Optimal. Leaf size=187 \[ \frac{482 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} \text{EllipticF}\left (\tan ^{-1}\left (\sqrt{x}\right ),-\frac{1}{2}\right )}{21 \sqrt{3 x^2+5 x+2}}-\frac{2 (5 x+14) \left (3 x^2+5 x+2\right )^{3/2}}{7 \sqrt{x}}+\frac{2}{105} \sqrt{x} (531 x+1045) \sqrt{3 x^2+5 x+2}+\frac{5848 \sqrt{x} (3 x+2)}{315 \sqrt{3 x^2+5 x+2}}-\frac{5848 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{315 \sqrt{3 x^2+5 x+2}} \]
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Rubi [A] time = 0.123554, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {812, 814, 839, 1189, 1100, 1136} \[ -\frac{2 (5 x+14) \left (3 x^2+5 x+2\right )^{3/2}}{7 \sqrt{x}}+\frac{2}{105} \sqrt{x} (531 x+1045) \sqrt{3 x^2+5 x+2}+\frac{5848 \sqrt{x} (3 x+2)}{315 \sqrt{3 x^2+5 x+2}}+\frac{482 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{21 \sqrt{3 x^2+5 x+2}}-\frac{5848 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{315 \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Rule 812
Rule 814
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^{3/2}} \, dx &=-\frac{2 (14+5 x) \left (2+5 x+3 x^2\right )^{3/2}}{7 \sqrt{x}}-\frac{6}{7} \int \frac{\left (-25-\frac{59 x}{2}\right ) \sqrt{2+5 x+3 x^2}}{\sqrt{x}} \, dx\\ &=\frac{2}{105} \sqrt{x} (1045+531 x) \sqrt{2+5 x+3 x^2}-\frac{2 (14+5 x) \left (2+5 x+3 x^2\right )^{3/2}}{7 \sqrt{x}}+\frac{4}{105} \int \frac{\frac{1205}{2}+731 x}{\sqrt{x} \sqrt{2+5 x+3 x^2}} \, dx\\ &=\frac{2}{105} \sqrt{x} (1045+531 x) \sqrt{2+5 x+3 x^2}-\frac{2 (14+5 x) \left (2+5 x+3 x^2\right )^{3/2}}{7 \sqrt{x}}+\frac{8}{105} \operatorname{Subst}\left (\int \frac{\frac{1205}{2}+731 x^2}{\sqrt{2+5 x^2+3 x^4}} \, dx,x,\sqrt{x}\right )\\ &=\frac{2}{105} \sqrt{x} (1045+531 x) \sqrt{2+5 x+3 x^2}-\frac{2 (14+5 x) \left (2+5 x+3 x^2\right )^{3/2}}{7 \sqrt{x}}+\frac{964}{21} \operatorname{Subst}\left (\int \frac{1}{\sqrt{2+5 x^2+3 x^4}} \, dx,x,\sqrt{x}\right )+\frac{5848}{105} \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{2+5 x^2+3 x^4}} \, dx,x,\sqrt{x}\right )\\ &=\frac{5848 \sqrt{x} (2+3 x)}{315 \sqrt{2+5 x+3 x^2}}+\frac{2}{105} \sqrt{x} (1045+531 x) \sqrt{2+5 x+3 x^2}-\frac{2 (14+5 x) \left (2+5 x+3 x^2\right )^{3/2}}{7 \sqrt{x}}-\frac{5848 \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{482 \sqrt{2} (1+x) \sqrt{\frac{2+3 x}{1+x}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{21 \sqrt{2+5 x+3 x^2}}\\ \end{align*}
Mathematica [C] time = 0.181008, size = 163, normalized size = 0.87 \[ \frac{1382 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{3/2} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right ),\frac{3}{2}\right )-2 \left (2025 x^5+7641 x^4+9855 x^3+177 x^2-7390 x-3328\right )+5848 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{3/2} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )}{315 \sqrt{x} \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 123, normalized size = 0.7 \begin{align*}{\frac{2}{945} \left ( -6075\,{x}^{5}+1462\,\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 ) -771\,\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 ) -22923\,{x}^{4}-29565\,{x}^{3}-26847\,{x}^{2}-21690\,x-7560 \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 (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}{\left (5 \, x - 2\right )}}{x^{\frac{3}{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{3}{2}}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \int - \frac{4 \sqrt{3 x^{2} + 5 x + 2}}{x^{\frac{3}{2}}}\, dx - \int 19 \sqrt{x} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int 15 x^{\frac{3}{2}} \sqrt{3 x^{2} + 5 x + 2}\, dx \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{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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