3.2896 \(\int \frac{(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^{3/2}} \, dx\)

Optimal. Leaf size=129 \[ \frac{11 (5 x+3)^{3/2}}{7 \sqrt{1-2 x} \sqrt{3 x+2}}+\frac{31 \sqrt{1-2 x} \sqrt{5 x+3}}{147 \sqrt{3 x+2}}+\frac{31}{147} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{1159}{147} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(31*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(147*Sqrt[2 + 3*x]) + (11*(3 + 5*x)^(3/2))/(7*S
qrt[1 - 2*x]*Sqrt[2 + 3*x]) + (1159*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1
- 2*x]], 35/33])/147 + (31*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]]
, 35/33])/147

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Rubi [A]  time = 0.260727, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{11 (5 x+3)^{3/2}}{7 \sqrt{1-2 x} \sqrt{3 x+2}}+\frac{31 \sqrt{1-2 x} \sqrt{5 x+3}}{147 \sqrt{3 x+2}}+\frac{31}{147} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{1159}{147} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(31*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(147*Sqrt[2 + 3*x]) + (11*(3 + 5*x)^(3/2))/(7*S
qrt[1 - 2*x]*Sqrt[2 + 3*x]) + (1159*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1
- 2*x]], 35/33])/147 + (31*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]]
, 35/33])/147

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Rubi in Sympy [A]  time = 24.1795, size = 114, normalized size = 0.88 \[ \frac{31 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{147 \sqrt{3 x + 2}} + \frac{1159 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{441} + \frac{341 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{5145} + \frac{11 \left (5 x + 3\right )^{\frac{3}{2}}}{7 \sqrt{- 2 x + 1} \sqrt{3 x + 2}} \]

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)**(3/2),x)

[Out]

31*sqrt(-2*x + 1)*sqrt(5*x + 3)/(147*sqrt(3*x + 2)) + 1159*sqrt(33)*elliptic_e(a
sin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/441 + 341*sqrt(35)*elliptic_f(asin(sqrt(5
5)*sqrt(-2*x + 1)/11), 33/35)/5145 + 11*(5*x + 3)**(3/2)/(7*sqrt(-2*x + 1)*sqrt(
3*x + 2))

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Mathematica [A]  time = 0.196196, size = 122, normalized size = 0.95 \[ \frac{6 \sqrt{3 x+2} \sqrt{5 x+3} (1093 x+724)+1295 \sqrt{2-4 x} (3 x+2) F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-2318 \sqrt{2-4 x} (3 x+2) E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{882 \sqrt{1-2 x} (3 x+2)} \]

Antiderivative was successfully verified.

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

[Out]

(6*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(724 + 1093*x) - 2318*Sqrt[2 - 4*x]*(2 + 3*x)*Ell
ipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 1295*Sqrt[2 - 4*x]*(2 + 3*x)*E
llipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(882*Sqrt[1 - 2*x]*(2 + 3*x))

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Maple [C]  time = 0.027, size = 159, normalized size = 1.2 \[ -{\frac{1}{26460\,{x}^{3}+20286\,{x}^{2}-6174\,x-5292}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 1295\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -2318\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +32790\,{x}^{2}+41394\,x+13032 \right ) } \]

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)^(3/2),x)

[Out]

-1/882*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(1295*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))-2318*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*El
lipticE(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))+3279
0*x^2+41394*x+13032)/(30*x^3+23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt{5 \, x + 3}}{{\left (6 \, x^{2} + x - 2\right )} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(25*x^2 + 30*x + 9)*sqrt(5*x + 3)/((6*x^2 + x - 2)*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((3+5*x)**(5/2)/(1-2*x)**(3/2)/(2+3*x)**(3/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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