Optimal. Leaf size=119 \[ \frac {4644}{5929 \sqrt {1-2 x}}-\frac {340}{77 \sqrt {1-2 x} (3+5 x)}+\frac {3}{7 \sqrt {1-2 x} (2+3 x) (3+5 x)}-\frac {1314}{49} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {3150}{121} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
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Rubi [A]
time = 0.03, 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 = {105, 156, 157,
162, 65, 212} \begin {gather*} \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 ) \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 105
Rule 156
Rule 157
Rule 162
Rule 212
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} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )+\frac {7875}{121} \text {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 [A]
time = 0.24, size = 94, normalized size = 0.79 \begin {gather*} \frac {-21955+9696 x+69660 x^2}{5929 \sqrt {1-2 x} \left (6+19 x+15 x^2\right )}-\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 {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.22, size = 79, normalized size = 0.66
method | result | size |
risch | \(\frac {69660 x^{2}+9696 x -21955}{5929 \left (15 x^{2}+19 x +6\right ) \sqrt {1-2 x}}+\frac {3150 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1331}-\frac {1314 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{343}\) | \(69\) |
derivativedivides | \(\frac {50 \sqrt {1-2 x}}{121 \left (-\frac {6}{5}-2 x \right )}+\frac {3150 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1331}+\frac {16}{5929 \sqrt {1-2 x}}+\frac {18 \sqrt {1-2 x}}{49 \left (-\frac {4}{3}-2 x \right )}-\frac {1314 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{343}\) | \(79\) |
default | \(\frac {50 \sqrt {1-2 x}}{121 \left (-\frac {6}{5}-2 x \right )}+\frac {3150 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1331}+\frac {16}{5929 \sqrt {1-2 x}}+\frac {18 \sqrt {1-2 x}}{49 \left (-\frac {4}{3}-2 x \right )}-\frac {1314 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{343}\) | \(79\) |
trager | \(-\frac {\left (69660 x^{2}+9696 x -21955\right ) \sqrt {1-2 x}}{5929 \left (30 x^{3}+23 x^{2}-7 x -6\right )}-\frac {657 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \RootOf \left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{343}-\frac {1575 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \RootOf \left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{1331}\) | \(126\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 119, normalized size = 1.00 \begin {gather*} -\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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.11, size = 142, normalized size = 1.19 \begin {gather*} \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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 8.61, size = 818, normalized size = 6.87 \begin {gather*} \frac {64827000 \sqrt {55} i \left (x - \frac {1}{2}\right )^{\frac {7}{2}} \operatorname {atan}{\left (\frac {\sqrt {110} \sqrt {x - \frac {1}{2}}}{11} \right )}}{27391980 \left (x - \frac {1}{2}\right )^{\frac {7}{2}} + 62088488 \left (x - \frac {1}{2}\right )^{\frac {5}{2}} + 35153041 \left (x - \frac {1}{2}\right )^{\frac {3}{2}}} - \frac {104936040 \sqrt {21} i \left (x - \frac {1}{2}\right )^{\frac {7}{2}} \operatorname {atan}{\left (\frac {\sqrt {42} \sqrt {x - \frac {1}{2}}}{7} \right )}}{27391980 \left (x - \frac {1}{2}\right )^{\frac {7}{2}} + 62088488 \left (x - \frac {1}{2}\right )^{\frac {5}{2}} + 35153041 \left (x - \frac {1}{2}\right )^{\frac {3}{2}}} - \frac {32413500 \sqrt {55} i \pi \left (x - \frac {1}{2}\right )^{\frac {7}{2}}}{27391980 \left (x - \frac {1}{2}\right )^{\frac {7}{2}} + 62088488 \left (x - \frac {1}{2}\right )^{\frac {5}{2}} + 