Optimal. Leaf size=160 \[ -\frac {1676975 \sqrt {1-2 x}}{7546 (3+5 x)}+\frac {\sqrt {1-2 x}}{7 (2+3 x)^3 (3+5 x)}+\frac {145 \sqrt {1-2 x}}{98 (2+3 x)^2 (3+5 x)}+\frac {7585 \sqrt {1-2 x}}{343 (2+3 x) (3+5 x)}-\frac {1051695}{343} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {32750}{11} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
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Rubi [A]
time = 0.04, antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps
used = 9, 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 {1676975 \sqrt {1-2 x}}{7546 (5 x+3)}+\frac {7585 \sqrt {1-2 x}}{343 (3 x+2) (5 x+3)}+\frac {145 \sqrt {1-2 x}}{98 (3 x+2)^2 (5 x+3)}+\frac {\sqrt {1-2 x}}{7 (3 x+2)^3 (5 x+3)}-\frac {1051695}{343} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {32750}{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)^4 (3+5 x)^2} \, dx &=\frac {\sqrt {1-2 x}}{7 (2+3 x)^3 (3+5 x)}+\frac {1}{21} \int \frac {75-105 x}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)^2} \, dx\\ &=\frac {\sqrt {1-2 x}}{7 (2+3 x)^3 (3+5 x)}+\frac {145 \sqrt {1-2 x}}{98 (2+3 x)^2 (3+5 x)}+\frac {1}{294} \int \frac {7920-10875 x}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^2} \, dx\\ &=\frac {\sqrt {1-2 x}}{7 (2+3 x)^3 (3+5 x)}+\frac {145 \sqrt {1-2 x}}{98 (2+3 x)^2 (3+5 x)}+\frac {7585 \sqrt {1-2 x}}{343 (2+3 x) (3+5 x)}+\frac {\int \frac {596595-682650 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^2} \, dx}{2058}\\ &=-\frac {1676975 \sqrt {1-2 x}}{7546 (3+5 x)}+\frac {\sqrt {1-2 x}}{7 (2+3 x)^3 (3+5 x)}+\frac {145 \sqrt {1-2 x}}{98 (2+3 x)^2 (3+5 x)}+\frac {7585 \sqrt {1-2 x}}{343 (2+3 x) (3+5 x)}-\frac {\int \frac {24644085-15092775 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx}{22638}\\ &=-\frac {1676975 \sqrt {1-2 x}}{7546 (3+5 x)}+\frac {\sqrt {1-2 x}}{7 (2+3 x)^3 (3+5 x)}+\frac {145 \sqrt {1-2 x}}{98 (2+3 x)^2 (3+5 x)}+\frac {7585 \sqrt {1-2 x}}{343 (2+3 x) (3+5 x)}+\frac {3155085}{686} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx-\frac {81875}{11} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=-\frac {1676975 \sqrt {1-2 x}}{7546 (3+5 x)}+\frac {\sqrt {1-2 x}}{7 (2+3 x)^3 (3+5 x)}+\frac {145 \sqrt {1-2 x}}{98 (2+3 x)^2 (3+5 x)}+\frac {7585 \sqrt {1-2 x}}{343 (2+3 x) (3+5 x)}-\frac {3155085}{686} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )+\frac {81875}{11} \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {1676975 \sqrt {1-2 x}}{7546 (3+5 x)}+\frac {\sqrt {1-2 x}}{7 (2+3 x)^3 (3+5 x)}+\frac {145 \sqrt {1-2 x}}{98 (2+3 x)^2 (3+5 x)}+\frac {7585 \sqrt {1-2 x}}{343 (2+3 x) (3+5 x)}-\frac {1051695}{343} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {32750}{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 = 0.30, size = 101, normalized size = 0.63 \begin {gather*} -\frac {\sqrt {1-2 x} \left (12724912+58335165 x+89054820 x^2+45278325 x^3\right )}{7546 (2+3 x)^3 (3+5 x)}-\frac {1051695}{343} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {32750}{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 = 0.22, size = 91, normalized size = 0.57
method | result | size |
risch | \(\frac {90556650 x^{4}+132831315 x^{3}+27615510 x^{2}-32885341 x -12724912}{7546 \left (3+5 x \right ) \sqrt {1-2 x}\, \left (2+3 x \right )^{3}}+\frac {32750 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{121}-\frac {1051695 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2401}\) | \(81\) |
derivativedivides | \(\frac {250 \sqrt {1-2 x}}{11 \left (-\frac {6}{5}-2 x \right )}+\frac {32750 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{121}+\frac {\frac {612765 \left (1-2 x \right )^{\frac {5}{2}}}{343}-\frac {412380 \left (1-2 x \right )^{\frac {3}{2}}}{49}+\frac {69399 \sqrt {1-2 x}}{7}}{\left (-4-6 x \right )^{3}}-\frac {1051695 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2401}\) | \(91\) |
