Optimal. Leaf size=119 \[ \frac {4644}{5929 \sqrt {1-2 x}}-\frac {340}{77 \sqrt {1-2 x} (5 x+3)}+\frac {3}{7 \sqrt {1-2 x} (3 x+2) (5 x+3)}-\frac {1314}{49} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {3150}{121} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {103, 151, 152, 156, 63, 206} \[ \frac {4644}{5929 \sqrt {1-2 x}}-\frac {340}{77 \sqrt {1-2 x} (5 x+3)}+\frac {3}{7 \sqrt {1-2 x} (3 x+2) (5 x+3)}-\frac {1314}{49} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {3150}{121} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
Antiderivative was successfully verified.
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Rule 63
Rule 103
Rule 151
Rule 152
Rule 156
Rule 206
Rubi steps
\begin {align*} \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^2} \, dx &=\frac {3}{7 \sqrt {1-2 x} (2+3 x) (3+5 x)}+\frac {1}{7} \int \frac {23-75 x}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^2} \, dx\\ &=-\frac {340}{77 \sqrt {1-2 x} (3+5 x)}+\frac {3}{7 \sqrt {1-2 x} (2+3 x) (3+5 x)}-\frac {1}{77} \int \frac {369-3060 x}{(1-2 x)^{3/2} (2+3 x) (3+5 x)} \, dx\\ &=\frac {4644}{5929 \sqrt {1-2 x}}-\frac {340}{77 \sqrt {1-2 x} (3+5 x)}+\frac {3}{7 \sqrt {1-2 x} (2+3 x) (3+5 x)}+\frac {2 \int \frac {-\frac {56277}{2}+17415 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx}{5929}\\ &=\frac {4644}{5929 \sqrt {1-2 x}}-\frac {340}{77 \sqrt {1-2 x} (3+5 x)}+\frac {3}{7 \sqrt {1-2 x} (2+3 x) (3+5 x)}+\frac {1971}{49} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx-\frac {7875}{121} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=\frac {4644}{5929 \sqrt {1-2 x}}-\frac {340}{77 \sqrt {1-2 x} (3+5 x)}+\frac {3}{7 \sqrt {1-2 x} (2+3 x) (3+5 x)}-\frac {1971}{49} \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )+\frac {7875}{121} \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {4644}{5929 \sqrt {1-2 x}}-\frac {340}{77 \sqrt {1-2 x} (3+5 x)}+\frac {3}{7 \sqrt {1-2 x} (2+3 x) (3+5 x)}-\frac {1314}{49} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {3150}{121} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\\ \end {align*}
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Mathematica [C] time = 0.05, size = 93, normalized size = 0.78 \[ \frac {158994 \left (15 x^2+19 x+6\right ) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {3}{7}-\frac {6 x}{7}\right )-7 \left (22050 \left (15 x^2+19 x+6\right ) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-\frac {5}{11} (2 x-1)\right )+11220 x+7117\right )}{5929 \sqrt {1-2 x} (3 x+2) (5 x+3)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.71, size = 142, normalized size = 1.19 \[ \frac {540225 \, \sqrt {11} \sqrt {5} {\left (30 \, x^{3} + 23 \, x^{2} - 7 \, x - 6\right )} \log \left (-\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} - 5 \, x + 8}{5 \, x + 3}\right ) + 874467 \, \sqrt {7} \sqrt {3} {\left (30 \, x^{3} + 23 \, x^{2} - 7 \, x - 6\right )} \log \left (\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) - 77 \, {\left (69660 \, x^{2} + 9696 \, x - 21955\right )} \sqrt {-2 \, x + 1}}{456533 \, {\left (30 \, x^{3} + 23 \, x^{2} - 7 \, x - 6\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.23, size = 132, normalized size = 1.11 \[ -\frac {1575}{1331} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {657}{343} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {4 \, {\left (17415 \, {\left (2 \, x - 1\right )}^{2} + 79356 \, x - 39370\right )}}{5929 \, {\left (15 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 68 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 77 \, \sqrt {-2 \, x + 1}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 79, normalized size = 0.66 \[ -\frac {1314 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{343}+\frac {3150 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{1331}+\frac {16}{5929 \sqrt {-2 x +1}}+\frac {50 \sqrt {-2 x +1}}{121 \left (-2 x -\frac {6}{5}\right )}+\frac {18 \sqrt {-2 x +1}}{49 \left (-2 x -\frac {4}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.23, size = 119, normalized size = 1.00 \[ -\frac {1575}{1331} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {657}{343} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {4 \, {\left (17415 \, {\left (2 \, x - 1\right )}^{2} + 79356 \, x - 39370\right )}}{5929 \, {\left (15 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 68 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 77 \, \sqrt {-2 \, x + 1}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.27, size = 80, normalized size = 0.67 \[ \frac {3150\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{1331}-\frac {1314\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{343}+\frac {\frac {105808\,x}{29645}+\frac {4644\,{\left (2\,x-1\right )}^2}{5929}-\frac {31496}{17787}}{\frac {77\,\sqrt {1-2\,x}}{15}-\frac {68\,{\left (1-2\,x\right )}^{3/2}}{15}+{\left (1-2\,x\right )}^{5/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 16.08, size = 894, normalized size = 7.51 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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