Optimal. Leaf size=182 \[ \frac{580 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} \text{EllipticF}\left (\tan ^{-1}\left (\sqrt{x}\right ),-\frac{1}{2}\right )}{27 \sqrt{3 x^2+5 x+2}}+\frac{2 (95 x+74) x^{3/2}}{3 \sqrt{3 x^2+5 x+2}}-\frac{580}{27} \sqrt{3 x^2+5 x+2} \sqrt{x}+\frac{1804 (3 x+2) \sqrt{x}}{81 \sqrt{3 x^2+5 x+2}}-\frac{1804 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{81 \sqrt{3 x^2+5 x+2}} \]
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Rubi [A] time = 0.120161, antiderivative size = 182, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {818, 832, 839, 1189, 1100, 1136} \[ \frac{2 (95 x+74) x^{3/2}}{3 \sqrt{3 x^2+5 x+2}}-\frac{580}{27} \sqrt{3 x^2+5 x+2} \sqrt{x}+\frac{1804 (3 x+2) \sqrt{x}}{81 \sqrt{3 x^2+5 x+2}}+\frac{580 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{27 \sqrt{3 x^2+5 x+2}}-\frac{1804 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{81 \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Rule 818
Rule 832
Rule 839
Rule 1189
Rule 1100
Rule 1136
Rubi steps
\begin{align*} \int \frac{(2-5 x) x^{5/2}}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx &=\frac{2 x^{3/2} (74+95 x)}{3 \sqrt{2+5 x+3 x^2}}+\frac{2}{3} \int \frac{(-111-145 x) \sqrt{x}}{\sqrt{2+5 x+3 x^2}} \, dx\\ &=\frac{2 x^{3/2} (74+95 x)}{3 \sqrt{2+5 x+3 x^2}}-\frac{580}{27} \sqrt{x} \sqrt{2+5 x+3 x^2}+\frac{4}{27} \int \frac{145+\frac{451 x}{2}}{\sqrt{x} \sqrt{2+5 x+3 x^2}} \, dx\\ &=\frac{2 x^{3/2} (74+95 x)}{3 \sqrt{2+5 x+3 x^2}}-\frac{580}{27} \sqrt{x} \sqrt{2+5 x+3 x^2}+\frac{8}{27} \operatorname{Subst}\left (\int \frac{145+\frac{451 x^2}{2}}{\sqrt{2+5 x^2+3 x^4}} \, dx,x,\sqrt{x}\right )\\ &=\frac{2 x^{3/2} (74+95 x)}{3 \sqrt{2+5 x+3 x^2}}-\frac{580}{27} \sqrt{x} \sqrt{2+5 x+3 x^2}+\frac{1160}{27} \operatorname{Subst}\left (\int \frac{1}{\sqrt{2+5 x^2+3 x^4}} \, dx,x,\sqrt{x}\right )+\frac{1804}{27} \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{2+5 x^2+3 x^4}} \, dx,x,\sqrt{x}\right )\\ &=\frac{1804 \sqrt{x} (2+3 x)}{81 \sqrt{2+5 x+3 x^2}}+\frac{2 x^{3/2} (74+95 x)}{3 \sqrt{2+5 x+3 x^2}}-\frac{580}{27} \sqrt{x} \sqrt{2+5 x+3 x^2}-\frac{1804 \sqrt{2} (1+x) \sqrt{\frac{2+3 x}{1+x}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{81 \sqrt{2+5 x+3 x^2}}+\frac{580 \sqrt{2} (1+x) \sqrt{\frac{2+3 x}{1+x}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{27 \sqrt{2+5 x+3 x^2}}\\ \end{align*}
Mathematica [C] time = 0.172862, size = 150, normalized size = 0.82 \[ \frac{-64 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{3/2} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right ),\frac{3}{2}\right )-90 x^3+708 x^2+1804 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{3/2} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )+5540 x+3608}{81 \sqrt{x} \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.018, size = 112, normalized size = 0.6 \begin{align*} -{\frac{2}{243} \left ( 483\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{6}\sqrt{-x}{\it EllipticF} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) -451\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{6}\sqrt{-x}{\it EllipticE} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) +135\,{x}^{3}+7056\,{x}^{2}+5220\,x \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}} \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 (5 \, x - 2\right )} x^{\frac{5}{2}}}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (5 \, x^{3} - 2 \, x^{2}\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} \sqrt{x}}{9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4}, x\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{{\left (5 \, x - 2\right )} x^{\frac{5}{2}}}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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