Optimal. Leaf size=93 \[ \frac {3 (1-2 x)^{3/2} \sqrt {3+5 x}}{14 (2+3 x)^2}+\frac {107 \sqrt {1-2 x} \sqrt {3+5 x}}{28 (2+3 x)}-\frac {1177 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{28 \sqrt {7}} \]
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Rubi [A]
time = 0.02, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps
used = 4, 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 {1177 \text {ArcTan}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{28 \sqrt {7}}+\frac {3 \sqrt {5 x+3} (1-2 x)^{3/2}}{14 (3 x+2)^2}+\frac {107 \sqrt {5 x+3} \sqrt {1-2 x}}{28 (3 x+2)} \end {gather*}
Antiderivative was successfully verified.
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Rule 95
Rule 96
Rule 98
Rule 210
Rubi steps
\begin {align*} \int \frac {\sqrt {1-2 x}}{(2+3 x)^3 \sqrt {3+5 x}} \, dx &=\frac {3 (1-2 x)^{3/2} \sqrt {3+5 x}}{14 (2+3 x)^2}+\frac {107}{28} \int \frac {\sqrt {1-2 x}}{(2+3 x)^2 \sqrt {3+5 x}} \, dx\\ &=\frac {3 (1-2 x)^{3/2} \sqrt {3+5 x}}{14 (2+3 x)^2}+\frac {107 \sqrt {1-2 x} \sqrt {3+5 x}}{28 (2+3 x)}+\frac {1177}{56} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=\frac {3 (1-2 x)^{3/2} \sqrt {3+5 x}}{14 (2+3 x)^2}+\frac {107 \sqrt {1-2 x} \sqrt {3+5 x}}{28 (2+3 x)}+\frac {1177}{28} \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )\\ &=\frac {3 (1-2 x)^{3/2} \sqrt {3+5 x}}{14 (2+3 x)^2}+\frac {107 \sqrt {1-2 x} \sqrt {3+5 x}}{28 (2+3 x)}-\frac {1177 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{28 \sqrt {7}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.36, size = 89, normalized size = 0.96 \begin {gather*} \frac {1}{196} \left (\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} (220+309 x)}{(2+3 x)^2}-1177 i \sqrt {7} \tanh ^{-1}\left (\frac {1}{7} \left (2 \sqrt {70}+3 \sqrt {70} x+3 i \sqrt {7-14 x} \sqrt {3+5 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. \(153\) vs.
\(2(72)=144\).
time = 0.13, size = 154, normalized size = 1.66
method | result | size |
risch | \(-\frac {\sqrt {3+5 x}\, \left (-1+2 x \right ) \left (220+309 x \right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{28 \left (2+3 x \right )^{2} \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right )}\, \sqrt {1-2 x}}+\frac {1177 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{392 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(119\) |
default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (10593 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+14124 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +4708 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+4326 x \sqrt {-10 x^{2}-x +3}+3080 \sqrt {-10 x^{2}-x +3}\right )}{392 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{2}}\) | \(154\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 76, normalized size = 0.82 \begin {gather*} \frac {1177}{392} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {\sqrt {-10 \, x^{2} - x + 3}}{2 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {103 \, \sqrt {-10 \, x^{2} - x + 3}}{28 \, {\left (3 \, x + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.68, size = 86, normalized size = 0.92 \begin {gather*} -\frac {1177 \, \sqrt {7} {\left (9 \, x^{2} + 12 \, x + 4\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 (309 \, x + 220\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{392 \, {\left (9 \, x^{2} + 12 \, x + 4\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 {\sqrt {1 - 2 x}}{\left (3 x + 2\right )^{3} \sqrt {5 x + 3}}\, 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. 257 vs.
