3.2748 \(\int (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{3/2} \, dx\)

Optimal. Leaf size=249 \[ \frac{2}{65} (1-2 x)^{5/2} (3 x+2)^{3/2} (5 x+3)^{5/2}+\frac{106 (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{5/2}}{3575}+\frac{8318 \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}}{482625}+\frac{25603 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}}{1876875}-\frac{6794792 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{84459375}-\frac{923943703 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{1520268750}-\frac{923943703 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{691031250 \sqrt{33}}-\frac{30660308017 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{691031250 \sqrt{33}} \]

[Out]

(-923943703*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/1520268750 - (6794792*Sqr
t[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/84459375 + (25603*Sqrt[1 - 2*x]*Sqrt[2
+ 3*x]*(3 + 5*x)^(5/2))/1876875 + (8318*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 + 5*x)
^(5/2))/482625 + (106*(1 - 2*x)^(3/2)*(2 + 3*x)^(3/2)*(3 + 5*x)^(5/2))/3575 + (2
*(1 - 2*x)^(5/2)*(2 + 3*x)^(3/2)*(3 + 5*x)^(5/2))/65 - (30660308017*EllipticE[Ar
cSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(691031250*Sqrt[33]) - (923943703*Ellipti
cF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(691031250*Sqrt[33])

_______________________________________________________________________________________

Rubi [A]  time = 0.56147, 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 \[ \frac{2}{65} (1-2 x)^{5/2} (3 x+2)^{3/2} (5 x+3)^{5/2}+\frac{106 (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{5/2}}{3575}+\frac{8318 \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}}{482625}+\frac{25603 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}}{1876875}-\frac{6794792 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{84459375}-\frac{923943703 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{1520268750}-\frac{923943703 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{691031250 \sqrt{33}}-\frac{30660308017 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{691031250 \sqrt{33}} \]

Antiderivative was successfully verified.

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

[Out]

(-923943703*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/1520268750 - (6794792*Sqr
t[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/84459375 + (25603*Sqrt[1 - 2*x]*Sqrt[2
+ 3*x]*(3 + 5*x)^(5/2))/1876875 + (8318*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 + 5*x)
^(5/2))/482625 + (106*(1 - 2*x)^(3/2)*(2 + 3*x)^(3/2)*(3 + 5*x)^(5/2))/3575 + (2
*(1 - 2*x)^(5/2)*(2 + 3*x)^(3/2)*(3 + 5*x)^(5/2))/65 - (30660308017*EllipticE[Ar
cSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(691031250*Sqrt[33]) - (923943703*Ellipti
cF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(691031250*Sqrt[33])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 54.268, size = 230, normalized size = 0.92 \[ \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (3 x + 2\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{39} - \frac{181 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (3 x + 2\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{1287} + \frac{10496 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{57915} + \frac{1087234 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{6081075} - \frac{18399116 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{152026875} - \frac{880870681 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{1520268750} - \frac{30660308017 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{22804031250} - \frac{923943703 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{24186093750} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(-2*x + 1)**(5/2)*(3*x + 2)**(5/2)*(5*x + 3)**(3/2)/39 - 181*(-2*x + 1)**(5/2)
*(3*x + 2)**(5/2)*sqrt(5*x + 3)/1287 + 10496*(-2*x + 1)**(3/2)*(3*x + 2)**(5/2)*
sqrt(5*x + 3)/57915 + 1087234*sqrt(-2*x + 1)*(3*x + 2)**(5/2)*sqrt(5*x + 3)/6081
075 - 18399116*sqrt(-2*x + 1)*(3*x + 2)**(3/2)*sqrt(5*x + 3)/152026875 - 8808706
81*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/1520268750 - 30660308017*sqrt(33)*
elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/22804031250 - 923943703*sqrt(
35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/24186093750

_______________________________________________________________________________________

Mathematica [A]  time = 0.398959, size = 112, normalized size = 0.45 \[ \frac{15 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (14033250000 x^5+5400675000 x^4-13684072500 x^3-3707642250 x^2+5290733520 x+1020785999\right )-30830473835 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )[phone] E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{22804031250 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

(15*Sqrt[2 - 4*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(1020785999 + 5290733520*x - 37076
42250*x^2 - 13684072500*x^3 + 5400675000*x^4 + 14033250000*x^5) + 61320616034*El
lipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 30830473835*EllipticF[ArcSin[
Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(22804031250*Sqrt[2])

_______________________________________________________________________________________

Maple [C]  time = 0.016, size = 189, normalized size = 0.8 \[{\frac{1}{1368241875000\,{x}^{3}[phone]\,{x}^{2}-319256437500\,x-273648375000}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 12629925000000\,{x}^{8}[phone]\,{x}^{7}-11536182000000\,{x}^{6}[phone]\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -61320616034\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -16439014800000\,{x}^{5}[phone]\,{x}^{4}[phone]\,{x}^{3}[phone]\,{x}^{2}-1166697093390\,x-183741479820 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/45608062500*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(12629925000000*x^8+1454
3550000000*x^7-11536182000000*x^6+30830473835*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))-61320616034*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*Elli
pticE(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))-164390
14800000*x^5+4104920740500*x^4+7811051450400*x^3+260663905110*x^2-1166697093390*
x-183741479820)/(30*x^3+23*x^2-7*x-6)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (60 \, x^{4} + 16 \, x^{3} - 37 \, x^{2} - 5 \, x + 6\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((60*x^4 + 16*x^3 - 37*x^2 - 5*x + 6)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-
2*x + 1), x)

_______________________________________________________________________________________

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)**(5/2)*(2+3*x)**(3/2)*(3+5*x)**(3/2),x)

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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