Optimal. Leaf size=191 \[ -\frac {1048 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{1323}-\frac {2 \sqrt {5 x+3} (1-2 x)^{5/2}}{21 (3 x+2)^{7/2}}+\frac {2 \sqrt {5 x+3} (1-2 x)^{3/2}}{7 (3 x+2)^{5/2}}+\frac {36052 \sqrt {5 x+3} \sqrt {1-2 x}}{1323 \sqrt {3 x+2}}+\frac {524 \sqrt {5 x+3} \sqrt {1-2 x}}{189 (3 x+2)^{3/2}}-\frac {36052 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1323} \]
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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 = {97, 150, 152, 158, 113, 119} \[ -\frac {2 \sqrt {5 x+3} (1-2 x)^{5/2}}{21 (3 x+2)^{7/2}}+\frac {2 \sqrt {5 x+3} (1-2 x)^{3/2}}{7 (3 x+2)^{5/2}}+\frac {36052 \sqrt {5 x+3} \sqrt {1-2 x}}{1323 \sqrt {3 x+2}}+\frac {524 \sqrt {5 x+3} \sqrt {1-2 x}}{189 (3 x+2)^{3/2}}-\frac {1048 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1323}-\frac {36052 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1323} \]
Antiderivative was successfully verified.
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Rule 97
Rule 113
Rule 119
Rule 150
Rule 152
Rule 158
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{(2+3 x)^{9/2}} \, dx &=-\frac {2 (1-2 x)^{5/2} \sqrt {3+5 x}}{21 (2+3 x)^{7/2}}+\frac {2}{21} \int \frac {\left (-\frac {25}{2}-30 x\right ) (1-2 x)^{3/2}}{(2+3 x)^{7/2} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} \sqrt {3+5 x}}{21 (2+3 x)^{7/2}}+\frac {2 (1-2 x)^{3/2} \sqrt {3+5 x}}{7 (2+3 x)^{5/2}}-\frac {4}{315} \int \frac {\left (-\frac {705}{2}-\frac {75 x}{2}\right ) \sqrt {1-2 x}}{(2+3 x)^{5/2} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} \sqrt {3+5 x}}{21 (2+3 x)^{7/2}}+\frac {2 (1-2 x)^{3/2} \sqrt {3+5 x}}{7 (2+3 x)^{5/2}}+\frac {524 \sqrt {1-2 x} \sqrt {3+5 x}}{189 (2+3 x)^{3/2}}+\frac {8 \int \frac {\frac {31665}{4}-5025 x}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{2835}\\ &=-\frac {2 (1-2 x)^{5/2} \sqrt {3+5 x}}{21 (2+3 x)^{7/2}}+\frac {2 (1-2 x)^{3/2} \sqrt {3+5 x}}{7 (2+3 x)^{5/2}}+\frac {524 \sqrt {1-2 x} \sqrt {3+5 x}}{189 (2+3 x)^{3/2}}+\frac {36052 \sqrt {1-2 x} \sqrt {3+5 x}}{1323 \sqrt {2+3 x}}+\frac {16 \int \frac {106800+\frac {675975 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{19845}\\ &=-\frac {2 (1-2 x)^{5/2} \sqrt {3+5 x}}{21 (2+3 x)^{7/2}}+\frac {2 (1-2 x)^{3/2} \sqrt {3+5 x}}{7 (2+3 x)^{5/2}}+\frac {524 \sqrt {1-2 x} \sqrt {3+5 x}}{189 (2+3 x)^{3/2}}+\frac {36052 \sqrt {1-2 x} \sqrt {3+5 x}}{1323 \sqrt {2+3 x}}+\frac {5764 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{1323}+\frac {36052 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{1323}\\ &=-\frac {2 (1-2 x)^{5/2} \sqrt {3+5 x}}{21 (2+3 x)^{7/2}}+\frac {2 (1-2 x)^{3/2} \sqrt {3+5 x}}{7 (2+3 x)^{5/2}}+\frac {524 \sqrt {1-2 x} \sqrt {3+5 x}}{189 (2+3 x)^{3/2}}+\frac {36052 \sqrt {1-2 x} \sqrt {3+5 x}}{1323 \sqrt {2+3 x}}-\frac {36052 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1323}-\frac {1048 \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 = 0.34, size = 106, normalized size = 0.55 \[ \frac {4 \left (\sqrt {2} \left (9013 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )-4690 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )\right )+\frac {3 \sqrt {1-2 x} \sqrt {5 x+3} \left (486702 x^3+988524 x^2+671007 x+151859\right )}{2 (3 x+2)^{7/2}}\right )}{3969} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.98, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (4 \, x^{2} - 4 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32}, x\right ) \]
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 \[ \int \frac {\sqrt {5 \, x + 3} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {9}{2}}}\,{d x} \]
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 \[ \frac {2 \left (14601060 x^{5}+31115826 x^{4}-486702 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{3} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+253260 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{3} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+18715464 x^{3}-973404 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+506520 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-2327925 x^{2}-648936 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+337680 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-5583486 x -144208 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+75040 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-1366731\right ) \sqrt {5 x +3}\, \sqrt {-2 x +1}}{3969 \left (10 x^{2}+x -3\right ) \left (3 x +2\right )^{\frac {7}{2}}} \]
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 \[ \int \frac {\sqrt {5 \, x + 3} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {9}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (1-2\,x\right )}^{5/2}\,\sqrt {5\,x+3}}{{\left (3\,x+2\right )}^{9/2}} \,d x \]
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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