Optimal. Leaf size=199 \[ \frac{c^2 d^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{4 e^{3/2} \left (c d^2-a e^2\right )^{3/2}}+\frac{c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 e (d+e x)^{3/2} \left (c d^2-a e^2\right )}-\frac{\sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 e (d+e x)^{5/2}} \]
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Rubi [A] time = 0.131257, antiderivative size = 199, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {662, 672, 660, 205} \[ \frac{c^2 d^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{4 e^{3/2} \left (c d^2-a e^2\right )^{3/2}}+\frac{c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 e (d+e x)^{3/2} \left (c d^2-a e^2\right )}-\frac{\sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 e (d+e x)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 662
Rule 672
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)^{7/2}} \, dx &=-\frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 e (d+e x)^{5/2}}+\frac{(c d) \int \frac{1}{(d+e x)^{3/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{4 e}\\ &=-\frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 e (d+e x)^{5/2}}+\frac{c d \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 e \left (c d^2-a e^2\right ) (d+e x)^{3/2}}+\frac{\left (c^2 d^2\right ) \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}{8 e \left (c d^2-a e^2\right )}\\ &=-\frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 e (d+e x)^{5/2}}+\frac{c d \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 e \left (c d^2-a e^2\right ) (d+e x)^{3/2}}+\frac{\left (c^2 d^2\right ) \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 )}{4 \left (c d^2-a e^2\right )}\\ &=-\frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 e (d+e x)^{5/2}}+\frac{c d \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 e \left (c d^2-a e^2\right ) (d+e x)^{3/2}}+\frac{c^2 d^2 \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 )}{4 e^{3/2} \left (c d^2-a e^2\right )^{3/2}}\\ \end{align*}
Mathematica [C] time = 0.0355337, size = 83, normalized size = 0.42 \[ \frac{2 c^2 d^2 ((d+e x) (a e+c d x))^{3/2} \, _2F_1\left (\frac{3}{2},3;\frac{5}{2};\frac{e (a e+c d x)}{a e^2-c d^2}\right )}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.261, size = 292, normalized size = 1.5 \begin{align*}{\frac{1}{ \left ( 4\,a{e}^{2}-4\,c{d}^{2} \right ) e}\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade} \left ({\it Artanh} \left ({e\sqrt{cdx+ae}{\frac{1}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}}} \right ){x}^{2}{c}^{2}{d}^{2}{e}^{2}+2\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ) x{c}^{2}{d}^{3}e+{\it Artanh} \left ({e\sqrt{cdx+ae}{\frac{1}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}}} \right ){c}^{2}{d}^{4}-xcde\sqrt{cdx+ae}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}-2\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}\sqrt{cdx+ae}a{e}^{2}+\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}\sqrt{cdx+ae}c{d}^{2} \right ) \left ( ex+d \right ) ^{-{\frac{5}{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{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.13503, size = 1528, normalized size = 7.68 \begin{align*} \left [-\frac{{\left (c^{2} d^{2} e^{3} x^{3} + 3 \, c^{2} d^{3} e^{2} x^{2} + 3 \, c^{2} d^{4} e x + c^{2} d^{5}\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 \,{\left (c^{2} d^{4} e - 3 \, a c d^{2} e^{3} + 2 \, a^{2} e^{5} -{\left (c^{2} d^{3} e^{2} - a 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}}{8 \,{\left (c^{2} d^{7} e^{2} - 2 \, a c d^{5} e^{4} + a^{2} d^{3} e^{6} +{\left (c^{2} d^{4} e^{5} - 2 \, a c d^{2} e^{7} + a^{2} e^{9}\right )} x^{3} + 3 \,{\left (c^{2} d^{5} e^{4} - 2 \, a c d^{3} e^{6} + a^{2} d e^{8}\right )} x^{2} + 3 \,{\left (c^{2} d^{6} e^{3} - 2 \, a c d^{4} e^{5} + a^{2} d^{2} e^{7}\right )} x\right )}}, -\frac{{\left (c^{2} d^{2} e^{3} x^{3} + 3 \, c^{2} d^{3} e^{2} x^{2} + 3 \, c^{2} d^{4} e x + c^{2} d^{5}\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 ) +{\left (c^{2} d^{4} e - 3 \, a c d^{2} e^{3} + 2 \, a^{2} e^{5} -{\left (c^{2} d^{3} e^{2} - a 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}}{4 \,{\left (c^{2} d^{7} e^{2} - 2 \, a c d^{5} e^{4} + a^{2} d^{3} e^{6} +{\left (c^{2} d^{4} e^{5} - 2 \, a c d^{2} e^{7} + a^{2} e^{9}\right )} x^{3} + 3 \,{\left (c^{2} d^{5} e^{4} - 2 \, a c d^{3} e^{6} + a^{2} d e^{8}\right )} x^{2} + 3 \,{\left (c^{2} d^{6} e^{3} - 2 \, a c d^{4} e^{5} + a^{2} d^{2} e^{7}\right )} x\right )}}\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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