Optimal. Leaf size=115 \[ -\frac {3895 \sqrt {5 x+3}}{7546 \sqrt {1-2 x}}+\frac {345 \sqrt {5 x+3}}{196 \sqrt {1-2 x} (3 x+2)}+\frac {3 \sqrt {5 x+3}}{14 \sqrt {1-2 x} (3 x+2)^2}-\frac {12465 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{1372 \sqrt {7}} \]
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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} \[ -\frac {3895 \sqrt {5 x+3}}{7546 \sqrt {1-2 x}}+\frac {345 \sqrt {5 x+3}}{196 \sqrt {1-2 x} (3 x+2)}+\frac {3 \sqrt {5 x+3}}{14 \sqrt {1-2 x} (3 x+2)^2}-\frac {12465 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{1372 \sqrt {7}} \]
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}{(1-2 x)^{3/2} (2+3 x)^3 \sqrt {3+5 x}} \, dx &=\frac {3 \sqrt {3+5 x}}{14 \sqrt {1-2 x} (2+3 x)^2}+\frac {1}{14} \int \frac {\frac {35}{2}-60 x}{(1-2 x)^{3/2} (2+3 x)^2 \sqrt {3+5 x}} \, dx\\ &=\frac {3 \sqrt {3+5 x}}{14 \sqrt {1-2 x} (2+3 x)^2}+\frac {345 \sqrt {3+5 x}}{196 \sqrt {1-2 x} (2+3 x)}+\frac {1}{98} \int \frac {-\frac {445}{4}-1725 x}{(1-2 x)^{3/2} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {3895 \sqrt {3+5 x}}{7546 \sqrt {1-2 x}}+\frac {3 \sqrt {3+5 x}}{14 \sqrt {1-2 x} (2+3 x)^2}+\frac {345 \sqrt {3+5 x}}{196 \sqrt {1-2 x} (2+3 x)}-\frac {\int -\frac {137115}{8 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{3773}\\ &=-\frac {3895 \sqrt {3+5 x}}{7546 \sqrt {1-2 x}}+\frac {3 \sqrt {3+5 x}}{14 \sqrt {1-2 x} (2+3 x)^2}+\frac {345 \sqrt {3+5 x}}{196 \sqrt {1-2 x} (2+3 x)}+\frac {12465 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{2744}\\ &=-\frac {3895 \sqrt {3+5 x}}{7546 \sqrt {1-2 x}}+\frac {3 \sqrt {3+5 x}}{14 \sqrt {1-2 x} (2+3 x)^2}+\frac {345 \sqrt {3+5 x}}{196 \sqrt {1-2 x} (2+3 x)}+\frac {12465 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{1372}\\ &=-\frac {3895 \sqrt {3+5 x}}{7546 \sqrt {1-2 x}}+\frac {3 \sqrt {3+5 x}}{14 \sqrt {1-2 x} (2+3 x)^2}+\frac {345 \sqrt {3+5 x}}{196 \sqrt {1-2 x} (2+3 x)}-\frac {12465 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{1372 \sqrt {7}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 85, normalized size = 0.74 \[ \frac {-7 \sqrt {5 x+3} \left (70110 x^2+13785 x-25204\right )-137115 \sqrt {7-14 x} (3 x+2)^2 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{105644 \sqrt {1-2 x} (3 x+2)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.91, size = 101, normalized size = 0.88 \[ -\frac {137115 \, \sqrt {7} {\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (70110 \, x^{2} + 13785 \, x - 25204\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{211288 \, {\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.60, size = 278, normalized size = 2.42 \[ \frac {2493}{38416} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {16 \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{18865 \, {\left (2 \, x - 1\right )}} + \frac {297 \, \sqrt {10} {\left (9 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {1640 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {6560 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{98 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 209, normalized size = 1.82 \[ \frac {\left (2468070 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+2056725 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+981540 \sqrt {-10 x^{2}-x +3}\, x^{2}-548460 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+192990 \sqrt {-10 x^{2}-x +3}\, x -548460 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-352856 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {5 x +3}\, \sqrt {-2 x +1}}{211288 \left (3 x +2\right )^{2} \left (2 x -1\right ) \sqrt {-10 x^{2}-x +3}} \]
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 \[ \int \frac {1}{\sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{3} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^3\,\sqrt {5\,x+3}} \,d x \]
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 \[ \int \frac {1}{\left (1 - 2 x\right )^{\frac {3}{2}} \left (3 x + 2\right )^{3} \sqrt {5 x + 3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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