Optimal. Leaf size=218 \[ -\frac {2930159 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{2806650}-\frac {22576 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{155925}-\frac {2377 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{155925}+\frac {194 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{7/2}}{7425}+\frac {2}{55} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{7/2}-\frac {97540001 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1275750 \sqrt {33}}-\frac {2930159 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1275750 \sqrt {33}} \]
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Rubi [A]
time = 0.05, 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 {2930159 F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1275750 \sqrt {33}}-\frac {97540001 E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1275750 \sqrt {33}}+\frac {2}{55} (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{7/2}+\frac {194 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}}{7425}-\frac {2377 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}}{155925}-\frac {22576 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}{155925}-\frac {2930159 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}{2806650} \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)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2} \, dx &=\frac {2}{55} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{7/2}-\frac {2}{55} \int \frac {\left (-\frac {67}{2}-\frac {97 x}{2}\right ) \sqrt {1-2 x} (3+5 x)^{5/2}}{\sqrt {2+3 x}} \, dx\\ &=\frac {194 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{7/2}}{7425}+\frac {2}{55} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{7/2}-\frac {4 \int \frac {\left (-\frac {1273}{2}-\frac {2377 x}{4}\right ) (3+5 x)^{5/2}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{7425}\\ &=-\frac {2377 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{155925}+\frac {194 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{7/2}}{7425}+\frac {2}{55} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{7/2}+\frac {4 \int \frac {(3+5 x)^{3/2} \left (\frac {432515}{8}+84660 x\right )}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{155925}\\ &=-\frac {22576 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{155925}-\frac {2377 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{155925}+\frac {194 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{7/2}}{7425}+\frac {2}{55} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{7/2}-\frac {4 \int \frac {\left (-\frac {28606455}{8}-\frac {43952385 x}{8}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{2338875}\\ &=-\frac {2930159 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{2806650}-\frac {22576 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{155925}-\frac {2377 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{155925}+\frac {194 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{7/2}}{7425}+\frac {2}{55} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{7/2}+\frac {4 \int \frac {\frac {1852415265}{16}+\frac {1463100015 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{21049875}\\ &=-\frac {2930159 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{2806650}-\frac {22576 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{155925}-\frac {2377 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{155925}+\frac {194 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{7/2}}{7425}+\frac {2}{55} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{7/2}+\frac {2930159 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{2551500}+\frac {97540001 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{14033250}\\ &=-\frac {2930159 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{2806650}-\frac {22576 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{155925}-\frac {2377 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{155925}+\frac {194 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{7/2}}{7425}+\frac {2}{55} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{7/2}-\frac {97540001 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1275750 \sqrt {33}}-\frac {2930159 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1275750 \sqrt {33}}\\ \end {align*}
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Mathematica [A]
time = 6.67, size = 107, normalized size = 0.49 \begin {gather*} \frac {-15 \sqrt {2-4 x} \sqrt {2+3 x} \sqrt {3+5 x} \left (-201247-14851260 x-10837350 x^2+24003000 x^3+25515000 x^4\right )+195080002 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-98384755 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{42099750 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 158, normalized size = 0.72
method | result | size |
default | \(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {2+3 x}\, \left (22963500000 x^{7}+39208050000 x^{6}+96695247 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-195080002 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+1450305000 x^{5}-30477235500 x^{4}-12473188200 x^{3}+4930627170 x^{2}+2715488670 x +36224460\right )}{84199500 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(158\) |
risch | \(\frac {\left (25515000 x^{4}+24003000 x^{3}-10837350 x^{2}-14851260 x -201247\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {2+3 x}\, \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{2806650 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \sqrt {1-2 x}}+\frac {\left (\frac {123494351 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {55-110 x}\, \EllipticF \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{308731500 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {97540001 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {55-110 x}\, \left (\frac {\EllipticE \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{15}-\frac {2 \EllipticF \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{3}\right )}{154365750 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right ) \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(261\) |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {100 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}\, x^{4}}{11}-\frac {2540 x^{3} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{297}+\frac {24083 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{6237}+\frac {165014 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{31185}+\frac {201247 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{2806650}+\frac {123494351 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{117879300 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {97540001 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{58939650 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(278\) |
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 = 0.23, size = 43, normalized size = 0.20 \begin {gather*} -\frac {1}{2806650} \, {\left (25515000 \, x^{4} + 24003000 \, x^{3} - 10837350 \, x^{2} - 14851260 \, x - 201247\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 )}^{3/2}\,\sqrt {3\,x+2}\,{\left (5\,x+3\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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