Optimal. Leaf size=76 \[ -\frac {15 \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{7/2}}-\frac {15}{8 a^3 x}+\frac {5}{8 a^2 x \left (a+b x^2\right )}+\frac {1}{4 a x \left (a+b x^2\right )^2} \]
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Rubi [A] time = 0.03, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {290, 325, 205} \[ \frac {5}{8 a^2 x \left (a+b x^2\right )}-\frac {15 \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{7/2}}-\frac {15}{8 a^3 x}+\frac {1}{4 a x \left (a+b x^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 205
Rule 290
Rule 325
Rubi steps
\begin {align*} \int \frac {1}{x^2 \left (a+b x^2\right )^3} \, dx &=\frac {1}{4 a x \left (a+b x^2\right )^2}+\frac {5 \int \frac {1}{x^2 \left (a+b x^2\right )^2} \, dx}{4 a}\\ &=\frac {1}{4 a x \left (a+b x^2\right )^2}+\frac {5}{8 a^2 x \left (a+b x^2\right )}+\frac {15 \int \frac {1}{x^2 \left (a+b x^2\right )} \, dx}{8 a^2}\\ &=-\frac {15}{8 a^3 x}+\frac {1}{4 a x \left (a+b x^2\right )^2}+\frac {5}{8 a^2 x \left (a+b x^2\right )}-\frac {(15 b) \int \frac {1}{a+b x^2} \, dx}{8 a^3}\\ &=-\frac {15}{8 a^3 x}+\frac {1}{4 a x \left (a+b x^2\right )^2}+\frac {5}{8 a^2 x \left (a+b x^2\right )}-\frac {15 \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 68, normalized size = 0.89 \[ -\frac {15 \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{7/2}}-\frac {8 a^2+25 a b x^2+15 b^2 x^4}{8 a^3 x \left (a+b x^2\right )^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.94, size = 202, normalized size = 2.66 \[ \left [-\frac {30 \, b^{2} x^{4} + 50 \, a b x^{2} - 15 \, {\left (b^{2} x^{5} + 2 \, a b x^{3} + a^{2} x\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} - 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right ) + 16 \, a^{2}}{16 \, {\left (a^{3} b^{2} x^{5} + 2 \, a^{4} b x^{3} + a^{5} x\right )}}, -\frac {15 \, b^{2} x^{4} + 25 \, a b x^{2} + 15 \, {\left (b^{2} x^{5} + 2 \, a b x^{3} + a^{2} x\right )} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right ) + 8 \, a^{2}}{8 \, {\left (a^{3} b^{2} x^{5} + 2 \, a^{4} b x^{3} + a^{5} x\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.64, size = 57, normalized size = 0.75 \[ -\frac {15 \, b \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{3}} - \frac {7 \, b^{2} x^{3} + 9 \, a b x}{8 \, {\left (b x^{2} + a\right )}^{2} a^{3}} - \frac {1}{a^{3} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 66, normalized size = 0.87 \[ -\frac {7 b^{2} x^{3}}{8 \left (b \,x^{2}+a \right )^{2} a^{3}}-\frac {9 b x}{8 \left (b \,x^{2}+a \right )^{2} a^{2}}-\frac {15 b \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}\, a^{3}}-\frac {1}{a^{3} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.98, size = 71, normalized size = 0.93 \[ -\frac {15 \, b^{2} x^{4} + 25 \, a b x^{2} + 8 \, a^{2}}{8 \, {\left (a^{3} b^{2} x^{5} + 2 \, a^{4} b x^{3} + a^{5} x\right )}} - \frac {15 \, b \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.67, size = 66, normalized size = 0.87 \[ -\frac {\frac {1}{a}+\frac {25\,b\,x^2}{8\,a^2}+\frac {15\,b^2\,x^4}{8\,a^3}}{a^2\,x+2\,a\,b\,x^3+b^2\,x^5}-\frac {15\,\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{8\,a^{7/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.42, size = 116, normalized size = 1.53 \[ \frac {15 \sqrt {- \frac {b}{a^{7}}} \log {\left (- \frac {a^{4} \sqrt {- \frac {b}{a^{7}}}}{b} + x \right )}}{16} - \frac {15 \sqrt {- \frac {b}{a^{7}}} \log {\left (\frac {a^{4} \sqrt {- \frac {b}{a^{7}}}}{b} + x \right )}}{16} + \frac {- 8 a^{2} - 25 a b x^{2} - 15 b^{2} x^{4}}{8 a^{5} x + 16 a^{4} b x^{3} + 8 a^{3} b^{2} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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