Optimal. Leaf size=205 \[ \frac{43 (x+1) \sqrt{\frac{3 x+2}{x+1}} \text{EllipticF}\left (\tan ^{-1}\left (\sqrt{x}\right ),-\frac{1}{2}\right )}{7 \sqrt{2} \sqrt{3 x^2+5 x+2}}-\frac{4 \sqrt{3 x^2+5 x+2} (1-3 x)}{7 x^{7/2}}-\frac{62 \sqrt{3 x^2+5 x+2}}{21 \sqrt{x}}+\frac{43 \sqrt{3 x^2+5 x+2}}{21 x^{3/2}}+\frac{62 \sqrt{x} (3 x+2)}{21 \sqrt{3 x^2+5 x+2}}-\frac{62 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{21 \sqrt{3 x^2+5 x+2}} \]
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Rubi [A] time = 0.133239, antiderivative size = 205, 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 = {810, 834, 839, 1189, 1100, 1136} \[ -\frac{4 \sqrt{3 x^2+5 x+2} (1-3 x)}{7 x^{7/2}}-\frac{62 \sqrt{3 x^2+5 x+2}}{21 \sqrt{x}}+\frac{43 \sqrt{3 x^2+5 x+2}}{21 x^{3/2}}+\frac{62 \sqrt{x} (3 x+2)}{21 \sqrt{3 x^2+5 x+2}}+\frac{43 (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{7 \sqrt{2} \sqrt{3 x^2+5 x+2}}-\frac{62 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{21 \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Rule 810
Rule 834
Rule 839
Rule 1189
Rule 1100
Rule 1136
Rubi steps
\begin{align*} \int \frac{(2-5 x) \sqrt{2+5 x+3 x^2}}{x^{9/2}} \, dx &=-\frac{4 (1-3 x) \sqrt{2+5 x+3 x^2}}{7 x^{7/2}}-\frac{1}{35} \int \frac{215+255 x}{x^{5/2} \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{4 (1-3 x) \sqrt{2+5 x+3 x^2}}{7 x^{7/2}}+\frac{43 \sqrt{2+5 x+3 x^2}}{21 x^{3/2}}+\frac{1}{105} \int \frac{310+\frac{645 x}{2}}{x^{3/2} \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{4 (1-3 x) \sqrt{2+5 x+3 x^2}}{7 x^{7/2}}+\frac{43 \sqrt{2+5 x+3 x^2}}{21 x^{3/2}}-\frac{62 \sqrt{2+5 x+3 x^2}}{21 \sqrt{x}}-\frac{1}{105} \int \frac{-\frac{645}{2}-465 x}{\sqrt{x} \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{4 (1-3 x) \sqrt{2+5 x+3 x^2}}{7 x^{7/2}}+\frac{43 \sqrt{2+5 x+3 x^2}}{21 x^{3/2}}-\frac{62 \sqrt{2+5 x+3 x^2}}{21 \sqrt{x}}-\frac{2}{105} \operatorname{Subst}\left (\int \frac{-\frac{645}{2}-465 x^2}{\sqrt{2+5 x^2+3 x^4}} \, dx,x,\sqrt{x}\right )\\ &=-\frac{4 (1-3 x) \sqrt{2+5 x+3 x^2}}{7 x^{7/2}}+\frac{43 \sqrt{2+5 x+3 x^2}}{21 x^{3/2}}-\frac{62 \sqrt{2+5 x+3 x^2}}{21 \sqrt{x}}+\frac{43}{7} \operatorname{Subst}\left (\int \frac{1}{\sqrt{2+5 x^2+3 x^4}} \, dx,x,\sqrt{x}\right )+\frac{62}{7} \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{2+5 x^2+3 x^4}} \, dx,x,\sqrt{x}\right )\\ &=\frac{62 \sqrt{x} (2+3 x)}{21 \sqrt{2+5 x+3 x^2}}-\frac{4 (1-3 x) \sqrt{2+5 x+3 x^2}}{7 x^{7/2}}+\frac{43 \sqrt{2+5 x+3 x^2}}{21 x^{3/2}}-\frac{62 \sqrt{2+5 x+3 x^2}}{21 \sqrt{x}}-\frac{62 \sqrt{2} (1+x) \sqrt{\frac{2+3 x}{1+x}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{21 \sqrt{2+5 x+3 x^2}}+\frac{43 (1+x) \sqrt{\frac{2+3 x}{1+x}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{7 \sqrt{2} \sqrt{2+5 x+3 x^2}}\\ \end{align*}
Mathematica [C] time = 0.16888, size = 155, normalized size = 0.76 \[ \frac{5 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{9/2} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right ),\frac{3}{2}\right )+258 x^4+646 x^3+460 x^2+124 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{9/2} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )+24 x-48}{42 x^{7/2} \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 129, normalized size = 0.6 \begin{align*} -{\frac{1}{126} \left ( 57\,\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}-62\,\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}+1116\,{x}^{5}+1086\,{x}^{4}-1194\,{x}^{3}-1380\,{x}^{2}-72\,x+144 \right ){x}^{-{\frac{7}{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{\sqrt{3 \, x^{2} + 5 \, x + 2}{\left (5 \, x - 2\right )}}{x^{\frac{9}{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 )}}{x^{\frac{9}{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{\sqrt{3 \, x^{2} + 5 \, x + 2}{\left (5 \, x - 2\right )}}{x^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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