Optimal. Leaf size=160 \[ \frac {823 \sqrt {\frac {3}{11}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{1375}+\frac {7 (3 x+2)^{5/2}}{11 \sqrt {1-2 x} \sqrt {5 x+3}}-\frac {37 \sqrt {1-2 x} (3 x+2)^{3/2}}{605 \sqrt {5 x+3}}+\frac {2388 \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}}{3025}+\frac {55019 \sqrt {\frac {3}{11}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2750} \]
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Rubi [A] time = 0.05, antiderivative size = 160, 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, 154, 158, 113, 119} \[ \frac {7 (3 x+2)^{5/2}}{11 \sqrt {1-2 x} \sqrt {5 x+3}}-\frac {37 \sqrt {1-2 x} (3 x+2)^{3/2}}{605 \sqrt {5 x+3}}+\frac {2388 \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}}{3025}+\frac {823 \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1375}+\frac {55019 \sqrt {\frac {3}{11}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2750} \]
Antiderivative was successfully verified.
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Rule 98
Rule 113
Rule 119
Rule 150
Rule 154
Rule 158
Rubi steps
\begin {align*} \int \frac {(2+3 x)^{7/2}}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx &=\frac {7 (2+3 x)^{5/2}}{11 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {1}{11} \int \frac {(2+3 x)^{3/2} \left (\frac {241}{2}+207 x\right )}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx\\ &=-\frac {37 \sqrt {1-2 x} (2+3 x)^{3/2}}{605 \sqrt {3+5 x}}+\frac {7 (2+3 x)^{5/2}}{11 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {2}{605} \int \frac {\sqrt {2+3 x} \left (\frac {8775}{4}+3582 x\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {37 \sqrt {1-2 x} (2+3 x)^{3/2}}{605 \sqrt {3+5 x}}+\frac {7 (2+3 x)^{5/2}}{11 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {2388 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{3025}+\frac {2 \int \frac {-\frac {156699}{2}-\frac {495171 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{9075}\\ &=-\frac {37 \sqrt {1-2 x} (2+3 x)^{3/2}}{605 \sqrt {3+5 x}}+\frac {7 (2+3 x)^{5/2}}{11 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {2388 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{3025}-\frac {2469 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{2750}-\frac {165057 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{30250}\\ &=-\frac {37 \sqrt {1-2 x} (2+3 x)^{3/2}}{605 \sqrt {3+5 x}}+\frac {7 (2+3 x)^{5/2}}{11 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {2388 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{3025}+\frac {55019 \sqrt {\frac {3}{11}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2750}+\frac {823 \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1375}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 127, normalized size = 0.79 \[ \frac {27860 \sqrt {2-4 x} (5 x+3) \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )+10 \sqrt {3 x+2} \sqrt {5 x+3} \left (-5445 x^2+20897 x+14494\right )-55019 \sqrt {2-4 x} (5 x+3) E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )}{30250 \sqrt {1-2 x} (5 x+3)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.67, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9}, x\right ) \]
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 \[ \int \frac {{\left (3 \, x + 2\right )}^{\frac {7}{2}}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 140, normalized size = 0.88 \[ -\frac {\sqrt {3 x +2}\, \sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (-163350 x^{3}+518010 x^{2}+852760 x -55019 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+27860 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+289880\right )}{30250 \left (30 x^{3}+23 x^{2}-7 x -6\right )} \]
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 \[ \int \frac {{\left (3 \, x + 2\right )}^{\frac {7}{2}}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (3\,x+2\right )}^{7/2}}{{\left (1-2\,x\right )}^{3/2}\,{\left (5\,x+3\right )}^{3/2}} \,d x \]
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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