Optimal. Leaf size=156 \[ -\frac {41 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{605 \sqrt {33}}+\frac {7 (3 x+2)^{3/2}}{33 (1-2 x)^{3/2} \sqrt {5 x+3}}+\frac {974 \sqrt {1-2 x} \sqrt {3 x+2}}{3993 \sqrt {5 x+3}}-\frac {203 \sqrt {3 x+2}}{363 \sqrt {1-2 x} \sqrt {5 x+3}}-\frac {974 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{605 \sqrt {33}} \]
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Rubi [A] time = 0.05, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 6, 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} \[ \frac {7 (3 x+2)^{3/2}}{33 (1-2 x)^{3/2} \sqrt {5 x+3}}+\frac {974 \sqrt {1-2 x} \sqrt {3 x+2}}{3993 \sqrt {5 x+3}}-\frac {203 \sqrt {3 x+2}}{363 \sqrt {1-2 x} \sqrt {5 x+3}}-\frac {41 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{605 \sqrt {33}}-\frac {974 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{605 \sqrt {33}} \]
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 {(2+3 x)^{5/2}}{(1-2 x)^{5/2} (3+5 x)^{3/2}} \, dx &=\frac {7 (2+3 x)^{3/2}}{33 (1-2 x)^{3/2} \sqrt {3+5 x}}-\frac {1}{33} \int \frac {\sqrt {2+3 x} \left (\frac {107}{2}+96 x\right )}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx\\ &=-\frac {203 \sqrt {2+3 x}}{363 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {7 (2+3 x)^{3/2}}{33 (1-2 x)^{3/2} \sqrt {3+5 x}}-\frac {1}{363} \int \frac {\frac {121}{2}-\frac {123 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx\\ &=-\frac {203 \sqrt {2+3 x}}{363 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {974 \sqrt {1-2 x} \sqrt {2+3 x}}{3993 \sqrt {3+5 x}}+\frac {7 (2+3 x)^{3/2}}{33 (1-2 x)^{3/2} \sqrt {3+5 x}}+\frac {2 \int \frac {\frac {3777}{4}+1461 x}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{3993}\\ &=-\frac {203 \sqrt {2+3 x}}{363 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {974 \sqrt {1-2 x} \sqrt {2+3 x}}{3993 \sqrt {3+5 x}}+\frac {7 (2+3 x)^{3/2}}{33 (1-2 x)^{3/2} \sqrt {3+5 x}}+\frac {41 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{1210}+\frac {974 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{6655}\\ &=-\frac {203 \sqrt {2+3 x}}{363 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {974 \sqrt {1-2 x} \sqrt {2+3 x}}{3993 \sqrt {3+5 x}}+\frac {7 (2+3 x)^{3/2}}{33 (1-2 x)^{3/2} \sqrt {3+5 x}}-\frac {974 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{605 \sqrt {33}}-\frac {41 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{605 \sqrt {33}}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 102, normalized size = 0.65 \[ \frac {-595 \sqrt {2} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )+\frac {10 \sqrt {3 x+2} \left (3896 x^2+3111 x+435\right )}{(1-2 x)^{3/2} \sqrt {5 x+3}}+1948 \sqrt {2} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )}{39930} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.68, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (9 \, x^{2} + 12 \, x + 4\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{200 \, x^{5} - 60 \, x^{4} - 138 \, x^{3} + 47 \, x^{2} + 24 \, x - 9}, x\right ) \]
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 \[ \int \frac {{\left (3 \, x + 2\right )}^{\frac {5}{2}}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.03, size = 228, normalized size = 1.46 \[ \frac {\left (116880 x^{3}+171250 x^{2}-3896 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+1190 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+75270 x +1948 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-595 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+8700\right ) \sqrt {5 x +3}\, \sqrt {-2 x +1}\, \sqrt {3 x +2}}{39930 \left (15 x^{2}+19 x +6\right ) \left (2 x -1\right )^{2}} \]
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 \[ \int \frac {{\left (3 \, x + 2\right )}^{\frac {5}{2}}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (3\,x+2\right )}^{5/2}}{{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{3/2}} \,d x \]
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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