Optimal. Leaf size=56 \[ \frac{a \tan ^{-1}\left (\frac{\sqrt{d x-c} \sqrt{c+d x}}{c}\right )}{c}+\frac{b \sqrt{d x-c} \sqrt{c+d x}}{d^2} \]
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Rubi [A] time = 0.0687923, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {460, 92, 205} \[ \frac{a \tan ^{-1}\left (\frac{\sqrt{d x-c} \sqrt{c+d x}}{c}\right )}{c}+\frac{b \sqrt{d x-c} \sqrt{c+d x}}{d^2} \]
Antiderivative was successfully verified.
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Rule 460
Rule 92
Rule 205
Rubi steps
\begin{align*} \int \frac{a+b x^2}{x \sqrt{-c+d x} \sqrt{c+d x}} \, dx &=\frac{b \sqrt{-c+d x} \sqrt{c+d x}}{d^2}+a \int \frac{1}{x \sqrt{-c+d x} \sqrt{c+d x}} \, dx\\ &=\frac{b \sqrt{-c+d x} \sqrt{c+d x}}{d^2}+(a d) \operatorname{Subst}\left (\int \frac{1}{c^2 d+d x^2} \, dx,x,\sqrt{-c+d x} \sqrt{c+d x}\right )\\ &=\frac{b \sqrt{-c+d x} \sqrt{c+d x}}{d^2}+\frac{a \tan ^{-1}\left (\frac{\sqrt{-c+d x} \sqrt{c+d x}}{c}\right )}{c}\\ \end{align*}
Mathematica [A] time = 0.0324644, size = 87, normalized size = 1.55 \[ \frac{a d^2 \sqrt{d^2 x^2-c^2} \tan ^{-1}\left (\frac{\sqrt{d^2 x^2-c^2}}{c}\right )-b c^3+b c d^2 x^2}{c d^2 \sqrt{d x-c} \sqrt{c+d x}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.019, size = 108, normalized size = 1.9 \begin{align*}{\frac{1}{{d}^{2}} \left ( -\ln \left ( -2\,{\frac{{c}^{2}-\sqrt{-{c}^{2}}\sqrt{{d}^{2}{x}^{2}-{c}^{2}}}{x}} \right ) a{d}^{2}+b\sqrt{-{c}^{2}}\sqrt{{d}^{2}{x}^{2}-{c}^{2}} \right ) \sqrt{dx-c}\sqrt{dx+c}{\frac{1}{\sqrt{-{c}^{2}}}}{\frac{1}{\sqrt{{d}^{2}{x}^{2}-{c}^{2}}}}} \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 = 1.5188, size = 135, normalized size = 2.41 \begin{align*} \frac{2 \, a d^{2} \arctan \left (-\frac{d x - \sqrt{d x + c} \sqrt{d x - c}}{c}\right ) + \sqrt{d x + c} \sqrt{d x - c} b c}{c d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 21.1566, size = 178, normalized size = 3.18 \begin{align*} - \frac{a{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{3}{4}, \frac{5}{4}, 1 & 1, 1, \frac{3}{2} \\\frac{1}{2}, \frac{3}{4}, 1, \frac{5}{4}, \frac{3}{2} & 0 \end{matrix} \middle |{\frac{c^{2}}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c} + \frac{i a{G_{6, 6}^{2, 6}\left (\begin{matrix} 0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, 1 & \\\frac{1}{4}, \frac{3}{4} & 0, \frac{1}{2}, \frac{1}{2}, 0 \end{matrix} \middle |{\frac{c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c} + \frac{b c{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{1}{4}, \frac{1}{4} & 0, 0, \frac{1}{2}, 1 \\- \frac{1}{2}, - \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, 0 & \end{matrix} \middle |{\frac{c^{2}}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{2}} + \frac{i b c{G_{6, 6}^{2, 6}\left (\begin{matrix} -1, - \frac{3}{4}, - \frac{1}{2}, - \frac{1}{4}, 0, 1 & \\- \frac{3}{4}, - \frac{1}{4} & -1, - \frac{1}{2}, - \frac{1}{2}, 0 \end{matrix} \middle |{\frac{c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16348, size = 74, normalized size = 1.32 \begin{align*} -\frac{2 \, a \arctan \left (\frac{{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{2}}{2 \, c}\right )}{c} + \frac{\sqrt{d x + c} \sqrt{d x - c} b}{d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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