Optimal. Leaf size=187 \[ \frac{3340 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} \text{EllipticF}\left (\tan ^{-1}\left (\sqrt{x}\right ),-\frac{1}{2}\right )}{27 \sqrt{3 x^2+5 x+2}}+\frac{2 (95 x+74) x^{5/2}}{9 \left (3 x^2+5 x+2\right )^{3/2}}+\frac{8020 (3 x+2) \sqrt{x}}{81 \sqrt{3 x^2+5 x+2}}-\frac{40 (206 x+167) \sqrt{x}}{27 \sqrt{3 x^2+5 x+2}}-\frac{8020 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{81 \sqrt{3 x^2+5 x+2}} \]
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Rubi [A] time = 0.121806, antiderivative size = 187, 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 = {818, 839, 1189, 1100, 1136} \[ \frac{2 (95 x+74) x^{5/2}}{9 \left (3 x^2+5 x+2\right )^{3/2}}+\frac{8020 (3 x+2) \sqrt{x}}{81 \sqrt{3 x^2+5 x+2}}-\frac{40 (206 x+167) \sqrt{x}}{27 \sqrt{3 x^2+5 x+2}}+\frac{3340 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{27 \sqrt{3 x^2+5 x+2}}-\frac{8020 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{81 \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Rule 818
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 x^{5/2} (74+95 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac{2}{9} \int \frac{(-185-55 x) x^{3/2}}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx\\ &=\frac{2 x^{5/2} (74+95 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}-\frac{40 \sqrt{x} (167+206 x)}{27 \sqrt{2+5 x+3 x^2}}+\frac{4}{27} \int \frac{835+\frac{2005 x}{2}}{\sqrt{x} \sqrt{2+5 x+3 x^2}} \, dx\\ &=\frac{2 x^{5/2} (74+95 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}-\frac{40 \sqrt{x} (167+206 x)}{27 \sqrt{2+5 x+3 x^2}}+\frac{8}{27} \operatorname{Subst}\left (\int \frac{835+\frac{2005 x^2}{2}}{\sqrt{2+5 x^2+3 x^4}} \, dx,x,\sqrt{x}\right )\\ &=\frac{2 x^{5/2} (74+95 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}-\frac{40 \sqrt{x} (167+206 x)}{27 \sqrt{2+5 x+3 x^2}}+\frac{6680}{27} \operatorname{Subst}\left (\int \frac{1}{\sqrt{2+5 x^2+3 x^4}} \, dx,x,\sqrt{x}\right )+\frac{8020}{27} \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{2+5 x^2+3 x^4}} \, dx,x,\sqrt{x}\right )\\ &=\frac{2 x^{5/2} (74+95 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac{8020 \sqrt{x} (2+3 x)}{81 \sqrt{2+5 x+3 x^2}}-\frac{40 \sqrt{x} (167+206 x)}{27 \sqrt{2+5 x+3 x^2}}-\frac{8020 \sqrt{2} (1+x) \sqrt{\frac{2+3 x}{1+x}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{81 \sqrt{2+5 x+3 x^2}}+\frac{3340 \sqrt{2} (1+x) \sqrt{\frac{2+3 x}{1+x}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{27 \sqrt{2+5 x+3 x^2}}\\ \end{align*}
Mathematica [C] time = 0.257583, size = 169, normalized size = 0.9 \[ \frac{2000 i \sqrt{\frac{2}{x}+2} \sqrt{\frac{2}{x}+3} \left (3 x^2+5 x+2\right ) x^{3/2} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right ),\frac{3}{2}\right )-270 x^4+58212 x^3+147100 x^2+8020 i \sqrt{\frac{2}{x}+2} \sqrt{\frac{2}{x}+3} \left (3 x^2+5 x+2\right ) x^{3/2} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )+120320 x+32080}{81 \sqrt{x} \left (3 x^2+5 x+2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.022, size = 297, normalized size = 1.6 \begin{align*} -{\frac{2}{243\, \left ( 1+x \right ) ^{2} \left ( 2+3\,x \right ) ^{2}} \left ( 3015\,\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}-6015\,\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}+5025\,\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-10025\,\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+2010\,\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 ) -4010\,\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 ) +108675\,{x}^{4}+273582\,{x}^{3}+224460\,{x}^{2}+60120\,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{{\left (5 \, x - 2\right )} x^{\frac{7}{2}}}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\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{{\left (5 \, x^{4} - 2 \, x^{3}\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} \sqrt{x}}{27 \, x^{6} + 135 \, x^{5} + 279 \, x^{4} + 305 \, x^{3} + 186 \, x^{2} + 60 \, x + 8}, 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{{\left (5 \, x - 2\right )} x^{\frac{7}{2}}}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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