Optimal. Leaf size=156 \[ -\frac {16 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{121 \sqrt {33}}-\frac {490 \sqrt {1-2 x} \sqrt {3 x+2}}{3993 \sqrt {5 x+3}}-\frac {40 \sqrt {1-2 x} \sqrt {3 x+2}}{363 (5 x+3)^{3/2}}+\frac {2 \sqrt {3 x+2}}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}+\frac {98 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{121 \sqrt {33}} \]
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Rubi [A] time = 0.05, antiderivative size = 156, normalized size of antiderivative = 1.00, 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 = {99, 152, 158, 113, 119} \[ -\frac {490 \sqrt {1-2 x} \sqrt {3 x+2}}{3993 \sqrt {5 x+3}}-\frac {40 \sqrt {1-2 x} \sqrt {3 x+2}}{363 (5 x+3)^{3/2}}+\frac {2 \sqrt {3 x+2}}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}-\frac {16 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{121 \sqrt {33}}+\frac {98 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{121 \sqrt {33}} \]
Antiderivative was successfully verified.
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Rule 99
Rule 113
Rule 119
Rule 152
Rule 158
Rubi steps
\begin {align*} \int \frac {\sqrt {2+3 x}}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx &=\frac {2 \sqrt {2+3 x}}{11 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {2}{11} \int \frac {-\frac {31}{2}-\frac {45 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}} \, dx\\ &=\frac {2 \sqrt {2+3 x}}{11 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {40 \sqrt {1-2 x} \sqrt {2+3 x}}{363 (3+5 x)^{3/2}}+\frac {4}{363} \int \frac {\frac {121}{4}+30 x}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx\\ &=\frac {2 \sqrt {2+3 x}}{11 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {40 \sqrt {1-2 x} \sqrt {2+3 x}}{363 (3+5 x)^{3/2}}-\frac {490 \sqrt {1-2 x} \sqrt {2+3 x}}{3993 \sqrt {3+5 x}}-\frac {8 \int \frac {\frac {309}{4}+\frac {735 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{3993}\\ &=\frac {2 \sqrt {2+3 x}}{11 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {40 \sqrt {1-2 x} \sqrt {2+3 x}}{363 (3+5 x)^{3/2}}-\frac {490 \sqrt {1-2 x} \sqrt {2+3 x}}{3993 \sqrt {3+5 x}}+\frac {8}{121} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx-\frac {98 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{1331}\\ &=\frac {2 \sqrt {2+3 x}}{11 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {40 \sqrt {1-2 x} \sqrt {2+3 x}}{363 (3+5 x)^{3/2}}-\frac {490 \sqrt {1-2 x} \sqrt {2+3 x}}{3993 \sqrt {3+5 x}}+\frac {98 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{121 \sqrt {33}}-\frac {16 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{121 \sqrt {33}}\\ \end {align*}
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Mathematica [A] time = 0.26, size = 96, normalized size = 0.62 \[ \frac {\sqrt {2} \left (362 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )+\frac {\sqrt {6 x+4} \left (2450 x^2+685 x-592\right )}{\sqrt {1-2 x} (5 x+3)^{3/2}}-98 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )\right )}{3993} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.77, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{500 \, x^{5} + 400 \, x^{4} - 235 \, x^{3} - 207 \, x^{2} + 27 \, x + 27}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {3 \, x + 2}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 219, normalized size = 1.40 \[ -\frac {2 \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \left (7350 x^{3}+6955 x^{2}-245 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+905 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-406 x -147 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+543 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-1184\right )}{3993 \left (5 x +3\right )^{\frac {3}{2}} \left (6 x^{2}+x -2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {3 \, x + 2}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {3\,x+2}}{{\left (1-2\,x\right )}^{3/2}\,{\left (5\,x+3\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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