Optimal. Leaf size=120 \[ -\frac {5293 \sqrt {1-2 x}}{18522 (2+3 x)^2}-\frac {5293 \sqrt {1-2 x}}{43218 (2+3 x)}+\frac {\sqrt {1-2 x} (3+5 x)^2}{105 (2+3 x)^5}+\frac {\sqrt {1-2 x} (1255+1971 x)}{6615 (2+3 x)^4}-\frac {5293 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{21609 \sqrt {21}} \]
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Rubi [A]
time = 0.02, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {100, 150, 44,
65, 212} \begin {gather*} \frac {\sqrt {1-2 x} (5 x+3)^2}{105 (3 x+2)^5}+\frac {\sqrt {1-2 x} (1971 x+1255)}{6615 (3 x+2)^4}-\frac {5293 \sqrt {1-2 x}}{43218 (3 x+2)}-\frac {5293 \sqrt {1-2 x}}{18522 (3 x+2)^2}-\frac {5293 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{21609 \sqrt {21}} \end {gather*}
Antiderivative was successfully verified.
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Rule 44
Rule 65
Rule 100
Rule 150
Rule 212
Rubi steps
\begin {align*} \int \frac {(3+5 x)^3}{\sqrt {1-2 x} (2+3 x)^6} \, dx &=\frac {\sqrt {1-2 x} (3+5 x)^2}{105 (2+3 x)^5}-\frac {1}{105} \int \frac {(-488-850 x) (3+5 x)}{\sqrt {1-2 x} (2+3 x)^5} \, dx\\ &=\frac {\sqrt {1-2 x} (3+5 x)^2}{105 (2+3 x)^5}+\frac {\sqrt {1-2 x} (1255+1971 x)}{6615 (2+3 x)^4}+\frac {5293 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3} \, dx}{1323}\\ &=-\frac {5293 \sqrt {1-2 x}}{18522 (2+3 x)^2}+\frac {\sqrt {1-2 x} (3+5 x)^2}{105 (2+3 x)^5}+\frac {\sqrt {1-2 x} (1255+1971 x)}{6615 (2+3 x)^4}+\frac {5293 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2} \, dx}{6174}\\ &=-\frac {5293 \sqrt {1-2 x}}{18522 (2+3 x)^2}-\frac {5293 \sqrt {1-2 x}}{43218 (2+3 x)}+\frac {\sqrt {1-2 x} (3+5 x)^2}{105 (2+3 x)^5}+\frac {\sqrt {1-2 x} (1255+1971 x)}{6615 (2+3 x)^4}+\frac {5293 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{43218}\\ &=-\frac {5293 \sqrt {1-2 x}}{18522 (2+3 x)^2}-\frac {5293 \sqrt {1-2 x}}{43218 (2+3 x)}+\frac {\sqrt {1-2 x} (3+5 x)^2}{105 (2+3 x)^5}+\frac {\sqrt {1-2 x} (1255+1971 x)}{6615 (2+3 x)^4}-\frac {5293 \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{43218}\\ &=-\frac {5293 \sqrt {1-2 x}}{18522 (2+3 x)^2}-\frac {5293 \sqrt {1-2 x}}{43218 (2+3 x)}+\frac {\sqrt {1-2 x} (3+5 x)^2}{105 (2+3 x)^5}+\frac {\sqrt {1-2 x} (1255+1971 x)}{6615 (2+3 x)^4}-\frac {5293 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{21609 \sqrt {21}}\\ \end {align*}
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Mathematica [A]
time = 0.29, size = 70, normalized size = 0.58 \begin {gather*} \frac {-\frac {21 \sqrt {1-2 x} \left (816938+4450198 x+8806422 x^2+7383735 x^3+2143665 x^4\right )}{2 (2+3 x)^5}-26465 \sqrt {21} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{2268945} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 75, normalized size = 0.62
method | result | size |
risch | \(\frac {4287330 x^{5}+12623805 x^{4}+10229109 x^{3}+93974 x^{2}-2816322 x -816938}{216090 \left (2+3 x \right )^{5} \sqrt {1-2 x}}-\frac {5293 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{453789}\) | \(61\) |
derivativedivides | \(\frac {\frac {47637 \left (1-2 x \right )^{\frac {9}{2}}}{2401}-\frac {10586 \left (1-2 x \right )^{\frac {7}{2}}}{49}+\frac {628504 \left (1-2 x \right )^{\frac {5}{2}}}{735}-\frac {648694 \left (1-2 x \right )^{\frac {3}{2}}}{441}+\frac {58781 \sqrt {1-2 x}}{63}}{\left (-4-6 x \right )^{5}}-\frac {5293 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{453789}\) | \(75\) |
default | \(\frac {\frac {47637 \left (1-2 x \right )^{\frac {9}{2}}}{2401}-\frac {10586 \left (1-2 x \right )^{\frac {7}{2}}}{49}+\frac {628504 \left (1-2 x \right )^{\frac {5}{2}}}{735}-\frac {648694 \left (1-2 x \right )^{\frac {3}{2}}}{441}+\frac {58781 \sqrt {1-2 x}}{63}}{\left (-4-6 x \right )^{5}}-\frac {5293 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{453789}\) | \(75\) |
trager | \(-\frac {\left (2143665 x^{4}+7383735 x^{3}+8806422 x^{2}+4450198 x +816938\right ) \sqrt {1-2 x}}{216090 \left (2+3 x \right )^{5}}-\frac {5293 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \RootOf \left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{907578}\) | \(82\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 128, normalized size = 1.07 \begin {gather*} \frac {5293}{907578} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {2143665 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 23342130 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 92390088 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 158930030 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 100809415 \, \sqrt {-2 \, x + 1}}{108045 \, {\left (243 \, {\left (2 \, x - 1\right )}^{5} + 2835 \, {\left (2 \, x - 1\right )}^{4} + 13230 \, {\left (2 \, x - 1\right )}^{3} + 30870 \, {\left (2 \, x - 1\right )}^{2} + 72030 \, x - 19208\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.29, size = 114, normalized size = 0.95 \begin {gather*} \frac {26465 \, \sqrt {21} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (2143665 \, x^{4} + 7383735 \, x^{3} + 8806422 \, x^{2} + 4450198 \, x + 816938\right )} \sqrt {-2 \, x + 1}}{4537890 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\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 = 0.72, size = 116, normalized size = 0.97 \begin {gather*} \frac {5293}{907578} \, \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 {2143665 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + 23342130 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 92390088 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 158930030 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 100809415 \, \sqrt {-2 \, x + 1}}{3457440 \, {\left (3 \, x + 2\right )}^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 108, normalized size = 0.90 \begin {gather*} -\frac {5293\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{453789}-\frac {\frac {58781\,\sqrt {1-2\,x}}{15309}-\frac {648694\,{\left (1-2\,x\right )}^{3/2}}{107163}+\frac {628504\,{\left (1-2\,x\right )}^{5/2}}{178605}-\frac {10586\,{\left (1-2\,x\right )}^{7/2}}{11907}+\frac {5293\,{\left (1-2\,x\right )}^{9/2}}{64827}}{\frac {24010\,x}{81}+\frac {3430\,{\left (2\,x-1\right )}^2}{27}+\frac {490\,{\left (2\,x-1\right )}^3}{9}+\frac {35\,{\left (2\,x-1\right )}^4}{3}+{\left (2\,x-1\right )}^5-\frac {19208}{243}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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