Optimal. Leaf size=129 \[ \frac{67}{63} \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )+\frac{11 \sqrt{3 x+2} (5 x+3)^{3/2}}{7 \sqrt{1-2 x}}+\frac{335}{63} \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}+\frac{4451}{126} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0380897, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {98, 154, 158, 113, 119} \[ \frac{11 \sqrt{3 x+2} (5 x+3)^{3/2}}{7 \sqrt{1-2 x}}+\frac{335}{63} \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}+\frac{67}{63} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{4451}{126} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 98
Rule 154
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(3+5 x)^{5/2}}{(1-2 x)^{3/2} \sqrt{2+3 x}} \, dx &=\frac{11 \sqrt{2+3 x} (3+5 x)^{3/2}}{7 \sqrt{1-2 x}}-\frac{1}{7} \int \frac{\sqrt{3+5 x} \left (\frac{435}{2}+335 x\right )}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx\\ &=\frac{335}{63} \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}+\frac{11 \sqrt{2+3 x} (3+5 x)^{3/2}}{7 \sqrt{1-2 x}}+\frac{1}{63} \int \frac{-7045-\frac{22255 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx\\ &=\frac{335}{63} \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}+\frac{11 \sqrt{2+3 x} (3+5 x)^{3/2}}{7 \sqrt{1-2 x}}-\frac{737}{126} \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx-\frac{4451}{126} \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx\\ &=\frac{335}{63} \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}+\frac{11 \sqrt{2+3 x} (3+5 x)^{3/2}}{7 \sqrt{1-2 x}}+\frac{4451}{126} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{67}{63} \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 = 0.0897087, size = 105, normalized size = 0.81 \[ \frac{2240 \sqrt{2-4 x} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+6 \sqrt{3 x+2} \sqrt{5 x+3} (632-175 x)-4451 \sqrt{2-4 x} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{378 \sqrt{1-2 x}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.017, size = 140, normalized size = 1.1 \begin{align*} -{\frac{1}{11340\,{x}^{3}+8694\,{x}^{2}-2646\,x-2268}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 2240\,\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 ) -4451\,\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 ) -15750\,{x}^{3}+36930\,{x}^{2}+65748\,x+22752 \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}}}{\sqrt{3 \, x + 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 (25 \, x^{2} + 30 \, x + 9\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2}, 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}}}{\sqrt{3 \, x + 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]