Optimal. Leaf size=251 \[ \frac{349240 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} \text{EllipticF}\left (\tan ^{-1}\left (\sqrt{x}\right ),-\frac{1}{2}\right )}{2189187 \sqrt{3 x^2+5 x+2}}-\frac{10}{39} \left (3 x^2+5 x+2\right )^{3/2} x^{7/2}+\frac{656 \left (3 x^2+5 x+2\right )^{3/2} x^{5/2}}{1287}-\frac{21620 \left (3 x^2+5 x+2\right )^{3/2} x^{3/2}}{34749}+\frac{157160 \left (3 x^2+5 x+2\right )^{3/2} \sqrt{x}}{243243}-\frac{8 (502911 x+397265) \sqrt{3 x^2+5 x+2} \sqrt{x}}{2189187}+\frac{1543648 (3 x+2) \sqrt{x}}{6567561 \sqrt{3 x^2+5 x+2}}-\frac{1543648 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{6567561 \sqrt{3 x^2+5 x+2}} \]
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Rubi [A] time = 0.201053, antiderivative size = 251, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {832, 814, 839, 1189, 1100, 1136} \[ -\frac{10}{39} \left (3 x^2+5 x+2\right )^{3/2} x^{7/2}+\frac{656 \left (3 x^2+5 x+2\right )^{3/2} x^{5/2}}{1287}-\frac{21620 \left (3 x^2+5 x+2\right )^{3/2} x^{3/2}}{34749}+\frac{157160 \left (3 x^2+5 x+2\right )^{3/2} \sqrt{x}}{243243}-\frac{8 (502911 x+397265) \sqrt{3 x^2+5 x+2} \sqrt{x}}{2189187}+\frac{1543648 (3 x+2) \sqrt{x}}{6567561 \sqrt{3 x^2+5 x+2}}+\frac{349240 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{2189187 \sqrt{3 x^2+5 x+2}}-\frac{1543648 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{6567561 \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Rule 832
Rule 814
Rule 839
Rule 1189
Rule 1100
Rule 1136
Rubi steps
\begin{align*} \int (2-5 x) x^{7/2} \sqrt{2+5 x+3 x^2} \, dx &=-\frac{10}{39} x^{7/2} \left (2+5 x+3 x^2\right )^{3/2}+\frac{2}{39} \int x^{5/2} (35+164 x) \sqrt{2+5 x+3 x^2} \, dx\\ &=\frac{656 x^{5/2} \left (2+5 x+3 x^2\right )^{3/2}}{1287}-\frac{10}{39} x^{7/2} \left (2+5 x+3 x^2\right )^{3/2}+\frac{4 \int \left (-820-\frac{5405 x}{2}\right ) x^{3/2} \sqrt{2+5 x+3 x^2} \, dx}{1287}\\ &=-\frac{21620 x^{3/2} \left (2+5 x+3 x^2\right )^{3/2}}{34749}+\frac{656 x^{5/2} \left (2+5 x+3 x^2\right )^{3/2}}{1287}-\frac{10}{39} x^{7/2} \left (2+5 x+3 x^2\right )^{3/2}+\frac{8 \int \sqrt{x} \left (\frac{16215}{2}+\frac{58935 x}{2}\right ) \sqrt{2+5 x+3 x^2} \, dx}{34749}\\ &=\frac{157160 \sqrt{x} \left (2+5 x+3 x^2\right )^{3/2}}{243243}-\frac{21620 x^{3/2} \left (2+5 x+3 x^2\right )^{3/2}}{34749}+\frac{656 x^{5/2} \left (2+5 x+3 x^2\right )^{3/2}}{1287}-\frac{10}{39} x^{7/2} \left (2+5 x+3 x^2\right )^{3/2}+\frac{16 \int \frac{\left (-\frac{58935}{2}-\frac{838185 x}{4}\right ) \sqrt{2+5 x+3 x^2}}{\sqrt{x}} \, dx}{729729}\\ &=-\frac{8 \sqrt{x} (397265+502911 x) \sqrt{2+5 x+3 x^2}}{2189187}+\frac{157160 \sqrt{x} \left (2+5 x+3 x^2\right )^{3/2}}{243243}-\frac{21620 x^{3/2} \left (2+5 x+3 x^2\right )^{3/2}}{34749}+\frac{656 x^{5/2} \left (2+5 x+3 x^2\right )^{3/2}}{1287}-\frac{10}{39} x^{7/2} \left (2+5 x+3 x^2\right )^{3/2}-\frac{32 \int \frac{-\frac{654825}{4}-\frac{723585 x}{2}}{\sqrt{x} \sqrt{2+5 