Optimal. Leaf size=249 \[ -\frac {325796 \sqrt {1-2 x} \sqrt {3+5 x}}{130977 (2+3 x)^{5/2}}+\frac {11243972 \sqrt {1-2 x} \sqrt {3+5 x}}{2750517 (2+3 x)^{3/2}}+\frac {780320008 \sqrt {1-2 x} \sqrt {3+5 x}}{19253619 \sqrt {2+3 x}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{33 (2+3 x)^{11/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{891 (2+3 x)^{9/2}}+\frac {12280 \sqrt {1-2 x} (3+5 x)^{3/2}}{6237 (2+3 x)^{7/2}}-\frac {780320008 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1750329 \sqrt {33}}-\frac {23441272 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1750329 \sqrt {33}} \]
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Rubi [A]
time = 0.07, 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 = {99, 155, 157,
164, 114, 120} \begin {gather*} -\frac {23441272 F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1750329 \sqrt {33}}-\frac {780320008 E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1750329 \sqrt {33}}-\frac {2 (5 x+3)^{3/2} (1-2 x)^{5/2}}{33 (3 x+2)^{11/2}}+\frac {230 (5 x+3)^{3/2} (1-2 x)^{3/2}}{891 (3 x+2)^{9/2}}+\frac {12280 (5 x+3)^{3/2} \sqrt {1-2 x}}{6237 (3 x+2)^{7/2}}+\frac {780320008 \sqrt {5 x+3} \sqrt {1-2 x}}{19253619 \sqrt {3 x+2}}+\frac {11243972 \sqrt {5 x+3} \sqrt {1-2 x}}{2750517 (3 x+2)^{3/2}}-\frac {325796 \sqrt {5 x+3} \sqrt {1-2 x}}{130977 (3 x+2)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 99
Rule 114
Rule 120
Rule 155
Rule 157
Rule 164
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^{13/2}} \, dx &=-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{33 (2+3 x)^{11/2}}+\frac {2}{33} \int \frac {\left (-\frac {15}{2}-40 x\right ) (1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^{11/2}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{33 (2+3 x)^{11/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{891 (2+3 x)^{9/2}}-\frac {4}{891} \int \frac {\sqrt {1-2 x} \sqrt {3+5 x} \left (-1200+\frac {1005 x}{2}\right )}{(2+3 x)^{9/2}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{33 (2+3 x)^{11/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{891 (2+3 x)^{9/2}}+\frac {12280 \sqrt {1-2 x} (3+5 x)^{3/2}}{6237 (2+3 x)^{7/2}}+\frac {8 \int \frac {\left (\frac {232425}{4}-\frac {131115 x}{2}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^{7/2}} \, dx}{18711}\\ &=-\frac {325796 \sqrt {1-2 x} \sqrt {3+5 x}}{130977 (2+3 x)^{5/2}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{33 (2+3 x)^{11/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{891 (2+3 x)^{9/2}}+\frac {12280 \sqrt {1-2 x} (3+5 x)^{3/2}}{6237 (2+3 x)^{7/2}}+\frac {16 \int \frac {\frac {7896165}{8}-1154775 x}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx}{1964655}\\ &=-\frac {325796 \sqrt {1-2 x} \sqrt {3+5 x}}{130977 (2+3 x)^{5/2}}+\frac {11243972 \sqrt {1-2 x} \sqrt {3+5 x}}{2750517 (2+3 x)^{3/2}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{33 (2+3 x)^{11/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{891 (2+3 x)^{9/2}}+\frac {12280 \sqrt {1-2 x} (3+5 x)^{3/2}}{6237 (2+3 x)^{7/2}}+\frac {32 \int \frac {\frac {347150355}{8}-\frac {210824475 x}{8}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{41257755}\\ &=-\frac {325796 \sqrt {1-2 x} \sqrt {3+5 x}}{130977 (2+3 x)^{5/2}}+\frac {11243972 \sqrt {1-2 x} \sqrt {3+5 x}}{2750517 (2+3 x)^{3/2}}+\frac {780320008 \sqrt {1-2 x} \sqrt {3+5 x}}{19253619 \sqrt {2+3 x}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{33 (2+3 x)^{11/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{891 (2+3 x)^{9/2}}+\frac {12280 \sqrt {1-2 x} (3+5 x)^{3/2}}{6237 (2+3 x)^{7/2}}+\frac {64 \int \frac {\frac {9262076325}{16}+\frac {7315500075 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{288804285}\\ &=-\frac {325796 \sqrt {1-2 x} \sqrt {3+5 x}}{130977 (2+3 x)^{5/2}}+\frac {11243972 \sqrt {1-2 x} \sqrt {3+5 x}}{2750517 (2+3 x)^{3/2}}+\frac {780320008 \sqrt {1-2 x} \sqrt {3+5 x}}{19253619 \sqrt {2+3 x}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{33 (2+3 x)^{11/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{891 (2+3 x)^{9/2}}+\frac {12280 \sqrt {1-2 x} (3+5 x)^{3/2}}{6237 (2+3 x)^{7/2}}+\frac {11720636 