3.32.54 \(\int \frac {(c_6+x c_7){}^2}{\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}} \, dx\)

Optimal. Leaf size=3329 \[ \text {result too large to display} \]

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Rubi [A]  time = 20.94, antiderivative size = 1806, normalized size of antiderivative = 0.54, number of steps used = 8, number of rules used = 5, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.122, Rules used = {1649, 823, 827, 1166, 208}

result too large to display

Antiderivative was successfully verified.

[In]

Int[(C[6] + x*C[7])^2/Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]],x]

[Out]

((C[1]*C[2] - C[0]*C[3])^3*(C[4] - Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5])*Sqrt[C[4] + Sqrt[(C[0] + x*C[1]
)/(C[2] + x*C[3])]*C[5]]*C[7]^2)/(3*C[3]^2*(C[1] - ((C[0] + x*C[1])*C[3])/(C[2] + x*C[3]))^3*(C[3]*C[4]^2 - C[
1]*C[5]^2)) - (ArcTanh[(C[3]^(1/4)*Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]])/Sqrt[Sqrt[C[3]]*C[
4] - Sqrt[C[1]]*C[5]]]*(C[1]*C[2] - C[0]*C[3])*C[5]*(4*C[0]^2*C[3]^3*C[4]^2*C[7]^2 - 14*C[0]^2*Sqrt[C[1]]*C[3]
^(5/2)*C[4]*C[5]*C[7]^2 + 28*C[0]*C[1]^(3/2)*C[3]^(3/2)*C[4]*C[5]*C[7]*(2*C[3]*C[6] - C[2]*C[7]) - C[0]*C[1]*C
[3]^2*C[7]*(16*C[3]*C[4]^2*C[6] - (8*C[2]*C[4]^2 + 15*C[0]*C[5]^2)*C[7]) + C[1]^3*C[5]^2*(32*C[3]^2*C[6]^2 - 2
4*C[2]*C[3]*C[6]*C[7] + 7*C[2]^2*C[7]^2) - 2*C[1]^(5/2)*Sqrt[C[3]]*C[4]*C[5]*(32*C[3]^2*C[6]^2 - 36*C[2]*C[3]*
C[6]*C[7] + 11*C[2]^2*C[7]^2) + 2*C[1]^2*C[3]*(16*C[3]^2*C[4]^2*C[6]^2 - 4*C[3]*(6*C[2]*C[4]^2 + 5*C[0]*C[5]^2
)*C[6]*C[7] + 5*C[2]*(2*C[2]*C[4]^2 + C[0]*C[5]^2)*C[7]^2)))/(64*C[1]^(5/2)*C[3]^(11/4)*(Sqrt[C[3]]*C[4] - Sqr
t[C[1]]*C[5])^(7/2)) + (ArcTanh[(C[3]^(1/4)*Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]])/Sqrt[Sqrt
[C[3]]*C[4] + Sqrt[C[1]]*C[5]]]*(C[1]*C[2] - C[0]*C[3])*C[5]*(4*C[0]^2*C[3]^3*C[4]^2*C[7]^2 + 14*C[0]^2*Sqrt[C
[1]]*C[3]^(5/2)*C[4]*C[5]*C[7]^2 - 28*C[0]*C[1]^(3/2)*C[3]^(3/2)*C[4]*C[5]*C[7]*(2*C[3]*C[6] - C[2]*C[7]) - C[
0]*C[1]*C[3]^2*C[7]*(16*C[3]*C[4]^2*C[6] - (8*C[2]*C[4]^2 + 15*C[0]*C[5]^2)*C[7]) + C[1]^3*C[5]^2*(32*C[3]^2*C
