Optimal. Leaf size=233 \[ \frac{32 \left (c d^2-a e^2\right )^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{35 c^4 d^4 \sqrt{d+e x}}+\frac{16 \sqrt{d+e x} \left (c d^2-a e^2\right )^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{35 c^3 d^3}+\frac{12 (d+e x)^{3/2} \left (c d^2-a e^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{35 c^2 d^2}+\frac{2 (d+e x)^{5/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{7 c d} \]
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Rubi [A] time = 0.178182, antiderivative size = 233, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.051, Rules used = {656, 648} \[ \frac{32 \left (c d^2-a e^2\right )^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{35 c^4 d^4 \sqrt{d+e x}}+\frac{16 \sqrt{d+e x} \left (c d^2-a e^2\right )^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{35 c^3 d^3}+\frac{12 (d+e x)^{3/2} \left (c d^2-a e^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{35 c^2 d^2}+\frac{2 (d+e x)^{5/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{7 c d} \]
Antiderivative was successfully verified.
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Rule 656
Rule 648
Rubi steps
\begin{align*} \int \frac{(d+e x)^{7/2}}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx &=\frac{2 (d+e x)^{5/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7 c d}+\frac{\left (6 \left (d^2-\frac{a e^2}{c}\right )\right ) \int \frac{(d+e x)^{5/2}}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{7 d}\\ &=\frac{12 \left (c d^2-a e^2\right ) (d+e x)^{3/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 c^2 d^2}+\frac{2 (d+e x)^{5/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7 c d}+\frac{\left (24 \left (d^2-\frac{a e^2}{c}\right )^2\right ) \int \frac{(d+e x)^{3/2}}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{35 d^2}\\ &=\frac{16 \left (c d^2-a e^2\right )^2 \sqrt{d+e x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 c^3 d^3}+\frac{12 \left (c d^2-a e^2\right ) (d+e x)^{3/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 c^2 d^2}+\frac{2 (d+e x)^{5/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7 c d}+\frac{\left (16 \left (d^2-\frac{a e^2}{c}\right )^3\right ) \int \frac{\sqrt{d+e x}}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{35 d^3}\\ &=\frac{32 \left (c d^2-a e^2\right )^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 c^4 d^4 \sqrt{d+e x}}+\frac{16 \left (c d^2-a e^2\right )^2 \sqrt{d+e x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 c^3 d^3}+\frac{12 \left (c d^2-a e^2\right ) (d+e x)^{3/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 c^2 d^2}+\frac{2 (d+e x)^{5/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7 c d}\\ \end{align*}
Mathematica [A] time = 0.105574, size = 131, normalized size = 0.56 \[ \frac{2 \sqrt{(d+e x) (a e+c d x)} \left (8 a^2 c d e^4 (7 d+e x)-16 a^3 e^6-2 a c^2 d^2 e^2 \left (35 d^2+14 d e x+3 e^2 x^2\right )+c^3 d^3 \left (35 d^2 e x+35 d^3+21 d e^2 x^2+5 e^3 x^3\right )\right )}{35 c^4 d^4 \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 168, normalized size = 0.7 \begin{align*} -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( -5\,{e}^{3}{x}^{3}{c}^{3}{d}^{3}+6\,a{c}^{2}{d}^{2}{e}^{4}{x}^{2}-21\,{c}^{3}{d}^{4}{e}^{2}{x}^{2}-8\,{a}^{2}cd{e}^{5}x+28\,a{c}^{2}{d}^{3}{e}^{3}x-35\,{c}^{3}{d}^{5}ex+16\,{a}^{3}{e}^{6}-56\,{a}^{2}c{d}^{2}{e}^{4}+70\,a{c}^{2}{d}^{4}{e}^{2}-35\,{c}^{3}{d}^{6} \right ) }{35\,{c}^{4}{d}^{4}}\sqrt{ex+d}{\frac{1}{\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03328, size = 259, normalized size = 1.11 \begin{align*} \frac{2 \,{\left (5 \, c^{4} d^{4} e^{3} x^{4} + 35 \, a c^{3} d^{6} e - 70 \, a^{2} c^{2} d^{4} e^{3} + 56 \, a^{3} c d^{2} e^{5} - 16 \, a^{4} e^{7} +{\left (21 \, c^{4} d^{5} e^{2} - a c^{3} d^{3} e^{4}\right )} x^{3} +{\left (35 \, c^{4} d^{6} e - 7 \, a c^{3} d^{4} e^{3} + 2 \, a^{2} c^{2} d^{2} e^{5}\right )} x^{2} +{\left (35 \, c^{4} d^{7} - 35 \, a c^{3} d^{5} e^{2} + 28 \, a^{2} c^{2} d^{3} e^{4} - 8 \, a^{3} c d e^{6}\right )} x\right )}}{35 \, \sqrt{c d x + a e} c^{4} d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.81674, size = 362, normalized size = 1.55 \begin{align*} \frac{2 \,{\left (5 \, c^{3} d^{3} e^{3} x^{3} + 35 \, c^{3} d^{6} - 70 \, a c^{2} d^{4} e^{2} + 56 \, a^{2} c d^{2} e^{4} - 16 \, a^{3} e^{6} + 3 \,{\left (7 \, c^{3} d^{4} e^{2} - 2 \, a c^{2} d^{2} e^{4}\right )} x^{2} +{\left (35 \, c^{3} d^{5} e - 28 \, a c^{2} d^{3} e^{3} + 8 \, a^{2} c d e^{5}\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}}{35 \,{\left (c^{4} d^{4} e x + c^{4} d^{5}\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*} \int \frac{{\left (e x + d\right )}^{\frac{7}{2}}}{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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