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

Optimal. Leaf size=101 \[ \frac{62 (2-3 x) \sqrt{\frac{5-2 x}{2-3 x}} \sqrt{-\frac{4 x+1}{2-3 x}} \Pi \left (-\frac{69}{55};\sin ^{-1}\left (\frac{\sqrt{\frac{11}{23}} \sqrt{5 x+7}}{\sqrt{2-3 x}}\right )|-\frac{23}{39}\right )}{5 \sqrt{429} \sqrt{2 x-5} \sqrt{4 x+1}} \]

[Out]

(62*(2 - 3*x)*Sqrt[(5 - 2*x)/(2 - 3*x)]*Sqrt[-((1 + 4*x)/(2 - 3*x))]*EllipticPi[
-69/55, ArcSin[(Sqrt[11/23]*Sqrt[7 + 5*x])/Sqrt[2 - 3*x]], -23/39])/(5*Sqrt[429]
*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])

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Rubi [A]  time = 0.199089, antiderivative size = 101, 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 \[ \frac{62 (2-3 x) \sqrt{\frac{5-2 x}{2-3 x}} \sqrt{-\frac{4 x+1}{2-3 x}} \Pi \left (-\frac{69}{55};\sin ^{-1}\left (\frac{\sqrt{\frac{11}{23}} \sqrt{5 x+7}}{\sqrt{2-3 x}}\right )|-\frac{23}{39}\right )}{5 \sqrt{429} \sqrt{2 x-5} \sqrt{4 x+1}} \]

Antiderivative was successfully verified.

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

[Out]

(62*(2 - 3*x)*Sqrt[(5 - 2*x)/(2 - 3*x)]*Sqrt[-((1 + 4*x)/(2 - 3*x))]*EllipticPi[
-69/55, ArcSin[(Sqrt[11/23]*Sqrt[7 + 5*x])/Sqrt[2 - 3*x]], -23/39])/(5*Sqrt[429]
*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])

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Rubi in Sympy [A]  time = 39.0575, size = 294, normalized size = 2.91 \[ - \frac{\sqrt{2} \sqrt{\frac{- 22 x + 55}{- 66 x + 44}} \sqrt{\frac{- 55 x - 77}{69 x - 46}} \sqrt{1 + \frac{31 \left (4 x + 1\right )}{23 \left (- 3 x + 2\right )}} \left (- 3 x + 2\right ) F\left (\operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{4 x + 1}}{2 \sqrt{- 3 x + 2}} \right )}\middle | - \frac{39}{23}\right )}{\sqrt{\frac{1 + \frac{31 \left (4 x + 1\right )}{23 \left (- 3 x + 2\right )}}{1 + \frac{4 x + 1}{2 \left (- 3 x + 2\right )}}} \sqrt{1 + \frac{4 x + 1}{2 \left (- 3 x + 2\right )}} \sqrt{2 x - 5} \sqrt{5 x + 7}} + \frac{3 \sqrt{2} \sqrt{\frac{- 22 x + 55}{- 66 x + 44}} \sqrt{\frac{- 55 x - 77}{69 x - 46}} \sqrt{1 + \frac{31 \left (4 x + 1\right )}{23 \left (- 3 x + 2\right )}} \left (- 3 x + 2\right ) \Pi \left (- \frac{1}{2}; \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{4 x + 1}}{2 \sqrt{- 3 x + 2}} \right )}\middle | - \frac{39}{23}\right )}{2 \sqrt{\frac{1 + \frac{31 \left (4 x + 1\right )}{23 \left (- 3 x + 2\right )}}{1 + \frac{4 x + 1}{2 \left (- 3 x + 2\right )}}} \sqrt{1 + \frac{4 x + 1}{2 \left (- 3 x + 2\right )}} \sqrt{2 x - 5} \sqrt{5 x + 7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(2)*sqrt((-22*x + 55)/(-66*x + 44))*sqrt((-55*x - 77)/(69*x - 46))*sqrt(1 +
31*(4*x + 1)/(23*(-3*x + 2)))*(-3*x + 2)*elliptic_f(atan(sqrt(2)*sqrt(4*x + 1)/
(2*sqrt(-3*x + 2))), -39/23)/(sqrt((1 + 31*(4*x + 1)/(23*(-3*x + 2)))/(1 + (4*x
+ 1)/(2*(-3*x + 2))))*sqrt(1 + (4*x + 1)/(2*(-3*x + 2)))*sqrt(2*x - 5)*sqrt(5*x
+ 7)) + 3*sqrt(2)*sqrt((-22*x + 55)/(-66*x + 44))*sqrt((-55*x - 77)/(69*x - 46))
*sqrt(1 + 31*(4*x + 1)/(23*(-3*x + 2)))*(-3*x + 2)*elliptic_pi(-1/2, atan(sqrt(2
)*sqrt(4*x + 1)/(2*sqrt(-3*x + 2))), -39/23)/(2*sqrt((1 + 31*(4*x + 1)/(23*(-3*x
+ 2)))/(1 + (4*x + 1)/(2*(-3*x + 2))))*sqrt(1 + (4*x + 1)/(2*(-3*x + 2)))*sqrt(
2*x - 5)*sqrt(5*x + 7))

