Optimal. Leaf size=104 \[ \frac{1}{2} x \sqrt{d x-c} \sqrt{c+d x} \left (b-\frac{2 a d^2}{c^2}\right )-\frac{\left (b c^2-2 a d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d x-c}}{\sqrt{c+d x}}\right )}{d}+\frac{a (d x-c)^{3/2} (c+d x)^{3/2}}{c^2 x} \]
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Rubi [A] time = 0.0869517, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161, Rules used = {454, 38, 63, 217, 206} \[ \frac{1}{2} x \sqrt{d x-c} \sqrt{c+d x} \left (b-\frac{2 a d^2}{c^2}\right )-\frac{\left (b c^2-2 a d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d x-c}}{\sqrt{c+d x}}\right )}{d}+\frac{a (d x-c)^{3/2} (c+d x)^{3/2}}{c^2 x} \]
Antiderivative was successfully verified.
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Rule 454
Rule 38
Rule 63
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{\sqrt{-c+d x} \sqrt{c+d x} \left (a+b x^2\right )}{x^2} \, dx &=\frac{a (-c+d x)^{3/2} (c+d x)^{3/2}}{c^2 x}+\left (b-\frac{2 a d^2}{c^2}\right ) \int \sqrt{-c+d x} \sqrt{c+d x} \, dx\\ &=\frac{1}{2} \left (b-\frac{2 a d^2}{c^2}\right ) x \sqrt{-c+d x} \sqrt{c+d x}+\frac{a (-c+d x)^{3/2} (c+d x)^{3/2}}{c^2 x}+\frac{1}{2} \left (-b c^2+2 a d^2\right ) \int \frac{1}{\sqrt{-c+d x} \sqrt{c+d x}} \, dx\\ &=\frac{1}{2} \left (b-\frac{2 a d^2}{c^2}\right ) x \sqrt{-c+d x} \sqrt{c+d x}+\frac{a (-c+d x)^{3/2} (c+d x)^{3/2}}{c^2 x}+\frac{\left (-b c^2+2 a d^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{2 c+x^2}} \, dx,x,\sqrt{-c+d x}\right )}{d}\\ &=\frac{1}{2} \left (b-\frac{2 a d^2}{c^2}\right ) x \sqrt{-c+d x} \sqrt{c+d x}+\frac{a (-c+d x)^{3/2} (c+d x)^{3/2}}{c^2 x}+\frac{\left (-b c^2+2 a d^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\frac{\sqrt{-c+d x}}{\sqrt{c+d x}}\right )}{d}\\ &=\frac{1}{2} \left (b-\frac{2 a d^2}{c^2}\right ) x \sqrt{-c+d x} \sqrt{c+d x}+\frac{a (-c+d x)^{3/2} (c+d x)^{3/2}}{c^2 x}-\frac{\left (b c^2-2 a d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{-c+d x}}{\sqrt{c+d x}}\right )}{d}\\ \end{align*}
Mathematica [A] time = 0.0802886, size = 101, normalized size = 0.97 \[ \frac{\sqrt{d x-c} \sqrt{c+d x} \left (c d \left (b x^2-2 a\right ) \sqrt{1-\frac{d^2 x^2}{c^2}}+x \left (b c^2-2 a d^2\right ) \sin ^{-1}\left (\frac{d x}{c}\right )\right )}{2 c d x \sqrt{1-\frac{d^2 x^2}{c^2}}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.014, size = 153, normalized size = 1.5 \begin{align*}{\frac{{\it csgn} \left ( d \right ) }{2\,dx}\sqrt{dx-c}\sqrt{dx+c} \left ({\it csgn} \left ( d \right ){x}^{2}bd\sqrt{{d}^{2}{x}^{2}-{c}^{2}}+2\,\ln \left ( \left ( \sqrt{{d}^{2}{x}^{2}-{c}^{2}}{\it csgn} \left ( d \right ) +dx \right ){\it csgn} \left ( d \right ) \right ) xa{d}^{2}-\ln \left ( \left ( \sqrt{{d}^{2}{x}^{2}-{c}^{2}}{\it csgn} \left ( d \right ) +dx \right ){\it csgn} \left ( d \right ) \right ) xb{c}^{2}-2\,{\it csgn} \left ( d \right ) d\sqrt{{d}^{2}{x}^{2}-{c}^{2}}a \right ){\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 = 1.57855, size = 181, normalized size = 1.74 \begin{align*} -\frac{2 \, a d^{2} x -{\left (b c^{2} - 2 \, a d^{2}\right )} x \log \left (-d x + \sqrt{d x + c} \sqrt{d x - c}\right ) -{\left (b d x^{2} - 2 \, a d\right )} \sqrt{d x + c} \sqrt{d x - c}}{2 \, d x} \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 \frac{\left (a + b x^{2}\right ) \sqrt{- c + d x} \sqrt{c + d x}}{x^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24287, size = 149, normalized size = 1.43 \begin{align*} -\frac{\frac{6144 \, a c^{2} d^{2}}{{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{4} + 4 \, c^{2}} - 2 \,{\left ({\left (d x + c\right )} b - b c\right )} \sqrt{d x + c} \sqrt{d x - c} -{\left (b c^{2} - 2 \, a d^{2}\right )} \log \left ({\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{4}\right )}{768 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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