Optimal. Leaf size=191 \[ -\frac{29357 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{85050}+\frac{2}{45} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{7/2}-\frac{31}{945} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}-\frac{223}{945} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}-\frac{29357 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{17010}-\frac{488149 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{42525} \]
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Rubi [A] time = 0.0676175, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {101, 154, 158, 113, 119} \[ \frac{2}{45} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{7/2}-\frac{31}{945} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}-\frac{223}{945} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}-\frac{29357 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{17010}-\frac{29357 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{85050}-\frac{488149 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{42525} \]
Antiderivative was successfully verified.
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Rule 101
Rule 154
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2} \, dx &=\frac{2}{45} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{7/2}-\frac{2}{45} \int \frac{\left (-\frac{23}{2}-\frac{31 x}{2}\right ) (3+5 x)^{5/2}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx\\ &=-\frac{31}{945} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}+\frac{2}{45} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{7/2}+\frac{2}{945} \int \frac{(3+5 x)^{3/2} \left (\frac{4355}{4}+\frac{3345 x}{2}\right )}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx\\ &=-\frac{223}{945} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}-\frac{31}{945} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}+\frac{2}{45} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{7/2}-\frac{2 \int \frac{\left (-\frac{143145}{2}-\frac{440355 x}{4}\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{14175}\\ &=-\frac{29357 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{17010}-\frac{223}{945} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}-\frac{31}{945} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}+\frac{2}{45} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{7/2}+\frac{2 \int \frac{\frac{18542145}{8}+\frac{7322235 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{127575}\\ &=-\frac{29357 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{17010}-\frac{223}{945} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}-\frac{31}{945} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}+\frac{2}{45} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{7/2}+\frac{322927 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{170100}+\frac{488149 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{42525}\\ &=-\frac{29357 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{17010}-\frac{223}{945} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}-\frac{31}{945} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}+\frac{2}{45} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{7/2}-\frac{488149 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{42525}-\frac{29357 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{85050}\\ \end{align*}
Mathematica [A] time = 0.237443, size = 102, normalized size = 0.53 \[ \frac{-983815 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+15 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (94500 x^3+156150 x^2+65250 x-26009\right )+1952596 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{255150 \sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.01, size = 155, normalized size = 0.8 \begin{align*}{\frac{1}{15309000\,{x}^{3}+11736900\,{x}^{2}-3572100\,x-3061800}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 85050000\,{x}^{6}+983815\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) -1952596\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) +205740000\,{x}^{5}+146623500\,{x}^{4}-28187100\,{x}^{3}-59755710\,{x}^{2}-6283110\,x+4681620 \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{\left (5 \, x + 3\right )}^{\frac{5}{2}} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}\,{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 (25 \, x^{2} + 30 \, x + 9\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}, 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{\left (5 \, x + 3\right )}^{\frac{5}{2}} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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