Optimal. Leaf size=102 \[ -\frac {68 \sqrt {1-2 x}}{3 (3+5 x)}+\frac {7 \sqrt {1-2 x}}{3 (2+3 x) (3+5 x)}-134 \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+138 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 102, 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 = {100, 156, 162,
65, 212} \begin {gather*} -\frac {68 \sqrt {1-2 x}}{3 (5 x+3)}+\frac {7 \sqrt {1-2 x}}{3 (3 x+2) (5 x+3)}-134 \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+138 \sqrt {\frac {11}{5}} \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)^2 (3+5 x)^2} \, dx &=\frac {7 \sqrt {1-2 x}}{3 (2+3 x) (3+5 x)}+\frac {1}{3} \int \frac {89-101 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^2} \, dx\\ &=-\frac {68 \sqrt {1-2 x}}{3 (3+5 x)}+\frac {7 \sqrt {1-2 x}}{3 (2+3 x) (3+5 x)}-\frac {1}{33} \int \frac {3663-2244 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx\\ &=-\frac {68 \sqrt {1-2 x}}{3 (3+5 x)}+\frac {7 \sqrt {1-2 x}}{3 (2+3 x) (3+5 x)}+469 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx-759 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=-\frac {68 \sqrt {1-2 x}}{3 (3+5 x)}+\frac {7 \sqrt {1-2 x}}{3 (2+3 x) (3+5 x)}-469 \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )+759 \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {68 \sqrt {1-2 x}}{3 (3+5 x)}+\frac {7 \sqrt {1-2 x}}{3 (2+3 x) (3+5 x)}-134 \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+138 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.19, size = 83, normalized size = 0.81 \begin {gather*} -\frac {\sqrt {1-2 x} (43+68 x)}{6+19 x+15 x^2}-134 \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+138 \sqrt {\frac {11}{5}} \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 = 0.69
method | result | size |
risch | \(\frac {\left (43+68 x \right ) \left (-1+2 x \right )}{\left (15 x^{2}+19 x +6\right ) \sqrt {1-2 x}}+\frac {138 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{5}-\frac {134 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{3}\) | \(68\) |
derivativedivides | \(\frac {22 \sqrt {1-2 x}}{5 \left (-\frac {6}{5}-2 x \right )}+\frac {138 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{5}+\frac {14 \sqrt {1-2 x}}{3 \left (-\frac {4}{3}-2 x \right )}-\frac {134 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{3}\) | \(70\) |
default | \(\frac {22 \sqrt {1-2 x}}{5 \left (-\frac {6}{5}-2 x \right )}+\frac {138 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{5}+\frac {14 \sqrt {1-2 x}}{3 \left (-\frac {4}{3}-2 x \right )}-\frac {134 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{3}\) | \(70\) |
trager | \(-\frac {\left (43+68 x \right ) \sqrt {1-2 x}}{15 x^{2}+19 x +6}+\frac {69 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (-\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x -8 \RootOf \left (\textit {\_Z}^{2}-55\right )-55 \sqrt {1-2 x}}{3+5 x}\right )}{5}+\frac {67 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x -5 \RootOf \left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{3}\) | \(117\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 110, normalized size = 1.08 \begin {gather*} -\frac {69}{5} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {67}{3} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {4 \, {\left (34 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 77 \, \sqrt {-2 \, x + 1}\right )}}{15 \, {\left (2 \, x - 1\right )}^{2} + 136 \, x + 9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.08, size = 122, normalized size = 1.20 \begin {gather*} \frac {207 \, \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 ) + 335 \, \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 ) - 15 \, {\left (68 \, x + 43\right )} \sqrt {-2 \, x + 1}}{15 \, {\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 [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 = 0.59, size = 116, normalized size = 1.14 \begin {gather*} -\frac {69}{5} \, \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 {67}{3} \, \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 (34 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 77 \, \sqrt {-2 \, x + 1}\right )}}{15 \, {\left (2 \, x - 1\right )}^{2} + 136 \, x + 9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.21, size = 72, normalized size = 0.71 \begin {gather*} \frac {138\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{5}-\frac {134\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{3}-\frac {\frac {308\,\sqrt {1-2\,x}}{15}-\frac {136\,{\left (1-2\,x\right )}^{3/2}}{15}}{\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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