3.2682 \(\int (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt{3+5 x} \, dx\)

Optimal. Leaf size=218 \[ \frac{2}{55} (1-2 x)^{3/2} (5 x+3)^{3/2} (3 x+2)^{5/2}+\frac{178 \sqrt{1-2 x} (5 x+3)^{3/2} (3 x+2)^{5/2}}{7425}+\frac{1103 \sqrt{1-2 x} (5 x+3)^{3/2} (3 x+2)^{3/2}}{259875}-\frac{124891 \sqrt{1-2 x} (5 x+3)^{3/2} \sqrt{3 x+2}}{2165625}-\frac{18177329 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{38981250}-\frac{18177329 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{17718750 \sqrt{33}}-\frac{604915631 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{17718750 \sqrt{33}} \]

[Out]

(-18177329*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/38981250 - (124891*Sqrt[1
- 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/2165625 + (1103*Sqrt[1 - 2*x]*(2 + 3*x)^(3
/2)*(3 + 5*x)^(3/2))/259875 + (178*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*(3 + 5*x)^(3/2)
)/7425 + (2*(1 - 2*x)^(3/2)*(2 + 3*x)^(5/2)*(3 + 5*x)^(3/2))/55 - (604915631*Ell
ipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(17718750*Sqrt[33]) - (18177329*
EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(17718750*Sqrt[33])

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Rubi [A]  time = 0.479312, 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 \[ \frac{2}{55} (1-2 x)^{3/2} (5 x+3)^{3/2} (3 x+2)^{5/2}+\frac{178 \sqrt{1-2 x} (5 x+3)^{3/2} (3 x+2)^{5/2}}{7425}+\frac{1103 \sqrt{1-2 x} (5 x+3)^{3/2} (3 x+2)^{3/2}}{259875}-\frac{124891 \sqrt{1-2 x} (5 x+3)^{3/2} \sqrt{3 x+2}}{2165625}-\frac{18177329 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{38981250}-\frac{18177329 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{17718750 \sqrt{33}}-\frac{604915631 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{17718750 \sqrt{33}} \]

Antiderivative was successfully verified.

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

[Out]

(-18177329*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/38981250 - (124891*Sqrt[1
- 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/2165625 + (1103*Sqrt[1 - 2*x]*(2 + 3*x)^(3
/2)*(3 + 5*x)^(3/2))/259875 + (178*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*(3 + 5*x)^(3/2)
)/7425 + (2*(1 - 2*x)^(3/2)*(2 + 3*x)^(5/2)*(3 + 5*x)^(3/2))/55 - (604915631*Ell
ipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(17718750*Sqrt[33]) - (18177329*
EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(17718750*Sqrt[33])

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Rubi in Sympy [A]  time = 46.5695, size = 201, normalized size = 0.92 \[ \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{\frac{7}{2}} \sqrt{5 x + 3}}{33} - \frac{107 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{1485} - \frac{10831 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{51975} - \frac{505079 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{3 x + 2} \sqrt{5 x + 3}}{866250} + \frac{17802656 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{19490625} - \frac{604915631 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{584718750} - \frac{18177329 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{584718750} \]

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)**(1/2),x)

[Out]

2*(-2*x + 1)**(3/2)*(3*x + 2)**(7/2)*sqrt(5*x + 3)/33 - 107*(-2*x + 1)**(3/2)*(3
*x + 2)**(5/2)*sqrt(5*x + 3)/1485 - 10831*(-2*x + 1)**(3/2)*(3*x + 2)**(3/2)*sqr
t(5*x + 3)/51975 - 505079*(-2*x + 1)**(3/2)*sqrt(3*x + 2)*sqrt(5*x + 3)/866250 +
17802656*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/19490625 - 604915631*sqrt(3
3)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/584718750 - 18177329*sqrt(
33)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/584718750

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Mathematica [A]  time = 0.371743, size = 107, normalized size = 0.49 \[ \frac{15 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (-127575000 x^4-140805000 x^3+48345750 x^2+89595360 x+4295257\right )-609979405 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )[phone] E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{584718750 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

(15*Sqrt[2 - 4*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(4295257 + 89595360*x + 48345750*x
^2 - 140805000*x^3 - 127575000*x^4) + 1209831262*EllipticE[ArcSin[Sqrt[2/11]*Sqr
t[3 + 5*x]], -33/2] - 609979405*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/
2])/(584718750*Sqrt[2])

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Maple [C]  time = 0.016, size = 184, normalized size = 0.8 \[{\frac{1}{35083125000\,{x}^{3}[phone]\,{x}^{2}-8186062500\,x-7016625000}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( -114817500000\,{x}^{7}-214751250000\,{x}^{6}[phone]\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -1209831262\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -26853525000\,{x}^{5}[phone]\,{x}^{4}[phone]\,{x}^{3}-24553533270\,{x}^{2}-17029168770\,x-773146260 \right ) } \]

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)^(1/2),x)

[Out]

1/1169437500*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(-114817500000*x^7-214751
250000*x^6+609979405*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))-1209831262*
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))-26853525000*x^5+166526941500*x^4
[phone]*x^3-24553533270*x^2-17029168770*x-773146260)/(30*x^3+23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-{\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(18*x^3 + 15*x^2 - 4*x - 4)*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)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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