[6]^2 - 24*C[2]*C[3]*C[6]*C[7] + 7*C[2]^2*C[7]^2) + 2*C[1]^(5/2)*Sqrt[C[3]]*C[4]*C[5]*(32*C[3]^2*C[6]^2 - 36*C
[2]*C[3]*C[6]*C[7] + 11*C[2]^2*C[7]^2) + 2*C[1]^2*C[3]*(16*C[3]^2*C[4]^2*C[6]^2 - 4*C[3]*(6*C[2]*C[4]^2 + 5*C[
0]*C[5]^2)*C[6]*C[7] + 5*C[2]*(2*C[2]*C[4]^2 + C[0]*C[5]^2)*C[7]^2)))/(64*C[1]^(5/2)*C[3]^(11/4)*(Sqrt[C[3]]*C
[4] + Sqrt[C[1]]*C[5])^(7/2)) - ((C[1]*C[2] - C[0]*C[3])^2*Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C
[5]]*C[7]*(Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]*(C[0]*C[3]^2*C[4]^2*C[7] - 3*C[1]^2*C[5]^2*(8*C[3]*C[6]
- 5*C[2]*C[7]) + C[1]*C[3]*(24*C[3]*C[4]^2*C[6] - (25*C[2]*C[4]^2 - 9*C[0]*C[5]^2)*C[7])) - 2*C[1]*C[4]*(12*C[
3]^2*C[4]^2*C[6] + 7*C[1]*C[2]*C[5]^2*C[7] - C[3]*(12*C[1]*C[5]^2*C[6] + (12*C[2]*C[4]^2 - 5*C[0]*C[5]^2)*C[7]
))))/(24*C[1]*C[3]^2*(C[1] - ((C[0] + x*C[1])*C[3])/(C[2] + x*C[3]))^2*(C[3]*C[4]^2 - C[1]*C[5]^2)^2) + ((C[2]
+ x*C[3])*Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]]*(3*Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[
5]*(2*C[0]^2*C[3]^4*C[4]^4*C[7]^2 - C[0]*C[1]*C[3]^3*C[4]^2*C[7]*(8*C[3]*C[4]^2*C[6] - (4*C[2]*C[4]^2 - 7*C[0]
*C[5]^2)*C[7]) - C[1]^4*C[5]^4*(32*C[3]^2*C[6]^2 - 24*C[2]*C[3]*C[6]*C[7] + 7*C[2]^2*C[7]^2) + C[1]^3*C[3]*C[5
]^2*(64*C[3]^2*C[4]^2*C[6]^2 - 8*C[3]*(12*C[2]*C[4]^2 - 5*C[0]*C[5]^2)*C[6]*C[7] + 5*C[2]*(5*C[2]*C[4]^2 - 2*C
[0]*C[5]^2)*C[7]^2) - C[1]^2*C[3]^2*(32*C[3]^2*C[4]^4*C[6]^2 - 8*C[3]*C[4]^2*(9*C[2]*C[4]^2 - 4*C[0]*C[5]^2)*C
[6]*C[7] + (38*C[2]^2*C[4]^4 - 46*C[0]*C[2]*C[4]^2*C[5]^2 + 15*C[0]^2*C[5]^4)*C[7]^2)) - C[1]*C[4]*(C[0]^2*C[3
]^3*C[4]^2*C[5]^2*C[7]^2 - C[1]^3*C[5]^4*(96*C[3]^2*C[6]^2 - 48*C[2]*C[3]*C[6]*C[7] + 13*C[2]^2*C[7]^2) + C[1]
^2*C[3]*C[5]^2*(192*C[3]^2*C[4]^2*C[6]^2 - 48*C[3]*(5*C[2]*C[4]^2 - 3*C[0]*C[5]^2)*C[6]*C[7] + C[2]*(49*C[2]*C
[4]^2 - 22*C[0]*C[5]^2)*C[7]^2) - C[1]*C[3]^2*(96*C[3]^2*C[4]^4*C[6]^2 - 48*C[3]*C[4]^2*(4*C[2]*C[4]^2 - 3*C[0
]*C[5]^2)*C[6]*C[7] + (96*C[2]^2*C[4]^4 - 142*C[0]*C[2]*C[4]^2*C[5]^2 + 61*C[0]^2*C[5]^4)*C[7]^2))))/(96*C[1]^
2*C[3]^2*(C[3]*C[4]^2 - C[1]*C[5]^2)^3)

