Optimal. Leaf size=191 \[ -\frac{110014 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{328125}-\frac{2 (3 x+2)^{3/2} (1-2 x)^{5/2}}{5 \sqrt{5 x+3}}-\frac{32}{175} (3 x+2)^{3/2} \sqrt{5 x+3} (1-2 x)^{3/2}-\frac{1972 (3 x+2)^{3/2} \sqrt{5 x+3} \sqrt{1-2 x}}{4375}+\frac{106772 \sqrt{3 x+2} \sqrt{5 x+3} \sqrt{1-2 x}}{65625}+\frac{53279 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{328125} \]
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Rubi [A] time = 0.067703, 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 = {97, 154, 158, 113, 119} \[ -\frac{2 (3 x+2)^{3/2} (1-2 x)^{5/2}}{5 \sqrt{5 x+3}}-\frac{32}{175} (3 x+2)^{3/2} \sqrt{5 x+3} (1-2 x)^{3/2}-\frac{1972 (3 x+2)^{3/2} \sqrt{5 x+3} \sqrt{1-2 x}}{4375}+\frac{106772 \sqrt{3 x+2} \sqrt{5 x+3} \sqrt{1-2 x}}{65625}-\frac{110014 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{328125}+\frac{53279 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{328125} \]
Antiderivative was successfully verified.
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Rule 97
Rule 154
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{5/2} (2+3 x)^{3/2}}{(3+5 x)^{3/2}} \, dx &=-\frac{2 (1-2 x)^{5/2} (2+3 x)^{3/2}}{5 \sqrt{3+5 x}}+\frac{2}{5} \int \frac{\left (-\frac{11}{2}-24 x\right ) (1-2 x)^{3/2} \sqrt{2+3 x}}{\sqrt{3+5 x}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2} (2+3 x)^{3/2}}{5 \sqrt{3+5 x}}-\frac{32}{175} (1-2 x)^{3/2} (2+3 x)^{3/2} \sqrt{3+5 x}+\frac{4}{525} \int \frac{\left (-\frac{1107}{4}-\frac{4437 x}{2}\right ) \sqrt{1-2 x} \sqrt{2+3 x}}{\sqrt{3+5 x}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2} (2+3 x)^{3/2}}{5 \sqrt{3+5 x}}-\frac{1972 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{4375}-\frac{32}{175} (1-2 x)^{3/2} (2+3 x)^{3/2} \sqrt{3+5 x}+\frac{8 \int \frac{\left (\frac{138825}{8}-\frac{240237 x}{2}\right ) \sqrt{2+3 x}}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{39375}\\ &=-\frac{2 (1-2 x)^{5/2} (2+3 x)^{3/2}}{5 \sqrt{3+5 x}}+\frac{106772 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{65625}-\frac{1972 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{4375}-\frac{32}{175} (1-2 x)^{3/2} (2+3 x)^{3/2} \sqrt{3+5 x}-\frac{8 \int \frac{-100179+\frac{479511 x}{8}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{590625}\\ &=-\frac{2 (1-2 x)^{5/2} (2+3 x)^{3/2}}{5 \sqrt{3+5 x}}+\frac{106772 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{65625}-\frac{1972 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{4375}-\frac{32}{175} (1-2 x)^{3/2} (2+3 x)^{3/2} \sqrt{3+5 x}-\frac{53279 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{328125}+\frac{605077 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{328125}\\ &=-\frac{2 (1-2 x)^{5/2} (2+3 x)^{3/2}}{5 \sqrt{3+5 x}}+\frac{106772 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{65625}-\frac{1972 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{4375}-\frac{32}{175} (1-2 x)^{3/2} (2+3 x)^{3/2} \sqrt{3+5 x}+\frac{53279 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{328125}-\frac{110014 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{328125}\\ \end{align*}
Mathematica [A] time = 0.293876, size = 107, normalized size = 0.56 \[ \frac{1868510 \sqrt{2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+\frac{30 \sqrt{1-2 x} \sqrt{3 x+2} \left (22500 x^3-31350 x^2+9545 x+9168\right )}{\sqrt{5 x+3}}-53279 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{984375} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.016, size = 150, normalized size = 0.8 \begin{align*} -{\frac{1}{29531250\,{x}^{3}+22640625\,{x}^{2}-6890625\,x-5906250}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 1868510\,\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 ) -53279\,\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 ) -4050000\,{x}^{5}+4968000\,{x}^{4}+572400\,{x}^{3}-3817590\,{x}^{2}+297660\,x+550080 \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\right )}^{\frac{3}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (5 \, x + 3\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 (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{25 \, x^{2} + 30 \, x + 9}, 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 (3 \, x + 2\right )}^{\frac{3}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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