Optimal. Leaf size=122 \[ \frac {3895 \sqrt {1-2 x} \sqrt {5 x+3}}{8232 (3 x+2)}+\frac {25 \sqrt {1-2 x} \sqrt {5 x+3}}{588 (3 x+2)^2}-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^3}-\frac {15235 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{2744 \sqrt {7}} \]
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Rubi [A] time = 0.04, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {99, 151, 12, 93, 204} \begin {gather*} \frac {3895 \sqrt {1-2 x} \sqrt {5 x+3}}{8232 (3 x+2)}+\frac {25 \sqrt {1-2 x} \sqrt {5 x+3}}{588 (3 x+2)^2}-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^3}-\frac {15235 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{2744 \sqrt {7}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 93
Rule 99
Rule 151
Rule 204
Rubi steps
\begin {align*} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^4} \, dx &=-\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{21 (2+3 x)^3}+\frac {1}{21} \int \frac {\frac {35}{2}+20 x}{\sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}} \, dx\\ &=-\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{21 (2+3 x)^3}+\frac {25 \sqrt {1-2 x} \sqrt {3+5 x}}{588 (2+3 x)^2}+\frac {1}{294} \int \frac {\frac {965}{4}-125 x}{\sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}} \, dx\\ &=-\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{21 (2+3 x)^3}+\frac {25 \sqrt {1-2 x} \sqrt {3+5 x}}{588 (2+3 x)^2}+\frac {3895 \sqrt {1-2 x} \sqrt {3+5 x}}{8232 (2+3 x)}+\frac {\int \frac {45705}{8 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{2058}\\ &=-\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{21 (2+3 x)^3}+\frac {25 \sqrt {1-2 x} \sqrt {3+5 x}}{588 (2+3 x)^2}+\frac {3895 \sqrt {1-2 x} \sqrt {3+5 x}}{8232 (2+3 x)}+\frac {15235 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{5488}\\ &=-\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{21 (2+3 x)^3}+\frac {25 \sqrt {1-2 x} \sqrt {3+5 x}}{588 (2+3 x)^2}+\frac {3895 \sqrt {1-2 x} \sqrt {3+5 x}}{8232 (2+3 x)}+\frac {15235 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{2744}\\ &=-\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{21 (2+3 x)^3}+\frac {25 \sqrt {1-2 x} \sqrt {3+5 x}}{588 (2+3 x)^2}+\frac {3895 \sqrt {1-2 x} \sqrt {3+5 x}}{8232 (2+3 x)}-\frac {15235 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{2744 \sqrt {7}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 74, normalized size = 0.61 \begin {gather*} \frac {\frac {7 \sqrt {1-2 x} \sqrt {5 x+3} \left (11685 x^2+15930 x+5296\right )}{(3 x+2)^3}-15235 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{19208} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 2.83, size = 192, normalized size = 1.57 \begin {gather*} \frac {5 \sqrt {11-2 (5 x+3)} \left (2337 \sqrt {5} (5 x+3)^{5/2}+1908 \sqrt {5} (5 x+3)^{3/2}-277 \sqrt {5} \sqrt {5 x+3}\right )}{2744 (3 (5 x+3)+1)^3}-\frac {15235 \tan ^{-1}\left (\frac {\sqrt {\frac {2}{34+\sqrt {1155}}} \sqrt {5 x+3}}{\sqrt {11}-\sqrt {11-2 (5 x+3)}}\right )}{2744 \sqrt {7}}-\frac {15235 \tan ^{-1}\left (\frac {\sqrt {68+2 \sqrt {1155}} \sqrt {5 x+3}}{\sqrt {11}-\sqrt {11-2 (5 x+3)}}\right )}{2744 \sqrt {7}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.35, size = 101, normalized size = 0.83 \begin {gather*} -\frac {15235 \, \sqrt {7} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\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 (11685 \, x^{2} + 15930 \, x + 5296\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{38416 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.50, size = 310, normalized size = 2.54 \begin {gather*} \frac {3047}{76832} \, \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 {55 \, \sqrt {10} {\left (277 \, {\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 )}^{5} - 159040 \, {\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 {20713280 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {82853120 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{1372 \, {\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 )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 202, normalized size = 1.66 \begin {gather*} \frac {\sqrt {5 x +3}\, \sqrt {-2 x +1}\, \left (411345 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+822690 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+163590 \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 )+223020 \sqrt {-10 x^{2}-x +3}\, x +121880 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+74144 \sqrt {-10 x^{2}-x +3}\right )}{38416 \sqrt {-10 x^{2}-x +3}\, \left (3 x +2\right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.20, size = 107, normalized size = 0.88 \begin {gather*} \frac {15235}{38416} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {\sqrt {-10 \, x^{2} - x + 3}}{21 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {25 \, \sqrt {-10 \, x^{2} - x + 3}}{588 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {3895 \, \sqrt {-10 \, x^{2} - x + 3}}{8232 \, {\left (3 \, x + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 13.69, size = 1273, normalized size = 10.43
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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