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 823

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(
m + 1)*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*(a + c*x^2)^(p + 1))/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), x] + Di
st[1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Simp[f*(c^2*d^2*(2*p + 3) + a*c*e^2*
(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x]
&& NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 827

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2, Subst[Int[(e*f
- d*g + g*x^2)/(c*d^2 + a*e^2 - 2*c*d*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e, f, g}, x]
&& NeQ[c*d^2 + a*e^2, 0]

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rule 1649

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq,
a + c*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + c*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + c
*x^2, x], x, 1]}, -Simp[((d + e*x)^(m + 1)*(a + c*x^2)^(p + 1)*(a*(e*f - d*g) + (c*d*f + a*e*g)*x))/(2*a*(p +
1)*(c*d^2 + a*e^2)), x] + Dist[1/(2*a*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*ExpandToSu
m[2*a*(p + 1)*(c*d^2 + a*e^2)*Q + c*d^2*f*(2*p + 3) - a*e*(d*g*m - e*f*(m + 2*p + 3)) + e*(c*d*f + a*e*g)*(m +
2*p + 4)*x, x], x], x]] /; FreeQ[{a, c, d, e, m}, x] && PolyQ[Pq, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] &
&  !(IGtQ[m, 0] && RationalQ[a, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))

Rubi steps

\begin {align*} \int \frac {(c_6+x c_7){}^2}{\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}} \, dx &=(2 (c_1 c_2-c_0 c_3)) \operatorname {Subst}\left (\int \frac {x \left (-c_1 c_6+c_0 c_7+x^2 (c_3 c_6-c_2 c_7)\right ){}^2}{\left (c_1-x^2 c_3\right ){}^4 \sqrt {c_4+x c_5}} \, dx,x,\sqrt {\frac {c_0+x c_1}{c_2+x c_3}}\right )\\ &=\frac {(c_1 c_2-c_0 c_3){}^3 \left (c_4-\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5\right ) \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} c_7{}^2}{3 c_3{}^2 \left (c_1-\frac {(c_0+x c_1) c_3}{c_2+x c_3}\right ){}^3 \left (c_3 c_4{}^2-c_1 c_5{}^2\right )}+\frac {(c_1 c_2-c_0 c_3) \operatorname {Subst}\left (\int \frac {\frac {c_1 (c_1 c_2-c_0 c_3){}^2 c_4 c_5 c_7{}^2}{2 c_3{}^2}-\frac {6 x^3 c_1 \left (c_3 c_4{}^2-c_1 c_5{}^2\right ) (c_3 c_6-c_2 c_7){}^2}{c_3}-\frac {3 x c_1 \left (c_0 c_3{}^2 c_7 \left (8 c_3 c_4{}^2 c_6-\left (8 c_2 c_4{}^2-3 c_0 c_5{}^2\right ) c_7\right )+c_1{}^2 c_5{}^2 \left (4 c_3{}^2 c_6{}^2-c_2{}^2 c_7{}^2\right )-2 c_1 c_3 \left (2 c_3{}^2 