3.52 \(\int \frac{\sqrt{2-3 x} (7+5 x)^2}{\sqrt{-5+2 x} \sqrt{1+4 x}} \, dx\)

Optimal. Leaf size=167 \[ \frac{1}{4} \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1} (5 x+7)+\frac{68}{9} \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}-\frac{17533 \sqrt{\frac{11}{6}} \sqrt{5-2 x} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right )|\frac{1}{3}\right )}{72 \sqrt{2 x-5}}+\frac{44569 \sqrt{11} \sqrt{2 x-5} E\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{432 \sqrt{5-2 x}} \]

[Out]

(68*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/9 + (Sqrt[2 - 3*x]*Sqrt[-5 + 2*x
]*Sqrt[1 + 4*x]*(7 + 5*x))/4 + (44569*Sqrt[11]*Sqrt[-5 + 2*x]*EllipticE[ArcSin[(
2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2])/(432*Sqrt[5 - 2*x]) - (17533*Sqrt[11/6]*Sqrt[
5 - 2*x]*EllipticF[ArcSin[Sqrt[3/11]*Sqrt[1 + 4*x]], 1/3])/(72*Sqrt[-5 + 2*x])

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Rubi [A]  time = 0.496396, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{1}{4} \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1} (5 x+7)+\frac{68}{9} \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}-\frac{17533 \sqrt{\frac{11}{6}} \sqrt{5-2 x} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right )|\frac{1}{3}\right )}{72 \sqrt{2 x-5}}+\frac{44569 \sqrt{11} \sqrt{2 x-5} E\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{432 \sqrt{5-2 x}} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[2 - 3*x]*(7 + 5*x)^2)/(Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]),x]

[Out]

(68*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/9 + (Sqrt[2 - 3*x]*Sqrt[-5 + 2*x
]*Sqrt[1 + 4*x]*(7 + 5*x))/4 + (44569*Sqrt[11]*Sqrt[-5 + 2*x]*EllipticE[ArcSin[(
2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2])/(432*Sqrt[5 - 2*x]) - (17533*Sqrt[11/6]*Sqrt[
5 - 2*x]*EllipticF[ArcSin[Sqrt[3/11]*Sqrt[1 + 4*x]], 1/3])/(72*Sqrt[-5 + 2*x])

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Rubi in Sympy [A]  time = 60.2791, size = 196, normalized size = 1.17 \[ - \frac{5 \left (- 3 x + 2\right )^{\frac{3}{2}} \sqrt{2 x - 5} \sqrt{4 x + 1}}{12} + \frac{365 \sqrt{- 3 x + 2} \sqrt{2 x - 5} \sqrt{4 x + 1}}{36} - \frac{192863 \sqrt{33} \sqrt{- \frac{12 x}{11} + \frac{8}{11}} \sqrt{- \frac{4 x}{11} + \frac{10}{11}} F\left (\operatorname{asin}{\left (\frac{\sqrt{33} \sqrt{4 x + 1}}{11} \right )}\middle | \frac{1}{3}\right )}{864 \sqrt{- 3 x + 2} \sqrt{2 x - 5}} - \frac{44569 \sqrt{33} \sqrt{- \frac{12 x}{11} + \frac{8}{11}} \sqrt{2 x - 5} E\left (\operatorname{asin}{\left (\frac{\sqrt{33} \sqrt{4 x + 1}}{11} \right )}\middle | \frac{1}{3}\right )}{864 \sqrt{- 3 x + 2} \sqrt{- \frac{4 x}{11} + \frac{10}{11}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((7+5*x)**2*(2-3*x)**(1/2)/(-5+2*x)**(1/2)/(1+4*x)**(1/2),x)

[Out]

-5*(-3*x + 2)**(3/2)*sqrt(2*x - 5)*sqrt(4*x + 1)/12 + 365*sqrt(-3*x + 2)*sqrt(2*
x - 5)*sqrt(4*x + 1)/36 - 192863*sqrt(33)*sqrt(-12*x/11 + 8/11)*sqrt(-4*x/11 + 1
0/11)*elliptic_f(asin(sqrt(33)*sqrt(4*x + 1)/11), 1/3)/(864*sqrt(-3*x + 2)*sqrt(
2*x - 5)) - 44569*sqrt(33)*sqrt(-12*x/11 + 8/11)*sqrt(2*x - 5)*elliptic_e(asin(s
qrt(33)*sqrt(4*x + 1)/11), 1/3)/(864*sqrt(-3*x + 2)*sqrt(-4*x/11 + 10/11))

