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

Optimal. Leaf size=51 \[ -\frac{3 \sqrt{5-2 x} \Pi \left (\frac{55}{124};\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{31 \sqrt{11} \sqrt{2 x-5}} \]

[Out]

(-3*Sqrt[5 - 2*x]*EllipticPi[55/124, ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2])/
(31*Sqrt[11]*Sqrt[-5 + 2*x])

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Rubi [A]  time = 0.285143, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.086 \[ -\frac{3 \sqrt{5-2 x} \Pi \left (\frac{55}{124};\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{31 \sqrt{11} \sqrt{2 x-5}} \]

Antiderivative was successfully verified.

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

[Out]

(-3*Sqrt[5 - 2*x]*EllipticPi[55/124, ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2])/
(31*Sqrt[11]*Sqrt[-5 + 2*x])

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Rubi in Sympy [A]  time = 10.1836, size = 73, normalized size = 1.43 \[ \frac{\sqrt{22} i \sqrt{- \frac{6 x}{11} + \frac{15}{11}} \sqrt{\frac{12 x}{11} + \frac{3}{11}} \Pi \left (- \frac{55}{62}; i \operatorname{asinh}{\left (\frac{\sqrt{22} \sqrt{- 3 x + 2}}{11} \right )}\middle | -2\right )}{31 \sqrt{2 x - 5} \sqrt{4 x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(22)*I*sqrt(-6*x/11 + 15/11)*sqrt(12*x/11 + 3/11)*elliptic_pi(-55/62, I*asin
h(sqrt(22)*sqrt(-3*x + 2)/11), -2)/(31*sqrt(2*x - 5)*sqrt(4*x + 1))

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Mathematica [A]  time = 0.314477, size = 99, normalized size = 1.94 \[ -\frac{3 (3 x-2) \sqrt{\frac{8 x^2-18 x-5}{(2-3 x)^2}} \left (F\left (\left .\sin ^{-1}\left (\frac{\sqrt{11}}{2 \sqrt{2-3 x}}\right )\right |-2\right )+\Pi \left (\frac{124}{55};\left .-\sin ^{-1}\left (\frac{\sqrt{11}}{2 \sqrt{2-3 x}}\right )\right |-2\right )\right )}{31 \sqrt{4 x+1} \sqrt{11 x-\frac{55}{2}}} \]

Antiderivative was successfully verified.

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

[Out]

(-3*(-2 + 3*x)*Sqrt[(-5 - 18*x + 8*x^2)/(2 - 3*x)^2]*(EllipticF[ArcSin[Sqrt[11]/
(2*Sqrt[2 - 3*x])], -2] + EllipticPi[124/55, -ArcSin[Sqrt[11]/(2*Sqrt[2 - 3*x])]
, -2]))/(31*Sqrt[1 + 4*x]*Sqrt[-55/2 + 11*x])

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Maple [A]  time = 0.023, size = 40, normalized size = 0.8 \[ -{\frac{3\,\sqrt{11}}{341}{\it EllipticPi} \left ({\frac{2\,\sqrt{11}}{11}\sqrt{2-3\,x}},{\frac{55}{124}},{\frac{i}{2}}\sqrt{2} \right ) \sqrt{5-2\,x}{\frac{1}{\sqrt{-5+2\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/341*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. \[ \int \frac{1}{{\left (5 \, x + 7\right )} \sqrt{4 \, x + 1} \sqrt{2 \, x - 5} \sqrt{-3 \, x + 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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