Optimal. Leaf size=315 \[ \frac{3 c^3 d^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{64 e^2 (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}+\frac{c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{32 e^2 (d+e x)^{5/2} \left (c d^2-a e^2\right )}+\frac{3 c^4 d^4 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{64 e^{5/2} \left (c d^2-a e^2\right )^{5/2}}-\frac{c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 e^2 (d+e x)^{7/2}}-\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 e (d+e x)^{11/2}} \]
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Rubi [A] time = 0.22298, antiderivative size = 315, 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 = {662, 672, 660, 205} \[ \frac{3 c^3 d^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{64 e^2 (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}+\frac{c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{32 e^2 (d+e x)^{5/2} \left (c d^2-a e^2\right )}+\frac{3 c^4 d^4 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{64 e^{5/2} \left (c d^2-a e^2\right )^{5/2}}-\frac{c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 e^2 (d+e x)^{7/2}}-\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 e (d+e x)^{11/2}} \]
Antiderivative was successfully verified.
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Rule 662
Rule 672
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 )^{3/2}}{(d+e x)^{13/2}} \, dx &=-\frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{4 e (d+e x)^{11/2}}+\frac{(3 c d) \int \frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^{9/2}} \, dx}{8 e}\\ &=-\frac{c d \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 e^2 (d+e x)^{7/2}}-\frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{4 e (d+e x)^{11/2}}+\frac{\left (c^2 d^2\right ) \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}{16 e^2}\\ &=-\frac{c d \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 e^2 (d+e x)^{7/2}}+\frac{c^2 d^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{32 e^2 \left (c d^2-a e^2\right ) (d+e x)^{5/2}}-\frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{4 e (d+e x)^{11/2}}+\frac{\left (3 c^3 d^3\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}{64 e^2 \left (c d^2-a e^2\right )}\\ &=-\frac{c d \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 e^2 (d+e x)^{7/2}}+\frac{c^2 d^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{32 e^2 \left (c d^2-a e^2\right ) (d+e x)^{5/2}}+\frac{3 c^3 d^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 e^2 \left (c d^2-a e^2\right )^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 )^{3/2}}{4 e (d+e x)^{11/2}}+\frac{\left (3 c^4 d^4\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}{128 e^2 \left (c d^2-a e^2\right )^2}\\ &=-\frac{c d \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 e^2 (d+e x)^{7/2}}+\frac{c^2 d^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{32 e^2 \left (c d^2-a e^2\right ) (d+e x)^{5/2}}+\frac{3 c^3 d^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 e^2 \left (c d^2-a e^2\right )^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 )^{3/2}}{4 e (d+e x)^{11/2}}+\frac{\left (3 c^4 d^4\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 )}{64 e \left (c d^2-a e^2\right )^2}\\ &=-\frac{c d \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 e^2 (d+e x)^{7/2}}+\frac{c^2 d^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{32 e^2 \left (c d^2-a e^2\right ) (d+e x)^{5/2}}+\frac{3 c^3 d^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 e^2 \left (c d^2-a e^2\right )^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 )^{3/2}}{4 e (d+e x)^{11/2}}+\frac{3 c^4 d^4 \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 )}{64 e^{5/2} \left (c d^2-a e^2\right )^{5/2}}\\ \end{align*}
Mathematica [C] time = 0.0532264, size = 83, normalized size = 0.26 \[ \frac{2 c^4 d^4 ((d+e x) (a e+c d x))^{5/2} \, _2F_1\left (\frac{5}{2},5;\frac{7}{2};\frac{e (a e+c d x)}{a e^2-c d^2}\right )}{5 (d+e x)^{5/2} \left (c d^2-a e^2\right )^5} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.257, size = 662, normalized size = 2.1 \begin{align*} -{\frac{1}{64\,{e}^{2} \left ( a{e}^{2}-c{d}^{2} \right ) ^{2}}\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade} \left ( 3\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ){x}^{4}{c}^{4}{d}^{4}{e}^{4}+12\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ){x}^{3}{c}^{4}{d}^{5}{e}^{3}+18\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ){x}^{2}{c}^{4}{d}^{6}{e}^{2}+12\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ) x{c}^{4}{d}^{7}e-3\,{x}^{3}{c}^{3}{d}^{3}{e}^{3}\sqrt{cdx+ae}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}+3\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ){c}^{4}{d}^{8}+2\,{x}^{2}a{c}^{2}{d}^{2}{e}^{4}\sqrt{cdx+ae}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}-11\,{x}^{2}{c}^{3}{d}^{4}{e}^{2}\sqrt{cdx+ae}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}+24\,x{a}^{2}cd{e}^{5}\sqrt{cdx+ae}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}-44\,xa{c}^{2}{d}^{3}{e}^{3}\sqrt{cdx+ae}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}+11\,x{c}^{3}{d}^{5}e\sqrt{cdx+ae}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}+16\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}\sqrt{cdx+ae}{a}^{3}{e}^{6}-24\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}\sqrt{cdx+ae}{a}^{2}c{d}^{2}{e}^{4}+2\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}\sqrt{cdx+ae}a{c}^{2}{d}^{4}{e}^{2}+3\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}\sqrt{cdx+ae}{c}^{3}{d}^{6} \right ) \left ( ex+d \right ) ^{-{\frac{9}{2}}}{\frac{1}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}}{\frac{1}{\sqrt{cdx+ae}}}} \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{3}{2}}}{{\left (e x + d\right )}^{\frac{13}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.18837, size = 2807, normalized size = 8.91 \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(-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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