Optimal. Leaf size=74 \[ \frac {15 x}{8 a^3}-\frac {x^5}{4 a \left (b+a x^2\right )^2}-\frac {5 x^3}{8 a^2 \left (b+a x^2\right )}-\frac {15 \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{8 a^{7/2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {199, 294, 327,
211} \begin {gather*} -\frac {15 \sqrt {b} \text {ArcTan}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{8 a^{7/2}}+\frac {15 x}{8 a^3}-\frac {5 x^3}{8 a^2 \left (a x^2+b\right )}-\frac {x^5}{4 a \left (a x^2+b\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 199
Rule 211
Rule 294
Rule 327
Rubi steps
\begin {align*} \int \frac {1}{\left (a+\frac {b}{x^2}\right )^3} \, dx &=\int \frac {x^6}{\left (b+a x^2\right )^3} \, dx\\ &=-\frac {x^5}{4 a \left (b+a x^2\right )^2}+\frac {5 \int \frac {x^4}{\left (b+a x^2\right )^2} \, dx}{4 a}\\ &=-\frac {x^5}{4 a \left (b+a x^2\right )^2}-\frac {5 x^3}{8 a^2 \left (b+a x^2\right )}+\frac {15 \int \frac {x^2}{b+a x^2} \, dx}{8 a^2}\\ &=\frac {15 x}{8 a^3}-\frac {x^5}{4 a \left (b+a x^2\right )^2}-\frac {5 x^3}{8 a^2 \left (b+a x^2\right )}-\frac {(15 b) \int \frac {1}{b+a x^2} \, dx}{8 a^3}\\ &=\frac {15 x}{8 a^3}-\frac {x^5}{4 a \left (b+a x^2\right )^2}-\frac {5 x^3}{8 a^2 \left (b+a x^2\right )}-\frac {15 \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{8 a^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 66, normalized size = 0.89 \begin {gather*} \frac {15 b^2 x+25 a b x^3+8 a^2 x^5}{8 a^3 \left (b+a x^2\right )^2}-\frac {15 \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{8 a^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 51, normalized size = 0.69
method | result | size |
default | \(\frac {x}{a^{3}}-\frac {b \left (\frac {-\frac {9}{8} a \,x^{3}-\frac {7}{8} b x}{\left (a \,x^{2}+b \right )^{2}}+\frac {15 \arctan \left (\frac {a x}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{a^{3}}\) | \(51\) |
risch | \(\frac {x}{a^{3}}+\frac {\frac {9}{8} a b \,x^{3}+\frac {7}{8} b^{2} x}{a^{3} \left (a \,x^{2}+b \right )^{2}}+\frac {15 \sqrt {-a b}\, \ln \left (-\sqrt {-a b}\, x -b \right )}{16 a^{4}}-\frac {15 \sqrt {-a b}\, \ln \left (\sqrt {-a b}\, x -b \right )}{16 a^{4}}\) | \(83\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 68, normalized size = 0.92 \begin {gather*} \frac {9 \, a b x^{3} + 7 \, b^{2} x}{8 \, {\left (a^{5} x^{4} + 2 \, a^{4} b x^{2} + a^{3} b^{2}\right )}} - \frac {15 \, b \arctan \left (\frac {a x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{3}} + \frac {x}{a^{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 = 2.73 \begin {gather*} \left [\frac {16 \, a^{2} x^{5} + 50 \, a b x^{3} + 30 \, b^{2} x + 15 \, {\left (a^{2} x^{4} + 2 \, 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 )}{16 \, {\left (a^{5} x^{4} + 2 \, a^{4} b x^{2} + a^{3} b^{2}\right )}}, \frac {8 \, a^{2} x^{5} + 25 \, a b x^{3} + 15 \, b^{2} x - 15 \, {\left (a^{2} x^{4} + 2 \, a b x^{2} + b^{2}\right )} \sqrt {\frac {b}{a}} \arctan \left (\frac {a x \sqrt {\frac {b}{a}}}{b}\right )}{8 \, {\left (a^{5} x^{4} + 2 \, a^{4} b x^{2} + a^{3} b^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.19, size = 107, normalized size = 1.45 \begin {gather*} \frac {15 \sqrt {- \frac {b}{a^{7}}} \log {\left (- a^{3} \sqrt {- \frac {b}{a^{7}}} + x \right )}}{16} - \frac {15 \sqrt {- \frac {b}{a^{7}}} \log {\left (a^{3} \sqrt {- \frac {b}{a^{7}}} + x \right )}}{16} + \frac {9 a b x^{3} + 7 b^{2} x}{8 a^{5} x^{4} + 16 a^{4} b x^{2} + 8 a^{3} b^{2}} + \frac {x}{a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.11, size = 54, normalized size = 0.73 \begin {gather*} -\frac {15 \, b \arctan \left (\frac {a x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{3}} + \frac {x}{a^{3}} + \frac {9 \, a b x^{3} + 7 \, b^{2} x}{8 \, {\left (a x^{2} + b\right )}^{2} a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.36, size = 64, normalized size = 0.86 \begin {gather*} \frac {\frac {7\,b^2\,x}{8}+\frac {9\,a\,b\,x^3}{8}}{a^5\,x^4+2\,a^4\,b\,x^2+a^3\,b^2}+\frac {x}{a^3}-\frac {15\,\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {a}\,x}{\sqrt {b}}\right )}{8\,a^{7/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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