Optimal. Leaf size=109 \[ \frac{4 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{15 c^2 d^2 (d+e x)^{3/2}}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 c d \sqrt{d+e x}} \]
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Rubi [A] time = 0.0602184, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.051, Rules used = {656, 648} \[ \frac{4 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{15 c^2 d^2 (d+e x)^{3/2}}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 c d \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Rule 656
Rule 648
Rubi steps
\begin{align*} \int \sqrt{d+e x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx &=\frac{2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{5 c d \sqrt{d+e x}}+\frac{\left (2 \left (d^2-\frac{a e^2}{c}\right )\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}{5 d}\\ &=\frac{4 \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}}{15 c^2 d^2 (d+e x)^{3/2}}+\frac{2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{5 c d \sqrt{d+e x}}\\ \end{align*}
Mathematica [A] time = 0.0479747, size = 55, normalized size = 0.5 \[ \frac{2 ((d+e x) (a e+c d x))^{3/2} \left (c d (5 d+3 e x)-2 a e^2\right )}{15 c^2 d^2 (d+e x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 69, normalized size = 0.6 \begin{align*} -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( -3\,cdex+2\,a{e}^{2}-5\,c{d}^{2} \right ) }{15\,{c}^{2}{d}^{2}}\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 = 1.08072, size = 112, normalized size = 1.03 \begin{align*} \frac{2 \,{\left (3 \, c^{2} d^{2} e x^{2} + 5 \, a c d^{2} e - 2 \, a^{2} e^{3} +{\left (5 \, c^{2} d^{3} + a c d e^{2}\right )} x\right )} \sqrt{c d x + a e}{\left (e x + d\right )}}{15 \,{\left (c^{2} d^{2} e x + c^{2} d^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.80272, size = 216, normalized size = 1.98 \begin{align*} \frac{2 \,{\left (3 \, c^{2} d^{2} e x^{2} + 5 \, a c d^{2} e - 2 \, a^{2} e^{3} +{\left (5 \, c^{2} d^{3} + a c d e^{2}\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}}{15 \,{\left (c^{2} d^{2} e x + c^{2} d^{3}\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 \sqrt{\left (d + e x\right ) \left (a e + c d x\right )} \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 \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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