Optimal. Leaf size=269 \[ -\frac{15 c^2 d^2 \sqrt{d+e x}}{4 \left (c d^2-a e^2\right )^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{15 c^2 d^2 \sqrt{e} \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{4 \left (c d^2-a e^2\right )^{7/2}}+\frac{5 c d}{4 \sqrt{d+e x} \left (c d^2-a e^2\right )^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{1}{2 (d+e x)^{3/2} \left (c d^2-a e^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]
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Rubi [A] time = 0.17749, antiderivative size = 269, 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 = {672, 666, 660, 205} \[ -\frac{15 c^2 d^2 \sqrt{d+e x}}{4 \left (c d^2-a e^2\right )^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{15 c^2 d^2 \sqrt{e} \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{4 \left (c d^2-a e^2\right )^{7/2}}+\frac{5 c d}{4 \sqrt{d+e x} \left (c d^2-a e^2\right )^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{1}{2 (d+e x)^{3/2} \left (c d^2-a e^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]
Antiderivative was successfully verified.
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Rule 672
Rule 666
Rule 660
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{(d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx &=\frac{1}{2 \left (c d^2-a e^2\right ) (d+e x)^{3/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{(5 c d) \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}{4 \left (c d^2-a e^2\right )}\\ &=\frac{1}{2 \left (c d^2-a e^2\right ) (d+e x)^{3/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{5 c d}{4 \left (c d^2-a e^2\right )^2 \sqrt{d+e x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{\left (15 c^2 d^2\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 )^2}\\ &=\frac{1}{2 \left (c d^2-a e^2\right ) (d+e x)^{3/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{5 c d}{4 \left (c d^2-a e^2\right )^2 \sqrt{d+e x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac{15 c^2 d^2 \sqrt{d+e x}}{4 \left (c d^2-a e^2\right )^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac{\left (15 c^2 d^2 e\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 )^3}\\ &=\frac{1}{2 \left (c d^2-a e^2\right ) (d+e x)^{3/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{5 c d}{4 \left (c d^2-a e^2\right )^2 \sqrt{d+e x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac{15 c^2 d^2 \sqrt{d+e x}}{4 \left (c d^2-a e^2\right )^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac{\left (15 c^2 d^2 e^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 )^3}\\ &=\frac{1}{2 \left (c d^2-a e^2\right ) (d+e x)^{3/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{5 c d}{4 \left (c d^2-a e^2\right )^2 \sqrt{d+e x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac{15 c^2 d^2 \sqrt{d+e x}}{4 \left (c d^2-a e^2\right )^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac{15 c^2 d^2 \sqrt{e} \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 )^{7/2}}\\ \end{align*}
Mathematica [C] time = 0.0272629, size = 81, normalized size = 0.3 \[ -\frac{2 c^2 d^2 \sqrt{d+e x} \, _2F_1\left (-\frac{1}{2},3;\frac{1}{2};\frac{e (a e+c d x)}{a e^2-c d^2}\right )}{\left (c d^2-a e^2\right )^3 \sqrt{(d+e x) (a e+c d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.248, size = 384, normalized size = 1.4 \begin{align*} -{\frac{1}{ \left ( 4\,cdx+4\,ae \right ) \left ( a{e}^{2}-c{d}^{2} \right ) ^{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 ) \sqrt{cdx+ae}{x}^{2}{c}^{2}{d}^{2}{e}^{3}+30\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ) \sqrt{cdx+ae}x{c}^{2}{d}^{3}{e}^{2}+15\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ) \sqrt{cdx+ae}{c}^{2}{d}^{4}e-15\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}{x}^{2}{c}^{2}{d}^{2}{e}^{2}-5\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}xacd{e}^{3}-25\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}x{c}^{2}{d}^{3}e+2\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}{a}^{2}{e}^{4}-9\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}ac{d}^{2}{e}^{2}-8\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}{c}^{2}{d}^{4} \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{3}{2}}{\left (e x + d\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.47221, size = 2273, normalized size = 8.45 \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac{3}{2}} \left (d + e x\right )^{\frac{3}{2}}}\, dx \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*} \left [\mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, 2\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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