Optimal. Leaf size=280 \[ -\frac {2257166048 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{53093313 \sqrt {33}}-\frac {2 (5 x+3)^{3/2} (1-2 x)^{5/2}}{39 (3 x+2)^{13/2}}+\frac {230 (5 x+3)^{3/2} (1-2 x)^{3/2}}{1287 (3 x+2)^{11/2}}+\frac {1300 (5 x+3)^{3/2} \sqrt {1-2 x}}{891 (3 x+2)^{9/2}}+\frac {75041008472 \sqrt {5 x+3} \sqrt {1-2 x}}{584026443 \sqrt {3 x+2}}+\frac {1079936248 \sqrt {5 x+3} \sqrt {1-2 x}}{83432349 (3 x+2)^{3/2}}+\frac {23210828 \sqrt {5 x+3} \sqrt {1-2 x}}{11918907 (3 x+2)^{5/2}}-\frac {3347620 \sqrt {5 x+3} \sqrt {1-2 x}}{1702701 (3 x+2)^{7/2}}-\frac {75041008472 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{53093313 \sqrt {33}} \]
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Rubi [A] time = 0.12, antiderivative size = 280, normalized size of antiderivative = 1.00, number of steps used = 10, 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} \[ -\frac {2 (5 x+3)^{3/2} (1-2 x)^{5/2}}{39 (3 x+2)^{13/2}}+\frac {230 (5 x+3)^{3/2} (1-2 x)^{3/2}}{1287 (3 x+2)^{11/2}}+\frac {1300 (5 x+3)^{3/2} \sqrt {1-2 x}}{891 (3 x+2)^{9/2}}+\frac {75041008472 \sqrt {5 x+3} \sqrt {1-2 x}}{584026443 \sqrt {3 x+2}}+\frac {1079936248 \sqrt {5 x+3} \sqrt {1-2 x}}{83432349 (3 x+2)^{3/2}}+\frac {23210828 \sqrt {5 x+3} \sqrt {1-2 x}}{11918907 (3 x+2)^{5/2}}-\frac {3347620 \sqrt {5 x+3} \sqrt {1-2 x}}{1702701 (3 x+2)^{7/2}}-\frac {2257166048 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{53093313 \sqrt {33}}-\frac {75041008472 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{53093313 \sqrt {33}} \]
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 {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^{15/2}} \, dx &=-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{39 (2+3 x)^{13/2}}+\frac {2}{39} \int \frac {\left (-\frac {15}{2}-40 x\right ) (1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^{13/2}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{39 (2+3 x)^{13/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1287 (2+3 x)^{11/2}}-\frac {4 \int \frac {\sqrt {1-2 x} \sqrt {3+5 x} \left (-\frac {2895}{2}+\frac {1995 x}{2}\right )}{(2+3 x)^{11/2}} \, dx}{1287}\\ &=-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{39 (2+3 x)^{13/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1287 (2+3 x)^{11/2}}+\frac {1300 \sqrt {1-2 x} (3+5 x)^{3/2}}{891 (2+3 x)^{9/2}}+\frac {8 \int \frac {\left (\frac {438345}{4}-149460 x\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^{9/2}} \, dx}{34749}\\ &=-\frac {3347620 \sqrt {1-2 x} \sqrt {3+5 x}}{1702701 (2+3 x)^{7/2}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{39 (2+3 x)^{13/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1287 (2+3 x)^{11/2}}+\frac {1300 \sqrt {1-2 x} (3+5 x)^{3/2}}{891 (2+3 x)^{9/2}}+\frac {16 \int \frac {\frac {15056835}{8}-\frac {10467525 x}{4}}{\sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}} \, dx}{5108103}\\ &=-\frac {3347620 \sqrt {1-2 x} \sqrt {3+5 x}}{1702701 (2+3 x)^{7/2}}+\frac {23210828 \sqrt {1-2 x} \sqrt {3+5 x}}{11918907 (2+3 x)^{5/2}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{39 (2+3 x)^{13/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1287 (2+3 x)^{11/2}}+\frac {1300 \sqrt {1-2 x} (3+5 x)^{3/2}}{891 (2+3 x)^{9/2}}+\frac {32 \int \frac {\frac {1154474415}{8}-\frac {1305609075 x}{8}}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx}{178783605}\\ &=-\frac {3347620 \sqrt {1-2 x} \sqrt {3+5 x}}{1702701 (2+3 x)^{7/2}}+\frac {23210828 \sqrt {1-2 x} \sqrt {3+5 x}}{11918907 (2+3 x)^{5/2}}+\frac {1079936248 \sqrt {1-2 x} \sqrt {3+5 x}}{83432349 (2+3 x)^{3/2}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{39 (2+3 x)^{13/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1287 (2+3 x)^{11/2}}+\frac {1300 \sqrt {1-2 x} (3+5 x)^{3/2}}{891 (2+3 x)^{9/2}}+\frac {64 \int \frac {\frac {100204281585}{16}-\frac {30373206975 x}{8}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{3754455705}\\ &=-\frac {3347620 \sqrt {1-2 x} \sqrt {3+5 x}}{1702701 (2+3 x)^{7/2}}+\frac {23210828 \sqrt {1-2 x} \sqrt {3+5 x}}{11918907 (2+3 x)^{5/2}}+\frac {1079936248 \sqrt {1-2 x} \sqrt {3+5 x}}{83432349 (2+3 x)^{3/2}}+\frac {75041008472 \sqrt {1-2 x} \sqrt {3+5 x}}{584026443 \sqrt {2+3 x}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{39 (2+3 x)^{13/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1287 (2+3 x)^{11/2}}+\frac {1300 \sqrt {1-2 x} (3+5 x)^{3/2}}{891 (2+3 x)^{9/2}}+\frac {128 \int \frac {\frac {1336148092575}{16}+\frac {2110528363275 x}{16}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{26281189935}\\ &=-\frac {3347620 \sqrt {1-2 x} \sqrt {3+5 x}}{1702701 (2+3 x)^{7/2}}+\frac {23210828 \sqrt {1-2 x} \sqrt {3+5 x}}{11918907 (2+3 x)^{5/2}}+\frac {1079936248 \sqrt {1-2 x} \sqrt {3+5 x}}{83432349 (2+3 x)^{3/2}}+\frac {75041008472 \sqrt {1-2 x} \sqrt {3+5 x}}{584026443 \sqrt {2+3 x}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{39 (2+3 x)^{13/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1287 (2+3 x)^{11/2}}+\frac {1300 \sqrt {1-2 x} (3+5 x)^{3/2}}{891 (2+3 x)^{9/2}}+\frac {1128583024 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{53093313}+\frac {75041008472 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{584026443}\\ &=-\frac {3347620 \sqrt {1-2 x} \sqrt {3+5 x}}{1702701 (2+3 x)^{7/2}}+\frac {23210828 \sqrt {1-2 x} \sqrt {3+5 x}}{11918907 (2+3 x)^{5/2}}+\frac {1079936248 \sqrt {1-2 x} \sqrt {3+5 x}}{83432349 (2+3 x)^{3/2}}+\frac {75041008472 \sqrt {1-2 x} \sqrt {3+5 x}}{584026443 \sqrt {2+3 x}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{39 (2+3 x)^{13/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1287 (2+3 x)^{11/2}}+\frac {1300 \sqrt {1-2 x} (3+5 x)^{3/2}}{891 (2+3 x)^{9/2}}-\frac {75041008472 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{53093313 \sqrt {33}}-\frac {2257166048 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{53093313 \sqrt {33}}\\ \end {align*}
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Mathematica [A] time = 0.41, size = 117, normalized size = 0.42 \[ \frac {-604764298880 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )+\frac {48 \sqrt {2-4 x} \sqrt {5 x+3} \left (27352447588044 x^6+110328276131100 x^5+185457331738206 x^4+166295375376786 x^3+83893544414217 x^2+22577209892436 x+2532151719515\right )}{(3 x+2)^{13/2}}+1200656135552 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )}{14016634632 \sqrt {2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.07, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{6561 \, x^{8} + 34992 \, x^{7} + 81648 \, x^{6} + 108864 \, x^{5} + 90720 \, x^{4} + 48384 \, x^{3} + 16128 \, x^{2} + 3072 \, x + 256}, x\right ) \]
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 \[ \int \frac {{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {15}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.03, size = 694, normalized size = 2.48 \[ \frac {2 \left (820573427641320 x^{8}+3391905626697132 x^{7}-27352447588044 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{6} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+13777286683860 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{6} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+5648532752247084 x^{6}-109409790352176 \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 )+55109146735440 \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 )+4552278771338298 x^{5}-182349650586960 \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 )+91848577892400 \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 )+1346576472913014 x^{4}-162088578299520 \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 )+81643180348800 \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 )-567661448375343 x^{3}-81044289149760 \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 )+40821590174400 \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 )-611345718465195 x^{2}-21611810439936 \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 )+10885757379840 \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 )-195598433873379 x -2401312271104 \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 )+1209528597760 \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 )-22789365475635\right ) \sqrt {5 x +3}\, \sqrt {-2 x +1}}{1752079329 \left (10 x^{2}+x -3\right ) \left (3 x +2\right )^{\frac {13}{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 \[ \int \frac {{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {15}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{3/2}}{{\left (3\,x+2\right )}^{15/2}} \,d x \]
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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