Optimal. Leaf size=191 \[ -\frac {10628 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{324135}-\frac {2 \sqrt {1-2 x} (5 x+3)^{3/2}}{21 (3 x+2)^{7/2}}+\frac {475592 \sqrt {1-2 x} \sqrt {5 x+3}}{324135 \sqrt {3 x+2}}+\frac {8578 \sqrt {1-2 x} \sqrt {5 x+3}}{46305 (3 x+2)^{3/2}}-\frac {214 \sqrt {1-2 x} \sqrt {5 x+3}}{2205 (3 x+2)^{5/2}}-\frac {475592 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{324135} \]
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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 = {97, 150, 152, 158, 113, 119} \[ -\frac {2 \sqrt {1-2 x} (5 x+3)^{3/2}}{21 (3 x+2)^{7/2}}+\frac {475592 \sqrt {1-2 x} \sqrt {5 x+3}}{324135 \sqrt {3 x+2}}+\frac {8578 \sqrt {1-2 x} \sqrt {5 x+3}}{46305 (3 x+2)^{3/2}}-\frac {214 \sqrt {1-2 x} \sqrt {5 x+3}}{2205 (3 x+2)^{5/2}}-\frac {10628 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{324135}-\frac {475592 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{324135} \]
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)^{3/2}}{(2+3 x)^{9/2}} \, dx &=-\frac {2 \sqrt {1-2 x} (3+5 x)^{3/2}}{21 (2+3 x)^{7/2}}+\frac {2}{21} \int \frac {\left (\frac {9}{2}-20 x\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^{7/2}} \, dx\\ &=-\frac {214 \sqrt {1-2 x} \sqrt {3+5 x}}{2205 (2+3 x)^{5/2}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{3/2}}{21 (2+3 x)^{7/2}}+\frac {4 \int \frac {-\frac {1097}{4}-\frac {1895 x}{2}}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx}{2205}\\ &=-\frac {214 \sqrt {1-2 x} \sqrt {3+5 x}}{2205 (2+3 x)^{5/2}}+\frac {8578 \sqrt {1-2 x} \sqrt {3+5 x}}{46305 (2+3 x)^{3/2}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{3/2}}{21 (2+3 x)^{7/2}}+\frac {8 \int \frac {6334-\frac {21445 x}{4}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{46305}\\ &=-\frac {214 \sqrt {1-2 x} \sqrt {3+5 x}}{2205 (2+3 x)^{5/2}}+\frac {8578 \sqrt {1-2 x} \sqrt {3+5 x}}{46305 (2+3 x)^{3/2}}+\frac {475592 \sqrt {1-2 x} \sqrt {3+5 x}}{324135 \sqrt {2+3 x}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{3/2}}{21 (2+3 x)^{7/2}}+\frac {16 \int \frac {\frac {742615}{8}+\frac {297245 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{324135}\\ &=-\frac {214 \sqrt {1-2 x} \sqrt {3+5 x}}{2205 (2+3 x)^{5/2}}+\frac {8578 \sqrt {1-2 x} \sqrt {3+5 x}}{46305 (2+3 x)^{3/2}}+\frac {475592 \sqrt {1-2 x} \sqrt {3+5 x}}{324135 \sqrt {2+3 x}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{3/2}}{21 (2+3 x)^{7/2}}+\frac {58454 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{324135}+\frac {475592 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{324135}\\ &=-\frac {214 \sqrt {1-2 x} \sqrt {3+5 x}}{2205 (2+3 x)^{5/2}}+\frac {8578 \sqrt {1-2 x} \sqrt {3+5 x}}{46305 (2+3 x)^{3/2}}+\frac {475592 \sqrt {1-2 x} \sqrt {3+5 x}}{324135 \sqrt {2+3 x}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{3/2}}{21 (2+3 x)^{7/2}}-\frac {475592 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{324135}-\frac {10628 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{324135}\\ \end {align*}
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Mathematica [A] time = 0.34, size = 104, normalized size = 0.54 \[ \frac {2 \left (\sqrt {2} \left (237796 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )-150115 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )\right )+\frac {3 \sqrt {1-2 x} \sqrt {5 x+3} \left (6420492 x^3+13111191 x^2+8796570 x+1944697\right )}{(3 x+2)^{7/2}}\right )}{972405} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.33, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (5 \, x + 3\right )}^{\frac {3}{2}} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32}, x\right ) \]
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 \[ \int \frac {{\left (5 \, x + 3\right )}^{\frac {3}{2}} \sqrt {-2 \, x + 1}}{{\left (3 \, x + 2\right )}^{\frac {9}{2}}}\,{d x} \]
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 \[ \frac {2 \left (192614760 x^{5}+412597206 x^{4}-6420492 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{3} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+4053105 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{3} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+245446245 x^{3}-12840984 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+8106210 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-33270099 x^{2}-8560656 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+5404140 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-73335039 x -1902368 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+1200920 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-17502273\right ) \sqrt {-2 x +1}\, \sqrt {5 x +3}}{972405 \left (10 x^{2}+x -3\right ) \left (3 x +2\right )^{\frac {7}{2}}} \]
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 \[ \int \frac {{\left (5 \, x + 3\right )}^{\frac {3}{2}} \sqrt {-2 \, x + 1}}{{\left (3 \, x + 2\right )}^{\frac {9}{2}}}\,{d x} \]
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 \[ \int \frac {\sqrt {1-2\,x}\,{\left (5\,x+3\right )}^{3/2}}{{\left (3\,x+2\right )}^{9/2}} \,d x \]
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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