Optimal. Leaf size=187 \[ -\frac {494 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{6655 \sqrt {33}}+\frac {7 (3 x+2)^{3/2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac {2209 \sqrt {1-2 x} \sqrt {3 x+2}}{43923 \sqrt {5 x+3}}-\frac {247 \sqrt {1-2 x} \sqrt {3 x+2}}{3993 (5 x+3)^{3/2}}+\frac {14 \sqrt {3 x+2}}{121 \sqrt {1-2 x} (5 x+3)^{3/2}}+\frac {2209 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{6655 \sqrt {33}} \]
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Rubi [A] time = 0.07, antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {98, 150, 152, 158, 113, 119} \[ \frac {7 (3 x+2)^{3/2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac {2209 \sqrt {1-2 x} \sqrt {3 x+2}}{43923 \sqrt {5 x+3}}-\frac {247 \sqrt {1-2 x} \sqrt {3 x+2}}{3993 (5 x+3)^{3/2}}+\frac {14 \sqrt {3 x+2}}{121 \sqrt {1-2 x} (5 x+3)^{3/2}}-\frac {494 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{6655 \sqrt {33}}+\frac {2209 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{6655 \sqrt {33}} \]
Antiderivative was successfully verified.
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Rule 98
Rule 113
Rule 119
Rule 150
Rule 152
Rule 158
Rubi steps
\begin {align*} \int \frac {(2+3 x)^{5/2}}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx &=\frac {7 (2+3 x)^{3/2}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {1}{33} \int \frac {\left (-\frac {33}{2}-9 x\right ) \sqrt {2+3 x}}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx\\ &=\frac {14 \sqrt {2+3 x}}{121 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {7 (2+3 x)^{3/2}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {1}{363} \int \frac {-552-\frac {1593 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}} \, dx\\ &=\frac {14 \sqrt {2+3 x}}{121 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {247 \sqrt {1-2 x} \sqrt {2+3 x}}{3993 (3+5 x)^{3/2}}+\frac {7 (2+3 x)^{3/2}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {2 \int \frac {\frac {3993}{4}+\frac {2223 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx}{11979}\\ &=\frac {14 \sqrt {2+3 x}}{121 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {247 \sqrt {1-2 x} \sqrt {2+3 x}}{3993 (3+5 x)^{3/2}}+\frac {7 (2+3 x)^{3/2}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {2209 \sqrt {1-2 x} \sqrt {2+3 x}}{43923 \sqrt {3+5 x}}-\frac {4 \int \frac {\frac {3519}{2}+\frac {19881 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{131769}\\ &=\frac {14 \sqrt {2+3 x}}{121 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {247 \sqrt {1-2 x} \sqrt {2+3 x}}{3993 (3+5 x)^{3/2}}+\frac {7 (2+3 x)^{3/2}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {2209 \sqrt {1-2 x} \sqrt {2+3 x}}{43923 \sqrt {3+5 x}}-\frac {2209 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{73205}+\frac {247 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{6655}\\ &=\frac {14 \sqrt {2+3 x}}{121 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {247 \sqrt {1-2 x} \sqrt {2+3 x}}{3993 (3+5 x)^{3/2}}+\frac {7 (2+3 x)^{3/2}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {2209 \sqrt {1-2 x} \sqrt {2+3 x}}{43923 \sqrt {3+5 x}}+\frac {2209 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{6655 \sqrt {33}}-\frac {494 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{6655 \sqrt {33}}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 104, normalized size = 0.56 \[ \frac {\sqrt {2} \left (10360 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )-2209 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )\right )-\frac {10 \sqrt {3 x+2} \left (22090 x^3-3402 x^2-22059 x-7186\right )}{(1-2 x)^{3/2} (5 x+3)^{3/2}}}{219615} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.95, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (9 \, x^{2} + 12 \, x + 4\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{1000 \, x^{6} + 300 \, x^{5} - 870 \, x^{4} - 179 \, x^{3} + 261 \, x^{2} + 27 \, x - 27}, x\right ) \]
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 \[ \int \frac {{\left (3 \, x + 2\right )}^{\frac {5}{2}}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.03, size = 311, normalized size = 1.66 \[ \frac {\left (-662700 x^{4}-339740 x^{3}+22090 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-103600 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+729810 x^{2}+2209 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-10360 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+656760 x -6627 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+31080 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+143720\right ) \sqrt {-2 x +1}}{219615 \left (5 x +3\right )^{\frac {3}{2}} \left (2 x -1\right )^{2} \sqrt {3 x +2}} \]
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 \[ \int \frac {{\left (3 \, x + 2\right )}^{\frac {5}{2}}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (3\,x+2\right )}^{5/2}}{{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{5/2}} \,d x \]
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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