Optimal. Leaf size=233 \[ -\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{e (d+e x)^{7/2}}+\frac{5 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 e^2 (d+e x)^{3/2}}+\frac{5 c d \left (a-\frac{c d^2}{e^2}\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e \sqrt{d+e x}}+\frac{5 c d \left (c d^2-a e^2\right )^{3/2} \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{e^{7/2}} \]
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Rubi [A] time = 0.181769, antiderivative size = 233, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {662, 664, 660, 205} \[ -\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{e (d+e x)^{7/2}}+\frac{5 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 e^2 (d+e x)^{3/2}}+\frac{5 c d \left (a-\frac{c d^2}{e^2}\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e \sqrt{d+e x}}+\frac{5 c d \left (c d^2-a e^2\right )^{3/2} \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{e^{7/2}} \]
Antiderivative was successfully verified.
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Rule 662
Rule 664
Rule 660
Rule 205
Rubi steps
\begin{align*} \int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx &=-\frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{e (d+e x)^{7/2}}+\frac{(5 c d) \int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx}{2 e}\\ &=\frac{5 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 e^2 (d+e x)^{3/2}}-\frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{e (d+e x)^{7/2}}-\frac{\left (5 c d \left (c d^2-a e^2\right )\right ) \int \frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^{3/2}} \, dx}{2 e^2}\\ &=-\frac{5 c d \left (c d^2-a e^2\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{e^3 \sqrt{d+e x}}+\frac{5 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 e^2 (d+e x)^{3/2}}-\frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{e (d+e x)^{7/2}}+\frac{\left (5 c d \left (c d^2-a e^2\right )^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}{2 e^3}\\ &=-\frac{5 c d \left (c d^2-a e^2\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{e^3 \sqrt{d+e x}}+\frac{5 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 e^2 (d+e x)^{3/2}}-\frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{e (d+e x)^{7/2}}+\frac{\left (5 c d \left (c d^2-a e^2\right )^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 )}{e^2}\\ &=-\frac{5 c d \left (c d^2-a e^2\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{e^3 \sqrt{d+e x}}+\frac{5 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 e^2 (d+e x)^{3/2}}-\frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{e (d+e x)^{7/2}}+\frac{5 c d \left (c d^2-a e^2\right )^{3/2} \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 )}{e^{7/2}}\\ \end{align*}
Mathematica [C] time = 0.0704535, size = 79, normalized size = 0.34 \[ \frac{2 c d ((d+e x) (a e+c d x))^{7/2} \, _2F_1\left (2,\frac{7}{2};\frac{9}{2};\frac{e (a e+c d x)}{a e^2-c d^2}\right )}{7 (d+e x)^{7/2} \left (c d^2-a e^2\right )^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.281, size = 521, normalized size = 2.2 \begin{align*} -{\frac{1}{3\,{e}^{3}}\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade} \left ( 15\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ) x{a}^{2}cd{e}^{5}-30\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ) xa{c}^{2}{d}^{3}{e}^{3}+15\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ) x{c}^{3}{d}^{5}e+15\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ){a}^{2}c{d}^{2}{e}^{4}-30\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ) a{c}^{2}{d}^{4}{e}^{2}+15\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ){c}^{3}{d}^{6}-2\,{x}^{2}{c}^{2}{d}^{2}{e}^{2}\sqrt{cdx+ae}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}-14\,xacd{e}^{3}\sqrt{cdx+ae}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}+10\,x{c}^{2}{d}^{3}e\sqrt{cdx+ae}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}+3\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}\sqrt{cdx+ae}{a}^{2}{e}^{4}-20\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}\sqrt{cdx+ae}ac{d}^{2}{e}^{2}+15\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}\sqrt{cdx+ae}{c}^{2}{d}^{4} \right ) \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{{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac{5}{2}}}{{\left (e x + d\right )}^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.0712, size = 1195, normalized size = 5.13 \begin{align*} \left [\frac{15 \,{\left (c^{2} d^{5} - a c d^{3} e^{2} +{\left (c^{2} d^{3} e^{2} - a c d e^{4}\right )} x^{2} + 2 \,{\left (c^{2} d^{4} e - a c d^{2} e^{3}\right )} x\right )} \sqrt{-\frac{c d^{2} - a e^{2}}{e}} \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{e x + d} e \sqrt{-\frac{c d^{2} - a e^{2}}{e}}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) + 2 \,{\left (2 \, c^{2} d^{2} e^{2} x^{2} - 15 \, c^{2} d^{4} + 20 \, a c d^{2} e^{2} - 3 \, a^{2} e^{4} - 2 \,{\left (5 \, c^{2} d^{3} e - 7 \, a c d e^{3}\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}}{6 \,{\left (e^{5} x^{2} + 2 \, d e^{4} x + d^{2} e^{3}\right )}}, -\frac{15 \,{\left (c^{2} d^{5} - a c d^{3} e^{2} +{\left (c^{2} d^{3} e^{2} - a c d e^{4}\right )} x^{2} + 2 \,{\left (c^{2} d^{4} e - a c d^{2} e^{3}\right )} x\right )} \sqrt{\frac{c d^{2} - a e^{2}}{e}} \arctan \left (\frac{\sqrt{e x + d} \sqrt{\frac{c d^{2} - a e^{2}}{e}}}{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}\right ) -{\left (2 \, c^{2} d^{2} e^{2} x^{2} - 15 \, c^{2} d^{4} + 20 \, a c d^{2} e^{2} - 3 \, a^{2} e^{4} - 2 \,{\left (5 \, c^{2} d^{3} e - 7 \, a c d e^{3}\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}}{3 \,{\left (e^{5} x^{2} + 2 \, d e^{4} x + d^{2} e^{3}\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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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