Optimal. Leaf size=218 \[ -\frac{1282376 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{41503 \sqrt{33}}+\frac{213119320 \sqrt{1-2 x} \sqrt{3 x+2}}{1369599 \sqrt{5 x+3}}-\frac{3205940 \sqrt{1-2 x} \sqrt{3 x+2}}{124509 (5 x+3)^{3/2}}+\frac{14496 \sqrt{1-2 x}}{3773 \sqrt{3 x+2} (5 x+3)^{3/2}}+\frac{54 \sqrt{1-2 x}}{539 (3 x+2)^{3/2} (5 x+3)^{3/2}}+\frac{4}{77 \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}}-\frac{42623864 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{41503 \sqrt{33}} \]
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Rubi [A] time = 0.0794264, antiderivative size = 218, 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 = {104, 152, 158, 113, 119} \[ \frac{213119320 \sqrt{1-2 x} \sqrt{3 x+2}}{1369599 \sqrt{5 x+3}}-\frac{3205940 \sqrt{1-2 x} \sqrt{3 x+2}}{124509 (5 x+3)^{3/2}}+\frac{14496 \sqrt{1-2 x}}{3773 \sqrt{3 x+2} (5 x+3)^{3/2}}+\frac{54 \sqrt{1-2 x}}{539 (3 x+2)^{3/2} (5 x+3)^{3/2}}+\frac{4}{77 \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}}-\frac{1282376 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{41503 \sqrt{33}}-\frac{42623864 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{41503 \sqrt{33}} \]
Antiderivative was successfully verified.
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Rule 104
Rule 152
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{1}{(1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{5/2}} \, dx &=\frac{4}{77 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}-\frac{2}{77} \int \frac{-\frac{167}{2}-105 x}{\sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}} \, dx\\ &=\frac{4}{77 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac{54 \sqrt{1-2 x}}{539 (2+3 x)^{3/2} (3+5 x)^{3/2}}-\frac{4 \int \frac{-1137+\frac{2025 x}{2}}{\sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx}{1617}\\ &=\frac{4}{77 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac{54 \sqrt{1-2 x}}{539 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac{14496 \sqrt{1-2 x}}{3773 \sqrt{2+3 x} (3+5 x)^{3/2}}-\frac{8 \int \frac{-\frac{285195}{4}+81540 x}{\sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}} \, dx}{11319}\\ &=\frac{4}{77 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac{54 \sqrt{1-2 x}}{539 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac{14496 \sqrt{1-2 x}}{3773 \sqrt{2+3 x} (3+5 x)^{3/2}}-\frac{3205940 \sqrt{1-2 x} \sqrt{2+3 x}}{124509 (3+5 x)^{3/2}}+\frac{16 \int \frac{-\frac{5827965}{2}+\frac{7213365 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}} \, dx}{373527}\\ &=\frac{4}{77 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac{54 \sqrt{1-2 x}}{539 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac{14496 \sqrt{1-2 x}}{3773 \sqrt{2+3 x} (3+5 x)^{3/2}}-\frac{3205940 \sqrt{1-2 x} \sqrt{2+3 x}}{124509 (3+5 x)^{3/2}}+\frac{213119320 \sqrt{1-2 x} \sqrt{2+3 x}}{1369599 \sqrt{3+5 x}}-\frac{32 \int \frac{-\frac{303580485}{8}-\frac{239759235 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{4108797}\\ &=\frac{4}{77 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac{54 \sqrt{1-2 x}}{539 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac{14496 \sqrt{1-2 x}}{3773 \sqrt{2+3 x} (3+5 x)^{3/2}}-\frac{3205940 \sqrt{1-2 x} \sqrt{2+3 x}}{124509 (3+5 x)^{3/2}}+\frac{213119320 \sqrt{1-2 x} \sqrt{2+3 x}}{1369599 \sqrt{3+5 x}}+\frac{641188 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{41503}+\frac{42623864 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{456533}\\ &=\frac{4}{77 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac{54 \sqrt{1-2 x}}{539 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac{14496 \sqrt{1-2 x}}{3773 \sqrt{2+3 x} (3+5 x)^{3/2}}-\frac{3205940 \sqrt{1-2 x} \sqrt{2+3 x}}{124509 (3+5 x)^{3/2}}+\frac{213119320 \sqrt{1-2 x} \sqrt{2+3 x}}{1369599 \sqrt{3+5 x}}-\frac{42623864 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{41503 \sqrt{33}}-\frac{1282376 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{41503 \sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.193001, size = 109, normalized size = 0.5 \[ \frac{2 \left (2 \sqrt{2} \left (10655966 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-5366165 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )\right )+\frac{-9590369400 x^4-13428808080 x^3-2415287594 x^2+3336610202 x+1213551469}{\sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}}\right )}{1369599} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.026, size = 311, normalized size = 1.4 \begin{align*}{\frac{2}{2739198\,x-1369599}\sqrt{1-2\,x} \left ( 160984950\,\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}-319678980\,\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}+203914270\,\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}-404926708\,\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}+64393980\,\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 ) -127871592\,\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 ) +9590369400\,{x}^{4}+13428808080\,{x}^{3}+2415287594\,{x}^{2}-3336610202\,x-1213551469 \right ) \left ( 2+3\,x \right ) ^{-{\frac{3}{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{1}{{\left (5 \, x + 3\right )}^{\frac{5}{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{\sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{13500 \, x^{8} + 37800 \, x^{7} + 33255 \, x^{6} + 121 \, x^{5} - 15709 \, x^{4} - 7145 \, x^{3} + 774 \, x^{2} + 1188 \, x + 216}, 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{1}{{\left (5 \, x + 3\right )}^{\frac{5}{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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