Optimal. Leaf size=72 \[ \frac{a^{3/2} x \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{5/2} \sqrt{d x^2}}-\frac{a x^2}{b^2 \sqrt{d x^2}}+\frac{x^4}{3 b \sqrt{d x^2}} \]
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Rubi [A] time = 0.0256232, antiderivative size = 72, 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, 302, 205} \[ \frac{a^{3/2} x \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{5/2} \sqrt{d x^2}}-\frac{a x^2}{b^2 \sqrt{d x^2}}+\frac{x^4}{3 b \sqrt{d x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 302
Rule 205
Rubi steps
\begin{align*} \int \frac{x^5}{\sqrt{d x^2} \left (a+b x^2\right )} \, dx &=\frac{x \int \frac{x^4}{a+b x^2} \, dx}{\sqrt{d x^2}}\\ &=\frac{x \int \left (-\frac{a}{b^2}+\frac{x^2}{b}+\frac{a^2}{b^2 \left (a+b x^2\right )}\right ) \, dx}{\sqrt{d x^2}}\\ &=-\frac{a x^2}{b^2 \sqrt{d x^2}}+\frac{x^4}{3 b \sqrt{d x^2}}+\frac{\left (a^2 x\right ) \int \frac{1}{a+b x^2} \, dx}{b^2 \sqrt{d x^2}}\\ &=-\frac{a x^2}{b^2 \sqrt{d x^2}}+\frac{x^4}{3 b \sqrt{d x^2}}+\frac{a^{3/2} x \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{5/2} \sqrt{d x^2}}\\ \end{align*}
Mathematica [A] time = 0.0240915, size = 56, normalized size = 0.78 \[ \frac{x \left (3 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )+\sqrt{b} x \left (b x^2-3 a\right )\right )}{3 b^{5/2} \sqrt{d x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 53, normalized size = 0.7 \begin{align*}{\frac{x}{3\,{b}^{2}} \left ( \sqrt{ab}{x}^{3}b-3\,\sqrt{ab}xa+3\,{a}^{2}\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ) \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 = 1.33494, size = 315, normalized size = 4.38 \begin{align*} \left [\frac{3 \, a d \sqrt{-\frac{a}{b d}} \log \left (\frac{b x^{2} + 2 \, \sqrt{d x^{2}} b \sqrt{-\frac{a}{b d}} - a}{b x^{2} + a}\right ) + 2 \,{\left (b x^{2} - 3 \, a\right )} \sqrt{d x^{2}}}{6 \, b^{2} d}, \frac{3 \, a d \sqrt{\frac{a}{b d}} \arctan \left (\frac{\sqrt{d x^{2}} b \sqrt{\frac{a}{b d}}}{a}\right ) +{\left (b x^{2} - 3 \, a\right )} \sqrt{d x^{2}}}{3 \, b^{2} d}\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{x^{5}}{\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 = 1.12183, size = 95, normalized size = 1.32 \begin{align*} \frac{a^{2} \arctan \left (\frac{\sqrt{d x^{2}} b}{\sqrt{a b d}}\right )}{\sqrt{a b d} b^{2}} + \frac{\sqrt{d x^{2}} b^{2} d^{5} x^{2} - 3 \, \sqrt{d x^{2}} a b d^{5}}{3 \, b^{3} d^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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