Optimal. Leaf size=201 \[ -\frac{115 (x+1) \sqrt{\frac{3 x+2}{x+1}} \text{EllipticF}\left (\tan ^{-1}\left (\sqrt{x}\right ),-\frac{1}{2}\right )}{\sqrt{2} \sqrt{3 x^2+5 x+2}}-\frac{170 \sqrt{x} (3 x+2)}{3 \sqrt{3 x^2+5 x+2}}+\frac{170 \sqrt{3 x^2+5 x+2}}{3 \sqrt{x}}-\frac{115 \sqrt{3 x^2+5 x+2}}{3 x^{3/2}}+\frac{2 (45 x+38)}{x^{3/2} \sqrt{3 x^2+5 x+2}}+\frac{170 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{3 \sqrt{3 x^2+5 x+2}} \]
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Rubi [A] time = 0.122305, antiderivative size = 201, 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} \[ -\frac{170 \sqrt{x} (3 x+2)}{3 \sqrt{3 x^2+5 x+2}}+\frac{170 \sqrt{3 x^2+5 x+2}}{3 \sqrt{x}}-\frac{115 \sqrt{3 x^2+5 x+2}}{3 x^{3/2}}+\frac{2 (45 x+38)}{x^{3/2} \sqrt{3 x^2+5 x+2}}-\frac{115 (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{\sqrt{2} \sqrt{3 x^2+5 x+2}}+\frac{170 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{3 \sqrt{3 x^2+5 x+2}} \]
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^{5/2} \left (2+5 x+3 x^2\right )^{3/2}} \, dx &=\frac{2 (38+45 x)}{x^{3/2} \sqrt{2+5 x+3 x^2}}-\int \frac{-115-135 x}{x^{5/2} \sqrt{2+5 x+3 x^2}} \, dx\\ &=\frac{2 (38+45 x)}{x^{3/2} \sqrt{2+5 x+3 x^2}}-\frac{115 \sqrt{2+5 x+3 x^2}}{3 x^{3/2}}+\frac{1}{3} \int \frac{-170-\frac{345 x}{2}}{x^{3/2} \sqrt{2+5 x+3 x^2}} \, dx\\ &=\frac{2 (38+45 x)}{x^{3/2} \sqrt{2+5 x+3 x^2}}-\frac{115 \sqrt{2+5 x+3 x^2}}{3 x^{3/2}}+\frac{170 \sqrt{2+5 x+3 x^2}}{3 \sqrt{x}}-\frac{1}{3} \int \frac{\frac{345}{2}+255 x}{\sqrt{x} \sqrt{2+5 x+3 x^2}} \, dx\\ &=\frac{2 (38+45 x)}{x^{3/2} \sqrt{2+5 x+3 x^2}}-\frac{115 \sqrt{2+5 x+3 x^2}}{3 x^{3/2}}+\frac{170 \sqrt{2+5 x+3 x^2}}{3 \sqrt{x}}-\frac{2}{3} \operatorname{Subst}\left (\int \frac{\frac{345}{2}+255 x^2}{\sqrt{2+5 x^2+3 x^4}} \, dx,x,\sqrt{x}\right )\\ &=\frac{2 (38+45 x)}{x^{3/2} \sqrt{2+5 x+3 x^2}}-\frac{115 \sqrt{2+5 x+3 x^2}}{3 x^{3/2}}+\frac{170 \sqrt{2+5 x+3 x^2}}{3 \sqrt{x}}-115 \operatorname{Subst}\left (\int \frac{1}{\sqrt{2+5 x^2+3 x^4}} \, dx,x,\sqrt{x}\right )-170 \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{2+5 x^2+3 x^4}} \, dx,x,\sqrt{x}\right )\\ &=-\frac{170 \sqrt{x} (2+3 x)}{3 \sqrt{2+5 x+3 x^2}}+\frac{2 (38+45 x)}{x^{3/2} \sqrt{2+5 x+3 x^2}}-\frac{115 \sqrt{2+5 x+3 x^2}}{3 x^{3/2}}+\frac{170 \sqrt{2+5 x+3 x^2}}{3 \sqrt{x}}+\frac{170 \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{115 (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 = 0.175396, size = 145, normalized size = 0.72 \[ \frac{-5 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{5/2} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right ),\frac{3}{2}\right )-690 x^2-340 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{5/2} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )-610 x-4}{6 x^{3/2} \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 115, normalized size = 0.6 \begin{align*}{\frac{1}{18} \left ( 165\,\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-170\,\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+3060\,{x}^{3}+3030\,{x}^{2}+210\,x-12 \right ){x}^{-{\frac{3}{2}}}{\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{5}{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^{7} + 30 \, x^{6} + 37 \, x^{5} + 20 \, x^{4} + 4 \, x^{3}}, 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{3}{2}} x^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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