Optimal. Leaf size=115 \[ \frac {(1-2 x)^{5/2}}{2 (3 x+2)^2 \sqrt {5 x+3}}+\frac {55 (1-2 x)^{3/2}}{4 (3 x+2) \sqrt {5 x+3}}-\frac {1815 \sqrt {1-2 x}}{4 \sqrt {5 x+3}}+\frac {1815}{4} \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {94, 93, 204} \[ \frac {(1-2 x)^{5/2}}{2 (3 x+2)^2 \sqrt {5 x+3}}+\frac {55 (1-2 x)^{3/2}}{4 (3 x+2) \sqrt {5 x+3}}-\frac {1815 \sqrt {1-2 x}}{4 \sqrt {5 x+3}}+\frac {1815}{4} \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right ) \]
Antiderivative was successfully verified.
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Rule 93
Rule 94
Rule 204
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2}}{(2+3 x)^3 (3+5 x)^{3/2}} \, dx &=\frac {(1-2 x)^{5/2}}{2 (2+3 x)^2 \sqrt {3+5 x}}+\frac {55}{4} \int \frac {(1-2 x)^{3/2}}{(2+3 x)^2 (3+5 x)^{3/2}} \, dx\\ &=\frac {(1-2 x)^{5/2}}{2 (2+3 x)^2 \sqrt {3+5 x}}+\frac {55 (1-2 x)^{3/2}}{4 (2+3 x) \sqrt {3+5 x}}+\frac {1815}{8} \int \frac {\sqrt {1-2 x}}{(2+3 x) (3+5 x)^{3/2}} \, dx\\ &=-\frac {1815 \sqrt {1-2 x}}{4 \sqrt {3+5 x}}+\frac {(1-2 x)^{5/2}}{2 (2+3 x)^2 \sqrt {3+5 x}}+\frac {55 (1-2 x)^{3/2}}{4 (2+3 x) \sqrt {3+5 x}}-\frac {12705}{8} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {1815 \sqrt {1-2 x}}{4 \sqrt {3+5 x}}+\frac {(1-2 x)^{5/2}}{2 (2+3 x)^2 \sqrt {3+5 x}}+\frac {55 (1-2 x)^{3/2}}{4 (2+3 x) \sqrt {3+5 x}}-\frac {12705}{4} \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )\\ &=-\frac {1815 \sqrt {1-2 x}}{4 \sqrt {3+5 x}}+\frac {(1-2 x)^{5/2}}{2 (2+3 x)^2 \sqrt {3+5 x}}+\frac {55 (1-2 x)^{3/2}}{4 (2+3 x) \sqrt {3+5 x}}+\frac {1815}{4} \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 = 0.05, size = 74, normalized size = 0.64 \[ \frac {1815}{4} \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )-\frac {\sqrt {1-2 x} \left (16657 x^2+21843 x+7148\right )}{4 (3 x+2)^2 \sqrt {5 x+3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 101, normalized size = 0.88 \[ \frac {1815 \, \sqrt {7} {\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 2 \, {\left (16657 \, x^{2} + 21843 \, x + 7148\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{8 \, {\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.45, size = 311, normalized size = 2.70 \[ -\frac {363}{16} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {121}{10} \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )} - \frac {847 \, \sqrt {10} {\left (9 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {1960 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {7840 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{2 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 202, normalized size = 1.76 \[ -\frac {\left (81675 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+157905 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+33314 \sqrt {-10 x^{2}-x +3}\, x^{2}+101640 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+43686 \sqrt {-10 x^{2}-x +3}\, x +21780 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+14296 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}}{8 \left (3 x +2\right )^{2} \sqrt {-10 x^{2}-x +3}\, \sqrt {5 x +3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.43, size = 143, normalized size = 1.24 \[ -\frac {1815}{8} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {16657 \, x}{18 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {52169}{108 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {343}{54 \, {\left (9 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 12 \, \sqrt {-10 \, x^{2} - x + 3} x + 4 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} + \frac {833}{12 \, {\left (3 \, \sqrt {-10 \, x^{2} - x + 3} x + 2 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} \]
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 \[ \int \frac {{\left (1-2\,x\right )}^{5/2}}{{\left (3\,x+2\right )}^3\,{\left (5\,x+3\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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