Optimal. Leaf size=128 \[ \frac{(1-2 x)^{5/2}}{105 (3 x+2)^5}-\frac{17 (1-2 x)^{3/2}}{126 (3 x+2)^4}-\frac{17 \sqrt{1-2 x}}{12348 (3 x+2)}-\frac{17 \sqrt{1-2 x}}{5292 (3 x+2)^2}+\frac{17 \sqrt{1-2 x}}{378 (3 x+2)^3}-\frac{17 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{6174 \sqrt{21}} \]
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Rubi [A] time = 0.0367996, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {78, 47, 51, 63, 206} \[ \frac{(1-2 x)^{5/2}}{105 (3 x+2)^5}-\frac{17 (1-2 x)^{3/2}}{126 (3 x+2)^4}-\frac{17 \sqrt{1-2 x}}{12348 (3 x+2)}-\frac{17 \sqrt{1-2 x}}{5292 (3 x+2)^2}+\frac{17 \sqrt{1-2 x}}{378 (3 x+2)^3}-\frac{17 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{6174 \sqrt{21}} \]
Antiderivative was successfully verified.
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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+3 x)^6} \, dx &=\frac{(1-2 x)^{5/2}}{105 (2+3 x)^5}+\frac{34}{21} \int \frac{(1-2 x)^{3/2}}{(2+3 x)^5} \, dx\\ &=\frac{(1-2 x)^{5/2}}{105 (2+3 x)^5}-\frac{17 (1-2 x)^{3/2}}{126 (2+3 x)^4}-\frac{17}{42} \int \frac{\sqrt{1-2 x}}{(2+3 x)^4} \, dx\\ &=\frac{(1-2 x)^{5/2}}{105 (2+3 x)^5}-\frac{17 (1-2 x)^{3/2}}{126 (2+3 x)^4}+\frac{17 \sqrt{1-2 x}}{378 (2+3 x)^3}+\frac{17}{378} \int \frac{1}{\sqrt{1-2 x} (2+3 x)^3} \, dx\\ &=\frac{(1-2 x)^{5/2}}{105 (2+3 x)^5}-\frac{17 (1-2 x)^{3/2}}{126 (2+3 x)^4}+\frac{17 \sqrt{1-2 x}}{378 (2+3 x)^3}-\frac{17 \sqrt{1-2 x}}{5292 (2+3 x)^2}+\frac{17 \int \frac{1}{\sqrt{1-2 x} (2+3 x)^2} \, dx}{1764}\\ &=\frac{(1-2 x)^{5/2}}{105 (2+3 x)^5}-\frac{17 (1-2 x)^{3/2}}{126 (2+3 x)^4}+\frac{17 \sqrt{1-2 x}}{378 (2+3 x)^3}-\frac{17 \sqrt{1-2 x}}{5292 (2+3 x)^2}-\frac{17 \sqrt{1-2 x}}{12348 (2+3 x)}+\frac{17 \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx}{12348}\\ &=\frac{(1-2 x)^{5/2}}{105 (2+3 x)^5}-\frac{17 (1-2 x)^{3/2}}{126 (2+3 x)^4}+\frac{17 \sqrt{1-2 x}}{378 (2+3 x)^3}-\frac{17 \sqrt{1-2 x}}{5292 (2+3 x)^2}-\frac{17 \sqrt{1-2 x}}{12348 (2+3 x)}-\frac{17 \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{12348}\\ &=\frac{(1-2 x)^{5/2}}{105 (2+3 x)^5}-\frac{17 (1-2 x)^{3/2}}{126 (2+3 x)^4}+\frac{17 \sqrt{1-2 x}}{378 (2+3 x)^3}-\frac{17 \sqrt{1-2 x}}{5292 (2+3 x)^2}-\frac{17 \sqrt{1-2 x}}{12348 (2+3 x)}-\frac{17 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{6174 \sqrt{21}}\\ \end{align*}
Mathematica [C] time = 0.0153959, size = 42, normalized size = 0.33 \[ \frac{(1-2 x)^{5/2} \left (\frac{16807}{(3 x+2)^5}-1088 \, _2F_1\left (\frac{5}{2},5;\frac{7}{2};\frac{3}{7}-\frac{6 x}{7}\right )\right )}{1764735} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 75, normalized size = 0.6 \begin{align*} 7776\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{5}} \left ({\frac{17\, \left ( 1-2\,x \right ) ^{9/2}}{592704}}-{\frac{17\, \left ( 1-2\,x \right ) ^{7/2}}{54432}}-{\frac{ \left ( 1-2\,x \right ) ^{5/2}}{25515}}+{\frac{119\, \left ( 1-2\,x \right ) ^{3/2}}{69984}}-{\frac{119\,\sqrt{1-2\,x}}{139968}} \right ) }-{\frac{17\,\sqrt{21}}{129654}{\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 = 1.56127, size = 173, normalized size = 1.35 \begin{align*} \frac{17}{259308} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{6885 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - 74970 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 9408 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 408170 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 204085 \, \sqrt{-2 \, x + 1}}{30870 \,{\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 = 1.40322, size = 346, normalized size = 2.7 \begin{align*} \frac{85 \, \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 (6885 \, x^{4} + 23715 \, x^{3} - 48252 \, x^{2} - 23998 \, x + 7912\right )} \sqrt{-2 \, x + 1}}{1296540 \,{\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 = 2.02955, size = 157, normalized size = 1.23 \begin{align*} \frac{17}{259308} \, \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{6885 \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + 74970 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - 9408 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + 408170 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 204085 \, \sqrt{-2 \, x + 1}}{987840 \,{\left (3 \, x + 2\right )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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