Optimal. Leaf size=81 \[ -\frac {7}{10 a^2 x^5}+\frac {7 b}{6 a^3 x^3}-\frac {7 b^2}{2 a^4 x}+\frac {1}{2 a x^5 \left (a+b x^2\right )}-\frac {7 b^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{9/2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {296, 331, 211}
\begin {gather*} -\frac {7 b^{5/2} \text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{9/2}}-\frac {7 b^2}{2 a^4 x}+\frac {7 b}{6 a^3 x^3}-\frac {7}{10 a^2 x^5}+\frac {1}{2 a x^5 \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 296
Rule 331
Rubi steps
\begin {align*} \int \frac {1}{x^6 \left (a+b x^2\right )^2} \, dx &=\frac {1}{2 a x^5 \left (a+b x^2\right )}+\frac {7 \int \frac {1}{x^6 \left (a+b x^2\right )} \, dx}{2 a}\\ &=-\frac {7}{10 a^2 x^5}+\frac {1}{2 a x^5 \left (a+b x^2\right )}-\frac {(7 b) \int \frac {1}{x^4 \left (a+b x^2\right )} \, dx}{2 a^2}\\ &=-\frac {7}{10 a^2 x^5}+\frac {7 b}{6 a^3 x^3}+\frac {1}{2 a x^5 \left (a+b x^2\right )}+\frac {\left (7 b^2\right ) \int \frac {1}{x^2 \left (a+b x^2\right )} \, dx}{2 a^3}\\ &=-\frac {7}{10 a^2 x^5}+\frac {7 b}{6 a^3 x^3}-\frac {7 b^2}{2 a^4 x}+\frac {1}{2 a x^5 \left (a+b x^2\right )}-\frac {\left (7 b^3\right ) \int \frac {1}{a+b x^2} \, dx}{2 a^4}\\ &=-\frac {7}{10 a^2 x^5}+\frac {7 b}{6 a^3 x^3}-\frac {7 b^2}{2 a^4 x}+\frac {1}{2 a x^5 \left (a+b x^2\right )}-\frac {7 b^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{9/2}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 80, normalized size = 0.99 \begin {gather*} -\frac {1}{5 a^2 x^5}+\frac {2 b}{3 a^3 x^3}-\frac {3 b^2}{a^4 x}-\frac {b^3 x}{2 a^4 \left (a+b x^2\right )}-\frac {7 b^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 67, normalized size = 0.83
method | result | size |
default | \(-\frac {b^{3} \left (\frac {x}{2 b \,x^{2}+2 a}+\frac {7 \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{4}}-\frac {1}{5 a^{2} x^{5}}+\frac {2 b}{3 a^{3} x^{3}}-\frac {3 b^{2}}{a^{4} x}\) | \(67\) |
risch | \(\frac {-\frac {7 b^{3} x^{6}}{2 a^{4}}-\frac {7 b^{2} x^{4}}{3 a^{3}}+\frac {7 b \,x^{2}}{15 a^{2}}-\frac {1}{5 a}}{x^{5} \left (b \,x^{2}+a \right )}+\frac {7 \left (\munderset {\textit {\_R} =\RootOf \left (a^{9} \textit {\_Z}^{2}+b^{5}\right )}{\sum }\textit {\_R} \ln \left (\left (3 \textit {\_R}^{2} a^{9}+2 b^{5}\right ) x +a^{5} b^{2} \textit {\_R} \right )\right )}{4}\) | \(97\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 75, normalized size = 0.93 \begin {gather*} -\frac {105 \, b^{3} x^{6} + 70 \, a b^{2} x^{4} - 14 \, a^{2} b x^{2} + 6 \, a^{3}}{30 \, {\left (a^{4} b x^{7} + a^{5} x^{5}\right )}} - \frac {7 \, b^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.42, size = 198, normalized size = 2.44 \begin {gather*} \left [-\frac {210 \, b^{3} x^{6} + 140 \, a b^{2} x^{4} - 28 \, a^{2} b x^{2} + 12 \, a^{3} - 105 \, {\left (b^{3} x^{7} + a b^{2} x^{5}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} - 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right )}{60 \, {\left (a^{4} b x^{7} + a^{5} x^{5}\right )}}, -\frac {105 \, b^{3} x^{6} + 70 \, a b^{2} x^{4} - 14 \, a^{2} b x^{2} + 6 \, a^{3} + 105 \, {\left (b^{3} x^{7} + a b^{2} x^{5}\right )} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right )}{30 \, {\left (a^{4} b x^{7} + a^{5} x^{5}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.17, size = 126, normalized size = 1.56 \begin {gather*} \frac {7 \sqrt {- \frac {b^{5}}{a^{9}}} \log {\left (- \frac {a^{5} \sqrt {- \frac {b^{5}}{a^{9}}}}{b^{3}} + x \right )}}{4} - \frac {7 \sqrt {- \frac {b^{5}}{a^{9}}} \log {\left (\frac {a^{5} \sqrt {- \frac {b^{5}}{a^{9}}}}{b^{3}} + x \right )}}{4} + \frac {- 6 a^{3} + 14 a^{2} b x^{2} - 70 a b^{2} x^{4} - 105 b^{3} x^{6}}{30 a^{5} x^{5} + 30 a^{4} b x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.10, size = 70, normalized size = 0.86 \begin {gather*} -\frac {7 \, b^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a^{4}} - \frac {b^{3} x}{2 \, {\left (b x^{2} + a\right )} a^{4}} - \frac {45 \, b^{2} x^{4} - 10 \, a b x^{2} + 3 \, a^{2}}{15 \, a^{4} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.85, size = 70, normalized size = 0.86 \begin {gather*} -\frac {\frac {1}{5\,a}-\frac {7\,b\,x^2}{15\,a^2}+\frac {7\,b^2\,x^4}{3\,a^3}+\frac {7\,b^3\,x^6}{2\,a^4}}{b\,x^7+a\,x^5}-\frac {7\,b^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{2\,a^{9/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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