Optimal. Leaf size=140 \[ -\frac {463344 \sqrt {1-2 x} (2+3 x)^2}{166375}-\frac {10283 (2+3 x)^3}{6655 \sqrt {1-2 x}}-\frac {38 (2+3 x)^4}{1815 \sqrt {1-2 x} (3+5 x)}+\frac {7 (2+3 x)^5}{33 (1-2 x)^{3/2} (3+5 x)}-\frac {21 \sqrt {1-2 x} (4633904+1544625 x)}{831875}-\frac {406 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{831875 \sqrt {55}} \]
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Rubi [A]
time = 0.03, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {100, 154, 155,
158, 152, 65, 212} \begin {gather*} \frac {7 (3 x+2)^5}{33 (1-2 x)^{3/2} (5 x+3)}-\frac {38 (3 x+2)^4}{1815 \sqrt {1-2 x} (5 x+3)}-\frac {10283 (3 x+2)^3}{6655 \sqrt {1-2 x}}-\frac {463344 \sqrt {1-2 x} (3 x+2)^2}{166375}-\frac {21 \sqrt {1-2 x} (1544625 x+4633904)}{831875}-\frac {406 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{831875 \sqrt {55}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 100
Rule 152
Rule 154
Rule 155
Rule 158
Rule 212
Rubi steps
\begin {align*} \int \frac {(2+3 x)^6}{(1-2 x)^{5/2} (3+5 x)^2} \, dx &=\frac {7 (2+3 x)^5}{33 (1-2 x)^{3/2} (3+5 x)}-\frac {1}{33} \int \frac {(2+3 x)^4 (239+411 x)}{(1-2 x)^{3/2} (3+5 x)^2} \, dx\\ &=-\frac {38 (2+3 x)^4}{1815 \sqrt {1-2 x} (3+5 x)}+\frac {7 (2+3 x)^5}{33 (1-2 x)^{3/2} (3+5 x)}-\frac {\int \frac {(2+3 x)^3 (8358+14133 x)}{(1-2 x)^{3/2} (3+5 x)} \, dx}{1815}\\ &=-\frac {10283 (2+3 x)^3}{6655 \sqrt {1-2 x}}-\frac {38 (2+3 x)^4}{1815 \sqrt {1-2 x} (3+5 x)}+\frac {7 (2+3 x)^5}{33 (1-2 x)^{3/2} (3+5 x)}-\frac {\int \frac {(-834141-1390032 x) (2+3 x)^2}{\sqrt {1-2 x} (3+5 x)} \, dx}{19965}\\ &=-\frac {463344 \sqrt {1-2 x} (2+3 x)^2}{166375}-\frac {10283 (2+3 x)^3}{6655 \sqrt {1-2 x}}-\frac {38 (2+3 x)^4}{1815 \sqrt {1-2 x} (3+5 x)}+\frac {7 (2+3 x)^5}{33 (1-2 x)^{3/2} (3+5 x)}+\frac {\int \frac {(2+3 x) (58387434+97311375 x)}{\sqrt {1-2 x} (3+5 x)} \, dx}{499125}\\ &=-\frac {463344 \sqrt {1-2 x} (2+3 x)^2}{166375}-\frac {10283 (2+3 x)^3}{6655 \sqrt {1-2 x}}-\frac {38 (2+3 x)^4}{1815 \sqrt {1-2 x} (3+5 x)}+\frac {7 (2+3 x)^5}{33 (1-2 x)^{3/2} (3+5 x)}-\frac {21 \sqrt {1-2 x} (4633904+1544625 x)}{831875}+\frac {203 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{831875}\\ &=-\frac {463344 \sqrt {1-2 x} (2+3 x)^2}{166375}-\frac {10283 (2+3 x)^3}{6655 \sqrt {1-2 x}}-\frac {38 (2+3 x)^4}{1815 \sqrt {1-2 x} (3+5 x)}+\frac {7 (2+3 x)^5}{33 (1-2 x)^{3/2} (3+5 x)}-\frac {21 \sqrt {1-2 x} (4633904+1544625 x)}{831875}-\frac {203 \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{831875}\\ &=-\frac {463344 \sqrt {1-2 x} (2+3 x)^2}{166375}-\frac {10283 (2+3 x)^3}{6655 \sqrt {1-2 x}}-\frac {38 (2+3 x)^4}{1815 \sqrt {1-2 x} (3+5 x)}+\frac {7 (2+3 x)^5}{33 (1-2 x)^{3/2} (3+5 x)}-\frac {21 \sqrt {1-2 x} (4633904+1544625 x)}{831875}-\frac {406 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{831875 \sqrt {55}}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 90, normalized size = 0.64 \begin {gather*} -\frac {55 \left (1035652776-1434109759 x-3837745731 x^2+2644064775 x^3+480298005 x^4+72772425 x^5\right )-1218 \sqrt {55} \sqrt {1-2 x} \left (-3+x+10 x^2\right ) \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{137259375 (1-2 x)^{3/2} (3+5 x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 81, normalized size = 0.58
method | result | size |
