Optimal. Leaf size=137 \[ -\frac {95}{72} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {5}{6} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{3 (2+3 x)}+\frac {155}{216} \sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )-\frac {59 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{27 \sqrt {7}} \]
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Rubi [A]
time = 0.04, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {99, 159, 163,
56, 222, 95, 210} \begin {gather*} \frac {155}{216} \sqrt {\frac {5}{2}} \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )-\frac {59 \text {ArcTan}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{27 \sqrt {7}}-\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{3 (3 x+2)}+\frac {5}{6} \sqrt {1-2 x} (5 x+3)^{3/2}-\frac {95}{72} \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 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)^2} \, dx &=-\frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{3 (2+3 x)}+\frac {1}{3} \int \frac {\left (\frac {19}{2}-30 x\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=\frac {5}{6} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{3 (2+3 x)}-\frac {1}{36} \int \frac {(-72-285 x) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=-\frac {95}{72} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {5}{6} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{3 (2+3 x)}+\frac {1}{216} \int \frac {1011+\frac {2325 x}{2}}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {95}{72} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {5}{6} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{3 (2+3 x)}+\frac {59}{54} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx+\frac {775}{432} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {95}{72} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {5}{6} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{3 (2+3 x)}+\frac {59}{27} \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )+\frac {1}{216} \left (155 \sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )\\ &=-\frac {95}{72} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {5}{6} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{3 (2+3 x)}+\frac {155}{216} \sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )-\frac {59 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{27 \sqrt {7}}\\ \end {align*}
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Mathematica [A]
time = 0.24, size = 108, normalized size = 0.79 \begin {gather*} \frac {\frac {42 \sqrt {1-2 x} \left (-138+175 x+1575 x^2+1500 x^3\right )}{(2+3 x) \sqrt {3+5 x}}-1085 \sqrt {10} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )-944 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{3024} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 163, normalized size = 1.19
method | result | size |
risch | \(-\frac {\sqrt {3+5 x}\, \left (-1+2 x \right ) \left (300 x^{2}+135 x -46\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{72 \left (2+3 x \right ) \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right )}\, \sqrt {1-2 x}}-\frac {\left (-\frac {155 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )}{864}-\frac {59 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right )}{378}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(138\) |
default | \(\frac {\sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (3255 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x +2832 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +25200 x^{2} \sqrt {-10 x^{2}-x +3}+2170 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+1888 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+11340 x \sqrt {-10 x^{2}-x +3}-3864 \sqrt {-10 x^{2}-x +3}\right )}{6048 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )}\) | \(163\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 90, normalized size = 0.66 \begin {gather*} \frac {25}{18} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {155}{864} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {59}{378} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {65}{216} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {\sqrt {-10 \, x^{2} - x + 3}}{27 \, {\left (3 \, x + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.40, size = 132, normalized size = 0.96 \begin {gather*} -\frac {1085 \, \sqrt {5} \sqrt {2} {\left (3 \, x + 2\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 ) + 944 \, \sqrt {7} {\left (3 \, x + 2\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 ) - 84 \, {\left (300 \, x^{2} + 135 \, x - 46\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{6048 \, {\left (3 \, x + 2\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 )^{2}}\, 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. 292 vs.
\(2 (99) = 198\).
time = 1.43, size = 292, normalized size = 2.13 \begin {gather*} \frac {59}{3780} \, \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 {1}{216} \, {\left (12 \, \sqrt {5} {\left (5 \, x + 3\right )} - 49 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + \frac {155}{864} \, \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 {22 \, \sqrt {10} {\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 )}}{27 \, {\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 )}} \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 )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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