Optimal. Leaf size=120 \[ \frac {\sqrt {1-2 x} (5 x+3)^2}{105 (3 x+2)^5}+\frac {\sqrt {1-2 x} (1971 x+1255)}{6615 (3 x+2)^4}-\frac {5293 \sqrt {1-2 x}}{43218 (3 x+2)}-\frac {5293 \sqrt {1-2 x}}{18522 (3 x+2)^2}-\frac {5293 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{21609 \sqrt {21}} \]
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Rubi [A] time = 0.03, 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 = {98, 145, 51, 63, 206} \[ \frac {\sqrt {1-2 x} (5 x+3)^2}{105 (3 x+2)^5}+\frac {\sqrt {1-2 x} (1971 x+1255)}{6615 (3 x+2)^4}-\frac {5293 \sqrt {1-2 x}}{43218 (3 x+2)}-\frac {5293 \sqrt {1-2 x}}{18522 (3 x+2)^2}-\frac {5293 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{21609 \sqrt {21}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 98
Rule 145
Rule 206
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 \operatorname {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 [C] time = 0.04, size = 52, normalized size = 0.43 \[ \frac {\sqrt {1-2 x} \left (\frac {343 \left (18375 x^2+24371 x+8083\right )}{(3 x+2)^5}-84688 \, _2F_1\left (\frac {1}{2},4;\frac {3}{2};\frac {3}{7}-\frac {6 x}{7}\right )\right )}{756315} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.96, size = 114, normalized size = 0.95 \[ \frac {26465 \, \sqrt {21} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (2143665 \, x^{4} + 7383735 \, x^{3} + 8806422 \, x^{2} + 4450198 \, x + 816938\right )} \sqrt {-2 \, x + 1}}{4537890 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.31, size = 116, normalized size = 0.97 \[ \frac {5293}{907578} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {2143665 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + 23342130 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 92390088 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 158930030 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 100809415 \, \sqrt {-2 \, x + 1}}{3457440 \, {\left (3 \, x + 2\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 75, normalized size = 0.62 \[ -\frac {5293 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{453789}+\frac {\frac {47637 \left (-2 x +1\right )^{\frac {9}{2}}}{2401}-\frac {10586 \left (-2 x +1\right )^{\frac {7}{2}}}{49}+\frac {628504 \left (-2 x +1\right )^{\frac {5}{2}}}{735}-\frac {648694 \left (-2 x +1\right )^{\frac {3}{2}}}{441}+\frac {58781 \sqrt {-2 x +1}}{63}}{\left (-6 x -4\right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, size = 128, normalized size = 1.07 \[ \frac {5293}{907578} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {2143665 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 23342130 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 92390088 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 158930030 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 100809415 \, \sqrt {-2 \, x + 1}}{108045 \, {\left (243 \, {\left (2 \, x - 1\right )}^{5} + 2835 \, {\left (2 \, x - 1\right )}^{4} + 13230 \, {\left (2 \, x - 1\right )}^{3} + 30870 \, {\left (2 \, x - 1\right )}^{2} + 72030 \, x - 19208\right )}} \]
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 \[ -\frac {5293\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{453789}-\frac {\frac {58781\,\sqrt {1-2\,x}}{15309}-\frac {648694\,{\left (1-2\,x\right )}^{3/2}}{107163}+\frac {628504\,{\left (1-2\,x\right )}^{5/2}}{178605}-\frac {10586\,{\left (1-2\,x\right )}^{7/2}}{11907}+\frac {5293\,{\left (1-2\,x\right )}^{9/2}}{64827}}{\frac {24010\,x}{81}+\frac {3430\,{\left (2\,x-1\right )}^2}{27}+\frac {490\,{\left (2\,x-1\right )}^3}{9}+\frac {35\,{\left (2\,x-1\right )}^4}{3}+{\left (2\,x-1\right )}^5-\frac {19208}{243}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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