Optimal. Leaf size=115 \[ -\frac {90415 \sqrt {1-2 x}}{2156 \sqrt {5 x+3}}+\frac {543 \sqrt {1-2 x}}{196 (3 x+2) \sqrt {5 x+3}}+\frac {3 \sqrt {1-2 x}}{14 (3 x+2)^2 \sqrt {5 x+3}}+\frac {56421 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{196 \sqrt {7}} \]
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Rubi [A] time = 0.04, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {103, 151, 152, 12, 93, 204} \[ -\frac {90415 \sqrt {1-2 x}}{2156 \sqrt {5 x+3}}+\frac {543 \sqrt {1-2 x}}{196 (3 x+2) \sqrt {5 x+3}}+\frac {3 \sqrt {1-2 x}}{14 (3 x+2)^2 \sqrt {5 x+3}}+\frac {56421 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{196 \sqrt {7}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 93
Rule 103
Rule 151
Rule 152
Rule 204
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{3/2}} \, dx &=\frac {3 \sqrt {1-2 x}}{14 (2+3 x)^2 \sqrt {3+5 x}}+\frac {1}{14} \int \frac {\frac {101}{2}-60 x}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2}} \, dx\\ &=\frac {3 \sqrt {1-2 x}}{14 (2+3 x)^2 \sqrt {3+5 x}}+\frac {543 \sqrt {1-2 x}}{196 (2+3 x) \sqrt {3+5 x}}+\frac {1}{98} \int \frac {\frac {11567}{4}-2715 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx\\ &=-\frac {90415 \sqrt {1-2 x}}{2156 \sqrt {3+5 x}}+\frac {3 \sqrt {1-2 x}}{14 (2+3 x)^2 \sqrt {3+5 x}}+\frac {543 \sqrt {1-2 x}}{196 (2+3 x) \sqrt {3+5 x}}-\frac {1}{539} \int \frac {620631}{8 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {90415 \sqrt {1-2 x}}{2156 \sqrt {3+5 x}}+\frac {3 \sqrt {1-2 x}}{14 (2+3 x)^2 \sqrt {3+5 x}}+\frac {543 \sqrt {1-2 x}}{196 (2+3 x) \sqrt {3+5 x}}-\frac {56421}{392} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {90415 \sqrt {1-2 x}}{2156 \sqrt {3+5 x}}+\frac {3 \sqrt {1-2 x}}{14 (2+3 x)^2 \sqrt {3+5 x}}+\frac {543 \sqrt {1-2 x}}{196 (2+3 x) \sqrt {3+5 x}}-\frac {56421}{196} \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )\\ &=-\frac {90415 \sqrt {1-2 x}}{2156 \sqrt {3+5 x}}+\frac {3 \sqrt {1-2 x}}{14 (2+3 x)^2 \sqrt {3+5 x}}+\frac {543 \sqrt {1-2 x}}{196 (2+3 x) \sqrt {3+5 x}}+\frac {56421 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{196 \sqrt {7}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 74, normalized size = 0.64 \[ \frac {620631 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )-\frac {7 \sqrt {1-2 x} \left (813735 x^2+1067061 x+349252\right )}{(3 x+2)^2 \sqrt {5 x+3}}}{15092} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 101, normalized size = 0.88 \[ \frac {620631 \, \sqrt {7} {\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (813735 \, x^{2} + 1067061 \, x + 349252\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{30184 \, {\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.78, size = 316, normalized size = 2.75 \[ -\frac {56421}{27440} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {25}{22} \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )} - \frac {297 \, {\left (107 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + 23800 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}\right )}}{98 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 202, normalized size = 1.76 \[ -\frac {\left (27928395 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+53994897 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+11392290 \sqrt {-10 x^{2}-x +3}\, x^{2}+34755336 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+14938854 \sqrt {-10 x^{2}-x +3}\, x +7447572 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+4889528 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}}{30184 \left (3 x +2\right )^{2} \sqrt {-10 x^{2}-x +3}\, \sqrt {5 x +3}} \]
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 \[ \int \frac {1}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{3} \sqrt {-2 \, x + 1}}\,{d x} \]
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 \[ \int \frac {1}{\sqrt {1-2\,x}\,{\left (3\,x+2\right )}^3\,{\left (5\,x+3\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{\sqrt {1 - 2 x} \left (3 x + 2\right )^{3} \left (5 x + 3\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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