Optimal. Leaf size=191 \[ \frac{6584 \sqrt{\frac{3}{11}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{3773}-\frac{1100380 \sqrt{1-2 x} \sqrt{3 x+2}}{41503 \sqrt{5 x+3}}+\frac{9876 \sqrt{1-2 x}}{3773 \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{54 \sqrt{1-2 x}}{539 (3 x+2)^{3/2} \sqrt{5 x+3}}+\frac{4}{77 \sqrt{1-2 x} (3 x+2)^{3/2} \sqrt{5 x+3}}+\frac{220076 \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3773} \]
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Rubi [A] time = 0.0672827, 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 = {104, 152, 158, 113, 119} \[ -\frac{1100380 \sqrt{1-2 x} \sqrt{3 x+2}}{41503 \sqrt{5 x+3}}+\frac{9876 \sqrt{1-2 x}}{3773 \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{54 \sqrt{1-2 x}}{539 (3 x+2)^{3/2} \sqrt{5 x+3}}+\frac{4}{77 \sqrt{1-2 x} (3 x+2)^{3/2} \sqrt{5 x+3}}+\frac{6584 \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3773}+\frac{220076 \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3773} \]
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)^{3/2}} \, dx &=\frac{4}{77 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}-\frac{2}{77} \int \frac{-\frac{127}{2}-75 x}{\sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx\\ &=\frac{4}{77 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{54 \sqrt{1-2 x}}{539 (2+3 x)^{3/2} \sqrt{3+5 x}}-\frac{4 \int \frac{-\frac{1659}{2}+\frac{1215 x}{2}}{\sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx}{1617}\\ &=\frac{4}{77 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{54 \sqrt{1-2 x}}{539 (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{9876 \sqrt{1-2 x}}{3773 \sqrt{2+3 x} \sqrt{3+5 x}}-\frac{8 \int \frac{-\frac{120615}{4}+\frac{37035 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}} \, dx}{11319}\\ &=\frac{4}{77 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{54 \sqrt{1-2 x}}{539 (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{9876 \sqrt{1-2 x}}{3773 \sqrt{2+3 x} \sqrt{3+5 x}}-\frac{1100380 \sqrt{1-2 x} \sqrt{2+3 x}}{41503 \sqrt{3+5 x}}+\frac{16 \int \frac{-\frac{783495}{2}-\frac{2475855 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{124509}\\ &=\frac{4}{77 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{54 \sqrt{1-2 x}}{539 (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{9876 \sqrt{1-2 x}}{3773 \sqrt{2+3 x} \sqrt{3+5 x}}-\frac{1100380 \sqrt{1-2 x} \sqrt{2+3 x}}{41503 \sqrt{3+5 x}}-\frac{9876 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{3773}-\frac{660228 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{41503}\\ &=\frac{4}{77 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{54 \sqrt{1-2 x}}{539 (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{9876 \sqrt{1-2 x}}{3773 \sqrt{2+3 x} \sqrt{3+5 x}}-\frac{1100380 \sqrt{1-2 x} \sqrt{2+3 x}}{41503 \sqrt{3+5 x}}+\frac{220076 \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3773}+\frac{6584 \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3773}\\ \end{align*}
Mathematica [A] time = 0.140981, size = 104, normalized size = 0.54 \[ \frac{2 \left (\frac{9903420 x^3+7926942 x^2-2259236 x-2088967}{\sqrt{1-2 x} (3 x+2)^{3/2} \sqrt{5 x+3}}-2 \sqrt{2} \left (55019 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-27860 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )\right )\right )}{41503} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.024, size = 219, normalized size = 1.2 \begin{align*} -{\frac{2}{415030\,{x}^{2}+41503\,x-124509}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 167160\,\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}-330114\,\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}+111440\,\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 ) -220076\,\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 ) +9903420\,{x}^{3}+7926942\,{x}^{2}-2259236\,x-2088967 \right ) \left ( 2+3\,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{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{\sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{2700 \, x^{7} + 5940 \, x^{6} + 3087 \, x^{5} - 1828 \, x^{4} - 2045 \, x^{3} - 202 \, x^{2} + 276 \, x + 72}, 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{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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