Optimal. Leaf size=191 \[ -\frac {429479 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{637875}+\frac {14318 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{70875}+\frac {362 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{3/2}}{2835}+\frac {2}{27} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {4457606 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3189375}-\frac {429479 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3189375} \]
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Rubi [A]
time = 0.05, 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 {429479 \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3189375}-\frac {4457606 \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3189375}+\frac {2}{27} \sqrt {3 x+2} (5 x+3)^{3/2} (1-2 x)^{5/2}+\frac {362 \sqrt {3 x+2} (5 x+3)^{3/2} (1-2 x)^{3/2}}{2835}+\frac {14318 \sqrt {3 x+2} (5 x+3)^{3/2} \sqrt {1-2 x}}{70875}-\frac {429479 \sqrt {3 x+2} \sqrt {5 x+3} \sqrt {1-2 x}}{637875} \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)^{5/2} (3+5 x)^{3/2}}{\sqrt {2+3 x}} \, dx &=\frac {2}{27} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {2}{27} \int \frac {\left (-51-\frac {181 x}{2}\right ) (1-2 x)^{3/2} \sqrt {3+5 x}}{\sqrt {2+3 x}} \, dx\\ &=\frac {362 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{3/2}}{2835}+\frac {2}{27} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {4 \int \frac {\left (-\frac {10167}{4}-\frac {21477 x}{4}\right ) \sqrt {1-2 x} \sqrt {3+5 x}}{\sqrt {2+3 x}} \, dx}{2835}\\ &=\frac {14318 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{70875}+\frac {362 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{3/2}}{2835}+\frac {2}{27} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {8 \int \frac {\left (-\frac {91323}{4}-\frac {1288437 x}{8}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{212625}\\ &=-\frac {429479 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{637875}+\frac {14318 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{70875}+\frac {362 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{3/2}}{2835}+\frac {2}{27} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{3/2}+\frac {8 \int \frac {\frac {18881943}{16}+\frac {6686409 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{1913625}\\ &=-\frac {429479 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{637875}+\frac {14318 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{70875}+\frac {362 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{3/2}}{2835}+\frac {2}{27} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{3/2}+\frac {4724269 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{6378750}+\frac {4457606 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{3189375}\\ &=-\frac {429479 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{637875}+\frac {14318 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{70875}+\frac {362 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{3/2}}{2835}+\frac {2}{27} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {4457606 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3189375}-\frac {429479 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3189375}\\ \end {align*}
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Mathematica [A]
time = 6.61, size = 102, normalized size = 0.53 \begin {gather*} \frac {15 \sqrt {2-4 x} \sqrt {2+3 x} \sqrt {3+5 x} \left (343207+232110 x-1192500 x^2+945000 x^3\right )+8915212 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )+5257595 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{9568125 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 153, normalized size = 0.80
method | result | size |
default | \(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {2+3 x}\, \left (-850500000 x^{6}+14172807 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-8915212 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+421200000 x^{5}+812376000 x^{4}-549367200 x^{3}-402719730 x^{2}+113853270 x +61777260\right )}{19136250 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(153\) |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {40 x^{3} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{27}-\frac {1060 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{567}+\frac {5158 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{14175}+\frac {343207 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{637875}+\frac {6293981 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{26790750 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {4457606 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{13395375 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(256\) |
risch | \(-\frac {\left (945000 x^{3}-1192500 x^{2}+232110 x +343207\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {2+3 x}\, \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{637875 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \sqrt {1-2 x}}-\frac {\left (-\frac {6293981 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {55-110 x}\, \EllipticF \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{70166250 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {4457606 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {55-110 x}\, \left (\frac {\EllipticE \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{15}-\frac {2 \EllipticF \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{3}\right )}{35083125 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right ) \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(257\) |
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 = 0.27, size = 38, normalized size = 0.20 \begin {gather*} \frac {1}{637875} \, {\left (945000 \, x^{3} - 1192500 \, x^{2} + 232110 \, x + 343207\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 )}^{5/2}\,{\left (5\,x+3\right )}^{3/2}}{\sqrt {3\,x+2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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