Optimal. Leaf size=222 \[ -\frac {48478 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{3645}-\frac {6464 \sqrt {1-2 x} (5 x+3)^{5/2}}{81 \sqrt {3 x+2}}+\frac {74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{27 (3 x+2)^{3/2}}-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{15 (3 x+2)^{5/2}}+\frac {11036}{81} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}-\frac {48478}{729} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}+\frac {136028 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3645} \]
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Rubi [A] time = 0.08, antiderivative size = 222, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {97, 150, 154, 158, 113, 119} \[ -\frac {6464 \sqrt {1-2 x} (5 x+3)^{5/2}}{81 \sqrt {3 x+2}}+\frac {74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{27 (3 x+2)^{3/2}}-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{15 (3 x+2)^{5/2}}+\frac {11036}{81} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}-\frac {48478}{729} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}-\frac {48478 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3645}+\frac {136028 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3645} \]
Antiderivative was successfully verified.
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Rule 97
Rule 113
Rule 119
Rule 150
Rule 154
Rule 158
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{7/2}} \, dx &=-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{15 (2+3 x)^{5/2}}+\frac {2}{15} \int \frac {\left (-\frac {5}{2}-50 x\right ) (1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^{5/2}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{15 (2+3 x)^{5/2}}+\frac {74 (1-2 x)^{3/2} (3+5 x)^{5/2}}{27 (2+3 x)^{3/2}}-\frac {4}{135} \int \frac {\left (-295-\frac {4925 x}{2}\right ) \sqrt {1-2 x} (3+5 x)^{3/2}}{(2+3 x)^{3/2}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{15 (2+3 x)^{5/2}}+\frac {74 (1-2 x)^{3/2} (3+5 x)^{5/2}}{27 (2+3 x)^{3/2}}-\frac {6464 \sqrt {1-2 x} (3+5 x)^{5/2}}{81 \sqrt {2+3 x}}+\frac {8}{405} \int \frac {\left (\frac {118045}{4}-\frac {206925 x}{2}\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=\frac {11036}{81} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{15 (2+3 x)^{5/2}}+\frac {74 (1-2 x)^{3/2} (3+5 x)^{5/2}}{27 (2+3 x)^{3/2}}-\frac {6464 \sqrt {1-2 x} (3+5 x)^{5/2}}{81 \sqrt {2+3 x}}-\frac {8 \int \frac {\left (\frac {137475}{2}-\frac {1817925 x}{4}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{6075}\\ &=-\frac {48478}{729} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {11036}{81} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{15 (2+3 x)^{5/2}}+\frac {74 (1-2 x)^{3/2} (3+5 x)^{5/2}}{27 (2+3 x)^{3/2}}-\frac {6464 \sqrt {1-2 x} (3+5 x)^{5/2}}{81 \sqrt {2+3 x}}+\frac {8 \int \frac {-\frac {2121825}{8}-\frac {2550525 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{54675}\\ &=-\frac {48478}{729} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {11036}{81} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{15 (2+3 x)^{5/2}}+\frac {74 (1-2 x)^{3/2} (3+5 x)^{5/2}}{27 (2+3 x)^{3/2}}-\frac {6464 \sqrt {1-2 x} (3+5 x)^{5/2}}{81 \sqrt {2+3 x}}-\frac {136028 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{3645}+\frac {266629 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{3645}\\ &=-\frac {48478}{729} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {11036}{81} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{15 (2+3 x)^{5/2}}+\frac {74 (1-2 x)^{3/2} (3+5 x)^{5/2}}{27 (2+3 x)^{3/2}}-\frac {6464 \sqrt {1-2 x} (3+5 x)^{5/2}}{81 \sqrt {2+3 x}}+\frac {136028 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3645}-\frac {48478 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3645}\\ \end {align*}
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Mathematica [A] time = 0.32, size = 109, normalized size = 0.49 \[ \frac {\sqrt {2} \left (935915 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )-136028 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )\right )+\frac {6 \sqrt {1-2 x} \sqrt {5 x+3} \left (24300 x^4-45090 x^3-461043 x^2-517257 x-158237\right )}{(3 x+2)^{5/2}}}{10935} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.91, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16}, x\right ) \]
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 \[ \int \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 324, normalized size = 1.46 \[ -\frac {\left (-1458000 x^{6}+2559600 x^{5}+28370520 x^{4}+32990058 x^{3}-1224252 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+8423235 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+4298988 x^{2}-1632336 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+11230980 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-8361204 x -544112 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+3743660 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-2848266\right ) \sqrt {5 x +3}\, \sqrt {-2 x +1}}{10935 \left (10 x^{2}+x -3\right ) \left (3 x +2\right )^{\frac {5}{2}}} \]
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 \[ \int \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^{7/2}} \,d x \]
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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