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Mathematica [A]  time = 0.302837, size = 120, normalized size = 0.72 \[ \frac{120 \sqrt{2-3 x} \sqrt{4 x+1} \left (18 x^2+89 x-335\right )-35066 \sqrt{66} \sqrt{5-2 x} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right )|\frac{1}{3}\right )+44569 \sqrt{66} \sqrt{5-2 x} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right )|\frac{1}{3}\right )}{864 \sqrt{2 x-5}} \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[2 - 3*x]*(7 + 5*x)^2)/(Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]),x]

[Out]

(120*Sqrt[2 - 3*x]*Sqrt[1 + 4*x]*(-335 + 89*x + 18*x^2) + 44569*Sqrt[66]*Sqrt[5
- 2*x]*EllipticE[ArcSin[Sqrt[3/11]*Sqrt[1 + 4*x]], 1/3] - 35066*Sqrt[66]*Sqrt[5
- 2*x]*EllipticF[ArcSin[Sqrt[3/11]*Sqrt[1 + 4*x]], 1/3])/(864*Sqrt[-5 + 2*x])

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Maple [A]  time = 0.022, size = 151, normalized size = 0.9 \[ -{\frac{1}{10368\,{x}^{3}-30240\,{x}^{2}+9072\,x+4320}\sqrt{2-3\,x}\sqrt{-5+2\,x}\sqrt{1+4\,x} \left ( 52599\,\sqrt{11}\sqrt{2-3\,x}\sqrt{5-2\,x}\sqrt{1+4\,x}{\it EllipticF} \left ( 2/11\,\sqrt{2-3\,x}\sqrt{11},i/2\sqrt{2} \right ) -44569\,\sqrt{11}\sqrt{2-3\,x}\sqrt{5-2\,x}\sqrt{1+4\,x}{\it EllipticE} \left ( 2/11\,\sqrt{2-3\,x}\sqrt{11},i/2\sqrt{2} \right ) -12960\,{x}^{4}-58680\,{x}^{3}+270060\,{x}^{2}-89820\,x-40200 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((7+5*x)^2*(2-3*x)^(1/2)/(-5+2*x)^(1/2)/(1+4*x)^(1/2),x)

[Out]

-1/432*(2-3*x)^(1/2)*(-5+2*x)^(1/2)*(1+4*x)^(1/2)*(52599*11^(1/2)*(2-3*x)^(1/2)*
(5-2*x)^(1/2)*(1+4*x)^(1/2)*EllipticF(2/11*(2-3*x)^(1/2)*11^(1/2),1/2*I*2^(1/2))
-44569*11^(1/2)*(2-3*x)^(1/2)*(5-2*x)^(1/2)*(1+4*x)^(1/2)*EllipticE(2/11*(2-3*x)
^(1/2)*11^(1/2),1/2*I*2^(1/2))-12960*x^4-58680*x^3+270060*x^2-89820*x-40200)/(24
*x^3-70*x^2+21*x+10)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x + 7\right )}^{2} \sqrt{-3 \, x + 2}}{\sqrt{4 \, x + 1} \sqrt{2 \, x - 5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 7)^2*sqrt(-3*x + 2)/(sqrt(4*x + 1)*sqrt(2*x - 5)),x, algorithm="maxima")

[Out]

integrate((5*x + 7)^2*sqrt(-3*x + 2)/(sqrt(4*x + 1)*sqrt(2*x - 5)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (25 \, x^{2} + 70 \, x + 49\right )} \sqrt{-3 \, x + 2}}{\sqrt{4 \, x + 1} \sqrt{2 \, x - 5}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 7)^2*sqrt(-3*x + 2)/(sqrt(4*x + 1)*sqrt(2*x - 5)),x, algorithm="fricas")

[Out]

integral((25*x^2 + 70*x + 49)*sqrt(-3*x + 2)/(sqrt(4*x + 1)*sqrt(2*x - 5)), 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((7+5*x)**2*(2-3*x)**(1/2)/(-5+2*x)**(1/2)/(1+4*x)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x + 7\right )}^{2} \sqrt{-3 \, x + 2}}{\sqrt{4 \, x + 1} \sqrt{2 \, x - 5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 7)^2*sqrt(-3*x + 2)/(sqrt(4*x + 1)*sqrt(2*x - 5)),x, algorithm="giac")

[Out]

integrate((5*x + 7)^2*sqrt(-3*x + 2)/(sqrt(4*x + 1)*sqrt(2*x - 5)), x)