Optimal. Leaf size=218 \[ -\frac {12996374 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{35083125}-\frac {78797 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{3898125}+\frac {30362 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{779625}+\frac {326 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}}{7425}+\frac {2}{55} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {829177897 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{31893750 \sqrt {33}}-\frac {12996374 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{15946875 \sqrt {33}} \]
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Rubi [A]
time = 0.06, antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps
used = 8, 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 {12996374 F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{15946875 \sqrt {33}}-\frac {829177897 E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{31893750 \sqrt {33}}+\frac {2}{55} (1-2 x)^{5/2} \sqrt {3 x+2} (5 x+3)^{5/2}+\frac {326 (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{5/2}}{7425}+\frac {30362 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}}{779625}-\frac {78797 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}{3898125}-\frac {12996374 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}{35083125} \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 (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{3/2} \, dx &=\frac {2}{55} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {2}{55} \int \frac {\left (-\frac {111}{2}-\frac {163 x}{2}\right ) (1-2 x)^{3/2} (3+5 x)^{3/2}}{\sqrt {2+3 x}} \, dx\\ &=\frac {326 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}}{7425}+\frac {2}{55} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {4 \int \frac {\left (-2809-\frac {15181 x}{4}\right ) \sqrt {1-2 x} (3+5 x)^{3/2}}{\sqrt {2+3 x}} \, dx}{7425}\\ &=\frac {30362 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{779625}+\frac {326 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}}{7425}+\frac {2}{55} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {8 \int \frac {\left (-\frac {466273}{8}-\frac {236391 x}{8}\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{779625}\\ &=-\frac {78797 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{3898125}+\frac {30362 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{779625}+\frac {326 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}}{7425}+\frac {2}{55} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {8 \int \frac {\sqrt {3+5 x} \left (\frac {48347127}{16}+\frac {19494561 x}{4}\right )}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{11694375}\\ &=-\frac {12996374 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{35083125}-\frac {78797 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{3898125}+\frac {30362 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{779625}+\frac {326 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}}{7425}+\frac {2}{55} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {8 \int \frac {-\frac {1578296283}{16}-\frac {2487533691 x}{16}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{105249375}\\ &=-\frac {12996374 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{35083125}-\frac {78797 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{3898125}+\frac {30362 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{779625}+\frac {326 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}}{7425}+\frac {2}{55} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {6498187 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{15946875}+\frac {829177897 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{350831250}\\ &=-\frac {12996374 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{35083125}-\frac {78797 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{3898125}+\frac {30362 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{779625}+\frac {326 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}}{7425}+\frac {2}{55} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {829177897 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{31893750 \sqrt {33}}-\frac {12996374 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{15946875 \sqrt {33}}\\ \end {align*}
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Mathematica [A]
time = 6.49, size = 107, normalized size = 0.49 \begin {gather*} \frac {15 \sqrt {2-4 x} \sqrt {2+3 x} \sqrt {3+5 x} \left (22517617+48272535 x-95024250 x^2-51502500 x^3+127575000 x^4\right )+829177897 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-400297555 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{526246875 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 158, normalized size = 0.72
method | result | size |
default | \(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {2+3 x}\, \left (-114817500000 x^{7}-41674500000 x^{6}+428880342 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-829177897 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+147849300000 x^{5}+34269426000 x^{4}-82799446950 x^{3}-22504288380 x^{2}+13417755870 x +4053171060\right )}{1052493750 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(158\) |
risch | \(-\frac {\left (127575000 x^{4}-51502500 x^{3}-95024250 x^{2}+48272535 x +22517617\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {2+3 x}\, \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{35083125 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \sqrt {1-2 x}}-\frac {\left (-\frac {526098761 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {55-110 x}\, \EllipticF \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{3859143750 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {829177897 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {55-110 x}\, \left (\frac {\EllipticE \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{15}-\frac {2 \EllipticF \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{3}\right )}{3859143750 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right ) \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(262\) |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {40 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}\, x^{4}}{11}-\frac {436 x^{3} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{297}-\frac {84466 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{31185}+\frac {1072723 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{779625}+\frac {22517617 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{35083125}+\frac {526098761 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{1473491250 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {829177897 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{1473491250 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(278\) |
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 = 0.28, size = 43, normalized size = 0.20 \begin {gather*} \frac {1}{35083125} \, {\left (127575000 \, x^{4} - 51502500 \, x^{3} - 95024250 \, x^{2} + 48272535 \, x + 22517617\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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 = -0.00 \begin {gather*} \int {\left (1-2\,x\right )}^{5/2}\,\sqrt {3\,x+2}\,{\left (5\,x+3\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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