Optimal. Leaf size=84 \[ \frac{a (d x-c)^{3/2} (c+d x)^{3/2}}{3 c^2 x^3}-\frac{b \sqrt{d x-c} \sqrt{c+d x}}{x}+2 b d \tanh ^{-1}\left (\frac{\sqrt{d x-c}}{\sqrt{c+d x}}\right ) \]
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Rubi [A] time = 0.0803436, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194, Rules used = {454, 97, 12, 63, 217, 206} \[ \frac{a (d x-c)^{3/2} (c+d x)^{3/2}}{3 c^2 x^3}-\frac{b \sqrt{d x-c} \sqrt{c+d x}}{x}+2 b d \tanh ^{-1}\left (\frac{\sqrt{d x-c}}{\sqrt{c+d x}}\right ) \]
Antiderivative was successfully verified.
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Rule 454
Rule 97
Rule 12
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^4} \, dx &=\frac{a (-c+d x)^{3/2} (c+d x)^{3/2}}{3 c^2 x^3}+b \int \frac{\sqrt{-c+d x} \sqrt{c+d x}}{x^2} \, dx\\ &=-\frac{b \sqrt{-c+d x} \sqrt{c+d x}}{x}+\frac{a (-c+d x)^{3/2} (c+d x)^{3/2}}{3 c^2 x^3}+b \int \frac{d^2}{\sqrt{-c+d x} \sqrt{c+d x}} \, dx\\ &=-\frac{b \sqrt{-c+d x} \sqrt{c+d x}}{x}+\frac{a (-c+d x)^{3/2} (c+d x)^{3/2}}{3 c^2 x^3}+\left (b d^2\right ) \int \frac{1}{\sqrt{-c+d x} \sqrt{c+d x}} \, dx\\ &=-\frac{b \sqrt{-c+d x} \sqrt{c+d x}}{x}+\frac{a (-c+d x)^{3/2} (c+d x)^{3/2}}{3 c^2 x^3}+(2 b d) \operatorname{Subst}\left (\int \frac{1}{\sqrt{2 c+x^2}} \, dx,x,\sqrt{-c+d x}\right )\\ &=-\frac{b \sqrt{-c+d x} \sqrt{c+d x}}{x}+\frac{a (-c+d x)^{3/2} (c+d x)^{3/2}}{3 c^2 x^3}+(2 b d) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\frac{\sqrt{-c+d x}}{\sqrt{c+d x}}\right )\\ &=-\frac{b \sqrt{-c+d x} \sqrt{c+d x}}{x}+\frac{a (-c+d x)^{3/2} (c+d x)^{3/2}}{3 c^2 x^3}+2 b d \tanh ^{-1}\left (\frac{\sqrt{-c+d x}}{\sqrt{c+d x}}\right )\\ \end{align*}
Mathematica [A] time = 0.075709, size = 105, normalized size = 1.25 \[ -\frac{\sqrt{d x-c} \sqrt{c+d x} \left (\sqrt{1-\frac{d^2 x^2}{c^2}} \left (a \left (c^2-d^2 x^2\right )+3 b c^2 x^2\right )+3 b c d x^3 \sin ^{-1}\left (\frac{d x}{c}\right )\right )}{3 c^2 x^3 \sqrt{1-\frac{d^2 x^2}{c^2}}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.016, size = 153, normalized size = 1.8 \begin{align*}{\frac{{\it csgn} \left ( d \right ) }{3\,{c}^{2}{x}^{3}}\sqrt{dx-c}\sqrt{dx+c} \left ( 3\,\ln \left ( \left ( \sqrt{{d}^{2}{x}^{2}-{c}^{2}}{\it csgn} \left ( d \right ) +dx \right ){\it csgn} \left ( d \right ) \right ){x}^{3}b{c}^{2}d+{\it csgn} \left ( d \right ){x}^{2}a{d}^{2}\sqrt{{d}^{2}{x}^{2}-{c}^{2}}-3\,{\it csgn} \left ( d \right ){x}^{2}b{c}^{2}\sqrt{{d}^{2}{x}^{2}-{c}^{2}}-{\it csgn} \left ( d \right ) a{c}^{2}\sqrt{{d}^{2}{x}^{2}-{c}^{2}} \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 = 1.53373, size = 216, normalized size = 2.57 \begin{align*} -\frac{3 \, b c^{2} d x^{3} \log \left (-d x + \sqrt{d x + c} \sqrt{d x - c}\right ) +{\left (3 \, b c^{2} d - a d^{3}\right )} x^{3} +{\left (a c^{2} +{\left (3 \, b c^{2} - a d^{2}\right )} x^{2}\right )} \sqrt{d x + c} \sqrt{d x - c}}{3 \, c^{2} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: MellinTransformStripError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.32079, size = 231, normalized size = 2.75 \begin{align*} -\frac{3 \, b d^{2} \log \left ({\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{4}\right ) + \frac{16 \,{\left (3 \, b c^{2} d^{2}{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{8} - 3 \, a d^{4}{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{8} + 24 \, b c^{4} d^{2}{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{4} + 48 \, b c^{6} d^{2} - 16 \, a c^{4} d^{4}\right )}}{{\left ({\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{4} + 4 \, c^{2}\right )}^{3}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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