Optimal. Leaf size=172 \[ \frac{45 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} \text{EllipticF}\left (\tan ^{-1}\left (\sqrt{x}\right ),-\frac{1}{2}\right )}{\sqrt{3 x^2+5 x+2}}+\frac{39 \sqrt{x} (3 x+2)}{\sqrt{3 x^2+5 x+2}}-\frac{39 \sqrt{3 x^2+5 x+2}}{\sqrt{x}}+\frac{2 (45 x+38)}{\sqrt{x} \sqrt{3 x^2+5 x+2}}-\frac{39 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{\sqrt{3 x^2+5 x+2}} \]
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Rubi [A] time = 0.102168, antiderivative size = 172, 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 = {822, 834, 839, 1189, 1100, 1136} \[ \frac{39 \sqrt{x} (3 x+2)}{\sqrt{3 x^2+5 x+2}}-\frac{39 \sqrt{3 x^2+5 x+2}}{\sqrt{x}}+\frac{2 (45 x+38)}{\sqrt{x} \sqrt{3 x^2+5 x+2}}+\frac{45 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{\sqrt{3 x^2+5 x+2}}-\frac{39 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{\sqrt{3 x^2+5 x+2}} \]
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 )^{3/2}} \, dx &=\frac{2 (38+45 x)}{\sqrt{x} \sqrt{2+5 x+3 x^2}}-\int \frac{-39-45 x}{x^{3/2} \sqrt{2+5 x+3 x^2}} \, dx\\ &=\frac{2 (38+45 x)}{\sqrt{x} \sqrt{2+5 x+3 x^2}}-\frac{39 \sqrt{2+5 x+3 x^2}}{\sqrt{x}}+\int \frac{45+\frac{117 x}{2}}{\sqrt{x} \sqrt{2+5 x+3 x^2}} \, dx\\ &=\frac{2 (38+45 x)}{\sqrt{x} \sqrt{2+5 x+3 x^2}}-\frac{39 \sqrt{2+5 x+3 x^2}}{\sqrt{x}}+2 \operatorname{Subst}\left (\int \frac{45+\frac{117 x^2}{2}}{\sqrt{2+5 x^2+3 x^4}} \, dx,x,\sqrt{x}\right )\\ &=\frac{2 (38+45 x)}{\sqrt{x} \sqrt{2+5 x+3 x^2}}-\frac{39 \sqrt{2+5 x+3 x^2}}{\sqrt{x}}+90 \operatorname{Subst}\left (\int \frac{1}{\sqrt{2+5 x^2+3 x^4}} \, dx,x,\sqrt{x}\right )+117 \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{2+5 x^2+3 x^4}} \, dx,x,\sqrt{x}\right )\\ &=\frac{39 \sqrt{x} (2+3 x)}{\sqrt{2+5 x+3 x^2}}+\frac{2 (38+45 x)}{\sqrt{x} \sqrt{2+5 x+3 x^2}}-\frac{39 \sqrt{2+5 x+3 x^2}}{\sqrt{x}}-\frac{39 \sqrt{2} (1+x) \sqrt{\frac{2+3 x}{1+x}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{\sqrt{2+5 x+3 x^2}}+\frac{45 \sqrt{2} (1+x) \sqrt{\frac{2+3 x}{1+x}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{\sqrt{2+5 x+3 x^2}}\\ \end{align*}
Mathematica [C] time = 0.177587, size = 137, normalized size = 0.8 \[ \frac{6 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{3/2} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right ),\frac{3}{2}\right )+39 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{3/2} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )+90 x+76}{\sqrt{x} \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.021, size = 108, normalized size = 0.6 \begin{align*} -{\frac{1}{2} \left ( 9\,\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 ) -13\,\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 ) +234\,{x}^{2}+210\,x+4 \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{3}{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}}{9 \, x^{6} + 30 \, x^{5} + 37 \, x^{4} + 20 \, x^{3} + 4 \, x^{2}}, 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{5}{3 x^{\frac{5}{2}} \sqrt{3 x^{2} + 5 x + 2} + 5 x^{\frac{3}{2}} \sqrt{3 x^{2} + 5 x + 2} + 2 \sqrt{x} \sqrt{3 x^{2} + 5 x + 2}}\, dx - \int - \frac{2}{3 x^{\frac{7}{2}} \sqrt{3 x^{2} + 5 x + 2} + 5 x^{\frac{5}{2}} \sqrt{3 x^{2} + 5 x + 2} + 2 x^{\frac{3}{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{5 \, x - 2}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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