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

Optimal. Leaf size=227 \[ \frac {397 \sqrt {\frac {3}{22}} \sqrt {5-2 x} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right ),\frac {1}{3}\right )}{89125 \sqrt {2 x-5}}+\frac {8953 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{556140 (5 x+7)}-\frac {\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{10 (5 x+7)^2}-\frac {8953 \sqrt {11} \sqrt {2 x-5} E\left (\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{1390350 \sqrt {5-2 x}}-\frac {14832503 \sqrt {5-2 x} \Pi \left (\frac {55}{124};\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{287339000 \sqrt {11} \sqrt {2 x-5}} \]

[Out]

397/1960750*EllipticF(1/11*33^(1/2)*(1+4*x)^(1/2),1/3*3^(1/2))*66^(1/2)*(5-2*x)^(1/2)/(-5+2*x)^(1/2)-14832503/
3160729000*EllipticPi(2/11*(2-3*x)^(1/2)*11^(1/2),55/124,1/2*I*2^(1/2))*(5-2*x)^(1/2)*11^(1/2)/(-5+2*x)^(1/2)-
8953/1390350*EllipticE(2/11*(2-3*x)^(1/2)*11^(1/2),1/2*I*2^(1/2))*11^(1/2)*(-5+2*x)^(1/2)/(5-2*x)^(1/2)-1/10*(
2-3*x)^(1/2)*(-5+2*x)^(1/2)*(1+4*x)^(1/2)/(7+5*x)^2+8953/556140*(2-3*x)^(1/2)*(-5+2*x)^(1/2)*(1+4*x)^(1/2)/(7+
5*x)

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Rubi [A]  time = 0.31, antiderivative size = 227, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 11, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.314, Rules used = {160, 1604, 1607, 168, 538, 537, 158, 114, 113, 121, 119} \[ \frac {8953 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{556140 (5 x+7)}-\frac {\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{10 (5 x+7)^2}+\frac {397 \sqrt {\frac {3}{22}} \sqrt {5-2 x} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right )|\frac {1}{3}\right )}{89125 \sqrt {2 x-5}}-\frac {8953 \sqrt {11} \sqrt {2 x-5} E\left (\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{1390350 \sqrt {5-2 x}}-\frac {14832503 \sqrt {5-2 x} \Pi \left (\frac {55}{124};\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{287339000 \sqrt {11} \sqrt {2 x-5}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/(10*(7 + 5*x)^2) + (8953*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4
*x])/(556140*(7 + 5*x)) - (8953*Sqrt[11]*Sqrt[-5 + 2*x]*EllipticE[ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2])/(
1390350*Sqrt[5 - 2*x]) + (397*Sqrt[3/22]*Sqrt[5 - 2*x]*EllipticF[ArcSin[Sqrt[3/11]*Sqrt[1 + 4*x]], 1/3])/(8912
5*Sqrt[-5 + 2*x]) - (14832503*Sqrt[5 - 2*x]*EllipticPi[55/124, ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2])/(287
339000*Sqrt[11]*Sqrt[-5 + 2*x])

Rule 113

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

Rule 114

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

Rule 119

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

Rule 121

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

Rule 158

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

Rule 160

Int[((a_.) + (b_.)*(x_))^(m_)*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)], x_Sy
mbol] :> Simp[((a + b*x)^(m + 1)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(b*(m + 1)), x] - Dist[1/(2*b*(m +
1)), Int[((a + b*x)^(m + 1)*Simp[d*e*g + c*f*g + c*e*h + 2*(d*f*g + d*e*h + c*f*h)*x + 3*d*f*h*x^2, x])/(Sqrt
[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m}, x] && IntegerQ[2*m] && Lt
Q[m, -1]

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 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)
])

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 1604

Int[(((a_.) + (b_.)*(x_))^(m_)*((A_.) + (B_.)*(x_) + (C_.)*(x_)^2))/(Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_
.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_Symbol] :> Simp[((A*b^2 - a*b*B + a^2*C)*(a + b*x)^(m + 1)*Sqrt[c + d*x]
*Sqrt[e + f*x]*Sqrt[g + h*x])/((m + 1)*(b*c - a*d)*(b*e - a*f)*(b*g - a*h)), x] - Dist[1/(2*(m + 1)*(b*c - a*d
)*(b*e - a*f)*(b*g - a*h)), Int[((a + b*x)^(m + 1)/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]))*Simp[A*(2*a^2*
d*f*h*(m + 1) - 2*a*b*(m + 1)*(d*f*g + d*e*h + c*f*h) + b^2*(2*m + 3)*(d*e*g + c*f*g + c*e*h)) - (b*B - a*C)*(
a*(d*e*g + c*f*g + c*e*h) + 2*b*c*e*g*(m + 1)) - 2*((A*b - a*B)*(a*d*f*h*(m + 1) - b*(m + 2)*(d*f*g + d*e*h +
c*f*h)) - C*(a^2*(d*f*g + d*e*h + c*f*h) - b^2*c*e*g*(m + 1) + a*b*(m + 1)*(d*e*g + c*f*g + c*e*h)))*x + d*f*h
*(2*m + 5)*(A*b^2 - a*b*B + a^2*C)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, A, B, C}, x] && IntegerQ[
2*m] && LtQ[m, -1]

