3.2702 \(\int (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{5/2} \, dx\)

Optimal. Leaf size=249 \[ \frac{2}{65} (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{7/2}+\frac{62 \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}}{3575}-\frac{67 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{7/2}}{160875}-\frac{160084 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}}{3378375}-\frac{2133359 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{6756750}-\frac{70536439 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{30405375}-\frac{70536439 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{13820625 \sqrt{33}}-\frac{9380126059 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{55282500 \sqrt{33}} \]

[Out]

(-70536439*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/30405375 - (2133359*Sqrt[1
- 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/6756750 - (160084*Sqrt[1 - 2*x]*Sqrt[2 +
3*x]*(3 + 5*x)^(5/2))/3378375 - (67*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(7/2))
/160875 + (62*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 + 5*x)^(7/2))/3575 + (2*(1 - 2*x)
^(3/2)*(2 + 3*x)^(3/2)*(3 + 5*x)^(7/2))/65 - (9380126059*EllipticE[ArcSin[Sqrt[3
/7]*Sqrt[1 - 2*x]], 35/33])/(55282500*Sqrt[33]) - (70536439*EllipticF[ArcSin[Sqr
t[3/7]*Sqrt[1 - 2*x]], 35/33])/(13820625*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]

(-70536439*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/30405375 - (2133359*Sqrt[1
- 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/6756750 - (160084*Sqrt[1 - 2*x]*Sqrt[2 +
3*x]*(3 + 5*x)^(5/2))/3378375 - (67*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(7/2))
/160875 + (62*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 + 5*x)^(7/2))/3575 + (2*(1 - 2*x)
^(3/2)*(2 + 3*x)^(3/2)*(3 + 5*x)^(7/2))/65 - (9380126059*EllipticE[ArcSin[Sqrt[3
/7]*Sqrt[1 - 2*x]], 35/33])/(55282500*Sqrt[33]) - (70536439*EllipticF[ArcSin[Sqr
t[3/7]*Sqrt[1 - 2*x]], 35/33])/(13820625*Sqrt[33])

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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)

[Out]

2*(-2*x + 1)**(3/2)*(3*x + 2)**(5/2)*(5*x + 3)**(5/2)/39 - 115*(-2*x + 1)**(3/2)
*(3*x + 2)**(5/2)*(5*x + 3)**(3/2)/1287 + 130*sqrt(-2*x + 1)*(3*x + 2)**(5/2)*(5
*x + 3)**(3/2)/891 - 27814*sqrt(-2*x + 1)*(3*x + 2)**(3/2)*(5*x + 3)**(3/2)/4054
05 - 2026949*sqrt(-2*x + 1)*(3*x + 2)**(3/2)*sqrt(5*x + 3)/4054050 - 134992031*s
qrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/60810750 - 9380126059*sqrt(33)*ellipti
c_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/1824322500 - 70536439*sqrt(33)*ellip
tic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/456080625

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Mathematica [A]  time = 0.444095, size = 115, normalized size = 0.46 \[ \frac{9380126059 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-5 \left (3 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (1403325000 x^5+2364390000 x^4+496455750 x^3-1110242250 x^2-638983395 x+67302101\right )[phone] F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )}{912161250 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - 2*x)^(3/2)*(2 + 3*x)^(3/2)*(3 + 5*x)^(5/2),x]

[Out]

(9380126059*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 5*(3*Sqrt[2 - 4
*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(67302101 - 638983395*x - 1110242250*x^2 + 49645
5750*x^3 + 2364390000*x^4 + 1403325000*x^5) + 944944217*EllipticF[ArcSin[Sqrt[2/
11]*Sqrt[3 + 5*x]], -33/2]))/(912161250*Sqrt[2])

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Maple [C]  time = 0.016, size = 189, normalized size = 0.8 \[{\frac{1}{54729675000\,{x}^{3}[phone]\,{x}^{2}-12770257500\,x-10945935000}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( -1262992500000\,{x}^{8}-3096245250000\,{x}^{7}-1783541025000\,{x}^{6}[phone]\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -9380126059\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) [phone]\,{x}^{5}[phone]\,{x}^{4}[phone]\,{x}^{3}-380468567640\,{x}^{2}-100883569890\,x+12114378180 \right ) } \]

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]

1/1824322500*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(-1262992500000*x^8-30962
45250000*x^7-1783541025000*x^6+4724721085*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1
-2*x)^(1/2)*EllipticF(1/11*11^(1/2)*2^(1/2)*(3+5*x)^(1/2),1/2*I*11^(1/2)*3^(1/2)
*2^(1/2))-9380126059*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*EllipticE
(1/11*11^(1/2)*2^(1/2)*(3+5*x)^(1/2),1/2*I*11^(1/2)*3^(1/2)*2^(1/2))[phone]
00*x^5+1870998115500*x^4+236537814150*x^3-380468567640*x^2-100883569890*x+121143
78180)/(30*x^3+23*x^2-7*x-6)

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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]

integrate((5*x + 3)^(5/2)*(3*x + 2)^(3/2)*(-2*x + 1)^(3/2), x)

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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]

integral(-(150*x^4 + 205*x^3 + 34*x^2 - 51*x - 18)*sqrt(5*x + 3)*sqrt(3*x + 2)*s
qrt(-2*x + 1), x)

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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]

Timed 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]

integrate((5*x + 3)^(5/2)*(3*x + 2)^(3/2)*(-2*x + 1)^(3/2), x)