Optimal. Leaf size=87 \[ \frac{\left (\frac{2}{13}+\frac{3 i}{13}\right ) 2^{\frac{3}{2}-i} (1-i a x)^{\frac{3}{2}+i} \sqrt{a^2 c x^2+c} \, _2F_1\left (-\frac{1}{2}+i,\frac{3}{2}+i;\frac{5}{2}+i;\frac{1}{2} (1-i a x)\right )}{a \sqrt{a^2 x^2+1}} \]
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Rubi [A] time = 0.0744987, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {5076, 5073, 69} \[ \frac{\left (\frac{2}{13}+\frac{3 i}{13}\right ) 2^{\frac{3}{2}-i} (1-i a x)^{\frac{3}{2}+i} \sqrt{a^2 c x^2+c} \, _2F_1\left (-\frac{1}{2}+i,\frac{3}{2}+i;\frac{5}{2}+i;\frac{1}{2} (1-i a x)\right )}{a \sqrt{a^2 x^2+1}} \]
Antiderivative was successfully verified.
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Rule 5076
Rule 5073
Rule 69
Rubi steps
\begin{align*} \int e^{2 \tan ^{-1}(a x)} \sqrt{c+a^2 c x^2} \, dx &=\frac{\sqrt{c+a^2 c x^2} \int e^{2 \tan ^{-1}(a x)} \sqrt{1+a^2 x^2} \, dx}{\sqrt{1+a^2 x^2}}\\ &=\frac{\sqrt{c+a^2 c x^2} \int (1-i a x)^{\frac{1}{2}+i} (1+i a x)^{\frac{1}{2}-i} \, dx}{\sqrt{1+a^2 x^2}}\\ &=\frac{\left (\frac{2}{13}+\frac{3 i}{13}\right ) 2^{\frac{3}{2}-i} (1-i a x)^{\frac{3}{2}+i} \sqrt{c+a^2 c x^2} \, _2F_1\left (-\frac{1}{2}+i,\frac{3}{2}+i;\frac{5}{2}+i;\frac{1}{2} (1-i a x)\right )}{a \sqrt{1+a^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0178089, size = 87, normalized size = 1. \[ \frac{\left (\frac{2}{13}+\frac{3 i}{13}\right ) 2^{\frac{3}{2}-i} (1-i a x)^{\frac{3}{2}+i} \sqrt{a^2 c x^2+c} \, _2F_1\left (-\frac{1}{2}+i,\frac{3}{2}+i;\frac{5}{2}+i;\frac{1}{2} (1-i a x)\right )}{a \sqrt{a^2 x^2+1}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.276, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{2\,\arctan \left ( ax \right ) }}\sqrt{{a}^{2}c{x}^{2}+c}\, 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 \sqrt{a^{2} c x^{2} + c} e^{\left (2 \, \arctan \left (a x\right )\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{a^{2} c x^{2} + c} e^{\left (2 \, \arctan \left (a x\right )\right )}, x\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 \sqrt{c \left (a^{2} x^{2} + 1\right )} e^{2 \operatorname{atan}{\left (a x \right )}}\, dx \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: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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