Optimal. Leaf size=249 \[ -\frac {697995152 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{194481 \sqrt {33}}+\frac {14 \sqrt {5 x+3} (1-2 x)^{3/2}}{33 (3 x+2)^{11/2}}+\frac {23204503328 \sqrt {5 x+3} \sqrt {1-2 x}}{2139291 \sqrt {3 x+2}}+\frac {333930952 \sqrt {5 x+3} \sqrt {1-2 x}}{305613 (3 x+2)^{3/2}}+\frac {7173272 \sqrt {5 x+3} \sqrt {1-2 x}}{43659 (3 x+2)^{5/2}}+\frac {171004 \sqrt {5 x+3} \sqrt {1-2 x}}{6237 (3 x+2)^{7/2}}+\frac {4508 \sqrt {5 x+3} \sqrt {1-2 x}}{891 (3 x+2)^{9/2}}-\frac {23204503328 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{194481 \sqrt {33}} \]
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Rubi [A] time = 0.10, antiderivative size = 249, normalized size of antiderivative = 1.00, number of steps used = 9, 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} \[ \frac {14 \sqrt {5 x+3} (1-2 x)^{3/2}}{33 (3 x+2)^{11/2}}+\frac {23204503328 \sqrt {5 x+3} \sqrt {1-2 x}}{2139291 \sqrt {3 x+2}}+\frac {333930952 \sqrt {5 x+3} \sqrt {1-2 x}}{305613 (3 x+2)^{3/2}}+\frac {7173272 \sqrt {5 x+3} \sqrt {1-2 x}}{43659 (3 x+2)^{5/2}}+\frac {171004 \sqrt {5 x+3} \sqrt {1-2 x}}{6237 (3 x+2)^{7/2}}+\frac {4508 \sqrt {5 x+3} \sqrt {1-2 x}}{891 (3 x+2)^{9/2}}-\frac {697995152 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{194481 \sqrt {33}}-\frac {23204503328 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{194481 \sqrt {33}} \]
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 {(1-2 x)^{5/2}}{(2+3 x)^{13/2} \sqrt {3+5 x}} \, dx &=\frac {14 (1-2 x)^{3/2} \sqrt {3+5 x}}{33 (2+3 x)^{11/2}}+\frac {2}{33} \int \frac {(227-223 x) \sqrt {1-2 x}}{(2+3 x)^{11/2} \sqrt {3+5 x}} \, dx\\ &=\frac {14 (1-2 x)^{3/2} \sqrt {3+5 x}}{33 (2+3 x)^{11/2}}+\frac {4508 \sqrt {1-2 x} \sqrt {3+5 x}}{891 (2+3 x)^{9/2}}-\frac {4}{891} \int \frac {-\frac {49835}{2}+37438 x}{\sqrt {1-2 x} (2+3 x)^{9/2} \sqrt {3+5 x}} \, dx\\ &=\frac {14 (1-2 x)^{3/2} \sqrt {3+5 x}}{33 (2+3 x)^{11/2}}+\frac {4508 \sqrt {1-2 x} \sqrt {3+5 x}}{891 (2+3 x)^{9/2}}+\frac {171004 \sqrt {1-2 x} \sqrt {3+5 x}}{6237 (2+3 x)^{7/2}}-\frac {8 \int \frac {-\frac {5473405}{2}+\frac {7481425 x}{2}}{\sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}} \, dx}{43659}\\ &=\frac {14 (1-2 x)^{3/2} \sqrt {3+5 x}}{33 (2+3 x)^{11/2}}+\frac {4508 \sqrt {1-2 x} \sqrt {3+5 x}}{891 (2+3 x)^{9/2}}+\frac {171004 \sqrt {1-2 x} \sqrt {3+5 x}}{6237 (2+3 x)^{7/2}}+\frac {7173272 \sqrt {1-2 x} \sqrt {3+5 x}}{43659 (2+3 x)^{5/2}}-\frac {16 \int \frac {-\frac {833286615}{4}+\frac {470745975 x}{2}}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx}{1528065}\\ &=\frac {14 (1-2 x)^{3/2} \sqrt {3+5 x}}{33 (2+3 x)^{11/2}}+\frac {4508 \sqrt {1-2 x} \sqrt {3+5 x}}{891 (2+3 x)^{9/2}}+\frac {171004 \sqrt {1-2 x} \sqrt {3+5 x}}{6237 (2+3 x)^{7/2}}+\frac {7173272 \sqrt {1-2 x} \sqrt {3+5 x}}{43659 (2+3 x)^{5/2}}+\frac {333930952 \sqrt {1-2 x} \sqrt {3+5 x}}{305613 (2+3 x)^{3/2}}-\frac {32 \int \frac {-9037592970+\frac {21914218725 x}{4}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{32089365}\\ &=\frac {14 (1-2 x)^{3/2} \sqrt {3+5 x}}{33 (2+3 x)^{11/2}}+\frac {4508 \sqrt {1-2 x} \sqrt {3+5 x}}{891 (2+3 x)^{9/2}}+\frac {171004 \sqrt {1-2 x} \sqrt {3+5 x}}{6237 (2+3 x)^{7/2}}+\frac {7173272 \sqrt {1-2 x} \sqrt {3+5 x}}{43659 (2+3 x)^{5/2}}+\frac {333930952 \sqrt {1-2 x} \sqrt {3+5 x}}{305613 (2+3 x)^{3/2}}+\frac {23204503328 \sqrt {1-2 x} \sqrt {3+5 x}}{2139291 \sqrt {2+3 x}}-\frac {64 \int \frac {-\frac {964063843575}{8}-\frac {380698882725 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{224625555}\\ &=\frac {14 (1-2 x)^{3/2} \sqrt {3+5 x}}{33 (2+3 x)^{11/2}}+\frac {4508 \sqrt {1-2 x} \sqrt {3+5 x}}{891 (2+3 x)^{9/2}}+\frac {171004 \sqrt {1-2 x} \sqrt {3+5 x}}{6237 (2+3 x)^{7/2}}+\frac {7173272 \sqrt {1-2 x} \sqrt {3+5 x}}{43659 (2+3 x)^{5/2}}+\frac {333930952 \sqrt {1-2 x} \sqrt {3+5 x}}{305613 (2+3 x)^{3/2}}+\frac {23204503328 \sqrt {1-2 x} \sqrt {3+5 x}}{2139291 \sqrt {2+3 x}}+\frac {348997576 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{194481}+\frac {23204503328 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{2139291}\\ &=\frac {14 (1-2 x)^{3/2} \sqrt {3+5 x}}{33 (2+3 x)^{11/2}}+\frac {4508 \sqrt {1-2 x} \sqrt {3+5 x}}{891 (2+3 x)^{9/2}}+\frac {171004 \sqrt {1-2 x} \sqrt {3+5 x}}{6237 (2+3 x)^{7/2}}+\frac {7173272 \sqrt {1-2 x} \sqrt {3+5 x}}{43659 (2+3 x)^{5/2}}+\frac {333930952 \sqrt {1-2 x} \sqrt {3+5 x}}{305613 (2+3 x)^{3/2}}+\frac {23204503328 \sqrt {1-2 x} \sqrt {3+5 x}}{2139291 \sqrt {2+3 x}}-\frac {23204503328 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{194481 \sqrt {33}}-\frac {697995152 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{194481 \sqrt {33}}\\ \end {align*}
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Mathematica [A] time = 0.42, size = 115, normalized size = 0.46 \[ \frac {16 \sqrt {2} \left (2900562916 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )-1460947915 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )\right )+\frac {12 \sqrt {1-2 x} \sqrt {5 x+3} \left (2819347154352 x^5+9492493272732 x^4+12787628716260 x^3+8615827181322 x^2+2903435279352 x+391506734113\right )}{(3 x+2)^{11/2}}}{12835746} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.89, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (4 \, x^{2} - 4 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{10935 \, x^{8} + 57591 \, x^{7} + 132678 \, x^{6} + 174636 \, x^{5} + 143640 \, x^{4} + 75600 \, x^{3} + 24864 \, x^{2} + 4672 \, x + 384}, x\right ) \]
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 \[ \int \frac {{\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{\sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{\frac {13}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.03, size = 599, normalized size = 2.41 \[ \frac {2 \left (84580414630560 x^{7}+293232839645016 x^{6}-2819347154352 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{5} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+1420041373380 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{5} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+386732216916828 x^{5}-9397823847840 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{4} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+4733471244600 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{4} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+211405262133852 x^{4}-12530431797120 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{3} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+6311294992800 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{3} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-2138118521814 x^{3}-8353621198080 \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 )+4207529995200 \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 )-57086936770452 x^{2}-2784540399360 \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 )+1402509998400 \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 )-24956397311829 x -371272053248 \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 )+187001333120 \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 )-3523560607017\right ) \sqrt {5 x +3}\, \sqrt {-2 x +1}}{6417873 \left (10 x^{2}+x -3\right ) \left (3 x +2\right )^{\frac {11}{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 \[ \int \frac {{\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{\sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{\frac {13}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (1-2\,x\right )}^{5/2}}{{\left (3\,x+2\right )}^{13/2}\,\sqrt {5\,x+3}} \,d x \]
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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