Optimal. Leaf size=253 \[ -\frac {1199452 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{22235661}+\frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)^{9/2}}+\frac {6036028 \sqrt {1-2 x} \sqrt {5 x+3}}{22235661 \sqrt {3 x+2}}-\frac {392998 \sqrt {1-2 x} \sqrt {5 x+3}}{3176523 (3 x+2)^{3/2}}-\frac {167228 \sqrt {1-2 x} \sqrt {5 x+3}}{453789 (3 x+2)^{5/2}}-\frac {67345 \sqrt {1-2 x} \sqrt {5 x+3}}{64827 (3 x+2)^{7/2}}+\frac {295 \sqrt {1-2 x} \sqrt {5 x+3}}{1323 (3 x+2)^{9/2}}-\frac {6036028 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{22235661} \]
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Rubi [A] time = 0.10, antiderivative size = 253, 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} \[ \frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)^{9/2}}+\frac {6036028 \sqrt {1-2 x} \sqrt {5 x+3}}{22235661 \sqrt {3 x+2}}-\frac {392998 \sqrt {1-2 x} \sqrt {5 x+3}}{3176523 (3 x+2)^{3/2}}-\frac {167228 \sqrt {1-2 x} \sqrt {5 x+3}}{453789 (3 x+2)^{5/2}}-\frac {67345 \sqrt {1-2 x} \sqrt {5 x+3}}{64827 (3 x+2)^{7/2}}+\frac {295 \sqrt {1-2 x} \sqrt {5 x+3}}{1323 (3 x+2)^{9/2}}-\frac {1199452 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{22235661}-\frac {6036028 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{22235661} \]
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)^{11/2}} \, dx &=\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)^{9/2}}-\frac {1}{7} \int \frac {\left (-\frac {555}{2}-490 x\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^{11/2}} \, dx\\ &=\frac {295 \sqrt {1-2 x} \sqrt {3+5 x}}{1323 (2+3 x)^{9/2}}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)^{9/2}}-\frac {2 \int \frac {-\frac {169585}{4}-\frac {144025 x}{2}}{\sqrt {1-2 x} (2+3 x)^{9/2} \sqrt {3+5 x}} \, dx}{1323}\\ &=\frac {295 \sqrt {1-2 x} \sqrt {3+5 x}}{1323 (2+3 x)^{9/2}}-\frac {67345 \sqrt {1-2 x} \sqrt {3+5 x}}{64827 (2+3 x)^{7/2}}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)^{9/2}}-\frac {4 \int \frac {-245765-\frac {1683625 x}{4}}{\sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}} \, dx}{64827}\\ &=\frac {295 \sqrt {1-2 x} \sqrt {3+5 x}}{1323 (2+3 x)^{9/2}}-\frac {67345 \sqrt {1-2 x} \sqrt {3+5 x}}{64827 (2+3 x)^{7/2}}-\frac {167228 \sqrt {1-2 x} \sqrt {3+5 x}}{453789 (2+3 x)^{5/2}}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)^{9/2}}-\frac {8 \int \frac {-\frac {7378905}{8}-\frac {3135525 x}{2}}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx}{2268945}\\ &=\frac {295 \sqrt {1-2 x} \sqrt {3+5 x}}{1323 (2+3 x)^{9/2}}-\frac {67345 \sqrt {1-2 x} \sqrt {3+5 x}}{64827 (2+3 x)^{7/2}}-\frac {167228 \sqrt {1-2 x} \sqrt {3+5 x}}{453789 (2+3 x)^{5/2}}-\frac {392998 \sqrt {1-2 x} \sqrt {3+5 x}}{3176523 (2+3 x)^{3/2}}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)^{9/2}}-\frac {16 \int \frac {-\frac {17369985}{8}-\frac {14737425 x}{8}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{47647845}\\ &=\frac {295 \sqrt {1-2 x} \sqrt {3+5 x}}{1323 (2+3 x)^{9/2}}-\frac {67345 \sqrt {1-2 x} \sqrt {3+5 x}}{64827 (2+3 x)^{7/2}}-\frac {167228 \sqrt {1-2 x} \sqrt {3+5 x}}{453789 (2+3 x)^{5/2}}-\frac {392998 \sqrt {1-2 x} \sqrt {3+5 x}}{3176523 (2+3 x)^{3/2}}+\frac {6036028 \sqrt {1-2 x} \sqrt {3+5 x}}{22235661 \sqrt {2+3 x}}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)^{9/2}}-\frac {32 \int \frac {-\frac {185288025}{16}-\frac {113175525 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{333534915}\\ &=\frac {295 \sqrt {1-2 x} \sqrt {3+5 x}}{1323 (2+3 x)^{9/2}}-\frac {67345 \sqrt {1-2 x} \sqrt {3+5 x}}{64827 (2+3 x)^{7/2}}-\frac {167228 \sqrt {1-2 x} \sqrt {3+5 x}}{453789 (2+3 x)^{5/2}}-\frac {392998 \sqrt {1-2 x} \sqrt {3+5 x}}{3176523 (2+3 x)^{3/2}}+\frac {6036028 \sqrt {1-2 x} \sqrt {3+5 x}}{22235661 \sqrt {2+3 x}}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)^{9/2}}+\frac {6036028 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{22235661}+\frac {6596986 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{22235661}\\ &=\frac {295 \sqrt {1-2 x} \sqrt {3+5 x}}{1323 (2+3 x)^{9/2}}-\frac {67345 \sqrt {1-2 x} \sqrt {3+5 x}}{64827 (2+3 x)^{7/2}}-\frac {167228 \sqrt {1-2 x} \sqrt {3+5 x}}{453789 (2+3 x)^{5/2}}-\frac {392998 \sqrt {1-2 x} \sqrt {3+5 x}}{3176523 (2+3 x)^{3/2}}+\frac {6036028 \sqrt {1-2 x} \sqrt {3+5 x}}{22235661 \sqrt {2+3 x}}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)^{9/2}}-\frac {6036028 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{22235661}-\frac {1199452 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{22235661}\\ \end {align*}
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Mathematica [A] time = 0.30, size = 115, normalized size = 0.45 \[ \frac {8 \sqrt {2} \left (6877465 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )+3018014 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )\right )-\frac {24 \sqrt {5 x+3} \left (488918268 x^5+985046292 x^4+466728543 x^3-227945505 x^2-243200677 x-52688263\right )}{\sqrt {1-2 x} (3 x+2)^{9/2}}}{266827932} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.80, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{2916 \, x^{8} + 8748 \, x^{7} + 8505 \, x^{6} + 756 \, x^{5} - 3780 \, x^{4} - 2016 \, x^{3} + 112 \, x^{2} + 320 \, x + 64}, x\right ) \]
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 \[ \int \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {11}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.03, size = 504, normalized size = 1.99 \[ -\frac {2 \sqrt {5 x +3}\, \sqrt {-2 x +1}\, \left (-7333774020 x^{6}-19175958792 x^{5}+244459134 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{4} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+557074665 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{4} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-15866344773 x^{4}+651891024 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{3} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+1485532440 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{3} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-781374312 x^{3}+651891024 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+1485532440 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+5699519700 x^{2}+289729344 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+660236640 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+2979130038 x +48288224 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+110039440 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+474194367\right )}{66706983 \left (3 x +2\right )^{\frac {9}{2}} \left (10 x^{2}+x -3\right )} \]
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 \[ \int \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {11}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (5\,x+3\right )}^{5/2}}{{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^{11/2}} \,d x \]
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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