Optimal. Leaf size=233 \[ \frac{1357 (x+1) \sqrt{\frac{3 x+2}{x+1}} \text{EllipticF}\left (\tan ^{-1}\left (\sqrt{x}\right ),-\frac{1}{2}\right )}{231 \sqrt{2} \sqrt{3 x^2+5 x+2}}-\frac{4 (9-20 x) \left (3 x^2+5 x+2\right )^{3/2}}{99 x^{11/2}}+\frac{(1367 x+634) \sqrt{3 x^2+5 x+2}}{231 x^{7/2}}-\frac{3229 \sqrt{3 x^2+5 x+2}}{1386 \sqrt{x}}+\frac{1357 \sqrt{3 x^2+5 x+2}}{693 x^{3/2}}+\frac{3229 \sqrt{x} (3 x+2)}{1386 \sqrt{3 x^2+5 x+2}}-\frac{3229 (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{693 \sqrt{2} \sqrt{3 x^2+5 x+2}} \]
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Rubi [A] time = 0.152056, antiderivative size = 233, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ -\frac{4 (9-20 x) \left (3 x^2+5 x+2\right )^{3/2}}{99 x^{11/2}}+\frac{(1367 x+634) \sqrt{3 x^2+5 x+2}}{231 x^{7/2}}-\frac{3229 \sqrt{3 x^2+5 x+2}}{1386 \sqrt{x}}+\frac{1357 \sqrt{3 x^2+5 x+2}}{693 x^{3/2}}+\frac{3229 \sqrt{x} (3 x+2)}{1386 \sqrt{3 x^2+5 x+2}}+\frac{1357 (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{231 \sqrt{2} \sqrt{3 x^2+5 x+2}}-\frac{3229 (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{693 \sqrt{2} \sqrt{3 x^2+5 x+2}} \]
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^{13/2}} \, dx &=-\frac{4 (9-20 x) \left (2+5 x+3 x^2\right )^{3/2}}{99 x^{11/2}}-\frac{1}{33} \int \frac{(317+375 x) \sqrt{2+5 x+3 x^2}}{x^{9/2}} \, dx\\ &=\frac{(634+1367 x) \sqrt{2+5 x+3 x^2}}{231 x^{7/2}}-\frac{4 (9-20 x) \left (2+5 x+3 x^2\right )^{3/2}}{99 x^{11/2}}+\frac{\int \frac{-6785-\frac{17235 x}{2}}{x^{5/2} \sqrt{2+5 x+3 x^2}} \, dx}{1155}\\ &=\frac{1357 \sqrt{2+5 x+3 x^2}}{693 x^{3/2}}+\frac{(634+1367 x) \sqrt{2+5 x+3 x^2}}{231 x^{7/2}}-\frac{4 (9-20 x) \left (2+5 x+3 x^2\right )^{3/2}}{99 x^{11/2}}-\frac{\int \frac{-\frac{16145}{2}-\frac{20355 x}{2}}{x^{3/2} \sqrt{2+5 x+3 x^2}} \, dx}{3465}\\ &=\frac{1357 \sqrt{2+5 x+3 x^2}}{693 x^{3/2}}-\frac{3229 \sqrt{2+5 x+3 x^2}}{1386 \sqrt{x}}+\frac{(634+1367 x) \sqrt{2+5 x+3 x^2}}{231 x^{7/2}}-\frac{4 (9-20 x) \left (2+5 x+3 x^2\right )^{3/2}}{99 x^{11/2}}+\frac{\int \frac{\frac{20355}{2}+\frac{48435 x}{4}}{\sqrt{x} \sqrt{2+5 x+3 x^2}} \, dx}{3465}\\ &=\frac{1357 \sqrt{2+5 x+3 x^2}}{693 x^{3/2}}-\frac{3229 \sqrt{2+5 x+3 x^2}}{1386 \sqrt{x}}+\frac{(634+1367 x) \sqrt{2+5 x+3 x^2}}{231 x^{7/2}}-\frac{4 (9-20 x) \left (2+5 x+3 x^2\right )^{3/2}}{99 x^{11/2}}+\frac{2 \operatorname{Subst}\left (\int \frac{\frac{20355}{2}+\frac{48435 x^2}{4}}{\sqrt{2+5 x^2+3 x^4}} \, dx,x,\sqrt{x}\right )}{3465}\\ &=\frac{1357 \sqrt{2+5 x+3 x^2}}{693 x^{3/2}}-\frac{3229 \sqrt{2+5 x+3 x^2}}{1386 \sqrt{x}}+\frac{(634+1367 x) \sqrt{2+5 x+3 x^2}}{231 x^{7/2}}-\frac{4 (9-20 x) \left (2+5 x+3 x^2\right )^{3/2}}{99 x^{11/2}}+\frac{1357}{231} \operatorname{Subst}\left (\int \frac{1}{\sqrt{2+5 x^2+3 x^4}} \, dx,x,\sqrt{x}\right )+\frac{3229}{462} \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{2+5 x^2+3 x^4}} \, dx,x,\sqrt{x}\right )\\ &=\frac{3229 \sqrt{x} (2+3 x)}{1386 \sqrt{2+5 x+3 x^2}}+\frac{1357 \sqrt{2+5 x+3 x^2}}{693 x^{3/2}}-\frac{3229 \sqrt{2+5 x+3 x^2}}{1386 \sqrt{x}}+\frac{(634+1367 x) \sqrt{2+5 x+3 x^2}}{231 x^{7/2}}-\frac{4 (9-20 x) \left (2+5 x+3 x^2\right )^{3/2}}{99 x^{11/2}}-\frac{3229 (1+x) \sqrt{\frac{2+3 x}{1+x}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{693 \sqrt{2} \sqrt{2+5 x+3 x^2}}+\frac{1357 (1+x) \sqrt{\frac{2+3 x}{1+x}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{231 \sqrt{2} \sqrt{2+5 x+3 x^2}}\\ \end{align*}
Mathematica [C] time = 0.179422, size = 165, normalized size = 0.71 \[ \frac{842 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{13/2} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right ),\frac{3}{2}\right )+8142 x^6+48256 x^5+86914 x^4+61744 x^3+11360 x^2+3229 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{13/2} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )-5600 x-2016}{1386 x^{11/2} \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.027, size = 139, normalized size = 0.6 \begin{align*} -{\frac{1}{8316} \left ( 1545\,\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}^{5}-3229\,\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}^{5}+58122\,{x}^{7}+48018\,{x}^{6}-250788\,{x}^{5}-521484\,{x}^{4}-370464\,{x}^{3}-68160\,{x}^{2}+33600\,x+12096 \right ){\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}{x}^{-{\frac{11}{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{13}{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{13}{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{13}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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