Optimal. Leaf size=76 \[ \frac {60}{343 \sqrt {1-2 x}}+\frac {1}{21 (1-2 x)^{3/2} (3 x+2)}+\frac {20}{147 (1-2 x)^{3/2}}-\frac {60}{343} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 76, normalized size of antiderivative = 1.00, 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} \[ \frac {60}{343 \sqrt {1-2 x}}+\frac {1}{21 (1-2 x)^{3/2} (3 x+2)}+\frac {20}{147 (1-2 x)^{3/2}}-\frac {60}{343} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \]
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)^{5/2} (2+3 x)^2} \, dx &=\frac {1}{21 (1-2 x)^{3/2} (2+3 x)}+\frac {10}{7} \int \frac {1}{(1-2 x)^{5/2} (2+3 x)} \, dx\\ &=\frac {20}{147 (1-2 x)^{3/2}}+\frac {1}{21 (1-2 x)^{3/2} (2+3 x)}+\frac {30}{49} \int \frac {1}{(1-2 x)^{3/2} (2+3 x)} \, dx\\ &=\frac {20}{147 (1-2 x)^{3/2}}+\frac {60}{343 \sqrt {1-2 x}}+\frac {1}{21 (1-2 x)^{3/2} (2+3 x)}+\frac {90}{343} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=\frac {20}{147 (1-2 x)^{3/2}}+\frac {60}{343 \sqrt {1-2 x}}+\frac {1}{21 (1-2 x)^{3/2} (2+3 x)}-\frac {90}{343} \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {20}{147 (1-2 x)^{3/2}}+\frac {60}{343 \sqrt {1-2 x}}+\frac {1}{21 (1-2 x)^{3/2} (2+3 x)}-\frac {60}{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 = 46, normalized size = 0.61 \[ -\frac {-20 (3 x+2) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {3}{7}-\frac {6 x}{7}\right )-7}{147 (1-2 x)^{3/2} (3 x+2)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.81, size = 90, normalized size = 1.18 \[ \frac {90 \, \sqrt {7} \sqrt {3} {\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )} \log \left (\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) - 7 \, {\left (1080 \, x^{2} - 240 \, x - 689\right )} \sqrt {-2 \, x + 1}}{7203 \, {\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.26, size = 77, normalized size = 1.01 \[ \frac {30}{2401} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {4 \, {\left (93 \, x - 85\right )}}{1029 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} + \frac {3 \, \sqrt {-2 \, x + 1}}{343 \, {\left (3 \, x + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 54, normalized size = 0.71 \[ -\frac {60 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{2401}+\frac {22}{147 \left (-2 x +1\right )^{\frac {3}{2}}}+\frac {62}{343 \sqrt {-2 x +1}}-\frac {2 \sqrt {-2 x +1}}{343 \left (-2 x -\frac {4}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.20, size = 74, normalized size = 0.97 \[ \frac {30}{2401} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {2 \, {\left (270 \, {\left (2 \, x - 1\right )}^{2} + 840 \, x - 959\right )}}{1029 \, {\left (3 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 7 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.21, size = 56, normalized size = 0.74 \[ -\frac {\frac {80\,x}{147}+\frac {60\,{\left (2\,x-1\right )}^2}{343}-\frac {274}{441}}{\frac {7\,{\left (1-2\,x\right )}^{3/2}}{3}-{\left (1-2\,x\right )}^{5/2}}-\frac {60\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{2401} \]
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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