Optimal. Leaf size=32 \[ -\frac {2 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{\sqrt {7}} \]
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Rubi [A] time = 0.01, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {93, 204} \[ -\frac {2 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{\sqrt {7}} \]
Antiderivative was successfully verified.
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Rule 93
Rule 204
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx &=2 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )\\ &=-\frac {2 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{\sqrt {7}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 32, normalized size = 1.00 \[ -\frac {2 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{\sqrt {7}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.96, size = 40, normalized size = 1.25 \[ -\frac {1}{7} \, \sqrt {7} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.05, size = 73, normalized size = 2.28 \[ \frac {1}{70} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 55, normalized size = 1.72 \[ \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )}{7 \sqrt {-10 x^{2}-x +3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.11, size = 28, normalized size = 0.88 \[ \frac {1}{7} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.27, size = 145, normalized size = 4.53 \[ \frac {2\,\sqrt {7}\,\mathrm {atan}\left (\frac {20\,\sqrt {21}\,x-\sqrt {3}\,\sqrt {7}+\sqrt {7}\,\sqrt {5\,x+3}+10\,\sqrt {21}\,\sqrt {1-2\,x}+2\,\sqrt {21}+\sqrt {3}\,\sqrt {7}\,\sqrt {1-2\,x}-4\,\sqrt {3}\,\sqrt {21}\,\sqrt {5\,x+3}-\sqrt {7}\,\sqrt {1-2\,x}\,\sqrt {5\,x+3}}{200\,x-77\,\sqrt {3}\,\sqrt {5\,x+3}-11\,\sqrt {1-2\,x}+37\,\sqrt {3}\,\sqrt {1-2\,x}\,\sqrt {5\,x+3}+131}\right )}{7} \]
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 \[ \int \frac {1}{\sqrt {1 - 2 x} \left (3 x + 2\right ) \sqrt {5 x + 3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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