Optimal. Leaf size=188 \[ \frac{3683}{210} \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )+\frac{(3 x+2)^{3/2} (5 x+3)^{5/2}}{\sqrt{1-2 x}}+\frac{12}{7} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}+\frac{167}{14} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}+\frac{3683}{42} \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}+\frac{244879}{420} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0610606, antiderivative size = 188, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {97, 154, 158, 113, 119} \[ \frac{(3 x+2)^{3/2} (5 x+3)^{5/2}}{\sqrt{1-2 x}}+\frac{12}{7} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}+\frac{167}{14} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}+\frac{3683}{42} \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}+\frac{3683}{210} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{244879}{420} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 97
Rule 154
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(2+3 x)^{3/2} (3+5 x)^{5/2}}{(1-2 x)^{3/2}} \, dx &=\frac{(2+3 x)^{3/2} (3+5 x)^{5/2}}{\sqrt{1-2 x}}-\int \frac{\sqrt{2+3 x} (3+5 x)^{3/2} \left (\frac{77}{2}+60 x\right )}{\sqrt{1-2 x}} \, dx\\ &=\frac{12}{7} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}+\frac{(2+3 x)^{3/2} (3+5 x)^{5/2}}{\sqrt{1-2 x}}+\frac{1}{35} \int \frac{\left (-4105-\frac{12525 x}{2}\right ) (3+5 x)^{3/2}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx\\ &=\frac{167}{14} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}+\frac{12}{7} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}+\frac{(2+3 x)^{3/2} (3+5 x)^{5/2}}{\sqrt{1-2 x}}-\frac{1}{525} \int \frac{\sqrt{3+5 x} \left (\frac{1077075}{4}+\frac{828675 x}{2}\right )}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx\\ &=\frac{3683}{42} \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}+\frac{167}{14} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}+\frac{12}{7} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}+\frac{(2+3 x)^{3/2} (3+5 x)^{5/2}}{\sqrt{1-2 x}}+\frac{\int \frac{-\frac{17440875}{2}-\frac{55097775 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{4725}\\ &=\frac{3683}{42} \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}+\frac{167}{14} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}+\frac{12}{7} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}+\frac{(2+3 x)^{3/2} (3+5 x)^{5/2}}{\sqrt{1-2 x}}-\frac{40513}{420} \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx-\frac{244879}{420} \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx\\ &=\frac{3683}{42} \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}+\frac{167}{14} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}+\frac{12}{7} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}+\frac{(2+3 x)^{3/2} (3+5 x)^{5/2}}{\sqrt{1-2 x}}+\frac{244879}{420} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{3683}{210} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )\\ \end{align*}
Mathematica [A] time = 0.212382, size = 115, normalized size = 0.61 \[ \frac{123340 \sqrt{2-4 x} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )-30 \sqrt{3 x+2} \sqrt{5 x+3} \left (450 x^3+1650 x^2+3349 x-6590\right )-244879 \sqrt{2-4 x} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{1260 \sqrt{1-2 x}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.016, size = 150, normalized size = 0.8 \begin{align*} -{\frac{1}{37800\,{x}^{3}+28980\,{x}^{2}-8820\,x-7560}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 123340\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) -244879\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) -202500\,{x}^{5}-999000\,{x}^{4}-2528550\,{x}^{3}+759570\,{x}^{2}+3153480\,x+1186200 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (3 \, x + 2\right )}^{\frac{3}{2}}}{{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{4 \, x^{2} - 4 \, x + 1}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (3 \, x + 2\right )}^{\frac{3}{2}}}{{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]