Optimal. Leaf size=191 \[ -\frac {1061 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{2835}-\frac {32}{63} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}-\frac {2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{3 \sqrt {3 x+2}}+\frac {202}{63} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}-\frac {1061}{567} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}-\frac {2894 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2835} \]
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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 = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {97, 154, 158, 113, 119} \[ -\frac {32}{63} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}-\frac {2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{3 \sqrt {3 x+2}}+\frac {202}{63} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}-\frac {1061}{567} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}-\frac {1061 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2835}-\frac {2894 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2835} \]
Antiderivative was successfully verified.
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Rule 97
Rule 113
Rule 119
Rule 154
Rule 158
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^{3/2}} \, dx &=-\frac {2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{3 \sqrt {2+3 x}}+\frac {2}{3} \int \frac {\left (\frac {7}{2}-40 x\right ) \sqrt {1-2 x} (3+5 x)^{3/2}}{\sqrt {2+3 x}} \, dx\\ &=-\frac {2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{3 \sqrt {2+3 x}}-\frac {32}{63} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {4}{315} \int \frac {\left (\frac {4495}{4}-\frac {7575 x}{2}\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=\frac {202}{63} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{3 \sqrt {2+3 x}}-\frac {32}{63} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {4 \int \frac {\left (\frac {1125}{2}-\frac {79575 x}{4}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{4725}\\ &=-\frac {1061}{567} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {202}{63} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{3 \sqrt {2+3 x}}-\frac {32}{63} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {4 \int \frac {\frac {435525}{8}+\frac {108525 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{42525}\\ &=-\frac {1061}{567} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {202}{63} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{3 \sqrt {2+3 x}}-\frac {32}{63} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {2894 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{2835}+\frac {11671 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{5670}\\ &=-\frac {1061}{567} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {202}{63} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{3 \sqrt {2+3 x}}-\frac {32}{63} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {2894 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2835}-\frac {1061 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2835}\\ \end {align*}
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Mathematica [A] time = 0.35, size = 107, normalized size = 0.56 \[ \frac {29225 \sqrt {2} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )+\frac {30 \sqrt {1-2 x} \sqrt {5 x+3} \left (-2700 x^3+180 x^2+1767 x+200\right )}{\sqrt {3 x+2}}+5788 \sqrt {2} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )}{17010} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.10, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{9 \, x^{2} + 12 \, x + 4}, x\right ) \]
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 \[ \int \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (3 \, x + 2\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 150, normalized size = 0.79 \[ -\frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \left (810000 x^{5}+27000 x^{4}-778500 x^{3}-96810 x^{2}+153030 x +5788 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+29225 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+18000\right )}{17010 \left (30 x^{3}+23 x^{2}-7 x -6\right )} \]
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 \[ \int \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (3 \, x + 2\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (1-2\,x\right )}^{3/2}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^{3/2}} \,d x \]
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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