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

Optimal. Leaf size=71 \[ \frac{2 \sqrt{5 x+7} F\left (\tan ^{-1}\left (\frac{\sqrt{4 x+1}}{\sqrt{2} \sqrt{2-3 x}}\right )|-\frac{39}{23}\right )}{\sqrt{253} \sqrt{2 x-5} \sqrt{\frac{5 x+7}{5-2 x}}} \]

[Out]

(2*Sqrt[7 + 5*x]*EllipticF[ArcTan[Sqrt[1 + 4*x]/(Sqrt[2]*Sqrt[2 - 3*x])], -39/23
])/(Sqrt[253]*Sqrt[-5 + 2*x]*Sqrt[(7 + 5*x)/(5 - 2*x)])

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Rubi [A]  time = 0.165811, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.054 \[ \frac{2 \sqrt{5 x+7} F\left (\tan ^{-1}\left (\frac{\sqrt{4 x+1}}{\sqrt{2} \sqrt{2-3 x}}\right )|-\frac{39}{23}\right )}{\sqrt{253} \sqrt{2 x-5} \sqrt{\frac{5 x+7}{5-2 x}}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[7 + 5*x]*EllipticF[ArcTan[Sqrt[1 + 4*x]/(Sqrt[2]*Sqrt[2 - 3*x])], -39/23
])/(Sqrt[253]*Sqrt[-5 + 2*x]*Sqrt[(7 + 5*x)/(5 - 2*x)])

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Rubi in Sympy [A]  time = 16.5437, size = 148, normalized size = 2.08 \[ \frac{2 \sqrt{506} \sqrt{\frac{- 156 x - 39}{- 110 x - 154}} \sqrt{\frac{117 x - 78}{55 x + 77}} \left (5 x + 7\right ) \sqrt{\frac{31 \left (2 x - 5\right )}{11 \left (5 x + 7\right )} + 1} F\left (\operatorname{atan}{\left (\frac{\sqrt{506} \sqrt{2 x - 5}}{22 \sqrt{5 x + 7}} \right )}\middle | - \frac{39}{23}\right )}{897 \sqrt{\frac{\frac{31 \left (2 x - 5\right )}{11 \left (5 x + 7\right )} + 1}{\frac{23 \left (2 x - 5\right )}{22 \left (5 x + 7\right )} + 1}} \sqrt{- 3 x + 2} \sqrt{4 x + 1} \sqrt{\frac{23 \left (2 x - 5\right )}{22 \left (5 x + 7\right )} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(506)*sqrt((-156*x - 39)/(-110*x - 154))*sqrt((117*x - 78)/(55*x + 77))*(5
*x + 7)*sqrt(31*(2*x - 5)/(11*(5*x + 7)) + 1)*elliptic_f(atan(sqrt(506)*sqrt(2*x
- 5)/(22*sqrt(5*x + 7))), -39/23)/(897*sqrt((31*(2*x - 5)/(11*(5*x + 7)) + 1)/(
23*(2*x - 5)/(22*(5*x + 7)) + 1))*sqrt(-3*x + 2)*sqrt(4*x + 1)*sqrt(23*(2*x - 5)
/(22*(5*x + 7)) + 1))

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Mathematica [A]  time = 0.341588, size = 118, normalized size = 1.66 \[ -\frac{\sqrt{\frac{2}{341}} \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1} F\left (\sin ^{-1}\left (\sqrt{\frac{155-62 x}{55 x+77}}\right )|\frac{23}{62}\right )}{\sqrt{\frac{5-2 x}{5 x+7}} \sqrt{\frac{3 x-2}{5 x+7}} \sqrt{\frac{4 x+1}{5 x+7}} (5 x+7)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

-((Sqrt[2/341]*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]*EllipticF[ArcSin[Sqrt[
(155 - 62*x)/(77 + 55*x)]], 23/62])/(Sqrt[(5 - 2*x)/(7 + 5*x)]*Sqrt[(-2 + 3*x)/(
7 + 5*x)]*Sqrt[(1 + 4*x)/(7 + 5*x)]*(7 + 5*x)^(3/2)))

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Maple [A]  time = 0.033, size = 140, normalized size = 2. \[{\frac{2\,\sqrt{13}\sqrt{3}\sqrt{11}}{12870\,{x}^{3}-22737\,{x}^{2}-35607\,x+30030}{\it EllipticF} \left ({\frac{\sqrt{31}\sqrt{11}}{31}\sqrt{{\frac{7+5\,x}{1+4\,x}}}},{\frac{\sqrt{2}\sqrt{3}\sqrt{31}\sqrt{13}}{39}} \right ) \sqrt{{\frac{-2+3\,x}{1+4\,x}}}\sqrt{{\frac{-5+2\,x}{1+4\,x}}}\sqrt{{\frac{7+5\,x}{1+4\,x}}} \left ( 1+4\,x \right ) ^{{\frac{3}{2}}}\sqrt{-5+2\,x}\sqrt{2-3\,x}\sqrt{7+5\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/429*EllipticF(1/31*31^(1/2)*11^(1/2)*((7+5*x)/(1+4*x))^(1/2),1/39*2^(1/2)*3^(1
/2)*31^(1/2)*13^(1/2))*((-2+3*x)/(1+4*x))^(1/2)*((-5+2*x)/(1+4*x))^(1/2)*13^(1/2
)*3^(1/2)*((7+5*x)/(1+4*x))^(1/2)*11^(1/2)*(1+4*x)^(3/2)*(-5+2*x)^(1/2)*(2-3*x)^
(1/2)*(7+5*x)^(1/2)/(30*x^3-53*x^2-83*x+70)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(1/(sqrt(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. \[ \int \frac{1}{\sqrt{- 3 x + 2} \sqrt{2 x - 5} \sqrt{4 x + 1} \sqrt{5 x + 7}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(7+5*x)**(1/2)/(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)*sqrt(5*x + 7)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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