Optimal. Leaf size=150 \[ -\frac {15863 \sqrt {1-2 x} \sqrt {3+5 x}}{20736}-\frac {53}{192} \sqrt {1-2 x} (3+5 x)^{3/2}+\frac {23}{216} \sqrt {1-2 x} (3+5 x)^{5/2}+\frac {1}{12} (1-2 x)^{3/2} (3+5 x)^{5/2}+\frac {648919 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{62208 \sqrt {10}}+\frac {14}{243} \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right ) \]
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Rubi [A]
time = 0.04, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {103, 159, 163,
56, 222, 95, 210} \begin {gather*} \frac {648919 \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{62208 \sqrt {10}}+\frac {14}{243} \sqrt {7} \text {ArcTan}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )+\frac {1}{12} (1-2 x)^{3/2} (5 x+3)^{5/2}+\frac {23}{216} \sqrt {1-2 x} (5 x+3)^{5/2}-\frac {53}{192} \sqrt {1-2 x} (5 x+3)^{3/2}-\frac {15863 \sqrt {1-2 x} \sqrt {5 x+3}}{20736} \end {gather*}
Antiderivative was successfully verified.
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Rule 56
Rule 95
Rule 103
Rule 159
Rule 163
Rule 210
Rule 222
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{2+3 x} \, dx &=\frac {1}{12} (1-2 x)^{3/2} (3+5 x)^{5/2}-\frac {1}{12} \int \frac {\left (-29-\frac {115 x}{2}\right ) \sqrt {1-2 x} (3+5 x)^{3/2}}{2+3 x} \, dx\\ &=\frac {23}{216} \sqrt {1-2 x} (3+5 x)^{5/2}+\frac {1}{12} (1-2 x)^{3/2} (3+5 x)^{5/2}-\frac {1}{540} \int \frac {\left (-\frac {425}{2}-\frac {7155 x}{4}\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=-\frac {53}{192} \sqrt {1-2 x} (3+5 x)^{3/2}+\frac {23}{216} \sqrt {1-2 x} (3+5 x)^{5/2}+\frac {1}{12} (1-2 x)^{3/2} (3+5 x)^{5/2}+\frac {\int \frac {\sqrt {3+5 x} \left (\frac {94995}{4}+\frac {237945 x}{8}\right )}{\sqrt {1-2 x} (2+3 x)} \, dx}{6480}\\ &=-\frac {15863 \sqrt {1-2 x} \sqrt {3+5 x}}{20736}-\frac {53}{192} \sqrt {1-2 x} (3+5 x)^{3/2}+\frac {23}{216} \sqrt {1-2 x} (3+5 x)^{5/2}+\frac {1}{12} (1-2 x)^{3/2} (3+5 x)^{5/2}-\frac {\int \frac {-\frac {3181875}{8}-\frac {9733785 x}{16}}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{38880}\\ &=-\frac {15863 \sqrt {1-2 x} \sqrt {3+5 x}}{20736}-\frac {53}{192} \sqrt {1-2 x} (3+5 x)^{3/2}+\frac {23}{216} \sqrt {1-2 x} (3+5 x)^{5/2}+\frac {1}{12} (1-2 x)^{3/2} (3+5 x)^{5/2}-\frac {49}{243} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx+\frac {648919 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{124416}\\ &=-\frac {15863 \sqrt {1-2 x} \sqrt {3+5 x}}{20736}-\frac {53}{192} \sqrt {1-2 x} (3+5 x)^{3/2}+\frac {23}{216} \sqrt {1-2 x} (3+5 x)^{5/2}+\frac {1}{12} (1-2 x)^{3/2} (3+5 x)^{5/2}-\frac {98}{243} \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )+\frac {648919 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{62208 \sqrt {5}}\\ &=-\frac {15863 \sqrt {1-2 x} \sqrt {3+5 x}}{20736}-\frac {53}{192} \sqrt {1-2 x} (3+5 x)^{3/2}+\frac {23}{216} \sqrt {1-2 x} (3+5 x)^{5/2}+\frac {1}{12} (1-2 x)^{3/2} (3+5 x)^{5/2}+\frac {648919 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{62208 \sqrt {10}}+\frac {14}{243} \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.23, size = 119, normalized size = 0.79 \begin {gather*} \frac {30 \sqrt {1-2 x} \left (7167+187013 x+275940 x^2-285600 x^3-432000 x^4\right )-648919 \sqrt {30+50 x} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )+35840 \sqrt {7} \sqrt {3+5 x} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{622080 \sqrt {3+5 x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 132, normalized size = 0.88
method | result | size |
default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (-5184000 x^{3} \sqrt {-10 x^{2}-x +3}-316800 x^{2} \sqrt {-10 x^{2}-x +3}+648919 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-35840 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+3501360 x \sqrt {-10 x^{2}-x +3}+143340 \sqrt {-10 x^{2}-x +3}\right )}{1244160 \sqrt {-10 x^{2}-x +3}}\) | \(132\) |
risch | \(\frac {\left (86400 x^{3}+5280 x^{2}-58356 x -2389\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{20736 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right )}\, \sqrt {1-2 x}}+\frac {\left (\frac {648919 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )}{1244160}-\frac {7 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right )}{243}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(135\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.56, size = 98, normalized size = 0.65 \begin {gather*} \frac {5}{12} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {7}{432} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {2675}{1728} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {648919}{1244160} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {7}{243} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {3397}{20736} \, \sqrt {-10 \, x^{2} - x + 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.71, size = 112, normalized size = 0.75 \begin {gather*} -\frac {1}{20736} \, {\left (86400 \, x^{3} + 5280 \, x^{2} - 58356 \, x - 2389\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} + \frac {7}{243} \, \sqrt {7} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - \frac {648919}{1244160} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \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}} \left (5 x + 3\right )^{\frac {5}{2}}}{3 x + 2}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.45, size = 199, normalized size = 1.33 \begin {gather*} -\frac {7}{2430} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {1}{518400} \, {\left (12 \, {\left (8 \, {\left (36 \, \sqrt {5} {\left (5 \, x + 3\right )} - 313 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 2385 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 79315 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + \frac {648919}{1244160} \, \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} \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}\,{\left (5\,x+3\right )}^{5/2}}{3\,x+2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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