Optimal. Leaf size=147 \[ -\frac {1015 \sqrt {1-2 x}}{6 (3+5 x)^2}+\frac {7 \sqrt {1-2 x}}{6 (2+3 x)^2 (3+5 x)^2}+\frac {45 \sqrt {1-2 x}}{2 (2+3 x) (3+5 x)^2}+\frac {1020 \sqrt {1-2 x}}{3+5 x}+14073 \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-13665 \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
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Rubi [A]
time = 0.04, antiderivative size = 147, 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 = {100, 156, 162,
65, 212} \begin {gather*} \frac {1020 \sqrt {1-2 x}}{5 x+3}-\frac {1015 \sqrt {1-2 x}}{6 (5 x+3)^2}+\frac {45 \sqrt {1-2 x}}{2 (3 x+2) (5 x+3)^2}+\frac {7 \sqrt {1-2 x}}{6 (3 x+2)^2 (5 x+3)^2}+14073 \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-13665 \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 100
Rule 156
Rule 162
Rule 212
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2}}{(2+3 x)^3 (3+5 x)^3} \, dx &=\frac {7 \sqrt {1-2 x}}{6 (2+3 x)^2 (3+5 x)^2}+\frac {1}{6} \int \frac {157-237 x}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^3} \, dx\\ &=\frac {7 \sqrt {1-2 x}}{6 (2+3 x)^2 (3+5 x)^2}+\frac {45 \sqrt {1-2 x}}{2 (2+3 x) (3+5 x)^2}+\frac {1}{42} \int \frac {17087-23625 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^3} \, dx\\ &=-\frac {1015 \sqrt {1-2 x}}{6 (3+5 x)^2}+\frac {7 \sqrt {1-2 x}}{6 (2+3 x)^2 (3+5 x)^2}+\frac {45 \sqrt {1-2 x}}{2 (2+3 x) (3+5 x)^2}-\frac {1}{924} \int \frac {1229382-1406790 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^2} \, dx\\ &=-\frac {1015 \sqrt {1-2 x}}{6 (3+5 x)^2}+\frac {7 \sqrt {1-2 x}}{6 (2+3 x)^2 (3+5 x)^2}+\frac {45 \sqrt {1-2 x}}{2 (2+3 x) (3+5 x)^2}+\frac {1020 \sqrt {1-2 x}}{3+5 x}+\frac {\int \frac {50784426-31101840 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx}{10164}\\ &=-\frac {1015 \sqrt {1-2 x}}{6 (3+5 x)^2}+\frac {7 \sqrt {1-2 x}}{6 (2+3 x)^2 (3+5 x)^2}+\frac {45 \sqrt {1-2 x}}{2 (2+3 x) (3+5 x)^2}+\frac {1020 \sqrt {1-2 x}}{3+5 x}-\frac {42219}{2} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx+\frac {68325}{2} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=-\frac {1015 \sqrt {1-2 x}}{6 (3+5 x)^2}+\frac {7 \sqrt {1-2 x}}{6 (2+3 x)^2 (3+5 x)^2}+\frac {45 \sqrt {1-2 x}}{2 (2+3 x) (3+5 x)^2}+\frac {1020 \sqrt {1-2 x}}{3+5 x}+\frac {42219}{2} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )-\frac {68325}{2} \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {1015 \sqrt {1-2 x}}{6 (3+5 x)^2}+\frac {7 \sqrt {1-2 x}}{6 (2+3 x)^2 (3+5 x)^2}+\frac {45 \sqrt {1-2 x}}{2 (2+3 x) (3+5 x)^2}+\frac {1020 \sqrt {1-2 x}}{3+5 x}+14073 \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-13665 \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 = 0.33, size = 95, normalized size = 0.65 \begin {gather*} \frac {\sqrt {1-2 x} \left (23219+110315 x+174435 x^2+91800 x^3\right )}{2 \left (6+19 x+15 x^2\right )^2}+14073 \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-13665 \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 = 0.18, size = 94, normalized size = 0.64
method | result | size |
risch | \(-\frac {\left (-1+2 x \right ) \left (91800 x^{3}+174435 x^{2}+110315 x +23219\right )}{2 \left (15 x^{2}+19 x +6\right )^{2} \sqrt {1-2 x}}-\frac {13665 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{11}+\frac {14073 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{7}\) | \(79\) |