Rule 1607

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.)*((g_.) + (h_.)*(x_)
)^(q_.), x_Symbol] :> Dist[PolynomialRemainder[Px, a + b*x, x], Int[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*(g + h
*x)^q, x], x] + Int[PolynomialQuotient[Px, a + b*x, x]*(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*(g + h*x)^q,
x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n, p, q}, x] && PolyQ[Px, x] && EqQ[m, -1]

Rubi steps

\begin {align*} \int \frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{(7+5 x)^3} \, dx &=-\frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{10 (7+5 x)^2}+\frac {1}{20} \int \frac {-21+140 x-72 x^2}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2} \, dx\\ &=-\frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{10 (7+5 x)^2}+\frac {8953 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{556140 (7+5 x)}+\frac {\int \frac {-106729-199200 x+214872 x^2}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)} \, dx}{1112280}\\ &=-\frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{10 (7+5 x)^2}+\frac {8953 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{556140 (7+5 x)}+\frac {\int \frac {-\frac {2500104}{25}+\frac {214872 x}{5}}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx}{1112280}+\frac {14832503 \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)} \, dx}{27807000}\\ &=-\frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{10 (7+5 x)^2}+\frac {8953 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{556140 (7+5 x)}+\frac {1191 \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx}{178250}+\frac {8953 \int \frac {\sqrt {-5+2 x}}{\sqrt {2-3 x} \sqrt {1+4 x}} \, dx}{463450}-\frac {14832503 \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 )}{13903500}\\ &=-\frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{10 (7+5 x)^2}+\frac {8953 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{556140 (7+5 x)}+\frac {\left (1191 \sqrt {5-2 x}\right ) \int \frac {1}{\sqrt {2-3 x} \sqrt {\frac {10}{11}-\frac {4 x}{11}} \sqrt {1+4 x}} \, dx}{89125 \sqrt {22} \sqrt {-5+2 x}}-\frac {\left (14832503 \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 )}{4634500 \sqrt {33} \sqrt {-5+2 x}}+\frac {\left (8953 \sqrt {-5+2 x}\right ) \int \frac {\sqrt {\frac {15}{11}-\frac {6 x}{11}}}{\sqrt {2-3 x} \sqrt {\frac {3}{11}+\frac {12 x}{11}}} \, dx}{463450 \sqrt {5-2 x}}\\ &=-\frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{10 (7+5 x)^2}+\frac {8953 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{556140 (7+5 x)}-\frac {8953 \sqrt {11} \sqrt {-5+2 x} E\left (\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{1390350 \sqrt {5-2 x}}+\frac {397 \sqrt {\frac {3}{22}} \sqrt {5-2 x} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{11}} \sqrt {1+4 x}\right )|\frac {1}{3}\right )}{89125 \sqrt {-5+2 x}}-\frac {14832503 \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 )}{287339000 \sqrt {11} \sqrt {-5+2 x}}\\ \end {align*}

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Mathematica [A]  time = 0.75, size = 134, normalized size = 0.59 \[ \frac {\sqrt {2 x-5} \left (\frac {\sqrt {11} \left (5759676 \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right ),-\frac {1}{2}\right )-61059460 E\left (\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )[phone] \Pi \left (\frac {55}{124};\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )\right )}{\sqrt {5-2 x}}+\frac {17050 \sqrt {2-3 x} \sqrt {4 x+1} (44765 x+7057)}{(5 x+7)^2}\right )}{9482187000} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[-5 + 2*x]*((17050*Sqrt[2 - 3*x]*Sqrt[1 + 4*x]*(7057 + 44765*x))/(7 + 5*x)^2 + (Sqrt[11]*(-61059460*Ellip
ticE[ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2] + 5759676*EllipticF[ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2] +
44497509*EllipticPi[55/124, ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2]))/Sqrt[5 - 2*x]))/9482187000

