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

Optimal. Leaf size=51 \[ -\frac{3 \sqrt{5-2 x} \Pi \left (\frac{55}{124};\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{31 \sqrt{11} \sqrt{2 x-5}} \]

[Out]

(-3*Sqrt[5 - 2*x]*EllipticPi[55/124, ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2])/(31*Sqrt[11]*Sqrt[-5 + 2*x])

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Rubi [A]  time = 0.0700453, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.086, Rules used = {168, 538, 537} \[ -\frac{3 \sqrt{5-2 x} \Pi \left (\frac{55}{124};\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{31 \sqrt{11} \sqrt{2 x-5}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*Sqrt[5 - 2*x]*EllipticPi[55/124, ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2])/(31*Sqrt[11]*Sqrt[-5 + 2*x])

Rule 168

Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_Sym
bol] :> Dist[-2, Subst[Int[1/(Simp[b*c - a*d - b*x^2, x]*Sqrt[Simp[(d*e - c*f)/d + (f*x^2)/d, x]]*Sqrt[Simp[(d
*g - c*h)/d + (h*x^2)/d, x]]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && GtQ[(d*e - c
*f)/d, 0]

Rule 538

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

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

Mathematica [A]  time = 0.468092, size = 99, normalized size = 1.94 \[ -\frac{3 (3 x-2) \sqrt{\frac{8 x^2-18 x-5}{(2-3 x)^2}} \left (\text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{11}}{2 \sqrt{2-3 x}}\right ),-2\right )+\Pi \left (\frac{124}{55};\left .-\sin ^{-1}\left (\frac{\sqrt{11}}{2 \sqrt{2-3 x}}\right )\right |-2\right )\right )}{31 \sqrt{4 x+1} \sqrt{11 x-\frac{55}{2}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*(-2 + 3*x)*Sqrt[(-5 - 18*x + 8*x^2)/(2 - 3*x)^2]*(EllipticF[ArcSin[Sqrt[11]/(2*Sqrt[2 - 3*x])], -2] + Elli
pticPi[124/55, -ArcSin[Sqrt[11]/(2*Sqrt[2 - 3*x])], -2]))/(31*Sqrt[1 + 4*x]*Sqrt[-55/2 + 11*x])

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Maple [A]  time = 0.016, size = 37, normalized size = 0.7 \begin{align*} -{\frac{3\,\sqrt{11}}{341}{\it EllipticPi} \left ({\frac{2}{11}\sqrt{22-33\,x}},{\frac{55}{124}},{\frac{i}{2}}\sqrt{2} \right ) \sqrt{5-2\,x}{\frac{1}{\sqrt{2\,x-5}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/341*EllipticPi(2/11*(22-33*x)^(1/2),55/124,1/2*I*2^(1/2))*(5-2*x)^(1/2)*11^(1/2)/(2*x-5)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (5 \, x + 7\right )} \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(1/(7+5*x)/(2-3*x)^(1/2)/(-5+2*x)^(1/2)/(1+4*x)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/((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{4 \, x + 1} \sqrt{2 \, x - 5} \sqrt{-3 \, x + 2}}{120 \, x^{4} - 182 \, x^{3} - 385 \, x^{2} + 197 \, x + 70}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(7+5*x)/(2-3*x)^(1/2)/(-5+2*x)^(1/2)/(1+4*x)^(1/2),x, algorithm="fricas")

[Out]

integral(-sqrt(4*x + 1)*sqrt(2*x - 5)*sqrt(-3*x + 2)/(120*x^4 - 182*x^3 - 385*x^2 + 197*x + 70), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (5 \, x + 7\right )} \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(1/(7+5*x)/(2-3*x)^(1/2)/(-5+2*x)^(1/2)/(1+4*x)^(1/2),x, algorithm="giac")

[Out]

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