Optimal. Leaf size=180 \[ -\frac{11 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} \text{EllipticF}\left (\tan ^{-1}\left (\sqrt{x}\right ),-\frac{1}{2}\right )}{\sqrt{3 x^2+5 x+2}}-\frac{4 \sqrt{3 x^2+5 x+2} (3-10 x)}{15 x^{5/2}}+\frac{139 \sqrt{3 x^2+5 x+2}}{15 \sqrt{x}}-\frac{139 \sqrt{x} (3 x+2)}{15 \sqrt{3 x^2+5 x+2}}+\frac{139 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{15 \sqrt{3 x^2+5 x+2}} \]
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Rubi [A] time = 0.113624, antiderivative size = 180, 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, 834, 839, 1189, 1100, 1136} \[ -\frac{4 \sqrt{3 x^2+5 x+2} (3-10 x)}{15 x^{5/2}}+\frac{139 \sqrt{3 x^2+5 x+2}}{15 \sqrt{x}}-\frac{139 \sqrt{x} (3 x+2)}{15 \sqrt{3 x^2+5 x+2}}-\frac{11 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{\sqrt{3 x^2+5 x+2}}+\frac{139 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{15 \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Rule 810
Rule 834
Rule 839
Rule 1189
Rule 1100
Rule 1136
Rubi steps
\begin{align*} \int \frac{(2-5 x) \sqrt{2+5 x+3 x^2}}{x^{7/2}} \, dx &=-\frac{4 (3-10 x) \sqrt{2+5 x+3 x^2}}{15 x^{5/2}}-\frac{1}{15} \int \frac{139+165 x}{x^{3/2} \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{4 (3-10 x) \sqrt{2+5 x+3 x^2}}{15 x^{5/2}}+\frac{139 \sqrt{2+5 x+3 x^2}}{15 \sqrt{x}}+\frac{1}{15} \int \frac{-165-\frac{417 x}{2}}{\sqrt{x} \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{4 (3-10 x) \sqrt{2+5 x+3 x^2}}{15 x^{5/2}}+\frac{139 \sqrt{2+5 x+3 x^2}}{15 \sqrt{x}}+\frac{2}{15} \operatorname{Subst}\left (\int \frac{-165-\frac{417 x^2}{2}}{\sqrt{2+5 x^2+3 x^4}} \, dx,x,\sqrt{x}\right )\\ &=-\frac{4 (3-10 x) \sqrt{2+5 x+3 x^2}}{15 x^{5/2}}+\frac{139 \sqrt{2+5 x+3 x^2}}{15 \sqrt{x}}-22 \operatorname{Subst}\left (\int \frac{1}{\sqrt{2+5 x^2+3 x^4}} \, dx,x,\sqrt{x}\right )-\frac{139}{5} \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{2+5 x^2+3 x^4}} \, dx,x,\sqrt{x}\right )\\ &=-\frac{139 \sqrt{x} (2+3 x)}{15 \sqrt{2+5 x+3 x^2}}-\frac{4 (3-10 x) \sqrt{2+5 x+3 x^2}}{15 x^{5/2}}+\frac{139 \sqrt{2+5 x+3 x^2}}{15 \sqrt{x}}+\frac{139 \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{11 \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 = 0.177236, size = 153, normalized size = 0.85 \[ \frac{-26 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{7/2} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right ),\frac{3}{2}\right )+4 \left (30 x^3+41 x^2+5 x-6\right )-139 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{7/2} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )}{15 x^{5/2} \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 124, normalized size = 0.7 \begin{align*}{\frac{1}{90} \left ( 87\,\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}-139\,\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}+2502\,{x}^{4}+4890\,{x}^{3}+2652\,{x}^{2}+120\,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{\sqrt{3 \, x^{2} + 5 \, x + 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{\sqrt{3 \, x^{2} + 5 \, x + 2}{\left (5 \, x - 2\right )}}{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{2 \sqrt{3 x^{2} + 5 x + 2}}{x^{\frac{7}{2}}}\, dx - \int \frac{5 \sqrt{3 x^{2} + 5 x + 2}}{x^{\frac{5}{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{\sqrt{3 \, x^{2} + 5 \, x + 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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