Optimal. Leaf size=218 \[ -\frac {5442127 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{7796250}-\frac {40703 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{433125}-\frac {23 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{9625}+\frac {62 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{2475}+\frac {2}{55} (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{5/2}-\frac {90397364 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1771875 \sqrt {33}}-\frac {5442127 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3543750 \sqrt {33}} \]
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Rubi [A]
time = 0.06, antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {103, 159, 164,
114, 120} \begin {gather*} -\frac {5442127 F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3543750 \sqrt {33}}-\frac {90397364 E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1771875 \sqrt {33}}+\frac {2}{55} (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{5/2}+\frac {62 \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}}{2475}-\frac {23 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}}{9625}-\frac {40703 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}{433125}-\frac {5442127 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}{7796250} \end {gather*}
Antiderivative was successfully verified.
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Rule 103
Rule 114
Rule 120
Rule 159
Rule 164
Rubi steps
\begin {align*} \int (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{3/2} \, dx &=\frac {2}{55} (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{5/2}-\frac {2}{55} \int \left (-\frac {69}{2}-\frac {93 x}{2}\right ) \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2} \, dx\\ &=\frac {62 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{2475}+\frac {2}{55} (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{5/2}-\frac {4 \int \frac {\left (-\frac {1635}{2}-\frac {621 x}{4}\right ) \sqrt {2+3 x} (3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx}{7425}\\ &=-\frac {23 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{9625}+\frac {62 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{2475}+\frac {2}{55} (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{5/2}+\frac {4 \int \frac {(3+5 x)^{3/2} \left (\frac {486987}{8}+\frac {366327 x}{4}\right )}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{259875}\\ &=-\frac {40703 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{433125}-\frac {23 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{9625}+\frac {62 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{2475}+\frac {2}{55} (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{5/2}-\frac {4 \int \frac {\left (-\frac {7951311}{2}-\frac {48979143 x}{8}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{3898125}\\ &=-\frac {5442127 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{7796250}-\frac {40703 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{433125}-\frac {23 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{9625}+\frac {62 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{2475}+\frac {2}{55} (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{5/2}+\frac {4 \int \frac {\frac {2060337177}{16}+203394069 x}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{35083125}\\ &=-\frac {5442127 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{7796250}-\frac {40703 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{433125}-\frac {23 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{9625}+\frac {62 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{2475}+\frac {2}{55} (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{5/2}+\frac {5442127 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{7087500}+\frac {90397364 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{19490625}\\ &=-\frac {5442127 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{7796250}-\frac {40703 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{433125}-\frac {23 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{9625}+\frac {62 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{2475}+\frac {2}{55} (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{5/2}-\frac {90397364 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1771875 \sqrt {33}}-\frac {5442127 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3543750 \sqrt {33}}\\ \end {align*}
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Mathematica [A]
time = 6.65, size = 110, normalized size = 0.50 \begin {gather*} \frac {361589456 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-5 \left (3 \sqrt {2-4 x} \sqrt {2+3 x} \sqrt {3+5 x} \left (-810641-27227430 x-17237250 x^2+43470000 x^3+42525000 x^4\right )+36399853 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )}{116943750 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 158, normalized size = 0.72
method | result | size |
default | \(-\frac {\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}\, \left (38272500000 x^{7}+68465250000 x^{6}+179590191 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-361589456 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+5550525000 x^{5}-53181589500 x^{4}-23721281100 x^{3}+8261123010 x^{2}+5071172010 x +145915380\right )}{233887500 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(158\) |
risch | \(\frac {\left (42525000 x^{4}+43470000 x^{3}-17237250 x^{2}-27227430 x -810641\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {2+3 x}\, \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{7796250 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \sqrt {1-2 x}}+\frac {\left (\frac {228926353 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {55-110 x}\, \EllipticF \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{857587500 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {90397364 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {55-110 x}\, \left (\frac {\EllipticE \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{15}-\frac {2 \EllipticF \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{3}\right )}{214396875 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right ) \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(261\) |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \sqrt {3+5 x}\, \sqrt {2+3 x}\, \left (-\frac {60 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}\, x^{4}}{11}-\frac {184 x^{3} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{33}+\frac {7661 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{3465}+\frac {302527 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{86625}+\frac {810641 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{7796250}+\frac {228926353 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{327442500 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {90397364 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{81860625 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \left (15 x^{2}+19 x +6\right )}\) | \(290\) |
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 = 0.15, size = 43, normalized size = 0.20 \begin {gather*} -\frac {1}{7796250} \, {\left (42525000 \, x^{4} + 43470000 \, x^{3} - 17237250 \, x^{2} - 27227430 \, x - 810641\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} \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 {\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^{3/2}\,{\left (5\,x+3\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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