\(2 (72) = 144\).
time = 0.93, size = 257, normalized size = 2.76 \begin {gather*} \frac {11}{3920} \, \sqrt {5} {\left (107 \, \sqrt {70} \sqrt {2} {\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 {280 \, \sqrt {2} {\left (173 \, {\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 {29960 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {119840 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{{\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}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 12.75, size = 1037, normalized size = 11.15 \begin {gather*} \frac {\frac {18323\,{\left (\sqrt {1-2\,x}-1\right )}^5}{875\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^5}-\frac {36646\,{\left (\sqrt {1-2\,x}-1\right )}^3}{4375\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^3}-\frac {2326\,\left (\sqrt {1-2\,x}-1\right )}{4375\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}+\frac {1163\,{\left (\sqrt {1-2\,x}-1\right )}^7}{140\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^7}+\frac {10607\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^2}{4375\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}-\frac {3646\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^4}{875\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^4}+\frac {10607\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^6}{700\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^6}}{\frac {544\,{\left (\sqrt {1-2\,x}-1\right )}^2}{625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}-\frac {1764\,{\left (\sqrt {1-2\,x}-1\right )}^4}{625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^4}+\frac {136\,{\left (\sqrt {1-2\,x}-1\right )}^6}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^6}+\frac {{\left (\sqrt {1-2\,x}-1\right )}^8}{{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^8}-\frac {96\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^3}{625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^3}+\frac {48\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^5}{125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^5}+\frac {12\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^7}{5\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^7}-\frac {96\,\sqrt {3}\,\left (\sqrt {1-2\,x}-1\right )}{625\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}+\frac {16}{625}}-\frac {1177\,\sqrt {7}\,\mathrm {atan}\left (\frac {\frac {1177\,\sqrt {7}\,\left (\frac {7062\,\sqrt {3}}{875}+\frac {3531\,\left (\sqrt {1-2\,x}-1\right )}{875\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {3531\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^2}{175\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}-\frac {\sqrt {7}\,\left (\frac {212\,{\left (\sqrt {1-2\,x}-1\right )}^2}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {888\,\sqrt {3}\,\left (\sqrt {1-2\,x}-1\right )}{125\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {536}{125}\right )\,1177{}\mathrm {i}}{392}\right )}{392}+\frac {1177\,\sqrt {7}\,\left (\frac {7062\,\sqrt {3}}{875}+\frac {3531\,\left (\sqrt {1-2\,x}-1\right )}{875\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {3531\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^2}{175\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {\sqrt {7}\,\left (\frac {212\,{\left (\sqrt {1-2\,x}-1\right )}^2}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {888\,\sqrt {3}\,\left (\sqrt {1-2\,x}-1\right )}{125\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {536}{125}\right )\,1177{}\mathrm {i}}{392}\right )}{392}}{\frac {1385329\,{\left (\sqrt {1-2\,x}-1\right )}^2}{9800\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {1385329}{24500}+\frac {\sqrt {7}\,\left (\frac {7062\,\sqrt {3}}{875}+\frac {3531\,\left (\sqrt {1-2\,x}-1\right )}{875\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {3531\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^2}{175\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}-\frac {\sqrt {7}\,\left (\frac {212\,{\left (\sqrt {1-2\,x}-1\right )}^2}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {888\,\sqrt {3}\,\left (\sqrt {1-2\,x}-1\right )}{125\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {536}{125}\right )\,1177{}\mathrm {i}}{392}\right )\,1177{}\mathrm {i}}{392}-\frac {\sqrt {7}\,\left (\frac {7062\,\sqrt {3}}{875}+\frac {3531\,\left (\sqrt {1-2\,x}-1\right )}{875\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {3531\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^2}{175\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {\sqrt {7}\,\left (\frac {212\,{\left (\sqrt {1-2\,x}-1\right )}^2}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {888\,\sqrt {3}\,\left (\sqrt {1-2\,x}-1\right )}{125\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {536}{125}\right )\,1177{}\mathrm {i}}{392}\right )\,1177{}\mathrm {i}}{392}}\right )}{196} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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