Optimal. Leaf size=188 \[ \frac{9694}{875} \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )+\frac{(5 x+3)^{3/2} (3 x+2)^{5/2}}{\sqrt{1-2 x}}+\frac{12}{7} \sqrt{1-2 x} (5 x+3)^{3/2} (3 x+2)^{3/2}+\frac{2511}{350} \sqrt{1-2 x} (5 x+3)^{3/2} \sqrt{3 x+2}+\frac{9694}{175} \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}+\frac{1289089 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3500} \]
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Rubi [A] time = 0.0587716, antiderivative size = 188, 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 = {97, 154, 158, 113, 119} \[ \frac{(5 x+3)^{3/2} (3 x+2)^{5/2}}{\sqrt{1-2 x}}+\frac{12}{7} \sqrt{1-2 x} (5 x+3)^{3/2} (3 x+2)^{3/2}+\frac{2511}{350} \sqrt{1-2 x} (5 x+3)^{3/2} \sqrt{3 x+2}+\frac{9694}{175} \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}+\frac{9694}{875} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{1289089 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3500} \]
Antiderivative was successfully verified.
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Rule 97
Rule 154
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(2+3 x)^{5/2} (3+5 x)^{3/2}}{(1-2 x)^{3/2}} \, dx &=\frac{(2+3 x)^{5/2} (3+5 x)^{3/2}}{\sqrt{1-2 x}}-\int \frac{(2+3 x)^{3/2} \sqrt{3+5 x} \left (\frac{75}{2}+60 x\right )}{\sqrt{1-2 x}} \, dx\\ &=\frac{12}{7} \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}+\frac{(2+3 x)^{5/2} (3+5 x)^{3/2}}{\sqrt{1-2 x}}+\frac{1}{35} \int \frac{\left (-3975-\frac{12555 x}{2}\right ) \sqrt{2+3 x} \sqrt{3+5 x}}{\sqrt{1-2 x}} \, dx\\ &=\frac{2511}{350} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}+\frac{12}{7} \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}+\frac{(2+3 x)^{5/2} (3+5 x)^{3/2}}{\sqrt{1-2 x}}-\frac{1}{875} \int \frac{\sqrt{3+5 x} \left (\frac{1133985}{4}+436230 x\right )}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx\\ &=\frac{9694}{175} \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}+\frac{2511}{350} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}+\frac{12}{7} \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}+\frac{(2+3 x)^{5/2} (3+5 x)^{3/2}}{\sqrt{1-2 x}}+\frac{\int \frac{-\frac{36724815}{4}-\frac{58009005 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{7875}\\ &=\frac{9694}{175} \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}+\frac{2511}{350} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}+\frac{12}{7} \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}+\frac{(2+3 x)^{5/2} (3+5 x)^{3/2}}{\sqrt{1-2 x}}-\frac{53317}{875} \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx-\frac{1289089 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{3500}\\ &=\frac{9694}{175} \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}+\frac{2511}{350} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}+\frac{12}{7} \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}+\frac{(2+3 x)^{5/2} (3+5 x)^{3/2}}{\sqrt{1-2 x}}+\frac{1289089 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3500}+\frac{9694}{875} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )\\ \end{align*}
Mathematica [A] time = 0.211968, size = 115, normalized size = 0.61 \[ \frac{649285 \sqrt{2-4 x} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )-30 \sqrt{3 x+2} \sqrt{5 x+3} \left (2250 x^3+8460 x^2+17487 x-34721\right )-1289089 \sqrt{2-4 x} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{10500 \sqrt{1-2 x}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.016, size = 150, normalized size = 0.8 \begin{align*} -{\frac{1}{315000\,{x}^{3}+241500\,{x}^{2}-73500\,x-63000}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 649285\,\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 ) -1289089\,\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 ) -1012500\,{x}^{5}-5089500\,{x}^{4}-13096350\,{x}^{3}+4134060\,{x}^{2}+16643310\,x+6249780 \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 (5 \, x + 3\right )}^{\frac{3}{2}}{\left (3 \, x + 2\right )}^{\frac{5}{2}}}{{\left (-2 \, x + 1\right )}^{\frac{3}{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 (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{4 \, x^{2} - 4 \, 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 \frac{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (3 \, x + 2\right )}^{\frac{5}{2}}}{{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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