35153041 \left (x - \frac {1}{2}\right )^{\frac {3}{2}}} + \frac {52468020 \sqrt {21} i \pi \left (x - \frac {1}{2}\right )^{\frac {7}{2}}}{27391980 \left (x - \frac {1}{2}\right )^{\frac {7}{2}} + 62088488 \left (x - \frac {1}{2}\right )^{\frac {5}{2}} + 35153041 \left (x - \frac {1}{2}\right )^{\frac {3}{2}}} + \frac {146941200 \sqrt {55} i \left (x - \frac {1}{2}\right )^{\frac {5}{2}} \operatorname {atan}{\left (\frac {\sqrt {110} \sqrt {x - \frac {1}{2}}}{11} \right )}}{27391980 \left (x - \frac {1}{2}\right )^{\frac {7}{2}} + 62088488 \left (x - \frac {1}{2}\right )^{\frac {5}{2}} + 35153041 \left (x - \frac {1}{2}\right )^{\frac {3}{2}}} - \frac {237855024 \sqrt {21} i \left (x - \frac {1}{2}\right )^{\frac {5}{2}} \operatorname {atan}{\left (\frac {\sqrt {42} \sqrt {x - \frac {1}{2}}}{7} \right )}}{27391980 \left (x - \frac {1}{2}\right )^{\frac {7}{2}} + 62088488 \left (x - \frac {1}{2}\right )^{\frac {5}{2}} + 35153041 \left (x - \frac {1}{2}\right )^{\frac {3}{2}}} - \frac {73470600 \sqrt {55} i \pi \left (x - \frac {1}{2}\right )^{\frac {5}{2}}}{27391980 \left (x - \frac {1}{2}\right )^{\frac {7}{2}} + 62088488 \left (x - \frac {1}{2}\right )^{\frac {5}{2}} + 35153041 \left (x - \frac {1}{2}\right )^{\frac {3}{2}}} + \frac {118927512 \sqrt {21} i \pi \left (x - \frac {1}{2}\right )^{\frac {5}{2}}}{27391980 \left (x - \frac {1}{2}\right )^{\frac {7}{2}} + 62088488 \left (x - \frac {1}{2}\right )^{\frac {5}{2}} + 35153041 \left (x - \frac {1}{2}\right )^{\frac {3}{2}}} + \frac {83194650 \sqrt {55} i \left (x - \frac {1}{2}\right )^{\frac {3}{2}} \operatorname {atan}{\left (\frac {\sqrt {110} \sqrt {x - \frac {1}{2}}}{11} \right )}}{27391980 \left (x - \frac {1}{2}\right )^{\frac {7}{2}} + 62088488 \left (x - \frac {1}{2}\right )^{\frac {5}{2}} + 35153041 \left (x - \frac {1}{2}\right )^{\frac {3}{2}}} - \frac {134667918 \sqrt {21} i \left (x - \frac {1}{2}\right )^{\frac {3}{2}} \operatorname {atan}{\left (\frac {\sqrt {42} \sqrt {x - \frac {1}{2}}}{7} \right )}}{27391980 \left (x - \frac {1}{2}\right )^{\frac {7}{2}} + 62088488 \left (x - \frac {1}{2}\right )^{\frac {5}{2}} + 35153041 \left (x - \frac {1}{2}\right )^{\frac {3}{2}}} - \frac {41597325 \sqrt {55} i \pi \left (x - \frac {1}{2}\right )^{\frac {3}{2}}}{27391980 \left (x - \frac {1}{2}\right )^{\frac {7}{2}} + 62088488 \left (x - \frac {1}{2}\right )^{\frac {5}{2}} + 35153041 \left (x - \frac {1}{2}\right )^{\frac {3}{2}}} + \frac {67333959 \sqrt {21} i \pi \left (x - \frac {1}{2}\right )^{\frac {3}{2}}}{27391980 \left (x - \frac {1}{2}\right )^{\frac {7}{2}} + 62088488 \left (x - \frac {1}{2}\right )^{\frac {5}{2}} + 35153041 \left (x - \frac {1}{2}\right )^{\frac {3}{2}}} - \frac {10727640 \sqrt {2} i \left (x - \frac {1}{2}\right )^{3}}{27391980 \left (x - \frac {1}{2}\right )^{\frac {7}{2}} + 62088488 \left (x - \frac {1}{2}\right )^{\frac {5}{2}} + 35153041 \left (x - \frac {1}{2}\right )^{\frac {3}{2}}} - \frac {12220824 \sqrt {2} i \left (x - \frac {1}{2}\right )^{2}}{27391980 \left (x - \frac {1}{2}\right )^{\frac {7}{2}} + 62088488 \left (x - \frac {1}{2}\right )^{\frac {5}{2}} + 35153041 \left (x - \frac {1}{2}\right )^{\frac {3}{2}}} - \frac {47432 \sqrt {2} i \left (x - \frac {1}{2}\right )}{27391980 \left (x - \frac {1}{2}\right )^{\frac {7}{2}} + 62088488 \left (x - \frac {1}{2}\right )^{\frac {5}{2}} + 35153041 \left (x - \frac {1}{2}\right )^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.06, size = 132, normalized size = 1.11 \begin {gather*} -\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 )}} \end {gather*}
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 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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