Optimal. Leaf size=137 \[ -\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{3 (3 x+2)}+\frac {5}{6} \sqrt {1-2 x} (5 x+3)^{3/2}-\frac {95}{72} \sqrt {1-2 x} \sqrt {5 x+3}+\frac {155}{216} \sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )-\frac {59 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{27 \sqrt {7}} \]
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Rubi [A] time = 0.05, 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 = {97, 154, 157, 54, 216, 93, 204} \[ -\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{3 (3 x+2)}+\frac {5}{6} \sqrt {1-2 x} (5 x+3)^{3/2}-\frac {95}{72} \sqrt {1-2 x} \sqrt {5 x+3}+\frac {155}{216} \sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )-\frac {59 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{27 \sqrt {7}} \]
Antiderivative was successfully verified.
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Rule 54
Rule 93
Rule 97
Rule 154
Rule 157
Rule 204
Rule 216
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} \operatorname {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 ) \operatorname {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 = 0.19, size = 130, normalized size = 0.95 \[ \frac {42 \sqrt {-(1-2 x)^2} \sqrt {5 x+3} \left (300 x^2+135 x-46\right )-944 (3 x+2) \sqrt {14 x-7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )-1085 \sqrt {10-20 x} (3 x+2) \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{3024 \sqrt {2 x-1} (3 x+2)} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.74, size = 132, normalized size = 0.96 \[ -\frac {1085 \, \sqrt {5} \sqrt {2} {\left (3 \, x + 2\right )} \arctan \left (\frac {\sqrt {5} \sqrt {2} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 944 \, \sqrt {7} {\left (3 \, x + 2\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 84 \, {\left (300 \, x^{2} + 135 \, x - 46\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{6048 \, {\left (3 \, x + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.13, size = 292, normalized size = 2.13 \[ \frac {59}{3780} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {1}{216} \, {\left (12 \, \sqrt {5} {\left (5 \, x + 3\right )} - 49 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + \frac {155}{864} \, \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {22 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{27 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 163, normalized size = 1.19 \[ \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (25200 \sqrt {-10 x^{2}-x +3}\, x^{2}+3255 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+2832 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+11340 \sqrt {-10 x^{2}-x +3}\, x +2170 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+1888 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-3864 \sqrt {-10 x^{2}-x +3}\right )}{6048 \sqrt {-10 x^{2}-x +3}\, \left (3 x +2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 90, normalized size = 0.66 \[ \frac {25}{18} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {155}{864} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {59}{378} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {65}{216} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {\sqrt {-10 \, x^{2} - x + 3}}{27 \, {\left (3 \, x + 2\right )}} \]
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 \[ \int \frac {\sqrt {1-2\,x}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^2} \,d x \]
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 \[ \int \frac {\sqrt {1 - 2 x} \left (5 x + 3\right )^{\frac {5}{2}}}{\left (3 x + 2\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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