Optimal. Leaf size=76 \[ -\frac {15}{8 b^3 x}+\frac {1}{4 b x \left (b+a x^2\right )^2}+\frac {5}{8 b^2 x \left (b+a x^2\right )}-\frac {15 \sqrt {a} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{8 b^{7/2}} \]
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Rubi [A]
time = 0.04, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {269, 296, 331,
211} \begin {gather*} -\frac {15 \sqrt {a} \text {ArcTan}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{8 b^{7/2}}+\frac {5}{8 b^2 x \left (a x^2+b\right )}+\frac {1}{4 b x \left (a x^2+b\right )^2}-\frac {15}{8 b^3 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 269
Rule 296
Rule 331
Rubi steps
\begin {align*} \int \frac {1}{\left (a+\frac {b}{x^2}\right )^3 x^8} \, dx &=\int \frac {1}{x^2 \left (b+a x^2\right )^3} \, dx\\ &=\frac {1}{4 b x \left (b+a x^2\right )^2}+\frac {5 \int \frac {1}{x^2 \left (b+a x^2\right )^2} \, dx}{4 b}\\ &=\frac {1}{4 b x \left (b+a x^2\right )^2}+\frac {5}{8 b^2 x \left (b+a x^2\right )}+\frac {15 \int \frac {1}{x^2 \left (b+a x^2\right )} \, dx}{8 b^2}\\ &=-\frac {15}{8 b^3 x}+\frac {1}{4 b x \left (b+a x^2\right )^2}+\frac {5}{8 b^2 x \left (b+a x^2\right )}-\frac {(15 a) \int \frac {1}{b+a x^2} \, dx}{8 b^3}\\ &=-\frac {15}{8 b^3 x}+\frac {1}{4 b x \left (b+a x^2\right )^2}+\frac {5}{8 b^2 x \left (b+a x^2\right )}-\frac {15 \sqrt {a} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{8 b^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 68, normalized size = 0.89 \begin {gather*} -\frac {8 b^2+25 a b x^2+15 a^2 x^4}{8 b^3 x \left (b+a x^2\right )^2}-\frac {15 \sqrt {a} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{8 b^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 54, normalized size = 0.71
method | result | size |
default | \(-\frac {a \left (\frac {\frac {7}{8} a \,x^{3}+\frac {9}{8} b x}{\left (a \,x^{2}+b \right )^{2}}+\frac {15 \arctan \left (\frac {a x}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{b^{3}}-\frac {1}{b^{3} x}\) | \(54\) |
risch | \(\frac {-\frac {15 a^{2} x^{4}}{8 b^{3}}-\frac {25 a \,x^{2}}{8 b^{2}}-\frac {1}{b}}{x \left (a \,x^{2}+b \right )^{2}}+\frac {15 \left (\munderset {\textit {\_R} =\RootOf \left (b^{7} \textit {\_Z}^{2}+a \right )}{\sum }\textit {\_R} \ln \left (\left (3 \textit {\_R}^{2} b^{7}+2 a \right ) x +b^{4} \textit {\_R} \right )\right )}{16}\) | \(79\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 71, normalized size = 0.93 \begin {gather*} -\frac {15 \, a^{2} x^{4} + 25 \, a b x^{2} + 8 \, b^{2}}{8 \, {\left (a^{2} b^{3} x^{5} + 2 \, a b^{4} x^{3} + b^{5} x\right )}} - \frac {15 \, a \arctan \left (\frac {a x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 202, normalized size = 2.66 \begin {gather*} \left [-\frac {30 \, a^{2} x^{4} + 50 \, a b x^{2} - 15 \, {\left (a^{2} x^{5} + 2 \, a b x^{3} + b^{2} x\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {a x^{2} - 2 \, b x \sqrt {-\frac {a}{b}} - b}{a x^{2} + b}\right ) + 16 \, b^{2}}{16 \, {\left (a^{2} b^{3} x^{5} + 2 \, a b^{4} x^{3} + b^{5} x\right )}}, -\frac {15 \, a^{2} x^{4} + 25 \, a b x^{2} + 15 \, {\left (a^{2} x^{5} + 2 \, a b x^{3} + b^{2} x\right )} \sqrt {\frac {a}{b}} \arctan \left (x \sqrt {\frac {a}{b}}\right ) + 8 \, b^{2}}{8 \, {\left (a^{2} b^{3} x^{5} + 2 \, a b^{4} x^{3} + b^{5} x\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.19, size = 116, normalized size = 1.53 \begin {gather*} \frac {15 \sqrt {- \frac {a}{b^{7}}} \log {\left (x - \frac {b^{4} \sqrt {- \frac {a}{b^{7}}}}{a} \right )}}{16} - \frac {15 \sqrt {- \frac {a}{b^{7}}} \log {\left (x + \frac {b^{4} \sqrt {- \frac {a}{b^{7}}}}{a} \right )}}{16} + \frac {- 15 a^{2} x^{4} - 25 a b x^{2} - 8 b^{2}}{8 a^{2} b^{3} x^{5} + 16 a b^{4} x^{3} + 8 b^{5} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.98, size = 57, normalized size = 0.75 \begin {gather*} -\frac {15 \, a \arctan \left (\frac {a x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} b^{3}} - \frac {7 \, a^{2} x^{3} + 9 \, a b x}{8 \, {\left (a x^{2} + b\right )}^{2} b^{3}} - \frac {1}{b^{3} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.19, size = 66, normalized size = 0.87 \begin {gather*} -\frac {\frac {1}{b}+\frac {25\,a\,x^2}{8\,b^2}+\frac {15\,a^2\,x^4}{8\,b^3}}{a^2\,x^5+2\,a\,b\,x^3+b^2\,x}-\frac {15\,\sqrt {a}\,\mathrm {atan}\left (\frac {\sqrt {a}\,x}{\sqrt {b}}\right )}{8\,b^{7/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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