Optimal. Leaf size=121 \[ -\frac{243 (1-2 x)^{13/2}}{1040}+\frac{5751 (1-2 x)^{11/2}}{2200}-\frac{5673}{500} (1-2 x)^{9/2}+\frac{806121 (1-2 x)^{7/2}}{35000}-\frac{4774713 (1-2 x)^{5/2}}{250000}+\frac{2 (1-2 x)^{3/2}}{46875}+\frac{22 \sqrt{1-2 x}}{78125}-\frac{22 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{78125} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0370955, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {88, 50, 63, 206} \[ -\frac{243 (1-2 x)^{13/2}}{1040}+\frac{5751 (1-2 x)^{11/2}}{2200}-\frac{5673}{500} (1-2 x)^{9/2}+\frac{806121 (1-2 x)^{7/2}}{35000}-\frac{4774713 (1-2 x)^{5/2}}{250000}+\frac{2 (1-2 x)^{3/2}}{46875}+\frac{22 \sqrt{1-2 x}}{78125}-\frac{22 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{78125} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 88
Rule 50
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{3/2} (2+3 x)^5}{3+5 x} \, dx &=\int \left (\frac{4774713 (1-2 x)^{3/2}}{50000}-\frac{806121 (1-2 x)^{5/2}}{5000}+\frac{51057}{500} (1-2 x)^{7/2}-\frac{5751}{200} (1-2 x)^{9/2}+\frac{243}{80} (1-2 x)^{11/2}+\frac{(1-2 x)^{3/2}}{3125 (3+5 x)}\right ) \, dx\\ &=-\frac{4774713 (1-2 x)^{5/2}}{250000}+\frac{806121 (1-2 x)^{7/2}}{35000}-\frac{5673}{500} (1-2 x)^{9/2}+\frac{5751 (1-2 x)^{11/2}}{2200}-\frac{243 (1-2 x)^{13/2}}{1040}+\frac{\int \frac{(1-2 x)^{3/2}}{3+5 x} \, dx}{3125}\\ &=\frac{2 (1-2 x)^{3/2}}{46875}-\frac{4774713 (1-2 x)^{5/2}}{250000}+\frac{806121 (1-2 x)^{7/2}}{35000}-\frac{5673}{500} (1-2 x)^{9/2}+\frac{5751 (1-2 x)^{11/2}}{2200}-\frac{243 (1-2 x)^{13/2}}{1040}+\frac{11 \int \frac{\sqrt{1-2 x}}{3+5 x} \, dx}{15625}\\ &=\frac{22 \sqrt{1-2 x}}{78125}+\frac{2 (1-2 x)^{3/2}}{46875}-\frac{4774713 (1-2 x)^{5/2}}{250000}+\frac{806121 (1-2 x)^{7/2}}{35000}-\frac{5673}{500} (1-2 x)^{9/2}+\frac{5751 (1-2 x)^{11/2}}{2200}-\frac{243 (1-2 x)^{13/2}}{1040}+\frac{121 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx}{78125}\\ &=\frac{22 \sqrt{1-2 x}}{78125}+\frac{2 (1-2 x)^{3/2}}{46875}-\frac{4774713 (1-2 x)^{5/2}}{250000}+\frac{806121 (1-2 x)^{7/2}}{35000}-\frac{5673}{500} (1-2 x)^{9/2}+\frac{5751 (1-2 x)^{11/2}}{2200}-\frac{243 (1-2 x)^{13/2}}{1040}-\frac{121 \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{78125}\\ &=\frac{22 \sqrt{1-2 x}}{78125}+\frac{2 (1-2 x)^{3/2}}{46875}-\frac{4774713 (1-2 x)^{5/2}}{250000}+\frac{806121 (1-2 x)^{7/2}}{35000}-\frac{5673}{500} (1-2 x)^{9/2}+\frac{5751 (1-2 x)^{11/2}}{2200}-\frac{243 (1-2 x)^{13/2}}{1040}-\frac{22 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{78125}\\ \end{align*}
Mathematica [A] time = 0.0653369, size = 71, normalized size = 0.59 \[ \frac{-5 \sqrt{1-2 x} \left (3508312500 x^6+9100350000 x^5+6683000625 x^4-1659418875 x^3-4276774170 x^2-1321809935 x+1180568944\right )-66066 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1173046875} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.007, size = 83, normalized size = 0.7 \begin{align*}{\frac{2}{46875} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{4774713}{250000} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{806121}{35000} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}-{\frac{5673}{500} \left ( 1-2\,x \right ) ^{{\frac{9}{2}}}}+{\frac{5751}{2200} \left ( 1-2\,x \right ) ^{{\frac{11}{2}}}}-{\frac{243}{1040} \left ( 1-2\,x \right ) ^{{\frac{13}{2}}}}-{\frac{22\,\sqrt{55}}{390625}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) }+{\frac{22}{78125}\sqrt{1-2\,x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 2.03103, size = 135, normalized size = 1.12 \begin{align*} -\frac{243}{1040} \,{\left (-2 \, x + 1\right )}^{\frac{13}{2}} + \frac{5751}{2200} \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} - \frac{5673}{500} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} + \frac{806121}{35000} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{4774713}{250000} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{2}{46875} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{11}{390625} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{22}{78125} \, \sqrt{-2 \, x + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.29503, size = 309, normalized size = 2.55 \begin{align*} \frac{11}{390625} \, \sqrt{11} \sqrt{5} \log \left (\frac{\sqrt{11} \sqrt{5} \sqrt{-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) - \frac{1}{234609375} \,{\left (3508312500 \, x^{6} + 9100350000 \, x^{5} + 6683000625 \, x^{4} - 1659418875 \, x^{3} - 4276774170 \, x^{2} - 1321809935 \, x + 1180568944\right )} \sqrt{-2 \, x + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 74.9696, size = 150, normalized size = 1.24 \begin{align*} - \frac{243 \left (1 - 2 x\right )^{\frac{13}{2}}}{1040} + \frac{5751 \left (1 - 2 x\right )^{\frac{11}{2}}}{2200} - \frac{5673 \left (1 - 2 x\right )^{\frac{9}{2}}}{500} + \frac{806121 \left (1 - 2 x\right )^{\frac{7}{2}}}{35000} - \frac{4774713 \left (1 - 2 x\right )^{\frac{5}{2}}}{250000} + \frac{2 \left (1 - 2 x\right )^{\frac{3}{2}}}{46875} + \frac{22 \sqrt{1 - 2 x}}{78125} + \frac{242 \left (\begin{cases} - \frac{\sqrt{55} \operatorname{acoth}{\left (\frac{\sqrt{55} \sqrt{1 - 2 x}}{11} \right )}}{55} & \text{for}\: 2 x - 1 < - \frac{11}{5} \\- \frac{\sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{1 - 2 x}}{11} \right )}}{55} & \text{for}\: 2 x - 1 > - \frac{11}{5} \end{cases}\right )}{78125} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 2.28806, size = 186, normalized size = 1.54 \begin{align*} -\frac{243}{1040} \,{\left (2 \, x - 1\right )}^{6} \sqrt{-2 \, x + 1} - \frac{5751}{2200} \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} - \frac{5673}{500} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} - \frac{806121}{35000} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{4774713}{250000} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{2}{46875} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{11}{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{22}{78125} \, \sqrt{-2 \, x + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]