Optimal. Leaf size=127 \[ \frac{212 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{125 \sqrt{33}}-\frac{2 \sqrt{3 x+2} (1-2 x)^{3/2}}{15 (5 x+3)^{3/2}}-\frac{18 \sqrt{3 x+2} \sqrt{1-2 x}}{25 \sqrt{5 x+3}}+\frac{38}{125} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0395267, antiderivative size = 127, 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, 150, 158, 113, 119} \[ -\frac{2 \sqrt{3 x+2} (1-2 x)^{3/2}}{15 (5 x+3)^{3/2}}-\frac{18 \sqrt{3 x+2} \sqrt{1-2 x}}{25 \sqrt{5 x+3}}+\frac{212 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{125 \sqrt{33}}+\frac{38}{125} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 97
Rule 150
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{3/2} \sqrt{2+3 x}}{(3+5 x)^{5/2}} \, dx &=-\frac{2 (1-2 x)^{3/2} \sqrt{2+3 x}}{15 (3+5 x)^{3/2}}+\frac{2}{15} \int \frac{\left (-\frac{9}{2}-12 x\right ) \sqrt{1-2 x}}{\sqrt{2+3 x} (3+5 x)^{3/2}} \, dx\\ &=-\frac{2 (1-2 x)^{3/2} \sqrt{2+3 x}}{15 (3+5 x)^{3/2}}-\frac{18 \sqrt{1-2 x} \sqrt{2+3 x}}{25 \sqrt{3+5 x}}+\frac{4}{75} \int \frac{-33-\frac{57 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{2 (1-2 x)^{3/2} \sqrt{2+3 x}}{15 (3+5 x)^{3/2}}-\frac{18 \sqrt{1-2 x} \sqrt{2+3 x}}{25 \sqrt{3+5 x}}-\frac{38}{125} \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx-\frac{106}{125} \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{2 (1-2 x)^{3/2} \sqrt{2+3 x}}{15 (3+5 x)^{3/2}}-\frac{18 \sqrt{1-2 x} \sqrt{2+3 x}}{25 \sqrt{3+5 x}}+\frac{38}{125} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{212 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{125 \sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.205047, size = 97, normalized size = 0.76 \[ \frac{2}{375} \left (-140 \sqrt{2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )-\frac{5 \sqrt{1-2 x} \sqrt{3 x+2} (125 x+86)}{(5 x+3)^{3/2}}-19 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.019, size = 219, normalized size = 1.7 \begin{align*}{\frac{2}{2250\,{x}^{2}+375\,x-750} \left ( 700\,\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}+95\,\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}+420\,\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 ) +57\,\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 ) -3750\,{x}^{3}-3205\,{x}^{2}+820\,x+860 \right ) \sqrt{2+3\,x}\sqrt{1-2\,x} \left ( 3+5\,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{\sqrt{3 \, x + 2}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}{{\left (5 \, x + 3\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}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}{125 \, x^{3} + 225 \, x^{2} + 135 \, x + 27}, 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{\sqrt{3 \, x + 2}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}{{\left (5 \, x + 3\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]