Optimal. Leaf size=128 \[ \frac{23 (1-2 x)^{5/2}}{588 (3 x+2)^4}-\frac{(1-2 x)^{5/2}}{315 (3 x+2)^5}-\frac{4693 (1-2 x)^{3/2}}{15876 (3 x+2)^3}-\frac{4693 \sqrt{1-2 x}}{222264 (3 x+2)}+\frac{4693 \sqrt{1-2 x}}{31752 (3 x+2)^2}-\frac{4693 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{111132 \sqrt{21}} \]
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Rubi [A] time = 0.0384327, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {89, 78, 47, 51, 63, 206} \[ \frac{23 (1-2 x)^{5/2}}{588 (3 x+2)^4}-\frac{(1-2 x)^{5/2}}{315 (3 x+2)^5}-\frac{4693 (1-2 x)^{3/2}}{15876 (3 x+2)^3}-\frac{4693 \sqrt{1-2 x}}{222264 (3 x+2)}+\frac{4693 \sqrt{1-2 x}}{31752 (3 x+2)^2}-\frac{4693 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{111132 \sqrt{21}} \]
Antiderivative was successfully verified.
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Rule 89
Rule 78
Rule 47
Rule 51
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{3/2} (3+5 x)^2}{(2+3 x)^6} \, dx &=-\frac{(1-2 x)^{5/2}}{315 (2+3 x)^5}+\frac{1}{315} \int \frac{(1-2 x)^{3/2} (1405+2625 x)}{(2+3 x)^5} \, dx\\ &=-\frac{(1-2 x)^{5/2}}{315 (2+3 x)^5}+\frac{23 (1-2 x)^{5/2}}{588 (2+3 x)^4}+\frac{4693 \int \frac{(1-2 x)^{3/2}}{(2+3 x)^4} \, dx}{1764}\\ &=-\frac{(1-2 x)^{5/2}}{315 (2+3 x)^5}+\frac{23 (1-2 x)^{5/2}}{588 (2+3 x)^4}-\frac{4693 (1-2 x)^{3/2}}{15876 (2+3 x)^3}-\frac{4693 \int \frac{\sqrt{1-2 x}}{(2+3 x)^3} \, dx}{5292}\\ &=-\frac{(1-2 x)^{5/2}}{315 (2+3 x)^5}+\frac{23 (1-2 x)^{5/2}}{588 (2+3 x)^4}-\frac{4693 (1-2 x)^{3/2}}{15876 (2+3 x)^3}+\frac{4693 \sqrt{1-2 x}}{31752 (2+3 x)^2}+\frac{4693 \int \frac{1}{\sqrt{1-2 x} (2+3 x)^2} \, dx}{31752}\\ &=-\frac{(1-2 x)^{5/2}}{315 (2+3 x)^5}+\frac{23 (1-2 x)^{5/2}}{588 (2+3 x)^4}-\frac{4693 (1-2 x)^{3/2}}{15876 (2+3 x)^3}+\frac{4693 \sqrt{1-2 x}}{31752 (2+3 x)^2}-\frac{4693 \sqrt{1-2 x}}{222264 (2+3 x)}+\frac{4693 \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx}{222264}\\ &=-\frac{(1-2 x)^{5/2}}{315 (2+3 x)^5}+\frac{23 (1-2 x)^{5/2}}{588 (2+3 x)^4}-\frac{4693 (1-2 x)^{3/2}}{15876 (2+3 x)^3}+\frac{4693 \sqrt{1-2 x}}{31752 (2+3 x)^2}-\frac{4693 \sqrt{1-2 x}}{222264 (2+3 x)}-\frac{4693 \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{222264}\\ &=-\frac{(1-2 x)^{5/2}}{315 (2+3 x)^5}+\frac{23 (1-2 x)^{5/2}}{588 (2+3 x)^4}-\frac{4693 (1-2 x)^{3/2}}{15876 (2+3 x)^3}+\frac{4693 \sqrt{1-2 x}}{31752 (2+3 x)^2}-\frac{4693 \sqrt{1-2 x}}{222264 (2+3 x)}-\frac{4693 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{111132 \sqrt{21}}\\ \end{align*}
Mathematica [C] time = 0.0257082, size = 47, normalized size = 0.37 \[ \frac{(1-2 x)^{5/2} \left (\frac{2401 (1035 x+662)}{(3 x+2)^5}-75088 \, _2F_1\left (\frac{5}{2},4;\frac{7}{2};\frac{3}{7}-\frac{6 x}{7}\right )\right )}{21176820} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 75, normalized size = 0.6 \begin{align*} -3888\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{5}} \left ( -{\frac{4693\, \left ( 1-2\,x \right ) ^{9/2}}{5334336}}-{\frac{907\, \left ( 1-2\,x \right ) ^{7/2}}{489888}}+{\frac{6119\, \left ( 1-2\,x \right ) ^{5/2}}{229635}}-{\frac{32851\, \left ( 1-2\,x \right ) ^{3/2}}{629856}}+{\frac{32851\,\sqrt{1-2\,x}}{1259712}} \right ) }-{\frac{4693\,\sqrt{21}}{2333772}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.00369, size = 173, normalized size = 1.35 \begin{align*} \frac{4693}{4667544} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{1900665 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} + 3999870 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 57567552 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 112678930 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 56339465 \, \sqrt{-2 \, x + 1}}{555660 \,{\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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.32736, size = 366, normalized size = 2.86 \begin{align*} \frac{23465 \, \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 (1900665 \, x^{4} - 5801265 \, x^{3} - 8540988 \, x^{2} - 2143262 \, x + 292028\right )} \sqrt{-2 \, x + 1}}{23337720 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.60948, size = 157, normalized size = 1.23 \begin{align*} \frac{4693}{4667544} \, \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{1900665 \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} - 3999870 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - 57567552 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + 112678930 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 56339465 \, \sqrt{-2 \, x + 1}}{17781120 \,{\left (3 \, x + 2\right )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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