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]*Sq
rt[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 = 0.0371325, 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, Rules used = {165, 537} \[ \frac{62 (2-3 x) \sqrt{\frac{5-2 x}{2-3 x}} \sqrt{-\frac{4 x+1}{2-3 x}} \Pi \left (-\frac{69}{55};\sin ^{-1}\left (\frac{\sqrt{\frac{11}{23}} \sqrt{5 x+7}}{\sqrt{2-3 x}}\right )|-\frac{23}{39}\right )}{5 \sqrt{429} \sqrt{2 x-5} \sqrt{4 x+1}} \]

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]*Sq
rt[7 + 5*x])/Sqrt[2 - 3*x]], -23/39])/(5*Sqrt[429]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*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{2-3 x}}{\sqrt{-5+2 x} \sqrt{1+4 x} \sqrt{7+5 x}} \, dx &=\frac{\left (62 (2-3 x) \sqrt{-\frac{-5+2 x}{2-3 x}} \sqrt{-\frac{1+4 x}{2-3 x}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{11 x^2}{23}} \sqrt{1+\frac{11 x^2}{39}} \left (5+3 x^2\right )} \, dx,x,\frac{\sqrt{7+5 x}}{\sqrt{2-3 x}}\right )}{\sqrt{897} \sqrt{-5+2 x} \sqrt{1+4 x}}\\ &=\frac{62 (2-3 x) \sqrt{\frac{5-2 x}{2-3 x}} \sqrt{-\frac{1+4 x}{2-3 x}} \Pi \left (-\frac{69}{55};\sin ^{-1}\left (\frac{\sqrt{\frac{11}{23}} \sqrt{7+5 x}}{\sqrt{2-3 x}}\right )|-\frac{23}{39}\right )}{5 \sqrt{429} \sqrt{-5+2 x} \sqrt{1+4 x}}\\ \end{align*}

Mathematica [A]  time = 0.546155, size = 170, normalized size = 1.68 \[ \frac{\sqrt{\frac{4 x+1}{5 x+7}} (5 x+7)^{3/2} \left (117 \sqrt{\frac{-6 x^2+19 x-10}{(5 x+7)^2}} \Pi \left (-\frac{55}{62};\sin ^{-1}\left (\sqrt{\frac{155-62 x}{55 x+77}}\right )|\frac{23}{62}\right )-62 \sqrt{\frac{5-2 x}{5 x+7}} \sqrt{\frac{3 x-2}{5 x+7}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{155-62 x}{55 x+77}}\right ),\frac{23}{62}\right )\right )}{5 \sqrt{682} \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}} \]

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 = 0.025, size = 172, normalized size = 1.7 \begin{align*}{\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}{4\,x+1}}},1/39\,\sqrt{31}\sqrt{78} \right ) +69\,{\it EllipticPi} \left ( 1/31\,\sqrt{31}\sqrt{11}\sqrt{{\frac{7+5\,x}{4\,x+1}}},{\frac{124}{55}},1/39\,\sqrt{31}\sqrt{78} \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{7+5\,x}\sqrt{2-3\,x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4290*(55*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))+69*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)*(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. \begin{align*} \int \frac{\sqrt{-3 \, x + 2}}{\sqrt{5 \, x + 7} \sqrt{4 \, x + 1} \sqrt{2 \, x - 5}}\,{d x} \end{align*}

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, 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. \begin{align*}{\rm integral}\left (\frac{\sqrt{5 \, x + 7} \sqrt{4 \, x + 1} \sqrt{2 \, x - 5} \sqrt{-3 \, x + 2}}{40 \, x^{3} - 34 \, x^{2} - 151 \, x - 35}, x\right ) \end{align*}

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, algorithm="fricas")

[Out]

integral(sqrt(5*x + 7)*sqrt(4*x + 1)*sqrt(2*x - 5)*sqrt(-3*x + 2)/(40*x^3 - 34*x^2 - 151*x - 35), x)

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

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]

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

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

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, algorithm="giac")

[Out]

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