c_4{}^2 c_6{}^2+4 c_0 c_3 c_5{}^2 c_6 c_7-c_2 \left (2 c_2 c_4{}^2+c_0 c_5{}^2\right ) c_7{}^2\right )\right )}{2 c_3{}^2}}{\left (c_1-x^2 c_3\right ){}^3 \sqrt {c_4+x c_5}} \, dx,x,\sqrt {\frac {c_0+x c_1}{c_2+x c_3}}\right )}{3 c_1 \left (c_3 c_4{}^2-c_1 c_5{}^2\right )}\\ &=\frac {(c_1 c_2-c_0 c_3){}^3 \left (c_4-\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5\right ) \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} c_7{}^2}{3 c_3{}^2 \left (c_1-\frac {(c_0+x c_1) c_3}{c_2+x c_3}\right ){}^3 \left (c_3 c_4{}^2-c_1 c_5{}^2\right )}-\frac {(c_1 c_2-c_0 c_3){}^2 \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} c_7 \left (\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5 \left (c_0 c_3{}^2 c_4{}^2 c_7-3 c_1{}^2 c_5{}^2 (8 c_3 c_6-5 c_2 c_7)+c_1 c_3 \left (24 c_3 c_4{}^2 c_6-\left (25 c_2 c_4{}^2-9 c_0 c_5{}^2\right ) c_7\right )\right )-2 c_1 c_4 \left (12 c_3{}^2 c_4{}^2 c_6+7 c_1 c_2 c_5{}^2 c_7-c_3 \left (12 c_1 c_5{}^2 c_6+\left (12 c_2 c_4{}^2-5 c_0 c_5{}^2\right ) c_7\right )\right )\right )}{24 c_1 c_3{}^2 \left (c_1-\frac {(c_0+x c_1) c_3}{c_2+x c_3}\right ){}^2 \left (c_3 c_4{}^2-c_1 c_5{}^2\right ){}^2}+\frac {(c_1 c_2-c_0 c_3) \operatorname {Subst}\left (\int \frac {-\frac {c_1 (c_1 c_2-c_0 c_3) c_4 c_5 c_7 \left (3 c_0 c_3{}^2 c_4{}^2 c_7+4 c_1{}^2 c_5{}^2 (3 c_3 c_6-c_2 c_7)-c_1 c_3 \left (12 c_3 c_4{}^2 c_6-\left (9 c_2 c_4{}^2-8 c_0 c_5{}^2\right ) c_7\right )\right )}{2 c_3{}^2}+\frac {x c_1 \left (5 c_0{}^2 c_3{}^3 c_4{}^2 c_5{}^2 c_7{}^2+3 c_1{}^3 c_5{}^4 \left (32 c_3{}^2 c_6{}^2-24 c_2 c_3 c_6 c_7+7 c_2{}^2 c_7{}^2\right )-c_1{}^2 c_3 c_5{}^2 \left (192 c_3{}^2 c_4{}^2 c_6{}^2-24 c_3 \left (11 c_2 c_4{}^2-5 c_0 c_5{}^2\right ) c_6 c_7+c_2 \left (67 c_2 c_4{}^2-30 c_0 c_5{}^2\right ) c_7{}^2\right )+c_1 c_3{}^2 \left (96 c_3{}^2 c_4{}^4 c_6{}^2-24 c_3 c_4{}^2 \left (8 c_2 c_4{}^2-5 c_0 c_5{}^2\right ) c_6 c_7+\left (96 c_2{}^2 c_4{}^4-130 c_0 c_2 c_4{}^2 c_5{}^2+45 c_0{}^2 c_5{}^4\right ) c_7{}^2\right )\right )}{4 c_3{}^2}}{\left (c_1-x^2 c_3\right ){}^2 \sqrt {c_4+x c_5}} \, dx,x,\sqrt {\frac {c_0+x c_1}{c_2+x c_3}}\right )}{12 c_1{}^2 \left (c_3 c_4{}^2-c_1 c_5{}^2\right ){}^2}\\ &=\frac {(c_1 c_2-c_0 c_3){}^3 \left (c_4-\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5\right ) \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} c_7{}^2}{3 c_3{}^2 \left (c_1-\frac {(c_0+x c_1) c_3}{c_2+x c_3}\right ){}^3 \left (c_3 c_4{}^2-c_1 c_5{}^2\right )}-\frac {(c_1 c_2-c_0 c_3){}^2 \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} c_7 \left (\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5 \left (c_0 c_3{}^2 c_4{}^2 c_7-3 c_1{}^2 c_5{}^2 (8 c_3 c_6-5 c_2 c_7)+c_1 c_3 \left (24 c_3 c_4{}^2 c_6-\left (25 c_2 c_4{}^2-9 c_0 c_5{}^2\right ) c_7\right )\right )-2 c_1 c_4 \left (12 c_3{}^2 c_4{}^2 c_6+7 c_1 c_2 c_5{}^2 c_7-c_3 \left (12 c_1 c_5{}^2 c_6+\left (12 c_2 c_4{}^2-5 c_0 c_5{}^2\right ) c_7\right )\right )\right )}{24 c_1 c_3{}^2 \left (c_1-\frac {(c_0+x c_1) c_3}{c_2+x c_3}\right ){}^2 \left (c_3 c_4{}^2-c_1 c_5{}^2\right ){}^2}+\frac {(c_2+x c_3) \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} \left (3 \sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5 \left (2 c_0{}^2 c_3{}^4 c_4{}^4 c_7{}^2-c_0 c_1 c_3{}^3 c_4{}^2 c_7 \left (8 c_3 c_4{}^2 c_6-\left (4 c_2 c_4{}^2-7 c_0 c_5{}^2\right ) c_7\right )-c_1{}^4 c_5{}^4 \left (32 c_3{}^2 c_6{}^2-24 c_2 c_3 c_6 c_7+7 c_2{}^2 c_7{}^2\right )+c_1{}^3 c_3 c_5{}^2 \left (64 c_3{}^2 c_4{}^2 c_6{}^2-8 c_3 \left (12 c_2 c_4{}^2-5 c_0 c_5{}^2\right ) c_6 c_7+5 c_2 \left (5 c_2 c_4{}^2-2 c_0 c_5{}^2\right ) c_7{}^2\right )-c_1{}^2 c_3{}^2 \left (32 c_3{}^2 c_4{}^4 c_6{}^2-8 c_3 c_4{}^2 \left (9 c_2 c_4{}^2-4 c_0 c_5{}^2\right ) c_6 c_7+\left (38 c_2{}^2 c_4{}^4-46 c_0 c_2 c_4{}^2 c_5{}^2+15 c_0{}^2 c_5{}^4\right ) c_7{}^2\right )\right )-c_1 c_4 \left (c_0{}^2 c_3{}^3 c_4{}^2 c_5{}^2 c_7{}^2-c_1{}^3 c_5{}^4 \left (96 c_3{}^2 c_6{}^2-48 c_2 c_3 c_6 c_7+13 c_2{}^2 c_7{}^2\right )+c_1{}^2 c_3 c_5{}^2 \left (192 c_3{}^2 c_4{}^2 c_6{}^2-48 c_3 \left (5 c_2 c_4{}^2-3 c_0 c_5{}^2\right ) c_6 c_7+c_2 \left (49 c_2 c_4{}^2-22 c_0 c_5{}^2\right ) c_7{}^2\right )-c_1 c_3{}^2 \left (96 c_3{}^2 c_4{}^4 c_6{}^2-48 c_3 c_4{}^2 \left (4 c_2 c_4{}^2-3 c_0 c_5{}^2\right ) c_6 c_7+\left (96 c_2{}^2 c_4{}^4-142 c_0 c_2 c_4{}^2 c_5{}^2+61 c_0{}^2 c_5{}^4\right ) c_7{}^2\right )\right )\right )}{96 c_1{}^2 c_3{}^2 \left (c_3 c_4{}^2-c_1 c_5{}^2\right ){}^3}-\frac {(c_1 c_2-c_0 c_3) \operatorname {Subst}\left (\int \frac {-\frac {3 x c_1 c_5{}^2 \left (2 c_0{}^2 c_3{}^4 c_4{}^4 c_7{}^2-c_0 c_1 c_3{}^3 c_4{}^2 c_7 \left (8 c_3 c_4{}^2 c_6-\left (4 c_2 c_4{}^2-7 c_0 c_5{}^2\right ) c_7\right )-c_1{}^4 c_5{}^4 \left (32 c_3{}^2 c_6{}^2-24 c_2 c_3 c_6 c_7+7 c_2{}^2 c_7{}^2\right )+c_1{}^3 c_3 c_5{}^2 \left (64 c_3{}^2 c_4{}^2 c_6{}^2-8 c_3 \left (12 c_2 c_4{}^2-5 c_0 c_5{}^2\right ) c_6 c_7+5 c_2 \left (5 c_2 c_4{}^2-2 c_0 c_5{}^2\right ) c_7{}^2\right )-c_1{}^2 c_3{}^2 \left (32 c_3{}^2 c_4{}^4 c_6{}^2-8 c_3 c_4{}^2 \left (9 