Optimal. Leaf size=191 \[ -\frac{43214 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{8505}+\frac{362 \sqrt{1-2 x} (5 x+3)^{5/2}}{135 (3 x+2)^{3/2}}-\frac{2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{15 (3 x+2)^{5/2}}+\frac{9808 \sqrt{1-2 x} (5 x+3)^{3/2}}{945 \sqrt{3 x+2}}-\frac{43214 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{1701}+\frac{116854 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{8505} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0639246, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {97, 150, 154, 158, 113, 119} \[ \frac{362 \sqrt{1-2 x} (5 x+3)^{5/2}}{135 (3 x+2)^{3/2}}-\frac{2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{15 (3 x+2)^{5/2}}+\frac{9808 \sqrt{1-2 x} (5 x+3)^{3/2}}{945 \sqrt{3 x+2}}-\frac{43214 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{1701}-\frac{43214 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{8505}+\frac{116854 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{8505} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 97
Rule 150
Rule 154
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^{7/2}} \, dx &=-\frac{2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{15 (2+3 x)^{5/2}}+\frac{2}{15} \int \frac{\left (\frac{7}{2}-40 x\right ) \sqrt{1-2 x} (3+5 x)^{3/2}}{(2+3 x)^{5/2}} \, dx\\ &=-\frac{2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{15 (2+3 x)^{5/2}}+\frac{362 \sqrt{1-2 x} (3+5 x)^{5/2}}{135 (2+3 x)^{3/2}}-\frac{4}{135} \int \frac{\left (241-\frac{2955 x}{2}\right ) (3+5 x)^{3/2}}{\sqrt{1-2 x} (2+3 x)^{3/2}} \, dx\\ &=\frac{9808 \sqrt{1-2 x} (3+5 x)^{3/2}}{945 \sqrt{2+3 x}}-\frac{2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{15 (2+3 x)^{5/2}}+\frac{362 \sqrt{1-2 x} (3+5 x)^{5/2}}{135 (2+3 x)^{3/2}}-\frac{8 \int \frac{\left (\frac{48285}{4}-\frac{324105 x}{4}\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{2835}\\ &=-\frac{43214 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{1701}+\frac{9808 \sqrt{1-2 x} (3+5 x)^{3/2}}{945 \sqrt{2+3 x}}-\frac{2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{15 (2+3 x)^{5/2}}+\frac{362 \sqrt{1-2 x} (3+5 x)^{5/2}}{135 (2+3 x)^{3/2}}+\frac{8 \int \frac{-\frac{338655}{8}-\frac{876405 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{25515}\\ &=-\frac{43214 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{1701}+\frac{9808 \sqrt{1-2 x} (3+5 x)^{3/2}}{945 \sqrt{2+3 x}}-\frac{2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{15 (2+3 x)^{5/2}}+\frac{362 \sqrt{1-2 x} (3+5 x)^{5/2}}{135 (2+3 x)^{3/2}}-\frac{116854 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{8505}+\frac{237677 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{8505}\\ &=-\frac{43214 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{1701}+\frac{9808 \sqrt{1-2 x} (3+5 x)^{3/2}}{945 \sqrt{2+3 x}}-\frac{2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{15 (2+3 x)^{5/2}}+\frac{362 \sqrt{1-2 x} (3+5 x)^{5/2}}{135 (2+3 x)^{3/2}}+\frac{116854 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{8505}-\frac{43214 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{8505}\\ \end{align*}
Mathematica [A] time = 0.245227, size = 104, normalized size = 0.54 \[ \frac{\sqrt{2} \left (829885 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )-116854 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )-\frac{6 \sqrt{1-2 x} \sqrt{5 x+3} \left (47250 x^3+377793 x^2+432387 x+134497\right )}{(3 x+2)^{5/2}}}{25515} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.019, size = 319, normalized size = 1.7 \begin{align*} -{\frac{1}{255150\,{x}^{2}+25515\,x-76545} \left ( 7468965\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-1051686\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+9958620\,\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}-1402248\,\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}+3319540\,\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 ) -467416\,\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 ) +2835000\,{x}^{5}+22951080\,{x}^{4}+27359478\,{x}^{3}+3863868\,{x}^{2}-6975984\,x-2420946 \right ) \sqrt{3+5\,x}\sqrt{1-2\,x} \left ( 2+3\,x \right ) ^{-{\frac{5}{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{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}{{\left (3 \, x + 2\right )}^{\frac{7}{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 (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16}, 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 (-2 \, x + 1\right )}^{\frac{3}{2}}}{{\left (3 \, x + 2\right )}^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]