default | \(\frac {250 \sqrt {1-2 x}}{11 \left (-\frac {6}{5}-2 x \right )}+\frac {32750 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{121}+\frac {\frac {612765 \left (1-2 x \right )^{\frac {5}{2}}}{343}-\frac {412380 \left (1-2 x \right )^{\frac {3}{2}}}{49}+\frac {69399 \sqrt {1-2 x}}{7}}{\left (-4-6 x \right )^{3}}-\frac {1051695 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2401}\) | \(91\) |
trager | \(-\frac {\left (45278325 x^{3}+89054820 x^{2}+58335165 x +12724912\right ) \sqrt {1-2 x}}{7546 \left (2+3 x \right )^{3} \left (3+5 x \right )}+\frac {45 \RootOf \left (\textit {\_Z}^{2}-11470276461\right ) \ln \left (\frac {3 \RootOf \left (\textit {\_Z}^{2}-11470276461\right ) x -5 \RootOf \left (\textit {\_Z}^{2}-11470276461\right )+490791 \sqrt {1-2 x}}{2+3 x}\right )}{4802}+\frac {125 \RootOf \left (\textit {\_Z}^{2}-943855\right ) \ln \left (-\frac {5 \RootOf \left (\textit {\_Z}^{2}-943855\right ) x -8 \RootOf \left (\textit {\_Z}^{2}-943855\right )-7205 \sqrt {1-2 x}}{3+5 x}\right )}{121}\) | \(129\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 146, normalized size = 0.91 \begin {gather*} -\frac {16375}{121} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {1051695}{4802} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {45278325 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 313944615 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 725394915 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 558527921 \, \sqrt {-2 \, x + 1}}{3773 \, {\left (135 \, {\left (2 \, x - 1\right )}^{4} + 1242 \, {\left (2 \, x - 1\right )}^{3} + 4284 \, {\left (2 \, x - 1\right )}^{2} + 13132 \, x - 2793\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.93, size = 162, normalized size = 1.01 \begin {gather*} \frac {78632750 \, \sqrt {11} \sqrt {5} {\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )} \log \left (-\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} - 5 \, x + 8}{5 \, x + 3}\right ) + 127255095 \, \sqrt {7} \sqrt {3} {\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )} \log \left (\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) - 77 \, {\left (45278325 \, x^{3} + 89054820 \, x^{2} + 58335165 \, x + 12724912\right )} \sqrt {-2 \, x + 1}}{581042 \, {\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\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 = 18.19, size = 10610, normalized size = 66.31 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.68, size = 139, normalized size = 0.87 \begin {gather*} -\frac {16375}{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 {1051695}{4802} \, \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 {625 \, \sqrt {-2 \, x + 1}}{11 \, {\left (5 \, x + 3\right )}} - \frac {9 \, {\left (68085 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 320740 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 377839 \, \sqrt {-2 \, x + 1}\right )}}{2744 \, {\left (3 \, x + 2\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.10, size = 108, normalized size = 0.68 \begin {gather*} \frac {32750\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{121}-\frac {1051695\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{2401}-\frac {\frac {11398529\,\sqrt {1-2\,x}}{10395}-\frac {2302841\,{\left (1-2\,x\right )}^{3/2}}{1617}+\frac {6976547\,{\left (1-2\,x\right )}^{5/2}}{11319}-\frac {335395\,{\left (1-2\,x\right )}^{7/2}}{3773}}{\frac {13132\,x}{135}+\frac {476\,{\left (2\,x-1\right )}^2}{15}+\frac {46\,{\left (2\,x-1\right )}^3}{5}+{\left (2\,x-1\right )}^4-\frac {931}{45}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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