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Mathematica [A]  time = 0.547595, size = 170, normalized size = 1.68 \[ \frac{\sqrt{\frac{4 x+1}{5 x+7}} (5 x+7)^{3/2} \left (117 \sqrt{\frac{-6 x^2+19 x-10}{(5 x+7)^2}} \Pi \left (-\frac{55}{62};\sin ^{-1}\left (\sqrt{\frac{155-62 x}{55 x+77}}\right )|\frac{23}{62}\right )-62 \sqrt{\frac{5-2 x}{5 x+7}} \sqrt{\frac{3 x-2}{5 x+7}} F\left (\sin ^{-1}\left (\sqrt{\frac{155-62 x}{55 x+77}}\right )|\frac{23}{62}\right )\right )}{5 \sqrt{682} \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(Sqrt[(1 + 4*x)/(7 + 5*x)]*(7 + 5*x)^(3/2)*(-62*Sqrt[(5 - 2*x)/(7 + 5*x)]*Sqrt[(
-2 + 3*x)/(7 + 5*x)]*EllipticF[ArcSin[Sqrt[(155 - 62*x)/(77 + 55*x)]], 23/62] +
117*Sqrt[(-10 + 19*x - 6*x^2)/(7 + 5*x)^2]*EllipticPi[-55/62, ArcSin[Sqrt[(155 -
62*x)/(77 + 55*x)]], 23/62]))/(5*Sqrt[682]*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1
+ 4*x])

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Maple [B]  time = 0.038, size = 184, normalized size = 1.8 \[{\frac{\sqrt{13}\sqrt{3}\sqrt{11}}{128700\,{x}^{3}-227370\,{x}^{2}-356070\,x+300300} \left ( 55\,{\it EllipticF} \left ( 1/31\,\sqrt{31}\sqrt{11}\sqrt{{\frac{7+5\,x}{1+4\,x}}},1/39\,\sqrt{2}\sqrt{3}\sqrt{31}\sqrt{13} \right ) +69\,{\it EllipticPi} \left ( 1/31\,\sqrt{31}\sqrt{11}\sqrt{{\frac{7+5\,x}{1+4\,x}}},{\frac{124}{55}},1/39\,\sqrt{2}\sqrt{3}\sqrt{31}\sqrt{13} \right ) \right ) \sqrt{{\frac{-2+3\,x}{1+4\,x}}}\sqrt{{\frac{-5+2\,x}{1+4\,x}}}\sqrt{{\frac{7+5\,x}{1+4\,x}}} \left ( 1+4\,x \right ) ^{{\frac{3}{2}}}\sqrt{-5+2\,x}\sqrt{7+5\,x}\sqrt{2-3\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4290*(55*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))+69*EllipticPi(1/31*31^(1/2)*11^(1/2)*((7+5*x)/(1+4*x
))^(1/2),124/55,1/39*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)*(7+5*x)^(1/2)*(2-3*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. \[ \int \frac{\sqrt{-3 \, x + 2}}{\sqrt{5 \, x + 7} \sqrt{4 \, x + 1} \sqrt{2 \, x - 5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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