risch | \(\frac {72772425 x^{5}+480298005 x^{4}+2644064775 x^{3}-3837745731 x^{2}-1434109759 x +1035652776}{2495625 \sqrt {1-2 x}\, \left (3+5 x \right ) \left (-1+2 x \right )}-\frac {406 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{45753125}\) | \(68\) |
derivativedivides | \(-\frac {729 \left (1-2 x \right )^{\frac {5}{2}}}{2000}+\frac {729 \left (1-2 x \right )^{\frac {3}{2}}}{125}-\frac {315171 \sqrt {1-2 x}}{5000}+\frac {2 \sqrt {1-2 x}}{4159375 \left (-\frac {6}{5}-2 x \right )}-\frac {406 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{45753125}+\frac {117649}{5808 \left (1-2 x \right )^{\frac {3}{2}}}-\frac {134456}{1331 \sqrt {1-2 x}}\) | \(81\) |
default | \(-\frac {729 \left (1-2 x \right )^{\frac {5}{2}}}{2000}+\frac {729 \left (1-2 x \right )^{\frac {3}{2}}}{125}-\frac {315171 \sqrt {1-2 x}}{5000}+\frac {2 \sqrt {1-2 x}}{4159375 \left (-\frac {6}{5}-2 x \right )}-\frac {406 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{45753125}+\frac {117649}{5808 \left (1-2 x \right )^{\frac {3}{2}}}-\frac {134456}{1331 \sqrt {1-2 x}}\) | \(81\) |
trager | \(-\frac {\left (72772425 x^{5}+480298005 x^{4}+2644064775 x^{3}-3837745731 x^{2}-1434109759 x +1035652776\right ) \sqrt {1-2 x}}{2495625 \left (-1+2 x \right )^{2} \left (3+5 x \right )}+\frac {203 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \RootOf \left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{45753125}\) | \(94\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 101, normalized size = 0.72 \begin {gather*} -\frac {729}{2000} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {729}{125} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {203}{45753125} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {315171}{5000} \, \sqrt {-2 \, x + 1} - \frac {10084199952 \, {\left (2 \, x - 1\right )}^{2} + 48414664375 \, x - 19758729375}{19965000 \, {\left (5 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 11 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.20, size = 99, normalized size = 0.71 \begin {gather*} \frac {609 \, \sqrt {55} {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) - 55 \, {\left (72772425 \, x^{5} + 480298005 \, x^{4} + 2644064775 \, x^{3} - 3837745731 \, x^{2} - 1434109759 \, x + 1035652776\right )} \sqrt {-2 \, x + 1}}{137259375 \, {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\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 = 2.04, size = 111, normalized size = 0.79 \begin {gather*} -\frac {729}{2000} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {729}{125} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {203}{45753125} \, \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 {315171}{5000} \, \sqrt {-2 \, x + 1} - \frac {16807 \, {\left (768 \, x - 307\right )}}{63888 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} - \frac {\sqrt {-2 \, x + 1}}{831875 \, {\left (5 \, x + 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 84, normalized size = 0.60 \begin {gather*} \frac {\frac {7042133\,x}{14520}+\frac {420174998\,{\left (2\,x-1\right )}^2}{4159375}-\frac {957999}{4840}}{\frac {11\,{\left (1-2\,x\right )}^{3/2}}{5}-{\left (1-2\,x\right )}^{5/2}}-\frac {315171\,\sqrt {1-2\,x}}{5000}+\frac {729\,{\left (1-2\,x\right )}^{3/2}}{125}-\frac {729\,{\left (1-2\,x\right )}^{5/2}}{2000}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,406{}\mathrm {i}}{45753125} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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