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]*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x])

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Rubi [A]  time = 0.0383713, 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, Rules used = {165, 537} \[ \frac{23 \sqrt{\frac{2-3 x}{5 x+7}} \sqrt{\frac{5-2 x}{5 x+7}} (5 x+7) \Pi \left (\frac{55}{124};\sin ^{-1}\left (\frac{\sqrt{\frac{31}{11}} \sqrt{4 x+1}}{\sqrt{5 x+7}}\right )|\frac{39}{62}\right )}{2 \sqrt{682} \sqrt{2-3 x} \sqrt{2 x-5}} \]

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]*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x])

Rule 165

Int[Sqrt[(a_.) + (b_.)*(x_)]/(Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_S
ymbol] :> Dist[(2*(a + b*x)*Sqrt[((b*g - a*h)*(c + d*x))/((d*g - c*h)*(a + b*x))]*Sqrt[((b*g - a*h)*(e + f*x))
/((f*g - e*h)*(a + b*x))])/(Sqrt[c + d*x]*Sqrt[e + f*x]), Subst[Int[1/((h - b*x^2)*Sqrt[1 + ((b*c - a*d)*x^2)/
(d*g - c*h)]*Sqrt[1 + ((b*e - a*f)*x^2)/(f*g - e*h)]), x], x, Sqrt[g + h*x]/Sqrt[a + b*x]], x] /; FreeQ[{a, b,
c, d, e, f, g, h}, x]

Rule 537

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1*Ellipt
icPi[(b*c)/(a*d), ArcSin[Rt[-(d/c), 2]*x], (c*f)/(d*e)])/(a*Sqrt[c]*Sqrt[e]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b,
c, d, e, f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && SimplerSqrtQ[-(f/e), -(d/c)
])

Rubi steps

\begin{align*} \int \frac{\sqrt{7+5 x}}{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}} \, dx &=\frac{\left (23 \sqrt{2} \sqrt{\frac{2-3 x}{7+5 x}} \sqrt{-\frac{-5+2 x}{7+5 x}} (7+5 x)\right ) \operatorname{Subst}\left (\int \frac{1}{\left (4-5 x^2\right ) \sqrt{1-\frac{31 x^2}{11}} \sqrt{1-\frac{39 x^2}{22}}} \, dx,x,\frac{\sqrt{1+4 x}}{\sqrt{7+5 x}}\right )}{11 \sqrt{2-3 x} \sqrt{-5+2 x}}\\ &=\frac{23 \sqrt{\frac{2-3 x}{7+5 x}} \sqrt{\frac{5-2 x}{7+5 x}} (7+5 x) \Pi \left (\frac{55}{124};\sin ^{-1}\left (\frac{\sqrt{\frac{31}{11}} \sqrt{1+4 x}}{\sqrt{7+5 x}}\right )|\frac{39}{62}\right )}{2 \sqrt{682} \sqrt{2-3 x} \sqrt{-5+2 x}}\\ \end{align*}

Mathematica [A]  time = 0.158327, size = 95, normalized size = 0.95 \[ -\frac{62 \sqrt{4 x+1} \sqrt{\frac{5-2 x}{5 x+7}} \Pi \left (-\frac{55}{69};\sin ^{-1}\left (\frac{\sqrt{\frac{23}{11}} \sqrt{2-3 x}}{\sqrt{5 x+7}}\right )|-\frac{39}{23}\right )}{3 \sqrt{253} \sqrt{2 x-5} \sqrt{\frac{4 x+1}{5 x+7}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-62*Sqrt[1 + 4*x]*Sqrt[(5 - 2*x)/(7 + 5*x)]*EllipticPi[-55/69, ArcSin[(Sqrt[23/11]*Sqrt[2 - 3*x])/Sqrt[7 + 5*
x]], -39/23])/(3*Sqrt[253]*Sqrt[-5 + 2*x]*Sqrt[(1 + 4*x)/(7 + 5*x)])

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Maple [B]  time = 0.021, size = 170, normalized size = 1.7 \begin{align*}{\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}{4\,x+1}}}},{\frac{\sqrt{31}\sqrt{78}}{39}} \right ) -{\it EllipticPi} \left ({\frac{\sqrt{31}\sqrt{11}}{31}\sqrt{{\frac{7+5\,x}{4\,x+1}}}},{\frac{124}{55}},{\frac{\sqrt{31}\sqrt{78}}{39}} \right ) \right ) \sqrt{{\frac{-2+3\,x}{4\,x+1}}}\sqrt{{\frac{2\,x-5}{4\,x+1}}}\sqrt{{\frac{7+5\,x}{4\,x+1}}} \left ( 4\,x+1 \right ) ^{{\frac{3}{2}}}\sqrt{2\,x-5}\sqrt{2-3\,x}\sqrt{7+5\,x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

23/858*(EllipticF(1/31*31^(1/2)*11^(1/2)*((7+5*x)/(4*x+1))^(1/2),1/39*31^(1/2)*78^(1/2))-EllipticPi(1/31*31^(1
/2)*11^(1/2)*((7+5*x)/(4*x+1))^(1/2),124/55,1/39*31^(1/2)*78^(1/2)))*((-2+3*x)/(4*x+1))^(1/2)*((2*x-5)/(4*x+1)
)^(1/2)*13^(1/2)*3^(1/2)*((7+5*x)/(4*x+1))^(1/2)*11^(1/2)*(4*x+1)^(3/2)*(2*x-5)^(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. \begin{align*} \int \frac{\sqrt{5 \, x + 7}}{\sqrt{4 \, x + 1} \sqrt{2 \, x - 5} \sqrt{-3 \, x + 2}}\,{d x} \end{align*}

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, 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. \begin{align*}{\rm integral}\left (-\frac{\sqrt{5 \, x + 7} \sqrt{4 \, x + 1} \sqrt{2 \, x - 5} \sqrt{-3 \, x + 2}}{24 \, x^{3} - 70 \, x^{2} + 21 \, x + 10}, x\right ) \end{align*}

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, algorithm="fricas")

[Out]

integral(-sqrt(5*x + 7)*sqrt(4*x + 1)*sqrt(2*x - 5)*sqrt(-3*x + 2)/(24*x^3 - 70*x^2 + 21*x + 10), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{5 x + 7}}{\sqrt{2 - 3 x} \sqrt{2 x - 5} \sqrt{4 x + 1}}\, dx \end{align*}

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]

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

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

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, algorithm="giac")

[Out]

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