Optimal. Leaf size=133 \[ \frac {\sqrt {1-2 x}}{4 (2+3 x)^4}+\frac {139 \sqrt {1-2 x}}{84 (2+3 x)^3}+\frac {14555 \sqrt {1-2 x}}{1176 (2+3 x)^2}+\frac {337955 \sqrt {1-2 x}}{2744 (2+3 x)}+\frac {11656955 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1372 \sqrt {21}}-250 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
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Rubi [A]
time = 0.04, antiderivative size = 133, 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 = {101, 156, 162,
65, 212} \begin {gather*} \frac {337955 \sqrt {1-2 x}}{2744 (3 x+2)}+\frac {14555 \sqrt {1-2 x}}{1176 (3 x+2)^2}+\frac {139 \sqrt {1-2 x}}{84 (3 x+2)^3}+\frac {\sqrt {1-2 x}}{4 (3 x+2)^4}+\frac {11656955 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1372 \sqrt {21}}-250 \sqrt {55} \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 101
Rule 156
Rule 162
Rule 212
Rubi steps
\begin {align*} \int \frac {\sqrt {1-2 x}}{(2+3 x)^5 (3+5 x)} \, dx &=\frac {\sqrt {1-2 x}}{4 (2+3 x)^4}-\frac {1}{4} \int \frac {-23+35 x}{\sqrt {1-2 x} (2+3 x)^4 (3+5 x)} \, dx\\ &=\frac {\sqrt {1-2 x}}{4 (2+3 x)^4}+\frac {139 \sqrt {1-2 x}}{84 (2+3 x)^3}-\frac {1}{84} \int \frac {-2535+3475 x}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)} \, dx\\ &=\frac {\sqrt {1-2 x}}{4 (2+3 x)^4}+\frac {139 \sqrt {1-2 x}}{84 (2+3 x)^3}+\frac {14555 \sqrt {1-2 x}}{1176 (2+3 x)^2}-\frac {\int \frac {-192405+218325 x}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)} \, dx}{1176}\\ &=\frac {\sqrt {1-2 x}}{4 (2+3 x)^4}+\frac {139 \sqrt {1-2 x}}{84 (2+3 x)^3}+\frac {14555 \sqrt {1-2 x}}{1176 (2+3 x)^2}+\frac {337955 \sqrt {1-2 x}}{2744 (2+3 x)}-\frac {\int \frac {-8277405+5069325 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx}{8232}\\ &=\frac {\sqrt {1-2 x}}{4 (2+3 x)^4}+\frac {139 \sqrt {1-2 x}}{84 (2+3 x)^3}+\frac {14555 \sqrt {1-2 x}}{1176 (2+3 x)^2}+\frac {337955 \sqrt {1-2 x}}{2744 (2+3 x)}-\frac {11656955 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{2744}+6875 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=\frac {\sqrt {1-2 x}}{4 (2+3 x)^4}+\frac {139 \sqrt {1-2 x}}{84 (2+3 x)^3}+\frac {14555 \sqrt {1-2 x}}{1176 (2+3 x)^2}+\frac {337955 \sqrt {1-2 x}}{2744 (2+3 x)}+\frac {11656955 \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{2744}-6875 \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {\sqrt {1-2 x}}{4 (2+3 x)^4}+\frac {139 \sqrt {1-2 x}}{84 (2+3 x)^3}+\frac {14555 \sqrt {1-2 x}}{1176 (2+3 x)^2}+\frac {337955 \sqrt {1-2 x}}{2744 (2+3 x)}+\frac {11656955 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1372 \sqrt {21}}-250 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.44, size = 88, normalized size = 0.66 \begin {gather*} \frac {\sqrt {1-2 x} \left (2849254+12587542 x+18555225 x^2+9124785 x^3\right )}{2744 (2+3 x)^4}+\frac {11656955 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1372 \sqrt {21}}-250 \sqrt {55} \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 = 84, normalized size = 0.63
method | result | size |
risch | \(-\frac {18249570 x^{4}+27985665 x^{3}+6619859 x^{2}-6889034 x -2849254}{2744 \left (2+3 x \right )^{4} \sqrt {1-2 x}}-250 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}+\frac {11656955 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{28812}\) | \(74\) |
derivativedivides | \(-250 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}-\frac {162 \left (\frac {337955 \left (1-2 x \right )^{\frac {7}{2}}}{8232}-\frac {3070705 \left (1-2 x \right )^{\frac {5}{2}}}{10584}+\frac {3100927 \left (1-2 x \right )^{\frac {3}{2}}}{4536}-\frac {116015 \sqrt {1-2 x}}{216}\right )}{\left (-4-6 x \right )^{4}}+\frac {11656955 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{28812}\) | \(84\) |
default | \(-250 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}-\frac {162 \left (\frac {337955 \left (1-2 x \right )^{\frac {7}{2}}}{8232}-\frac {3070705 \left (1-2 x \right )^{\frac {5}{2}}}{10584}+\frac {3100927 \left (1-2 x \right )^{\frac {3}{2}}}{4536}-\frac {116015 \sqrt {1-2 x}}{216}\right )}{\left (-4-6 x \right )^{4}}+\frac {11656955 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{28812}\) | \(84\) |
trager | \(\frac {\left (9124785 x^{3}+18555225 x^{2}+12587542 x +2849254\right ) \sqrt {1-2 x}}{2744 \left (2+3 x \right )^{4}}+125 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \RootOf \left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )+\frac {11656955 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \RootOf \left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{57624}\) | \(121\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.61, size = 146, normalized size = 1.10 \begin {gather*} 125 \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {11656955}{57624} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {9124785 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 64484805 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 