Optimal. Leaf size=136 \[ \frac {215}{9604 (1-2 x)^{3/2}}+\frac {1935}{67228 \sqrt {1-2 x}}+\frac {1}{84 (1-2 x)^{3/2} (2+3 x)^4}-\frac {43}{588 (1-2 x)^{3/2} (2+3 x)^3}-\frac {129}{2744 (1-2 x)^{3/2} (2+3 x)^2}-\frac {129}{2744 (1-2 x)^{3/2} (2+3 x)}-\frac {1935 \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{67228} \]
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Rubi [A]
time = 0.03, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {79, 44, 53, 65,
212} \begin {gather*} \frac {1935}{67228 \sqrt {1-2 x}}-\frac {129}{2744 (1-2 x)^{3/2} (3 x+2)}+\frac {215}{9604 (1-2 x)^{3/2}}-\frac {129}{2744 (1-2 x)^{3/2} (3 x+2)^2}-\frac {43}{588 (1-2 x)^{3/2} (3 x+2)^3}+\frac {1}{84 (1-2 x)^{3/2} (3 x+2)^4}-\frac {1935 \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{67228} \end {gather*}
Antiderivative was successfully verified.
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Rule 44
Rule 53
Rule 65
Rule 79
Rule 212
Rubi steps
\begin {align*} \int \frac {3+5 x}{(1-2 x)^{5/2} (2+3 x)^5} \, dx &=\frac {1}{84 (1-2 x)^{3/2} (2+3 x)^4}+\frac {43}{28} \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^4} \, dx\\ &=\frac {1}{84 (1-2 x)^{3/2} (2+3 x)^4}+\frac {43}{294 (1-2 x)^{3/2} (2+3 x)^3}+\frac {387}{196} \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^4} \, dx\\ &=\frac {1}{84 (1-2 x)^{3/2} (2+3 x)^4}+\frac {43}{294 (1-2 x)^{3/2} (2+3 x)^3}+\frac {387}{686 \sqrt {1-2 x} (2+3 x)^3}+\frac {1161}{196} \int \frac {1}{\sqrt {1-2 x} (2+3 x)^4} \, dx\\ &=\frac {1}{84 (1-2 x)^{3/2} (2+3 x)^4}+\frac {43}{294 (1-2 x)^{3/2} (2+3 x)^3}+\frac {387}{686 \sqrt {1-2 x} (2+3 x)^3}-\frac {387 \sqrt {1-2 x}}{1372 (2+3 x)^3}+\frac {1935 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3} \, dx}{1372}\\ &=\frac {1}{84 (1-2 x)^{3/2} (2+3 x)^4}+\frac {43}{294 (1-2 x)^{3/2} (2+3 x)^3}+\frac {387}{686 \sqrt {1-2 x} (2+3 x)^3}-\frac {387 \sqrt {1-2 x}}{1372 (2+3 x)^3}-\frac {1935 \sqrt {1-2 x}}{19208 (2+3 x)^2}+\frac {5805 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2} \, dx}{19208}\\ &=\frac {1}{84 (1-2 x)^{3/2} (2+3 x)^4}+\frac {43}{294 (1-2 x)^{3/2} (2+3 x)^3}+\frac {387}{686 \sqrt {1-2 x} (2+3 x)^3}-\frac {387 \sqrt {1-2 x}}{1372 (2+3 x)^3}-\frac {1935 \sqrt {1-2 x}}{19208 (2+3 x)^2}-\frac {5805 \sqrt {1-2 x}}{134456 (2+3 x)}+\frac {5805 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{134456}\\ &=\frac {1}{84 (1-2 x)^{3/2} (2+3 x)^4}+\frac {43}{294 (1-2 x)^{3/2} (2+3 x)^3}+\frac {387}{686 \sqrt {1-2 x} (2+3 x)^3}-\frac {387 \sqrt {1-2 x}}{1372 (2+3 x)^3}-\frac {1935 \sqrt {1-2 x}}{19208 (2+3 x)^2}-\frac {5805 \sqrt {1-2 x}}{134456 (2+3 x)}-\frac {5805 \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{134456}\\ &=\frac {1}{84 (1-2 x)^{3/2} (2+3 x)^4}+\frac {43}{294 (1-2 x)^{3/2} (2+3 x)^3}+\frac {387}{686 \sqrt {1-2 x} (2+3 x)^3}-\frac {387 \sqrt {1-2 x}}{1372 (2+3 x)^3}-\frac {1935 \sqrt {1-2 x}}{19208 (2+3 x)^2}-\frac {5805 \sqrt {1-2 x}}{134456 (2+3 x)}-\frac {1935 \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{67228}\\ \end {align*}
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Mathematica [A]
time = 0.28, size = 75, normalized size = 0.55 \begin {gather*} \frac {-\frac {7 \left (-48490-611202 x-1034451 x^2+1069281 x^3+3343680 x^4+1880820 x^5\right )}{2 (1-2 x)^{3/2} (2+3 x)^4}-5805 \sqrt {21} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1411788} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.19, size = 84, normalized size = 0.62
method | result | size |
risch | \(\frac {1880820 x^{5}+3343680 x^{4}+1069281 x^{3}-1034451 x^{2}-611202 x -48490}{403368 \left (2+3 x \right )^{4} \sqrt {1-2 x}\, \left (-1+2 x \right )}-\frac {1935 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{470596}\) | \(68\) |
