Optimal. Leaf size=269 \[ \frac{5 c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 (d+e x)^{3/2} \left (c d^2-a e^2\right )^3}+\frac{5 c^3 d^3 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{8 \sqrt{e} \left (c d^2-a e^2\right )^{7/2}}+\frac{5 c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{12 (d+e x)^{5/2} \left (c d^2-a e^2\right )^2}+\frac{\sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 (d+e x)^{7/2} \left (c d^2-a e^2\right )} \]
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Rubi [A] time = 0.166055, antiderivative size = 269, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {672, 660, 205} \[ \frac{5 c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 (d+e x)^{3/2} \left (c d^2-a e^2\right )^3}+\frac{5 c^3 d^3 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{8 \sqrt{e} \left (c d^2-a e^2\right )^{7/2}}+\frac{5 c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{12 (d+e x)^{5/2} \left (c d^2-a e^2\right )^2}+\frac{\sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 (d+e x)^{7/2} \left (c d^2-a e^2\right )} \]
Antiderivative was successfully verified.
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Rule 672
Rule 660
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{(d+e x)^{7/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx &=\frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 \left (c d^2-a e^2\right ) (d+e x)^{7/2}}+\frac{(5 c d) \int \frac{1}{(d+e x)^{5/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{6 \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}}{3 \left (c d^2-a e^2\right ) (d+e x)^{7/2}}+\frac{5 c d \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 \left (c d^2-a e^2\right )^2 (d+e x)^{5/2}}+\frac{\left (5 c^2 d^2\right ) \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}{8 \left (c d^2-a e^2\right )^2}\\ &=\frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 \left (c d^2-a e^2\right ) (d+e x)^{7/2}}+\frac{5 c d \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 \left (c d^2-a e^2\right )^2 (d+e x)^{5/2}}+\frac{5 c^2 d^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 \left (c d^2-a e^2\right )^3 (d+e x)^{3/2}}+\frac{\left (5 c^3 d^3\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}{16 \left (c d^2-a e^2\right )^3}\\ &=\frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 \left (c d^2-a e^2\right ) (d+e x)^{7/2}}+\frac{5 c d \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 \left (c d^2-a e^2\right )^2 (d+e x)^{5/2}}+\frac{5 c^2 d^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 \left (c d^2-a e^2\right )^3 (d+e x)^{3/2}}+\frac{\left (5 c^3 d^3 e\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 )}{8 \left (c d^2-a e^2\right )^3}\\ &=\frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 \left (c d^2-a e^2\right ) (d+e x)^{7/2}}+\frac{5 c d \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 \left (c d^2-a e^2\right )^2 (d+e x)^{5/2}}+\frac{5 c^2 d^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 \left (c d^2-a e^2\right )^3 (d+e x)^{3/2}}+\frac{5 c^3 d^3 \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 )}{8 \sqrt{e} \left (c d^2-a e^2\right )^{7/2}}\\ \end{align*}
Mathematica [C] time = 0.0318227, size = 81, normalized size = 0.3 \[ \frac{2 c^3 d^3 \sqrt{(d+e x) (a e+c d x)} \, _2F_1\left (\frac{1}{2},4;\frac{3}{2};\frac{e (a e+c d x)}{a e^2-c d^2}\right )}{\sqrt{d+e x} \left (c d^2-a e^2\right )^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.243, size = 454, normalized size = 1.7 \begin{align*}{\frac{1}{24\, \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 ){x}^{3}{c}^{3}{d}^{3}{e}^{3}+45\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ){x}^{2}{c}^{3}{d}^{4}{e}^{2}+45\,{\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 ){c}^{3}{d}^{6}-15\,{x}^{2}{c}^{2}{d}^{2}{e}^{2}\sqrt{cdx+ae}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}+10\,xacd{e}^{3}\sqrt{cdx+ae}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}-40\,x{c}^{2}{d}^{3}e\sqrt{cdx+ae}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}-8\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}\sqrt{cdx+ae}{a}^{2}{e}^{4}+26\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}\sqrt{cdx+ae}ac{d}^{2}{e}^{2}-33\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}\sqrt{cdx+ae}{c}^{2}{d}^{4} \right ) \left ( ex+d \right ) ^{-{\frac{7}{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{1}{\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 = 2.4491, size = 2535, normalized size = 9.42 \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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