x+3 x^2}} \, dx}{32837805}\\ &=-\frac{8 \sqrt{x} (397265+502911 x) \sqrt{2+5 x+3 x^2}}{2189187}+\frac{157160 \sqrt{x} \left (2+5 x+3 x^2\right )^{3/2}}{243243}-\frac{21620 x^{3/2} \left (2+5 x+3 x^2\right )^{3/2}}{34749}+\frac{656 x^{5/2} \left (2+5 x+3 x^2\right )^{3/2}}{1287}-\frac{10}{39} x^{7/2} \left (2+5 x+3 x^2\right )^{3/2}-\frac{64 \operatorname{Subst}\left (\int \frac{-\frac{654825}{4}-\frac{723585 x^2}{2}}{\sqrt{2+5 x^2+3 x^4}} \, dx,x,\sqrt{x}\right )}{32837805}\\ &=-\frac{8 \sqrt{x} (397265+502911 x) \sqrt{2+5 x+3 x^2}}{2189187}+\frac{157160 \sqrt{x} \left (2+5 x+3 x^2\right )^{3/2}}{243243}-\frac{21620 x^{3/2} \left (2+5 x+3 x^2\right )^{3/2}}{34749}+\frac{656 x^{5/2} \left (2+5 x+3 x^2\right )^{3/2}}{1287}-\frac{10}{39} x^{7/2} \left (2+5 x+3 x^2\right )^{3/2}+\frac{698480 \operatorname{Subst}\left (\int \frac{1}{\sqrt{2+5 x^2+3 x^4}} \, dx,x,\sqrt{x}\right )}{2189187}+\frac{1543648 \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{2+5 x^2+3 x^4}} \, dx,x,\sqrt{x}\right )}{2189187}\\ &=\frac{1543648 \sqrt{x} (2+3 x)}{6567561 \sqrt{2+5 x+3 x^2}}-\frac{8 \sqrt{x} (397265+502911 x) \sqrt{2+5 x+3 x^2}}{2189187}+\frac{157160 \sqrt{x} \left (2+5 x+3 x^2\right )^{3/2}}{243243}-\frac{21620 x^{3/2} \left (2+5 x+3 x^2\right )^{3/2}}{34749}+\frac{656 x^{5/2} \left (2+5 x+3 x^2\right )^{3/2}}{1287}-\frac{10}{39} x^{7/2} \left (2+5 x+3 x^2\right )^{3/2}-\frac{1543648 \sqrt{2} (1+x) \sqrt{\frac{2+3 x}{1+x}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{6567561 \sqrt{2+5 x+3 x^2}}+\frac{349240 \sqrt{2} (1+x) \sqrt{\frac{2+3 x}{1+x}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{2189187 \sqrt{2+5 x+3 x^2}}\\ \end{align*}
Mathematica [C] time = 0.224669, size = 178, normalized size = 0.71 \[ \frac{-495928 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{3/2} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right ),\frac{3}{2}\right )+2 \left (-7577955 x^8-10195794 x^7+671895 x^6+2892348 x^5+58374 x^4-141444 x^3+670548 x^2+2811400 x+1543648\right )+1543648 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{3/2} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )}{6567561 \sqrt{x} \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.028, size = 137, normalized size = 0.6 \begin{align*} -{\frac{2}{19702683} \left ( 22733865\,{x}^{8}+30587382\,{x}^{7}-2015685\,{x}^{6}-8677044\,{x}^{5}+633876\,\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 ) -385912\,\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 ) -175122\,{x}^{4}+424332\,{x}^{3}+4934772\,{x}^{2}+3143160\,x \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 \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 (-{\left (5 \, x^{4} - 2 \, x^{3}\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} \sqrt{x}, 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 -\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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