Optimal. Leaf size=191 \[ \frac {2 (1-2 x)^{3/2} \sqrt {3+5 x}}{3 (2+3 x)^{7/2}}+\frac {76 \sqrt {1-2 x} \sqrt {3+5 x}}{9 (2+3 x)^{5/2}}+\frac {10124 \sqrt {1-2 x} \sqrt {3+5 x}}{189 (2+3 x)^{3/2}}+\frac {703480 \sqrt {1-2 x} \sqrt {3+5 x}}{1323 \sqrt {2+3 x}}-\frac {703480 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1323}-\frac {21160 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1323} \]
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Rubi [A]
time = 0.04, 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 = {100, 155, 157,
164, 114, 120} \begin {gather*} -\frac {21160 \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1323}-\frac {703480 \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 100
Rule 114
Rule 120
Rule 155
Rule 157
Rule 164
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 = 8.93, size = 107, normalized size = 0.56 \begin {gather*} \frac {4 \left (\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} \left (2967269+13103724 x+19312866 x^2+9496980 x^3\right )}{2 (2+3 x)^{7/2}}+5 \sqrt {2} \left (35174 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-17717 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )\right )}{3969} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(400\) vs.
\(2(139)=278\).
time = 0.10, size = 401, normalized size = 2.10
method | result | size |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {8 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{27 \left (\frac {2}{3}+x \right )^{3}}+\frac {10124 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1701 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {7034800}{1323} x^{2}-\frac {703480}{1323} x +\frac {703480}{441}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {2226820 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{27783 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {3517400 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{27783 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {14 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{729 \left (\frac {2}{3}+x \right )^{4}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(273\) |
default | \(-\frac {2 \left (4713390 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-9496980 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+9426780 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-18993960 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+6284520 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-12662640 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+1396560 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-2813920 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-284909400 x^{5}-607876920 x^{4}-365577498 x^{3}+45486552 x^{2}+109031709 x +26705421\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}}{3969 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {7}{2}}}\) | \(401\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.23, size = 60, normalized size = 0.31 \begin {gather*} \frac {2 \, {\left (9496980 \, x^{3} + 19312866 \, x^{2} + 13103724 \, x + 2967269\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{1323 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\left (1-2\,x\right )}^{5/2}}{{\left (3\,x+2\right )}^{9/2}\,\sqrt {5\,x+3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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