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 \[ \frac{24957247 \sqrt{5-2 x} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right ),\frac{1}{3}\right )}{4956597750 \sqrt{66} \sqrt{2 x-5}}+\frac{16830401 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{30929169960 (5 x+7)}+\frac{8953 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{1668420 (5 x+7)^2}-\frac{\sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{15 (5 x+7)^3}-\frac{16830401 \sqrt{11} \sqrt{2 x-5} E\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{77322924900 \sqrt{5-2 x}}+\frac{15664616449 \sqrt{5-2 x} \Pi \left (\frac{55}{124};\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{15980071146000 \sqrt{11} \sqrt{2 x-5}} \]
[Out]
-(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])
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Rubi [A] time = 0.397013, antiderivative size = 263, normalized size of antiderivative = 1.,
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 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 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]
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)
])
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 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 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 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 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)]))
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*}
Mathematica [A] time = 0.753849, size = 141, normalized size = 0.54 \[ \frac{\sqrt{2 x-5} \left (\frac{\sqrt{11} \left (120693246492 \text{EllipticF}\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right ),-\frac{1}{2}\right )-114783334820 E\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )+46993849347 \Pi \left (\frac{55}{124};-\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )\right )}{\sqrt{5-2 x}}+\frac{17050 \sqrt{2-3 x} \sqrt{4 x+1} \left (420760025 x^2+2007981640 x-75460017\right )}{(5 x+7)^3}\right )}{527342347818000} \]
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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Maple [B] time = 0.026, size = 602, normalized size = 2.3 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((2-3*x)^(1/2)*(2*x-5)^(1/2)*(4*x+1)^(1/2)/(7+5*x)^4,x)
[Out]
1/527342347818000*(2-3*x)^(1/2)*(2*x-5)^(1/2)*(4*x+1)^(1/2)*(15086655811500*11^(1/2)*(2-3*x)^(1/2)*(5-2*x)^(1/
2)*(4*x+1)^(1/2)*EllipticF(2/11*(22-33*x)^(1/2),1/2*I*2^(1/2))*x^3-14347916852500*11^(1/2)*(2-3*x)^(1/2)*(5-2*
x)^(1/2)*(4*x+1)^(1/2)*EllipticE(2/11*(22-33*x)^(1/2),1/2*I*2^(1/2))*x^3-5874231168375*11^(1/2)*(2-3*x)^(1/2)*
(5-2*x)^(1/2)*(4*x+1)^(1/2)*EllipticPi(2/11*(22-33*x)^(1/2),55/124,1/2*I*2^(1/2))*x^3+63363954408300*11^(1/2)*
(2-3*x)^(1/2)*(5-2*x)^(1/2)*(4*x+1)^(1/2)*EllipticF(2/11*(22-33*x)^(1/2),1/2*I*2^(1/2))*x^2-60261250780500*11^
(1/2)*(2-3*x)^(1/2)*(5-2*x)^(1/2)*(4*x+1)^(1/2)*EllipticE(2/11*(22-33*x)^(1/2),1/2*I*2^(1/2))*x^2-246717709071
75*11^(1/2)*(2-3*x)^(1/2)*(5-2*x)^(1/2)*(4*x+1)^(1/2)*EllipticPi(2/11*(22-33*x)^(1/2),55/124,1/2*I*2^(1/2))*x^
2+88709536171620*11^(1/2)*(2-3*x)^(1/2)*(5-2*x)^(1/2)*(4*x+1)^(1/2)*EllipticF(2/11*(22-33*x)^(1/2),1/2*I*2^(1/
2))*x-84365751092700*11^(1/2)*(2-3*x)^(1/2)*(5-2*x)^(1/2)*(4*x+1)^(1/2)*EllipticE(2/11*(22-33*x)^(1/2),1/2*I*2
^(1/2))*x-34540479270045*11^(1/2)*(2-3*x)^(1/2)*(5-2*x)^(1/2)*(4*x+1)^(1/2)*EllipticPi(2/11*(22-33*x)^(1/2),55
/124,1/2*I*2^(1/2))*x+41397783546756*11^(1/2)*(2-3*x)^(1/2)*(5-2*x)^(1/2)*(4*x+1)^(1/2)*EllipticF(2/11*(22-33*
x)^(1/2),1/2*I*2^(1/2))-39370683843260*11^(1/2)*(2-3*x)^(1/2)*(5-2*x)^(1/2)*(4*x+1)^(1/2)*EllipticE(2/11*(22-3
3*x)^(1/2),1/2*I*2^(1/2))-16118890326021*11^(1/2)*(2-3*x)^(1/2)*(5-2*x)^(1/2)*(4*x+1)^(1/2)*EllipticPi(2/11*(2
2-33*x)^(1/2),55/124,1/2*I*2^(1/2))+172175002230000*x^5+319488997250500*x^4-2276751199345150*x^3+8807589407540
00*x^2+315342410533150*x-12865932898500)/(24*x^3-70*x^2+21*x+10)/(7+5*x)^3
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{4 \, x + 1} \sqrt{2 \, x - 5} \sqrt{-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{4}}\,{d x} \end{align*}
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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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}}{625 \, x^{4} + 3500 \, x^{3} + 7350 \, x^{2} + 6860 \, x + 2401}, x\right ) \end{align*}
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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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}}{\left (5 x + 7\right )^{4}}\, dx \end{align*}
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]
Integral(sqrt(2 - 3*x)*sqrt(2*x - 5)*sqrt(4*x + 1)/(5*x + 7)**4, x)
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{4 \, x + 1} \sqrt{2 \, x - 5} \sqrt{-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{4}}\,{d x} \end{align*}
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)