151945423 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 119379435 \, \sqrt {-2 \, x + 1}}{1372 \, {\left (81 \, {\left (2 \, x - 1\right )}^{4} + 756 \, {\left (2 \, x - 1\right )}^{3} + 2646 \, {\left (2 \, x - 1\right )}^{2} + 8232 \, x - 1715\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.80, size = 150, normalized size = 1.13 \begin {gather*} \frac {7203000 \, \sqrt {55} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 11656955 \, \sqrt {21} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (9124785 \, x^{3} + 18555225 \, x^{2} + 12587542 \, x + 2849254\right )} \sqrt {-2 \, x + 1}}{57624 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 184.30, size = 869, normalized size = 6.53 \begin {gather*} 3300 \left (\begin {cases} \frac {\sqrt {21} \left (- \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )}\right )}{147} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right ) - 1320 \left (\begin {cases} \frac {\sqrt {21} \cdot \left (\frac {3 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{16} - \frac {3 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{16} + \frac {3}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} + \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )^{2}} + \frac {3}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )} - \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )^{2}}\right )}{1029} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right ) + 528 \left (\begin {cases} \frac {\sqrt {21} \left (- \frac {5 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{32} + \frac {5 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{32} - \frac {5}{32 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} - \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )^{2}} - \frac {1}{48 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )^{3}} - \frac {5}{32 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )} + \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )^{2}} - \frac {1}{48 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )^{3}}\right )}{7203} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right ) - 224 \left (\begin {cases} \frac {\sqrt {21} \cdot \left (\frac {35 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{256} - \frac {35 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{256} + \frac {35}{256 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} + \frac {15}{256 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )^{2}} + \frac {5}{192 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )^{3}} + \frac {1}{128 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )^{4}} + \frac {35}{256 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )} - \frac {15}{256 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )^{2}} + \frac {5}{192 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )^{3}} - \frac {1}{128 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )^{4}}\right )}{50421} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right ) - 8250 \left (\begin {cases} - \frac {\sqrt {21} \operatorname {acoth}{\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} \right )}}{21} & \text {for}\: x < - \frac {2}{3} \\- \frac {\sqrt {21} \operatorname {atanh}{\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} \right )}}{21} & \text {for}\: x > - \frac {2}{3} \end {cases}\right ) + 13750 \left (\begin {cases} - \frac {\sqrt {55} \operatorname {acoth}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{55} & \text {for}\: x < - \frac {3}{5} \\- \frac {\sqrt {55} \operatorname {atanh}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{55} & \text {for}\: x > - \frac {3}{5} \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.43, size = 139, normalized size = 1.05 \begin {gather*} 125 \, \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 {11656955}{57624} \, \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 {9124785 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 64484805 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 151945423 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 119379435 \, \sqrt {-2 \, x + 1}}{21952 \, {\left (3 \, x + 2\right )}^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.21, size = 107, normalized size = 0.80 \begin {gather*} \frac {11656955\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{28812}-250\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )+\frac {\frac {116015\,\sqrt {1-2\,x}}{108}-\frac {3100927\,{\left (1-2\,x\right )}^{3/2}}{2268}+\frac {3070705\,{\left (1-2\,x\right )}^{5/2}}{5292}-\frac {337955\,{\left (1-2\,x\right )}^{7/2}}{4116}}{\frac {2744\,x}{27}+\frac {98\,{\left (2\,x-1\right )}^2}{3}+\frac {28\,{\left (2\,x-1\right )}^3}{3}+{\left (2\,x-1\right )}^4-\frac {1715}{81}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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