Optimal. Leaf size=148 \[ -\frac{(1-2 x)^{3/2} (3 x+2)^5}{5 (5 x+3)}+\frac{39}{275} (1-2 x)^{3/2} (3 x+2)^4+\frac{38 (1-2 x)^{3/2} (3 x+2)^3}{4125}-\frac{4016 (1-2 x)^{3/2} (3 x+2)^2}{48125}-\frac{2 (1-2 x)^{3/2} (204777 x+298462)}{515625}+\frac{324 \sqrt{1-2 x}}{78125}-\frac{324 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{78125} \]
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Rubi [A] time = 0.0541112, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {97, 153, 147, 50, 63, 206} \[ -\frac{(1-2 x)^{3/2} (3 x+2)^5}{5 (5 x+3)}+\frac{39}{275} (1-2 x)^{3/2} (3 x+2)^4+\frac{38 (1-2 x)^{3/2} (3 x+2)^3}{4125}-\frac{4016 (1-2 x)^{3/2} (3 x+2)^2}{48125}-\frac{2 (1-2 x)^{3/2} (204777 x+298462)}{515625}+\frac{324 \sqrt{1-2 x}}{78125}-\frac{324 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{78125} \]
Antiderivative was successfully verified.
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Rule 97
Rule 153
Rule 147
Rule 50
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{3/2} (2+3 x)^5}{(3+5 x)^2} \, dx &=-\frac{(1-2 x)^{3/2} (2+3 x)^5}{5 (3+5 x)}+\frac{1}{5} \int \frac{(9-39 x) \sqrt{1-2 x} (2+3 x)^4}{3+5 x} \, dx\\ &=\frac{39}{275} (1-2 x)^{3/2} (2+3 x)^4-\frac{(1-2 x)^{3/2} (2+3 x)^5}{5 (3+5 x)}-\frac{1}{275} \int \frac{\sqrt{1-2 x} (2+3 x)^3 (-288+114 x)}{3+5 x} \, dx\\ &=\frac{38 (1-2 x)^{3/2} (2+3 x)^3}{4125}+\frac{39}{275} (1-2 x)^{3/2} (2+3 x)^4-\frac{(1-2 x)^{3/2} (2+3 x)^5}{5 (3+5 x)}+\frac{\int \frac{\sqrt{1-2 x} (2+3 x)^2 (24894+36144 x)}{3+5 x} \, dx}{12375}\\ &=-\frac{4016 (1-2 x)^{3/2} (2+3 x)^2}{48125}+\frac{38 (1-2 x)^{3/2} (2+3 x)^3}{4125}+\frac{39}{275} (1-2 x)^{3/2} (2+3 x)^4-\frac{(1-2 x)^{3/2} (2+3 x)^5}{5 (3+5 x)}-\frac{\int \frac{(-1742580-2866878 x) \sqrt{1-2 x} (2+3 x)}{3+5 x} \, dx}{433125}\\ &=-\frac{4016 (1-2 x)^{3/2} (2+3 x)^2}{48125}+\frac{38 (1-2 x)^{3/2} (2+3 x)^3}{4125}+\frac{39}{275} (1-2 x)^{3/2} (2+3 x)^4-\frac{(1-2 x)^{3/2} (2+3 x)^5}{5 (3+5 x)}-\frac{2 (1-2 x)^{3/2} (298462+204777 x)}{515625}+\frac{162 \int \frac{\sqrt{1-2 x}}{3+5 x} \, dx}{15625}\\ &=\frac{324 \sqrt{1-2 x}}{78125}-\frac{4016 (1-2 x)^{3/2} (2+3 x)^2}{48125}+\frac{38 (1-2 x)^{3/2} (2+3 x)^3}{4125}+\frac{39}{275} (1-2 x)^{3/2} (2+3 x)^4-\frac{(1-2 x)^{3/2} (2+3 x)^5}{5 (3+5 x)}-\frac{2 (1-2 x)^{3/2} (298462+204777 x)}{515625}+\frac{1782 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx}{78125}\\ &=\frac{324 \sqrt{1-2 x}}{78125}-\frac{4016 (1-2 x)^{3/2} (2+3 x)^2}{48125}+\frac{38 (1-2 x)^{3/2} (2+3 x)^3}{4125}+\frac{39}{275} (1-2 x)^{3/2} (2+3 x)^4-\frac{(1-2 x)^{3/2} (2+3 x)^5}{5 (3+5 x)}-\frac{2 (1-2 x)^{3/2} (298462+204777 x)}{515625}-\frac{1782 \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{78125}\\ &=\frac{324 \sqrt{1-2 x}}{78125}-\frac{4016 (1-2 x)^{3/2} (2+3 x)^2}{48125}+\frac{38 (1-2 x)^{3/2} (2+3 x)^3}{4125}+\frac{39}{275} (1-2 x)^{3/2} (2+3 x)^4-\frac{(1-2 x)^{3/2} (2+3 x)^5}{5 (3+5 x)}-\frac{2 (1-2 x)^{3/2} (298462+204777 x)}{515625}-\frac{324 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{78125}\\ \end{align*}
Mathematica [A] time = 0.0867688, size = 78, normalized size = 0.53 \[ \frac{-\frac{5 \sqrt{1-2 x} \left (106312500 x^6+270112500 x^5+181738125 x^4-76760550 x^3-135193430 x^2-2532130 x+23061496\right )}{5 x+3}-24948 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{30078125} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 90, normalized size = 0.6 \begin{align*}{\frac{243}{2200} \left ( 1-2\,x \right ) ^{{\frac{11}{2}}}}-{\frac{981}{1000} \left ( 1-2\,x \right ) ^{{\frac{9}{2}}}}+{\frac{107109}{35000} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}-{\frac{434043}{125000} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{2}{3125} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{326}{78125}\sqrt{1-2\,x}}+{\frac{22}{390625}\sqrt{1-2\,x} \left ( -2\,x-{\frac{6}{5}} \right ) ^{-1}}-{\frac{324\,\sqrt{55}}{390625}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.53019, size = 144, normalized size = 0.97 \begin{align*} \frac{243}{2200} \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} - \frac{981}{1000} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} + \frac{107109}{35000} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{434043}{125000} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{2}{3125} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{162}{390625} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{326}{78125} \, \sqrt{-2 \, x + 1} - \frac{11 \, \sqrt{-2 \, x + 1}}{78125 \,{\left (5 \, x + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55599, size = 320, normalized size = 2.16 \begin{align*} \frac{12474 \, \sqrt{11} \sqrt{5}{\left (5 \, x + 3\right )} \log \left (\frac{\sqrt{11} \sqrt{5} \sqrt{-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) - 5 \,{\left (106312500 \, x^{6} + 270112500 \, x^{5} + 181738125 \, x^{4} - 76760550 \, x^{3} - 135193430 \, x^{2} - 2532130 \, x + 23061496\right )} \sqrt{-2 \, x + 1}}{30078125 \,{\left (5 \, x + 3\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 = 2.24396, size = 186, normalized size = 1.26 \begin{align*} -\frac{243}{2200} \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} - \frac{981}{1000} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} - \frac{107109}{35000} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{434043}{125000} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{2}{3125} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{162}{390625} \, \sqrt{55} \log \left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{326}{78125} \, \sqrt{-2 \, x + 1} - \frac{11 \, \sqrt{-2 \, x + 1}}{78125 \,{\left (5 \, x + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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