Optimal. Leaf size=72 \[ \frac {49}{22 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {1229 \sqrt {1-2 x}}{1210 \sqrt {3+5 x}}-\frac {9 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{5 \sqrt {10}} \]
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Rubi [A]
time = 0.01, antiderivative size = 72, 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 = {91, 79, 56, 222}
\begin {gather*} -\frac {9 \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{5 \sqrt {10}}-\frac {1229 \sqrt {1-2 x}}{1210 \sqrt {5 x+3}}+\frac {49}{22 \sqrt {1-2 x} \sqrt {5 x+3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 56
Rule 79
Rule 91
Rule 222
Rubi steps
\begin {align*} \int \frac {(2+3 x)^2}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx &=\frac {49}{22 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {1}{22} \int \frac {-\frac {127}{2}+99 x}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx\\ &=\frac {49}{22 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {1229 \sqrt {1-2 x}}{1210 \sqrt {3+5 x}}-\frac {9}{10} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=\frac {49}{22 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {1229 \sqrt {1-2 x}}{1210 \sqrt {3+5 x}}-\frac {9 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{5 \sqrt {5}}\\ &=\frac {49}{22 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {1229 \sqrt {1-2 x}}{1210 \sqrt {3+5 x}}-\frac {9 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{5 \sqrt {10}}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 59, normalized size = 0.82 \begin {gather*} \frac {733+1229 x}{605 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {9 \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{5 \sqrt {10}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 103, normalized size = 1.43
method | result | size |
default | \(-\frac {\sqrt {1-2 x}\, \left (10890 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{2}+1089 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x -3267 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+24580 x \sqrt {-10 x^{2}-x +3}+14660 \sqrt {-10 x^{2}-x +3}\right )}{12100 \left (-1+2 x \right ) \sqrt {-10 x^{2}-x +3}\, \sqrt {3+5 x}}\) | \(103\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.69, size = 41, normalized size = 0.57 \begin {gather*} \frac {9}{100} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {1229 \, x}{605 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {733}{605 \, \sqrt {-10 \, x^{2} - x + 3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.49, size = 82, normalized size = 1.14 \begin {gather*} \frac {1089 \, \sqrt {10} {\left (10 \, x^{2} + x - 3\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 ) - 20 \, {\left (1229 \, x + 733\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{12100 \, {\left (10 \, x^{2} + x - 3\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 {\left (3 x + 2\right )^{2}}{\left (1 - 2 x\right )^{\frac {3}{2}} \left (5 x + 3\right )^{\frac {3}{2}}}\, 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. 105 vs.
\(2 (51) = 102\).
time = 0.69, size = 105, normalized size = 1.46 \begin {gather*} -\frac {9}{50} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {\sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{6050 \, \sqrt {5 \, x + 3}} - \frac {49 \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{605 \, {\left (2 \, x - 1\right )}} + \frac {2 \, \sqrt {10} \sqrt {5 \, x + 3}}{3025 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\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 = -0.01 \begin {gather*} \int \frac {{\left (3\,x+2\right )}^2}{{\left (1-2\,x\right )}^{3/2}\,{\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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