Optimal. Leaf size=191 \[ -\frac{110717 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{5315625}+\frac{2}{45} \sqrt{3 x+2} (5 x+3)^{3/2} (1-2 x)^{5/2}+\frac{326 \sqrt{3 x+2} (5 x+3)^{3/2} (1-2 x)^{3/2}}{4725}+\frac{10214 \sqrt{3 x+2} (5 x+3)^{3/2} \sqrt{1-2 x}}{118125}-\frac{110717 \sqrt{3 x+2} \sqrt{5 x+3} \sqrt{1-2 x}}{1063125}-\frac{6799613 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5315625} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0674677, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {101, 154, 158, 113, 119} \[ \frac{2}{45} \sqrt{3 x+2} (5 x+3)^{3/2} (1-2 x)^{5/2}+\frac{326 \sqrt{3 x+2} (5 x+3)^{3/2} (1-2 x)^{3/2}}{4725}+\frac{10214 \sqrt{3 x+2} (5 x+3)^{3/2} \sqrt{1-2 x}}{118125}-\frac{110717 \sqrt{3 x+2} \sqrt{5 x+3} \sqrt{1-2 x}}{1063125}-\frac{110717 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5315625}-\frac{6799613 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5315625} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 101
Rule 154
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int (1-2 x)^{5/2} \sqrt{2+3 x} \sqrt{3+5 x} \, dx &=\frac{2}{45} (1-2 x)^{5/2} \sqrt{2+3 x} (3+5 x)^{3/2}-\frac{2}{45} \int \frac{\left (-\frac{111}{2}-\frac{163 x}{2}\right ) (1-2 x)^{3/2} \sqrt{3+5 x}}{\sqrt{2+3 x}} \, dx\\ &=\frac{326 (1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{3/2}}{4725}+\frac{2}{45} (1-2 x)^{5/2} \sqrt{2+3 x} (3+5 x)^{3/2}-\frac{4 \int \frac{\left (-\frac{5583}{2}-\frac{15321 x}{4}\right ) \sqrt{1-2 x} \sqrt{3+5 x}}{\sqrt{2+3 x}} \, dx}{4725}\\ &=\frac{10214 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{118125}+\frac{326 (1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{3/2}}{4725}+\frac{2}{45} (1-2 x)^{5/2} \sqrt{2+3 x} (3+5 x)^{3/2}-\frac{8 \int \frac{\left (-\frac{423783}{8}-\frac{332151 x}{8}\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{354375}\\ &=-\frac{110717 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{1063125}+\frac{10214 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{118125}+\frac{326 (1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{3/2}}{4725}+\frac{2}{45} (1-2 x)^{5/2} \sqrt{2+3 x} (3+5 x)^{3/2}+\frac{8 \int \frac{\frac{25209339}{16}+\frac{20398839 x}{8}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{3189375}\\ &=-\frac{110717 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{1063125}+\frac{10214 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{118125}+\frac{326 (1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{3/2}}{4725}+\frac{2}{45} (1-2 x)^{5/2} \sqrt{2+3 x} (3+5 x)^{3/2}+\frac{1217887 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{10631250}+\frac{6799613 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{5315625}\\ &=-\frac{110717 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{1063125}+\frac{10214 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{118125}+\frac{326 (1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{3/2}}{4725}+\frac{2}{45} (1-2 x)^{5/2} \sqrt{2+3 x} (3+5 x)^{3/2}-\frac{6799613 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5315625}-\frac{110717 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5315625}\\ \end{align*}
Mathematica [A] time = 0.233557, size = 102, normalized size = 0.53 \[ \frac{-9945565 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+15 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (945000 x^3-1111500 x^2+55530 x+526861\right )+13599226 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{15946875 \sqrt{2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.011, size = 155, normalized size = 0.8 \begin{align*}{\frac{1}{956812500\,{x}^{3}+733556250\,{x}^{2}-223256250\,x-191362500}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 850500000\,{x}^{6}+9945565\,\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 ) -13599226\,\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 ) -348300000\,{x}^{5}-915408000\,{x}^{4}+575805600\,{x}^{3}+551942790\,{x}^{2}-120636210\,x-94834980 \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 \sqrt{5 \, x + 3} \sqrt{3 \, x + 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 ({\left (4 \, x^{2} - 4 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}, 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 \sqrt{5 \, x + 3} \sqrt{3 \, x + 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]