Optimal. Leaf size=122 \[ \frac {2 (5 x+3)^{5/2}}{7 \sqrt {1-2 x} (3 x+2)^2}+\frac {5 \sqrt {1-2 x} (5 x+3)^{3/2}}{98 (3 x+2)^2}+\frac {165 \sqrt {1-2 x} \sqrt {5 x+3}}{1372 (3 x+2)}+\frac {1815 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{1372 \sqrt {7}} \]
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Rubi [A] time = 0.03, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {94, 93, 204} \begin {gather*} \frac {2 (5 x+3)^{5/2}}{7 \sqrt {1-2 x} (3 x+2)^2}+\frac {5 \sqrt {1-2 x} (5 x+3)^{3/2}}{98 (3 x+2)^2}+\frac {165 \sqrt {1-2 x} \sqrt {5 x+3}}{1372 (3 x+2)}+\frac {1815 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{1372 \sqrt {7}} \end {gather*}
Antiderivative was successfully verified.
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Rule 93
Rule 94
Rule 204
Rubi steps
\begin {align*} \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^3} \, dx &=\frac {2 (3+5 x)^{5/2}}{7 \sqrt {1-2 x} (2+3 x)^2}-\frac {5}{7} \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^3} \, dx\\ &=\frac {5 \sqrt {1-2 x} (3+5 x)^{3/2}}{98 (2+3 x)^2}+\frac {2 (3+5 x)^{5/2}}{7 \sqrt {1-2 x} (2+3 x)^2}-\frac {165}{196} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^2} \, dx\\ &=\frac {165 \sqrt {1-2 x} \sqrt {3+5 x}}{1372 (2+3 x)}+\frac {5 \sqrt {1-2 x} (3+5 x)^{3/2}}{98 (2+3 x)^2}+\frac {2 (3+5 x)^{5/2}}{7 \sqrt {1-2 x} (2+3 x)^2}-\frac {1815 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{2744}\\ &=\frac {165 \sqrt {1-2 x} \sqrt {3+5 x}}{1372 (2+3 x)}+\frac {5 \sqrt {1-2 x} (3+5 x)^{3/2}}{98 (2+3 x)^2}+\frac {2 (3+5 x)^{5/2}}{7 \sqrt {1-2 x} (2+3 x)^2}-\frac {1815 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{1372}\\ &=\frac {165 \sqrt {1-2 x} \sqrt {3+5 x}}{1372 (2+3 x)}+\frac {5 \sqrt {1-2 x} (3+5 x)^{3/2}}{98 (2+3 x)^2}+\frac {2 (3+5 x)^{5/2}}{7 \sqrt {1-2 x} (2+3 x)^2}+\frac {1815 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{1372 \sqrt {7}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 85, normalized size = 0.70 \begin {gather*} \frac {7 \sqrt {5 x+3} \left (8110 x^2+11525 x+4068\right )+1815 \sqrt {7-14 x} (3 x+2)^2 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{9604 \sqrt {1-2 x} (3 x+2)^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.21, size = 106, normalized size = 0.87 \begin {gather*} \frac {121 \sqrt {5 x+3} \left (\frac {15 (1-2 x)^2}{(5 x+3)^2}+\frac {175 (1-2 x)}{5 x+3}+392\right )}{1372 \sqrt {1-2 x} \left (\frac {1-2 x}{5 x+3}+7\right )^2}+\frac {1815 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{1372 \sqrt {7}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.26, size = 101, normalized size = 0.83 \begin {gather*} \frac {1815 \, \sqrt {7} {\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )} \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 ) - 14 \, {\left (8110 \, x^{2} + 11525 \, x + 4068\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{19208 \, {\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.70, size = 276, normalized size = 2.26 \begin {gather*} -\frac {363}{38416} \, \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 {242 \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{1715 \, {\left (2 \, x - 1\right )}} + \frac {121 \, \sqrt {10} {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {360 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {1440 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{98 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 209, normalized size = 1.71 \begin {gather*} -\frac {\left (32670 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+27225 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+113540 \sqrt {-10 x^{2}-x +3}\, x^{2}-7260 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+161350 \sqrt {-10 x^{2}-x +3}\, x -7260 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+56952 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}\, \sqrt {5 x +3}}{19208 \left (3 x +2\right )^{2} \left (2 x -1\right ) \sqrt {-10 x^{2}-x +3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.18, size = 143, normalized size = 1.17 \begin {gather*} -\frac {1815}{19208} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {20275 \, x}{6174 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {83665}{37044 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {1}{378 \, {\left (9 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 12 \, \sqrt {-10 \, x^{2} - x + 3} x + 4 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} - \frac {125}{1764 \, {\left (3 \, \sqrt {-10 \, x^{2} - x + 3} x + 2 \, \sqrt {-10 \, x^{2} - x + 3}\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 (5\,x+3\right )}^{5/2}}{{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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