Optimal. Leaf size=171 \[ \frac{16 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{105 c^3 d^3 (d+e x)^{3/2}}+\frac{8 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{35 c^2 d^2 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{7 c d} \]
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Rubi [A] time = 0.112851, 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} \[ \frac{16 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{105 c^3 d^3 (d+e x)^{3/2}}+\frac{8 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{35 c^2 d^2 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{7 c d} \]
Antiderivative was successfully verified.
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Rule 656
Rule 648
Rubi steps
\begin{align*} \int (d+e x)^{3/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx &=\frac{2 \sqrt{d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{7 c d}+\frac{\left (4 \left (d^2-\frac{a e^2}{c}\right )\right ) \int \sqrt{d+e x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{7 d}\\ &=\frac{8 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{35 c^2 d^2 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{7 c d}+\frac{\left (8 \left (d^2-\frac{a e^2}{c}\right )^2\right ) \int \frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt{d+e x}} \, dx}{35 d^2}\\ &=\frac{16 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{105 c^3 d^3 (d+e x)^{3/2}}+\frac{8 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{35 c^2 d^2 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{7 c d}\\ \end{align*}
Mathematica [A] time = 0.0736661, size = 88, normalized size = 0.51 \[ \frac{2 ((d+e x) (a e+c d x))^{3/2} \left (8 a^2 e^4-4 a c d e^2 (7 d+3 e x)+c^2 d^2 \left (35 d^2+42 d e x+15 e^2 x^2\right )\right )}{105 c^3 d^3 (d+e x)^{3/2}} \]
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 ( 15\,{e}^{2}{x}^{2}{c}^{2}{d}^{2}-12\,acd{e}^{3}x+42\,{c}^{2}{d}^{3}ex+8\,{a}^{2}{e}^{4}-28\,ac{d}^{2}{e}^{2}+35\,{c}^{2}{d}^{4} \right ) }{105\,{c}^{3}{d}^{3}}\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade}{\frac{1}{\sqrt{ex+d}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08813, size = 189, normalized size = 1.11 \begin{align*} \frac{2 \,{\left (15 \, c^{3} d^{3} e^{2} x^{3} + 35 \, a c^{2} d^{4} e - 28 \, a^{2} c d^{2} e^{3} + 8 \, a^{3} e^{5} + 3 \,{\left (14 \, c^{3} d^{4} e + a c^{2} d^{2} e^{3}\right )} x^{2} +{\left (35 \, c^{3} d^{5} + 14 \, a c^{2} d^{3} e^{2} - 4 \, a^{2} c d e^{4}\right )} x\right )} \sqrt{c d x + a e}{\left (e x + d\right )}}{105 \,{\left (c^{3} d^{3} e x + c^{3} d^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.84812, size = 336, normalized size = 1.96 \begin{align*} \frac{2 \,{\left (15 \, c^{3} d^{3} e^{2} x^{3} + 35 \, a c^{2} d^{4} e - 28 \, a^{2} c d^{2} e^{3} + 8 \, a^{3} e^{5} + 3 \,{\left (14 \, c^{3} d^{4} e + a c^{2} d^{2} e^{3}\right )} x^{2} +{\left (35 \, c^{3} d^{5} + 14 \, a c^{2} d^{3} e^{2} - 4 \, a^{2} c d e^{4}\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}}{105 \,{\left (c^{3} d^{3} e x + c^{3} d^{4}\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(-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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