Optimal. Leaf size=191 \[ \frac {23612 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{108045}-\frac {2 \sqrt {1-2 x} (5 x+3)^{5/2}}{21 (3 x+2)^{7/2}}-\frac {118 \sqrt {1-2 x} (5 x+3)^{3/2}}{735 (3 x+2)^{5/2}}+\frac {173482 \sqrt {1-2 x} \sqrt {5 x+3}}{108045 \sqrt {3 x+2}}-\frac {4282 \sqrt {1-2 x} \sqrt {5 x+3}}{15435 (3 x+2)^{3/2}}-\frac {173482 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{108045} \]
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Rubi [A] time = 0.06, 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 = {97, 150, 152, 158, 113, 119} \[ -\frac {2 \sqrt {1-2 x} (5 x+3)^{5/2}}{21 (3 x+2)^{7/2}}-\frac {118 \sqrt {1-2 x} (5 x+3)^{3/2}}{735 (3 x+2)^{5/2}}+\frac {173482 \sqrt {1-2 x} \sqrt {5 x+3}}{108045 \sqrt {3 x+2}}-\frac {4282 \sqrt {1-2 x} \sqrt {5 x+3}}{15435 (3 x+2)^{3/2}}+\frac {23612 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{108045}-\frac {173482 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{108045} \]
Antiderivative was successfully verified.
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Rule 97
Rule 113
Rule 119
Rule 150
Rule 152
Rule 158
Rubi steps
\begin {align*} \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^{9/2}} \, dx &=-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{21 (2+3 x)^{7/2}}+\frac {2}{21} \int \frac {\left (\frac {19}{2}-30 x\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^{7/2}} \, dx\\ &=-\frac {118 \sqrt {1-2 x} (3+5 x)^{3/2}}{735 (2+3 x)^{5/2}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{21 (2+3 x)^{7/2}}+\frac {4 \int \frac {\left (\frac {603}{4}-\frac {4365 x}{2}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^{5/2}} \, dx}{2205}\\ &=-\frac {4282 \sqrt {1-2 x} \sqrt {3+5 x}}{15435 (2+3 x)^{3/2}}-\frac {118 \sqrt {1-2 x} (3+5 x)^{3/2}}{735 (2+3 x)^{5/2}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{21 (2+3 x)^{7/2}}+\frac {8 \int \frac {-\frac {222417}{8}-90495 x}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{138915}\\ &=-\frac {4282 \sqrt {1-2 x} \sqrt {3+5 x}}{15435 (2+3 x)^{3/2}}+\frac {173482 \sqrt {1-2 x} \sqrt {3+5 x}}{108045 \sqrt {2+3 x}}-\frac {118 \sqrt {1-2 x} (3+5 x)^{3/2}}{735 (2+3 x)^{5/2}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{21 (2+3 x)^{7/2}}+\frac {16 \int \frac {\frac {878805}{4}+\frac {3903345 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{972405}\\ &=-\frac {4282 \sqrt {1-2 x} \sqrt {3+5 x}}{15435 (2+3 x)^{3/2}}+\frac {173482 \sqrt {1-2 x} \sqrt {3+5 x}}{108045 \sqrt {2+3 x}}-\frac {118 \sqrt {1-2 x} (3+5 x)^{3/2}}{735 (2+3 x)^{5/2}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{21 (2+3 x)^{7/2}}-\frac {129866 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{108045}+\frac {173482 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{108045}\\ &=-\frac {4282 \sqrt {1-2 x} \sqrt {3+5 x}}{15435 (2+3 x)^{3/2}}+\frac {173482 \sqrt {1-2 x} \sqrt {3+5 x}}{108045 \sqrt {2+3 x}}-\frac {118 \sqrt {1-2 x} (3+5 x)^{3/2}}{735 (2+3 x)^{5/2}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{21 (2+3 x)^{7/2}}-\frac {173482 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{108045}+\frac {23612 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{108045}\\ \end {align*}
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Mathematica [A] time = 0.31, size = 104, normalized size = 0.54 \[ \frac {2 \left (\sqrt {2} \left (86741 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )-281540 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )\right )+\frac {3 \sqrt {1-2 x} \sqrt {5 x+3} \left (2342007 x^3+4290411 x^2+2623695 x+535637\right )}{(3 x+2)^{7/2}}\right )}{324135} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.31, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32}, x\right ) \]
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 \[ \int \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} \sqrt {-2 \, x + 1}}{{\left (3 \, x + 2\right )}^{\frac {9}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 409, normalized size = 2.14 \[ \frac {2 \left (70260210 x^{5}+135738351 x^{4}-2342007 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{3} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+7601580 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{3} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+70504020 x^{3}-4684014 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+15203160 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-14673504 x^{2}-3122676 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+10135440 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-22006344 x -693928 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+2252320 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-4820733\right ) \sqrt {-2 x +1}\, \sqrt {5 x +3}}{324135 \left (10 x^{2}+x -3\right ) \left (3 x +2\right )^{\frac {7}{2}}} \]
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 \[ \int \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} \sqrt {-2 \, x + 1}}{{\left (3 \, x + 2\right )}^{\frac {9}{2}}}\,{d x} \]
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 \[ \int \frac {\sqrt {1-2\,x}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^{9/2}} \,d x \]
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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