Optimal. Leaf size=150 \[ -\frac{3 \tanh ^{-1}\left (\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right )}{4 \sqrt{2} c^{3/2} d^{5/2} e}+\frac{3 \sqrt{d+e x}}{4 c d^2 e \sqrt{c d^2-c e^2 x^2}}-\frac{1}{2 c d e \sqrt{d+e x} \sqrt{c d^2-c e^2 x^2}} \]
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Rubi [A] time = 0.0727255, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138, Rules used = {673, 667, 661, 208} \[ -\frac{3 \tanh ^{-1}\left (\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right )}{4 \sqrt{2} c^{3/2} d^{5/2} e}+\frac{3 \sqrt{d+e x}}{4 c d^2 e \sqrt{c d^2-c e^2 x^2}}-\frac{1}{2 c d e \sqrt{d+e x} \sqrt{c d^2-c e^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 673
Rule 667
Rule 661
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{d+e x} \left (c d^2-c e^2 x^2\right )^{3/2}} \, dx &=-\frac{1}{2 c d e \sqrt{d+e x} \sqrt{c d^2-c e^2 x^2}}+\frac{3 \int \frac{\sqrt{d+e x}}{\left (c d^2-c e^2 x^2\right )^{3/2}} \, dx}{4 d}\\ &=-\frac{1}{2 c d e \sqrt{d+e x} \sqrt{c d^2-c e^2 x^2}}+\frac{3 \sqrt{d+e x}}{4 c d^2 e \sqrt{c d^2-c e^2 x^2}}+\frac{3 \int \frac{1}{\sqrt{d+e x} \sqrt{c d^2-c e^2 x^2}} \, dx}{8 c d^2}\\ &=-\frac{1}{2 c d e \sqrt{d+e x} \sqrt{c d^2-c e^2 x^2}}+\frac{3 \sqrt{d+e x}}{4 c d^2 e \sqrt{c d^2-c e^2 x^2}}+\frac{(3 e) \operatorname{Subst}\left (\int \frac{1}{-2 c d e^2+e^2 x^2} \, dx,x,\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{d+e x}}\right )}{4 c d^2}\\ &=-\frac{1}{2 c d e \sqrt{d+e x} \sqrt{c d^2-c e^2 x^2}}+\frac{3 \sqrt{d+e x}}{4 c d^2 e \sqrt{c d^2-c e^2 x^2}}-\frac{3 \tanh ^{-1}\left (\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right )}{4 \sqrt{2} c^{3/2} d^{5/2} e}\\ \end{align*}
Mathematica [A] time = 0.0873345, size = 128, normalized size = 0.85 \[ \frac{2 \sqrt{d} \sqrt{d+e x} (d+3 e x)-3 \sqrt{2} (d+e x) \sqrt{d^2-e^2 x^2} \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{\sqrt{2} \sqrt{d} \sqrt{d+e x}}\right )}{8 c d^{5/2} e (d+e x) \sqrt{c \left (d^2-e^2 x^2\right )}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.194, size = 152, normalized size = 1. \begin{align*}{\frac{1}{8\,{c}^{2} \left ( ex-d \right ) e{d}^{2}}\sqrt{-c \left ({e}^{2}{x}^{2}-{d}^{2} \right ) } \left ( 3\,\sqrt{- \left ( ex-d \right ) c}\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{- \left ( ex-d \right ) c}\sqrt{2}}{\sqrt{cd}}} \right ) xe+3\,\sqrt{- \left ( ex-d \right ) c}\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{- \left ( ex-d \right ) c}\sqrt{2}}{\sqrt{cd}}} \right ) d-6\,\sqrt{cd}xe-2\,\sqrt{cd}d \right ) \left ( ex+d \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{cd}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (-c e^{2} x^{2} + c d^{2}\right )}^{\frac{3}{2}} \sqrt{e x + d}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.2437, size = 805, normalized size = 5.37 \begin{align*} \left [\frac{3 \, \sqrt{2}{\left (e^{3} x^{3} + d e^{2} x^{2} - d^{2} e x - d^{3}\right )} \sqrt{c d} \log \left (-\frac{c e^{2} x^{2} - 2 \, c d e x - 3 \, c d^{2} + 2 \, \sqrt{2} \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{c d} \sqrt{e x + d}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) - 4 \, \sqrt{-c e^{2} x^{2} + c d^{2}}{\left (3 \, d e x + d^{2}\right )} \sqrt{e x + d}}{16 \,{\left (c^{2} d^{3} e^{4} x^{3} + c^{2} d^{4} e^{3} x^{2} - c^{2} d^{5} e^{2} x - c^{2} d^{6} e\right )}}, -\frac{3 \, \sqrt{2}{\left (e^{3} x^{3} + d e^{2} x^{2} - d^{2} e x - d^{3}\right )} \sqrt{-c d} \arctan \left (\frac{\sqrt{2} \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{-c d} \sqrt{e x + d}}{c e^{2} x^{2} - c d^{2}}\right ) + 2 \, \sqrt{-c e^{2} x^{2} + c d^{2}}{\left (3 \, d e x + d^{2}\right )} \sqrt{e x + d}}{8 \,{\left (c^{2} d^{3} e^{4} x^{3} + c^{2} d^{4} e^{3} x^{2} - c^{2} d^{5} e^{2} x - c^{2} d^{6} e\right )}}\right ] \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{1}{\left (- c \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{3}{2}} \sqrt{d + e x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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