c_2 c_4{}^2-4 c_0 c_5{}^2\right ) c_6 c_7+\left (38 c_2{}^2 c_4{}^4-46 c_0 c_2 c_4{}^2 c_5{}^2+15 c_0{}^2 c_5{}^4\right ) c_7{}^2\right )\right )}{8 c_3}-\frac {3 c_1 c_4 c_5 \left (4 c_0{}^2 c_3{}^4 c_4{}^4 c_7{}^2-c_0 c_1 c_3{}^3 c_4{}^2 c_7 \left (16 c_3 c_4{}^2 c_6-\left (8 c_2 c_4{}^2-15 c_0 c_5{}^2\right ) c_7\right )+c_1{}^4 c_5{}^4 \left (32 c_3{}^2 c_6{}^2-c_2{}^2 c_7{}^2\right )-c_1{}^3 c_3 c_5{}^2 \left (64 c_3{}^2 c_4{}^2 c_6{}^2-16 c_3 \left (3 c_2 c_4{}^2-4 c_0 c_5{}^2\right ) c_6 c_7-c_2 \left (c_2 c_4{}^2+2 c_0 c_5{}^2\right ) c_7{}^2\right )+c_1{}^2 c_3{}^2 \left (32 c_3{}^2 c_4{}^4 c_6{}^2-16 c_3 c_4{}^2 \left (3 c_2 c_4{}^2-5 c_0 c_5{}^2\right ) c_6 c_7+\left (20 c_2{}^2 c_4{}^4-50 c_0 c_2 c_4{}^2 c_5{}^2+31 c_0{}^2 c_5{}^4\right ) c_7{}^2\right )\right )}{8 c_3}}{\left (c_1-x^2 c_3\right ) \sqrt {c_4+x c_5}} \, dx,x,\sqrt {\frac {c_0+x c_1}{c_2+x c_3}}\right )}{24 c_1{}^3 c_3 \left (c_3 c_4{}^2-c_1 c_5{}^2\right ){}^3}\\ &=\frac {(c_1 c_2-c_0 c_3){}^3 \left (c_4-\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5\right ) \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} c_7{}^2}{3 c_3{}^2 \left (c_1-\frac {(c_0+x c_1) c_3}{c_2+x c_3}\right ){}^3 \left (c_3 c_4{}^2-c_1 c_5{}^2\right )}-\frac {(c_1 c_2-c_0 c_3){}^2 \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} c_7 \left (\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5 \left (c_0 c_3{}^2 c_4{}^2 c_7-3 c_1{}^2 c_5{}^2 (8 c_3 c_6-5 c_2 c_7)+c_1 c_3 \left (24 c_3 c_4{}^2 c_6-\left (25 c_2 c_4{}^2-9 c_0 c_5{}^2\right ) c_7\right )\right )-2 c_1 c_4 \left (12 c_3{}^2 c_4{}^2 c_6+7 c_1 c_2 c_5{}^2 c_7-c_3 \left (12 c_1 c_5{}^2 c_6+\left (12 c_2 c_4{}^2-5 c_0 c_5{}^2\right ) c_7\right )\right )\right )}{24 c_1 c_3{}^2 \left (c_1-\frac {(c_0+x c_1) c_3}{c_2+x c_3}\right ){}^2 \left (c_3 c_4{}^2-c_1 c_5{}^2\right ){}^2}+\frac {(c_2+x c_3) \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} \left (3 \sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5 \left (2 c_0{}^2 c_3{}^4 c_4{}^4 c_7{}^2-c_0 c_1 c_3{}^3 c_4{}^2 c_7 \left (8 c_3 c_4{}^2 c_6-\left (4 c_2 c_4{}^2-7 c_0 c_5{}^2\right ) c_7\right )-c_1{}^4 c_5{}^4 \left (32 c_3{}^2 c_6{}^2-24 c_2 c_3 c_6 c_7+7 c_2{}^2 c_7{}^2\right )+c_1{}^3 c_3 c_5{}^2 \left (64 c_3{}^2 c_4{}^2 c_6{}^2-8 c_3 \left (12 c_2 c_4{}^2-5 c_0 c_5{}^2\right ) c_6 c_7+5 c_2 \left (5 c_2 c_4{}^2-2 c_0 c_5{}^2\right ) c_7{}^2\right )-c_1{}^2 c_3{}^2 \left (32 c_3{}^2 c_4{}^4 c_6{}^2-8 c_3 c_4{}^2 \left (9 c_2 c_4{}^2-4 c_0 c_5{}^2\right ) c_6 