Optimal. Leaf size=129 \[ \frac {194 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{1125}+\frac {2}{25} (1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {2797 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{5625}-\frac {598 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{5625} \]
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Rubi [A]
time = 0.03, antiderivative size = 129, 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 {598 \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{5625}-\frac {2797 \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{5625}+\frac {2}{25} \sqrt {3 x+2} \sqrt {5 x+3} (1-2 x)^{3/2}+\frac {194 \sqrt {3 x+2} \sqrt {5 x+3} \sqrt {1-2 x}}{1125} \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} \sqrt {2+3 x}}{\sqrt {3+5 x}} \, dx &=\frac {2}{25} (1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {2}{25} \int \frac {\left (-\frac {67}{2}-\frac {97 x}{2}\right ) \sqrt {1-2 x}}{\sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=\frac {194 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{1125}+\frac {2}{25} (1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {4 \int \frac {-584-\frac {2797 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{1125}\\ &=\frac {194 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{1125}+\frac {2}{25} (1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {2797 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{5625}+\frac {3289 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{5625}\\ &=\frac {194 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{1125}+\frac {2}{25} (1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {2797 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{5625}-\frac {598 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{5625}\\ \end {align*}
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Mathematica [A]
time = 3.01, size = 97, normalized size = 0.75 \begin {gather*} \frac {60 (71-45 x) \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+2797 \sqrt {2} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )+7070 \sqrt {2} F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{16875} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 143, normalized size = 1.11
method | result | size |
default | \(-\frac {\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}\, \left (9867 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-2797 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+81000 x^{4}-65700 x^{3}-116880 x^{2}+13620 x +25560\right )}{16875 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(143\) |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {4 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{25}+\frac {284 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1125}+\frac {2336 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{23625 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {2797 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{23625 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(212\) |
risch | \(\frac {4 \left (-71+45 x \right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {2+3 x}\, \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{1125 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \sqrt {1-2 x}}+\frac {\left (\frac {2336 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {55-110 x}\, \EllipticF \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{61875 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {2797 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {55-110 x}\, \left (\frac {\EllipticE \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{15}-\frac {2 \EllipticF \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{3}\right )}{61875 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right ) \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(246\) |
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 = 0.26, size = 28, normalized size = 0.22 \begin {gather*} -\frac {4}{1125} \, {\left (45 \, x - 71\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 {\left (1 - 2 x\right )^{\frac {3}{2}} \sqrt {3 x + 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 {{\left (1-2\,x\right )}^{3/2}\,\sqrt {3\,x+2}}{\sqrt {5\,x+3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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