Optimal. Leaf size=175 \[ \frac{20501 \sqrt{-3 x^2-5 x-2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right ),-\frac{2}{3}\right )}{2268 \sqrt{3} \sqrt{3 x^2+5 x+2}}+\frac{1}{63} (52-7 x) \sqrt{2 x+3} \left (3 x^2+5 x+2\right )^{3/2}-\frac{\sqrt{2 x+3} (12429 x+107) \sqrt{3 x^2+5 x+2}}{5670}-\frac{11123 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{1620 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]
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Rubi [A] time = 0.105165, antiderivative size = 175, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {814, 843, 718, 424, 419} \[ \frac{1}{63} (52-7 x) \sqrt{2 x+3} \left (3 x^2+5 x+2\right )^{3/2}-\frac{\sqrt{2 x+3} (12429 x+107) \sqrt{3 x^2+5 x+2}}{5670}+\frac{20501 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{2268 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{11123 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{1620 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Rule 814
Rule 843
Rule 718
Rule 424
Rule 419
Rubi steps
\begin{align*} \int \frac{(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{\sqrt{3+2 x}} \, dx &=\frac{1}{63} (52-7 x) \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{3/2}-\frac{1}{126} \int \frac{(1204+1381 x) \sqrt{2+5 x+3 x^2}}{\sqrt{3+2 x}} \, dx\\ &=-\frac{\sqrt{3+2 x} (107+12429 x) \sqrt{2+5 x+3 x^2}}{5670}+\frac{1}{63} (52-7 x) \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{3/2}+\frac{\int \frac{-65539-77861 x}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx}{11340}\\ &=-\frac{\sqrt{3+2 x} (107+12429 x) \sqrt{2+5 x+3 x^2}}{5670}+\frac{1}{63} (52-7 x) \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{3/2}-\frac{11123 \int \frac{\sqrt{3+2 x}}{\sqrt{2+5 x+3 x^2}} \, dx}{3240}+\frac{20501 \int \frac{1}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx}{4536}\\ &=-\frac{\sqrt{3+2 x} (107+12429 x) \sqrt{2+5 x+3 x^2}}{5670}+\frac{1}{63} (52-7 x) \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{3/2}-\frac{\left (11123 \sqrt{-2-5 x-3 x^2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{2 x^2}{3}}}{\sqrt{1-x^2}} \, dx,x,\frac{\sqrt{6+6 x}}{\sqrt{2}}\right )}{1620 \sqrt{3} \sqrt{2+5 x+3 x^2}}+\frac{\left (20501 \sqrt{-2-5 x-3 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sqrt{1+\frac{2 x^2}{3}}} \, dx,x,\frac{\sqrt{6+6 x}}{\sqrt{2}}\right )}{2268 \sqrt{3} \sqrt{2+5 x+3 x^2}}\\ &=-\frac{\sqrt{3+2 x} (107+12429 x) \sqrt{2+5 x+3 x^2}}{5670}+\frac{1}{63} (52-7 x) \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{3/2}-\frac{11123 \sqrt{-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{1620 \sqrt{3} \sqrt{2+5 x+3 x^2}}+\frac{20501 \sqrt{-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{2268 \sqrt{3} \sqrt{2+5 x+3 x^2}}\\ \end{align*}
Mathematica [A] time = 0.345127, size = 203, normalized size = 1.16 \[ -\frac{-16358 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^2 \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right ),\frac{3}{5}\right )+2 \left (34020 x^6-88290 x^5-687798 x^4-1306791 x^3-1043385 x^2-312914 x-10832\right ) \sqrt{2 x+3}+77861 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^2 E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )}{34020 (2 x+3) \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 151, normalized size = 0.9 \begin{align*}{\frac{1}{2041200\,{x}^{3}+6463800\,{x}^{2}+6463800\,x+2041200}\sqrt{3+2\,x}\sqrt{3\,{x}^{2}+5\,x+2} \left ( -680400\,{x}^{6}+1765800\,{x}^{5}+24644\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-20-30\,x}{\it EllipticF} \left ( 1/5\,\sqrt{30\,x+45},1/3\,\sqrt{15} \right ) +77861\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-20-30\,x}{\it EllipticE} \left ( 1/5\,\sqrt{30\,x+45},1/3\,\sqrt{15} \right ) +13755960\,{x}^{4}+26135820\,{x}^{3}+25539360\,{x}^{2}+14044380\,x+3331080 \right ) } \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 (x - 5\right )}}{\sqrt{2 \, x + 3}}\,{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 (3 \, x^{3} - 10 \, x^{2} - 23 \, x - 10\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}}{\sqrt{2 \, x + 3}}, 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{10 \sqrt{3 x^{2} + 5 x + 2}}{\sqrt{2 x + 3}}\, dx - \int - \frac{23 x \sqrt{3 x^{2} + 5 x + 2}}{\sqrt{2 x + 3}}\, dx - \int - \frac{10 x^{2} \sqrt{3 x^{2} + 5 x + 2}}{\sqrt{2 x + 3}}\, dx - \int \frac{3 x^{3} \sqrt{3 x^{2} + 5 x + 2}}{\sqrt{2 x + 3}}\, 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 (x - 5\right )}}{\sqrt{2 \, x + 3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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