Optimal. Leaf size=62 \[ \frac {1}{3} \sqrt {10} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )+\frac {2 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{3 \sqrt {7}} \]
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Rubi [A]
time = 0.02, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {132, 56, 222,
12, 95, 210} \begin {gather*} \frac {1}{3} \sqrt {10} \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )+\frac {2 \text {ArcTan}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{3 \sqrt {7}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 56
Rule 95
Rule 132
Rule 210
Rule 222
Rubi steps
\begin {align*} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)} \, dx &=-\left (\frac {1}{3} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\right )+\frac {5}{3} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\left (\frac {2}{3} \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )\right )+\frac {1}{3} \left (2 \sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )\\ &=\frac {1}{3} \sqrt {10} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )+\frac {2 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{3 \sqrt {7}}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(138\) vs. \(2(62)=124\).
time = 0.86, size = 138, normalized size = 2.23 \begin {gather*} -\frac {2}{21} \left (\sqrt {7} \tan ^{-1}\left (\frac {\sqrt {2 \left (34+\sqrt {1155}\right )} \sqrt {3+5 x}}{-\sqrt {11}+\sqrt {5-10 x}}\right )+7 \sqrt {10} \tan ^{-1}\left (\frac {\sqrt {6+10 x}}{\sqrt {11}-\sqrt {5-10 x}}\right )+\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 [A]
time = 0.08, size = 69, normalized size = 1.11
method | result | size |
default | \(-\frac {\sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (2 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-7 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )\right )}{42 \sqrt {-10 x^{2}-x +3}}\) | \(69\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 40, normalized size = 0.65 \begin {gather*} \frac {1}{6} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {1}{21} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 81, normalized size = 1.31 \begin {gather*} \frac {1}{21} \, \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 {1}{6} \, \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 {\sqrt {5 x + 3}}{\sqrt {1 - 2 x} \left (3 x + 2\right )}\, 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. 141 vs.
\(2 (44) = 88\).
time = 1.68, size = 141, normalized size = 2.27 \begin {gather*} -\frac {1}{210} \, \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}{6} \, \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 [B]
time = 3.86, size = 138, normalized size = 2.23 \begin {gather*} -\frac {2\,\sqrt {7}\,\mathrm {atan}\left (\frac {5580\,\sqrt {21}\,x+2699\,\sqrt {21}-5489\,\sqrt {35\,x+21}+649\,\sqrt {21-42\,x}+2141\,\sqrt {7}\,\sqrt {1-2\,x}\,\sqrt {5\,x+3}}{7400\,x-5489\,\sqrt {1-2\,x}-4543\,\sqrt {15\,x+9}+3063\,\sqrt {3}\,\sqrt {1-2\,x}\,\sqrt {5\,x+3}+9929}\right )}{21}-\frac {2\,\sqrt {10}\,\mathrm {atan}\left (\frac {\sqrt {10}-\sqrt {10-20\,x}}{2\,\sqrt {3}-2\,\sqrt {5\,x+3}}\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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