Optimal. Leaf size=249 \[ \frac{2}{65} (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{7/2}+\frac{62 \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}}{3575}-\frac{67 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{7/2}}{160875}-\frac{160084 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}}{3378375}-\frac{2133359 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{6756750}-\frac{70536439 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{30405375}-\frac{70536439 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{13820625 \sqrt{33}}-\frac{9380126059 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{55282500 \sqrt{33}} \]
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Rubi [A] time = 0.558925, 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 \[ \frac{2}{65} (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{7/2}+\frac{62 \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}}{3575}-\frac{67 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{7/2}}{160875}-\frac{160084 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}}{3378375}-\frac{2133359 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{6756750}-\frac{70536439 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{30405375}-\frac{70536439 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{13820625 \sqrt{33}}-\frac{9380126059 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{55282500 \sqrt{33}} \]
Antiderivative was successfully verified.
[In] Int[(1 - 2*x)^(3/2)*(2 + 3*x)^(3/2)*(3 + 5*x)^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 53.9109, size = 230, normalized size = 0.92 \[ \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{39} - \frac{115 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{1287} + \frac{130 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{891} - \frac{27814 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{405405} - \frac{2026949 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{4054050} - \frac{134992031 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{60810750} - \frac{9380126059 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{1824322500} - \frac{70536439 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{456080625} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1-2*x)**(3/2)*(2+3*x)**(3/2)*(3+5*x)**(5/2),x)
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Mathematica [A] time = 0.444095, size = 115, normalized size = 0.46 \[ \frac{9380126059 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-5 \left (3 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (1403325000 x^5+2364390000 x^4+496455750 x^3-1110242250 x^2-638983395 x+67302101\right )+944944217 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )}{912161250 \sqrt{2}} \]
Antiderivative was successfully verified.
[In] Integrate[(1 - 2*x)^(3/2)*(2 + 3*x)^(3/2)*(3 + 5*x)^(5/2),x]
[Out]
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Maple [C] time = 0.016, size = 189, normalized size = 0.8 \[{\frac{1}{54729675000\,{x}^{3}+41959417500\,{x}^{2}-12770257500\,x-10945935000}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( -1262992500000\,{x}^{8}-3096245250000\,{x}^{7}-1783541025000\,{x}^{6}+4724721085\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -9380126059\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +1405783957500\,{x}^{5}+1870998115500\,{x}^{4}+236537814150\,{x}^{3}-380468567640\,{x}^{2}-100883569890\,x+12114378180 \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1-2*x)^(3/2)*(2+3*x)^(3/2)*(3+5*x)^(5/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (3 \, x + 2\right )}^{\frac{3}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^(5/2)*(3*x + 2)^(3/2)*(-2*x + 1)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-{\left (150 \, x^{4} + 205 \, x^{3} + 34 \, x^{2} - 51 \, x - 18\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^(5/2)*(3*x + 2)^(3/2)*(-2*x + 1)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((1-2*x)**(3/2)*(2+3*x)**(3/2)*(3+5*x)**(5/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (3 \, x + 2\right )}^{\frac{3}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^(5/2)*(3*x + 2)^(3/2)*(-2*x + 1)^(3/2),x, algorithm="giac")
[Out]