Optimal. Leaf size=218 \[ -\frac{465127 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{47250 \sqrt{33}}+\frac{2}{55} \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}-\frac{3}{275} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{7/2}-\frac{177 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}}{1925}-\frac{7031 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{11550}-\frac{465127 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{103950}-\frac{30926081 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{94500 \sqrt{33}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0830044, antiderivative size = 218, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ \frac{2}{55} \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}-\frac{3}{275} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{7/2}-\frac{177 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}}{1925}-\frac{7031 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{11550}-\frac{465127 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{103950}-\frac{465127 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{47250 \sqrt{33}}-\frac{30926081 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{94500 \sqrt{33}} \]
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)^{3/2} (3+5 x)^{5/2} \, dx &=\frac{2}{55} \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}-\frac{2}{55} \int \frac{\left (-\frac{25}{2}-\frac{27 x}{2}\right ) \sqrt{2+3 x} (3+5 x)^{5/2}}{\sqrt{1-2 x}} \, dx\\ &=-\frac{3}{275} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{7/2}+\frac{2}{55} \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}+\frac{2 \int \frac{(3+5 x)^{5/2} \left (\frac{6309}{4}+\frac{4779 x}{2}\right )}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{2475}\\ &=-\frac{177 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{1925}-\frac{3}{275} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{7/2}+\frac{2}{55} \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}-\frac{2 \int \frac{\left (-155520-\frac{949185 x}{4}\right ) (3+5 x)^{3/2}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{51975}\\ &=-\frac{7031 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{11550}-\frac{177 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{1925}-\frac{3}{275} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{7/2}+\frac{2}{55} \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}+\frac{2 \int \frac{\sqrt{3+5 x} \left (\frac{81615195}{8}+\frac{62792145 x}{4}\right )}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{779625}\\ &=-\frac{465127 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{103950}-\frac{7031 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{11550}-\frac{177 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{1925}-\frac{3}{275} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{7/2}+\frac{2}{55} \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}-\frac{2 \int \frac{-330394410-\frac{4175020935 x}{8}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{7016625}\\ &=-\frac{465127 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{103950}-\frac{7031 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{11550}-\frac{177 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{1925}-\frac{3}{275} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{7/2}+\frac{2}{55} \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}+\frac{465127 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{94500}+\frac{30926081 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{1039500}\\ &=-\frac{465127 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{103950}-\frac{7031 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{11550}-\frac{177 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{1925}-\frac{3}{275} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{7/2}+\frac{2}{55} \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}-\frac{30926081 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{94500 \sqrt{33}}-\frac{465127 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{47250 \sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.269682, size = 107, normalized size = 0.49 \[ \frac{-15576890 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+15 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (1417500 x^4+3354750 x^3+2737800 x^2+570555 x-567484\right )+30926081 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{1559250 \sqrt{2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.01, size = 160, normalized size = 0.7 \begin{align*}{\frac{1}{93555000\,{x}^{3}+71725500\,{x}^{2}-21829500\,x-18711000}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 1275750000\,{x}^{7}+3997350000\,{x}^{6}+15576890\,\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 ) -30926081\,\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 ) +4481122500\,{x}^{5}+1442934000\,{x}^{4}-1295845650\,{x}^{3}-1004184510\,{x}^{2}+16471740\,x+102147120 \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{3}{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 (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\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{3}{2}} \sqrt{-2 \, x + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]