Optimal. Leaf size=144 \[ -\frac {32735 \sqrt {5 x+3}}{15092 \sqrt {1-2 x}}+\frac {2865 \sqrt {5 x+3}}{392 \sqrt {1-2 x} (3 x+2)}+\frac {27 \sqrt {5 x+3}}{28 \sqrt {1-2 x} (3 x+2)^2}+\frac {\sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^3}-\frac {102345 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{2744 \sqrt {7}} \]
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Rubi [A] time = 0.05, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 7, 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} \[ -\frac {32735 \sqrt {5 x+3}}{15092 \sqrt {1-2 x}}+\frac {2865 \sqrt {5 x+3}}{392 \sqrt {1-2 x} (3 x+2)}+\frac {27 \sqrt {5 x+3}}{28 \sqrt {1-2 x} (3 x+2)^2}+\frac {\sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^3}-\frac {102345 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{2744 \sqrt {7}} \]
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)^4 \sqrt {3+5 x}} \, dx &=\frac {\sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^3}+\frac {1}{21} \int \frac {\frac {69}{2}-90 x}{(1-2 x)^{3/2} (2+3 x)^3 \sqrt {3+5 x}} \, dx\\ &=\frac {\sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^3}+\frac {27 \sqrt {3+5 x}}{28 \sqrt {1-2 x} (2+3 x)^2}+\frac {1}{294} \int \frac {\frac {4935}{4}-5670 x}{(1-2 x)^{3/2} (2+3 x)^2 \sqrt {3+5 x}} \, dx\\ &=\frac {\sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^3}+\frac {27 \sqrt {3+5 x}}{28 \sqrt {1-2 x} (2+3 x)^2}+\frac {2865 \sqrt {3+5 x}}{392 \sqrt {1-2 x} (2+3 x)}+\frac {\int \frac {-\frac {85785}{8}-\frac {300825 x}{2}}{(1-2 x)^{3/2} (2+3 x) \sqrt {3+5 x}} \, dx}{2058}\\ &=-\frac {32735 \sqrt {3+5 x}}{15092 \sqrt {1-2 x}}+\frac {\sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^3}+\frac {27 \sqrt {3+5 x}}{28 \sqrt {1-2 x} (2+3 x)^2}+\frac {2865 \sqrt {3+5 x}}{392 \sqrt {1-2 x} (2+3 x)}-\frac {\int -\frac {23641695}{16 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{79233}\\ &=-\frac {32735 \sqrt {3+5 x}}{15092 \sqrt {1-2 x}}+\frac {\sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^3}+\frac {27 \sqrt {3+5 x}}{28 \sqrt {1-2 x} (2+3 x)^2}+\frac {2865 \sqrt {3+5 x}}{392 \sqrt {1-2 x} (2+3 x)}+\frac {102345 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{5488}\\ &=-\frac {32735 \sqrt {3+5 x}}{15092 \sqrt {1-2 x}}+\frac {\sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^3}+\frac {27 \sqrt {3+5 x}}{28 \sqrt {1-2 x} (2+3 x)^2}+\frac {2865 \sqrt {3+5 x}}{392 \sqrt {1-2 x} (2+3 x)}+\frac {102345 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{2744}\\ &=-\frac {32735 \sqrt {3+5 x}}{15092 \sqrt {1-2 x}}+\frac {\sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^3}+\frac {27 \sqrt {3+5 x}}{28 \sqrt {1-2 x} (2+3 x)^2}+\frac {2865 \sqrt {3+5 x}}{392 \sqrt {1-2 x} (2+3 x)}-\frac {102345 \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 = 0.07, size = 90, normalized size = 0.62 \[ \frac {-7 \sqrt {5 x+3} \left (1767690 x^3+1549935 x^2-377658 x-421184\right )-1125795 \sqrt {7-14 x} (3 x+2)^3 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{211288 \sqrt {1-2 x} (3 x+2)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.05, size = 116, normalized size = 0.81 \[ -\frac {1125795 \, \sqrt {7} {\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (1767690 \, x^{3} + 1549935 \, x^{2} - 377658 \, x - 421184\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{422576 \, {\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.60, size = 336, normalized size = 2.33 \[ \frac {20469}{76832} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {32 \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{132055 \, {\left (2 \, x - 1\right )}} + \frac {297 \, \sqrt {10} {\left (4937 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} + 1785280 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {188708800 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {754835200 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{9604 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 257, normalized size = 1.78 \[ \frac {\left (60792930 \sqrt {7}\, x^{4} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+91189395 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+24747660 \sqrt {-10 x^{2}-x +3}\, x^{3}+20264310 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+21699090 \sqrt {-10 x^{2}-x +3}\, x^{2}-22515900 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-5287212 \sqrt {-10 x^{2}-x +3}\, x -9006360 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-5896576 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {5 x +3}\, \sqrt {-2 x +1}}{422576 \left (3 x +2\right )^{3} \left (2 x -1\right ) \sqrt {-10 x^{2}-x +3}} \]
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 \[ \int \frac {1}{\sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{4} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^4\,\sqrt {5\,x+3}} \,d x \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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