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fricas [F]  time = 0.72, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}}{125 \, x^{3} + 525 \, x^{2} + 735 \, x + 343}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(4*x + 1)*sqrt(2*x - 5)*sqrt(-3*x + 2)/(125*x^3 + 525*x^2 + 735*x + 343), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.03, size = 461, normalized size = 2.03 \[ \frac {\sqrt {-3 x +2}\, \sqrt {2 x -5}\, \sqrt {4 x +1}\, \left (18317838000 x^{4}-50539303100 x^{3}-1526486500 \sqrt {11}\, \sqrt {-3 x +2}\, \sqrt {-2 x +5}\, \sqrt {4 x +1}\, x^{2} \EllipticE \left (\frac {2 \sqrt {-33 x +22}}{11}, \frac {i \sqrt {2}}{2}\right )[phone] \sqrt {11}\, \sqrt {-3 x +2}\, \sqrt {-2 x +5}\, \sqrt {4 x +1}\, x^{2} \EllipticF \left (\frac {2 \sqrt {-33 x +22}}{11}, \frac {i \sqrt {2}}{2}\right )[phone] \sqrt {11}\, \sqrt {-3 x +2}\, \sqrt {-2 x +5}\, \sqrt {4 x +1}\, x^{2} \EllipticPi \left (\frac {2 \sqrt {-33 x +22}}{11}, \frac {55}{124}, \frac {i \sqrt {2}}{2}\right )[phone] x^{2}-4274162200 \sqrt {11}\, \sqrt {-3 x +2}\, \sqrt {-2 x +5}\, \sqrt {4 x +1}\, x \EllipticE \left (\frac {2 \sqrt {-33 x +22}}{11}, \frac {i \sqrt {2}}{2}\right )[phone] \sqrt {11}\, \sqrt {-3 x +2}\, \sqrt {-2 x +5}\, \sqrt {4 x +1}\, x \EllipticF \left (\frac {2 \sqrt {-33 x +22}}{11}, \frac {i \sqrt {2}}{2}\right )[phone] \sqrt {11}\, \sqrt {-3 x +2}\, \sqrt {-2 x +5}\, \sqrt {4 x +1}\, x \EllipticPi \left (\frac {2 \sqrt {-33 x +22}}{11}, \frac {55}{124}, \frac {i \sqrt {2}}{2}\right )[phone] x -2991913540 \sqrt {11}\, \sqrt {-3 x +2}\, \sqrt {-2 x +5}\, \sqrt {4 x +1}\, \EllipticE \left (\frac {2 \sqrt {-33 x +22}}{11}, \frac {i \sqrt {2}}{2}\right )[phone] \sqrt {11}\, \sqrt {-3 x +2}\, \sqrt {-2 x +5}\, \sqrt {4 x +1}\, \EllipticF \left (\frac {2 \sqrt {-33 x +22}}{11}, \frac {i \sqrt {2}}{2}\right )[phone] \sqrt {11}\, \sqrt {-3 x +2}\, \sqrt {-2 x +5}\, \sqrt {4 x +1}\, \EllipticPi \left (\frac {2 \sqrt {-33 x +22}}{11}, \frac {55}{124}, \frac {i \sqrt {2}}{2}\right )[phone]\right )}{9482187000 \left (24 x^{3}-70 x^{2}+21 x +10\right ) \left (5 x +7\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9482187000*(-3*x+2)^(1/2)*(2*x-5)^(1/2)*(4*x+1)^(1/2)*(143991900*11^(1/2)*(-3*x+2)^(1/2)*(-2*x+5)^(1/2)*(4*x
+1)^(1/2)*EllipticF(2/11*(-33*x+22)^(1/2),1/2*I*2^(1/2))*x^2-1526486500*11^(1/2)*(-3*x+2)^(1/2)*(-2*x+5)^(1/2)
*(4*x+1)^(1/2)*EllipticE(2/11*(-33*x+22)^(1/2),1/2*I*2^(1/2))*x^2+1112437725*11^(1/2)*(-3*x+2)^(1/2)*(-2*x+5)^
(1/2)*(4*x+1)^(1/2)*EllipticPi(2/11*(-33*x+22)^(1/2),55/124,1/2*I*2^(1/2))*x^2+403177320*11^(1/2)*(-3*x+2)^(1/
2)*(-2*x+5)^(1/2)*(4*x+1)^(1/2)*EllipticF(2/11*(-33*x+22)^(1/2),1/2*I*2^(1/2))*x-4274162200*11^(1/2)*(-3*x+2)^
(1/2)*(-2*x+5)^(1/2)*(4*x+1)^(1/2)*EllipticE(2/11*(-33*x+22)^(1/2),1/2*I*2^(1/2))*x+3114825630*11^(1/2)*(-3*x+
2)^(1/2)*(-2*x+5)^(1/2)*(4*x+1)^(1/2)*EllipticPi(2/11*(-33*x+22)^(1/2),55/124,1/2*I*2^(1/2))*x+282224124*11^(1
/2)*(-3*x+2)^(1/2)*(-2*x+5)^(1/2)*(4*x+1)^(1/2)*EllipticF(2/11*(-33*x+22)^(1/2),1/2*I*2^(1/2))-2991913540*11^(
1/2)*(-3*x+2)^(1/2)*(-2*x+5)^(1/2)*(4*x+1)^(1/2)*EllipticE(2/11*(-33*x+22)^(1/2),1/2*I*2^(1/2))[phone]*11^
(1/2)*(-3*x+2)^(1/2)*(-2*x+5)^(1/2)*(4*x+1)^(1/2)*EllipticPi(2/11*(-33*x+22)^(1/2),55/124,1/2*I*2^(1/2))+18317
838000*x^4-50539303100*x^3+7605578750*x^2+10159191350*x+1203218500)/(24*x^3-70*x^2+21*x+10)/(5*x+7)^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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