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]

23/1364*(7+5*x)*EllipticPi(1/11*341^(1/2)*(1+4*x)^(1/2)/(7+5*x)^(1/2),55/124,1/62*2418^(1/2))*682^(1/2)*((2-3*
x)/(7+5*x))^(1/2)*((5-2*x)/(7+5*x))^(1/2)/(2-3*x)^(1/2)/(-5+2*x)^(1/2)

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Rubi [A]  time = 0.04, antiderivative size = 100, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.17, 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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fricas [F]  time = 0.78, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {5 \, x + 7} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}}{24 \, x^{3} - 70 \, x^{2} + 21 \, x + 10}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {5 \, x + 7}}{\sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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)

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maple [B]  time = 0.03, size = 170, normalized size = 1.70 \[ \frac {23 \left (\EllipticF \left (\frac {\sqrt {31}\, \sqrt {11}\, \sqrt {\frac {5 x +7}{4 x +1}}}{31}, \frac {\sqrt {31}\, \sqrt {78}}{39}\right )-\EllipticPi \left (\frac {\sqrt {31}\, \sqrt {11}\, \sqrt {\frac {5 x +7}{4 x +1}}}{31}, \frac {124}{55}, \frac {\sqrt {31}\, \sqrt {78}}{39}\right )\right ) \sqrt {\frac {3 x -2}{4 x +1}}\, \sqrt {\frac {2 x -5}{4 x +1}}\, \sqrt {13}\, \sqrt {3}\, \sqrt {\frac {5 x +7}{4 x +1}}\, \sqrt {11}\, \left (4 x +1\right )^{\frac {3}{2}} \sqrt {2 x -5}\, \sqrt {-3 x +2}\, \sqrt {5 x +7}}{858 \left (30 x^{3}-53 x^{2}-83 x +70\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x+7)^(1/2)/(-3*x+2)^(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)*((5*x+7)/(4*x+1))^(1/2),1/39*31^(1/2)*78^(1/2))-EllipticPi(1/31*31^(1
/2)*11^(1/2)*((5*x+7)/(4*x+1))^(1/2),124/55,1/39*31^(1/2)*78^(1/2)))*((3*x-2)/(4*x+1))^(1/2)*((2*x-5)/(4*x+1))
^(1/2)*13^(1/2)*3^(1/2)*((5*x+7)/(4*x+1))^(1/2)*11^(1/2)*(4*x+1)^(3/2)*(2*x-5)^(1/2)*(-3*x+2)^(1/2)*(5*x+7)^(1
/2)/(30*x^3-53*x^2-83*x+70)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {5 \, x + 7}}{\sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {5\,x+7}}{\sqrt {2-3\,x}\,\sqrt {4\,x+1}\,\sqrt {2\,x-5}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {5 x + 7}}{\sqrt {2 - 3 x} \sqrt {2 x - 5} \sqrt {4 x + 1}}\, dx \]

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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