Optimal. Leaf size=160 \[ -\frac{256 d^3 \left (c d^2-c e^2 x^2\right )^{5/2}}{1155 c e (d+e x)^{5/2}}-\frac{64 d^2 \left (c d^2-c e^2 x^2\right )^{5/2}}{231 c e (d+e x)^{3/2}}-\frac{8 d \left (c d^2-c e^2 x^2\right )^{5/2}}{33 c e \sqrt{d+e x}}-\frac{2 \sqrt{d+e x} \left (c d^2-c e^2 x^2\right )^{5/2}}{11 c e} \]
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Rubi [A] time = 0.0700215, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {657, 649} \[ -\frac{256 d^3 \left (c d^2-c e^2 x^2\right )^{5/2}}{1155 c e (d+e x)^{5/2}}-\frac{64 d^2 \left (c d^2-c e^2 x^2\right )^{5/2}}{231 c e (d+e x)^{3/2}}-\frac{8 d \left (c d^2-c e^2 x^2\right )^{5/2}}{33 c e \sqrt{d+e x}}-\frac{2 \sqrt{d+e x} \left (c d^2-c e^2 x^2\right )^{5/2}}{11 c e} \]
Antiderivative was successfully verified.
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Rule 657
Rule 649
Rubi steps
\begin{align*} \int (d+e x)^{3/2} \left (c d^2-c e^2 x^2\right )^{3/2} \, dx &=-\frac{2 \sqrt{d+e x} \left (c d^2-c e^2 x^2\right )^{5/2}}{11 c e}+\frac{1}{11} (12 d) \int \sqrt{d+e x} \left (c d^2-c e^2 x^2\right )^{3/2} \, dx\\ &=-\frac{8 d \left (c d^2-c e^2 x^2\right )^{5/2}}{33 c e \sqrt{d+e x}}-\frac{2 \sqrt{d+e x} \left (c d^2-c e^2 x^2\right )^{5/2}}{11 c e}+\frac{1}{33} \left (32 d^2\right ) \int \frac{\left (c d^2-c e^2 x^2\right )^{3/2}}{\sqrt{d+e x}} \, dx\\ &=-\frac{64 d^2 \left (c d^2-c e^2 x^2\right )^{5/2}}{231 c e (d+e x)^{3/2}}-\frac{8 d \left (c d^2-c e^2 x^2\right )^{5/2}}{33 c e \sqrt{d+e x}}-\frac{2 \sqrt{d+e x} \left (c d^2-c e^2 x^2\right )^{5/2}}{11 c e}+\frac{1}{231} \left (128 d^3\right ) \int \frac{\left (c d^2-c e^2 x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx\\ &=-\frac{256 d^3 \left (c d^2-c e^2 x^2\right )^{5/2}}{1155 c e (d+e x)^{5/2}}-\frac{64 d^2 \left (c d^2-c e^2 x^2\right )^{5/2}}{231 c e (d+e x)^{3/2}}-\frac{8 d \left (c d^2-c e^2 x^2\right )^{5/2}}{33 c e \sqrt{d+e x}}-\frac{2 \sqrt{d+e x} \left (c d^2-c e^2 x^2\right )^{5/2}}{11 c e}\\ \end{align*}
Mathematica [A] time = 0.0567404, size = 73, normalized size = 0.46 \[ -\frac{2 c (d-e x)^2 \left (755 d^2 e x+533 d^3+455 d e^2 x^2+105 e^3 x^3\right ) \sqrt{c \left (d^2-e^2 x^2\right )}}{1155 e \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 66, normalized size = 0.4 \begin{align*} -{\frac{ \left ( -2\,ex+2\,d \right ) \left ( 105\,{e}^{3}{x}^{3}+455\,d{e}^{2}{x}^{2}+755\,{d}^{2}xe+533\,{d}^{3} \right ) }{1155\,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 = 1.28505, size = 130, normalized size = 0.81 \begin{align*} -\frac{2 \,{\left (105 \, c^{\frac{3}{2}} e^{5} x^{5} + 245 \, c^{\frac{3}{2}} d e^{4} x^{4} - 50 \, c^{\frac{3}{2}} d^{2} e^{3} x^{3} - 522 \, c^{\frac{3}{2}} d^{3} e^{2} x^{2} - 311 \, c^{\frac{3}{2}} d^{4} e x + 533 \, c^{\frac{3}{2}} d^{5}\right )}{\left (e x + d\right )} \sqrt{-e x + d}}{1155 \,{\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 = 2.46514, size = 220, normalized size = 1.38 \begin{align*} -\frac{2 \,{\left (105 \, c e^{5} x^{5} + 245 \, c d e^{4} x^{4} - 50 \, c d^{2} e^{3} x^{3} - 522 \, c d^{3} e^{2} x^{2} - 311 \, c d^{4} e x + 533 \, c d^{5}\right )} \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d}}{1155 \,{\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}} \left (d + e x\right )^{\frac{3}{2}}\, 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}}{\left (e x + d\right )}^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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