Optimal. Leaf size=48 \[ \frac {7 \sqrt {3+5 x}}{11 \sqrt {1-2 x}}-\frac {3 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{\sqrt {10}} \]
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Rubi [A]
time = 0.01, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {79, 56, 222}
\begin {gather*} \frac {7 \sqrt {5 x+3}}{11 \sqrt {1-2 x}}-\frac {3 \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{\sqrt {10}} \end {gather*}
Antiderivative was successfully verified.
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Rule 56
Rule 79
Rule 222
Rubi steps
\begin {align*} \int \frac {2+3 x}{(1-2 x)^{3/2} \sqrt {3+5 x}} \, dx &=\frac {7 \sqrt {3+5 x}}{11 \sqrt {1-2 x}}-\frac {3}{2} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=\frac {7 \sqrt {3+5 x}}{11 \sqrt {1-2 x}}-\frac {3 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{\sqrt {5}}\\ &=\frac {7 \sqrt {3+5 x}}{11 \sqrt {1-2 x}}-\frac {3 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{\sqrt {10}}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 52, normalized size = 1.08 \begin {gather*} \frac {7 \sqrt {3+5 x}}{11 \sqrt {1-2 x}}+\frac {3 \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{\sqrt {10}} \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. \(73\) vs.
\(2(35)=70\).
time = 0.09, size = 74, normalized size = 1.54
method | result | size |
default | \(-\frac {\left (66 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x -33 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+140 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}}{220 \left (-1+2 x \right ) \sqrt {-10 x^{2}-x +3}}\) | \(74\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.65, size = 36, normalized size = 0.75 \begin {gather*} -\frac {3}{20} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {7 \, \sqrt {-10 \, x^{2} - x + 3}}{11 \, {\left (2 \, x - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 71 vs.
\(2 (35) = 70\).
time = 0.78, size = 71, normalized size = 1.48 \begin {gather*} \frac {33 \, \sqrt {10} {\left (2 \, x - 1\right )} \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 ) - 140 \, \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{220 \, {\left (2 \, x - 1\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 {3 x + 2}{\left (1 - 2 x\right )^{\frac {3}{2}} \sqrt {5 x + 3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.64, size = 45, normalized size = 0.94 \begin {gather*} -\frac {3}{10} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {7 \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{55 \, {\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 = -0.02 \begin {gather*} \int \frac {3\,x+2}{{\left (1-2\,x\right )}^{3/2}\,\sqrt {5\,x+3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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