Optimal. Leaf size=185 \[ \frac{295 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} \text{EllipticF}\left (\tan ^{-1}\left (\sqrt{x}\right ),-\frac{1}{2}\right )}{\sqrt{3 x^2+5 x+2}}+\frac{715 \sqrt{x} (3 x+2)}{3 \sqrt{3 x^2+5 x+2}}-\frac{5 \sqrt{x} (429 x+361)}{3 \sqrt{3 x^2+5 x+2}}+\frac{2 \sqrt{x} (45 x+38)}{3 \left (3 x^2+5 x+2\right )^{3/2}}-\frac{715 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{3 \sqrt{3 x^2+5 x+2}} \]
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Rubi [A] time = 0.110436, antiderivative size = 185, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {822, 839, 1189, 1100, 1136} \[ \frac{715 \sqrt{x} (3 x+2)}{3 \sqrt{3 x^2+5 x+2}}-\frac{5 \sqrt{x} (429 x+361)}{3 \sqrt{3 x^2+5 x+2}}+\frac{2 \sqrt{x} (45 x+38)}{3 \left (3 x^2+5 x+2\right )^{3/2}}+\frac{295 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{\sqrt{3 x^2+5 x+2}}-\frac{715 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{3 \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Rule 822
Rule 839
Rule 1189
Rule 1100
Rule 1136
Rubi steps
\begin{align*} \int \frac{2-5 x}{\sqrt{x} \left (2+5 x+3 x^2\right )^{5/2}} \, dx &=\frac{2 \sqrt{x} (38+45 x)}{3 \left (2+5 x+3 x^2\right )^{3/2}}-\frac{1}{3} \int \frac{35-135 x}{\sqrt{x} \left (2+5 x+3 x^2\right )^{3/2}} \, dx\\ &=\frac{2 \sqrt{x} (38+45 x)}{3 \left (2+5 x+3 x^2\right )^{3/2}}-\frac{5 \sqrt{x} (361+429 x)}{3 \sqrt{2+5 x+3 x^2}}+\frac{1}{3} \int \frac{885+\frac{2145 x}{2}}{\sqrt{x} \sqrt{2+5 x+3 x^2}} \, dx\\ &=\frac{2 \sqrt{x} (38+45 x)}{3 \left (2+5 x+3 x^2\right )^{3/2}}-\frac{5 \sqrt{x} (361+429 x)}{3 \sqrt{2+5 x+3 x^2}}+\frac{2}{3} \operatorname{Subst}\left (\int \frac{885+\frac{2145 x^2}{2}}{\sqrt{2+5 x^2+3 x^4}} \, dx,x,\sqrt{x}\right )\\ &=\frac{2 \sqrt{x} (38+45 x)}{3 \left (2+5 x+3 x^2\right )^{3/2}}-\frac{5 \sqrt{x} (361+429 x)}{3 \sqrt{2+5 x+3 x^2}}+590 \operatorname{Subst}\left (\int \frac{1}{\sqrt{2+5 x^2+3 x^4}} \, dx,x,\sqrt{x}\right )+715 \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{2+5 x^2+3 x^4}} \, dx,x,\sqrt{x}\right )\\ &=\frac{2 \sqrt{x} (38+45 x)}{3 \left (2+5 x+3 x^2\right )^{3/2}}+\frac{715 \sqrt{x} (2+3 x)}{3 \sqrt{2+5 x+3 x^2}}-\frac{5 \sqrt{x} (361+429 x)}{3 \sqrt{2+5 x+3 x^2}}-\frac{715 \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{295 \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 = 0.251677, size = 167, normalized size = 0.9 \[ \frac{170 i \sqrt{\frac{2}{x}+2} \sqrt{\frac{2}{x}+3} \left (3 x^2+5 x+2\right ) x^{3/2} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right ),\frac{3}{2}\right )+2 \left (2655 x^3+6615 x^2+5383 x+1430\right )+715 i \sqrt{\frac{2}{x}+2} \sqrt{\frac{2}{x}+3} \left (3 x^2+5 x+2\right ) x^{3/2} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )}{3 \sqrt{x} \left (3 x^2+5 x+2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 297, normalized size = 1.6 \begin{align*} -{\frac{1}{18\, \left ( 1+x \right ) ^{2} \left ( 2+3\,x \right ) ^{2}} \left ( 1125\,\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}-2145\,\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}+1875\,\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-3575\,\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+750\,\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 ) -1430\,\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 ) +38610\,{x}^{4}+96840\,{x}^{3}+79350\,{x}^{2}+21204\,x \right ) \sqrt{3\,{x}^{2}+5\,x+2}{\frac{1}{\sqrt{x}}}} \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}} \sqrt{x}}\,{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^{7} + 135 \, x^{6} + 279 \, x^{5} + 305 \, x^{4} + 186 \, x^{3} + 60 \, x^{2} + 8 \, x}, 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}} \sqrt{x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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