Optimal. Leaf size=160 \[ \frac{494}{675} \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )-\frac{8}{15} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}-\frac{2 (1-2 x)^{3/2} (5 x+3)^{3/2}}{3 \sqrt{3 x+2}}+\frac{494}{135} \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}-\frac{2209}{675} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0531589, 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 = {97, 154, 158, 113, 119} \[ -\frac{8}{15} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}-\frac{2 (1-2 x)^{3/2} (5 x+3)^{3/2}}{3 \sqrt{3 x+2}}+\frac{494}{135} \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}+\frac{494}{675} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{2209}{675} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 97
Rule 154
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^{3/2}} \, dx &=-\frac{2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{3 \sqrt{2+3 x}}+\frac{2}{3} \int \frac{\left (-\frac{3}{2}-30 x\right ) \sqrt{1-2 x} \sqrt{3+5 x}}{\sqrt{2+3 x}} \, dx\\ &=-\frac{2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{3 \sqrt{2+3 x}}-\frac{8}{15} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}+\frac{4}{225} \int \frac{\left (\frac{1395}{4}-\frac{3705 x}{2}\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx\\ &=\frac{494}{135} \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}-\frac{2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{3 \sqrt{2+3 x}}-\frac{8}{15} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}-\frac{4 \int \frac{-\frac{5865}{2}-\frac{33135 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{2025}\\ &=\frac{494}{135} \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}-\frac{2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{3 \sqrt{2+3 x}}-\frac{8}{15} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}+\frac{2209}{675} \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx-\frac{2717}{675} \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx\\ &=\frac{494}{135} \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}-\frac{2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{3 \sqrt{2+3 x}}-\frac{8}{15} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}-\frac{2209}{675} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{494}{675} \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 = 0.198173, size = 102, normalized size = 0.64 \[ \frac{-10360 \sqrt{2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )-\frac{30 \sqrt{1-2 x} \sqrt{5 x+3} \left (90 x^2-102 x-143\right )}{\sqrt{3 x+2}}+2209 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{2025} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.016, size = 145, normalized size = 0.9 \begin{align*}{\frac{1}{60750\,{x}^{3}+46575\,{x}^{2}-14175\,x-12150}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 10360\,\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 ) -2209\,\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 ) -27000\,{x}^{4}+27900\,{x}^{3}+54060\,{x}^{2}-4890\,x-12870 \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{3}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}{{\left (3 \, x + 2\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 (10 \, x^{2} + x - 3\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{9 \, x^{2} + 12 \, x + 4}, 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{3}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}{{\left (3 \, x + 2\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]