Optimal. Leaf size=160 \[ \frac{192}{49} \sqrt{\frac{3}{11}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )-\frac{31940 \sqrt{1-2 x} \sqrt{3 x+2}}{539 \sqrt{5 x+3}}+\frac{288 \sqrt{1-2 x}}{49 \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{2 \sqrt{1-2 x}}{7 (3 x+2)^{3/2} \sqrt{5 x+3}}+\frac{6388}{49} \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
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Rubi [A] time = 0.055133, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 6, 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} \[ -\frac{31940 \sqrt{1-2 x} \sqrt{3 x+2}}{539 \sqrt{5 x+3}}+\frac{288 \sqrt{1-2 x}}{49 \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{2 \sqrt{1-2 x}}{7 (3 x+2)^{3/2} \sqrt{5 x+3}}+\frac{192}{49} \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{6388}{49} \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
Antiderivative was successfully verified.
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Rule 104
Rule 152
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx &=\frac{2 \sqrt{1-2 x}}{7 (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{2}{21} \int \frac{42-45 x}{\sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx\\ &=\frac{2 \sqrt{1-2 x}}{7 (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{288 \sqrt{1-2 x}}{49 \sqrt{2+3 x} \sqrt{3+5 x}}+\frac{4}{147} \int \frac{\frac{3495}{2}-1080 x}{\sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}} \, dx\\ &=\frac{2 \sqrt{1-2 x}}{7 (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{288 \sqrt{1-2 x}}{49 \sqrt{2+3 x} \sqrt{3+5 x}}-\frac{31940 \sqrt{1-2 x} \sqrt{2+3 x}}{539 \sqrt{3+5 x}}-\frac{8 \int \frac{\frac{45495}{2}+\frac{71865 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{1617}\\ &=\frac{2 \sqrt{1-2 x}}{7 (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{288 \sqrt{1-2 x}}{49 \sqrt{2+3 x} \sqrt{3+5 x}}-\frac{31940 \sqrt{1-2 x} \sqrt{2+3 x}}{539 \sqrt{3+5 x}}-\frac{288}{49} \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx-\frac{19164}{539} \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx\\ &=\frac{2 \sqrt{1-2 x}}{7 (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{288 \sqrt{1-2 x}}{49 \sqrt{2+3 x} \sqrt{3+5 x}}-\frac{31940 \sqrt{1-2 x} \sqrt{2+3 x}}{539 \sqrt{3+5 x}}+\frac{6388}{49} \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{192}{49} \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 = 0.119627, size = 100, normalized size = 0.62 \[ \frac{2}{539} \left (-2 \sqrt{2} \left (1597 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-805 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )\right )-\frac{\sqrt{1-2 x} \left (143730 x^2+186888 x+60635\right )}{(3 x+2)^{3/2} \sqrt{5 x+3}}\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.024, size = 219, normalized size = 1.4 \begin{align*} -{\frac{2}{5390\,{x}^{2}+539\,x-1617}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 4830\,\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}-9582\,\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}+3220\,\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 ) -6388\,\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 ) +287460\,{x}^{3}+230046\,{x}^{2}-65618\,x-60635 \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}} \sqrt{-2 \, x + 1}}\,{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}}{1350 \, x^{6} + 3645 \, x^{5} + 3366 \, x^{4} + 769 \, x^{3} - 638 \, x^{2} - 420 \, 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}} \sqrt{-2 \, x + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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