Optimal. Leaf size=93 \[ \frac {4 (5 x+3)^{3/2}}{77 \sqrt {1-2 x} (3 x+2)}-\frac {29 \sqrt {1-2 x} \sqrt {5 x+3}}{539 (3 x+2)}-\frac {29 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{49 \sqrt {7}} \]
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Rubi [A] time = 0.02, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {96, 94, 93, 204} \[ \frac {4 (5 x+3)^{3/2}}{77 \sqrt {1-2 x} (3 x+2)}-\frac {29 \sqrt {1-2 x} \sqrt {5 x+3}}{539 (3 x+2)}-\frac {29 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{49 \sqrt {7}} \]
Antiderivative was successfully verified.
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Rule 93
Rule 94
Rule 96
Rule 204
Rubi steps
\begin {align*} \int \frac {\sqrt {3+5 x}}{(1-2 x)^{3/2} (2+3 x)^2} \, dx &=\frac {4 (3+5 x)^{3/2}}{77 \sqrt {1-2 x} (2+3 x)}+\frac {29}{77} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^2} \, dx\\ &=-\frac {29 \sqrt {1-2 x} \sqrt {3+5 x}}{539 (2+3 x)}+\frac {4 (3+5 x)^{3/2}}{77 \sqrt {1-2 x} (2+3 x)}+\frac {29}{98} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {29 \sqrt {1-2 x} \sqrt {3+5 x}}{539 (2+3 x)}+\frac {4 (3+5 x)^{3/2}}{77 \sqrt {1-2 x} (2+3 x)}+\frac {29}{49} \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )\\ &=-\frac {29 \sqrt {1-2 x} \sqrt {3+5 x}}{539 (2+3 x)}+\frac {4 (3+5 x)^{3/2}}{77 \sqrt {1-2 x} (2+3 x)}-\frac {29 \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 = 0.04, size = 78, normalized size = 0.84 \[ \frac {7 \sqrt {5 x+3} (18 x+5)-29 \sqrt {7-14 x} (3 x+2) \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{343 \sqrt {1-2 x} (3 x+2)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 82, normalized size = 0.88 \[ -\frac {29 \, \sqrt {7} {\left (6 \, x^{2} + x - 2\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 14 \, {\left (18 \, x + 5\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{686 \, {\left (6 \, x^{2} + x - 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.66, size = 219, normalized size = 2.35 \[ \frac {29}{6860} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {4 \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{245 \, {\left (2 \, x - 1\right )}} - \frac {66 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{49 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 161, normalized size = 1.73 \[ \frac {\left (174 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+29 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-252 \sqrt {-10 x^{2}-x +3}\, x -58 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-70 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}\, \sqrt {5 x +3}}{686 \left (3 x +2\right ) \left (2 x -1\right ) \sqrt {-10 x^{2}-x +3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.21, size = 92, normalized size = 0.99 \[ \frac {29}{686} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {30 \, x}{49 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {19}{147 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {1}{21 \, {\left (3 \, \sqrt {-10 \, x^{2} - x + 3} x + 2 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} \]
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 \[ \int \frac {\sqrt {5\,x+3}}{{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^2} \,d x \]
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 \[ \int \frac {\sqrt {5 x + 3}}{\left (1 - 2 x\right )^{\frac {3}{2}} \left (3 x + 2\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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