Optimal. Leaf size=156 \[ -\frac{824 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{5929 \sqrt{33}}-\frac{41570 \sqrt{1-2 x} \sqrt{3 x+2}}{195657 \sqrt{5 x+3}}+\frac{824 \sqrt{3 x+2}}{17787 \sqrt{1-2 x} \sqrt{5 x+3}}+\frac{4 \sqrt{3 x+2}}{231 (1-2 x)^{3/2} \sqrt{5 x+3}}+\frac{8314 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5929 \sqrt{33}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0540794, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {104, 152, 158, 113, 119} \[ -\frac{41570 \sqrt{1-2 x} \sqrt{3 x+2}}{195657 \sqrt{5 x+3}}+\frac{824 \sqrt{3 x+2}}{17787 \sqrt{1-2 x} \sqrt{5 x+3}}+\frac{4 \sqrt{3 x+2}}{231 (1-2 x)^{3/2} \sqrt{5 x+3}}-\frac{824 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5929 \sqrt{33}}+\frac{8314 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5929 \sqrt{33}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 104
Rule 152
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{1}{(1-2 x)^{5/2} \sqrt{2+3 x} (3+5 x)^{3/2}} \, dx &=\frac{4 \sqrt{2+3 x}}{231 (1-2 x)^{3/2} \sqrt{3+5 x}}-\frac{2}{231} \int \frac{-\frac{161}{2}-45 x}{(1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{3/2}} \, dx\\ &=\frac{4 \sqrt{2+3 x}}{231 (1-2 x)^{3/2} \sqrt{3+5 x}}+\frac{824 \sqrt{2+3 x}}{17787 \sqrt{1-2 x} \sqrt{3+5 x}}+\frac{4 \int \frac{\frac{7865}{4}+1545 x}{\sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}} \, dx}{17787}\\ &=\frac{4 \sqrt{2+3 x}}{231 (1-2 x)^{3/2} \sqrt{3+5 x}}+\frac{824 \sqrt{2+3 x}}{17787 \sqrt{1-2 x} \sqrt{3+5 x}}-\frac{41570 \sqrt{1-2 x} \sqrt{2+3 x}}{195657 \sqrt{3+5 x}}-\frac{8 \int \frac{\frac{30615}{4}+\frac{62355 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{195657}\\ &=\frac{4 \sqrt{2+3 x}}{231 (1-2 x)^{3/2} \sqrt{3+5 x}}+\frac{824 \sqrt{2+3 x}}{17787 \sqrt{1-2 x} \sqrt{3+5 x}}-\frac{41570 \sqrt{1-2 x} \sqrt{2+3 x}}{195657 \sqrt{3+5 x}}+\frac{412 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{5929}-\frac{8314 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{65219}\\ &=\frac{4 \sqrt{2+3 x}}{231 (1-2 x)^{3/2} \sqrt{3+5 x}}+\frac{824 \sqrt{2+3 x}}{17787 \sqrt{1-2 x} \sqrt{3+5 x}}-\frac{41570 \sqrt{1-2 x} \sqrt{2+3 x}}{195657 \sqrt{3+5 x}}+\frac{8314 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5929 \sqrt{33}}-\frac{824 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5929 \sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.147971, size = 98, normalized size = 0.63 \[ \frac{2 \left (\sqrt{2} \left (10955 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )-4157 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )+\frac{\sqrt{3 x+2} \left (-83140 x^2+74076 x-14559\right )}{(1-2 x)^{3/2} \sqrt{5 x+3}}\right )}{195657} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.024, size = 228, normalized size = 1.5 \begin{align*} -{\frac{2}{195657\, \left ( 2\,x-1 \right ) ^{2} \left ( 15\,{x}^{2}+19\,x+6 \right ) }\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 21910\,\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}-8314\,\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}-10955\,\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 ) +4157\,\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 ) +249420\,{x}^{3}-55948\,{x}^{2}-104475\,x+29118 \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{1}{{\left (5 \, x + 3\right )}^{\frac{3}{2}} \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}}{600 \, x^{6} + 220 \, x^{5} - 534 \, x^{4} - 135 \, x^{3} + 166 \, x^{2} + 21 \, x - 18}, 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{1}{{\left (5 \, x + 3\right )}^{\frac{3}{2}} \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]