Optimal. Leaf size=101 \[ \frac {31030 \sqrt {1-2 x}}{27951 \sqrt {5 x+3}}-\frac {410 \sqrt {1-2 x}}{2541 (5 x+3)^{3/2}}+\frac {4}{77 \sqrt {1-2 x} (5 x+3)^{3/2}}-\frac {54 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{7 \sqrt {7}} \]
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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} \[ \frac {31030 \sqrt {1-2 x}}{27951 \sqrt {5 x+3}}-\frac {410 \sqrt {1-2 x}}{2541 (5 x+3)^{3/2}}+\frac {4}{77 \sqrt {1-2 x} (5 x+3)^{3/2}}-\frac {54 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{7 \sqrt {7}} \]
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)^{3/2} (2+3 x) (3+5 x)^{5/2}} \, dx &=\frac {4}{77 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {2}{77} \int \frac {-\frac {113}{2}-60 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^{5/2}} \, dx\\ &=\frac {4}{77 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {410 \sqrt {1-2 x}}{2541 (3+5 x)^{3/2}}+\frac {4 \int \frac {-\frac {1627}{4}+615 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx}{2541}\\ &=\frac {4}{77 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {410 \sqrt {1-2 x}}{2541 (3+5 x)^{3/2}}+\frac {31030 \sqrt {1-2 x}}{27951 \sqrt {3+5 x}}-\frac {8 \int -\frac {107811}{8 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{27951}\\ &=\frac {4}{77 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {410 \sqrt {1-2 x}}{2541 (3+5 x)^{3/2}}+\frac {31030 \sqrt {1-2 x}}{27951 \sqrt {3+5 x}}+\frac {27}{7} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=\frac {4}{77 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {410 \sqrt {1-2 x}}{2541 (3+5 x)^{3/2}}+\frac {31030 \sqrt {1-2 x}}{27951 \sqrt {3+5 x}}+\frac {54}{7} \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )\\ &=\frac {4}{77 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {410 \sqrt {1-2 x}}{2541 (3+5 x)^{3/2}}+\frac {31030 \sqrt {1-2 x}}{27951 \sqrt {3+5 x}}-\frac {54 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{7 \sqrt {7}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 67, normalized size = 0.66 \[ -\frac {2 \left (155150 x^2+11005 x-45016\right )}{27951 \sqrt {1-2 x} (5 x+3)^{3/2}}-\frac {54 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{7 \sqrt {7}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.45, size = 101, normalized size = 1.00 \[ -\frac {107811 \, \sqrt {7} {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (155150 \, x^{2} + 11005 \, x - 45016\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{195657 \, {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.31, size = 216, normalized size = 2.14 \[ \frac {27}{490} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {5}{63888} \, \sqrt {10} {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {696 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {2784 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )} - \frac {16 \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{46585 \, {\left (2 \, x - 1\right )}} \]
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 \[ \frac {\sqrt {-2 x +1}\, \left (5390550 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+3773385 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+2172100 \sqrt {-10 x^{2}-x +3}\, x^{2}-1293732 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+154070 \sqrt {-10 x^{2}-x +3}\, x -970299 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-630224 \sqrt {-10 x^{2}-x +3}\right )}{195657 \left (2 x -1\right ) \sqrt {-10 x^{2}-x +3}\, \left (5 x +3\right )^{\frac {3}{2}}} \]
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 \[ \int \frac {1}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (1-2\,x\right )}^{3/2}\,\left (3\,x+2\right )\,{\left (5\,x+3\right )}^{5/2}} \,d x \]
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 \[ \int \frac {1}{\left (1 - 2 x\right )^{\frac {3}{2}} \left (3 x + 2\right ) \left (5 x + 3\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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