3.2759 \(\int (1-2 x)^{5/2} \sqrt{2+3 x} (3+5 x)^{5/2} \, dx\)

Optimal. Leaf size=249 \[ \frac{2}{65} (1-2 x)^{5/2} \sqrt{3 x+2} (5 x+3)^{7/2}+\frac{326 (1-2 x)^{3/2} \sqrt{3 x+2} (5 x+3)^{7/2}}{10725}+\frac{2314 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{7/2}}{111375}-\frac{121031 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}}{30405375}-\frac{3872003 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{30405375}-\frac{486785077 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{547296750}-\frac{486785077 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{248771250 \sqrt{33}}-\frac{8120161139 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{124385625 \sqrt{33}} \]

[Out]

(-486785077*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/547296750 - (3872003*Sqrt
[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/30405375 - (121031*Sqrt[1 - 2*x]*Sqrt[2
+ 3*x]*(3 + 5*x)^(5/2))/30405375 + (2314*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^
(7/2))/111375 + (326*(1 - 2*x)^(3/2)*Sqrt[2 + 3*x]*(3 + 5*x)^(7/2))/10725 + (2*(
1 - 2*x)^(5/2)*Sqrt[2 + 3*x]*(3 + 5*x)^(7/2))/65 - (8120161139*EllipticE[ArcSin[
Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(124385625*Sqrt[33]) - (486785077*EllipticF[Ar
cSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(248771250*Sqrt[33])

_______________________________________________________________________________________

Rubi [A]  time = 0.571533, 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 \[ \frac{2}{65} (1-2 x)^{5/2} \sqrt{3 x+2} (5 x+3)^{7/2}+\frac{326 (1-2 x)^{3/2} \sqrt{3 x+2} (5 x+3)^{7/2}}{10725}+\frac{2314 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{7/2}}{111375}-\frac{121031 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}}{30405375}-\frac{3872003 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{30405375}-\frac{486785077 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{547296750}-\frac{486785077 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{248771250 \sqrt{33}}-\frac{8120161139 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{124385625 \sqrt{33}} \]

Antiderivative was successfully verified.

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

[Out]

(-486785077*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/547296750 - (3872003*Sqrt
[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/30405375 - (121031*Sqrt[1 - 2*x]*Sqrt[2
+ 3*x]*(3 + 5*x)^(5/2))/30405375 + (2314*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^
(7/2))/111375 + (326*(1 - 2*x)^(3/2)*Sqrt[2 + 3*x]*(3 + 5*x)^(7/2))/10725 + (2*(
1 - 2*x)^(5/2)*Sqrt[2 + 3*x]*(3 + 5*x)^(7/2))/65 - (8120161139*EllipticE[ArcSin[
Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(124385625*Sqrt[33]) - (486785077*EllipticF[Ar
cSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(248771250*Sqrt[33])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 57.9022, size = 230, normalized size = 0.92 \[ \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (3 x + 2\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{39} - \frac{185 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (3 x + 2\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{1287} + \frac{6008 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{34749} + \frac{200318 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{1216215} - \frac{2955908 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{18243225} - \frac{469049629 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{547296750} - \frac{8120161139 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{4104725625} - \frac{486785077 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{8209451250} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(-2*x + 1)**(5/2)*(3*x + 2)**(3/2)*(5*x + 3)**(5/2)/39 - 185*(-2*x + 1)**(5/2)
*(3*x + 2)**(3/2)*(5*x + 3)**(3/2)/1287 + 6008*(-2*x + 1)**(3/2)*(3*x + 2)**(3/2
)*(5*x + 3)**(3/2)/34749 + 200318*sqrt(-2*x + 1)*(3*x + 2)**(3/2)*(5*x + 3)**(3/
2)/1216215 - 2955908*sqrt(-2*x + 1)*(3*x + 2)**(3/2)*sqrt(5*x + 3)/18243225 - 46
9049629*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/547296750 - 8120161139*sqrt(3
3)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/4104725625 - 486785077*sqr
t(33)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/8209451250

_______________________________________________________________________________________

Mathematica [A]  time = 0.422342, size = 112, normalized size = 0.45 \[ \frac{15 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (8419950000 x^5+2577015000 x^4-7942630500 x^3-1730459250 x^2+2923422930 x+495379991\right )-16416737015 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )[phone] E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{8209451250 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

(15*Sqrt[2 - 4*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(495379991 + 2923422930*x - 173045
9250*x^2 - 7942630500*x^3 + 2577015000*x^4 + 8419950000*x^5) + 32480644556*Ellip
ticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 16416737015*EllipticF[ArcSin[Sqr
t[2/11]*Sqrt[3 + 5*x]], -33/2])/(8209451250*Sqrt[2])

_______________________________________________________________________________________

Maple [C]  time = 0.017, size = 189, normalized size = 0.8 \[{\frac{1}{492567075000\,{x}^{3}[phone]\,{x}^{2}-114932317500\,x-98513415000}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 7577955000000\,{x}^{8}[phone]\,{x}^{7}-7138416600000\,{x}^{6}[phone]\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -32480644556\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -9094592520000\,{x}^{5}[phone]\,{x}^{4}[phone]\,{x}^{3}[phone]\,{x}^{2}-630245925510\,x-89168398380 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/16418902500*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(7577955000000*x^8+81290
79000000*x^7-7138416600000*x^6+16416737015*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))-32480644556*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*Ellipti
cE(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))-909459252
0000*x^5+2641153459500*x^4+4256073746100*x^3+39376043490*x^2-630245925510*x-8916
8398380)/(30*x^3+23*x^2-7*x-6)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)*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)*(3+5*x)**(5/2)*(2+3*x)**(1/2),x)

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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