b^{3} x^{6} - 35 \, a b^{2} x^{4} + 21 \, a^{2} b x^{2} - 15 \, a^{3}}{105 \, a^{4} x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 59, normalized size = 0.86 \[ \frac {b^{7/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{a^{9/2}}-\frac {\frac {1}{7\,a}-\frac {b\,x^2}{5\,a^2}+\frac {b^2\,x^4}{3\,a^3}-\frac {b^3\,x^6}{a^4}}{x^7} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.30, size = 112, normalized size = 1.62 \[ - \frac {\sqrt {- \frac {b^{7}}{a^{9}}} \log {\left (- \frac {a^{5} \sqrt {- \frac {b^{7}}{a^{9}}}}{b^{4}} + x \right )}}{2} + \frac {\sqrt {- \frac {b^{7}}{a^{9}}} \log {\left (\frac {a^{5} \sqrt {- \frac {b^{7}}{a^{9}}}}{b^{4}} + x \right )}}{2} + \frac {- 15 a^{3} + 21 a^{2} b x^{2} - 35 a b^{2} x^{4} + 105 b^{3} x^{6}}{105 a^{4} x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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