Optimal. Leaf size=129 \[ \frac{c d \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{e^{3/2} \sqrt{c d^2-a e^2}}-\frac{\sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e (d+e x)^{3/2}} \]
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Rubi [A] time = 0.0732543, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {662, 660, 205} \[ \frac{c d \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{e^{3/2} \sqrt{c d^2-a e^2}}-\frac{\sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e (d+e x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 662
Rule 660
Rule 205
Rubi steps
\begin{align*} \int \frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^{5/2}} \, dx &=-\frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{e (d+e x)^{3/2}}+\frac{(c d) \int \frac{1}{\sqrt{d+e x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{2 e}\\ &=-\frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{e (d+e x)^{3/2}}+(c d) \operatorname{Subst}\left (\int \frac{1}{2 c d^2 e-e \left (c d^2+a e^2\right )+e^2 x^2} \, dx,x,\frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt{d+e x}}\right )\\ &=-\frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{e (d+e x)^{3/2}}+\frac{c d \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt{c d^2-a e^2} \sqrt{d+e x}}\right )}{e^{3/2} \sqrt{c d^2-a e^2}}\\ \end{align*}
Mathematica [A] time = 0.159372, size = 112, normalized size = 0.87 \[ \frac{\sqrt{(d+e x) (a e+c d x)} \left (\frac{c d (d+e x) \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d^2-a e^2}}\right )}{\sqrt{c d^2-a e^2} \sqrt{a e+c d x}}-\sqrt{e}\right )}{e^{3/2} (d+e x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.244, size = 163, normalized size = 1.3 \begin{align*}{\frac{1}{e} \left ( -{\it Artanh} \left ({e\sqrt{cdx+ae}{\frac{1}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}}} \right ) xcde-{\it Artanh} \left ({e\sqrt{cdx+ae}{\frac{1}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}}} \right ) c{d}^{2}-\sqrt{cdx+ae}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e} \right ) \sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade} \left ( ex+d \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{cdx+ae}}}{\frac{1}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}{{\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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Fricas [B] time = 1.87339, size = 1004, normalized size = 7.78 \begin{align*} \left [-\frac{{\left (c d e^{2} x^{2} + 2 \, c d^{2} e x + c d^{3}\right )} \sqrt{-c d^{2} e + a e^{3}} \log \left (-\frac{c d e^{2} x^{2} + 2 \, a e^{3} x - c d^{3} + 2 \, a d e^{2} - 2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{-c d^{2} e + a e^{3}} \sqrt{e x + d}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) + 2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (c d^{2} e - a e^{3}\right )} \sqrt{e x + d}}{2 \,{\left (c d^{4} e^{2} - a d^{2} e^{4} +{\left (c d^{2} e^{4} - a e^{6}\right )} x^{2} + 2 \,{\left (c d^{3} e^{3} - a d e^{5}\right )} x\right )}}, -\frac{{\left (c d e^{2} x^{2} + 2 \, c d^{2} e x + c d^{3}\right )} \sqrt{c d^{2} e - a e^{3}} \arctan \left (\frac{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{c d^{2} e - a e^{3}} \sqrt{e x + d}}{c d e^{2} x^{2} + a d e^{2} +{\left (c d^{2} e + a e^{3}\right )} x}\right ) + \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (c d^{2} e - a e^{3}\right )} \sqrt{e x + d}}{c d^{4} e^{2} - a d^{2} e^{4} +{\left (c d^{2} e^{4} - a e^{6}\right )} x^{2} + 2 \,{\left (c d^{3} e^{3} - a d e^{5}\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(-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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