Optimal. Leaf size=185 \[ -\frac{117 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} \text{EllipticF}\left (\tan ^{-1}\left (\sqrt{x}\right ),-\frac{1}{2}\right )}{\sqrt{3 x^2+5 x+2}}-\frac{4 (3-5 x) \left (3 x^2+5 x+2\right )^{3/2}}{15 x^{5/2}}+\frac{2 (89-35 x) \sqrt{3 x^2+5 x+2}}{5 \sqrt{x}}-\frac{1418 \sqrt{x} (3 x+2)}{15 \sqrt{3 x^2+5 x+2}}+\frac{1418 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{15 \sqrt{3 x^2+5 x+2}} \]
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Rubi [A] time = 0.121017, antiderivative size = 185, 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 = {810, 812, 839, 1189, 1100, 1136} \[ -\frac{4 (3-5 x) \left (3 x^2+5 x+2\right )^{3/2}}{15 x^{5/2}}+\frac{2 (89-35 x) \sqrt{3 x^2+5 x+2}}{5 \sqrt{x}}-\frac{1418 \sqrt{x} (3 x+2)}{15 \sqrt{3 x^2+5 x+2}}-\frac{117 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{\sqrt{3 x^2+5 x+2}}+\frac{1418 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{15 \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Rule 810
Rule 812
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^{7/2}} \, dx &=-\frac{4 (3-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{15 x^{5/2}}-\frac{1}{5} \int \frac{(89+105 x) \sqrt{2+5 x+3 x^2}}{x^{3/2}} \, dx\\ &=\frac{2 (89-35 x) \sqrt{2+5 x+3 x^2}}{5 \sqrt{x}}-\frac{4 (3-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{15 x^{5/2}}+\frac{2}{15} \int \frac{-\frac{1755}{2}-\frac{2127 x}{2}}{\sqrt{x} \sqrt{2+5 x+3 x^2}} \, dx\\ &=\frac{2 (89-35 x) \sqrt{2+5 x+3 x^2}}{5 \sqrt{x}}-\frac{4 (3-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{15 x^{5/2}}+\frac{4}{15} \operatorname{Subst}\left (\int \frac{-\frac{1755}{2}-\frac{2127 x^2}{2}}{\sqrt{2+5 x^2+3 x^4}} \, dx,x,\sqrt{x}\right )\\ &=\frac{2 (89-35 x) \sqrt{2+5 x+3 x^2}}{5 \sqrt{x}}-\frac{4 (3-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{15 x^{5/2}}-234 \operatorname{Subst}\left (\int \frac{1}{\sqrt{2+5 x^2+3 x^4}} \, dx,x,\sqrt{x}\right )-\frac{1418}{5} \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{2+5 x^2+3 x^4}} \, dx,x,\sqrt{x}\right )\\ &=-\frac{1418 \sqrt{x} (2+3 x)}{15 \sqrt{2+5 x+3 x^2}}+\frac{2 (89-35 x) \sqrt{2+5 x+3 x^2}}{5 \sqrt{x}}-\frac{4 (3-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{15 x^{5/2}}+\frac{1418 \sqrt{2} (1+x) \sqrt{\frac{2+3 x}{1+x}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{15 \sqrt{2+5 x+3 x^2}}-\frac{117 \sqrt{2} (1+x) \sqrt{\frac{2+3 x}{1+x}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{\sqrt{2+5 x+3 x^2}}\\ \end{align*}
Mathematica [C] time = 0.189335, size = 163, normalized size = 0.88 \[ \frac{-337 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{7/2} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right ),\frac{3}{2}\right )-2 \left (225 x^5+1605 x^4+2230 x^3+906 x^2+80 x+24\right )-1418 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{7/2} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )}{15 x^{5/2} \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 129, normalized size = 0.7 \begin{align*}{\frac{1}{45} \left ( 372\,\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}^{2}-709\,\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}^{2}-1350\,{x}^{5}+3132\,{x}^{4}+7890\,{x}^{3}+3072\,{x}^{2}-480\,x-144 \right ){x}^{-{\frac{5}{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{7}{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{7}{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{7}{2}}}\, dx - \int \frac{19 \sqrt{3 x^{2} + 5 x + 2}}{x^{\frac{3}{2}}}\, dx - \int \frac{15 \sqrt{3 x^{2} + 5 x + 2}}{\sqrt{x}}\, 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{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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