Optimal. Leaf size=187 \[ -\frac{783 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} \text{EllipticF}\left (\tan ^{-1}\left (\sqrt{x}\right ),-\frac{1}{2}\right )}{7 \sqrt{3 x^2+5 x+2}}-\frac{4 (1-2 x) \left (3 x^2+5 x+2\right )^{3/2}}{7 x^{7/2}}+\frac{3 (133 x+22) \sqrt{3 x^2+5 x+2}}{7 x^{3/2}}-\frac{633 \sqrt{x} (3 x+2)}{7 \sqrt{3 x^2+5 x+2}}+\frac{633 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{7 \sqrt{3 x^2+5 x+2}} \]
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Rubi [A] time = 0.118924, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {810, 839, 1189, 1100, 1136} \[ -\frac{4 (1-2 x) \left (3 x^2+5 x+2\right )^{3/2}}{7 x^{7/2}}+\frac{3 (133 x+22) \sqrt{3 x^2+5 x+2}}{7 x^{3/2}}-\frac{633 \sqrt{x} (3 x+2)}{7 \sqrt{3 x^2+5 x+2}}-\frac{783 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{7 \sqrt{3 x^2+5 x+2}}+\frac{633 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{7 \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Rule 810
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^{9/2}} \, dx &=-\frac{4 (1-2 x) \left (2+5 x+3 x^2\right )^{3/2}}{7 x^{7/2}}-\frac{3}{35} \int \frac{(165+195 x) \sqrt{2+5 x+3 x^2}}{x^{5/2}} \, dx\\ &=\frac{3 (22+133 x) \sqrt{2+5 x+3 x^2}}{7 x^{3/2}}-\frac{4 (1-2 x) \left (2+5 x+3 x^2\right )^{3/2}}{7 x^{7/2}}+\frac{1}{35} \int \frac{-3915-\frac{9495 x}{2}}{\sqrt{x} \sqrt{2+5 x+3 x^2}} \, dx\\ &=\frac{3 (22+133 x) \sqrt{2+5 x+3 x^2}}{7 x^{3/2}}-\frac{4 (1-2 x) \left (2+5 x+3 x^2\right )^{3/2}}{7 x^{7/2}}+\frac{2}{35} \operatorname{Subst}\left (\int \frac{-3915-\frac{9495 x^2}{2}}{\sqrt{2+5 x^2+3 x^4}} \, dx,x,\sqrt{x}\right )\\ &=\frac{3 (22+133 x) \sqrt{2+5 x+3 x^2}}{7 x^{3/2}}-\frac{4 (1-2 x) \left (2+5 x+3 x^2\right )^{3/2}}{7 x^{7/2}}-\frac{1566}{7} \operatorname{Subst}\left (\int \frac{1}{\sqrt{2+5 x^2+3 x^4}} \, dx,x,\sqrt{x}\right )-\frac{1899}{7} \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{2+5 x^2+3 x^4}} \, dx,x,\sqrt{x}\right )\\ &=-\frac{633 \sqrt{x} (2+3 x)}{7 \sqrt{2+5 x+3 x^2}}+\frac{3 (22+133 x) \sqrt{2+5 x+3 x^2}}{7 x^{3/2}}-\frac{4 (1-2 x) \left (2+5 x+3 x^2\right )^{3/2}}{7 x^{7/2}}+\frac{633 \sqrt{2} (1+x) \sqrt{\frac{2+3 x}{1+x}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{7 \sqrt{2+5 x+3 x^2}}-\frac{783 \sqrt{2} (1+x) \sqrt{\frac{2+3 x}{1+x}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{7 \sqrt{2+5 x+3 x^2}}\\ \end{align*}
Mathematica [C] time = 0.175263, size = 163, normalized size = 0.87 \[ \frac{-150 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{9/2} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right ),\frac{3}{2}\right )-2 \left (315 x^5+384 x^4-19 x^3-72 x^2+24 x+8\right )-633 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{9/2} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )}{7 x^{7/2} \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 129, normalized size = 0.7 \begin{align*}{\frac{1}{14} \left ( 111\,\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}-211\,\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}+2538\,{x}^{5}+4794\,{x}^{4}+2608\,{x}^{3}+288\,{x}^{2}-96\,x-32 \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{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{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{{\left (15 \, x^{3} + 19 \, x^{2} - 4\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}}{x^{\frac{9}{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{9}{2}}}\, dx - \int \frac{19 \sqrt{3 x^{2} + 5 x + 2}}{x^{\frac{5}{2}}}\, dx - \int \frac{15 \sqrt{3 x^{2} + 5 x + 2}}{x^{\frac{3}{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{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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