Optimal. Leaf size=249 \[ -\frac {37904696 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{47647845 \sqrt {33}}-\frac {2 \sqrt {1-2 x} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}-\frac {118 \sqrt {1-2 x} (5 x+3)^{3/2}}{2079 (3 x+2)^{9/2}}+\frac {1305025844 \sqrt {1-2 x} \sqrt {5 x+3}}{524126295 \sqrt {3 x+2}}+\frac {19417096 \sqrt {1-2 x} \sqrt {5 x+3}}{74875185 (3 x+2)^{3/2}}+\frac {627806 \sqrt {1-2 x} \sqrt {5 x+3}}{10696455 (3 x+2)^{5/2}}-\frac {13022 \sqrt {1-2 x} \sqrt {5 x+3}}{305613 (3 x+2)^{7/2}}-\frac {1305025844 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{47647845 \sqrt {33}} \]
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Rubi [A] time = 0.10, antiderivative size = 249, normalized size of antiderivative = 1.00, number of steps used = 9, 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} \[ -\frac {2 \sqrt {1-2 x} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}-\frac {118 \sqrt {1-2 x} (5 x+3)^{3/2}}{2079 (3 x+2)^{9/2}}+\frac {1305025844 \sqrt {1-2 x} \sqrt {5 x+3}}{524126295 \sqrt {3 x+2}}+\frac {19417096 \sqrt {1-2 x} \sqrt {5 x+3}}{74875185 (3 x+2)^{3/2}}+\frac {627806 \sqrt {1-2 x} \sqrt {5 x+3}}{10696455 (3 x+2)^{5/2}}-\frac {13022 \sqrt {1-2 x} \sqrt {5 x+3}}{305613 (3 x+2)^{7/2}}-\frac {37904696 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{47647845 \sqrt {33}}-\frac {1305025844 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{47647845 \sqrt {33}} \]
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)^{13/2}} \, dx &=-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac {2}{33} \int \frac {\left (\frac {19}{2}-30 x\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^{11/2}} \, dx\\ &=-\frac {118 \sqrt {1-2 x} (3+5 x)^{3/2}}{2079 (2+3 x)^{9/2}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac {4 \int \frac {\left (-\frac {189}{4}-\frac {5025 x}{2}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^{9/2}} \, dx}{6237}\\ &=-\frac {13022 \sqrt {1-2 x} \sqrt {3+5 x}}{305613 (2+3 x)^{7/2}}-\frac {118 \sqrt {1-2 x} (3+5 x)^{3/2}}{2079 (2+3 x)^{9/2}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac {8 \int \frac {-\frac {676497}{8}-185700 x}{\sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}} \, dx}{916839}\\ &=-\frac {13022 \sqrt {1-2 x} \sqrt {3+5 x}}{305613 (2+3 x)^{7/2}}+\frac {627806 \sqrt {1-2 x} \sqrt {3+5 x}}{10696455 (2+3 x)^{5/2}}-\frac {118 \sqrt {1-2 x} (3+5 x)^{3/2}}{2079 (2+3 x)^{9/2}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac {16 \int \frac {\frac {1286433}{2}-\frac {14125635 x}{8}}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx}{32089365}\\ &=-\frac {13022 \sqrt {1-2 x} \sqrt {3+5 x}}{305613 (2+3 x)^{7/2}}+\frac {627806 \sqrt {1-2 x} \sqrt {3+5 x}}{10696455 (2+3 x)^{5/2}}+\frac {19417096 \sqrt {1-2 x} \sqrt {3+5 x}}{74875185 (2+3 x)^{3/2}}-\frac {118 \sqrt {1-2 x} (3+5 x)^{3/2}}{2079 (2+3 x)^{9/2}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac {32 \int \frac {\frac {687512943}{16}-\frac {109221165 x}{4}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{673876665}\\ &=-\frac {13022 \sqrt {1-2 x} \sqrt {3+5 x}}{305613 (2+3 x)^{7/2}}+\frac {627806 \sqrt {1-2 x} \sqrt {3+5 x}}{10696455 (2+3 x)^{5/2}}+\frac {19417096 \sqrt {1-2 x} \sqrt {3+5 x}}{74875185 (2+3 x)^{3/2}}+\frac {1305025844 \sqrt {1-2 x} \sqrt {3+5 x}}{524126295 \sqrt {2+3 x}}-\frac {118 \sqrt {1-2 x} (3+5 x)^{3/2}}{2079 (2+3 x)^{9/2}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac {64 \int \frac {\frac {2319498765}{4}+\frac {14681540745 x}{16}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{4717136655}\\ &=-\frac {13022 \sqrt {1-2 x} \sqrt {3+5 x}}{305613 (2+3 x)^{7/2}}+\frac {627806 \sqrt {1-2 x} \sqrt {3+5 x}}{10696455 (2+3 x)^{5/2}}+\frac {19417096 \sqrt {1-2 x} \sqrt {3+5 x}}{74875185 (2+3 x)^{3/2}}+\frac {1305025844 \sqrt {1-2 x} \sqrt {3+5 x}}{524126295 \sqrt {2+3 x}}-\frac {118 \sqrt {1-2 x} (3+5 x)^{3/2}}{2079 (2+3 x)^{9/2}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac {18952348 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{47647845}+\frac {1305025844 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{524126295}\\ &=-\frac {13022 \sqrt {1-2 x} \sqrt {3+5 x}}{305613 (2+3 x)^{7/2}}+\frac {627806 \sqrt {1-2 x} \sqrt {3+5 x}}{10696455 (2+3 x)^{5/2}}+\frac {19417096 \sqrt {1-2 x} \sqrt {3+5 x}}{74875185 (2+3 x)^{3/2}}+\frac {1305025844 \sqrt {1-2 x} \sqrt {3+5 x}}{524126295 \sqrt {2+3 x}}-\frac {118 \sqrt {1-2 x} (3+5 x)^{3/2}}{2079 (2+3 x)^{9/2}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}-\frac {1305025844 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{47647845 \sqrt {33}}-\frac {37904696 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{47647845 \sqrt {33}}\\ \end {align*}
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Mathematica [A] time = 0.43, size = 112, normalized size = 0.45 \[ \frac {-10873573760 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )+\frac {48 \sqrt {2-4 x} \sqrt {5 x+3} \left (158560640046 x^5+534040213536 x^4+719808574005 x^3+484598540169 x^2+162787885893 x+21813966691\right )}{(3 x+2)^{11/2}}+20880413504 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )}{12579031080 \sqrt {2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.24, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128}, x\right ) \]
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 \[ \int \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} \sqrt {-2 \, x + 1}}{{\left (3 \, x + 2\right )}^{\frac {13}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.04, size = 599, normalized size = 2.41 \[ \frac {2 \left (4756819201380 x^{7}+16496888326218 x^{6}-158560640046 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{5} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+82571200740 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{5} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+21769332100344 x^{5}-528535466820 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{4} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+275237335800 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{4} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+11891020005261 x^{4}-704713955760 \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 )+366983114400 \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 )-140844968748 x^{3}-469809303840 \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 )+244655409600 \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 )-3218604203112 x^{2}-156603101280 \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 )+81551803200 \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 )-1399649072964 x -20880413504 \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 )+10873573760 \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 )-196325700219\right ) \sqrt {-2 x +1}\, \sqrt {5 x +3}}{1572378885 \left (10 x^{2}+x -3\right ) \left (3 x +2\right )^{\frac {11}{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 \[ \int \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} \sqrt {-2 \, x + 1}}{{\left (3 \, x + 2\right )}^{\frac {13}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\sqrt {1-2\,x}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^{13/2}} \,d x \]
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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