Optimal. Leaf size=222 \[ -\frac{33232}{35} \sqrt{\frac{3}{11}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )+\frac{301304 \sqrt{1-2 x} \sqrt{3 x+2}}{21 \sqrt{5 x+3}}-\frac{16616 \sqrt{1-2 x} \sqrt{3 x+2}}{7 (5 x+3)^{3/2}}+\frac{111884 \sqrt{1-2 x}}{315 \sqrt{3 x+2} (5 x+3)^{3/2}}+\frac{536 \sqrt{1-2 x}}{45 (3 x+2)^{3/2} (5 x+3)^{3/2}}+\frac{14 \sqrt{1-2 x}}{15 (3 x+2)^{5/2} (5 x+3)^{3/2}}-\frac{301304}{35} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
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Rubi [A] time = 0.0813914, antiderivative size = 222, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {98, 152, 158, 113, 119} \[ \frac{301304 \sqrt{1-2 x} \sqrt{3 x+2}}{21 \sqrt{5 x+3}}-\frac{16616 \sqrt{1-2 x} \sqrt{3 x+2}}{7 (5 x+3)^{3/2}}+\frac{111884 \sqrt{1-2 x}}{315 \sqrt{3 x+2} (5 x+3)^{3/2}}+\frac{536 \sqrt{1-2 x}}{45 (3 x+2)^{3/2} (5 x+3)^{3/2}}+\frac{14 \sqrt{1-2 x}}{15 (3 x+2)^{5/2} (5 x+3)^{3/2}}-\frac{33232}{35} \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{301304}{35} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
Antiderivative was successfully verified.
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Rule 98
Rule 152
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{3/2}}{(2+3 x)^{7/2} (3+5 x)^{5/2}} \, dx &=\frac{14 \sqrt{1-2 x}}{15 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac{2}{15} \int \frac{156-235 x}{\sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}} \, dx\\ &=\frac{14 \sqrt{1-2 x}}{15 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac{536 \sqrt{1-2 x}}{45 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac{4}{315} \int \frac{\frac{33999}{2}-23450 x}{\sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx\\ &=\frac{14 \sqrt{1-2 x}}{15 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac{536 \sqrt{1-2 x}}{45 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac{111884 \sqrt{1-2 x}}{315 \sqrt{2+3 x} (3+5 x)^{3/2}}+\frac{8 \int \frac{1277955-\frac{2936955 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}} \, dx}{2205}\\ &=\frac{14 \sqrt{1-2 x}}{15 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac{536 \sqrt{1-2 x}}{45 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac{111884 \sqrt{1-2 x}}{315 \sqrt{2+3 x} (3+5 x)^{3/2}}-\frac{16616 \sqrt{1-2 x} \sqrt{2+3 x}}{7 (3+5 x)^{3/2}}-\frac{16 \int \frac{\frac{209379555}{4}-\frac{64771245 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}} \, dx}{72765}\\ &=\frac{14 \sqrt{1-2 x}}{15 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac{536 \sqrt{1-2 x}}{45 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac{111884 \sqrt{1-2 x}}{315 \sqrt{2+3 x} (3+5 x)^{3/2}}-\frac{16616 \sqrt{1-2 x} \sqrt{2+3 x}}{7 (3+5 x)^{3/2}}+\frac{301304 \sqrt{1-2 x} \sqrt{2+3 x}}{21 \sqrt{3+5 x}}+\frac{32 \int \frac{681610545+\frac{4306575735 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{800415}\\ &=\frac{14 \sqrt{1-2 x}}{15 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac{536 \sqrt{1-2 x}}{45 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac{111884 \sqrt{1-2 x}}{315 \sqrt{2+3 x} (3+5 x)^{3/2}}-\frac{16616 \sqrt{1-2 x} \sqrt{2+3 x}}{7 (3+5 x)^{3/2}}+\frac{301304 \sqrt{1-2 x} \sqrt{2+3 x}}{21 \sqrt{3+5 x}}+\frac{49848}{35} \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx+\frac{301304}{35} \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx\\ &=\frac{14 \sqrt{1-2 x}}{15 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac{536 \sqrt{1-2 x}}{45 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac{111884 \sqrt{1-2 x}}{315 \sqrt{2+3 x} (3+5 x)^{3/2}}-\frac{16616 \sqrt{1-2 x} \sqrt{2+3 x}}{7 (3+5 x)^{3/2}}+\frac{301304 \sqrt{1-2 x} \sqrt{2+3 x}}{21 \sqrt{3+5 x}}-\frac{301304}{35} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{33232}{35} \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )\\ \end{align*}
Mathematica [A] time = 0.258877, size = 109, normalized size = 0.49 \[ \frac{2}{105} \left (4 \sqrt{2} \left (37663 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-18970 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )\right )+\frac{\sqrt{1-2 x} \left (101690100 x^4+261029520 x^3+251053266 x^2+107221804 x+17157169\right )}{(3 x+2)^{5/2} (5 x+3)^{3/2}}\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.024, size = 406, normalized size = 1.8 \begin{align*}{\frac{2}{210\,x-105}\sqrt{1-2\,x} \left ( 3414600\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{3}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-6779340\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{3}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+6601560\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-13106724\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+4249280\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-8436512\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+910560\,\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 ) -1807824\,\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 ) +203380200\,{x}^{5}+420368940\,{x}^{4}+241077012\,{x}^{3}-36609658\,{x}^{2}-72907466\,x-17157169 \right ) \left ( 2+3\,x \right ) ^{-{\frac{5}{2}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{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 (-2 \, x + 1\right )}^{\frac{3}{2}}}{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (3 \, x + 2\right )}^{\frac{7}{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{\sqrt{5 \, x + 3} \sqrt{3 \, x + 2}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}{10125 \, x^{7} + 45225 \, x^{6} + 86535 \, x^{5} + 91947 \, x^{4} + 58592 \, x^{3} + 22392 \, x^{2} + 4752 \, x + 432}, 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 (-2 \, x + 1\right )}^{\frac{3}{2}}}{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (3 \, x + 2\right )}^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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