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 = 0.0240175, antiderivative size = 68, normalized size of antiderivative = 1., 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 325
Rule 205
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*}
Mathematica [A] time = 0.0215465, 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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Maple [A] time = 0.01, size = 58, normalized size = 0.9 \begin{align*}{\frac{1}{3\,{a}^{2}{x}^{2}} \left ( 3\,{b}^{2}\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ){x}^{3}+3\,b{x}^{2}\sqrt{ab}-a\sqrt{ab} \right ){\frac{1}{\sqrt{d{x}^{2}}}}{\frac{1}{\sqrt{ab}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.36295, size = 331, normalized size = 4.87 \begin{align*} \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 ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{3} \sqrt{d x^{2}} \left (a + b x^{2}\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11755, size = 92, normalized size = 1.35 \begin{align*} \frac{1}{3} \, d^{2}{\left (\frac{3 \, b^{2} \arctan \left (\frac{\sqrt{d x^{2}} b}{\sqrt{a b d}}\right )}{\sqrt{a b d} a^{2} d^{2}} + \frac{3 \, b d x^{2} - a d}{\sqrt{d x^{2}} a^{2} d^{3} x^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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