Optimal. Leaf size=191 \[ \frac {175111 \sqrt {\frac {3}{11}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{68750}+\frac {7 (3 x+2)^{7/2}}{11 \sqrt {1-2 x} \sqrt {5 x+3}}-\frac {37 \sqrt {1-2 x} (3 x+2)^{5/2}}{605 \sqrt {5 x+3}}+\frac {10851 \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}}{15125}+\frac {502941 \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}}{151250}+\frac {2911577 \sqrt {\frac {3}{11}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{34375} \]
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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 = {98, 150, 154, 158, 113, 119} \[ \frac {7 (3 x+2)^{7/2}}{11 \sqrt {1-2 x} \sqrt {5 x+3}}-\frac {37 \sqrt {1-2 x} (3 x+2)^{5/2}}{605 \sqrt {5 x+3}}+\frac {10851 \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}}{15125}+\frac {502941 \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}}{151250}+\frac {175111 \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{68750}+\frac {2911577 \sqrt {\frac {3}{11}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{34375} \]
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)^{9/2}}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx &=\frac {7 (2+3 x)^{7/2}}{11 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {1}{11} \int \frac {(2+3 x)^{5/2} \left (\frac {367}{2}+312 x\right )}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx\\ &=-\frac {37 \sqrt {1-2 x} (2+3 x)^{5/2}}{605 \sqrt {3+5 x}}+\frac {7 (2+3 x)^{7/2}}{11 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {2}{605} \int \frac {(2+3 x)^{3/2} \left (\frac {13173}{4}+\frac {10851 x}{2}\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {37 \sqrt {1-2 x} (2+3 x)^{5/2}}{605 \sqrt {3+5 x}}+\frac {7 (2+3 x)^{7/2}}{11 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {10851 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{15125}+\frac {2 \int \frac {\left (-\frac {929925}{4}-\frac {1508823 x}{4}\right ) \sqrt {2+3 x}}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{15125}\\ &=-\frac {37 \sqrt {1-2 x} (2+3 x)^{5/2}}{605 \sqrt {3+5 x}}+\frac {7 (2+3 x)^{7/2}}{11 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {502941 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{151250}+\frac {10851 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{15125}-\frac {2 \int \frac {\frac {66357261}{8}+\frac {26204193 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{226875}\\ &=-\frac {37 \sqrt {1-2 x} (2+3 x)^{5/2}}{605 \sqrt {3+5 x}}+\frac {7 (2+3 x)^{7/2}}{11 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {502941 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{151250}+\frac {10851 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{15125}-\frac {525333 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{137500}-\frac {8734731 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{378125}\\ &=-\frac {37 \sqrt {1-2 x} (2+3 x)^{5/2}}{605 \sqrt {3+5 x}}+\frac {7 (2+3 x)^{7/2}}{11 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {502941 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{151250}+\frac {10851 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{15125}+\frac {2911577 \sqrt {\frac {3}{11}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{34375}+\frac {175111 \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{68750}\\ \end {align*}
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Mathematica [A] time = 0.32, size = 132, normalized size = 0.69 \[ \frac {5867645 \sqrt {2-4 x} (5 x+3) \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )+10 \sqrt {3 x+2} \sqrt {5 x+3} \left (-490050 x^3-2188890 x^2+3684629 x+2892883\right )-11646308 \sqrt {2-4 x} (5 x+3) E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )}{1512500 \sqrt {1-2 x} (5 x+3)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.78, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9}, x\right ) \]
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 \[ \int \frac {{\left (3 \, x + 2\right )}^{\frac {9}{2}}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 145, normalized size = 0.76 \[ -\frac {\sqrt {3 x +2}\, \sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (-14701500 x^{4}-75467700 x^{3}+66761070 x^{2}+160479070 x -11646308 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+5867645 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+57857660\right )}{1512500 \left (30 x^{3}+23 x^{2}-7 x -6\right )} \]
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 \[ \int \frac {{\left (3 \, x + 2\right )}^{\frac {9}{2}}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (3\,x+2\right )}^{9/2}}{{\left (1-2\,x\right )}^{3/2}\,{\left (5\,x+3\right )}^{3/2}} \,d x \]
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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