Optimal. Leaf size=119 \[ -\frac{2 \left (c d^2-c e^2 x^2\right )^{5/2}}{9 c e \sqrt{d+e x}}-\frac{16 d \left (c d^2-c e^2 x^2\right )^{5/2}}{63 c e (d+e x)^{3/2}}-\frac{64 d^2 \left (c d^2-c e^2 x^2\right )^{5/2}}{315 c e (d+e x)^{5/2}} \]
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Rubi [A] time = 0.0498016, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {657, 649} \[ -\frac{2 \left (c d^2-c e^2 x^2\right )^{5/2}}{9 c e \sqrt{d+e x}}-\frac{16 d \left (c d^2-c e^2 x^2\right )^{5/2}}{63 c e (d+e x)^{3/2}}-\frac{64 d^2 \left (c d^2-c e^2 x^2\right )^{5/2}}{315 c e (d+e x)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 657
Rule 649
Rubi steps
\begin{align*} \int \sqrt{d+e x} \left (c d^2-c e^2 x^2\right )^{3/2} \, dx &=-\frac{2 \left (c d^2-c e^2 x^2\right )^{5/2}}{9 c e \sqrt{d+e x}}+\frac{1}{9} (8 d) \int \frac{\left (c d^2-c e^2 x^2\right )^{3/2}}{\sqrt{d+e x}} \, dx\\ &=-\frac{16 d \left (c d^2-c e^2 x^2\right )^{5/2}}{63 c e (d+e x)^{3/2}}-\frac{2 \left (c d^2-c e^2 x^2\right )^{5/2}}{9 c e \sqrt{d+e x}}+\frac{1}{63} \left (32 d^2\right ) \int \frac{\left (c d^2-c e^2 x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx\\ &=-\frac{64 d^2 \left (c d^2-c e^2 x^2\right )^{5/2}}{315 c e (d+e x)^{5/2}}-\frac{16 d \left (c d^2-c e^2 x^2\right )^{5/2}}{63 c e (d+e x)^{3/2}}-\frac{2 \left (c d^2-c e^2 x^2\right )^{5/2}}{9 c e \sqrt{d+e x}}\\ \end{align*}
Mathematica [A] time = 0.0475711, size = 62, normalized size = 0.52 \[ -\frac{2 c (d-e x)^2 \left (107 d^2+110 d e x+35 e^2 x^2\right ) \sqrt{c \left (d^2-e^2 x^2\right )}}{315 e \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 55, normalized size = 0.5 \begin{align*} -{\frac{ \left ( -2\,ex+2\,d \right ) \left ( 35\,{e}^{2}{x}^{2}+110\,dex+107\,{d}^{2} \right ) }{315\,e} \left ( -c{e}^{2}{x}^{2}+c{d}^{2} \right ) ^{{\frac{3}{2}}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.21692, size = 111, normalized size = 0.93 \begin{align*} -\frac{2 \,{\left (35 \, c^{\frac{3}{2}} e^{4} x^{4} + 40 \, c^{\frac{3}{2}} d e^{3} x^{3} - 78 \, c^{\frac{3}{2}} d^{2} e^{2} x^{2} - 104 \, c^{\frac{3}{2}} d^{3} e x + 107 \, c^{\frac{3}{2}} d^{4}\right )}{\left (e x + d\right )} \sqrt{-e x + d}}{315 \,{\left (e^{2} x + d e\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.25751, size = 189, normalized size = 1.59 \begin{align*} -\frac{2 \,{\left (35 \, c e^{4} x^{4} + 40 \, c d e^{3} x^{3} - 78 \, c d^{2} e^{2} x^{2} - 104 \, c d^{3} e x + 107 \, c d^{4}\right )} \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d}}{315 \,{\left (e^{2} x + d e\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 \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*} \int{\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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