Optimal. Leaf size=135 \[ -\frac {\sqrt {5 x+3} (1-2 x)^{5/2}}{3 (3 x+2)}-\frac {1}{3} \sqrt {5 x+3} (1-2 x)^{3/2}-\frac {43}{30} \sqrt {5 x+3} \sqrt {1-2 x}-\frac {2119 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{90 \sqrt {10}}-\frac {35}{9} \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 135, 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} \[ -\frac {\sqrt {5 x+3} (1-2 x)^{5/2}}{3 (3 x+2)}-\frac {1}{3} \sqrt {5 x+3} (1-2 x)^{3/2}-\frac {43}{30} \sqrt {5 x+3} \sqrt {1-2 x}-\frac {2119 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{90 \sqrt {10}}-\frac {35}{9} \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right ) \]
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 {(1-2 x)^{5/2} \sqrt {3+5 x}}{(2+3 x)^2} \, dx &=-\frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{3 (2+3 x)}+\frac {1}{3} \int \frac {\left (-\frac {25}{2}-30 x\right ) (1-2 x)^{3/2}}{(2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {1}{3} (1-2 x)^{3/2} \sqrt {3+5 x}-\frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{3 (2+3 x)}+\frac {1}{90} \int \frac {(-765-1935 x) \sqrt {1-2 x}}{(2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {43}{30} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {1}{3} (1-2 x)^{3/2} \sqrt {3+5 x}-\frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{3 (2+3 x)}+\frac {\int \frac {-13410-\frac {95355 x}{2}}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{1350}\\ &=-\frac {43}{30} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {1}{3} (1-2 x)^{3/2} \sqrt {3+5 x}-\frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{3 (2+3 x)}-\frac {2119}{180} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx+\frac {245}{18} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {43}{30} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {1}{3} (1-2 x)^{3/2} \sqrt {3+5 x}-\frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{3 (2+3 x)}+\frac {245}{9} \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )-\frac {2119 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{90 \sqrt {5}}\\ &=-\frac {43}{30} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {1}{3} (1-2 x)^{3/2} \sqrt {3+5 x}-\frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{3 (2+3 x)}-\frac {2119 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{90 \sqrt {10}}-\frac {35}{9} \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )\\ \end {align*}
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Mathematica [A] time = 0.24, size = 130, normalized size = 0.96 \[ \frac {30 \sqrt {-(1-2 x)^2} \sqrt {5 x+3} \left (20 x^2-79 x-116\right )-3500 (3 x+2) \sqrt {14 x-7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )+2119 \sqrt {10-20 x} (3 x+2) \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{900 \sqrt {2 x-1} (3 x+2)} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 1.01, size = 126, normalized size = 0.93 \[ -\frac {3500 \, \sqrt {7} {\left (3 \, x + 2\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 2119 \, \sqrt {10} {\left (3 \, x + 2\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 60 \, {\left (20 \, x^{2} - 79 \, x - 116\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{1800 \, {\left (3 \, x + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.11, size = 292, normalized size = 2.16 \[ \frac {7}{36} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {1}{1350} \, {\left (12 \, \sqrt {5} {\left (5 \, x + 3\right )} - 313 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - \frac {2119}{1800} \, \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {1078 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{27 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 163, normalized size = 1.21 \[ -\frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (-1200 \sqrt {-10 x^{2}-x +3}\, x^{2}+6357 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-10500 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+4740 \sqrt {-10 x^{2}-x +3}\, x +4238 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-7000 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+6960 \sqrt {-10 x^{2}-x +3}\right )}{1800 \sqrt {-10 x^{2}-x +3}\, \left (3 x +2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.19, size = 90, normalized size = 0.67 \[ \frac {2}{9} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {2119}{1800} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {35}{18} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {277}{270} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {49 \, \sqrt {-10 \, x^{2} - x + 3}}{27 \, {\left (3 \, x + 2\right )}} \]
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 \[ \int \frac {{\left (1-2\,x\right )}^{5/2}\,\sqrt {5\,x+3}}{{\left (3\,x+2\right )}^2} \,d x \]
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 \[ \int \frac {\left (1 - 2 x\right )^{\frac {5}{2}} \sqrt {5 x + 3}}{\left (3 x + 2\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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