c_7+\left (38 c_2{}^2 c_4{}^4-46 c_0 c_2 c_4{}^2 c_5{}^2+15 c_0{}^2 c_5{}^4\right ) c_7{}^2\right )\right )-c_1 c_4 \left (c_0{}^2 c_3{}^3 c_4{}^2 c_5{}^2 c_7{}^2-c_1{}^3 c_5{}^4 \left (96 c_3{}^2 c_6{}^2-48 c_2 c_3 c_6 c_7+13 c_2{}^2 c_7{}^2\right )+c_1{}^2 c_3 c_5{}^2 \left (192 c_3{}^2 c_4{}^2 c_6{}^2-48 c_3 \left (5 c_2 c_4{}^2-3 c_0 c_5{}^2\right ) c_6 c_7+c_2 \left (49 c_2 c_4{}^2-22 c_0 c_5{}^2\right ) c_7{}^2\right )-c_1 c_3{}^2 \left (96 c_3{}^2 c_4{}^4 c_6{}^2-48 c_3 c_4{}^2 \left (4 c_2 c_4{}^2-3 c_0 c_5{}^2\right ) c_6 c_7+\left (96 c_2{}^2 c_4{}^4-142 c_0 c_2 c_4{}^2 c_5{}^2+61 c_0{}^2 c_5{}^4\right ) c_7{}^2\right )\right )\right )}{96 c_1{}^2 c_3{}^2 \left (c_3 c_4{}^2-c_1 c_5{}^2\right ){}^3}-\frac {(c_1 c_2-c_0 c_3) \operatorname {Subst}\left (\int \frac {-\frac {3 x^2 c_1 c_5{}^2 \left (2 c_0{}^2 c_3{}^4 c_4{}^4 c_7{}^2-c_0 c_1 c_3{}^3 c_4{}^2 c_7 \left (8 c_3 c_4{}^2 c_6-\left (4 c_2 c_4{}^2-7 c_0 c_5{}^2\right ) c_7\right )-c_1{}^4 c_5{}^4 \left (32 c_3{}^2 c_6{}^2-24 c_2 c_3 c_6 c_7+7 c_2{}^2 c_7{}^2\right )+c_1{}^3 c_3 c_5{}^2 \left (64 c_3{}^2 c_4{}^2 c_6{}^2-8 c_3 \left (12 c_2 c_4{}^2-5 c_0 c_5{}^2\right ) c_6 c_7+5 c_2 \left (5 c_2 c_4{}^2-2 c_0 c_5{}^2\right ) c_7{}^2\right )-c_1{}^2 c_3{}^2 \left (32 c_3{}^2 c_4{}^4 c_6{}^2-8 c_3 c_4{}^2 \left (9 c_2 c_4{}^2-4 c_0 c_5{}^2\right ) c_6 c_7+\left (38 c_2{}^2 c_4{}^4-46 c_0 c_2 c_4{}^2 c_5{}^2+15 c_0{}^2 c_5{}^4\right ) c_7{}^2\right )\right )}{8 c_3}+\frac {3 c_1 c_4 c_5{}^2 \left (2 c_0{}^2 c_3{}^4 c_4{}^4 c_7{}^2-c_0 c_1 c_3{}^3 c_4{}^2 c_7 \left (8 c_3 c_4{}^2 c_6-\left (4 c_2 c_4{}^2-7 c_0 c_5{}^2\right ) c_7\right )-c_1{}^4 c_5{}^4 \left (32 c_3{}^2 c_6{}^2-24 c_2 c_3 c_6 c_7+7 c_2{}^2 c_7{}^2\right )+c_1{}^3 c_3 c_5{}^2 \left (64 c_3{}^2 c_4{}^2 c_6{}^2-8 c_3 \left (12 c_2 c_4{}^2-5 c_0 c_5{}^2\right ) c_6 c_7+5 c_2 \left (5 c_2 c_4{}^2-2 c_0 c_5{}^2\right ) c_7{}^2\right )-c_1{}^2 c_3{}^2 \left (32 c_3{}^2 c_4{}^4 c_6{}^2-8 c_3 c_4{}^2 \left (9 c_2 c_4{}^2-4 c_0 c_5{}^2\right ) c_6 c_7+\left (38 c_2{}^2 c_4{}^4-46 c_0 c_2 c_4{}^2 c_5{}^2+15 c_0{}^2 c_5{}^4\right ) c_7{}^2\right )\right )}{8 c_3}-\frac {3 c_1 c_4 c_5{}^2 \left (4 c_0{}^2 c_3{}^4 c_4{}^4 c_7{}^2-c_0 c_1 c_3{}^3 c_4{}^2 c_7 \left (16 c_3 c_4{}^2 c_6-\left (8 c_2 c_4{}^2-15 c_0 c_5{}^2\right ) c_7\right )+c_1{}^4 c_5{}^4 \left (32 c_3{}^2 c_6{}^2-c_2{}^2 c_7{}^2\right )-c_1{}^3 c_3 c_5{}^2 \left (64 c_3{}^2 c_4{}^2 c_6{}^2-16 c_3 \left (3 c_2 c_4{}^2-4 c_0 c_5{}^2\right ) c_6 c_7-c_2 \left (c_2 c_4{}^2+2 c_0 