Optimal. Leaf size=160 \[ \frac {30922 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{118125}+\frac {326 (1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}}{2625}+\frac {2}{35} (1-2 x)^{5/2} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {408311 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{590625}-\frac {132824 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{590625} \]
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Rubi [A]
time = 0.04, antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps
used = 6, 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 {132824 \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{590625}-\frac {408311 \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{590625}+\frac {2}{35} \sqrt {3 x+2} \sqrt {5 x+3} (1-2 x)^{5/2}+\frac {326 \sqrt {3 x+2} \sqrt {5 x+3} (1-2 x)^{3/2}}{2625}+\frac {30922 \sqrt {3 x+2} \sqrt {5 x+3} \sqrt {1-2 x}}{118125} \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} \sqrt {2+3 x}}{\sqrt {3+5 x}} \, dx &=\frac {2}{35} (1-2 x)^{5/2} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {2}{35} \int \frac {\left (-\frac {111}{2}-\frac {163 x}{2}\right ) (1-2 x)^{3/2}}{\sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=\frac {326 (1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}}{2625}+\frac {2}{35} (1-2 x)^{5/2} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {4 \int \frac {\left (-2774-\frac {15461 x}{4}\right ) \sqrt {1-2 x}}{\sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{2625}\\ &=\frac {30922 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{118125}+\frac {326 (1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}}{2625}+\frac {2}{35} (1-2 x)^{5/2} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {8 \int \frac {-\frac {391093}{8}-\frac {408311 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{118125}\\ &=\frac {30922 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{118125}+\frac {326 (1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}}{2625}+\frac {2}{35} (1-2 x)^{5/2} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {408311 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{590625}+\frac {730532 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{590625}\\ &=\frac {30922 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{118125}+\frac {326 (1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}}{2625}+\frac {2}{35} (1-2 x)^{5/2} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {408311 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{590625}-\frac {132824 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{590625}\\ \end {align*}
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Mathematica [A]
time = 3.52, size = 102, normalized size = 0.64 \begin {gather*} \frac {30 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x} \left (26171-28170 x+13500 x^2\right )+408311 \sqrt {2} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )+1783285 \sqrt {2} F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{1771875} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 148, normalized size = 0.92
method | result | size |
default | \(-\frac {\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}\, \left (2191596 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-408311 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-12150000 x^{5}+16038000 x^{4}-1281600 x^{3}-21543690 x^{2}+425310 x +4710780\right )}{1771875 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(148\) |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {8 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{35}-\frac {1252 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{2625}+\frac {52342 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{118125}+\frac {391093 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{2480625 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {408311 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{2480625 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(234\) |
risch | \(-\frac {2 \left (13500 x^{2}-28170 x +26171\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {2+3 x}\, \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{118125 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \sqrt {1-2 x}}-\frac {\left (-\frac {391093 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {55-110 x}\, \EllipticF \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{6496875 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {408311 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {55-110 x}\, \left (\frac {\EllipticE \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{15}-\frac {2 \EllipticF \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{3}\right )}{6496875 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right ) \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(252\) |
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 = 0.18, size = 33, normalized size = 0.21 \begin {gather*} \frac {2}{118125} \, {\left (13500 \, x^{2} - 28170 \, x + 26171\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}\,\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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