Optimal. Leaf size=101 \[ -\frac {42230 \sqrt {1-2 x}}{195657 \sqrt {5 x+3}}+\frac {956}{17787 \sqrt {1-2 x} \sqrt {5 x+3}}+\frac {4}{231 (1-2 x)^{3/2} \sqrt {5 x+3}}+\frac {54 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{49 \sqrt {7}} \]
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Rubi [A] time = 0.04, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {104, 152, 12, 93, 204} \[ -\frac {42230 \sqrt {1-2 x}}{195657 \sqrt {5 x+3}}+\frac {956}{17787 \sqrt {1-2 x} \sqrt {5 x+3}}+\frac {4}{231 (1-2 x)^{3/2} \sqrt {5 x+3}}+\frac {54 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{49 \sqrt {7}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 93
Rule 104
Rule 152
Rule 204
Rubi steps
\begin {align*} \int \frac {1}{(1-2 x)^{5/2} (2+3 x) (3+5 x)^{3/2}} \, dx &=\frac {4}{231 (1-2 x)^{3/2} \sqrt {3+5 x}}-\frac {2}{231} \int \frac {-\frac {179}{2}-60 x}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2}} \, dx\\ &=\frac {4}{231 (1-2 x)^{3/2} \sqrt {3+5 x}}+\frac {956}{17787 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {4 \int \frac {\frac {12827}{4}+3585 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx}{17787}\\ &=\frac {4}{231 (1-2 x)^{3/2} \sqrt {3+5 x}}+\frac {956}{17787 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {42230 \sqrt {1-2 x}}{195657 \sqrt {3+5 x}}-\frac {8 \int \frac {107811}{8 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{195657}\\ &=\frac {4}{231 (1-2 x)^{3/2} \sqrt {3+5 x}}+\frac {956}{17787 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {42230 \sqrt {1-2 x}}{195657 \sqrt {3+5 x}}-\frac {27}{49} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=\frac {4}{231 (1-2 x)^{3/2} \sqrt {3+5 x}}+\frac {956}{17787 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {42230 \sqrt {1-2 x}}{195657 \sqrt {3+5 x}}-\frac {54}{49} \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )\\ &=\frac {4}{231 (1-2 x)^{3/2} \sqrt {3+5 x}}+\frac {956}{17787 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {42230 \sqrt {1-2 x}}{195657 \sqrt {3+5 x}}+\frac {54 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{49 \sqrt {7}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 67, normalized size = 0.66 \[ \frac {54 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{49 \sqrt {7}}-\frac {2 \left (84460 x^2-73944 x+14163\right )}{195657 (1-2 x)^{3/2} \sqrt {5 x+3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.05, size = 101, normalized size = 1.00 \[ \frac {107811 \, \sqrt {7} {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (84460 \, x^{2} - 73944 \, x + 14163\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{1369599 \, {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.38, size = 172, normalized size = 1.70 \[ -\frac {27}{3430} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {25}{2662} \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )} - \frac {8 \, {\left (548 \, \sqrt {5} {\left (5 \, x + 3\right )} - 3399 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{4891425 \, {\left (2 \, x - 1\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 202, normalized size = 2.00 \[ -\frac {\sqrt {-2 x +1}\, \left (2156220 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-862488 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+1182440 \sqrt {-10 x^{2}-x +3}\, x^{2}-754677 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-1035216 \sqrt {-10 x^{2}-x +3}\, x +323433 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+198282 \sqrt {-10 x^{2}-x +3}\right )}{1369599 \left (2 x -1\right )^{2} \sqrt {-10 x^{2}-x +3}\, \sqrt {5 x +3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (1-2\,x\right )}^{5/2}\,\left (3\,x+2\right )\,{\left (5\,x+3\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{\left (1 - 2 x\right )^{\frac {5}{2}} \left (3 x + 2\right ) \left (5 x + 3\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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