Optimal. Leaf size=66 \[ -\frac {5 b x}{2 a^3}+\frac {5 x^3}{6 a^2}-\frac {x^5}{2 a \left (b+a x^2\right )}+\frac {5 b^{3/2} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{2 a^{7/2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 66, 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, 294, 308,
211} \begin {gather*} \frac {5 b^{3/2} \text {ArcTan}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{2 a^{7/2}}-\frac {5 b x}{2 a^3}+\frac {5 x^3}{6 a^2}-\frac {x^5}{2 a \left (a x^2+b\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 269
Rule 294
Rule 308
Rubi steps
\begin {align*} \int \frac {x^2}{\left (a+\frac {b}{x^2}\right )^2} \, dx &=\int \frac {x^6}{\left (b+a x^2\right )^2} \, dx\\ &=-\frac {x^5}{2 a \left (b+a x^2\right )}+\frac {5 \int \frac {x^4}{b+a x^2} \, dx}{2 a}\\ &=-\frac {x^5}{2 a \left (b+a x^2\right )}+\frac {5 \int \left (-\frac {b}{a^2}+\frac {x^2}{a}+\frac {b^2}{a^2 \left (b+a x^2\right )}\right ) \, dx}{2 a}\\ &=-\frac {5 b x}{2 a^3}+\frac {5 x^3}{6 a^2}-\frac {x^5}{2 a \left (b+a x^2\right )}+\frac {\left (5 b^2\right ) \int \frac {1}{b+a x^2} \, dx}{2 a^3}\\ &=-\frac {5 b x}{2 a^3}+\frac {5 x^3}{6 a^2}-\frac {x^5}{2 a \left (b+a x^2\right )}+\frac {5 b^{3/2} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{2 a^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 60, normalized size = 0.91 \begin {gather*} \frac {x \left (-12 b+2 a x^2-\frac {3 b^2}{b+a x^2}\right )}{6 a^3}+\frac {5 b^{3/2} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{2 a^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 53, normalized size = 0.80
method | result | size |
default | \(\frac {\frac {1}{3} a \,x^{3}-2 b x}{a^{3}}+\frac {b^{2} \left (-\frac {x}{2 \left (a \,x^{2}+b \right )}+\frac {5 \arctan \left (\frac {a x}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{3}}\) | \(53\) |
risch | \(\frac {x^{3}}{3 a^{2}}-\frac {2 b x}{a^{3}}-\frac {b^{2} x}{2 a^{3} \left (a \,x^{2}+b \right )}+\frac {5 \sqrt {-a b}\, b \ln \left (-\sqrt {-a b}\, x +b \right )}{4 a^{4}}-\frac {5 \sqrt {-a b}\, b \ln \left (\sqrt {-a b}\, x +b \right )}{4 a^{4}}\) | \(82\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 59, normalized size = 0.89 \begin {gather*} -\frac {b^{2} x}{2 \, {\left (a^{4} x^{2} + a^{3} b\right )}} + \frac {5 \, b^{2} \arctan \left (\frac {a x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a^{3}} + \frac {a x^{3} - 6 \, b x}{3 \, a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 164, normalized size = 2.48 \begin {gather*} \left [\frac {4 \, a^{2} x^{5} - 20 \, a b x^{3} - 30 \, b^{2} x + 15 \, {\left (a b x^{2} + b^{2}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {a x^{2} + 2 \, a x \sqrt {-\frac {b}{a}} - b}{a x^{2} + b}\right )}{12 \, {\left (a^{4} x^{2} + a^{3} b\right )}}, \frac {2 \, a^{2} x^{5} - 10 \, a b x^{3} - 15 \, b^{2} x + 15 \, {\left (a b x^{2} + b^{2}\right )} \sqrt {\frac {b}{a}} \arctan \left (\frac {a x \sqrt {\frac {b}{a}}}{b}\right )}{6 \, {\left (a^{4} x^{2} + a^{3} b\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.13, size = 107, normalized size = 1.62 \begin {gather*} - \frac {b^{2} x}{2 a^{4} x^{2} + 2 a^{3} b} - \frac {5 \sqrt {- \frac {b^{3}}{a^{7}}} \log {\left (- \frac {a^{3} \sqrt {- \frac {b^{3}}{a^{7}}}}{b} + x \right )}}{4} + \frac {5 \sqrt {- \frac {b^{3}}{a^{7}}} \log {\left (\frac {a^{3} \sqrt {- \frac {b^{3}}{a^{7}}}}{b} + x \right )}}{4} + \frac {x^{3}}{3 a^{2}} - \frac {2 b x}{a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.78, size = 61, normalized size = 0.92 \begin {gather*} \frac {5 \, b^{2} \arctan \left (\frac {a x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a^{3}} - \frac {b^{2} x}{2 \, {\left (a x^{2} + b\right )} a^{3}} + \frac {a^{4} x^{3} - 6 \, a^{3} b x}{3 \, a^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 56, normalized size = 0.85 \begin {gather*} \frac {x^3}{3\,a^2}+\frac {5\,b^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {a}\,x}{\sqrt {b}}\right )}{2\,a^{7/2}}-\frac {b^2\,x}{2\,\left (a^4\,x^2+b\,a^3\right )}-\frac {2\,b\,x}{a^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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