Optimal. Leaf size=106 \[ \frac {107}{180} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {1}{6} (1-2 x)^{3/2} \sqrt {3+5 x}+\frac {4091 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{540 \sqrt {10}}+\frac {14}{27} \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps
used = 7, 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 {4091 \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{540 \sqrt {10}}+\frac {14}{27} \sqrt {7} \text {ArcTan}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )+\frac {1}{6} \sqrt {5 x+3} (1-2 x)^{3/2}+\frac {107}{180} \sqrt {5 x+3} \sqrt {1-2 x} \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} \sqrt {3+5 x}}{2+3 x} \, dx &=\frac {1}{6} (1-2 x)^{3/2} \sqrt {3+5 x}-\frac {1}{6} \int \frac {\left (-31-\frac {107 x}{2}\right ) \sqrt {1-2 x}}{(2+3 x) \sqrt {3+5 x}} \, dx\\ &=\frac {107}{180} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {1}{6} (1-2 x)^{3/2} \sqrt {3+5 x}-\frac {1}{90} \int \frac {-\frac {1037}{2}-\frac {4091 x}{4}}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=\frac {107}{180} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {1}{6} (1-2 x)^{3/2} \sqrt {3+5 x}-\frac {49}{27} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx+\frac {4091 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{1080}\\ &=\frac {107}{180} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {1}{6} (1-2 x)^{3/2} \sqrt {3+5 x}-\frac {98}{27} \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )+\frac {4091 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{540 \sqrt {5}}\\ &=\frac {107}{180} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {1}{6} (1-2 x)^{3/2} \sqrt {3+5 x}+\frac {4091 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{540 \sqrt {10}}+\frac {14}{27} \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 = 0.17, size = 109, normalized size = 1.03 \begin {gather*} \frac {30 \sqrt {1-2 x} \left (411+505 x-300 x^2\right )-4091 \sqrt {30+50 x} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )+2800 \sqrt {7} \sqrt {3+5 x} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{5400 \sqrt {3+5 x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 98, normalized size = 0.92
method | result | size |
default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (4091 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-2800 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-3600 x \sqrt {-10 x^{2}-x +3}+8220 \sqrt {-10 x^{2}-x +3}\right )}{10800 \sqrt {-10 x^{2}-x +3}}\) | \(98\) |
risch | \(\frac {\left (-137+60 x \right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{180 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right )}\, \sqrt {1-2 x}}+\frac {\left (\frac {4091 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )}{10800}-\frac {7 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right )}{27}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(125\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 69, normalized size = 0.65 \begin {gather*} -\frac {1}{3} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {4091}{10800} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {7}{27} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {137}{180} \, \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 = 2.04, size = 102, normalized size = 0.96 \begin {gather*} -\frac {1}{180} \, {\left (60 \, x - 137\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} + \frac {7}{27} \, \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 {4091}{10800} \, \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}} \sqrt {5 x + 3}}{3 x + 2}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 173 vs.
\(2 (76) = 152\).
time = 2.67, size = 173, normalized size = 1.63 \begin {gather*} -\frac {7}{270} \, \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}{900} \, {\left (12 \, \sqrt {5} {\left (5 \, x + 3\right )} - 173 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + \frac {4091}{10800} \, \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}\,\sqrt {5\,x+3}}{3\,x+2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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