Optimal. Leaf size=256 \[ -\frac{1733 (x+1) \sqrt{\frac{3 x+2}{x+1}} \text{EllipticF}\left (\tan ^{-1}\left (\sqrt{x}\right ),-\frac{1}{2}\right )}{2 \sqrt{2} \sqrt{3 x^2+5 x+2}}-\frac{9521 \sqrt{x} (3 x+2)}{30 \sqrt{3 x^2+5 x+2}}+\frac{9521 \sqrt{3 x^2+5 x+2}}{30 \sqrt{x}}-\frac{1733 \sqrt{3 x^2+5 x+2}}{6 x^{3/2}}+\frac{1252 \sqrt{3 x^2+5 x+2}}{5 x^{5/2}}-\frac{1965 x+1541}{3 x^{5/2} \sqrt{3 x^2+5 x+2}}+\frac{2 (45 x+38)}{3 x^{5/2} \left (3 x^2+5 x+2\right )^{3/2}}+\frac{9521 (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{15 \sqrt{2} \sqrt{3 x^2+5 x+2}} \]
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Rubi [A] time = 0.167411, antiderivative size = 256, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ -\frac{9521 \sqrt{x} (3 x+2)}{30 \sqrt{3 x^2+5 x+2}}+\frac{9521 \sqrt{3 x^2+5 x+2}}{30 \sqrt{x}}-\frac{1733 \sqrt{3 x^2+5 x+2}}{6 x^{3/2}}+\frac{1252 \sqrt{3 x^2+5 x+2}}{5 x^{5/2}}-\frac{1965 x+1541}{3 x^{5/2} \sqrt{3 x^2+5 x+2}}+\frac{2 (45 x+38)}{3 x^{5/2} \left (3 x^2+5 x+2\right )^{3/2}}-\frac{1733 (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{2 \sqrt{2} \sqrt{3 x^2+5 x+2}}+\frac{9521 (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{15 \sqrt{2} \sqrt{3 x^2+5 x+2}} \]
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^{7/2} \left (2+5 x+3 x^2\right )^{5/2}} \, dx &=\frac{2 (38+45 x)}{3 x^{5/2} \left (2+5 x+3 x^2\right )^{3/2}}-\frac{1}{3} \int \frac{-193-405 x}{x^{7/2} \left (2+5 x+3 x^2\right )^{3/2}} \, dx\\ &=\frac{2 (38+45 x)}{3 x^{5/2} \left (2+5 x+3 x^2\right )^{3/2}}-\frac{1541+1965 x}{3 x^{5/2} \sqrt{2+5 x+3 x^2}}+\frac{1}{3} \int \frac{-3756-\frac{9825 x}{2}}{x^{7/2} \sqrt{2+5 x+3 x^2}} \, dx\\ &=\frac{2 (38+45 x)}{3 x^{5/2} \left (2+5 x+3 x^2\right )^{3/2}}-\frac{1541+1965 x}{3 x^{5/2} \sqrt{2+5 x+3 x^2}}+\frac{1252 \sqrt{2+5 x+3 x^2}}{5 x^{5/2}}-\frac{1}{15} \int \frac{-\frac{25995}{2}-16902 x}{x^{5/2} \sqrt{2+5 x+3 x^2}} \, dx\\ &=\frac{2 (38+45 x)}{3 x^{5/2} \left (2+5 x+3 x^2\right )^{3/2}}-\frac{1541+1965 x}{3 x^{5/2} \sqrt{2+5 x+3 x^2}}+\frac{1252 \sqrt{2+5 x+3 x^2}}{5 x^{5/2}}-\frac{1733 \sqrt{2+5 x+3 x^2}}{6 x^{3/2}}+\frac{1}{45} \int \frac{-\frac{28563}{2}-\frac{77985 x}{4}}{x^{3/2} \sqrt{2+5 x+3 x^2}} \, dx\\ &=\frac{2 (38+45 x)}{3 x^{5/2} \left (2+5 x+3 x^2\right )^{3/2}}-\frac{1541+1965 x}{3 x^{5/2} \sqrt{2+5 x+3 x^2}}+\frac{1252 \sqrt{2+5 x+3 x^2}}{5 x^{5/2}}-\frac{1733 \sqrt{2+5 x+3 x^2}}{6 x^{3/2}}+\frac{9521 \sqrt{2+5 x+3 x^2}}{30 \sqrt{x}}-\frac{1}{45} \int \frac{\frac{77985}{4}+\frac{85689 x}{4}}{\sqrt{x} \sqrt{2+5 x+3 x^2}} \, dx\\ &=\frac{2 (38+45 x)}{3 x^{5/2} \left (2+5 x+3 x^2\right )^{3/2}}-\frac{1541+1965 x}{3 x^{5/2} \sqrt{2+5 x+3 x^2}}+\frac{1252 \sqrt{2+5 x+3 x^2}}{5 x^{5/2}}-\frac{1733 \sqrt{2+5 x+3 x^2}}{6 x^{3/2}}+\frac{9521 \sqrt{2+5 x+3 x^2}}{30 \sqrt{x}}-\frac{2}{45} \operatorname{Subst}\left (\int \frac{\frac{77985}{4}+\frac{85689 x^2}{4}}{\sqrt{2+5 x^2+3 x^4}} \, dx,x,\sqrt{x}\right )\\ &=\frac{2 (38+45 x)}{3 x^{5/2} \left (2+5 x+3 x^2\right )^{3/2}}-\frac{1541+1965 x}{3 x^{5/2} \sqrt{2+5 x+3 x^2}}+\frac{1252 \sqrt{2+5 x+3 x^2}}{5 x^{5/2}}-\frac{1733 \sqrt{2+5 x+3 x^2}}{6 x^{3/2}}+\frac{9521 \sqrt{2+5 x+3 x^2}}{30 \sqrt{x}}-\frac{1733}{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{2+5 x^2+3 x^4}} \, dx,x,\sqrt{x}\right )-\frac{9521}{10} \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 x^{5/2} \left (2+5 x+3 x^2\right )^{3/2}}-\frac{9521 \sqrt{x} (2+3 x)}{30 \sqrt{2+5 x+3 x^2}}-\frac{1541+1965 x}{3 x^{5/2} \sqrt{2+5 x+3 x^2}}+\frac{1252 \sqrt{2+5 x+3 x^2}}{5 x^{5/2}}-\frac{1733 \sqrt{2+5 x+3 x^2}}{6 x^{3/2}}+\frac{9521 \sqrt{2+5 x+3 x^2}}{30 \sqrt{x}}+\frac{9521 (1+x) \sqrt{\frac{2+3 x}{1+x}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{15 \sqrt{2} \sqrt{2+5 x+3 x^2}}-\frac{1733 (1+x) \sqrt{\frac{2+3 x}{1+x}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{2 \sqrt{2} \sqrt{2+5 x+3 x^2}}\\ \end{align*}
Mathematica [C] time = 0.270886, size = 177, normalized size = 0.69 \[ \frac{-6953 i \sqrt{\frac{2}{x}+2} \sqrt{\frac{2}{x}+3} \left (3 x^2+5 x+2\right ) x^{7/2} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right ),\frac{3}{2}\right )-2 \left (77985 x^5+192342 x^4+154195 x^3+39836 x^2-130 x+12\right )-19042 i \sqrt{\frac{2}{x}+2} \sqrt{\frac{2}{x}+3} \left (3 x^2+5 x+2\right ) x^{7/2} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )}{60 x^{5/2} \left (3 x^2+5 x+2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 318, normalized size = 1.2 \begin{align*}{\frac{1}{ \left ( 360+540\,x \right ) \left ( 1+x \right ) } \left ( 7704\,\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}^{4}-28563\,\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}^{4}+12840\,\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}^{3}-47605\,\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}^{3}+5136\,\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}-19042\,\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}+514134\,{x}^{6}+1245870\,{x}^{5}+959610\,{x}^{4}+217350\,{x}^{3}-10512\,{x}^{2}+780\,x-72 \right ){x}^{-{\frac{5}{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{5}{2}} x^{\frac{7}{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^{10} + 135 \, x^{9} + 279 \, x^{8} + 305 \, x^{7} + 186 \, x^{6} + 60 \, x^{5} + 8 \, x^{4}}, 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{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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