Optimal. Leaf size=200 \[ \frac{412 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} \text{EllipticF}\left (\tan ^{-1}\left (\sqrt{x}\right ),-\frac{1}{2}\right )}{189 \sqrt{3 x^2+5 x+2}}-\frac{10}{21} \sqrt{3 x^2+5 x+2} x^{5/2}+\frac{128}{105} \sqrt{3 x^2+5 x+2} x^{3/2}-\frac{412}{189} \sqrt{3 x^2+5 x+2} \sqrt{x}+\frac{13688 (3 x+2) \sqrt{x}}{2835 \sqrt{3 x^2+5 x+2}}-\frac{13688 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{2835 \sqrt{3 x^2+5 x+2}} \]
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Rubi [A] time = 0.143127, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {832, 839, 1189, 1100, 1136} \[ -\frac{10}{21} \sqrt{3 x^2+5 x+2} x^{5/2}+\frac{128}{105} \sqrt{3 x^2+5 x+2} x^{3/2}-\frac{412}{189} \sqrt{3 x^2+5 x+2} \sqrt{x}+\frac{13688 (3 x+2) \sqrt{x}}{2835 \sqrt{3 x^2+5 x+2}}+\frac{412 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{189 \sqrt{3 x^2+5 x+2}}-\frac{13688 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{2835 \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Rule 832
Rule 839
Rule 1189
Rule 1100
Rule 1136
Rubi steps
\begin{align*} \int \frac{(2-5 x) x^{5/2}}{\sqrt{2+5 x+3 x^2}} \, dx &=-\frac{10}{21} x^{5/2} \sqrt{2+5 x+3 x^2}+\frac{2}{21} \int \frac{x^{3/2} (25+96 x)}{\sqrt{2+5 x+3 x^2}} \, dx\\ &=\frac{128}{105} x^{3/2} \sqrt{2+5 x+3 x^2}-\frac{10}{21} x^{5/2} \sqrt{2+5 x+3 x^2}+\frac{4}{315} \int \frac{\left (-288-\frac{1545 x}{2}\right ) \sqrt{x}}{\sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{412}{189} \sqrt{x} \sqrt{2+5 x+3 x^2}+\frac{128}{105} x^{3/2} \sqrt{2+5 x+3 x^2}-\frac{10}{21} x^{5/2} \sqrt{2+5 x+3 x^2}+\frac{8 \int \frac{\frac{1545}{2}+\frac{5133 x}{2}}{\sqrt{x} \sqrt{2+5 x+3 x^2}} \, dx}{2835}\\ &=-\frac{412}{189} \sqrt{x} \sqrt{2+5 x+3 x^2}+\frac{128}{105} x^{3/2} \sqrt{2+5 x+3 x^2}-\frac{10}{21} x^{5/2} \sqrt{2+5 x+3 x^2}+\frac{16 \operatorname{Subst}\left (\int \frac{\frac{1545}{2}+\frac{5133 x^2}{2}}{\sqrt{2+5 x^2+3 x^4}} \, dx,x,\sqrt{x}\right )}{2835}\\ &=-\frac{412}{189} \sqrt{x} \sqrt{2+5 x+3 x^2}+\frac{128}{105} x^{3/2} \sqrt{2+5 x+3 x^2}-\frac{10}{21} x^{5/2} \sqrt{2+5 x+3 x^2}+\frac{824}{189} \operatorname{Subst}\left (\int \frac{1}{\sqrt{2+5 x^2+3 x^4}} \, dx,x,\sqrt{x}\right )+\frac{13688}{945} \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{2+5 x^2+3 x^4}} \, dx,x,\sqrt{x}\right )\\ &=\frac{13688 \sqrt{x} (2+3 x)}{2835 \sqrt{2+5 x+3 x^2}}-\frac{412}{189} \sqrt{x} \sqrt{2+5 x+3 x^2}+\frac{128}{105} x^{3/2} \sqrt{2+5 x+3 x^2}-\frac{10}{21} x^{5/2} \sqrt{2+5 x+3 x^2}-\frac{13688 \sqrt{2} (1+x) \sqrt{\frac{2+3 x}{1+x}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{2835 \sqrt{2+5 x+3 x^2}}+\frac{412 \sqrt{2} (1+x) \sqrt{\frac{2+3 x}{1+x}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{189 \sqrt{2+5 x+3 x^2}}\\ \end{align*}
Mathematica [C] time = 0.175812, size = 160, normalized size = 0.8 \[ \frac{-7508 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{3/2} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right ),\frac{3}{2}\right )-4050 x^5+3618 x^4-3960 x^3+17076 x^2+13688 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{3/2} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )+56080 x+27376}{2835 \sqrt{x} \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 122, normalized size = 0.6 \begin{align*} -{\frac{2}{8505} \left ( 6075\,{x}^{5}+7176\,\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 ) -3422\,\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 ) -5427\,{x}^{4}+5940\,{x}^{3}+35982\,{x}^{2}+18540\,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 \frac{{\left (5 \, x - 2\right )} x^{\frac{5}{2}}}{\sqrt{3 \, x^{2} + 5 \, x + 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 (5 \, x^{3} - 2 \, x^{2}\right )} \sqrt{x}}{\sqrt{3 \, x^{2} + 5 \, x + 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 x^{\frac{5}{2}}}{\sqrt{3 x^{2} + 5 x + 2}}\, dx - \int \frac{5 x^{\frac{7}{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 (5 \, x - 2\right )} x^{\frac{5}{2}}}{\sqrt{3 \, x^{2} + 5 \, x + 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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