Optimal. Leaf size=92 \[ -\frac {58}{539 \sqrt {1-2 x}}+\frac {3}{7 \sqrt {1-2 x} (2+3 x)}+\frac {228}{49} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {50}{11} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {105, 157, 162,
65, 212} \begin {gather*} -\frac {58}{539 \sqrt {1-2 x}}+\frac {3}{7 \sqrt {1-2 x} (3 x+2)}+\frac {228}{49} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {50}{11} \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 157
Rule 162
Rule 212
Rubi steps
\begin {align*} \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^2 (3+5 x)} \, dx &=\frac {3}{7 \sqrt {1-2 x} (2+3 x)}+\frac {1}{7} \int \frac {8-45 x}{(1-2 x)^{3/2} (2+3 x) (3+5 x)} \, dx\\ &=-\frac {58}{539 \sqrt {1-2 x}}+\frac {3}{7 \sqrt {1-2 x} (2+3 x)}-\frac {2}{539} \int \frac {-482+\frac {435 x}{2}}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx\\ &=-\frac {58}{539 \sqrt {1-2 x}}+\frac {3}{7 \sqrt {1-2 x} (2+3 x)}-\frac {342}{49} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx+\frac {125}{11} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=-\frac {58}{539 \sqrt {1-2 x}}+\frac {3}{7 \sqrt {1-2 x} (2+3 x)}+\frac {342}{49} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )-\frac {125}{11} \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {58}{539 \sqrt {1-2 x}}+\frac {3}{7 \sqrt {1-2 x} (2+3 x)}+\frac {228}{49} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {50}{11} \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 = 0.20, size = 84, normalized size = 0.91 \begin {gather*} \frac {115-174 x}{539 \sqrt {1-2 x} (2+3 x)}+\frac {228}{49} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {50}{11} \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 = 0.17, size = 63, normalized size = 0.68
method | result | size |
risch | \(-\frac {174 x -115}{539 \left (2+3 x \right ) \sqrt {1-2 x}}-\frac {50 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{121}+\frac {228 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{343}\) | \(59\) |
derivativedivides | \(-\frac {50 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{121}+\frac {8}{539 \sqrt {1-2 x}}-\frac {6 \sqrt {1-2 x}}{49 \left (-\frac {4}{3}-2 x \right )}+\frac {228 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{343}\) | \(63\) |
default | \(-\frac {50 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{121}+\frac {8}{539 \sqrt {1-2 x}}-\frac {6 \sqrt {1-2 x}}{49 \left (-\frac {4}{3}-2 x \right )}+\frac {228 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{343}\) | \(63\) |
trager | \(\frac {\left (174 x -115\right ) \sqrt {1-2 x}}{3234 x^{2}+539 x -1078}-\frac {25 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (-\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x -8 \RootOf \left (\textit {\_Z}^{2}-55\right )-55 \sqrt {1-2 x}}{3+5 x}\right )}{121}+\frac {114 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \RootOf \left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{343}\) | \(115\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 101, normalized size = 1.10 \begin {gather*} \frac {25}{121} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {114}{343} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {2 \, {\left (174 \, x - 115\right )}}{539 \, {\left (3 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 7 \, \sqrt {-2 \, x + 1}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.96, size = 116, normalized size = 1.26 \begin {gather*} \frac {8575 \, \sqrt {11} \sqrt {5} {\left (6 \, x^{2} + x - 2\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 13794 \, \sqrt {7} \sqrt {3} {\left (6 \, x^{2} + x - 2\right )} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 77 \, {\left (174 \, x - 115\right )} \sqrt {-2 \, x + 1}}{41503 \, {\left (6 \, x^{2} + x - 2\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 = 5.90, size = 376, normalized size = 4.09 \begin {gather*} \frac {13398 \sqrt {2} i \left (x - \frac {1}{2}\right )^{\frac {3}{2}}}{290521 x + 249018 \left (x - \frac {1}{2}\right )^{2} - \frac {290521}{2}} - \frac {2156 \sqrt {2} i \sqrt {x - \frac {1}{2}}}{290521 x + 249018 \left (x - \frac {1}{2}\right )^{2} - \frac {290521}{2}} - \frac {102900 \sqrt {55} i \left (x - \frac {1}{2}\right )^{2} \operatorname {atan}{\left (\frac {\sqrt {110} \sqrt {x - \frac {1}{2}}}{11} \right )}}{290521 x + 249018 \left (x - \frac {1}{2}\right )^{2} - \frac {290521}{2}} + \frac {165528 \sqrt {21} i \left (x - \frac {1}{2}\right )^{2} \operatorname {atan}{\left (\frac {\sqrt {42} \sqrt {x - \frac {1}{2}}}{7} \right )}}{290521 x + 249018 \left (x - \frac {1}{2}\right )^{2} - \frac {290521}{2}} - \frac {82764 \sqrt {21} i \pi \left (x - \frac {1}{2}\right )^{2}}{290521 x + 249018 \left (x - \frac {1}{2}\right )^{2} - \frac {290521}{2}} + \frac {51450 \sqrt {55} i \pi \left (x - \frac {1}{2}\right )^{2}}{290521 x + 249018 \left (x - \frac {1}{2}\right )^{2} - \frac {290521}{2}} - \frac {120050 \sqrt {55} i \left (x - \frac {1}{2}\right ) \operatorname {atan}{\left (\frac {\sqrt {110} \sqrt {x - \frac {1}{2}}}{11} \right )}}{290521 x + 249018 \left (x - \frac {1}{2}\right )^{2} - \frac {290521}{2}} + \frac {193116 \sqrt {21} i \left (x - \frac {1}{2}\right ) \operatorname {atan}{\left (\frac {\sqrt {42} \sqrt {x - \frac {1}{2}}}{7} \right )}}{290521 x + 249018 \left (x - \frac {1}{2}\right )^{2} - \frac {290521}{2}} - \frac {96558 \sqrt {21} i \pi \left (x - \frac {1}{2}\right )}{290521 x + 249018 \left (x - \frac {1}{2}\right )^{2} - \frac {290521}{2}} + \frac {60025 \sqrt {55} i \pi \left (x - \frac {1}{2}\right )}{290521 x + 249018 \left (x - \frac {1}{2}\right )^{2} - \frac {290521}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.85, size = 107, normalized size = 1.16 \begin {gather*} \frac {25}{121} \, \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 {114}{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 {2 \, {\left (174 \, x - 115\right )}}{539 \, {\left (3 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 7 \, \sqrt {-2 \, x + 1}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.28, size = 65, normalized size = 0.71 \begin {gather*} \frac {228\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{343}-\frac {\frac {116\,x}{539}-\frac {230}{1617}}{\frac {7\,\sqrt {1-2\,x}}{3}-{\left (1-2\,x\right )}^{3/2}}-\frac {50\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{121} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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