Optimal. Leaf size=158 \[ -\frac{17}{343} \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )+\frac{11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} \sqrt{3 x+2}}+\frac{438 \sqrt{1-2 x} \sqrt{5 x+3}}{343 \sqrt{3 x+2}}-\frac{143 \sqrt{5 x+3}}{49 \sqrt{1-2 x} \sqrt{3 x+2}}-\frac{146}{343} \sqrt{33} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0506717, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {98, 150, 152, 158, 113, 119} \[ \frac{11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} \sqrt{3 x+2}}+\frac{438 \sqrt{1-2 x} \sqrt{5 x+3}}{343 \sqrt{3 x+2}}-\frac{143 \sqrt{5 x+3}}{49 \sqrt{1-2 x} \sqrt{3 x+2}}-\frac{17}{343} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{146}{343} \sqrt{33} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 98
Rule 150
Rule 152
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)^{3/2}} \, dx &=\frac{11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} \sqrt{2+3 x}}-\frac{1}{21} \int \frac{\sqrt{3+5 x} \left (\frac{249}{2}+180 x\right )}{(1-2 x)^{3/2} (2+3 x)^{3/2}} \, dx\\ &=-\frac{143 \sqrt{3+5 x}}{49 \sqrt{1-2 x} \sqrt{2+3 x}}+\frac{11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} \sqrt{2+3 x}}-\frac{1}{147} \int \frac{-174+\frac{135 x}{2}}{\sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}} \, dx\\ &=-\frac{143 \sqrt{3+5 x}}{49 \sqrt{1-2 x} \sqrt{2+3 x}}+\frac{438 \sqrt{1-2 x} \sqrt{3+5 x}}{343 \sqrt{2+3 x}}+\frac{11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} \sqrt{2+3 x}}-\frac{2 \int \frac{-\frac{8445}{4}-3285 x}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{1029}\\ &=-\frac{143 \sqrt{3+5 x}}{49 \sqrt{1-2 x} \sqrt{2+3 x}}+\frac{438 \sqrt{1-2 x} \sqrt{3+5 x}}{343 \sqrt{2+3 x}}+\frac{11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} \sqrt{2+3 x}}+\frac{187}{686} \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx+\frac{438}{343} \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx\\ &=-\frac{143 \sqrt{3+5 x}}{49 \sqrt{1-2 x} \sqrt{2+3 x}}+\frac{438 \sqrt{1-2 x} \sqrt{3+5 x}}{343 \sqrt{2+3 x}}+\frac{11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} \sqrt{2+3 x}}-\frac{146}{343} \sqrt{33} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{17}{343} \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 = 0.141324, size = 102, normalized size = 0.65 \[ \frac{-315 \sqrt{2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+\frac{2 \sqrt{5 x+3} \left (5256 x^2+3445 x-72\right )}{(1-2 x)^{3/2} \sqrt{3 x+2}}+876 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{2058} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.023, size = 228, normalized size = 1.4 \begin{align*}{\frac{1}{2058\, \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 ( 630\,\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}-1752\,\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}-315\,\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 ) +876\,\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 ) +52560\,{x}^{3}+65986\,{x}^{2}+19950\,x-432 \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{5}{2}}}{{\left (3 \, x + 2\right )}^{\frac{3}{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{{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{72 \, x^{5} - 12 \, x^{4} - 58 \, x^{3} + 15 \, 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{5}{2}}}{{\left (3 \, x + 2\right )}^{\frac{3}{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]