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

Optimal. Leaf size=100 \[ \frac{23 \sqrt{\frac{2-3 x}{5 x+7}} \sqrt{\frac{5-2 x}{5 x+7}} (5 x+7) \Pi \left (\frac{55}{124};\sin ^{-1}\left (\frac{\sqrt{\frac{31}{11}} \sqrt{4 x+1}}{\sqrt{5 x+7}}\right )|\frac{39}{62}\right )}{2 \sqrt{682} \sqrt{2-3 x} \sqrt{2 x-5}} \]

[Out]

(23*Sqrt[(2 - 3*x)/(7 + 5*x)]*Sqrt[(5 - 2*x)/(7 + 5*x)]*(7 + 5*x)*EllipticPi[55/
124, ArcSin[(Sqrt[31/11]*Sqrt[1 + 4*x])/Sqrt[7 + 5*x]], 39/62])/(2*Sqrt[682]*Sqr
t[2 - 3*x]*Sqrt[-5 + 2*x])

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Rubi [A]  time = 0.196189, antiderivative size = 100, 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 \[ \frac{23 \sqrt{\frac{2-3 x}{5 x+7}} \sqrt{\frac{5-2 x}{5 x+7}} (5 x+7) \Pi \left (\frac{55}{124};\sin ^{-1}\left (\frac{\sqrt{\frac{31}{11}} \sqrt{4 x+1}}{\sqrt{5 x+7}}\right )|\frac{39}{62}\right )}{2 \sqrt{682} \sqrt{2-3 x} \sqrt{2 x-5}} \]

Antiderivative was successfully verified.

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

[Out]

(23*Sqrt[(2 - 3*x)/(7 + 5*x)]*Sqrt[(5 - 2*x)/(7 + 5*x)]*(7 + 5*x)*EllipticPi[55/
124, ArcSin[(Sqrt[31/11]*Sqrt[1 + 4*x])/Sqrt[7 + 5*x]], 39/62])/(2*Sqrt[682]*Sqr
t[2 - 3*x]*Sqrt[-5 + 2*x])

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Rubi in Sympy [A]  time = 22.3187, size = 87, normalized size = 0.87 \[ \frac{\sqrt{341} \sqrt{\frac{69 x - 46}{- 55 x - 77}} \sqrt{\frac{- 46 x + 115}{110 x + 154}} \left (5 x + 7\right ) \Pi \left (\frac{55}{124}; \operatorname{asin}{\left (\frac{\sqrt{341} \sqrt{4 x + 1}}{11 \sqrt{5 x + 7}} \right )}\middle | \frac{39}{62}\right )}{62 \sqrt{- 3 x + 2} \sqrt{2 x - 5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(341)*sqrt((69*x - 46)/(-55*x - 77))*sqrt((-46*x + 115)/(110*x + 154))*(5*x
+ 7)*elliptic_pi(55/124, asin(sqrt(341)*sqrt(4*x + 1)/(11*sqrt(5*x + 7))), 39/62
)/(62*sqrt(-3*x + 2)*sqrt(2*x - 5))

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Mathematica [A]  time = 0.616849, size = 185, normalized size = 1.85 \[ -\frac{23 \sqrt{\frac{3 x-2}{4 x+1}} (4 x+1)^{3/2} \left (\sqrt{\frac{2 x-5}{4 x+1}} \sqrt{\frac{5 x+7}{4 x+1}} F\left (\sin ^{-1}\left (\sqrt{\frac{22}{39}} \sqrt{\frac{5 x+7}{4 x+1}}\right )|\frac{39}{62}\right )-\sqrt{\frac{10 x^2-11 x-35}{(4 x+1)^2}} \Pi \left (\frac{78}{55};\sin ^{-1}\left (\sqrt{\frac{22}{39}} \sqrt{\frac{5 x+7}{4 x+1}}\right )|\frac{39}{62}\right )\right )}{2 \sqrt{682} \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{5 x+7}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-23*Sqrt[(-2 + 3*x)/(1 + 4*x)]*(1 + 4*x)^(3/2)*(Sqrt[(-5 + 2*x)/(1 + 4*x)]*Sqrt
[(7 + 5*x)/(1 + 4*x)]*EllipticF[ArcSin[Sqrt[22/39]*Sqrt[(7 + 5*x)/(1 + 4*x)]], 3
9/62] - Sqrt[(-35 - 11*x + 10*x^2)/(1 + 4*x)^2]*EllipticPi[78/55, ArcSin[Sqrt[22
/39]*Sqrt[(7 + 5*x)/(1 + 4*x)]], 39/62]))/(2*Sqrt[682]*Sqrt[2 - 3*x]*Sqrt[-5 + 2
*x]*Sqrt[7 + 5*x])

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Maple [B]  time = 0.031, size = 182, normalized size = 1.8 \[{\frac{23\,\sqrt{13}\sqrt{3}\sqrt{11}}{25740\,{x}^{3}-45474\,{x}^{2}-71214\,x+60060} \left ({\it EllipticF} \left ({\frac{\sqrt{31}\sqrt{11}}{31}\sqrt{{\frac{7+5\,x}{1+4\,x}}}},{\frac{\sqrt{2}\sqrt{3}\sqrt{31}\sqrt{13}}{39}} \right ) -{\it EllipticPi} \left ({\frac{\sqrt{31}\sqrt{11}}{31}\sqrt{{\frac{7+5\,x}{1+4\,x}}}},{\frac{124}{55}},{\frac{\sqrt{2}\sqrt{3}\sqrt{31}\sqrt{13}}{39}} \right ) \right ) \sqrt{{\frac{-2+3\,x}{1+4\,x}}}\sqrt{{\frac{-5+2\,x}{1+4\,x}}}\sqrt{{\frac{7+5\,x}{1+4\,x}}} \left ( 1+4\,x \right ) ^{{\frac{3}{2}}}\sqrt{-5+2\,x}\sqrt{2-3\,x}\sqrt{7+5\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

23/858*(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))-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)*(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. \[ \int \frac{\sqrt{5 \, x + 7}}{\sqrt{4 \, x + 1} \sqrt{2 \, x - 5} \sqrt{-3 \, x + 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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