Optimal. Leaf size=159 \[ -\frac{74 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} \text{EllipticF}\left (\tan ^{-1}\left (\sqrt{x}\right ),-\frac{1}{2}\right )}{3 \sqrt{3 x^2+5 x+2}}-\frac{200 \sqrt{x} (3 x+2)}{9 \sqrt{3 x^2+5 x+2}}+\frac{2 \sqrt{x} (95 x+74)}{3 \sqrt{3 x^2+5 x+2}}+\frac{200 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{9 \sqrt{3 x^2+5 x+2}} \]
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Rubi [A] time = 0.0998932, antiderivative size = 159, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {818, 839, 1189, 1100, 1136} \[ -\frac{200 \sqrt{x} (3 x+2)}{9 \sqrt{3 x^2+5 x+2}}+\frac{2 \sqrt{x} (95 x+74)}{3 \sqrt{3 x^2+5 x+2}}-\frac{74 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{3 \sqrt{3 x^2+5 x+2}}+\frac{200 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{9 \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Rule 818
Rule 839
Rule 1189
Rule 1100
Rule 1136
Rubi steps
\begin{align*} \int \frac{(2-5 x) x^{3/2}}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx &=\frac{2 \sqrt{x} (74+95 x)}{3 \sqrt{2+5 x+3 x^2}}+\frac{2}{3} \int \frac{-37-50 x}{\sqrt{x} \sqrt{2+5 x+3 x^2}} \, dx\\ &=\frac{2 \sqrt{x} (74+95 x)}{3 \sqrt{2+5 x+3 x^2}}+\frac{4}{3} \operatorname{Subst}\left (\int \frac{-37-50 x^2}{\sqrt{2+5 x^2+3 x^4}} \, dx,x,\sqrt{x}\right )\\ &=\frac{2 \sqrt{x} (74+95 x)}{3 \sqrt{2+5 x+3 x^2}}-\frac{148}{3} \operatorname{Subst}\left (\int \frac{1}{\sqrt{2+5 x^2+3 x^4}} \, dx,x,\sqrt{x}\right )-\frac{200}{3} \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{2+5 x^2+3 x^4}} \, dx,x,\sqrt{x}\right )\\ &=-\frac{200 \sqrt{x} (2+3 x)}{9 \sqrt{2+5 x+3 x^2}}+\frac{2 \sqrt{x} (74+95 x)}{3 \sqrt{2+5 x+3 x^2}}+\frac{200 \sqrt{2} (1+x) \sqrt{\frac{2+3 x}{1+x}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{9 \sqrt{2+5 x+3 x^2}}-\frac{74 \sqrt{2} (1+x) \sqrt{\frac{2+3 x}{1+x}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{3 \sqrt{2+5 x+3 x^2}}\\ \end{align*}
Mathematica [C] time = 0.168898, size = 145, normalized size = 0.91 \[ \frac{-22 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{3/2} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right ),\frac{3}{2}\right )-30 x^2-200 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{3/2} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )-556 x-400}{9 \sqrt{x} \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 107, normalized size = 0.7 \begin{align*}{\frac{2}{27} \left ( 39\,\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 ) -50\,\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 ) +855\,{x}^{2}+666\,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{3}{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^{2} - 2 \, x\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] time = 0., size = 0, normalized size = 0. \begin{align*} - \int - \frac{2 x^{\frac{3}{2}}}{3 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 5 x \sqrt{3 x^{2} + 5 x + 2} + 2 \sqrt{3 x^{2} + 5 x + 2}}\, dx - \int \frac{5 x^{\frac{5}{2}}}{3 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 5 x \sqrt{3 x^{2} + 5 x + 2} + 2 \sqrt{3 x^{2} + 5 x + 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*} \int -\frac{{\left (5 \, x - 2\right )} x^{\frac{3}{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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