Optimal. Leaf size=180 \[ \frac{2 \sqrt{d+e x} \left (c d^2-a e^2\right )^3}{c^4 d^4}+\frac{2 (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}{3 c^3 d^3}+\frac{2 (d+e x)^{5/2} \left (c d^2-a e^2\right )}{5 c^2 d^2}-\frac{2 \left (c d^2-a e^2\right )^{7/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{c^{9/2} d^{9/2}}+\frac{2 (d+e x)^{7/2}}{7 c d} \]
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Rubi [A] time = 0.237441, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.108, Rules used = {626, 50, 63, 208} \[ \frac{2 \sqrt{d+e x} \left (c d^2-a e^2\right )^3}{c^4 d^4}+\frac{2 (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}{3 c^3 d^3}+\frac{2 (d+e x)^{5/2} \left (c d^2-a e^2\right )}{5 c^2 d^2}-\frac{2 \left (c d^2-a e^2\right )^{7/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{c^{9/2} d^{9/2}}+\frac{2 (d+e x)^{7/2}}{7 c d} \]
Antiderivative was successfully verified.
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Rule 626
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{(d+e x)^{9/2}}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx &=\int \frac{(d+e x)^{7/2}}{a e+c d x} \, dx\\ &=\frac{2 (d+e x)^{7/2}}{7 c d}+\frac{\left (c d^2-a e^2\right ) \int \frac{(d+e x)^{5/2}}{a e+c d x} \, dx}{c d}\\ &=\frac{2 \left (c d^2-a e^2\right ) (d+e x)^{5/2}}{5 c^2 d^2}+\frac{2 (d+e x)^{7/2}}{7 c d}+\frac{\left (c d^2-a e^2\right )^2 \int \frac{(d+e x)^{3/2}}{a e+c d x} \, dx}{c^2 d^2}\\ &=\frac{2 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}{3 c^3 d^3}+\frac{2 \left (c d^2-a e^2\right ) (d+e x)^{5/2}}{5 c^2 d^2}+\frac{2 (d+e x)^{7/2}}{7 c d}+\frac{\left (c d^2-a e^2\right )^3 \int \frac{\sqrt{d+e x}}{a e+c d x} \, dx}{c^3 d^3}\\ &=\frac{2 \left (c d^2-a e^2\right )^3 \sqrt{d+e x}}{c^4 d^4}+\frac{2 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}{3 c^3 d^3}+\frac{2 \left (c d^2-a e^2\right ) (d+e x)^{5/2}}{5 c^2 d^2}+\frac{2 (d+e x)^{7/2}}{7 c d}+\frac{\left (c d^2-a e^2\right )^4 \int \frac{1}{(a e+c d x) \sqrt{d+e x}} \, dx}{c^4 d^4}\\ &=\frac{2 \left (c d^2-a e^2\right )^3 \sqrt{d+e x}}{c^4 d^4}+\frac{2 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}{3 c^3 d^3}+\frac{2 \left (c d^2-a e^2\right ) (d+e x)^{5/2}}{5 c^2 d^2}+\frac{2 (d+e x)^{7/2}}{7 c d}+\frac{\left (2 \left (c d^2-a e^2\right )^4\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{c d^2}{e}+a e+\frac{c d x^2}{e}} \, dx,x,\sqrt{d+e x}\right )}{c^4 d^4 e}\\ &=\frac{2 \left (c d^2-a e^2\right )^3 \sqrt{d+e x}}{c^4 d^4}+\frac{2 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}{3 c^3 d^3}+\frac{2 \left (c d^2-a e^2\right ) (d+e x)^{5/2}}{5 c^2 d^2}+\frac{2 (d+e x)^{7/2}}{7 c d}-\frac{2 \left (c d^2-a e^2\right )^{7/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{c^{9/2} d^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.289746, size = 175, normalized size = 0.97 \[ \frac{2 \left (c d^2-a e^2\right ) \left (5 \left (c d^2-a e^2\right ) \left (\sqrt{c} \sqrt{d} \sqrt{d+e x} \left (c d (4 d+e x)-3 a e^2\right )-3 \left (c d^2-a e^2\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )\right )+3 c^{5/2} d^{5/2} (d+e