Optimal. Leaf size=93 \[ \frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{2 (3 x+2)^2}-\frac {11 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)}-\frac {121 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{28 \sqrt {7}} \]
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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 = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {94, 93, 204} \[ \frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{2 (3 x+2)^2}-\frac {11 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)}-\frac {121 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{28 \sqrt {7}} \]
Antiderivative was successfully verified.
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Rule 93
Rule 94
Rule 204
Rubi steps
\begin {align*} \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^3} \, dx &=\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{2 (2+3 x)^2}+\frac {11}{4} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^2} \, dx\\ &=-\frac {11 \sqrt {1-2 x} \sqrt {3+5 x}}{28 (2+3 x)}+\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{2 (2+3 x)^2}+\frac {121}{56} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {11 \sqrt {1-2 x} \sqrt {3+5 x}}{28 (2+3 x)}+\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{2 (2+3 x)^2}+\frac {121}{28} \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )\\ &=-\frac {11 \sqrt {1-2 x} \sqrt {3+5 x}}{28 (2+3 x)}+\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{2 (2+3 x)^2}-\frac {121 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{28 \sqrt {7}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 69, normalized size = 0.74 \[ \frac {1}{196} \left (\frac {7 \sqrt {1-2 x} \sqrt {5 x+3} (37 x+20)}{(3 x+2)^2}-121 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.08, size = 86, normalized size = 0.92 \[ -\frac {121 \, \sqrt {7} {\left (9 \, x^{2} + 12 \, x + 4\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{392 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.51, size = 250, normalized size = 2.69 \[ \frac {121}{3920} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {121 \, \sqrt {10} {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {280 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {1120 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{14 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 154, normalized size = 1.66 \[ \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (1089 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+1452 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+518 \sqrt {-10 x^{2}-x +3}\, x +484 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+280 \sqrt {-10 x^{2}-x +3}\right )}{392 \sqrt {-10 x^{2}-x +3}\, \left (3 x +2\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.46, size = 90, normalized size = 0.97 \[ \frac {121}{392} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {5}{21} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{14 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac {37 \, \sqrt {-10 \, x^{2} - x + 3}}{84 \, {\left (3 \, x + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 12.62, size = 1037, normalized size = 11.15 \[ \text {result too large to display} \]
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 \[ \int \frac {\sqrt {1 - 2 x} \sqrt {5 x + 3}}{\left (3 x + 2\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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