Optimal. Leaf size=125 \[ -\frac{\text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{7 \sqrt{33}}-\frac{68 \sqrt{3 x+2} \sqrt{5 x+3}}{231 \sqrt{1-2 x}}+\frac{\sqrt{3 x+2} \sqrt{5 x+3}}{3 (1-2 x)^{3/2}}-\frac{34 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{7 \sqrt{33}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0403479, antiderivative size = 125, 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 = {97, 152, 158, 113, 119} \[ -\frac{68 \sqrt{3 x+2} \sqrt{5 x+3}}{231 \sqrt{1-2 x}}+\frac{\sqrt{3 x+2} \sqrt{5 x+3}}{3 (1-2 x)^{3/2}}-\frac{F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{7 \sqrt{33}}-\frac{34 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{7 \sqrt{33}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 97
Rule 152
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{\sqrt{2+3 x} \sqrt{3+5 x}}{(1-2 x)^{5/2}} \, dx &=\frac{\sqrt{2+3 x} \sqrt{3+5 x}}{3 (1-2 x)^{3/2}}-\frac{1}{3} \int \frac{\frac{19}{2}+15 x}{(1-2 x)^{3/2} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx\\ &=\frac{\sqrt{2+3 x} \sqrt{3+5 x}}{3 (1-2 x)^{3/2}}-\frac{68 \sqrt{2+3 x} \sqrt{3+5 x}}{231 \sqrt{1-2 x}}+\frac{2}{231} \int \frac{\frac{645}{4}+255 x}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx\\ &=\frac{\sqrt{2+3 x} \sqrt{3+5 x}}{3 (1-2 x)^{3/2}}-\frac{68 \sqrt{2+3 x} \sqrt{3+5 x}}{231 \sqrt{1-2 x}}+\frac{1}{14} \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx+\frac{34}{77} \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx\\ &=\frac{\sqrt{2+3 x} \sqrt{3+5 x}}{3 (1-2 x)^{3/2}}-\frac{68 \sqrt{2+3 x} \sqrt{3+5 x}}{231 \sqrt{1-2 x}}-\frac{34 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{7 \sqrt{33}}-\frac{F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{7 \sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.14535, size = 115, normalized size = 0.92 \[ \frac{35 \sqrt{2-4 x} (2 x-1) \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+2 \sqrt{3 x+2} \sqrt{5 x+3} (136 x+9)-68 \sqrt{2-4 x} (2 x-1) E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{462 (1-2 x)^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.016, size = 228, normalized size = 1.8 \begin{align*}{\frac{1}{462\, \left ( 2\,x-1 \right ) ^{2} \left ( 15\,{x}^{2}+19\,x+6 \right ) } \left ( 70\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-136\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-35\,\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 ) +68\,\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 ) +4080\,{x}^{3}+5438\,{x}^{2}+1974\,x+108 \right ) \sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x}} \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{\sqrt{5 \, x + 3} \sqrt{3 \, x + 2}}{{\left (-2 \, x + 1\right )}^{\frac{5}{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{\sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{3 x + 2} \sqrt{5 x + 3}}{\left (1 - 2 x\right )^{\frac{5}{2}}}\, dx \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{\sqrt{5 \, x + 3} \sqrt{3 \, x + 2}}{{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]