Optimal. Leaf size=79 \[ -\frac {370 \sqrt {1-2 x}}{847 \sqrt {5 x+3}}+\frac {4}{77 \sqrt {1-2 x} \sqrt {5 x+3}}+\frac {18 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{7 \sqrt {7}} \]
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Rubi [A] time = 0.03, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {104, 152, 12, 93, 204} \[ -\frac {370 \sqrt {1-2 x}}{847 \sqrt {5 x+3}}+\frac {4}{77 \sqrt {1-2 x} \sqrt {5 x+3}}+\frac {18 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{7 \sqrt {7}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 93
Rule 104
Rule 152
Rule 204
Rubi steps
\begin {align*} \int \frac {1}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2}} \, dx &=\frac {4}{77 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {2}{77} \int \frac {-\frac {73}{2}-30 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx\\ &=\frac {4}{77 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {370 \sqrt {1-2 x}}{847 \sqrt {3+5 x}}+\frac {4}{847} \int -\frac {1089}{4 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=\frac {4}{77 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {370 \sqrt {1-2 x}}{847 \sqrt {3+5 x}}-\frac {9}{7} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=\frac {4}{77 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {370 \sqrt {1-2 x}}{847 \sqrt {3+5 x}}-\frac {18}{7} \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )\\ &=\frac {4}{77 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {370 \sqrt {1-2 x}}{847 \sqrt {3+5 x}}+\frac {18 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{7 \sqrt {7}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 72, normalized size = 0.91 \[ \frac {5180 x+2178 \sqrt {7-14 x} \sqrt {5 x+3} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )-2282}{5929 \sqrt {1-2 x} \sqrt {5 x+3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.21, size = 82, normalized size = 1.04 \[ \frac {1089 \, \sqrt {7} {\left (10 \, x^{2} + x - 3\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (370 \, x - 163\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{5929 \, {\left (10 \, x^{2} + x - 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.21, size = 159, normalized size = 2.01 \[ -\frac {9}{490} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {5}{242} \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )} - \frac {8 \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{4235 \, {\left (2 \, x - 1\right )}} \]
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 \[ -\frac {\sqrt {-2 x +1}\, \left (10890 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+1089 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+5180 \sqrt {-10 x^{2}-x +3}\, x -3267 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-2282 \sqrt {-10 x^{2}-x +3}\right )}{5929 \left (2 x -1\right ) \sqrt {-10 x^{2}-x +3}\, \sqrt {5 x +3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.46, size = 58, normalized size = 0.73 \[ -\frac {9}{49} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {740 \, x}{847 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {326}{847 \, \sqrt {-10 \, x^{2} - x + 3}} \]
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 \[ \int \frac {1}{{\left (1-2\,x\right )}^{3/2}\,\left (3\,x+2\right )\,{\left (5\,x+3\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (1 - 2 x\right )^{\frac {3}{2}} \left (3 x + 2\right ) \left (5 x + 3\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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