Optimal. Leaf size=171 \[ \frac{8 (d+e x)^{3/2} \left (c d^2-a e^2\right )}{3 c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{16 \sqrt{d+e x} \left (c d^2-a e^2\right )^2}{3 c^3 d^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{2 (d+e x)^{5/2}}{3 c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]
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Rubi [A] time = 0.116254, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.051, Rules used = {656, 648} \[ \frac{8 (d+e x)^{3/2} \left (c d^2-a e^2\right )}{3 c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{16 \sqrt{d+e x} \left (c d^2-a e^2\right )^2}{3 c^3 d^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{2 (d+e x)^{5/2}}{3 c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]
Antiderivative was successfully verified.
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Rule 656
Rule 648
Rubi steps
\begin{align*} \int \frac{(d+e x)^{7/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx &=\frac{2 (d+e x)^{5/2}}{3 c d \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{\left (4 \left (d^2-\frac{a e^2}{c}\right )\right ) \int \frac{(d+e x)^{5/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx}{3 d}\\ &=\frac{8 \left (c d^2-a e^2\right ) (d+e x)^{3/2}}{3 c^2 d^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{2 (d+e x)^{5/2}}{3 c d \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{\left (8 \left (c d^2-a e^2\right )^2\right ) \int \frac{(d+e x)^{3/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx}{3 c^2 d^2}\\ &=-\frac{16 \left (c d^2-a e^2\right )^2 \sqrt{d+e x}}{3 c^3 d^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{8 \left (c d^2-a e^2\right ) (d+e x)^{3/2}}{3 c^2 d^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{2 (d+e x)^{5/2}}{3 c d \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ \end{align*}
Mathematica [A] time = 0.0599896, size = 87, normalized size = 0.51 \[ -\frac{2 \sqrt{d+e x} \left (8 a^2 e^4+4 a c d e^2 (e x-3 d)+c^2 d^2 \left (3 d^2-6 d e x-e^2 x^2\right )\right )}{3 c^3 d^3 \sqrt{(d+e x) (a e+c d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 110, normalized size = 0.6 \begin{align*} -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( -{e}^{2}{x}^{2}{c}^{2}{d}^{2}+4\,acd{e}^{3}x-6\,{c}^{2}{d}^{3}ex+8\,{a}^{2}{e}^{4}-12\,ac{d}^{2}{e}^{2}+3\,{c}^{2}{d}^{4} \right ) }{3\,{c}^{3}{d}^{3}} \left ( ex+d \right ) ^{{\frac{3}{2}}} \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05948, size = 107, normalized size = 0.63 \begin{align*} \frac{2 \,{\left (c^{2} d^{2} e^{2} x^{2} - 3 \, c^{2} d^{4} + 12 \, a c d^{2} e^{2} - 8 \, a^{2} e^{4} + 2 \,{\left (3 \, c^{2} d^{3} e - 2 \, a c d e^{3}\right )} x\right )}}{3 \, \sqrt{c d x + a e} c^{3} d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.18425, size = 292, normalized size = 1.71 \begin{align*} \frac{2 \,{\left (c^{2} d^{2} e^{2} x^{2} - 3 \, c^{2} d^{4} + 12 \, a c d^{2} e^{2} - 8 \, a^{2} e^{4} + 2 \,{\left (3 \, c^{2} d^{3} e - 2 \, a c d e^{3}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d}}{3 \,{\left (c^{4} d^{4} e x^{2} + a c^{3} d^{4} e +{\left (c^{4} d^{5} + a c^{3} d^{3} e^{2}\right )} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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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