Optimal. Leaf size=187 \[ -\frac {104663 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{171875 \sqrt {33}}-\frac {2 \sqrt {1-2 x} (3 x+2)^{7/2}}{165 (5 x+3)^{3/2}}-\frac {668 \sqrt {1-2 x} (3 x+2)^{5/2}}{9075 \sqrt {5 x+3}}+\frac {403 \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}}{75625}-\frac {87476 \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}}{378125}-\frac {6515539 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{343750 \sqrt {33}} \]
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Rubi [A] time = 0.07, antiderivative size = 187, 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} \[ -\frac {2 \sqrt {1-2 x} (3 x+2)^{7/2}}{165 (5 x+3)^{3/2}}-\frac {668 \sqrt {1-2 x} (3 x+2)^{5/2}}{9075 \sqrt {5 x+3}}+\frac {403 \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}}{75625}-\frac {87476 \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}}{378125}-\frac {104663 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{171875 \sqrt {33}}-\frac {6515539 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{343750 \sqrt {33}} \]
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}}{\sqrt {1-2 x} (3+5 x)^{5/2}} \, dx &=-\frac {2 \sqrt {1-2 x} (2+3 x)^{7/2}}{165 (3+5 x)^{3/2}}-\frac {2}{165} \int \frac {\left (-\frac {227}{2}-\frac {267 x}{2}\right ) (2+3 x)^{5/2}}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^{7/2}}{165 (3+5 x)^{3/2}}-\frac {668 \sqrt {1-2 x} (2+3 x)^{5/2}}{9075 \sqrt {3+5 x}}-\frac {4 \int \frac {(2+3 x)^{3/2} \left (-2721+\frac {1209 x}{4}\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{9075}\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^{7/2}}{165 (3+5 x)^{3/2}}-\frac {668 \sqrt {1-2 x} (2+3 x)^{5/2}}{9075 \sqrt {3+5 x}}+\frac {403 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{75625}+\frac {4 \int \frac {\sqrt {2+3 x} \left (\frac {1058175}{8}+196821 x\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{226875}\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^{7/2}}{165 (3+5 x)^{3/2}}-\frac {668 \sqrt {1-2 x} (2+3 x)^{5/2}}{9075 \sqrt {3+5 x}}-\frac {87476 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{378125}+\frac {403 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{75625}-\frac {4 \int \frac {-\frac {18628119}{4}-\frac {58639851 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{3403125}\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^{7/2}}{165 (3+5 x)^{3/2}}-\frac {668 \sqrt {1-2 x} (2+3 x)^{5/2}}{9075 \sqrt {3+5 x}}-\frac {87476 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{378125}+\frac {403 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{75625}+\frac {104663 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{343750}+\frac {6515539 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{3781250}\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^{7/2}}{165 (3+5 x)^{3/2}}-\frac {668 \sqrt {1-2 x} (2+3 x)^{5/2}}{9075 \sqrt {3+5 x}}-\frac {87476 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{378125}+\frac {403 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{75625}-\frac {6515539 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{343750 \sqrt {33}}-\frac {104663 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{171875 \sqrt {33}}\\ \end {align*}
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Mathematica [A] time = 0.31, size = 107, normalized size = 0.57 \[ \frac {-3061660 \sqrt {2} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )-\frac {10 \sqrt {1-2 x} \sqrt {3 x+2} \left (3675375 x^3+13721400 x^2+12517925 x+3365042\right )}{(5 x+3)^{3/2}}+6515539 \sqrt {2} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )}{11343750} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{250 \, x^{4} + 325 \, x^{3} + 45 \, x^{2} - 81 \, x - 27}, x\right ) \]
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 \[ \int \frac {{\left (3 \, x + 2\right )}^{\frac {9}{2}}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} \sqrt {-2 \, x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 229, normalized size = 1.22 \[ \frac {\left (-220522500 x^{5}-860037750 x^{4}-814782000 x^{3}-52653770 x^{2}-32577695 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+15308300 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+216708080 x -19546617 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+9184980 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+67300840\right ) \sqrt {-2 x +1}\, \sqrt {3 x +2}}{11343750 \left (6 x^{2}+x -2\right ) \left (5 x +3\right )^{\frac {3}{2}}} \]
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 \[ \int \frac {{\left (3 \, x + 2\right )}^{\frac {9}{2}}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} \sqrt {-2 \, x + 1}}\,{d x} \]
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 \[ \int \frac {{\left (3\,x+2\right )}^{9/2}}{\sqrt {1-2\,x}\,{\left (5\,x+3\right )}^{5/2}} \,d x \]
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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