Optimal. Leaf size=17 \[ \frac {x \sqrt {\tan \left (a+b x^2\right )}}{b} \]
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Rubi [F]
time = 0.03, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \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 \end {gather*}
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*}
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Mathematica [A]
time = 0.51, size = 17, normalized size = 1.00 \begin {gather*} \frac {x \sqrt {\tan \left (a+b x^2\right )}}{b} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.43, size = 0, normalized size = 0.00 \[\int \frac {x^{2}}{\sqrt {\tan \left (b \,x^{2}+a \right )}}+\frac {\sqrt {\tan }\left (b \,x^{2}+a \right )}{b}+x^{2} \left (\tan ^{\frac {3}{2}}\left (b \,x^{2}+a \right )\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 15, normalized size = 0.88 \begin {gather*} \frac {x \sqrt {\tan \left (b x^{2} + a\right )}}{b} \end {gather*}
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 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.17, size = 45, normalized size = 2.65 \begin {gather*} \frac {x\,\sqrt {-\frac {{\mathrm {e}}^{2{}\mathrm {i}\,b\,x^2+a\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}}{{\mathrm {e}}^{2{}\mathrm {i}\,b\,x^2+a\,2{}\mathrm {i}}+1}}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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