3.2701 \(\int (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{5/2} \, dx\)

Optimal. Leaf size=280 \[ \frac{2}{75} (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{7/2}+\frac{178 \sqrt{1-2 x} (3 x+2)^{5/2} (5 x+3)^{7/2}}{14625}+\frac{2503 \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}}{804375}-\frac{199721 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{7/2}}{12065625}-\frac{57509209 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}}{506756250}-\frac{380132617 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{506756250}-\frac{50299451003 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{9121612500}-\frac{50299451003 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{4146187500 \sqrt{33}}-\frac{836091184171 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2073093750 \sqrt{33}} \]

[Out]

(-50299451003*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/9121612500 - (380132617
*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/506756250 - (57509209*Sqrt[1 - 2*x
]*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2))/506756250 - (199721*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]
*(3 + 5*x)^(7/2))/12065625 + (2503*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 + 5*x)^(7/2)
)/804375 + (178*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*(3 + 5*x)^(7/2))/14625 + (2*(1 - 2
*x)^(3/2)*(2 + 3*x)^(5/2)*(3 + 5*x)^(7/2))/75 - (836091184171*EllipticE[ArcSin[S
qrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(2073093750*Sqrt[33]) - (50299451003*EllipticF[
ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(4146187500*Sqrt[33])

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Rubi [A]  time = 0.646294, antiderivative size = 280, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{2}{75} (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{7/2}+\frac{178 \sqrt{1-2 x} (3 x+2)^{5/2} (5 x+3)^{7/2}}{14625}+\frac{2503 \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}}{804375}-\frac{199721 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{7/2}}{12065625}-\frac{57509209 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}}{506756250}-\frac{380132617 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{506756250}-\frac{50299451003 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{9121612500}-\frac{50299451003 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{4146187500 \sqrt{33}}-\frac{836091184171 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2073093750 \sqrt{33}} \]

Antiderivative was successfully verified.

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

[Out]

(-50299451003*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/9121612500 - (380132617
*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/506756250 - (57509209*Sqrt[1 - 2*x
]*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2))/506756250 - (199721*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]
*(3 + 5*x)^(7/2))/12065625 + (2503*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 + 5*x)^(7/2)
)/804375 + (178*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*(3 + 5*x)^(7/2))/14625 + (2*(1 - 2
*x)^(3/2)*(2 + 3*x)^(5/2)*(3 + 5*x)^(7/2))/75 - (836091184171*EllipticE[ArcSin[S
qrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(2073093750*Sqrt[33]) - (50299451003*EllipticF[
ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(4146187500*Sqrt[33])

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Rubi in Sympy [A]  time = 62.163, size = 258, normalized size = 0.92 \[ \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{\frac{7}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{45} - \frac{23 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{\frac{7}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{351} + \frac{2152 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{7}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{19305} - \frac{35767 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{868725} - \frac{10362379 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{36486450} - \frac{1033872841 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{912161250} - \frac{48128081531 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{9121612500} - \frac{836091184171 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{68412093750} - \frac{50299451003 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{136824187500} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(-2*x + 1)**(3/2)*(3*x + 2)**(7/2)*(5*x + 3)**(5/2)/45 - 23*(-2*x + 1)**(3/2)*
(3*x + 2)**(7/2)*(5*x + 3)**(3/2)/351 + 2152*sqrt(-2*x + 1)*(3*x + 2)**(7/2)*(5*
x + 3)**(3/2)/19305 - 35767*sqrt(-2*x + 1)*(3*x + 2)**(5/2)*(5*x + 3)**(3/2)/868
725 - 10362379*sqrt(-2*x + 1)*(3*x + 2)**(5/2)*sqrt(5*x + 3)/36486450 - 10338728
41*sqrt(-2*x + 1)*(3*x + 2)**(3/2)*sqrt(5*x + 3)/912161250 - 48128081531*sqrt(-2
*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/9121612500 - 836091184171*sqrt(33)*elliptic_
e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/68412093750 - 50299451003*sqrt(33)*ell
iptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/136824187500

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Mathematica [A]  time = 0.373274, size = 119, normalized size = 0.42 \[ \frac{\sqrt{2} \left (3344364736684 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-1684482853585 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )-30 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (547296750000 x^6+1316318850000 x^5+888419542500 x^4-227285730000 x^3-522917547750 x^2-177853891770 x+44426819351\right )}{273648375000} \]

Antiderivative was successfully verified.

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

[Out]

(-30*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(44426819351 - 177853891770*x - 5
22917547750*x^2 - 227285730000*x^3 + 888419542500*x^4 + 1316318850000*x^5 + 5472
96750000*x^6) + Sqrt[2]*(3344364736684*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]
], -33/2] - 1684482853585*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]))/2
73648375000

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Maple [C]  time = 0.032, size = 194, normalized size = 0.7 \[{\frac{1}{8209451250000\,{x}^{3}[phone]\,{x}^{2}-1915538625000\,x-1641890250000}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( -492567075000000\,{x}^{9}-1562321722500000\,{x}^{8}-1592905277250000\,{x}^{7}-33511953825000\,{x}^{6}[phone]\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -3344364736684\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +1050958443600000\,{x}^{5}[phone]\,{x}^{4}-67989068522100\,{x}^{3}-162128981218890\,{x}^{2}-22684068454890\,x+7996827483180 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/273648375000*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(-492567075000000*x^9-1
562321722500000*x^8-1592905277250000*x^7-33511953825000*x^6+1684482853585*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))-3344364736684*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))+1050958443600000*x^5+633067124890500*x^4-67989068522100*
x^3-162128981218890*x^2-22684068454890*x+7996827483180)/(30*x^3+23*x^2-7*x-6)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (3 \, x + 2\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(5/2)*(3*x + 2)^(5/2)*(-2*x + 1)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-{\left (450 \, x^{5} + 915 \, x^{4} + 512 \, x^{3} - 85 \, x^{2} - 156 \, x - 36\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(5/2)*(3*x + 2)^(5/2)*(-2*x + 1)^(3/2),x, algorithm="fricas")

[Out]

integral(-(450*x^5 + 915*x^4 + 512*x^3 - 85*x^2 - 156*x - 36)*sqrt(5*x + 3)*sqrt
(3*x + 2)*sqrt(-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)**(5/2)*(3+5*x)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.486771, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(5/2)*(3*x + 2)^(5/2)*(-2*x + 1)^(3/2),x, algorithm="giac")

[Out]

Done