derivativedivides | \(\frac {176}{50421 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {2080}{117649 \sqrt {1-2 x}}+\frac {\frac {423225 \left (1-2 x \right )^{\frac {7}{2}}}{470596}-\frac {461403 \left (1-2 x \right )^{\frac {5}{2}}}{67228}+\frac {169335 \left (1-2 x \right )^{\frac {3}{2}}}{9604}-\frac {20805 \sqrt {1-2 x}}{1372}}{\left (-4-6 x \right )^{4}}-\frac {1935 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{470596}\) | \(84\) |
default | \(\frac {176}{50421 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {2080}{117649 \sqrt {1-2 x}}+\frac {\frac {423225 \left (1-2 x \right )^{\frac {7}{2}}}{470596}-\frac {461403 \left (1-2 x \right )^{\frac {5}{2}}}{67228}+\frac {169335 \left (1-2 x \right )^{\frac {3}{2}}}{9604}-\frac {20805 \sqrt {1-2 x}}{1372}}{\left (-4-6 x \right )^{4}}-\frac {1935 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{470596}\) | \(84\) |
trager | \(-\frac {\left (1880820 x^{5}+3343680 x^{4}+1069281 x^{3}-1034451 x^{2}-611202 x -48490\right ) \sqrt {1-2 x}}{403368 \left (2+3 x \right )^{4} \left (-1+2 x \right )^{2}}-\frac {1935 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \RootOf \left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{941192}\) | \(94\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.55, size = 128, normalized size = 0.94 \begin {gather*} \frac {1935}{941192} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {470205 \, {\left (2 \, x - 1\right )}^{5} + 4022865 \, {\left (2 \, x - 1\right )}^{4} + 12458691 \, {\left (2 \, x - 1\right )}^{3} + 15872031 \, {\left (2 \, x - 1\right )}^{2} + 11327232 \, x - 7353920}{201684 \, {\left (81 \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - 756 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + 2646 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 4116 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 2401 \, {\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 = 1.10, size = 135, normalized size = 0.99 \begin {gather*} \frac {5805 \, \sqrt {7} \sqrt {3} {\left (324 \, x^{6} + 540 \, x^{5} + 81 \, x^{4} - 264 \, x^{3} - 104 \, x^{2} + 32 \, x + 16\right )} \log \left (\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) - 7 \, {\left (1880820 \, x^{5} + 3343680 \, x^{4} + 1069281 \, x^{3} - 1034451 \, x^{2} - 611202 \, x - 48490\right )} \sqrt {-2 \, x + 1}}{2823576 \, {\left (324 \, x^{6} + 540 \, x^{5} + 81 \, x^{4} - 264 \, x^{3} - 104 \, x^{2} + 32 \, x + 16\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 = 1.56, size = 121, normalized size = 0.89 \begin {gather*} \frac {1935}{941192} \, \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 {16 \, {\left (780 \, x - 467\right )}}{352947 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} - \frac {3 \, {\left (141075 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 1076607 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 2765805 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 2378705 \, \sqrt {-2 \, x + 1}\right )}}{7529536 \, {\left (3 \, x + 2\right )}^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.23, size = 108, normalized size = 0.79 \begin {gather*} -\frac {1935\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{470596}-\frac {\frac {2752\,x}{3969}+\frac {1333\,{\left (2\,x-1\right )}^2}{1372}+\frac {3139\,{\left (2\,x-1\right )}^3}{4116}+\frac {2365\,{\left (2\,x-1\right )}^4}{9604}+\frac {1935\,{\left (2\,x-1\right )}^5}{67228}-\frac {5360}{11907}}{\frac {2401\,{\left (1-2\,x\right )}^{3/2}}{81}-\frac {1372\,{\left (1-2\,x\right )}^{5/2}}{27}+\frac {98\,{\left (1-2\,x\right )}^{7/2}}{3}-\frac {28\,{\left (1-2\,x\right )}^{9/2}}{3}+{\left (1-2\,x\right )}^{11/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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