Optimal. Leaf size=144 \[ \frac {215}{84} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {59 \sqrt {1-2 x} (3+5 x)^{3/2}}{84 (2+3 x)}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{6 (2+3 x)^2}+\frac {25}{9} \sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )+\frac {2119 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{252 \sqrt {7}} \]
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Rubi [A]
time = 0.04, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {99, 154, 159,
163, 56, 222, 95, 210} \begin {gather*} \frac {25}{9} \sqrt {\frac {5}{2}} \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )+\frac {2119 \text {ArcTan}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{252 \sqrt {7}}-\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{6 (3 x+2)^2}-\frac {59 \sqrt {1-2 x} (5 x+3)^{3/2}}{84 (3 x+2)}+\frac {215}{84} \sqrt {1-2 x} \sqrt {5 x+3} \end {gather*}
Antiderivative was successfully verified.
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Rule 56
Rule 95
Rule 99
Rule 154
Rule 159
Rule 163
Rule 210
Rule 222
Rubi steps
\begin {align*} \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^3} \, dx &=-\frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{6 (2+3 x)^2}+\frac {1}{6} \int \frac {\left (\frac {19}{2}-30 x\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^2} \, dx\\ &=-\frac {59 \sqrt {1-2 x} (3+5 x)^{3/2}}{84 (2+3 x)}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{6 (2+3 x)^2}+\frac {1}{126} \int \frac {\left (\frac {1197}{4}-1935 x\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=\frac {215}{84} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {59 \sqrt {1-2 x} (3+5 x)^{3/2}}{84 (2+3 x)}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{6 (2+3 x)^2}-\frac {1}{756} \int \frac {-\frac {14643}{2}-15750 x}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=\frac {215}{84} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {59 \sqrt {1-2 x} (3+5 x)^{3/2}}{84 (2+3 x)}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{6 (2+3 x)^2}-\frac {2119}{504} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx+\frac {125}{18} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=\frac {215}{84} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {59 \sqrt {1-2 x} (3+5 x)^{3/2}}{84 (2+3 x)}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{6 (2+3 x)^2}-\frac {2119}{252} \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )+\frac {1}{9} \left (25 \sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )\\ &=\frac {215}{84} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {59 \sqrt {1-2 x} (3+5 x)^{3/2}}{84 (2+3 x)}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{6 (2+3 x)^2}+\frac {25}{9} \sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )+\frac {2119 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{252 \sqrt {7}}\\ \end {align*}
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Mathematica [A]
time = 0.23, size = 103, normalized size = 0.72 \begin {gather*} \frac {\frac {21 \sqrt {1-2 x} \sqrt {3+5 x} \left (380+1039 x+700 x^2\right )}{(2+3 x)^2}-2450 \sqrt {10} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )+2119 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{1764} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 208, normalized size = 1.44
method | result | size |
risch | \(-\frac {\sqrt {3+5 x}\, \left (-1+2 x \right ) \left (700 x^{2}+1039 x +380\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{84 \left (2+3 x \right )^{2} \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right )}\, \sqrt {1-2 x}}-\frac {\left (-\frac {25 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )}{36}+\frac {2119 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right )}{3528}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(138\) |
default | \(-\frac {\sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (19071 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}-22050 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{2}+25428 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x -29400 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x -29400 x^{2} \sqrt {-10 x^{2}-x +3}+8476 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-9800 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-43638 x \sqrt {-10 x^{2}-x +3}-15960 \sqrt {-10 x^{2}-x +3}\right )}{3528 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{2}}\) | \(208\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 101, normalized size = 0.70 \begin {gather*} \frac {25}{36} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {2119}{3528} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {20}{21} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{42 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {9 \, \sqrt {-10 \, x^{2} - x + 3}}{28 \, {\left (3 \, x + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.97, size = 147, normalized size = 1.02 \begin {gather*} -\frac {2450 \, \sqrt {5} \sqrt {2} {\left (9 \, x^{2} + 12 \, x + 4\right )} \arctan \left (\frac {\sqrt {5} \sqrt {2} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 2119 \, \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 ) - 42 \, {\left (700 \, x^{2} + 1039 \, x + 380\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{3528 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {1 - 2 x} \left (5 x + 3\right )^{\frac {5}{2}}}{\left (3 x + 2\right )^{3}}\, dx \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. 338 vs.
\(2 (106) = 212\).
time = 2.05, size = 338, normalized size = 2.35 \begin {gather*} -\frac {2119}{35280} \, \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 {25}{36} \, \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {5}{27} \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + \frac {11 \, \sqrt {10} {\left (247 \, {\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 {87640 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {350560 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{378 \, {\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 {\sqrt {1-2\,x}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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