Optimal. Leaf size=72 \[ \frac{\sqrt{d x-c} \sqrt{c+d x} \left (3 a d^2+2 b c^2\right )}{3 d^4}+\frac{b x^2 \sqrt{d x-c} \sqrt{c+d x}}{3 d^2} \]
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Rubi [A] time = 0.0463676, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {460, 74} \[ \frac{\sqrt{d x-c} \sqrt{c+d x} \left (3 a d^2+2 b c^2\right )}{3 d^4}+\frac{b x^2 \sqrt{d x-c} \sqrt{c+d x}}{3 d^2} \]
Antiderivative was successfully verified.
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Rule 460
Rule 74
Rubi steps
\begin{align*} \int \frac{x \left (a+b x^2\right )}{\sqrt{-c+d x} \sqrt{c+d x}} \, dx &=\frac{b x^2 \sqrt{-c+d x} \sqrt{c+d x}}{3 d^2}-\frac{1}{3} \left (-3 a-\frac{2 b c^2}{d^2}\right ) \int \frac{x}{\sqrt{-c+d x} \sqrt{c+d x}} \, dx\\ &=\frac{\left (2 b c^2+3 a d^2\right ) \sqrt{-c+d x} \sqrt{c+d x}}{3 d^4}+\frac{b x^2 \sqrt{-c+d x} \sqrt{c+d x}}{3 d^2}\\ \end{align*}
Mathematica [A] time = 0.0327275, size = 61, normalized size = 0.85 \[ \frac{\left (d^2 x^2-c^2\right ) \left (3 a d^2+2 b c^2+b d^2 x^2\right )}{3 d^4 \sqrt{d x-c} \sqrt{c+d x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 43, normalized size = 0.6 \begin{align*}{\frac{b{d}^{2}{x}^{2}+3\,a{d}^{2}+2\,b{c}^{2}}{3\,{d}^{4}}\sqrt{dx+c}\sqrt{dx-c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.981241, size = 93, normalized size = 1.29 \begin{align*} \frac{\sqrt{d^{2} x^{2} - c^{2}} b x^{2}}{3 \, d^{2}} + \frac{2 \, \sqrt{d^{2} x^{2} - c^{2}} b c^{2}}{3 \, d^{4}} + \frac{\sqrt{d^{2} x^{2} - c^{2}} a}{d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.4751, size = 93, normalized size = 1.29 \begin{align*} \frac{{\left (b d^{2} x^{2} + 2 \, b c^{2} + 3 \, a d^{2}\right )} \sqrt{d x + c} \sqrt{d x - c}}{3 \, d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 33.112, size = 223, normalized size = 3.1 \begin{align*} \frac{a 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 a 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}} + \frac{b c^{3}{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{5}{4}, - \frac{3}{4} & -1, -1, - \frac{1}{2}, 1 \\- \frac{3}{2}, - \frac{5}{4}, -1, - \frac{3}{4}, - \frac{1}{2}, 0 & \end{matrix} \middle |{\frac{c^{2}}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{4}} + \frac{i b c^{3}{G_{6, 6}^{2, 6}\left (\begin{matrix} -2, - \frac{7}{4}, - \frac{3}{2}, - \frac{5}{4}, -1, 1 & \\- \frac{7}{4}, - \frac{5}{4} & -2, - \frac{3}{2}, - \frac{3}{2}, 0 \end{matrix} \middle |{\frac{c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14379, size = 82, normalized size = 1.14 \begin{align*} \frac{{\left (3 \, b c^{2} d^{9} + 3 \, a d^{11} +{\left ({\left (d x + c\right )} b d^{9} - 2 \, b c d^{9}\right )}{\left (d x + c\right )}\right )} \sqrt{d x + c} \sqrt{d x - c}}{1920 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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