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{1750329}+\frac {780320008 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{19253619}\\ &=-\frac {325796 \sqrt {1-2 x} \sqrt {3+5 x}}{130977 (2+3 x)^{5/2}}+\frac {11243972 \sqrt {1-2 x} \sqrt {3+5 x}}{2750517 (2+3 x)^{3/2}}+\frac {780320008 \sqrt {1-2 x} \sqrt {3+5 x}}{19253619 \sqrt {2+3 x}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{33 (2+3 x)^{11/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{891 (2+3 x)^{9/2}}+\frac {12280 \sqrt {1-2 x} (3+5 x)^{3/2}}{6237 (2+3 x)^{7/2}}-\frac {780320008 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1750329 \sqrt {33}}-\frac {23441272 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1750329 \sqrt {33}}\\ \end {align*}
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Mathematica [A]
time = 8.95, size = 115, normalized size = 0.46 \begin {gather*} \frac {\frac {24 \sqrt {1-2 x} \sqrt {3+5 x} \left (13163824553+97637232762 x+289719086787 x^2+429993423180 x^3+319217269302 x^4+94808880972 x^5\right )}{(2+3 x)^{11/2}}+16 \sqrt {2} \left (195080002 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-98384755 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )}{231043428} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(586\) vs.
\(2(185)=370\).
time = 0.10, size = 587, normalized size = 2.36
method | result | size |
elliptic | \(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {6524 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1948617 \left (\frac {2}{3}+x \right )^{5}}+\frac {98 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{649539 \left (\frac {2}{3}+x \right )^{6}}+\frac {181672 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{10609137 \left (\frac {2}{3}+x \right )^{3}}+\frac {40874 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{4546773 \left (\frac {2}{3}+x \right )^{4}}+\frac {-\frac {7803200080}{19253619} x^{2}-\frac {780320008}{19253619} x +\frac {780320008}{6417873}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {11243972 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{24754653 \left (\frac {2}{3}+x \right )^{2}}+\frac {2469887020 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{404325999 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {3901600040 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{404325999 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\left (10 x^{2}+x -3\right ) \sqrt {2+3 x}}\) | \(332\) |
default | \(-\frac {2 \left (46993890042 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{5} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-94808880972 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{5} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+156646300140 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-316029603240 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+208861733520 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-421372804320 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+139241155680 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-280915202880 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-2844266429160 x^{7}+46413718560 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-93638400960 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-9860944721976 x^{6}+6188495808 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-12485120128 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-13004174574558 x^{5}-7108597449432 x^{4}+71666565399 x^{3}+1919645346207 x^{2}+839243621199 x +118474420977\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}}{57760857 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {11}{2}}}\) | \(587\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 80, normalized size = 0.32 \begin {gather*} \frac {2 \, {\left (94808880972 \, x^{5} + 319217269302 \, x^{4} + 429993423180 \, x^{3} + 289719086787 \, x^{2} + 97637232762 \, x + 13163824553\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{19253619 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{3/2}}{{\left (3\,x+2\right )}^{13/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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