Optimal. Leaf size=210 \[ -\frac{899 (x+1) \sqrt{\frac{3 x+2}{x+1}} \text{EllipticF}\left (\tan ^{-1}\left (\sqrt{x}\right ),-\frac{1}{2}\right )}{21 \sqrt{2} \sqrt{3 x^2+5 x+2}}-\frac{4 (7-15 x) \left (3 x^2+5 x+2\right )^{3/2}}{63 x^{9/2}}+\frac{(4055 x+1446) \sqrt{3 x^2+5 x+2}}{315 x^{5/2}}+\frac{5438 \sqrt{3 x^2+5 x+2}}{315 \sqrt{x}}-\frac{5438 \sqrt{x} (3 x+2)}{315 \sqrt{3 x^2+5 x+2}}+\frac{5438 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{315 \sqrt{3 x^2+5 x+2}} \]
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Rubi [A] time = 0.132194, antiderivative size = 210, 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} \[ -\frac{4 (7-15 x) \left (3 x^2+5 x+2\right )^{3/2}}{63 x^{9/2}}+\frac{(4055 x+1446) \sqrt{3 x^2+5 x+2}}{315 x^{5/2}}+\frac{5438 \sqrt{3 x^2+5 x+2}}{315 \sqrt{x}}-\frac{5438 \sqrt{x} (3 x+2)}{315 \sqrt{3 x^2+5 x+2}}-\frac{899 (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{21 \sqrt{2} \sqrt{3 x^2+5 x+2}}+\frac{5438 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{315 \sqrt{3 x^2+5 x+2}} \]
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) \left (2+5 x+3 x^2\right )^{3/2}}{x^{11/2}} \, dx &=-\frac{4 (7-15 x) \left (2+5 x+3 x^2\right )^{3/2}}{63 x^{9/2}}-\frac{1}{21} \int \frac{(241+285 x) \sqrt{2+5 x+3 x^2}}{x^{7/2}} \, dx\\ &=\frac{(1446+4055 x) \sqrt{2+5 x+3 x^2}}{315 x^{5/2}}-\frac{4 (7-15 x) \left (2+5 x+3 x^2\right )^{3/2}}{63 x^{9/2}}+\frac{1}{315} \int \frac{-5438-\frac{13485 x}{2}}{x^{3/2} \sqrt{2+5 x+3 x^2}} \, dx\\ &=\frac{5438 \sqrt{2+5 x+3 x^2}}{315 \sqrt{x}}+\frac{(1446+4055 x) \sqrt{2+5 x+3 x^2}}{315 x^{5/2}}-\frac{4 (7-15 x) \left (2+5 x+3 x^2\right )^{3/2}}{63 x^{9/2}}-\frac{1}{315} \int \frac{\frac{13485}{2}+8157 x}{\sqrt{x} \sqrt{2+5 x+3 x^2}} \, dx\\ &=\frac{5438 \sqrt{2+5 x+3 x^2}}{315 \sqrt{x}}+\frac{(1446+4055 x) \sqrt{2+5 x+3 x^2}}{315 x^{5/2}}-\frac{4 (7-15 x) \left (2+5 x+3 x^2\right )^{3/2}}{63 x^{9/2}}-\frac{2}{315} \operatorname{Subst}\left (\int \frac{\frac{13485}{2}+8157 x^2}{\sqrt{2+5 x^2+3 x^4}} \, dx,x,\sqrt{x}\right )\\ &=\frac{5438 \sqrt{2+5 x+3 x^2}}{315 \sqrt{x}}+\frac{(1446+4055 x) \sqrt{2+5 x+3 x^2}}{315 x^{5/2}}-\frac{4 (7-15 x) \left (2+5 x+3 x^2\right )^{3/2}}{63 x^{9/2}}-\frac{899}{21} \operatorname{Subst}\left (\int \frac{1}{\sqrt{2+5 x^2+3 x^4}} \, dx,x,\sqrt{x}\right )-\frac{5438}{105} \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{2+5 x^2+3 x^4}} \, dx,x,\sqrt{x}\right )\\ &=-\frac{5438 \sqrt{x} (2+3 x)}{315 \sqrt{2+5 x+3 x^2}}+\frac{5438 \sqrt{2+5 x+3 x^2}}{315 \sqrt{x}}+\frac{(1446+4055 x) \sqrt{2+5 x+3 x^2}}{315 x^{5/2}}-\frac{4 (7-15 x) \left (2+5 x+3 x^2\right )^{3/2}}{63 x^{9/2}}+\frac{5438 \sqrt{2} (1+x) \sqrt{\frac{2+3 x}{1+x}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{315 \sqrt{2+5 x+3 x^2}}-\frac{899 (1+x) \sqrt{\frac{2+3 x}{1+x}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{21 \sqrt{2} \sqrt{2+5 x+3 x^2}}\\ \end{align*}
Mathematica [C] time = 0.174783, size = 160, normalized size = 0.76 \[ \frac{-2609 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{11/2} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right ),\frac{3}{2}\right )+29730 x^5+64706 x^4+44480 x^3+7424 x^2-10876 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{11/2} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )-3200 x-1120}{630 x^{9/2} \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 134, normalized size = 0.6 \begin{align*}{\frac{1}{1890} \left ( 2829\,\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}-5438\,\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}+97884\,{x}^{6}+252330\,{x}^{5}+259374\,{x}^{4}+133440\,{x}^{3}+22272\,{x}^{2}-9600\,x-3360 \right ){\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}{x}^{-{\frac{9}{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 (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}{\left (5 \, x - 2\right )}}{x^{\frac{11}{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 (15 \, x^{3} + 19 \, x^{2} - 4\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}}{x^{\frac{11}{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 (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}{\left (5 \, x - 2\right )}}{x^{\frac{11}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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