Optimal. Leaf size=191 \[ -\frac {21160 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{1323}+\frac {2 \sqrt {5 x+3} (1-2 x)^{3/2}}{3 (3 x+2)^{7/2}}+\frac {703480 \sqrt {5 x+3} \sqrt {1-2 x}}{1323 \sqrt {3 x+2}}+\frac {10124 \sqrt {5 x+3} \sqrt {1-2 x}}{189 (3 x+2)^{3/2}}+\frac {76 \sqrt {5 x+3} \sqrt {1-2 x}}{9 (3 x+2)^{5/2}}-\frac {703480 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1323} \]
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Rubi [A] time = 0.07, antiderivative size = 191, 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} \[ \frac {2 \sqrt {5 x+3} (1-2 x)^{3/2}}{3 (3 x+2)^{7/2}}+\frac {703480 \sqrt {5 x+3} \sqrt {1-2 x}}{1323 \sqrt {3 x+2}}+\frac {10124 \sqrt {5 x+3} \sqrt {1-2 x}}{189 (3 x+2)^{3/2}}+\frac {76 \sqrt {5 x+3} \sqrt {1-2 x}}{9 (3 x+2)^{5/2}}-\frac {21160 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1323}-\frac {703480 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1323} \]
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)^{9/2} \sqrt {3+5 x}} \, dx &=\frac {2 (1-2 x)^{3/2} \sqrt {3+5 x}}{3 (2+3 x)^{7/2}}+\frac {2}{21} \int \frac {(161-91 x) \sqrt {1-2 x}}{(2+3 x)^{7/2} \sqrt {3+5 x}} \, dx\\ &=\frac {2 (1-2 x)^{3/2} \sqrt {3+5 x}}{3 (2+3 x)^{7/2}}+\frac {76 \sqrt {1-2 x} \sqrt {3+5 x}}{9 (2+3 x)^{5/2}}-\frac {4}{315} \int \frac {-\frac {16835}{2}+9520 x}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx\\ &=\frac {2 (1-2 x)^{3/2} \sqrt {3+5 x}}{3 (2+3 x)^{7/2}}+\frac {76 \sqrt {1-2 x} \sqrt {3+5 x}}{9 (2+3 x)^{5/2}}+\frac {10124 \sqrt {1-2 x} \sqrt {3+5 x}}{189 (2+3 x)^{3/2}}-\frac {8 \int \frac {-\frac {730625}{2}+\frac {442925 x}{2}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{6615}\\ &=\frac {2 (1-2 x)^{3/2} \sqrt {3+5 x}}{3 (2+3 x)^{7/2}}+\frac {76 \sqrt {1-2 x} \sqrt {3+5 x}}{9 (2+3 x)^{5/2}}+\frac {10124 \sqrt {1-2 x} \sqrt {3+5 x}}{189 (2+3 x)^{3/2}}+\frac {703480 \sqrt {1-2 x} \sqrt {3+5 x}}{1323 \sqrt {2+3 x}}-\frac {16 \int \frac {-\frac {19484675}{4}-\frac {15388625 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{46305}\\ &=\frac {2 (1-2 x)^{3/2} \sqrt {3+5 x}}{3 (2+3 x)^{7/2}}+\frac {76 \sqrt {1-2 x} \sqrt {3+5 x}}{9 (2+3 x)^{5/2}}+\frac {10124 \sqrt {1-2 x} \sqrt {3+5 x}}{189 (2+3 x)^{3/2}}+\frac {703480 \sqrt {1-2 x} \sqrt {3+5 x}}{1323 \sqrt {2+3 x}}+\frac {116380 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{1323}+\frac {703480 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{1323}\\ &=\frac {2 (1-2 x)^{3/2} \sqrt {3+5 x}}{3 (2+3 x)^{7/2}}+\frac {76 \sqrt {1-2 x} \sqrt {3+5 x}}{9 (2+3 x)^{5/2}}+\frac {10124 \sqrt {1-2 x} \sqrt {3+5 x}}{189 (2+3 x)^{3/2}}+\frac {703480 \sqrt {1-2 x} \sqrt {3+5 x}}{1323 \sqrt {2+3 x}}-\frac {703480 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1323}-\frac {21160 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1323}\\ \end {align*}
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Mathematica [A] time = 0.32, size = 107, normalized size = 0.56 \[ \frac {4 \left (5 \sqrt {2} \left (35174 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )-17717 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )\right )+\frac {3 \sqrt {1-2 x} \sqrt {5 x+3} \left (9496980 x^3+19312866 x^2+13103724 x+2967269\right )}{2 (3 x+2)^{7/2}}\right )}{3969} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.20, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (4 \, x^{2} - 4 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{1215 \, x^{6} + 4779 \, x^{5} + 7830 \, x^{4} + 6840 \, x^{3} + 3360 \, x^{2} + 880 \, x + 96}, x\right ) \]
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 \[ \int \frac {{\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{\sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{\frac {9}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 409, normalized size = 2.14 \[ \frac {2 \left (284909400 x^{5}+607876920 x^{4}-9496980 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{3} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+4783590 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{3} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+365577498 x^{3}-18993960 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+9567180 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-45486552 x^{2}-12662640 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+6378120 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-109031709 x -2813920 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+1417360 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-26705421\right ) \sqrt {5 x +3}\, \sqrt {-2 x +1}}{3969 \left (10 x^{2}+x -3\right ) \left (3 x +2\right )^{\frac {7}{2}}} \]
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 \[ \int \frac {{\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{\sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{\frac {9}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (1-2\,x\right )}^{5/2}}{{\left (3\,x+2\right )}^{9/2}\,\sqrt {5\,x+3}} \,d x \]
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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