Optimal. Leaf size=223 \[ -\frac{11320 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} \text{EllipticF}\left (\tan ^{-1}\left (\sqrt{x}\right ),-\frac{1}{2}\right )}{5103 \sqrt{3 x^2+5 x+2}}-\frac{10}{27} \sqrt{3 x^2+5 x+2} x^{7/2}+\frac{508}{567} \sqrt{3 x^2+5 x+2} x^{5/2}-\frac{820}{567} \sqrt{3 x^2+5 x+2} x^{3/2}+\frac{11320 \sqrt{3 x^2+5 x+2} \sqrt{x}}{5103}-\frac{68920 (3 x+2) \sqrt{x}}{15309 \sqrt{3 x^2+5 x+2}}+\frac{68920 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{15309 \sqrt{3 x^2+5 x+2}} \]
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Rubi [A] time = 0.162616, antiderivative size = 223, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {832, 839, 1189, 1100, 1136} \[ -\frac{10}{27} \sqrt{3 x^2+5 x+2} x^{7/2}+\frac{508}{567} \sqrt{3 x^2+5 x+2} x^{5/2}-\frac{820}{567} \sqrt{3 x^2+5 x+2} x^{3/2}+\frac{11320 \sqrt{3 x^2+5 x+2} \sqrt{x}}{5103}-\frac{68920 (3 x+2) \sqrt{x}}{15309 \sqrt{3 x^2+5 x+2}}-\frac{11320 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{5103 \sqrt{3 x^2+5 x+2}}+\frac{68920 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{15309 \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Rule 832
Rule 839
Rule 1189
Rule 1100
Rule 1136
Rubi steps
\begin{align*} \int \frac{(2-5 x) x^{7/2}}{\sqrt{2+5 x+3 x^2}} \, dx &=-\frac{10}{27} x^{7/2} \sqrt{2+5 x+3 x^2}+\frac{2}{27} \int \frac{x^{5/2} (35+127 x)}{\sqrt{2+5 x+3 x^2}} \, dx\\ &=\frac{508}{567} x^{5/2} \sqrt{2+5 x+3 x^2}-\frac{10}{27} x^{7/2} \sqrt{2+5 x+3 x^2}+\frac{4}{567} \int \frac{\left (-635-\frac{3075 x}{2}\right ) x^{3/2}}{\sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{820}{567} x^{3/2} \sqrt{2+5 x+3 x^2}+\frac{508}{567} x^{5/2} \sqrt{2+5 x+3 x^2}-\frac{10}{27} x^{7/2} \sqrt{2+5 x+3 x^2}+\frac{8 \int \frac{\sqrt{x} \left (\frac{9225}{2}+\frac{21225 x}{2}\right )}{\sqrt{2+5 x+3 x^2}} \, dx}{8505}\\ &=\frac{11320 \sqrt{x} \sqrt{2+5 x+3 x^2}}{5103}-\frac{820}{567} x^{3/2} \sqrt{2+5 x+3 x^2}+\frac{508}{567} x^{5/2} \sqrt{2+5 x+3 x^2}-\frac{10}{27} x^{7/2} \sqrt{2+5 x+3 x^2}+\frac{16 \int \frac{-\frac{21225}{2}-\frac{129225 x}{4}}{\sqrt{x} \sqrt{2+5 x+3 x^2}} \, dx}{76545}\\ &=\frac{11320 \sqrt{x} \sqrt{2+5 x+3 x^2}}{5103}-\frac{820}{567} x^{3/2} \sqrt{2+5 x+3 x^2}+\frac{508}{567} x^{5/2} \sqrt{2+5 x+3 x^2}-\frac{10}{27} x^{7/2} \sqrt{2+5 x+3 x^2}+\frac{32 \operatorname{Subst}\left (\int \frac{-\frac{21225}{2}-\frac{129225 x^2}{4}}{\sqrt{2+5 x^2+3 x^4}} \, dx,x,\sqrt{x}\right )}{76545}\\ &=\frac{11320 \sqrt{x} \sqrt{2+5 x+3 x^2}}{5103}-\frac{820}{567} x^{3/2} \sqrt{2+5 x+3 x^2}+\frac{508}{567} x^{5/2} \sqrt{2+5 x+3 x^2}-\frac{10}{27} x^{7/2} \sqrt{2+5 x+3 x^2}-\frac{22640 \operatorname{Subst}\left (\int \frac{1}{\sqrt{2+5 x^2+3 x^4}} \, dx,x,\sqrt{x}\right )}{5103}-\frac{68920 \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{2+5 x^2+3 x^4}} \, dx,x,\sqrt{x}\right )}{5103}\\ &=-\frac{68920 \sqrt{x} (2+3 x)}{15309 \sqrt{2+5 x+3 x^2}}+\frac{11320 \sqrt{x} \sqrt{2+5 x+3 x^2}}{5103}-\frac{820}{567} x^{3/2} \sqrt{2+5 x+3 x^2}+\frac{508}{567} x^{5/2} \sqrt{2+5 x+3 x^2}-\frac{10}{27} x^{7/2} \sqrt{2+5 x+3 x^2}+\frac{68920 \sqrt{2} (1+x) \sqrt{\frac{2+3 x}{1+x}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{15309 \sqrt{2+5 x+3 x^2}}-\frac{11320 \sqrt{2} (1+x) \sqrt{\frac{2+3 x}{1+x}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{5103 \sqrt{2+5 x+3 x^2}}\\ \end{align*}
Mathematica [C] time = 0.198227, size = 168, normalized size = 0.75 \[ \frac{34960 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{3/2} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right ),\frac{3}{2}\right )-2 \left (8505 x^6-6399 x^5+4590 x^4-9306 x^3+40620 x^2+138340 x+68920\right )-68920 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{3/2} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )}{15309 \sqrt{x} \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.029, size = 127, normalized size = 0.6 \begin{align*}{\frac{2}{45927} \left ( -25515\,{x}^{6}+19197\,{x}^{5}+34710\,\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 ) -17230\,\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 ) -13770\,{x}^{4}+27918\,{x}^{3}+188280\,{x}^{2}+101880\,x \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{{\left (5 \, x - 2\right )} x^{\frac{7}{2}}}{\sqrt{3 \, x^{2} + 5 \, x + 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{x}}{\sqrt{3 \, x^{2} + 5 \, x + 2}}, 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}}}{\sqrt{3 \, x^{2} + 5 \, x + 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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