Optimal. Leaf size=222 \[ -\frac {220028 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{1764735}+\frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)^{7/2}}-\frac {106558 \sqrt {1-2 x} \sqrt {5 x+3}}{1764735 \sqrt {3 x+2}}-\frac {106772 \sqrt {1-2 x} \sqrt {5 x+3}}{252105 (3 x+2)^{3/2}}-\frac {37117 \sqrt {1-2 x} \sqrt {5 x+3}}{36015 (3 x+2)^{5/2}}+\frac {229 \sqrt {1-2 x} \sqrt {5 x+3}}{1029 (3 x+2)^{7/2}}+\frac {106558 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1764735} \]
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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 = {98, 150, 152, 158, 113, 119} \[ \frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)^{7/2}}-\frac {106558 \sqrt {1-2 x} \sqrt {5 x+3}}{1764735 \sqrt {3 x+2}}-\frac {106772 \sqrt {1-2 x} \sqrt {5 x+3}}{252105 (3 x+2)^{3/2}}-\frac {37117 \sqrt {1-2 x} \sqrt {5 x+3}}{36015 (3 x+2)^{5/2}}+\frac {229 \sqrt {1-2 x} \sqrt {5 x+3}}{1029 (3 x+2)^{7/2}}-\frac {220028 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1764735}+\frac {106558 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1764735} \]
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 {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^{9/2}} \, dx &=\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)^{7/2}}-\frac {1}{7} \int \frac {\left (-\frac {357}{2}-325 x\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^{9/2}} \, dx\\ &=\frac {229 \sqrt {1-2 x} \sqrt {3+5 x}}{1029 (2+3 x)^{7/2}}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)^{7/2}}-\frac {2 \int \frac {-\frac {86161}{4}-36950 x}{\sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}} \, dx}{1029}\\ &=\frac {229 \sqrt {1-2 x} \sqrt {3+5 x}}{1029 (2+3 x)^{7/2}}-\frac {37117 \sqrt {1-2 x} \sqrt {3+5 x}}{36015 (2+3 x)^{5/2}}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)^{7/2}}-\frac {4 \int \frac {-79446-\frac {556755 x}{4}}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx}{36015}\\ &=\frac {229 \sqrt {1-2 x} \sqrt {3+5 x}}{1029 (2+3 x)^{7/2}}-\frac {37117 \sqrt {1-2 x} \sqrt {3+5 x}}{36015 (2+3 x)^{5/2}}-\frac {106772 \sqrt {1-2 x} \sqrt {3+5 x}}{252105 (2+3 x)^{3/2}}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)^{7/2}}-\frac {8 \int \frac {-\frac {1014441}{8}-\frac {400395 x}{2}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{756315}\\ &=\frac {229 \sqrt {1-2 x} \sqrt {3+5 x}}{1029 (2+3 x)^{7/2}}-\frac {37117 \sqrt {1-2 x} \sqrt {3+5 x}}{36015 (2+3 x)^{5/2}}-\frac {106772 \sqrt {1-2 x} \sqrt {3+5 x}}{252105 (2+3 x)^{3/2}}-\frac {106558 \sqrt {1-2 x} \sqrt {3+5 x}}{1764735 \sqrt {2+3 x}}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)^{7/2}}-\frac {16 \int \frac {-166965+\frac {799185 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{5294205}\\ &=\frac {229 \sqrt {1-2 x} \sqrt {3+5 x}}{1029 (2+3 x)^{7/2}}-\frac {37117 \sqrt {1-2 x} \sqrt {3+5 x}}{36015 (2+3 x)^{5/2}}-\frac {106772 \sqrt {1-2 x} \sqrt {3+5 x}}{252105 (2+3 x)^{3/2}}-\frac {106558 \sqrt {1-2 x} \sqrt {3+5 x}}{1764735 \sqrt {2+3 x}}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)^{7/2}}-\frac {106558 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{1764735}+\frac {1210154 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{1764735}\\ &=\frac {229 \sqrt {1-2 x} \sqrt {3+5 x}}{1029 (2+3 x)^{7/2}}-\frac {37117 \sqrt {1-2 x} \sqrt {3+5 x}}{36015 (2+3 x)^{5/2}}-\frac {106772 \sqrt {1-2 x} \sqrt {3+5 x}}{252105 (2+3 x)^{3/2}}-\frac {106558 \sqrt {1-2 x} \sqrt {3+5 x}}{1764735 \sqrt {2+3 x}}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)^{7/2}}+\frac {106558 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1764735}-\frac {220028 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1764735}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 109, normalized size = 0.49 \[ \frac {2 \left (\sqrt {2} \left (1868510 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )-53279 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )\right )+\frac {3 \sqrt {5 x+3} \left (2877066 x^4+11042235 x^3+12020751 x^2+4889131 x+616327\right )}{\sqrt {1-2 x} (3 x+2)^{7/2}}\right )}{5294205} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.75, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{972 \, x^{7} + 2268 \, x^{6} + 1323 \, x^{5} - 630 \, x^{4} - 840 \, x^{3} - 112 \, x^{2} + 112 \, x + 32}, x\right ) \]
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 \[ \int \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {9}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.03, size = 409, normalized size = 1.84 \[ -\frac {2 \sqrt {5 x +3}\, \sqrt {-2 x +1}\, \left (43155990 x^{5}+191527119 x^{4}-1438533 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{3} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+50449770 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{3} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+279691380 x^{3}-2877066 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+100899540 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+181523724 x^{2}-1918044 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+67266360 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+53247084 x -426232 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+14948080 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+5546943\right )}{5294205 \left (3 x +2\right )^{\frac {7}{2}} \left (10 x^{2}+x -3\right )} \]
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 \[ \int \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {9}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (5\,x+3\right )}^{5/2}}{{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^{9/2}} \,d x \]
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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