Optimal. Leaf size=86 \[ -\frac {5}{6} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {29}{18} \sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )-\frac {2 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{9 \sqrt {7}} \]
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Rubi [A]
time = 0.02, antiderivative size = 86, 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 = {104, 163, 56,
222, 95, 210} \begin {gather*} \frac {29}{18} \sqrt {\frac {5}{2}} \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 )}{9 \sqrt {7}}-\frac {5}{6} \sqrt {1-2 x} \sqrt {5 x+3} \end {gather*}
Antiderivative was successfully verified.
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Rule 56
Rule 95
Rule 104
Rule 163
Rule 210
Rule 222
Rubi steps
\begin {align*} \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)} \, dx &=-\frac {5}{6} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {1}{6} \int \frac {-49-\frac {145 x}{2}}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {5}{6} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {1}{9} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx+\frac {145}{36} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {5}{6} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {2}{9} \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )+\frac {1}{18} \left (29 \sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )\\ &=-\frac {5}{6} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {29}{18} \sqrt {\frac {5}{2}} \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 )}{9 \sqrt {7}}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 86, normalized size = 1.00 \begin {gather*} \frac {1}{252} \left (-210 \sqrt {1-2 x} \sqrt {3+5 x}-203 \sqrt {10} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )-8 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 83, normalized size = 0.97
method | result | size |
default | \(\frac {\sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (203 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+8 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-420 \sqrt {-10 x^{2}-x +3}\right )}{504 \sqrt {-10 x^{2}-x +3}}\) | \(83\) |
risch | \(\frac {5 \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{6 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right )}\, \sqrt {1-2 x}}+\frac {\left (\frac {29 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )}{72}+\frac {\sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right )}{63}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(120\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.88, size = 54, normalized size = 0.63 \begin {gather*} \frac {29}{72} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {1}{63} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {5}{6} \, \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 = 0.41, size = 103, normalized size = 1.20 \begin {gather*} -\frac {29}{72} \, \sqrt {5} \sqrt {2} \arctan \left (\frac {\sqrt {5} \sqrt {2} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - \frac {1}{63} \, \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 {5}{6} \, \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} \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 (5 x + 3\right )^{\frac {3}{2}}}{\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. 160 vs.
\(2 (60) = 120\).
time = 1.76, size = 160, normalized size = 1.86 \begin {gather*} \frac {1}{630} \, \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 {29}{72} \, \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 )} - \frac {1}{6} \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} \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 (5\,x+3\right )}^{3/2}}{\sqrt {1-2\,x}\,\left (3\,x+2\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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