x)^{5/2}\right )}{15 c^{9/2} d^{9/2}}+\frac{2 (d+e x)^{7/2}}{7 c d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.213, size = 455, normalized size = 2.5 \begin{align*}{\frac{2}{7\,cd} \left ( ex+d \right ) ^{{\frac{7}{2}}}}-{\frac{2\,a{e}^{2}}{5\,{c}^{2}{d}^{2}} \left ( ex+d \right ) ^{{\frac{5}{2}}}}+{\frac{2}{5\,c} \left ( ex+d \right ) ^{{\frac{5}{2}}}}+{\frac{2\,{a}^{2}{e}^{4}}{3\,{c}^{3}{d}^{3}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{4\,a{e}^{2}}{3\,{c}^{2}d} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+{\frac{2\,d}{3\,c} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-2\,{\frac{{a}^{3}{e}^{6}\sqrt{ex+d}}{{c}^{4}{d}^{4}}}+6\,{\frac{{a}^{2}{e}^{4}\sqrt{ex+d}}{{c}^{3}{d}^{2}}}-6\,{\frac{a{e}^{2}\sqrt{ex+d}}{{c}^{2}}}+2\,{\frac{{d}^{2}\sqrt{ex+d}}{c}}+2\,{\frac{{a}^{4}{e}^{8}}{{c}^{4}{d}^{4}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}\arctan \left ({\frac{\sqrt{ex+d}cd}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}} \right ) }-8\,{\frac{{a}^{3}{e}^{6}}{{c}^{3}{d}^{2}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}\arctan \left ({\frac{\sqrt{ex+d}cd}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}} \right ) }+12\,{\frac{{a}^{2}{e}^{4}}{{c}^{2}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}\arctan \left ({\frac{\sqrt{ex+d}cd}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}} \right ) }-8\,{\frac{a{d}^{2}{e}^{2}}{c\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}\arctan \left ({\frac{\sqrt{ex+d}cd}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}} \right ) }+2\,{\frac{{d}^{4}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}\arctan \left ({\frac{\sqrt{ex+d}cd}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.96166, size = 1081, normalized size = 6.01 \begin{align*} \left [\frac{105 \,{\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} \sqrt{\frac{c d^{2} - a e^{2}}{c d}} \log \left (\frac{c d e x + 2 \, c d^{2} - a e^{2} - 2 \, \sqrt{e x + d} c d \sqrt{\frac{c d^{2} - a e^{2}}{c d}}}{c d x + a e}\right ) + 2 \,{\left (15 \, c^{3} d^{3} e^{3} x^{3} + 176 \, c^{3} d^{6} - 406 \, a c^{2} d^{4} e^{2} + 350 \, a^{2} c d^{2} e^{4} - 105 \, a^{3} e^{6} + 3 \,{\left (22 \, c^{3} d^{4} e^{2} - 7 \, a c^{2} d^{2} e^{4}\right )} x^{2} +{\left (122 \, c^{3} d^{5} e - 112 \, a c^{2} d^{3} e^{3} + 35 \, a^{2} c d e^{5}\right )} x\right )} \sqrt{e x + d}}{105 \, c^{4} d^{4}}, -\frac{2 \,{\left (105 \,{\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} \sqrt{-\frac{c d^{2} - a e^{2}}{c d}} \arctan \left (-\frac{\sqrt{e x + d} c d \sqrt{-\frac{c d^{2} - a e^{2}}{c d}}}{c d^{2} - a e^{2}}\right ) -{\left (15 \, c^{3} d^{3} e^{3} x^{3} + 176 \, c^{3} d^{6} - 406 \, a c^{2} d^{4} e^{2} + 350 \, a^{2} c d^{2} e^{4} - 105 \, a^{3} e^{6} + 3 \,{\left (22 \, c^{3} d^{4} e^{2} - 7 \, a c^{2} d^{2} e^{4}\right )} x^{2} +{\left (122 \, c^{3} d^{5} e - 112 \, a c^{2} d^{3} e^{3} + 35 \, a^{2} c d e^{5}\right )} x\right )} \sqrt{e x + d}\right )}}{105 \, c^{4} 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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