Optimal. Leaf size=83 \[ \frac {65}{343 \sqrt {1-2 x}}-\frac {65}{294 \sqrt {1-2 x} (3 x+2)}+\frac {1}{42 \sqrt {1-2 x} (3 x+2)^2}-\frac {65}{343} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 90, normalized size of antiderivative = 1.08, number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {78, 51, 63, 206} \[ -\frac {195 \sqrt {1-2 x}}{686 (3 x+2)}+\frac {65}{147 \sqrt {1-2 x} (3 x+2)}+\frac {1}{42 \sqrt {1-2 x} (3 x+2)^2}-\frac {65}{343} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \]
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)^3} \, dx &=\frac {1}{42 \sqrt {1-2 x} (2+3 x)^2}+\frac {65}{42} \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^2} \, dx\\ &=\frac {1}{42 \sqrt {1-2 x} (2+3 x)^2}+\frac {65}{147 \sqrt {1-2 x} (2+3 x)}+\frac {195}{98} \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2} \, dx\\ &=\frac {1}{42 \sqrt {1-2 x} (2+3 x)^2}+\frac {65}{147 \sqrt {1-2 x} (2+3 x)}-\frac {195 \sqrt {1-2 x}}{686 (2+3 x)}+\frac {195}{686} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=\frac {1}{42 \sqrt {1-2 x} (2+3 x)^2}+\frac {65}{147 \sqrt {1-2 x} (2+3 x)}-\frac {195 \sqrt {1-2 x}}{686 (2+3 x)}-\frac {195}{686} \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {1}{42 \sqrt {1-2 x} (2+3 x)^2}+\frac {65}{147 \sqrt {1-2 x} (2+3 x)}-\frac {195 \sqrt {1-2 x}}{686 (2+3 x)}-\frac {65}{343} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\\ \end {align*}
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Mathematica [C] time = 0.02, size = 48, normalized size = 0.58 \[ \frac {260 (3 x+2)^2 \, _2F_1\left (-\frac {1}{2},2;\frac {1}{2};\frac {3}{7}-\frac {6 x}{7}\right )+49}{2058 \sqrt {1-2 x} (3 x+2)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.88, size = 90, normalized size = 1.08 \[ \frac {65 \, \sqrt {7} \sqrt {3} {\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )} \log \left (\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) - 7 \, {\left (1170 \, x^{2} + 1105 \, x + 233\right )} \sqrt {-2 \, x + 1}}{4802 \, {\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.17, size = 77, normalized size = 0.93 \[ \frac {65}{4802} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {44}{343 \, \sqrt {-2 \, x + 1}} + \frac {27 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 61 \, \sqrt {-2 \, x + 1}}{196 \, {\left (3 \, x + 2\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 57, normalized size = 0.69 \[ -\frac {65 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{2401}+\frac {44}{343 \sqrt {-2 x +1}}+\frac {\frac {27 \left (-2 x +1\right )^{\frac {3}{2}}}{49}-\frac {61 \sqrt {-2 x +1}}{49}}{\left (-6 x -4\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 83, normalized size = 1.00 \[ \frac {65}{4802} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {585 \, {\left (2 \, x - 1\right )}^{2} + 4550 \, x - 119}{343 \, {\left (9 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 42 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 49 \, \sqrt {-2 \, x + 1}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 62, normalized size = 0.75 \[ \frac {\frac {650\,x}{441}+\frac {65\,{\left (2\,x-1\right )}^2}{343}-\frac {17}{441}}{\frac {49\,\sqrt {1-2\,x}}{9}-\frac {14\,{\left (1-2\,x\right )}^{3/2}}{3}+{\left (1-2\,x\right )}^{5/2}}-\frac {65\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{2401} \]
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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