Optimal. Leaf size=79 \[ \frac {4}{77 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {370 \sqrt {1-2 x}}{847 \sqrt {3+5 x}}+\frac {18 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{7 \sqrt {7}} \]
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Rubi [A]
time = 0.02, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {106, 157, 12,
95, 210} \begin {gather*} \frac {18 \text {ArcTan}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{7 \sqrt {7}}-\frac {370 \sqrt {1-2 x}}{847 \sqrt {5 x+3}}+\frac {4}{77 \sqrt {1-2 x} \sqrt {5 x+3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 95
Rule 106
Rule 157
Rule 210
Rubi steps
\begin {align*} \int \frac {1}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2}} \, dx &=\frac {4}{77 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {2}{77} \int \frac {-\frac {73}{2}-30 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx\\ &=\frac {4}{77 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {370 \sqrt {1-2 x}}{847 \sqrt {3+5 x}}+\frac {4}{847} \int -\frac {1089}{4 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=\frac {4}{77 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {370 \sqrt {1-2 x}}{847 \sqrt {3+5 x}}-\frac {9}{7} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=\frac {4}{77 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {370 \sqrt {1-2 x}}{847 \sqrt {3+5 x}}-\frac {18}{7} \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )\\ &=\frac {4}{77 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {370 \sqrt {1-2 x}}{847 \sqrt {3+5 x}}+\frac {18 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{7 \sqrt {7}}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 62, normalized size = 0.78 \begin {gather*} \frac {2 (-163+370 x)}{847 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {18 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{7 \sqrt {7}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(153\) vs.
\(2(58)=116\).
time = 0.08, size = 154, normalized size = 1.95
method | result | size |
default | \(-\frac {\sqrt {1-2 x}\, \left (10890 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+1089 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x -3267 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+5180 x \sqrt {-10 x^{2}-x +3}-2282 \sqrt {-10 x^{2}-x +3}\right )}{5929 \left (-1+2 x \right ) \sqrt {-10 x^{2}-x +3}\, \sqrt {3+5 x}}\) | \(154\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 58, normalized size = 0.73 \begin {gather*} -\frac {9}{49} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {740 \, x}{847 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {326}{847 \, \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 = 0.46, size = 82, normalized size = 1.04 \begin {gather*} \frac {1089 \, \sqrt {7} {\left (10 \, x^{2} + x - 3\right )} \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 ) - 14 \, {\left (370 \, x - 163\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{5929 \, {\left (10 \, x^{2} + x - 3\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 {1}{\left (1 - 2 x\right )^{\frac {3}{2}} \cdot \left (3 x + 2\right ) \left (5 x + 3\right )^{\frac {3}{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. 159 vs.
\(2 (58) = 116\).
time = 0.51, size = 159, normalized size = 2.01 \begin {gather*} -\frac {9}{490} \, \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 {5}{242} \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )} - \frac {8 \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{4235 \, {\left (2 \, x - 1\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 {1}{{\left (1-2\,x\right )}^{3/2}\,\left (3\,x+2\right )\,{\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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