Optimal. Leaf size=132 \[ \frac{\sqrt{3 \sqrt{a}+2 \sqrt{b}} \tanh ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{3 x+2}}{\sqrt{3 \sqrt{a}+2 \sqrt{b}}}\right )}{\sqrt{a} b^{3/4}}-\frac{\sqrt{3 \sqrt{a}-2 \sqrt{b}} \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{3 x+2}}{\sqrt{3 \sqrt{a}-2 \sqrt{b}}}\right )}{\sqrt{a} b^{3/4}} \]
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Rubi [A] time = 0.132167, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {700, 1130, 208, 205} \[ \frac{\sqrt{3 \sqrt{a}+2 \sqrt{b}} \tanh ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{3 x+2}}{\sqrt{3 \sqrt{a}+2 \sqrt{b}}}\right )}{\sqrt{a} b^{3/4}}-\frac{\sqrt{3 \sqrt{a}-2 \sqrt{b}} \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{3 x+2}}{\sqrt{3 \sqrt{a}-2 \sqrt{b}}}\right )}{\sqrt{a} b^{3/4}} \]
Antiderivative was successfully verified.
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Rule 700
Rule 1130
Rule 208
Rule 205
Rubi steps
\begin{align*} \int \frac{\sqrt{2+3 x}}{a-b x^2} \, dx &=6 \operatorname{Subst}\left (\int \frac{x^2}{9 a-4 b+4 b x^2-b x^4} \, dx,x,\sqrt{2+3 x}\right )\\ &=\left (3-\frac{2 \sqrt{b}}{\sqrt{a}}\right ) \operatorname{Subst}\left (\int \frac{1}{-3 \sqrt{a} \sqrt{b}+2 b-b x^2} \, dx,x,\sqrt{2+3 x}\right )+\left (3+\frac{2 \sqrt{b}}{\sqrt{a}}\right ) \operatorname{Subst}\left (\int \frac{1}{3 \sqrt{a} \sqrt{b}+2 b-b x^2} \, dx,x,\sqrt{2+3 x}\right )\\ &=-\frac{\sqrt{3 \sqrt{a}-2 \sqrt{b}} \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{2+3 x}}{\sqrt{3 \sqrt{a}-2 \sqrt{b}}}\right )}{\sqrt{a} b^{3/4}}+\frac{\sqrt{3 \sqrt{a}+2 \sqrt{b}} \tanh ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{2+3 x}}{\sqrt{3 \sqrt{a}+2 \sqrt{b}}}\right )}{\sqrt{a} b^{3/4}}\\ \end{align*}
Mathematica [A] time = 0.0932695, size = 123, normalized size = 0.93 \[ \frac{\sqrt{3 \sqrt{a}+2 \sqrt{b}} \tanh ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{3 x+2}}{\sqrt{3 \sqrt{a}+2 \sqrt{b}}}\right )-\sqrt{3 \sqrt{a}-2 \sqrt{b}} \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{3 x+2}}{\sqrt{3 \sqrt{a}-2 \sqrt{b}}}\right )}{\sqrt{a} b^{3/4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.158, size = 182, normalized size = 1.4 \begin{align*} 3\,{\frac{1}{\sqrt{ \left ( 3\,\sqrt{ab}+2\,b \right ) b}}{\it Artanh} \left ({\frac{b\sqrt{2+3\,x}}{\sqrt{ \left ( 3\,\sqrt{ab}+2\,b \right ) b}}} \right ) }+2\,{\frac{b}{\sqrt{ab}\sqrt{ \left ( 3\,\sqrt{ab}+2\,b \right ) b}}{\it Artanh} \left ({\frac{b\sqrt{2+3\,x}}{\sqrt{ \left ( 3\,\sqrt{ab}+2\,b \right ) b}}} \right ) }-3\,{\frac{1}{\sqrt{ \left ( 3\,\sqrt{ab}-2\,b \right ) b}}\arctan \left ({\frac{b\sqrt{2+3\,x}}{\sqrt{ \left ( 3\,\sqrt{ab}-2\,b \right ) b}}} \right ) }+2\,{\frac{b}{\sqrt{ab}\sqrt{ \left ( 3\,\sqrt{ab}-2\,b \right ) b}}\arctan \left ({\frac{b\sqrt{2+3\,x}}{\sqrt{ \left ( 3\,\sqrt{ab}-2\,b \right ) b}}} \right ) } \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 \frac{\sqrt{3 \, x + 2}}{b x^{2} - a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.2393, size = 703, normalized size = 5.33 \begin{align*} \frac{1}{2} \, \sqrt{\frac{3 \, a b \sqrt{\frac{1}{a b^{3}}} + 2}{a b}} \log \left (a b^{2} \sqrt{\frac{3 \, a b \sqrt{\frac{1}{a b^{3}}} + 2}{a b}} \sqrt{\frac{1}{a b^{3}}} + \sqrt{3 \, x + 2}\right ) - \frac{1}{2} \, \sqrt{\frac{3 \, a b \sqrt{\frac{1}{a b^{3}}} + 2}{a b}} \log \left (-a b^{2} \sqrt{\frac{3 \, a b \sqrt{\frac{1}{a b^{3}}} + 2}{a b}} \sqrt{\frac{1}{a b^{3}}} + \sqrt{3 \, x + 2}\right ) - \frac{1}{2} \, \sqrt{-\frac{3 \, a b \sqrt{\frac{1}{a b^{3}}} - 2}{a b}} \log \left (a b^{2} \sqrt{-\frac{3 \, a b \sqrt{\frac{1}{a b^{3}}} - 2}{a b}} \sqrt{\frac{1}{a b^{3}}} + \sqrt{3 \, x + 2}\right ) + \frac{1}{2} \, \sqrt{-\frac{3 \, a b \sqrt{\frac{1}{a b^{3}}} - 2}{a b}} \log \left (-a b^{2} \sqrt{-\frac{3 \, a b \sqrt{\frac{1}{a b^{3}}} - 2}{a b}} \sqrt{\frac{1}{a b^{3}}} + \sqrt{3 \, x + 2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.56618, size = 58, normalized size = 0.44 \begin{align*} - 6 \operatorname{RootSum}{\left (20736 t^{4} a^{2} b^{3} - 576 t^{2} a b^{2} - 9 a + 4 b, \left ( t \mapsto t \log{\left (- 576 t^{3} a b^{2} + 8 t b + \sqrt{3 x + 2} \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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