Optimal. Leaf size=130 \[ \frac {2841815 \sqrt {1-2 x}}{195657 \sqrt {5 x+3}}-\frac {28705 \sqrt {1-2 x}}{17787 (5 x+3)^{3/2}}-\frac {58}{539 \sqrt {1-2 x} (5 x+3)^{3/2}}+\frac {3}{7 \sqrt {1-2 x} (3 x+2) (5 x+3)^{3/2}}-\frac {4887 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{49 \sqrt {7}} \]
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Rubi [A] time = 0.05, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {103, 152, 12, 93, 204} \[ \frac {2841815 \sqrt {1-2 x}}{195657 \sqrt {5 x+3}}-\frac {28705 \sqrt {1-2 x}}{17787 (5 x+3)^{3/2}}-\frac {58}{539 \sqrt {1-2 x} (5 x+3)^{3/2}}+\frac {3}{7 \sqrt {1-2 x} (3 x+2) (5 x+3)^{3/2}}-\frac {4887 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{49 \sqrt {7}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 93
Rule 103
Rule 152
Rule 204
Rubi steps
\begin {align*} \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{5/2}} \, dx &=\frac {3}{7 \sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}}+\frac {1}{7} \int \frac {\frac {61}{2}-90 x}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^{5/2}} \, dx\\ &=-\frac {58}{539 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {3}{7 \sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}}-\frac {2}{539} \int \frac {-\frac {3653}{4}+870 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^{5/2}} \, dx\\ &=-\frac {58}{539 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {28705 \sqrt {1-2 x}}{17787 (3+5 x)^{3/2}}+\frac {3}{7 \sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}}+\frac {4 \int \frac {-\frac {361687}{8}+\frac {86115 x}{2}}{\sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx}{17787}\\ &=-\frac {58}{539 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {28705 \sqrt {1-2 x}}{17787 (3+5 x)^{3/2}}+\frac {3}{7 \sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}}+\frac {2841815 \sqrt {1-2 x}}{195657 \sqrt {3+5 x}}-\frac {8 \int -\frac {19513791}{16 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{195657}\\ &=-\frac {58}{539 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {28705 \sqrt {1-2 x}}{17787 (3+5 x)^{3/2}}+\frac {3}{7 \sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}}+\frac {2841815 \sqrt {1-2 x}}{195657 \sqrt {3+5 x}}+\frac {4887}{98} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {58}{539 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {28705 \sqrt {1-2 x}}{17787 (3+5 x)^{3/2}}+\frac {3}{7 \sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}}+\frac {2841815 \sqrt {1-2 x}}{195657 \sqrt {3+5 x}}+\frac {4887}{49} \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )\\ &=-\frac {58}{539 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {28705 \sqrt {1-2 x}}{17787 (3+5 x)^{3/2}}+\frac {3}{7 \sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}}+\frac {2841815 \sqrt {1-2 x}}{195657 \sqrt {3+5 x}}-\frac {4887 \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 = 0.07, size = 102, normalized size = 0.78 \[ \frac {-19513791 \sqrt {7-14 x} \sqrt {5 x+3} \left (15 x^2+19 x+6\right ) \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )-7 \left (85254450 x^3+63467215 x^2-20145298 x-16461125\right )}{1369599 \sqrt {1-2 x} (3 x+2) (5 x+3)^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.40, size = 116, normalized size = 0.89 \[ -\frac {19513791 \, \sqrt {7} {\left (150 \, x^{4} + 205 \, x^{3} + 34 \, x^{2} - 51 \, x - 18\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (85254450 \, x^{3} + 63467215 \, x^{2} - 20145298 \, x - 16461125\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{2739198 \, {\left (150 \, x^{4} + 205 \, x^{3} + 34 \, x^{2} - 51 \, x - 18\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.22, size = 335, normalized size = 2.58 \[ \frac {4887}{6860} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {25}{63888} \, \sqrt {10} {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {1488 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {5952 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )} - \frac {32 \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{326095 \, {\left (2 \, x - 1\right )}} + \frac {1782 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{49 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 257, normalized size = 1.98 \[ \frac {\sqrt {-2 x +1}\, \left (2927068650 \sqrt {7}\, x^{4} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+4000327155 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+1193562300 \sqrt {-10 x^{2}-x +3}\, x^{3}+663468894 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+888541010 \sqrt {-10 x^{2}-x +3}\, x^{2}-995203341 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-282034172 \sqrt {-10 x^{2}-x +3}\, x -351248238 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-230455750 \sqrt {-10 x^{2}-x +3}\right )}{2739198 \left (3 x +2\right ) \left (2 x -1\right ) \sqrt {-10 x^{2}-x +3}\, \left (5 x +3\right )^{\frac {3}{2}}} \]
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 \[ \int \frac {1}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{2} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^2\,{\left (5\,x+3\right )}^{5/2}} \,d x \]
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 \[ \int \frac {1}{\left (1 - 2 x\right )^{\frac {3}{2}} \left (3 x + 2\right )^{2} \left (5 x + 3\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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