Optimal. Leaf size=160 \[ -\frac{68}{343} \sqrt{\frac{3}{11}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )+\frac{5256 \sqrt{1-2 x} \sqrt{5 x+3}}{3773 \sqrt{3 x+2}}+\frac{54 \sqrt{1-2 x} \sqrt{5 x+3}}{539 (3 x+2)^{3/2}}+\frac{4 \sqrt{5 x+3}}{77 \sqrt{1-2 x} (3 x+2)^{3/2}}-\frac{1752}{343} \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0518215, antiderivative size = 160, 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} \[ \frac{5256 \sqrt{1-2 x} \sqrt{5 x+3}}{3773 \sqrt{3 x+2}}+\frac{54 \sqrt{1-2 x} \sqrt{5 x+3}}{539 (3 x+2)^{3/2}}+\frac{4 \sqrt{5 x+3}}{77 \sqrt{1-2 x} (3 x+2)^{3/2}}-\frac{68}{343} \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{1752}{343} \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 104
Rule 152
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{1}{(1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt{3+5 x}} \, dx &=\frac{4 \sqrt{3+5 x}}{77 \sqrt{1-2 x} (2+3 x)^{3/2}}-\frac{2}{77} \int \frac{-\frac{87}{2}-45 x}{\sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}} \, dx\\ &=\frac{4 \sqrt{3+5 x}}{77 \sqrt{1-2 x} (2+3 x)^{3/2}}+\frac{54 \sqrt{1-2 x} \sqrt{3+5 x}}{539 (2+3 x)^{3/2}}-\frac{4 \int \frac{-522+\frac{405 x}{2}}{\sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}} \, dx}{1617}\\ &=\frac{4 \sqrt{3+5 x}}{77 \sqrt{1-2 x} (2+3 x)^{3/2}}+\frac{54 \sqrt{1-2 x} \sqrt{3+5 x}}{539 (2+3 x)^{3/2}}+\frac{5256 \sqrt{1-2 x} \sqrt{3+5 x}}{3773 \sqrt{2+3 x}}-\frac{8 \int \frac{-\frac{25335}{4}-9855 x}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{11319}\\ &=\frac{4 \sqrt{3+5 x}}{77 \sqrt{1-2 x} (2+3 x)^{3/2}}+\frac{54 \sqrt{1-2 x} \sqrt{3+5 x}}{539 (2+3 x)^{3/2}}+\frac{5256 \sqrt{1-2 x} \sqrt{3+5 x}}{3773 \sqrt{2+3 x}}+\frac{102}{343} \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx+\frac{5256 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{3773}\\ &=\frac{4 \sqrt{3+5 x}}{77 \sqrt{1-2 x} (2+3 x)^{3/2}}+\frac{54 \sqrt{1-2 x} \sqrt{3+5 x}}{539 (2+3 x)^{3/2}}+\frac{5256 \sqrt{1-2 x} \sqrt{3+5 x}}{3773 \sqrt{2+3 x}}-\frac{1752}{343} \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{68}{343} \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )\\ \end{align*}
Mathematica [A] time = 0.137747, size = 99, normalized size = 0.62 \[ \frac{2 \left (3 \sqrt{2} \left (292 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-105 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )\right )+\frac{\sqrt{5 x+3} \left (-15768 x^2-3006 x+5543\right )}{\sqrt{1-2 x} (3 x+2)^{3/2}}\right )}{3773} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.023, size = 219, normalized size = 1.4 \begin{align*}{\frac{2}{37730\,{x}^{2}+3773\,x-11319}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 945\,\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}-2628\,\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}+630\,\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 ) -1752\,\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 ) +78840\,{x}^{3}+62334\,{x}^{2}-18697\,x-16629 \right ) \left ( 2+3\,x \right ) ^{-{\frac{3}{2}}}} \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}{\sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{\frac{5}{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{\sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{540 \, x^{6} + 864 \, x^{5} + 99 \, x^{4} - 425 \, x^{3} - 154 \, x^{2} + 52 \, x + 24}, 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}{\sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{\frac{5}{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]