Optimal. Leaf size=115 \[ -\frac {5709 \sqrt {1-2 x}}{28 \sqrt {3+5 x}}+\frac {3 (1-2 x)^{5/2}}{14 (2+3 x)^2 \sqrt {3+5 x}}+\frac {173 (1-2 x)^{3/2}}{28 (2+3 x) \sqrt {3+5 x}}+\frac {5709 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{4 \sqrt {7}} \]
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Rubi [A]
time = 0.02, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {98, 96, 95, 210}
\begin {gather*} \frac {5709 \text {ArcTan}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{4 \sqrt {7}}+\frac {3 (1-2 x)^{5/2}}{14 (3 x+2)^2 \sqrt {5 x+3}}+\frac {173 (1-2 x)^{3/2}}{28 (3 x+2) \sqrt {5 x+3}}-\frac {5709 \sqrt {1-2 x}}{28 \sqrt {5 x+3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 95
Rule 96
Rule 98
Rule 210
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2}}{(2+3 x)^3 (3+5 x)^{3/2}} \, dx &=\frac {3 (1-2 x)^{5/2}}{14 (2+3 x)^2 \sqrt {3+5 x}}+\frac {173}{28} \int \frac {(1-2 x)^{3/2}}{(2+3 x)^2 (3+5 x)^{3/2}} \, dx\\ &=\frac {3 (1-2 x)^{5/2}}{14 (2+3 x)^2 \sqrt {3+5 x}}+\frac {173 (1-2 x)^{3/2}}{28 (2+3 x) \sqrt {3+5 x}}+\frac {5709}{56} \int \frac {\sqrt {1-2 x}}{(2+3 x) (3+5 x)^{3/2}} \, dx\\ &=-\frac {5709 \sqrt {1-2 x}}{28 \sqrt {3+5 x}}+\frac {3 (1-2 x)^{5/2}}{14 (2+3 x)^2 \sqrt {3+5 x}}+\frac {173 (1-2 x)^{3/2}}{28 (2+3 x) \sqrt {3+5 x}}-\frac {5709}{8} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {5709 \sqrt {1-2 x}}{28 \sqrt {3+5 x}}+\frac {3 (1-2 x)^{5/2}}{14 (2+3 x)^2 \sqrt {3+5 x}}+\frac {173 (1-2 x)^{3/2}}{28 (2+3 x) \sqrt {3+5 x}}-\frac {5709}{4} \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )\\ &=-\frac {5709 \sqrt {1-2 x}}{28 \sqrt {3+5 x}}+\frac {3 (1-2 x)^{5/2}}{14 (2+3 x)^2 \sqrt {3+5 x}}+\frac {173 (1-2 x)^{3/2}}{28 (2+3 x) \sqrt {3+5 x}}+\frac {5709 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{4 \sqrt {7}}\\ \end {align*}
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Mathematica [A]
time = 1.86, size = 140, normalized size = 1.22 \begin {gather*} \frac {1}{28} \left (-\frac {7 \sqrt {1-2 x} \left (3212+9815 x+7485 x^2\right )}{(2+3 x)^2 \sqrt {3+5 x}}-5709 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {2 \left (34+\sqrt {1155}\right )} \sqrt {3+5 x}}{-\sqrt {11}+\sqrt {5-10 x}}\right )-5709 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {6+10 x}}{\sqrt {34+\sqrt {1155}} \left (-\sqrt {11}+\sqrt {5-10 x}\right )}\right )\right ) \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. \(201\) vs.
\(2(88)=176\).
time = 0.10, size = 202, normalized size = 1.76
method | result | size |
default | \(-\frac {\left (256905 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+496683 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+319704 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +104790 x^{2} \sqrt {-10 x^{2}-x +3}+68508 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+137410 x \sqrt {-10 x^{2}-x +3}+44968 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}}{56 \left (2+3 x \right )^{2} \sqrt {-10 x^{2}-x +3}\, \sqrt {3+5 x}}\) | \(202\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 143, normalized size = 1.24 \begin {gather*} -\frac {5709}{56} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {2495 \, x}{6 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {2605}{12 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {49}{18 \, {\left (9 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 12 \, \sqrt {-10 \, x^{2} - x + 3} x + 4 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} + \frac {1127}{36 \, {\left (3 \, \sqrt {-10 \, x^{2} - x + 3} x + 2 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.24, size = 101, normalized size = 0.88 \begin {gather*} \frac {5709 \, \sqrt {7} {\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\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 (7485 \, x^{2} + 9815 \, x + 3212\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{56 \, {\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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. 311 vs.
\(2 (88) = 176\).
time = 0.59, size = 311, normalized size = 2.70 \begin {gather*} -\frac {5709}{560} \, \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 {11}{2} \, \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 {55 \, \sqrt {10} {\left (61 \, {\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 )}^{3} + \frac {13384 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {53536 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{2 \, {\left ({\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 )}^{2} + 280\right )}^{2}} \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 (3\,x+2\right )}^3\,{\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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