Optimal. Leaf size=67 \[ \frac{a}{c^2 x \sqrt{d x-c} \sqrt{c+d x}}-\frac{x \left (2 a d^2+b c^2\right )}{c^4 \sqrt{d x-c} \sqrt{c+d x}} \]
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Rubi [A] time = 0.076683, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {454, 39} \[ \frac{a}{c^2 x \sqrt{d x-c} \sqrt{c+d x}}-\frac{x \left (2 a d^2+b c^2\right )}{c^4 \sqrt{d x-c} \sqrt{c+d x}} \]
Antiderivative was successfully verified.
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Rule 454
Rule 39
Rubi steps
\begin{align*} \int \frac{a+b x^2}{x^2 (-c+d x)^{3/2} (c+d x)^{3/2}} \, dx &=\frac{a}{c^2 x \sqrt{-c+d x} \sqrt{c+d x}}+\left (b+\frac{2 a d^2}{c^2}\right ) \int \frac{1}{(-c+d x)^{3/2} (c+d x)^{3/2}} \, dx\\ &=\frac{a}{c^2 x \sqrt{-c+d x} \sqrt{c+d x}}-\frac{\left (b c^2+2 a d^2\right ) x}{c^4 \sqrt{-c+d x} \sqrt{c+d x}}\\ \end{align*}
Mathematica [A] time = 0.0241339, size = 51, normalized size = 0.76 \[ \frac{a \left (c^2-2 d^2 x^2\right )-b c^2 x^2}{c^4 x \sqrt{d x-c} \sqrt{c+d x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 48, normalized size = 0.7 \begin{align*}{\frac{-2\,a{d}^{2}{x}^{2}-b{c}^{2}{x}^{2}+a{c}^{2}}{x{c}^{4}}{\frac{1}{\sqrt{dx+c}}}{\frac{1}{\sqrt{dx-c}}}} \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 = 1.49661, size = 197, normalized size = 2.94 \begin{align*} -\frac{{\left (b c^{2} d^{2} + 2 \, a d^{4}\right )} x^{3} -{\left (a c^{2} d -{\left (b c^{2} d + 2 \, a d^{3}\right )} x^{2}\right )} \sqrt{d x + c} \sqrt{d x - c} -{\left (b c^{4} + 2 \, a c^{2} d^{2}\right )} x}{c^{4} d^{3} x^{3} - c^{6} d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 58.0321, size = 165, normalized size = 2.46 \begin{align*} a \left (- \frac{d{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{7}{4}, \frac{9}{4}, 1 & \frac{3}{2}, \frac{5}{2}, 3 \\\frac{7}{4}, 2, \frac{9}{4}, \frac{5}{2}, 3 & 0 \end{matrix} \middle |{\frac{c^{2}}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac{3}{2}} c^{4}} + \frac{i d{G_{6, 6}^{2, 6}\left (\begin{matrix} \frac{1}{2}, 1, \frac{5}{4}, \frac{3}{2}, \frac{7}{4}, 1 & \\\frac{5}{4}, \frac{7}{4} & \frac{1}{2}, 1, 2, 0 \end{matrix} \middle |{\frac{c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac{3}{2}} c^{4}}\right ) + b \left (- \frac{{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{3}{4}, \frac{5}{4}, 1 & \frac{1}{2}, \frac{3}{2}, 2 \\\frac{3}{4}, 1, \frac{5}{4}, \frac{3}{2}, 2 & 0 \end{matrix} \middle |{\frac{c^{2}}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac{3}{2}} c^{2} d} + \frac{i{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{1}{2}, 0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1 & \\\frac{1}{4}, \frac{3}{4} & - \frac{1}{2}, 0, 1, 0 \end{matrix} \middle |{\frac{c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac{3}{2}} c^{2} d}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.36761, size = 296, normalized size = 4.42 \begin{align*} -\frac{{\left (b c^{2} + a d^{2}\right )} \sqrt{d x + c}}{2 \, \sqrt{d x - c} c^{4} d} - \frac{2 \,{\left (b c^{2}{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{4} + a d^{2}{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{4} + 4 \, a c d^{2}{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{2} + 4 \, b c^{4} + 12 \, a c^{2} d^{2}\right )}}{{\left ({\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{6} + 2 \, c{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{4} + 4 \, c^{2}{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{2} + 8 \, c^{3}\right )} c^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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