Optimal. Leaf size=68 \[ \frac {b^{3/2} x \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{5/2} \sqrt {d x^2}}+\frac {b}{a^2 \sqrt {d x^2}}-\frac {1}{3 a x^2 \sqrt {d x^2}} \]
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Rubi [A] time = 0.02, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {15, 325, 205} \[ \frac {b^{3/2} x \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{5/2} \sqrt {d x^2}}+\frac {b}{a^2 \sqrt {d x^2}}-\frac {1}{3 a x^2 \sqrt {d x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 205
Rule 325
Rubi steps
\begin {align*} \int \frac {1}{x^3 \sqrt {d x^2} \left (a+b x^2\right )} \, dx &=\frac {x \int \frac {1}{x^4 \left (a+b x^2\right )} \, dx}{\sqrt {d x^2}}\\ &=-\frac {1}{3 a x^2 \sqrt {d x^2}}-\frac {(b x) \int \frac {1}{x^2 \left (a+b x^2\right )} \, dx}{a \sqrt {d x^2}}\\ &=\frac {b}{a^2 \sqrt {d x^2}}-\frac {1}{3 a x^2 \sqrt {d x^2}}+\frac {\left (b^2 x\right ) \int \frac {1}{a+b x^2} \, dx}{a^2 \sqrt {d x^2}}\\ &=\frac {b}{a^2 \sqrt {d x^2}}-\frac {1}{3 a x^2 \sqrt {d x^2}}+\frac {b^{3/2} x \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{5/2} \sqrt {d x^2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 58, normalized size = 0.85 \[ \frac {d \left (3 b^{3/2} x^3 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )-\sqrt {a} \left (a-3 b x^2\right )\right )}{3 a^{5/2} \left (d x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 157, normalized size = 2.31 \[ \left [\frac {3 \, b d x^{4} \sqrt {-\frac {b}{a d}} \log \left (\frac {b x^{2} + 2 \, \sqrt {d x^{2}} a \sqrt {-\frac {b}{a d}} - a}{b x^{2} + a}\right ) + 2 \, {\left (3 \, b x^{2} - a\right )} \sqrt {d x^{2}}}{6 \, a^{2} d x^{4}}, \frac {3 \, b d x^{4} \sqrt {\frac {b}{a d}} \arctan \left (\sqrt {d x^{2}} \sqrt {\frac {b}{a d}}\right ) + {\left (3 \, b x^{2} - a\right )} \sqrt {d x^{2}}}{3 \, a^{2} d x^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 60, normalized size = 0.88 \[ \frac {b^{2} \arctan \left (\frac {\sqrt {d x^{2}} b}{\sqrt {a b d}}\right )}{\sqrt {a b d} a^{2}} + \frac {3 \, b d x^{2} - a d}{3 \, \sqrt {d x^{2}} a^{2} d x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 58, normalized size = 0.85 \[ \frac {3 b^{2} x^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )+3 \sqrt {a b}\, b \,x^{2}-\sqrt {a b}\, a}{3 \sqrt {d \,x^{2}}\, \sqrt {a b}\, a^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.01, size = 52, normalized size = 0.76 \[ \frac {b^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a^{2} \sqrt {d}} + \frac {3 \, b \sqrt {d} x^{2} - a \sqrt {d}}{3 \, a^{2} d x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.63, size = 53, normalized size = 0.78 \[ \frac {b^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x^2}}{\sqrt {a}}\right )}{a^{5/2}\,\sqrt {d}}-\frac {1}{3\,a\,\sqrt {d}\,{\left (x^2\right )}^{3/2}}+\frac {b\,x^2}{a^2\,\sqrt {d}\,{\left (x^2\right )}^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{3} \sqrt {d x^{2}} \left (a + b x^{2}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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