Optimal. Leaf size=329 \[ \frac{35 c^2 d^2 e \sqrt{d+e x}}{4 \left (c d^2-a e^2\right )^4 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{35 c^2 d^2 e^{3/2} \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{4 \left (c d^2-a e^2\right )^{9/2}}-\frac{35 c d e}{12 \sqrt{d+e x} \left (c d^2-a e^2\right )^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{7 c d \sqrt{d+e x}}{6 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{1}{2 \sqrt{d+e x} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]
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Rubi [A] time = 0.280659, antiderivative size = 329, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {672, 666, 660, 205} \[ \frac{35 c^2 d^2 e \sqrt{d+e x}}{4 \left (c d^2-a e^2\right )^4 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{35 c^2 d^2 e^{3/2} \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{4 \left (c d^2-a e^2\right )^{9/2}}-\frac{35 c d e}{12 \sqrt{d+e x} \left (c d^2-a e^2\right )^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{7 c d \sqrt{d+e x}}{6 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{1}{2 \sqrt{d+e x} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 672
Rule 666
Rule 660
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx &=\frac{1}{2 \left (c d^2-a e^2\right ) \sqrt{d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac{(7 c d) \int \frac{\sqrt{d+e x}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx}{4 \left (c d^2-a e^2\right )}\\ &=\frac{1}{2 \left (c d^2-a e^2\right ) \sqrt{d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac{7 c d \sqrt{d+e x}}{6 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac{(35 c d e) \int \frac{1}{\sqrt{d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx}{12 \left (c d^2-a e^2\right )^2}\\ &=\frac{1}{2 \left (c d^2-a e^2\right ) \sqrt{d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac{7 c d \sqrt{d+e x}}{6 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac{35 c d e}{12 \left (c d^2-a e^2\right )^3 \sqrt{d+e x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac{\left (35 c^2 d^2 e\right ) \int \frac{\sqrt{d+e x}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx}{8 \left (c d^2-a e^2\right )^3}\\ &=\frac{1}{2 \left (c d^2-a e^2\right ) \sqrt{d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac{7 c d \sqrt{d+e x}}{6 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac{35 c d e}{12 \left (c d^2-a e^2\right )^3 \sqrt{d+e x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{35 c^2 d^2 e \sqrt{d+e x}}{4 \left (c d^2-a e^2\right )^4 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{\left (35 c^2 d^2 e^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 \left (c d^2-a e^2\right )^4}\\ &=\frac{1}{2 \left (c d^2-a e^2\right ) \sqrt{d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac{7 c d \sqrt{d+e x}}{6 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac{35 c d e}{12 \left (c d^2-a e^2\right )^3 \sqrt{d+e x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{35 c^2 d^2 e \sqrt{d+e x}}{4 \left (c d^2-a e^2\right )^4 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{\left (35 c^2 d^2 e^3\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 )^4}\\ &=\frac{1}{2 \left (c d^2-a e^2\right ) \sqrt{d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac{7 c d \sqrt{d+e x}}{6 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac{35 c d e}{12 \left (c d^2-a e^2\right )^3 \sqrt{d+e x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{35 c^2 d^2 e \sqrt{d+e x}}{4 \left (c d^2-a e^2\right )^4 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{35 c^2 d^2 e^{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 )}{4 \left (c d^2-a e^2\right )^{9/2}}\\ \end{align*}
Mathematica [C] time = 0.0323592, size = 83, normalized size = 0.25 \[ -\frac{2 c^2 d^2 (d+e x)^{3/2} \, _2F_1\left (-\frac{3}{2},3;-\frac{1}{2};\frac{e (a e+c d x)}{a e^2-c d^2}\right )}{3 \left (c d^2-a e^2\right )^3 ((d+e x) (a e+c d x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.22, size = 668, normalized size = 2. \begin{align*} -{\frac{1}{12\, \left ( cdx+ae \right ) ^{2} \left ( a{e}^{2}-c{d}^{2} \right ) ^{4}}\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade} \left ( 105\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ) \sqrt{cdx+ae}{x}^{3}{c}^{3}{d}^{3}{e}^{4}+105\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ){x}^{2}a{c}^{2}{d}^{2}{e}^{5}\sqrt{cdx+ae}+210\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ) \sqrt{cdx+ae}{x}^{2}{c}^{3}{d}^{4}{e}^{3}+210\,{\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}^{4}\sqrt{cdx+ae}+105\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ) \sqrt{cdx+ae}x{c}^{3}{d}^{5}{e}^{2}-105\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}{x}^{3}{c}^{3}{d}^{3}{e}^{3}+105\,{\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}^{3}\sqrt{cdx+ae}-140\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}{x}^{2}a{c}^{2}{d}^{2}{e}^{4}-175\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}{x}^{2}{c}^{3}{d}^{4}{e}^{2}-21\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}x{a}^{2}cd{e}^{5}-238\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}xa{c}^{2}{d}^{3}{e}^{3}-56\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}x{c}^{3}{d}^{5}e+6\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}{a}^{3}{e}^{6}-39\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}{a}^{2}c{d}^{2}{e}^{4}-80\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}a{c}^{2}{d}^{4}{e}^{2}+8\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}{c}^{3}{d}^{6} \right ) \left ( ex+d \right ) ^{-{\frac{5}{2}}}{\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{1}{{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac{5}{2}} \sqrt{e x + d}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.079, size = 3603, normalized size = 10.95 \begin{align*} \text{result too large to display} \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*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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