3.2897 \(\int \frac{(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^{5/2}} \, dx\)

Optimal. Leaf size=160 \[ \frac{11 (5 x+3)^{3/2}}{7 \sqrt{1-2 x} (3 x+2)^{3/2}}-\frac{2797 \sqrt{1-2 x} \sqrt{5 x+3}}{3087 \sqrt{3 x+2}}+\frac{97 \sqrt{1-2 x} \sqrt{5 x+3}}{441 (3 x+2)^{3/2}}+\frac{598 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3087}+\frac{2797 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3087} \]

[Out]

(97*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(441*(2 + 3*x)^(3/2)) - (2797*Sqrt[1 - 2*x]*Sqr
t[3 + 5*x])/(3087*Sqrt[2 + 3*x]) + (11*(3 + 5*x)^(3/2))/(7*Sqrt[1 - 2*x]*(2 + 3*
x)^(3/2)) + (2797*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/
3087 + (598*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/3087

_______________________________________________________________________________________

Rubi [A]  time = 0.336866, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{11 (5 x+3)^{3/2}}{7 \sqrt{1-2 x} (3 x+2)^{3/2}}-\frac{2797 \sqrt{1-2 x} \sqrt{5 x+3}}{3087 \sqrt{3 x+2}}+\frac{97 \sqrt{1-2 x} \sqrt{5 x+3}}{441 (3 x+2)^{3/2}}+\frac{598 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3087}+\frac{2797 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3087} \]

Antiderivative was successfully verified.

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

[Out]

(97*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(441*(2 + 3*x)^(3/2)) - (2797*Sqrt[1 - 2*x]*Sqr
t[3 + 5*x])/(3087*Sqrt[2 + 3*x]) + (11*(3 + 5*x)^(3/2))/(7*Sqrt[1 - 2*x]*(2 + 3*
x)^(3/2)) + (2797*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/
3087 + (598*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/3087

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 30.7608, size = 143, normalized size = 0.89 \[ - \frac{2797 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{3087 \sqrt{3 x + 2}} + \frac{97 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{441 \left (3 x + 2\right )^{\frac{3}{2}}} + \frac{2797 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{9261} + \frac{6578 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{108045} + \frac{11 \left (5 x + 3\right )^{\frac{3}{2}}}{7 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2797*sqrt(-2*x + 1)*sqrt(5*x + 3)/(3087*sqrt(3*x + 2)) + 97*sqrt(-2*x + 1)*sqrt
(5*x + 3)/(441*(3*x + 2)**(3/2)) + 2797*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-
2*x + 1)/7), 35/33)/9261 + 6578*sqrt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)
/11), 33/35)/108045 + 11*(5*x + 3)**(3/2)/(7*sqrt(-2*x + 1)*(3*x + 2)**(3/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.201984, size = 100, normalized size = 0.62 \[ \frac{\frac{6 \sqrt{5 x+3} \left (8391 x^2+12847 x+4819\right )}{\sqrt{1-2 x} (3 x+2)^{3/2}}-\sqrt{2} \left (7070 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+2797 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )}{9261} \]

Antiderivative was successfully verified.

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

[Out]

((6*Sqrt[3 + 5*x]*(4819 + 12847*x + 8391*x^2))/(Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)) -
Sqrt[2]*(2797*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 7070*Ellipti
cF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]))/9261

_______________________________________________________________________________________

Maple [C]  time = 0.034, size = 267, normalized size = 1.7 \[{\frac{1}{92610\,{x}^{2}+9261\,x-27783}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 21210\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+8391\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+14140\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +5594\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -251730\,{x}^{3}-536448\,{x}^{2}-375816\,x-86742 \right ) \left ( 2+3\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/9261*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(21210*2^(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))*x*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1
-2*x)^(1/2)+8391*2^(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))*x*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)+14140*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))+5594*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))-251730*x^3-536448*x^2-375816*x-86742)/(2+3*x)^(3/2)/(10*x^2+x-3)

_______________________________________________________________________________________

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

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)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt{5 \, x + 3}}{{\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}, x\right ) \]

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(-(25*x^2 + 30*x + 9)*sqrt(5*x + 3)/((18*x^3 + 15*x^2 - 4*x - 4)*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((3+5*x)**(5/2)/(1-2*x)**(3/2)/(2+3*x)**(5/2),x)

[Out]

Timed out

_______________________________________________________________________________________

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

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="giac")

[Out]

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