Optimal. Leaf size=191 \[ -\frac {11908 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{25515}-\frac {499 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{2835}+\frac {46}{567} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {2}{27} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {886499 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{255150}-\frac {11908 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{127575} \]
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Rubi [A]
time = 0.04, antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps
used = 7, 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 {11908 \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{127575}-\frac {886499 \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{255150}+\frac {2}{27} (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{5/2}+\frac {46}{567} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}-\frac {499 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}{2835}-\frac {11908 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}{25515} \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 \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{\sqrt {2+3 x}} \, dx &=\frac {2}{27} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {2}{27} \int \frac {\left (-29-\frac {115 x}{2}\right ) \sqrt {1-2 x} (3+5 x)^{3/2}}{\sqrt {2+3 x}} \, dx\\ &=\frac {46}{567} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {2}{27} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {4 \int \frac {\left (-\frac {685}{4}-\frac {7485 x}{4}\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{2835}\\ &=-\frac {499 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{2835}+\frac {46}{567} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {2}{27} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {4 \int \frac {\sqrt {3+5 x} \left (\frac {263745}{8}+44655 x\right )}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{42525}\\ &=-\frac {11908 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{25515}-\frac {499 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{2835}+\frac {46}{567} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {2}{27} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {4 \int \frac {-\frac {8371455}{8}-\frac {13297485 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{382725}\\ &=-\frac {11908 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{25515}-\frac {499 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{2835}+\frac {46}{567} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {2}{27} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {65494 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{127575}+\frac {886499 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{255150}\\ &=-\frac {11908 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{25515}-\frac {499 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{2835}+\frac {46}{567} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {2}{27} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {886499 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{255150}-\frac {11908 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{127575}\\ \end {align*}
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Mathematica [A]
time = 6.80, size = 105, normalized size = 0.55 \begin {gather*} \frac {886499 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 (-10259-62325 x+14400 x^2+94500 x^3\right )+98707 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )}{382725 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 153, normalized size = 0.80
method | result | size |
default | \(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {2+3 x}\, \left (85050000 x^{6}+392964 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-886499 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+78165000 x^{5}-66001500 x^{4}-72271350 x^{3}+3417540 x^{2}+13372890 x +1846620\right )}{765450 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(153\) |
risch | \(\frac {\left (94500 x^{3}+14400 x^{2}-62325 x -10259\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {2+3 x}\, \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{25515 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \sqrt {1-2 x}}+\frac {\left (\frac {558097 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {55-110 x}\, \EllipticF \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{2806650 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {886499 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {55-110 x}\, \left (\frac {\EllipticE \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{15}-\frac {2 \EllipticF \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{3}\right )}{2806650 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right ) \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(256\) |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {100 x^{3} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{27}-\frac {320 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{567}+\frac {1385 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{567}+\frac {10259 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{25515}+\frac {558097 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{1071630 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {886499 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{1071630 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(256\) |
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 = 0.32, size = 38, normalized size = 0.20 \begin {gather*} -\frac {1}{25515} \, {\left (94500 \, x^{3} + 14400 \, x^{2} - 62325 \, x - 10259\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 = -0.01 \begin {gather*} \int \frac {{\left (1-2\,x\right )}^{3/2}\,{\left (5\,x+3\right )}^{5/2}}{\sqrt {3\,x+2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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