Optimal. Leaf size=122 \[ \frac {2 (5 x+3)^{5/2}}{21 (1-2 x)^{3/2} (3 x+2)}-\frac {10 (5 x+3)^{3/2}}{147 \sqrt {1-2 x} (3 x+2)}-\frac {5 \sqrt {1-2 x} \sqrt {5 x+3}}{343 (3 x+2)}-\frac {55 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{343 \sqrt {7}} \]
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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} \[ \frac {2 (5 x+3)^{5/2}}{21 (1-2 x)^{3/2} (3 x+2)}-\frac {10 (5 x+3)^{3/2}}{147 \sqrt {1-2 x} (3 x+2)}-\frac {5 \sqrt {1-2 x} \sqrt {5 x+3}}{343 (3 x+2)}-\frac {55 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{343 \sqrt {7}} \]
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)^{5/2} (2+3 x)^2} \, dx &=\frac {2 (3+5 x)^{5/2}}{21 (1-2 x)^{3/2} (2+3 x)}-\frac {5}{21} \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{3/2} (2+3 x)^2} \, dx\\ &=-\frac {10 (3+5 x)^{3/2}}{147 \sqrt {1-2 x} (2+3 x)}+\frac {2 (3+5 x)^{5/2}}{21 (1-2 x)^{3/2} (2+3 x)}+\frac {5}{49} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^2} \, dx\\ &=-\frac {5 \sqrt {1-2 x} \sqrt {3+5 x}}{343 (2+3 x)}-\frac {10 (3+5 x)^{3/2}}{147 \sqrt {1-2 x} (2+3 x)}+\frac {2 (3+5 x)^{5/2}}{21 (1-2 x)^{3/2} (2+3 x)}+\frac {55}{686} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {5 \sqrt {1-2 x} \sqrt {3+5 x}}{343 (2+3 x)}-\frac {10 (3+5 x)^{3/2}}{147 \sqrt {1-2 x} (2+3 x)}+\frac {2 (3+5 x)^{5/2}}{21 (1-2 x)^{3/2} (2+3 x)}+\frac {55}{343} \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )\\ &=-\frac {5 \sqrt {1-2 x} \sqrt {3+5 x}}{343 (2+3 x)}-\frac {10 (3+5 x)^{3/2}}{147 \sqrt {1-2 x} (2+3 x)}+\frac {2 (3+5 x)^{5/2}}{21 (1-2 x)^{3/2} (2+3 x)}-\frac {55 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{343 \sqrt {7}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 86, normalized size = 0.70 \[ -\frac {-7 \sqrt {5 x+3} \left (3090 x^2+3070 x+657\right )-165 \sqrt {7-14 x} \left (6 x^2+x-2\right ) \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{7203 (1-2 x)^{3/2} (3 x+2)} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.23, size = 101, normalized size = 0.83 \[ -\frac {165 \, \sqrt {7} {\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (3090 \, x^{2} + 3070 \, x + 657\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14406 \, {\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.91, size = 232, normalized size = 1.90 \[ \frac {11}{9604} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {22 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{343 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}} + \frac {22 \, {\left (47 \, \sqrt {5} {\left (5 \, x + 3\right )} - 66 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{25725 \, {\left (2 \, x - 1\right )}^{2}} \]
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 \[ \frac {\left (1980 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-660 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+43260 \sqrt {-10 x^{2}-x +3}\, x^{2}-825 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+42980 \sqrt {-10 x^{2}-x +3}\, x +330 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+9198 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}\, \sqrt {5 x +3}}{14406 \left (3 x +2\right ) \left (2 x -1\right )^{2} \sqrt {-10 x^{2}-x +3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.14, size = 138, normalized size = 1.13 \[ \frac {55}{4802} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {2575 \, x}{1029 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {625 \, x^{2}}{18 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {135}{1372 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {138125 \, x}{5292 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {1}{567 \, {\left (3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 2 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} + \frac {50315}{15876 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \]
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 \[ \int \frac {{\left (5\,x+3\right )}^{5/2}}{{\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^2} \,d x \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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