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

Optimal. Leaf size=263 \[ \frac {24957247 \sqrt {5-2 x} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right ),\frac {1}{3}\right )}{4956597750 \sqrt {66} \sqrt {2 x-5}}+\frac {16830401 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{30929169960 (5 x+7)}+\frac {8953 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{1668420 (5 x+7)^2}-\frac {\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{15 (5 x+7)^3}-\frac {16830401 \sqrt {11} \sqrt {2 x-5} E\left (\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{77322924900 \sqrt {5-2 x}}+\frac {15664616449 \sqrt {5-2 x} \Pi \left (\frac {55}{124};\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{15980071146000 \sqrt {11} \sqrt {2 x-5}} \]

[Out]

15664616449/175780782606000*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)[phone]/327135451500*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)-16830401/77322924900*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)-1/15*(2-3*x)^(1/2)*(-5+2*x)^(1/2)*(1+4*x)^(1/2)/(7+5*x)^3+8953/1668420*(2-3*x)^(1/2)*(-5+2
*x)^(1/2)*(1+4*x)^(1/2)/(7+5*x)^2+16830401/30929169960*(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 = 0.40, antiderivative size = 263, normalized size of antiderivative = 1.00, number of steps used = 12, 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} \[ \frac {16830401 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{30929169960 (5 x+7)}+\frac {8953 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{1668420 (5 x+7)^2}-\frac {\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{15 (5 x+7)^3}+\frac {24957247 \sqrt {5-2 x} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right )|\frac {1}{3}\right )}{4956597750 \sqrt {66} \sqrt {2 x-5}}-\frac {16830401 \sqrt {11} \sqrt {2 x-5} E\left (\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{77322924900 \sqrt {5-2 x}}+\frac {15664616449 \sqrt {5-2 x} \Pi \left (\frac {55}{124};\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{15980071146000 \sqrt {11} \sqrt {2 x-5}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/(15*(7 + 5*x)^3) + (8953*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4
*x])/(1668420*(7 + 5*x)^2) + (16830401*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/(30929169960*(7 + 5*x)) - (
16830401*Sqrt[11]*Sqrt[-5 + 2*x]*EllipticE[ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2])/(77322924900*Sqrt[5 - 2*
x]) + (24957247*Sqrt[5 - 2*x]*EllipticF[ArcSin[Sqrt[3/11]*Sqrt[1 + 4*x]], 1/3])/(4956597750*Sqrt[66]*Sqrt[-5 +
2*x]) + (15664616449*Sqrt[5 - 2*x]*EllipticPi[55/124, ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2])/(15980071146
000*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)^4} \, dx &=-\frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{15 (7+5 x)^3}+\frac {1}{30} \int \frac {-21+140 x-72 x^2}{\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}}{15 (7+5 x)^3}+\frac {8953 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{1668420 (7+5 x)^2}+\frac {\int \frac {-401471+855020 x-214872 x^2}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2} \, dx}{3336840}\\ &=-\frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{15 (7+5 x)^3}+\frac {8953 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{1668420 (7+5 x)^2}+\frac {16830401 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{30929169960 (7+5 x)}+\frac {\int \frac {-2850617379+1003030560 x+1211788872 x^2}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)} \, dx}{185575019760}\\ &=-\frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{15 (7+5 x)^3}+\frac {8953 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{1668420 (7+5 x)^2}+\frac {16830401 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{30929169960 (7+5 x)}+\frac {\int \frac {-\frac {3467369304}{25}+\frac {1211788872 x}{5}}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx}{185575019760}-\frac {15664616449 \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)} \, dx}{1546458498000}\\ &=-\frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{15 (7+5 x)^3}+\frac {8953 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{1668420 (7+5 x)^2}+\frac {16830401 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{30929169960 (7+5 x)}+\frac {16830401 \int \frac {\sqrt {-5+2 x}}{\sqrt {2-3 x} \sqrt {1+4 x}} \, dx}{25774308300}+\frac {24957247 \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx}{9913195500}+\frac {15664616449 \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 )}{773229249000}\\ &=-\frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{15 (7+5 x)^3}+\frac {8953 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{1668420 (7+5 x)^2}+\frac {16830401 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{30929169960 (7+5 x)}+\frac {\left (24957247 \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}{4956597750 \sqrt {22} \sqrt {-5+2 x}}+\frac {\left (15664616449 \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 )}{257743083000 \sqrt {33} \sqrt {-5+2 x}}+\frac {\left (16830401 \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}{25774308300 \sqrt {5-2 x}}\\ &=-\frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{15 (7+5 x)^3}+\frac {8953 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{1668420 (7+5 x)^2}+\frac {16830401 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{30929169960 (7+5 x)}-\frac {16830401 \sqrt {11} \sqrt {-5+2 x} E\left (\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{77322924900 \sqrt {5-2 x}}+\frac {24957247 \sqrt {5-2 x} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{11}} \sqrt {1+4 x}\right )|\frac {1}{3}\right )}{4956597750 \sqrt {66} \sqrt {-5+2 x}}+\frac {15664616449 \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 )}{15980071146000 \sqrt {11} \sqrt {-5+2 x}}\\ \end {align*}

