Optimal. Leaf size=249 \[ -\frac {2295970088 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{10135125}-\frac {138809831 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{4504500}-\frac {221673 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{50050}-\frac {14303 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{12870}-\frac {41}{143} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac {1}{13} \sqrt {1-2 x} (2+3 x)^{7/2} (3+5 x)^{5/2}-\frac {610627101631 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{36855000 \sqrt {33}}-\frac {2295970088 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{4606875 \sqrt {33}} \]
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Rubi [A]
time = 0.07, antiderivative size = 249, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {103, 159, 164,
114, 120} \begin {gather*} -\frac {2295970088 F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{4606875 \sqrt {33}}-\frac {610627101631 E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{36855000 \sqrt {33}}-\frac {1}{13} \sqrt {1-2 x} (5 x+3)^{5/2} (3 x+2)^{7/2}-\frac {41}{143} \sqrt {1-2 x} (5 x+3)^{5/2} (3 x+2)^{5/2}-\frac {14303 \sqrt {1-2 x} (5 x+3)^{5/2} (3 x+2)^{3/2}}{12870}-\frac {221673 \sqrt {1-2 x} (5 x+3)^{5/2} \sqrt {3 x+2}}{50050}-\frac {138809831 \sqrt {1-2 x} (5 x+3)^{3/2} \sqrt {3 x+2}}{4504500}-\frac {2295970088 \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}}{10135125} \end {gather*}
Antiderivative was successfully verified.
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Rule 103
Rule 114
Rule 120
Rule 159
Rule 164
Rubi steps
\begin {align*} \int \frac {(2+3 x)^{7/2} (3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx &=-\frac {1}{13} \sqrt {1-2 x} (2+3 x)^{7/2} (3+5 x)^{5/2}+\frac {1}{13} \int \frac {(2+3 x)^{5/2} (3+5 x)^{3/2} \left (\frac {257}{2}+205 x\right )}{\sqrt {1-2 x}} \, dx\\ &=-\frac {41}{143} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac {1}{13} \sqrt {1-2 x} (2+3 x)^{7/2} (3+5 x)^{5/2}-\frac {1}{715} \int \frac {\left (-\frac {45285}{2}-\frac {71515 x}{2}\right ) (2+3 x)^{3/2} (3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx\\ &=-\frac {14303 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{12870}-\frac {41}{143} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac {1}{13} \sqrt {1-2 x} (2+3 x)^{7/2} (3+5 x)^{5/2}+\frac {\int \frac {\sqrt {2+3 x} (3+5 x)^{3/2} \left (\frac {12799775}{4}+\frac {9975285 x}{2}\right )}{\sqrt {1-2 x}} \, dx}{32175}\\ &=-\frac {221673 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{50050}-\frac {14303 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{12870}-\frac {41}{143} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac {1}{13} \sqrt {1-2 x} (2+3 x)^{7/2} (3+5 x)^{5/2}-\frac {\int \frac {\left (-\frac {1364822645}{4}-\frac {2082147465 x}{4}\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{1126125}\\ &=-\frac {138809831 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{4504500}-\frac {221673 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{50050}-\frac {14303 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{12870}-\frac {41}{143} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac {1}{13} \sqrt {1-2 x} (2+3 x)^{7/2} (3+5 x)^{5/2}+\frac {\int \frac {\sqrt {3+5 x} \left (\frac {179052019605}{8}+34439551320 x\right )}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{16891875}\\ &=-\frac {2295970088 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{10135125}-\frac {138809831 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{4504500}-\frac {221673 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{50050}-\frac {14303 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{12870}-\frac {41}{143} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac {1}{13} \sqrt {1-2 x} (2+3 x)^{7/2} (3+5 x)^{5/2}-\frac {\int \frac {-\frac {5798711966295}{8}-\frac {9159406524465 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{152026875}\\ &=-\frac {2295970088 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{10135125}-\frac {138809831 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{4504500}-\frac {221673 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{50050}-\frac {14303 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{12870}-\frac {41}{143} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac {1}{13} \sqrt {1-2 x} (2+3 x)^{7/2} (3+5 x)^{5/2}+\frac {1147985044 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{4606875}+\frac {610627101631 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{405405000}\\ &=-\frac {2295970088 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{10135125}-\frac {138809831 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{4504500}-\frac {221673 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{50050}-\frac {14303 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{12870}-\frac {41}{143} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac {1}{13} \sqrt {1-2 x} (2+3 x)^{7/2} (3+5 x)^{5/2}-\frac {610627101631 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{36855000 \sqrt {33}}-\frac {2295970088 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{4606875 \sqrt {33}}\\ \end {align*}
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Mathematica [A]
time = 5.25, size = 115, normalized size = 0.46 \begin {gather*} \frac {610627101631 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-5 \left (3 \sqrt {2-4 x} \sqrt {2+3 x} \sqrt {3+5 x} \left (16001700059+19961825445 x+22592085750 x^2+18620894250 x^3+9351247500 x^4+2104987500 x^5\right )+61511810003 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )}{608107500 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 163, normalized size = 0.65
method | result | size |
default | \(-\frac {\sqrt {2+3 x}\, \sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (1894488750000 x^{8}+9868564125000 x^{7}+22769118225000 x^{6}+303068051616 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-610627101631 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+30838634482500 x^{5}+27960569725500 x^{4}+20079090637650 x^{3}+2782614262260 x^{2}-6953485592490 x -2880306010620\right )}{1216215000 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(163\) |
risch | \(\frac {\left (2104987500 x^{5}+9351247500 x^{4}+18620894250 x^{3}+22592085750 x^{2}+19961825445 x +16001700059\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {2+3 x}\, \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{40540500 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \sqrt {1-2 x}}+\frac {\left (\frac {386580797753 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {55-110 x}\, \EllipticF \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{4459455000 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {610627101631 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {55-110 x}\, \left (\frac {\EllipticE \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{15}-\frac {2 \EllipticF \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{3}\right )}{4459455000 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right ) \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(266\) |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \sqrt {3+5 x}\, \sqrt {2+3 x}\, \left (-\frac {675 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}\, x^{5}}{13}-\frac {32985 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}\, x^{4}}{143}-\frac {394093 x^{3} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{858}-\frac {772379 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1386}-\frac {443596121 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{900900}-\frac {16001700059 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{40540500}+\frac {386580797753 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{1702701000 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {610627101631 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{1702701000 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \left (15 x^{2}+19 x +6\right )}\) | \(312\) |
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 = 0.29, size = 48, normalized size = 0.19 \begin {gather*} -\frac {1}{40540500} \, {\left (2104987500 \, x^{5} + 9351247500 \, x^{4} + 18620894250 \, x^{3} + 22592085750 \, x^{2} + 19961825445 \, x + 16001700059\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} \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 = -0.00 \begin {gather*} \int \frac {{\left (3\,x+2\right )}^{7/2}\,{\left (5\,x+3\right )}^{5/2}}{\sqrt {1-2\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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