Optimal. Leaf size=79 \[ \frac {(1-2 x)^{3/2}}{(3 x+2) \sqrt {5 x+3}}-\frac {33 \sqrt {1-2 x}}{\sqrt {5 x+3}}+33 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {94, 93, 204} \[ \frac {(1-2 x)^{3/2}}{(3 x+2) \sqrt {5 x+3}}-\frac {33 \sqrt {1-2 x}}{\sqrt {5 x+3}}+33 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right ) \]
Antiderivative was successfully verified.
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Rule 93
Rule 94
Rule 204
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2}}{(2+3 x)^2 (3+5 x)^{3/2}} \, dx &=\frac {(1-2 x)^{3/2}}{(2+3 x) \sqrt {3+5 x}}+\frac {33}{2} \int \frac {\sqrt {1-2 x}}{(2+3 x) (3+5 x)^{3/2}} \, dx\\ &=-\frac {33 \sqrt {1-2 x}}{\sqrt {3+5 x}}+\frac {(1-2 x)^{3/2}}{(2+3 x) \sqrt {3+5 x}}-\frac {231}{2} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {33 \sqrt {1-2 x}}{\sqrt {3+5 x}}+\frac {(1-2 x)^{3/2}}{(2+3 x) \sqrt {3+5 x}}-231 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )\\ &=-\frac {33 \sqrt {1-2 x}}{\sqrt {3+5 x}}+\frac {(1-2 x)^{3/2}}{(2+3 x) \sqrt {3+5 x}}+33 \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 = 0.03, size = 65, normalized size = 0.82 \[ 33 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )-\frac {\sqrt {1-2 x} (101 x+65)}{(3 x+2) \sqrt {5 x+3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.18, size = 86, normalized size = 1.09 \[ \frac {33 \, \sqrt {7} {\left (15 \, x^{2} + 19 \, x + 6\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 2 \, {\left (101 \, x + 65\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{2 \, {\left (15 \, x^{2} + 19 \, x + 6\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.37, size = 252, normalized size = 3.19 \[ -\frac {33}{20} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {11}{10} \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )} - \frac {154 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{{\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 154, normalized size = 1.95 \[ -\frac {\left (495 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+627 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+202 \sqrt {-10 x^{2}-x +3}\, x +198 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+130 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}}{2 \left (3 x +2\right ) \sqrt {-10 x^{2}-x +3}\, \sqrt {5 x +3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.39, size = 92, normalized size = 1.16 \[ -\frac {33}{2} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {202 \, x}{3 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {317}{9 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {49}{9 \, {\left (3 \, \sqrt {-10 \, x^{2} - x + 3} x + 2 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} \]
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 \[ \int \frac {{\left (1-2\,x\right )}^{3/2}}{{\left (3\,x+2\right )}^2\,{\left (5\,x+3\right )}^{3/2}} \,d x \]
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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