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Mathematica [A]  time = 0.79, size = 139, normalized size = 0.53 \[ \frac {\sqrt {2 x-5} \left (\frac {\sqrt {11} \left (120693246492 \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right ),-\frac {1}{2}\right )-114783334820 E\left (\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )-46993849347 \Pi \left (\frac {55}{124};\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )\right )}{\sqrt {5-2 x}}+\frac {17050 \sqrt {2-3 x} \sqrt {4 x+1} \left (420760025 x^2+2007981640 x-75460017\right )}{(5 x+7)^3}\right )}{527342347818000} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[-5 + 2*x]*((17050*Sqrt[2 - 3*x]*Sqrt[1 + 4*x]*(-75460017 + 2007981640*x + 420760025*x^2))/(7 + 5*x)^3 +
(Sqrt[11]*(-114783334820*EllipticE[ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2] + 120693246492*EllipticF[ArcSin[(
2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2] - 46993849347*EllipticPi[55/124, ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2]))
/Sqrt[5 - 2*x]))/527342347818000

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fricas [F]  time = 1.10, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}}{625 \, x^{4} + 3500 \, x^{3} + 7350 \, x^{2} + 6860 \, x + 2401}, x\right ) \]

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)^4,x, algorithm="fricas")

[Out]

integral(sqrt(4*x + 1)*sqrt(2*x - 5)*sqrt(-3*x + 2)/(625*x^4 + 3500*x^3 + 7350*x^2 + 6860*x + 2401), x)

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

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)^4,x, algorithm="giac")

[Out]

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

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maple [B]  time = 0.03, size = 602, normalized size = 2.29 \[ \frac {\sqrt {-3 x +2}\, \sqrt {2 x -5}\, \sqrt {4 x +1}\, \left (172175002230000 x^{5}[phone] x^{4}-14347916852500 \sqrt {11}\, \sqrt {-3 x +2}\, \sqrt {-2 x +5}\, \sqrt {4 x +1}\, x^{3} \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^{3} \EllipticF \left (\frac {2 \sqrt {-33 x +22}}{11}, \frac {i \sqrt {2}}{2}\right )-5874231168375 \sqrt {11}\, \sqrt {-3 x +2}\, \sqrt {-2 x +5}\, \sqrt {4 x +1}\, x^{3} \EllipticPi \left (\frac {2 \sqrt {-33 x +22}}{11}, \frac {55}{124}, \frac {i \sqrt {2}}{2}\right )-2276751199345150 x^{3}-60261250780500 \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 )-24671770907175 \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}-84365751092700 \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 )-34540479270045 \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 -39370683843260 \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 )-16118890326021 \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 )-12865932898500\right )}{527342347818000 \left (24 x^{3}-70 x^{2}+21 x +10\right ) \left (5 x +7\right )^{3}} \]

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)^4,x)

[Out]

1/527342347818000*(-3*x+2)^(1/2)*(2*x-5)^(1/2)*(4*x+1)^(1/2)*(15086655811500*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^3-14347916852500*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^3-5874231168375*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^3+63363954408300
*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^2-60261
250780500*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^2-24671770907175*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),55/12
4,1/2*I*2^(1/2))*x^2+88709536171620*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-84365751092700*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-34540479270045*11^(1/2)*(-3*x+2)^(1/2)*(-2*x+5)^(1/2)*(4*x+1)^(1/2)*Ellipti
cPi(2/11*(-33*x+22)^(1/2),55/124,1/2*I*2^(1/2))*x+41397783546756*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))-39370683843260*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))-16118890326021*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))[phone]*x^5+319488997250500*x
^4-2276751199345150*x^3+880758940754000*x^2+315342410533150*x-12865932898500)/(24*x^3-70*x^2+21*x+10)/(5*x+7)^
3

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

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)^4,x, algorithm="maxima")

[Out]

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

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

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)^4,x)

[Out]

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

[Out]

Timed out

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