Optimal. Leaf size=191 \[ -\frac {663409 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{47250}-\frac {9547 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{5250}-\frac {137}{315} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}-\frac {1}{9} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}-\frac {44109377 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{472500}-\frac {663409 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{236250} \]
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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 {663409 \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{236250}-\frac {44109377 \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{472500}-\frac {1}{9} \sqrt {1-2 x} (5 x+3)^{3/2} (3 x+2)^{5/2}-\frac {137}{315} \sqrt {1-2 x} (5 x+3)^{3/2} (3 x+2)^{3/2}-\frac {9547 \sqrt {1-2 x} (5 x+3)^{3/2} \sqrt {3 x+2}}{5250}-\frac {663409 \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}}{47250} \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 {(2+3 x)^{5/2} (3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx &=-\frac {1}{9} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}+\frac {1}{9} \int \frac {(2+3 x)^{3/2} \sqrt {3+5 x} \left (\frac {171}{2}+137 x\right )}{\sqrt {1-2 x}} \, dx\\ &=-\frac {137}{315} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}-\frac {1}{9} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}-\frac {1}{315} \int \frac {\left (-\frac {18135}{2}-\frac {28641 x}{2}\right ) \sqrt {2+3 x} \sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx\\ &=-\frac {9547 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{5250}-\frac {137}{315} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}-\frac {1}{9} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}+\frac {\int \frac {\sqrt {3+5 x} \left (\frac {2586807}{4}+\frac {1990227 x}{2}\right )}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{7875}\\ &=-\frac {663409 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{47250}-\frac {9547 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{5250}-\frac {137}{315} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}-\frac {1}{9} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}-\frac {\int \frac {-\frac {41887689}{2}-\frac {132328131 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{70875}\\ &=-\frac {663409 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{47250}-\frac {9547 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{5250}-\frac {137}{315} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}-\frac {1}{9} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}+\frac {7297499 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{472500}+\frac {44109377 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{472500}\\ &=-\frac {663409 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{47250}-\frac {9547 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{5250}-\frac {137}{315} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}-\frac {1}{9} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}-\frac {44109377 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{472500}-\frac {663409 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{236250}\\ \end {align*}
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Mathematica [A]
time = 4.44, size = 105, normalized size = 0.55 \begin {gather*} \frac {44109377 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 (1107478+1114065 x+765000 x^2+236250 x^3\right )+4443376 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )}{708750 \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 | \(\frac {\sqrt {2+3 x}\, \sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (-212625000 x^{6}+44109377 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-21892497 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-851512500 x^{5}-1480896000 x^{4}-1562260050 x^{3}-392506170 x^{2}+433102080 x +199346040\right )}{42525000 x^{3}+32602500 x^{2}-9922500 x -8505000}\) | \(153\) |
risch | \(\frac {\left (236250 x^{3}+765000 x^{2}+1114065 x +1107478\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {2+3 x}\, \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{47250 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \sqrt {1-2 x}}+\frac {\left (\frac {13962563 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {55-110 x}\, \EllipticF \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{2598750 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {44109377 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {55-110 x}\, \left (\frac {\EllipticE \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{15}-\frac {2 \EllipticF \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{3}\right )}{5197500 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right ) \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(256\) |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \sqrt {3+5 x}\, \sqrt {2+3 x}\, \left (-5 x^{3} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}-\frac {340 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{21}-\frac {24757 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1050}-\frac {553739 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{23625}+\frac {13962563 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{992250 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {44109377 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{1984500 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \left (15 x^{2}+19 x +6\right )}\) | \(268\) |
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 = 38, normalized size = 0.20 \begin {gather*} -\frac {1}{47250} \, {\left (236250 \, x^{3} + 765000 \, x^{2} + 1114065 \, x + 1107478\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 (3\,x+2\right )}^{5/2}\,{\left (5\,x+3\right )}^{3/2}}{\sqrt {1-2\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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