Optimal. Leaf size=249 \[ -\frac{493825477 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{18427500 \sqrt{33}}+\frac{2}{65} \sqrt{1-2 x} (3 x+2)^{5/2} (5 x+3)^{7/2}-\frac{23 \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}}{3575}-\frac{2014 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{7/2}}{53625}-\frac{564731 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}}{2252250}-\frac{1865989 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{1126125}-\frac{493825477 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{40540500}-\frac{16416987253 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{18427500 \sqrt{33}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.09979, antiderivative size = 249, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ \frac{2}{65} \sqrt{1-2 x} (3 x+2)^{5/2} (5 x+3)^{7/2}-\frac{23 \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}}{3575}-\frac{2014 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{7/2}}{53625}-\frac{564731 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}}{2252250}-\frac{1865989 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{1126125}-\frac{493825477 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{40540500}-\frac{493825477 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{18427500 \sqrt{33}}-\frac{16416987253 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{18427500 \sqrt{33}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 101
Rule 154
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2} \, dx &=\frac{2}{65} \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{7/2}-\frac{2}{65} \int \frac{\left (-\frac{27}{2}-\frac{23 x}{2}\right ) (2+3 x)^{3/2} (3+5 x)^{5/2}}{\sqrt{1-2 x}} \, dx\\ &=-\frac{23 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{3575}+\frac{2}{65} \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{7/2}+\frac{2 \int \frac{\sqrt{2+3 x} (3+5 x)^{5/2} \left (\frac{7895}{4}+3021 x\right )}{\sqrt{1-2 x}} \, dx}{3575}\\ &=-\frac{2014 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{7/2}}{53625}-\frac{23 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{3575}+\frac{2}{65} \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{7/2}-\frac{2 \int \frac{\left (-278841-\frac{1694193 x}{4}\right ) (3+5 x)^{5/2}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{160875}\\ &=-\frac{564731 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{2252250}-\frac{2014 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{7/2}}{53625}-\frac{23 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{3575}+\frac{2}{65} \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{7/2}+\frac{2 \int \frac{(3+5 x)^{3/2} \left (\frac{220162935}{8}+\frac{83969505 x}{2}\right )}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{3378375}\\ &=-\frac{1865989 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{1126125}-\frac{564731 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{2252250}-\frac{2014 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{7/2}}{53625}-\frac{23 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{3575}+\frac{2}{65} \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{7/2}-\frac{2 \int \frac{\left (-\frac{14441685345}{8}-\frac{22222146465 x}{8}\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{50675625}\\ &=-\frac{493825477 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{40540500}-\frac{1865989 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{1126125}-\frac{564731 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{2252250}-\frac{2014 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{7/2}}{53625}-\frac{23 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{3575}+\frac{2}{65} \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{7/2}+\frac{2 \int \frac{\frac{935406033885}{16}+\frac{738764426385 x}{8}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{456080625}\\ &=-\frac{493825477 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{40540500}-\frac{1865989 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{1126125}-\frac{564731 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{2252250}-\frac{2014 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{7/2}}{53625}-\frac{23 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{3575}+\frac{2}{65} \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{7/2}+\frac{493825477 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{36855000}+\frac{16416987253 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{202702500}\\ &=-\frac{493825477 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{40540500}-\frac{1865989 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{1126125}-\frac{564731 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{2252250}-\frac{2014 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{7/2}}{53625}-\frac{23 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{3575}+\frac{2}{65} \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{7/2}-\frac{16416987253 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{18427500 \sqrt{33}}-\frac{493825477 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{18427500 \sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.294723, size = 112, normalized size = 0.45 \[ \frac{-16537733765 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+15 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (1403325000 x^5+4299277500 x^4+5075689500 x^3+2626854750 x^2+139824180 x-707313559\right )+32833974506 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{608107500 \sqrt{2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.02, size = 165, normalized size = 0.7 \begin{align*}{\frac{1}{36486450000\,{x}^{3}+27972945000\,{x}^{2}-8513505000\,x-7297290000}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 1262992500000\,{x}^{8}+4837644000000\,{x}^{7}+7239923775000\,{x}^{6}+16537733765\,\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 ) -32833974506\,\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 ) +4710948255000\,{x}^{5}+98606794500\,{x}^{4}-2005367126400\,{x}^{3}-990243288510\,{x}^{2}+123367494990\,x+127316440620 \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{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (3 \, x + 2\right )}^{\frac{5}{2}} \sqrt{-2 \, x + 1}\,{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 (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\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{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (3 \, x + 2\right )}^{\frac{5}{2}} \sqrt{-2 \, x + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]