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

Optimal. Leaf size=103 \[ -\frac{\sqrt{\frac{6}{11}} \sqrt{5-2 x} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right )|\frac{1}{3}\right )}{5 \sqrt{2 x-5}}-\frac{3 \sqrt{5-2 x} \Pi \left (\frac{55}{124};\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{5 \sqrt{11} \sqrt{2 x-5}} \]

[Out]

-(Sqrt[6/11]*Sqrt[5 - 2*x]*EllipticF[ArcSin[Sqrt[3/11]*Sqrt[1 + 4*x]], 1/3])/(5*
Sqrt[-5 + 2*x]) - (3*Sqrt[5 - 2*x]*EllipticPi[55/124, ArcSin[(2*Sqrt[2 - 3*x])/S
qrt[11]], -1/2])/(5*Sqrt[11]*Sqrt[-5 + 2*x])

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Rubi [A]  time = 0.476034, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171 \[ -\frac{\sqrt{\frac{6}{11}} \sqrt{5-2 x} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right )|\frac{1}{3}\right )}{5 \sqrt{2 x-5}}-\frac{3 \sqrt{5-2 x} \Pi \left (\frac{55}{124};\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{5 \sqrt{11} \sqrt{2 x-5}} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[6/11]*Sqrt[5 - 2*x]*EllipticF[ArcSin[Sqrt[3/11]*Sqrt[1 + 4*x]], 1/3])/(5*
Sqrt[-5 + 2*x]) - (3*Sqrt[5 - 2*x]*EllipticPi[55/124, ArcSin[(2*Sqrt[2 - 3*x])/S
qrt[11]], -1/2])/(5*Sqrt[11]*Sqrt[-5 + 2*x])

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Rubi in Sympy [A]  time = 25.5643, size = 138, normalized size = 1.34 \[ - \frac{3 \sqrt{11} \sqrt{- \frac{12 x}{11} + \frac{8}{11}} \sqrt{- \frac{4 x}{11} + \frac{10}{11}} F\left (\operatorname{asin}{\left (\frac{\sqrt{11} \sqrt{4 x + 1}}{11} \right )}\middle | 3\right )}{10 \sqrt{- 3 x + 2} \sqrt{2 x - 5}} + \frac{62 \sqrt{11} \sqrt{- \frac{12 x}{11} + \frac{8}{11}} \sqrt{- \frac{4 x}{11} + \frac{10}{11}} \Pi \left (- \frac{55}{23}; \operatorname{asin}{\left (\frac{\sqrt{11} \sqrt{4 x + 1}}{11} \right )}\middle | 3\right )}{115 \sqrt{- 3 x + 2} \sqrt{2 x - 5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*sqrt(11)*sqrt(-12*x/11 + 8/11)*sqrt(-4*x/11 + 10/11)*elliptic_f(asin(sqrt(11)
*sqrt(4*x + 1)/11), 3)/(10*sqrt(-3*x + 2)*sqrt(2*x - 5)) + 62*sqrt(11)*sqrt(-12*
x/11 + 8/11)*sqrt(-4*x/11 + 10/11)*elliptic_pi(-55/23, asin(sqrt(11)*sqrt(4*x +
1)/11), 3)/(115*sqrt(-3*x + 2)*sqrt(2*x - 5))

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Mathematica [A]  time = 0.220793, size = 70, normalized size = 0.68 \[ \frac{3 \sqrt{5-2 x} \left (F\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )+\Pi \left (\frac{55}{124};-\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )\right )}{5 \sqrt{22 x-55}} \]

Antiderivative was successfully verified.

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

[Out]

(3*Sqrt[5 - 2*x]*(EllipticF[ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2] + Elliptic
Pi[55/124, -ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2]))/(5*Sqrt[-55 + 22*x])

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Maple [A]  time = 0.021, size = 62, normalized size = 0.6 \[{\frac{3\,\sqrt{11}}{55} \left ({\it EllipticF} \left ({\frac{2\,\sqrt{11}}{11}\sqrt{2-3\,x}},{\frac{i}{2}}\sqrt{2} \right ) -{\it EllipticPi} \left ({\frac{2\,\sqrt{11}}{11}\sqrt{2-3\,x}},{\frac{55}{124}},{\frac{i}{2}}\sqrt{2} \right ) \right ) \sqrt{5-2\,x}{\frac{1}{\sqrt{-5+2\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/55*(EllipticF(2/11*(2-3*x)^(1/2)*11^(1/2),1/2*I*2^(1/2))-EllipticPi(2/11*(2-3*
x)^(1/2)*11^(1/2),55/124,1/2*I*2^(1/2)))*(5-2*x)^(1/2)*11^(1/2)/(-5+2*x)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(sqrt(-3*x + 2)/((5*x + 7)*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((2-3*x)**(1/2)/(7+5*x)/(-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. \[ \int \frac{\sqrt{-3 \, x + 2}}{{\left (5 \, x + 7\right )} \sqrt{4 \, x + 1} \sqrt{2 \, x - 5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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