Optimal. Leaf size=127 \[ -\frac {9}{125} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {2}{25} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}-\frac {146}{625} \sqrt {33} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {17}{625} \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps
used = 5, 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 {17}{625} \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {146}{625} \sqrt {33} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {2}{25} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}-\frac {9}{125} \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2} \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 {\sqrt {1-2 x} (2+3 x)^{3/2}}{\sqrt {3+5 x}} \, dx &=\frac {2}{25} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}-\frac {2}{25} \int \frac {\left (-\frac {25}{2}-\frac {27 x}{2}\right ) \sqrt {2+3 x}}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {9}{125} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {2}{25} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}+\frac {2}{375} \int \frac {\frac {1689}{4}+657 x}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {9}{125} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {2}{25} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}+\frac {187 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{1250}+\frac {438}{625} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=-\frac {9}{125} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {2}{25} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}-\frac {146}{625} \sqrt {33} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {17}{625} \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\\ \end {align*}
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Mathematica [A]
time = 3.00, size = 97, normalized size = 0.76 \begin {gather*} \frac {10 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x} (11+30 x)+292 \sqrt {2} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-105 \sqrt {2} F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{1250} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 143, normalized size = 1.13
method | result | size |
default | \(-\frac {\sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (187 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-292 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-9000 x^{4}-10200 x^{3}-430 x^{2}+2570 x +660\right )}{1250 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(143\) |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {6 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{25}+\frac {11 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{125}+\frac {563 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5250 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {146 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{875 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(212\) |
risch | \(-\frac {\left (11+30 x \right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {2+3 x}\, \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{125 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \sqrt {1-2 x}}-\frac {\left (-\frac {563 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {55-110 x}\, \EllipticF \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{13750 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {438 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {55-110 x}\, \left (\frac {\EllipticE \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{15}-\frac {2 \EllipticF \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{3}\right )}{6875 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right ) \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(247\) |
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 = 0.12, size = 28, normalized size = 0.22 \begin {gather*} \frac {1}{125} \, {\left (30 \, x + 11\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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {1 - 2 x} \left (3 x + 2\right )^{\frac {3}{2}}}{\sqrt {5 x + 3}}\, dx \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 {\sqrt {1-2\,x}\,{\left (3\,x+2\right )}^{3/2}}{\sqrt {5\,x+3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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