Optimal. Leaf size=208 \[ -\frac{695 (x+1) \sqrt{\frac{3 x+2}{x+1}} \text{EllipticF}\left (\tan ^{-1}\left (\sqrt{x}\right ),-\frac{1}{2}\right )}{\sqrt{2} \sqrt{3 x^2+5 x+2}}-\frac{838 \sqrt{x} (3 x+2)}{3 \sqrt{3 x^2+5 x+2}}+\frac{838 \sqrt{3 x^2+5 x+2}}{3 \sqrt{x}}-\frac{2085 x+1717}{3 \sqrt{x} \sqrt{3 x^2+5 x+2}}+\frac{2 (45 x+38)}{3 \sqrt{x} \left (3 x^2+5 x+2\right )^{3/2}}+\frac{838 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{3 \sqrt{3 x^2+5 x+2}} \]
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Rubi [A] time = 0.123528, antiderivative size = 208, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {822, 834, 839, 1189, 1100, 1136} \[ -\frac{838 \sqrt{x} (3 x+2)}{3 \sqrt{3 x^2+5 x+2}}+\frac{838 \sqrt{3 x^2+5 x+2}}{3 \sqrt{x}}-\frac{2085 x+1717}{3 \sqrt{x} \sqrt{3 x^2+5 x+2}}+\frac{2 (45 x+38)}{3 \sqrt{x} \left (3 x^2+5 x+2\right )^{3/2}}-\frac{695 (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{\sqrt{2} \sqrt{3 x^2+5 x+2}}+\frac{838 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{3 \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Rule 822
Rule 834
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 )^{5/2}} \, dx &=\frac{2 (38+45 x)}{3 \sqrt{x} \left (2+5 x+3 x^2\right )^{3/2}}-\frac{1}{3} \int \frac{-41-225 x}{x^{3/2} \left (2+5 x+3 x^2\right )^{3/2}} \, dx\\ &=\frac{2 (38+45 x)}{3 \sqrt{x} \left (2+5 x+3 x^2\right )^{3/2}}-\frac{1717+2085 x}{3 \sqrt{x} \sqrt{2+5 x+3 x^2}}+\frac{1}{3} \int \frac{-838-\frac{2085 x}{2}}{x^{3/2} \sqrt{2+5 x+3 x^2}} \, dx\\ &=\frac{2 (38+45 x)}{3 \sqrt{x} \left (2+5 x+3 x^2\right )^{3/2}}-\frac{1717+2085 x}{3 \sqrt{x} \sqrt{2+5 x+3 x^2}}+\frac{838 \sqrt{2+5 x+3 x^2}}{3 \sqrt{x}}-\frac{1}{3} \int \frac{\frac{2085}{2}+1257 x}{\sqrt{x} \sqrt{2+5 x+3 x^2}} \, dx\\ &=\frac{2 (38+45 x)}{3 \sqrt{x} \left (2+5 x+3 x^2\right )^{3/2}}-\frac{1717+2085 x}{3 \sqrt{x} \sqrt{2+5 x+3 x^2}}+\frac{838 \sqrt{2+5 x+3 x^2}}{3 \sqrt{x}}-\frac{2}{3} \operatorname{Subst}\left (\int \frac{\frac{2085}{2}+1257 x^2}{\sqrt{2+5 x^2+3 x^4}} \, dx,x,\sqrt{x}\right )\\ &=\frac{2 (38+45 x)}{3 \sqrt{x} \left (2+5 x+3 x^2\right )^{3/2}}-\frac{1717+2085 x}{3 \sqrt{x} \sqrt{2+5 x+3 x^2}}+\frac{838 \sqrt{2+5 x+3 x^2}}{3 \sqrt{x}}-695 \operatorname{Subst}\left (\int \frac{1}{\sqrt{2+5 x^2+3 x^4}} \, dx,x,\sqrt{x}\right )-838 \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{2+5 x^2+3 x^4}} \, dx,x,\sqrt{x}\right )\\ &=\frac{2 (38+45 x)}{3 \sqrt{x} \left (2+5 x+3 x^2\right )^{3/2}}-\frac{838 \sqrt{x} (2+3 x)}{3 \sqrt{2+5 x+3 x^2}}-\frac{1717+2085 x}{3 \sqrt{x} \sqrt{2+5 x+3 x^2}}+\frac{838 \sqrt{2+5 x+3 x^2}}{3 \sqrt{x}}+\frac{838 \sqrt{2} (1+x) \sqrt{\frac{2+3 x}{1+x}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{3 \sqrt{2+5 x+3 x^2}}-\frac{695 (1+x) \sqrt{\frac{2+3 x}{1+x}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{\sqrt{2} \sqrt{2+5 x+3 x^2}}\\ \end{align*}
Mathematica [C] time = 0.25811, size = 167, normalized size = 0.8 \[ \frac{-409 i \sqrt{\frac{2}{x}+2} \sqrt{\frac{2}{x}+3} \left (3 x^2+5 x+2\right ) x^{3/2} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right ),\frac{3}{2}\right )-2 \left (6255 x^3+15576 x^2+12665 x+3358\right )-1676 i \sqrt{\frac{2}{x}+2} \sqrt{\frac{2}{x}+3} \left (3 x^2+5 x+2\right ) x^{3/2} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )}{6 \sqrt{x} \left (3 x^2+5 x+2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 298, normalized size = 1.4 \begin{align*}{\frac{1}{ \left ( 36+54\,x \right ) \left ( 1+x \right ) } \left ( 1287\,\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 ){x}^{2}-2514\,\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 ){x}^{2}+2145\,\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 ) x-4190\,\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 ) x+858\,\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 ) -1676\,\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 ) +45252\,{x}^{4}+113310\,{x}^{3}+92580\,{x}^{2}+24570\,x-36 \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{5 \, x - 2}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} x^{\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{\sqrt{3 \, x^{2} + 5 \, x + 2}{\left (5 \, x - 2\right )} \sqrt{x}}{27 \, x^{8} + 135 \, x^{7} + 279 \, x^{6} + 305 \, x^{5} + 186 \, x^{4} + 60 \, x^{3} + 8 \, x^{2}}, 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{5 \, x - 2}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} x^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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