Optimal. Leaf size=17 \[ \frac{x \sqrt{\tan \left (a+b x^2\right )}}{b} \]
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Rubi [F] time = 0.0346331, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \left (\frac{x^2}{\sqrt{\tan \left (a+b x^2\right )}}+\frac{\sqrt{\tan \left (a+b x^2\right )}}{b}+x^2 \tan ^{\frac{3}{2}}\left (a+b x^2\right )\right ) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \left (\frac{x^2}{\sqrt{\tan \left (a+b x^2\right )}}+\frac{\sqrt{\tan \left (a+b x^2\right )}}{b}+x^2 \tan ^{\frac{3}{2}}\left (a+b x^2\right )\right ) \, dx &=\frac{\int \sqrt{\tan \left (a+b x^2\right )} \, dx}{b}+\int \frac{x^2}{\sqrt{\tan \left (a+b x^2\right )}} \, dx+\int x^2 \tan ^{\frac{3}{2}}\left (a+b x^2\right ) \, dx\\ \end{align*}
Mathematica [A] time = 0.559597, size = 17, normalized size = 1. \[ \frac{x \sqrt{\tan \left (a+b x^2\right )}}{b} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.253, size = 0, normalized size = 0. \begin{align*} \int{{x}^{2}{\frac{1}{\sqrt{\tan \left ( b{x}^{2}+a \right ) }}}}+{\frac{1}{b}\sqrt{\tan \left ( b{x}^{2}+a \right ) }}+{x}^{2} \left ( \tan \left ( b{x}^{2}+a \right ) \right ) ^{{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \tan \left (b x^{2} + a\right )^{\frac{3}{2}} + \frac{x^{2}}{\sqrt{\tan \left (b x^{2} + a\right )}} + \frac{\sqrt{\tan \left (b x^{2} + a\right )}}{b}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55773, size = 35, normalized size = 2.06 \begin{align*} \frac{x \sqrt{\tan \left (b x^{2} + a\right )}}{b} \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*} \frac{\int \frac{b x^{2}}{\sqrt{\tan{\left (a + b x^{2} \right )}}}\, dx + \int b x^{2} \tan ^{\frac{3}{2}}{\left (a + b x^{2} \right )}\, dx + \int \sqrt{\tan{\left (a + b x^{2} \right )}}\, dx}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \tan \left (b x^{2} + a\right )^{\frac{3}{2}} + \frac{x^{2}}{\sqrt{\tan \left (b x^{2} + a\right )}} + \frac{\sqrt{\tan \left (b x^{2} + a\right )}}{b}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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