derivativedivides | \(\frac {-5075 \left (1-2 x \right )^{\frac {3}{2}}+11055 \sqrt {1-2 x}}{\left (-6-10 x \right )^{2}}-\frac {13665 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{11}-\frac {324 \left (\frac {205 \left (1-2 x \right )^{\frac {3}{2}}}{36}-\frac {161 \sqrt {1-2 x}}{12}\right )}{\left (-4-6 x \right )^{2}}+\frac {14073 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{7}\) | \(94\) |
default | \(\frac {-5075 \left (1-2 x \right )^{\frac {3}{2}}+11055 \sqrt {1-2 x}}{\left (-6-10 x \right )^{2}}-\frac {13665 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{11}-\frac {324 \left (\frac {205 \left (1-2 x \right )^{\frac {3}{2}}}{36}-\frac {161 \sqrt {1-2 x}}{12}\right )}{\left (-4-6 x \right )^{2}}+\frac {14073 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{7}\) | \(94\) |
trager | \(\frac {\left (91800 x^{3}+174435 x^{2}+110315 x +23219\right ) \sqrt {1-2 x}}{2 \left (15 x^{2}+19 x +6\right )^{2}}-\frac {15 \RootOf \left (\textit {\_Z}^{2}-45645655\right ) \ln \left (\frac {-5 \RootOf \left (\textit {\_Z}^{2}-45645655\right ) x +50105 \sqrt {1-2 x}+8 \RootOf \left (\textit {\_Z}^{2}-45645655\right )}{3+5 x}\right )}{22}-\frac {14073 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x -5 \RootOf \left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{14}\) | \(126\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 146, normalized size = 0.99 \begin {gather*} \frac {13665}{22} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {14073}{14} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {2 \, {\left (45900 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 312135 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 707200 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 533841 \, \sqrt {-2 \, x + 1}\right )}}{225 \, {\left (2 \, x - 1\right )}^{4} + 2040 \, {\left (2 \, x - 1\right )}^{3} + 6934 \, {\left (2 \, x - 1\right )}^{2} + 20944 \, x - 4543} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.87, size = 162, normalized size = 1.10 \begin {gather*} \frac {95655 \, \sqrt {11} \sqrt {5} {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 154803 \, \sqrt {7} \sqrt {3} {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 77 \, {\left (91800 \, x^{3} + 174435 \, x^{2} + 110315 \, x + 23219\right )} \sqrt {-2 \, x + 1}}{154 \, {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.58, size = 148, normalized size = 1.01 \begin {gather*} \frac {13665}{22} \, \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 {14073}{14} \, \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 (45900 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 312135 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 707200 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 533841 \, \sqrt {-2 \, x + 1}\right )}}{{\left (15 \, {\left (2 \, x - 1\right )}^{2} + 136 \, x + 9\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.22, size = 107, normalized size = 0.73 \begin {gather*} \frac {14073\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{7}-\frac {13665\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{11}+\frac {\frac {355894\,\sqrt {1-2\,x}}{75}-\frac {56576\,{\left (1-2\,x\right )}^{3/2}}{9}+\frac {41618\,{\left (1-2\,x\right )}^{5/2}}{15}-408\,{\left (1-2\,x\right )}^{7/2}}{\frac {20944\,x}{225}+\frac {6934\,{\left (2\,x-1\right )}^2}{225}+\frac {136\,{\left (2\,x-1\right )}^3}{15}+{\left (2\,x-1\right )}^4-\frac {4543}{225}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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