Optimal. Leaf size=106 \[ -\frac {340 \sqrt {1-2 x}}{77 (3+5 x)}+\frac {3 \sqrt {1-2 x}}{7 (2+3 x) (3+5 x)}-\frac {426}{7} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {650}{11} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 106, 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, 156, 162,
65, 212} \begin {gather*} -\frac {340 \sqrt {1-2 x}}{77 (5 x+3)}+\frac {3 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)}-\frac {426}{7} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {650}{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 156
Rule 162
Rule 212
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^2} \, dx &=\frac {3 \sqrt {1-2 x}}{7 (2+3 x) (3+5 x)}+\frac {1}{7} \int \frac {41-45 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^2} \, dx\\ &=-\frac {340 \sqrt {1-2 x}}{77 (3+5 x)}+\frac {3 \sqrt {1-2 x}}{7 (2+3 x) (3+5 x)}-\frac {1}{77} \int \frac {1663-1020 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx\\ &=-\frac {340 \sqrt {1-2 x}}{77 (3+5 x)}+\frac {3 \sqrt {1-2 x}}{7 (2+3 x) (3+5 x)}+\frac {639}{7} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx-\frac {1625}{11} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=-\frac {340 \sqrt {1-2 x}}{77 (3+5 x)}+\frac {3 \sqrt {1-2 x}}{7 (2+3 x) (3+5 x)}-\frac {639}{7} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )+\frac {1625}{11} \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {340 \sqrt {1-2 x}}{77 (3+5 x)}+\frac {3 \sqrt {1-2 x}}{7 (2+3 x) (3+5 x)}-\frac {426}{7} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {650}{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 = 0.21, size = 89, normalized size = 0.84 \begin {gather*} -\frac {\sqrt {1-2 x} (647+1020 x)}{77 \left (6+19 x+15 x^2\right )}-\frac {426}{7} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {650}{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 = 0.15, size = 70, normalized size = 0.66
method | result | size |
risch | \(\frac {\left (647+1020 x \right ) \left (-1+2 x \right )}{77 \left (15 x^{2}+19 x +6\right ) \sqrt {1-2 x}}+\frac {650 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{121}-\frac {426 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{49}\) | \(69\) |
derivativedivides | \(\frac {10 \sqrt {1-2 x}}{11 \left (-\frac {6}{5}-2 x \right )}+\frac {650 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{121}+\frac {6 \sqrt {1-2 x}}{7 \left (-\frac {4}{3}-2 x \right )}-\frac {426 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{49}\) | \(70\) |
default | \(\frac {10 \sqrt {1-2 x}}{11 \left (-\frac {6}{5}-2 x \right )}+\frac {650 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{121}+\frac {6 \sqrt {1-2 x}}{7 \left (-\frac {4}{3}-2 x \right )}-\frac {426 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{49}\) | \(70\) |
trager | \(-\frac {\left (647+1020 x \right ) \sqrt {1-2 x}}{77 \left (15 x^{2}+19 x +6\right )}-\frac {325 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \RootOf \left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{121}+\frac {213 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x -5 \RootOf \left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{49}\) | \(116\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 110, normalized size = 1.04 \begin {gather*} -\frac {325}{121} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {213}{49} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {4 \, {\left (510 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 1157 \, \sqrt {-2 \, x + 1}\right )}}{77 \, {\left (15 \, {\left (2 \, x - 1\right )}^{2} + 136 \, x + 9\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.59, size = 122, normalized size = 1.15 \begin {gather*} \frac {15925 \, \sqrt {11} \sqrt {5} {\left (15 \, x^{2} + 19 \, x + 6\right )} \log \left (-\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} - 5 \, x + 8}{5 \, x + 3}\right ) + 25773 \, \sqrt {7} \sqrt {3} {\left (15 \, x^{2} + 19 \, x + 6\right )} \log \left (\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) - 77 \, {\left (1020 \, x + 647\right )} \sqrt {-2 \, x + 1}}{5929 \, {\left (15 \, x^{2} + 19 \, 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 = 8.09, size = 988, normalized size = 9.32 \begin {gather*} - \frac {314160 \sqrt {2} i \left (x - \frac {1}{2}\right )^{\frac {5}{2}}}{456533 x + 355740 \left (x - \frac {1}{2}\right )^{3} + 806344 \left (x - \frac {1}{2}\right )^{2} - \frac {456533}{2}} - \frac {356356 \sqrt {2} i \left (x - \frac {1}{2}\right )^{\frac {3}{2}}}{456533 x + 355740 \left (x - \frac {1}{2}\right )^{3} + 806344 \left (x - \frac {1}{2}\right )^{2} - \frac {456533}{2}} - \frac {29400 \sqrt {55} i \left (x - \frac {1}{2}\right )^{3} \operatorname {atan}{\left (\frac {\sqrt {110}}{10 \sqrt {x - \frac {1}{2}}} \right )}}{456533 x + 355740 \left (x - \frac {1}{2}\right )^{3} + 806344 \left (x - \frac {1}{2}\right )^{2} - \frac {456533}{2}} + \frac {1881600 \sqrt {55} i \left (x - \frac {1}{2}\right )^{3} \operatorname {atan}{\left (\frac {\sqrt {110} \sqrt {x - \frac {1}{2}}}{11} \right )}}{456533 x + 355740 \left (x - \frac {1}{2}\right )^{3} + 806344 \left (x - \frac {1}{2}\right )^{2} - \frac {456533}{2}} - \frac {43560 \sqrt {21} i \left (x - \frac {1}{2}\right )^{3} \operatorname {atan}{\left (\frac {\sqrt {42}}{6 \sqrt {x - \frac {1}{2}}} \right )}}{456533 x + 355740 \left (x - \frac {1}{2}\right )^{3} + 806344 \left (x - \frac {1}{2}\right )^{2} - \frac {456533}{2}} - \frac {3136320 \sqrt {21} i \left (x - \frac {1}{2}\right )^{3} \operatorname {atan}{\left (\frac {\sqrt {42} \sqrt {x - \frac {1}{2}}}{7} \right )}}{456533 x + 355740 \left (x - \frac {1}{2}\right )^{3} + 806344 \left (x - \frac {1}{2}\right )^{2} - \frac {456533}{2}} - \frac {940800 \sqrt {55} i \pi \left (x - \frac {1}{2}\right )^{3}}{456533 x + 355740 \left (x - \frac {1}{2}\right )^{3} + 806344 \left (x - \frac {1}{2}\right )^{2} - \frac {456533}{2}} + \frac {1568160 \sqrt {21} i \pi \left (x - \frac {1}{2}\right )^{3}}{456533 x + 355740 \left (x - \frac {1}{2}\right )^{3} + 806344 \left (x - \frac {1}{2}\right )^{2} - \frac {456533}{2}} - \frac {66640 \sqrt {55} i \left (x - \frac {1}{2}\right )^{2} \operatorname {atan}{\left (\frac {\sqrt {110}}{10 \sqrt {x - \frac {1}{2}}} \right )}}{456533 x + 355740 \left (x - \frac {1}{2}\right )^{3} + 806344 \left (x - \frac {1}{2}\right )^{2} - \frac {456533}{2}} + \frac {4264960 \sqrt {55} i \left (x - \frac {1}{2}\right )^{2} \operatorname {atan}{\left (\frac {\sqrt {110} \sqrt {x - \frac {1}{2}}}{11} \right )}}{456533 x + 355740 \left (x - \frac {1}{2}\right )^{3} + 806344 \left (x - \frac {1}{2}\right )^{2} - \frac {456533}{2}} - \frac {98736 \sqrt {21} i \left (x - \frac {1}{2}\right )^{2} \operatorname {atan}{\left (\frac {\sqrt {42}}{6 \sqrt {x - \frac {1}{2}}} \right )}}{456533 x + 355740 \left (x - \frac {1}{2}\right )^{3} + 806344 \left (x - \frac {1}{2}\right )^{2} - \frac {456533}{2}} - \frac {7108992 \sqrt {21} i \left (x - \frac {1}{2}\right )^{2} \operatorname {atan}{\left (\frac {\sqrt {42} \sqrt {x - \frac {1}{2}}}{7} \right )}}{456533 x + 355740 \left (x - \frac {1}{2}\right )^{3} + 806344 \left (x - \frac {1}{2}\right )^{2} - \frac {456533}{2}} - \frac {2132480 \sqrt {55} i \pi \left (x - \frac {1}{2}\right )^{2}}{456533 x + 355740 \left (x - \frac {1}{2}\right )^{3} + 806344 \left (x - \frac {1}{2}\right )^{2} - \frac {456533}{2}} + \frac {3554496 \sqrt {21} i \pi \left (x - \frac {1}{2}\right )^{2}}{456533 x + 355740 \left (x - \frac {1}{2}\right )^{3} + 806344 \left (x - \frac {1}{2}\right )^{2} - \frac {456533}{2}} - \frac {37730 \sqrt {55} i \left (x - \frac {1}{2}\right ) \operatorname {atan}{\left (\frac {\sqrt {110}}{10 \sqrt {x - \frac {1}{2}}} \right )}}{456533 x + 355740 \left (x - \frac {1}{2}\right )^{3} + 806344 \left (x - \frac {1}{2}\right )^{2} - \frac {456533}{2}} + \frac {2414720 \sqrt {55} i \left (x - \frac {1}{2}\right ) \operatorname {atan}{\left (\frac {\sqrt {110} \sqrt {x - \frac {1}{2}}}{11} \right )}}{456533 x + 355740 \left (x - \frac {1}{2}\right )^{3} + 806344 \left (x - \frac {1}{2}\right )^{2} - \frac {456533}{2}} - \frac {55902 \sqrt {21} i \left (x - \frac {1}{2}\right ) \operatorname {atan}{\left (\frac {\sqrt {42}}{6 \sqrt {x - \frac {1}{2}}} \right )}}{456533 x + 355740 \left (x - \frac {1}{2}\right )^{3} + 806344 \left (x - \frac {1}{2}\right )^{2} - \frac {456533}{2}} - \frac {4024944 \sqrt {21} i \left (x - \frac {1}{2}\right ) \operatorname {atan}{\left (\frac {\sqrt {42} \sqrt {x - \frac {1}{2}}}{7} \right )}}{456533 x + 355740 \left (x - \frac {1}{2}\right )^{3} + 806344 \left (x - \frac {1}{2}\right )^{2} - \frac {456533}{2}} - \frac {1207360 \sqrt {55} i \pi \left (x - \frac {1}{2}\right )}{456533 x + 355740 \left (x - \frac {1}{2}\right )^{3} + 806344 \left (x - \frac {1}{2}\right )^{2} - \frac {456533}{2}} + \frac {2012472 \sqrt {21} i \pi \left (x - \frac {1}{2}\right )}{456533 x + 355740 \left (x - \frac {1}{2}\right )^{3} + 806344 \left (x - \frac {1}{2}\right )^{2} - \frac {456533}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.03, size = 116, normalized size = 1.09 \begin {gather*} -\frac {325}{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 {213}{49} \, \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 (510 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 1157 \, \sqrt {-2 \, x + 1}\right )}}{77 \, {\left (15 \, {\left (2 \, x - 1\right )}^{2} + 136 \, x + 9\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.10, size = 72, normalized size = 0.68 \begin {gather*} \frac {650\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{121}-\frac {426\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{49}-\frac {\frac {4628\,\sqrt {1-2\,x}}{1155}-\frac {136\,{\left (1-2\,x\right )}^{3/2}}{77}}{\frac {136\,x}{15}+{\left (2\,x-1\right )}^2+\frac {3}{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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