Optimal. Leaf size=201 \[ -\frac{4096 d^4 \left (c d^2-c e^2 x^2\right )^{5/2}}{15015 c e (d+e x)^{5/2}}-\frac{1024 d^3 \left (c d^2-c e^2 x^2\right )^{5/2}}{3003 c e (d+e x)^{3/2}}-\frac{128 d^2 \left (c d^2-c e^2 x^2\right )^{5/2}}{429 c e \sqrt{d+e x}}-\frac{32 d \sqrt{d+e x} \left (c d^2-c e^2 x^2\right )^{5/2}}{143 c e}-\frac{2 (d+e x)^{3/2} \left (c d^2-c e^2 x^2\right )^{5/2}}{13 c e} \]
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Rubi [A] time = 0.0996525, antiderivative size = 201, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {657, 649} \[ -\frac{4096 d^4 \left (c d^2-c e^2 x^2\right )^{5/2}}{15015 c e (d+e x)^{5/2}}-\frac{1024 d^3 \left (c d^2-c e^2 x^2\right )^{5/2}}{3003 c e (d+e x)^{3/2}}-\frac{128 d^2 \left (c d^2-c e^2 x^2\right )^{5/2}}{429 c e \sqrt{d+e x}}-\frac{32 d \sqrt{d+e x} \left (c d^2-c e^2 x^2\right )^{5/2}}{143 c e}-\frac{2 (d+e x)^{3/2} \left (c d^2-c e^2 x^2\right )^{5/2}}{13 c e} \]
Antiderivative was successfully verified.
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Rule 657
Rule 649
Rubi steps
\begin{align*} \int (d+e x)^{5/2} \left (c d^2-c e^2 x^2\right )^{3/2} \, dx &=-\frac{2 (d+e x)^{3/2} \left (c d^2-c e^2 x^2\right )^{5/2}}{13 c e}+\frac{1}{13} (16 d) \int (d+e x)^{3/2} \left (c d^2-c e^2 x^2\right )^{3/2} \, dx\\ &=-\frac{32 d \sqrt{d+e x} \left (c d^2-c e^2 x^2\right )^{5/2}}{143 c e}-\frac{2 (d+e x)^{3/2} \left (c d^2-c e^2 x^2\right )^{5/2}}{13 c e}+\frac{1}{143} \left (192 d^2\right ) \int \sqrt{d+e x} \left (c d^2-c e^2 x^2\right )^{3/2} \, dx\\ &=-\frac{128 d^2 \left (c d^2-c e^2 x^2\right )^{5/2}}{429 c e \sqrt{d+e x}}-\frac{32 d \sqrt{d+e x} \left (c d^2-c e^2 x^2\right )^{5/2}}{143 c e}-\frac{2 (d+e x)^{3/2} \left (c d^2-c e^2 x^2\right )^{5/2}}{13 c e}+\frac{1}{429} \left (512 d^3\right ) \int \frac{\left (c d^2-c e^2 x^2\right )^{3/2}}{\sqrt{d+e x}} \, dx\\ &=-\frac{1024 d^3 \left (c d^2-c e^2 x^2\right )^{5/2}}{3003 c e (d+e x)^{3/2}}-\frac{128 d^2 \left (c d^2-c e^2 x^2\right )^{5/2}}{429 c e \sqrt{d+e x}}-\frac{32 d \sqrt{d+e x} \left (c d^2-c e^2 x^2\right )^{5/2}}{143 c e}-\frac{2 (d+e x)^{3/2} \left (c d^2-c e^2 x^2\right )^{5/2}}{13 c e}+\frac{\left (2048 d^4\right ) \int \frac{\left (c d^2-c e^2 x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx}{3003}\\ &=-\frac{4096 d^4 \left (c d^2-c e^2 x^2\right )^{5/2}}{15015 c e (d+e x)^{5/2}}-\frac{1024 d^3 \left (c d^2-c e^2 x^2\right )^{5/2}}{3003 c e (d+e x)^{3/2}}-\frac{128 d^2 \left (c d^2-c e^2 x^2\right )^{5/2}}{429 c e \sqrt{d+e x}}-\frac{32 d \sqrt{d+e x} \left (c d^2-c e^2 x^2\right )^{5/2}}{143 c e}-\frac{2 (d+e x)^{3/2} \left (c d^2-c e^2 x^2\right )^{5/2}}{13 c e}\\ \end{align*}
Mathematica [A] time = 0.0673561, size = 84, normalized size = 0.42 \[ -\frac{2 c (d-e x)^2 \left (14210 d^2 e^2 x^2+16700 d^3 e x+9683 d^4+6300 d e^3 x^3+1155 e^4 x^4\right ) \sqrt{c \left (d^2-e^2 x^2\right )}}{15015 e \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 77, normalized size = 0.4 \begin{align*} -{\frac{ \left ( -2\,ex+2\,d \right ) \left ( 1155\,{e}^{4}{x}^{4}+6300\,d{e}^{3}{x}^{3}+14210\,{d}^{2}{e}^{2}{x}^{2}+16700\,{d}^{3}xe+9683\,{d}^{4} \right ) }{15015\,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 = 1.27707, size = 149, normalized size = 0.74 \begin{align*} -\frac{2 \,{\left (1155 \, c^{\frac{3}{2}} e^{6} x^{6} + 3990 \, c^{\frac{3}{2}} d e^{5} x^{5} + 2765 \, c^{\frac{3}{2}} d^{2} e^{4} x^{4} - 5420 \, c^{\frac{3}{2}} d^{3} e^{3} x^{3} - 9507 \, c^{\frac{3}{2}} d^{4} e^{2} x^{2} - 2666 \, c^{\frac{3}{2}} d^{5} e x + 9683 \, c^{\frac{3}{2}} d^{6}\right )}{\left (e x + d\right )} \sqrt{-e x + d}}{15015 \,{\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 = 2.34063, size = 259, normalized size = 1.29 \begin{align*} -\frac{2 \,{\left (1155 \, c e^{6} x^{6} + 3990 \, c d e^{5} x^{5} + 2765 \, c d^{2} e^{4} x^{4} - 5420 \, c d^{3} e^{3} x^{3} - 9507 \, c d^{4} e^{2} x^{2} - 2666 \, c d^{5} e x + 9683 \, c d^{6}\right )} \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d}}{15015 \,{\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{5}{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{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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