Optimal. Leaf size=122 \[ -\frac {75 \sqrt {1-2 x} \sqrt {3+5 x}}{1372 (2+3 x)}-\frac {25 \sqrt {1-2 x} (3+5 x)^{3/2}}{1078 (2+3 x)^2}+\frac {4 (3+5 x)^{5/2}}{77 \sqrt {1-2 x} (2+3 x)^2}-\frac {825 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{1372 \sqrt {7}} \]
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Rubi [A]
time = 0.02, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {98, 96, 95, 210}
\begin {gather*} -\frac {825 \text {ArcTan}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{1372 \sqrt {7}}+\frac {4 (5 x+3)^{5/2}}{77 \sqrt {1-2 x} (3 x+2)^2}-\frac {25 \sqrt {1-2 x} (5 x+3)^{3/2}}{1078 (3 x+2)^2}-\frac {75 \sqrt {1-2 x} \sqrt {5 x+3}}{1372 (3 x+2)} \end {gather*}
Antiderivative was successfully verified.
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Rule 95
Rule 96
Rule 98
Rule 210
Rubi steps
\begin {align*} \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{3/2} (2+3 x)^3} \, dx &=\frac {4 (3+5 x)^{5/2}}{77 \sqrt {1-2 x} (2+3 x)^2}+\frac {25}{77} \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^3} \, dx\\ &=-\frac {25 \sqrt {1-2 x} (3+5 x)^{3/2}}{1078 (2+3 x)^2}+\frac {4 (3+5 x)^{5/2}}{77 \sqrt {1-2 x} (2+3 x)^2}+\frac {75}{196} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^2} \, dx\\ &=-\frac {75 \sqrt {1-2 x} \sqrt {3+5 x}}{1372 (2+3 x)}-\frac {25 \sqrt {1-2 x} (3+5 x)^{3/2}}{1078 (2+3 x)^2}+\frac {4 (3+5 x)^{5/2}}{77 \sqrt {1-2 x} (2+3 x)^2}+\frac {825 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{2744}\\ &=-\frac {75 \sqrt {1-2 x} \sqrt {3+5 x}}{1372 (2+3 x)}-\frac {25 \sqrt {1-2 x} (3+5 x)^{3/2}}{1078 (2+3 x)^2}+\frac {4 (3+5 x)^{5/2}}{77 \sqrt {1-2 x} (2+3 x)^2}+\frac {825 \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{1372}\\ &=-\frac {75 \sqrt {1-2 x} \sqrt {3+5 x}}{1372 (2+3 x)}-\frac {25 \sqrt {1-2 x} (3+5 x)^{3/2}}{1078 (2+3 x)^2}+\frac {4 (3+5 x)^{5/2}}{77 \sqrt {1-2 x} (2+3 x)^2}-\frac {825 \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 = 2.05, size = 142, normalized size = 1.16 \begin {gather*} \frac {25 \left (\frac {7 \sqrt {3+5 x} \left (396+2245 x+2550 x^2\right )}{25 \sqrt {1-2 x} (2+3 x)^2}+33 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {2 \left (34+\sqrt {1155}\right )} \sqrt {3+5 x}}{-\sqrt {11}+\sqrt {5-10 x}}\right )+33 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {6+10 x}}{\sqrt {34+\sqrt {1155}} \left (-\sqrt {11}+\sqrt {5-10 x}\right )}\right )\right )}{9604} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(208\) vs.
\(2(95)=190\).
time = 0.09, size = 209, normalized size = 1.71
method | result | size |
default | \(\frac {\left (14850 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+12375 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}-3300 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x -35700 x^{2} \sqrt {-10 x^{2}-x +3}-3300 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-31430 x \sqrt {-10 x^{2}-x +3}-5544 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}}{19208 \left (2+3 x \right )^{2} \left (-1+2 x \right ) \sqrt {-10 x^{2}-x +3}}\) | \(209\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 143, normalized size = 1.17 \begin {gather*} \frac {825}{19208} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {2125 \, x}{2058 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {625}{4116 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {1}{126 \, {\left (9 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 12 \, \sqrt {-10 \, x^{2} - x + 3} x + 4 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} + \frac {235}{1764 \, {\left (3 \, \sqrt {-10 \, x^{2} - x + 3} x + 2 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.07, size = 101, normalized size = 0.83 \begin {gather*} -\frac {825 \, \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 (2550 \, x^{2} + 2245 \, x + 396\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{19208 \, {\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 278 vs.
\(2 (95) = 190\).
time = 1.12, size = 278, normalized size = 2.28 \begin {gather*} \frac {165}{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 {44 \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{1715 \, {\left (2 \, x - 1\right )}} - \frac {11 \, \sqrt {10} {\left (13 \, {\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 {6280 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {25120 \, \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}} \end {gather*}
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 \begin {gather*} \int \frac {{\left (5\,x+3\right )}^{3/2}}{{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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