Optimal. Leaf size=143 \[ \frac {123}{16807 \sqrt {1-2 x}}-\frac {41}{4802 \sqrt {1-2 x} (3 x+2)}-\frac {41}{3430 \sqrt {1-2 x} (3 x+2)^2}-\frac {41}{1715 \sqrt {1-2 x} (3 x+2)^3}-\frac {41}{735 \sqrt {1-2 x} (3 x+2)^4}+\frac {1}{105 \sqrt {1-2 x} (3 x+2)^5}-\frac {123 \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{16807} \]
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Rubi [A] time = 0.05, antiderivative size = 150, normalized size of antiderivative = 1.05, number of steps used = 8, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {78, 51, 63, 206} \[ -\frac {369 \sqrt {1-2 x}}{33614 (3 x+2)}-\frac {123 \sqrt {1-2 x}}{4802 (3 x+2)^2}-\frac {123 \sqrt {1-2 x}}{1715 (3 x+2)^3}-\frac {369 \sqrt {1-2 x}}{1715 (3 x+2)^4}+\frac {328}{735 \sqrt {1-2 x} (3 x+2)^4}+\frac {1}{105 \sqrt {1-2 x} (3 x+2)^5}-\frac {123 \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{16807} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 78
Rule 206
Rubi steps
\begin {align*} \int \frac {3+5 x}{(1-2 x)^{3/2} (2+3 x)^6} \, dx &=\frac {1}{105 \sqrt {1-2 x} (2+3 x)^5}+\frac {164}{105} \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^5} \, dx\\ &=\frac {1}{105 \sqrt {1-2 x} (2+3 x)^5}+\frac {328}{735 \sqrt {1-2 x} (2+3 x)^4}+\frac {1476}{245} \int \frac {1}{\sqrt {1-2 x} (2+3 x)^5} \, dx\\ &=\frac {1}{105 \sqrt {1-2 x} (2+3 x)^5}+\frac {328}{735 \sqrt {1-2 x} (2+3 x)^4}-\frac {369 \sqrt {1-2 x}}{1715 (2+3 x)^4}+\frac {369}{245} \int \frac {1}{\sqrt {1-2 x} (2+3 x)^4} \, dx\\ &=\frac {1}{105 \sqrt {1-2 x} (2+3 x)^5}+\frac {328}{735 \sqrt {1-2 x} (2+3 x)^4}-\frac {369 \sqrt {1-2 x}}{1715 (2+3 x)^4}-\frac {123 \sqrt {1-2 x}}{1715 (2+3 x)^3}+\frac {123}{343} \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3} \, dx\\ &=\frac {1}{105 \sqrt {1-2 x} (2+3 x)^5}+\frac {328}{735 \sqrt {1-2 x} (2+3 x)^4}-\frac {369 \sqrt {1-2 x}}{1715 (2+3 x)^4}-\frac {123 \sqrt {1-2 x}}{1715 (2+3 x)^3}-\frac {123 \sqrt {1-2 x}}{4802 (2+3 x)^2}+\frac {369 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2} \, dx}{4802}\\ &=\frac {1}{105 \sqrt {1-2 x} (2+3 x)^5}+\frac {328}{735 \sqrt {1-2 x} (2+3 x)^4}-\frac {369 \sqrt {1-2 x}}{1715 (2+3 x)^4}-\frac {123 \sqrt {1-2 x}}{1715 (2+3 x)^3}-\frac {123 \sqrt {1-2 x}}{4802 (2+3 x)^2}-\frac {369 \sqrt {1-2 x}}{33614 (2+3 x)}+\frac {369 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{33614}\\ &=\frac {1}{105 \sqrt {1-2 x} (2+3 x)^5}+\frac {328}{735 \sqrt {1-2 x} (2+3 x)^4}-\frac {369 \sqrt {1-2 x}}{1715 (2+3 x)^4}-\frac {123 \sqrt {1-2 x}}{1715 (2+3 x)^3}-\frac {123 \sqrt {1-2 x}}{4802 (2+3 x)^2}-\frac {369 \sqrt {1-2 x}}{33614 (2+3 x)}-\frac {369 \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{33614}\\ &=\frac {1}{105 \sqrt {1-2 x} (2+3 x)^5}+\frac {328}{735 \sqrt {1-2 x} (2+3 x)^4}-\frac {369 \sqrt {1-2 x}}{1715 (2+3 x)^4}-\frac {123 \sqrt {1-2 x}}{1715 (2+3 x)^3}-\frac {123 \sqrt {1-2 x}}{4802 (2+3 x)^2}-\frac {369 \sqrt {1-2 x}}{33614 (2+3 x)}-\frac {123 \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{16807}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 42, normalized size = 0.29 \[ \frac {5248 \, _2F_1\left (-\frac {1}{2},5;\frac {1}{2};\frac {3}{7}-\frac {6 x}{7}\right )+\frac {16807}{(3 x+2)^5}}{1764735 \sqrt {1-2 x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.78, size = 135, normalized size = 0.94 \[ \frac {615 \, \sqrt {7} \sqrt {3} {\left (486 \, x^{6} + 1377 \, x^{5} + 1350 \, x^{4} + 360 \, x^{3} - 240 \, x^{2} - 176 \, x - 32\right )} \log \left (\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) - 7 \, {\left (298890 \, x^{5} + 880065 \, x^{4} + 964197 \, x^{3} + 430992 \, x^{2} + 8774 \, x - 32894\right )} \sqrt {-2 \, x + 1}}{1176490 \, {\left (486 \, x^{6} + 1377 \, x^{5} + 1350 \, x^{4} + 360 \, x^{3} - 240 \, x^{2} - 176 \, x - 32\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.23, size = 125, normalized size = 0.87 \[ \frac {123}{235298} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {352}{117649 \, \sqrt {-2 \, x + 1}} - \frac {618435 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + 6401430 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 25316928 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 45656730 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 31609165 \, \sqrt {-2 \, x + 1}}{18823840 \, {\left (3 \, x + 2\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 84, normalized size = 0.59 \[ -\frac {123 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{117649}+\frac {352}{117649 \sqrt {-2 x +1}}+\frac {\frac {123687 \left (-2 x +1\right )^{\frac {9}{2}}}{117649}-\frac {182898 \left (-2 x +1\right )^{\frac {7}{2}}}{16807}+\frac {516672 \left (-2 x +1\right )^{\frac {5}{2}}}{12005}-\frac {26622 \left (-2 x +1\right )^{\frac {3}{2}}}{343}+\frac {2633 \sqrt {-2 x +1}}{49}}{\left (-6 x -4\right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.19, size = 137, normalized size = 0.96 \[ \frac {123}{235298} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {149445 \, {\left (2 \, x - 1\right )}^{5} + 1627290 \, {\left (2 \, x - 1\right )}^{4} + 6943104 \, {\left (2 \, x - 1\right )}^{3} + 14283990 \, {\left (2 \, x - 1\right )}^{2} + 27141590 \, x - 9345035}{84035 \, {\left (243 \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - 2835 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + 13230 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 30870 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 36015 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 16807 \, \sqrt {-2 \, x + 1}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 118, normalized size = 0.83 \[ \frac {\frac {15826\,x}{11907}+\frac {6478\,{\left (2\,x-1\right )}^2}{9261}+\frac {5248\,{\left (2\,x-1\right )}^3}{15435}+\frac {82\,{\left (2\,x-1\right )}^4}{1029}+\frac {123\,{\left (2\,x-1\right )}^5}{16807}-\frac {5449}{11907}}{\frac {16807\,\sqrt {1-2\,x}}{243}-\frac {12005\,{\left (1-2\,x\right )}^{3/2}}{81}+\frac {3430\,{\left (1-2\,x\right )}^{5/2}}{27}-\frac {490\,{\left (1-2\,x\right )}^{7/2}}{9}+\frac {35\,{\left (1-2\,x\right )}^{9/2}}{3}-{\left (1-2\,x\right )}^{11/2}}-\frac {123\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{117649} \]
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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