c_5{}^2\right ) c_7{}^2\right )+c_1{}^2 c_3{}^2 \left (32 c_3{}^2 c_4{}^4 c_6{}^2-16 c_3 c_4{}^2 \left (3 c_2 c_4{}^2-5 c_0 c_5{}^2\right ) c_6 c_7+\left (20 c_2{}^2 c_4{}^4-50 c_0 c_2 c_4{}^2 c_5{}^2+31 c_0{}^2 c_5{}^4\right ) c_7{}^2\right )\right )}{8 c_3}}{-x^4 c_3+2 x^2 c_3 c_4-c_3 c_4{}^2+c_1 c_5{}^2} \, dx,x,\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}\right )}{12 c_1{}^3 c_3 \left (c_3 c_4{}^2-c_1 c_5{}^2\right ){}^3}\\ &=\frac {(c_1 c_2-c_0 c_3){}^3 \left (c_4-\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5\right ) \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} c_7{}^2}{3 c_3{}^2 \left (c_1-\frac {(c_0+x c_1) c_3}{c_2+x c_3}\right ){}^3 \left (c_3 c_4{}^2-c_1 c_5{}^2\right )}-\frac {(c_1 c_2-c_0 c_3){}^2 \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} c_7 \left (\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5 \left (c_0 c_3{}^2 c_4{}^2 c_7-3 c_1{}^2 c_5{}^2 (8 c_3 c_6-5 c_2 c_7)+c_1 c_3 \left (24 c_3 c_4{}^2 c_6-\left (25 c_2 c_4{}^2-9 c_0 c_5{}^2\right ) c_7\right )\right )-2 c_1 c_4 \left (12 c_3{}^2 c_4{}^2 c_6+7 c_1 c_2 c_5{}^2 c_7-c_3 \left (12 c_1 c_5{}^2 c_6+\left (12 c_2 c_4{}^2-5 c_0 c_5{}^2\right ) c_7\right )\right )\right )}{24 c_1 c_3{}^2 \left (c_1-\frac {(c_0+x c_1) c_3}{c_2+x c_3}\right ){}^2 \left (c_3 c_4{}^2-c_1 c_5{}^2\right ){}^2}+\frac {(c_2+x c_3) \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} \left (3 \sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5 \left (2 c_0{}^2 c_3{}^4 c_4{}^4 c_7{}^2-c_0 c_1 c_3{}^3 c_4{}^2 c_7 \left (8 c_3 c_4{}^2 c_6-\left (4 c_2 c_4{}^2-7 c_0 c_5{}^2\right ) c_7\right )-c_1{}^4 c_5{}^4 \left (32 c_3{}^2 c_6{}^2-24 c_2 c_3 c_6 c_7+7 c_2{}^2 c_7{}^2\right )+c_1{}^3 c_3 c_5{}^2 \left (64 c_3{}^2 c_4{}^2 c_6{}^2-8 c_3 \left (12 c_2 c_4{}^2-5 c_0 c_5{}^2\right ) c_6 c_7+5 c_2 \left (5 c_2 c_4{}^2-2 c_0 c_5{}^2\right ) c_7{}^2\right )-c_1{}^2 c_3{}^2 \left (32 c_3{}^2 c_4{}^4 c_6{}^2-8 c_3 c_4{}^2 \left (9 c_2 c_4{}^2-4 c_0 c_5{}^2\right ) c_6 c_7+\left (38 c_2{}^2 c_4{}^4-46 c_0 c_2 c_4{}^2 c_5{}^2+15 c_0{}^2 c_5{}^4\right ) c_7{}^2\right )\right )-c_1 c_4 \left (c_0{}^2 c_3{}^3 c_4{}^2 c_5{}^2 c_7{}^2-c_1{}^3 c_5{}^4 \left (96 c_3{}^2 c_6{}^2-48 c_2 c_3 c_6 c_7+13 c_2{}^2 c_7{}^2\right )+c_1{}^2 c_3 c_5{}^2 \left (192 c_3{}^2 c_4{}^2 c_6{}^2-48 c_3 \left (5 c_2 c_4{}^2-3 c_0 c_5{}^2\right ) c_6 c_7+c_2 \left (49 c_2 c_4{}^2-22 c_0 c_5{}^2\right ) c_7{}^2\right )-c_1 c_3{}^2 \left (96 c_3{}^2 c_4{}^4 c_6{}^2-48 c_3 c_4{}^2 \left (4 c_2 c_4{}^2-3 c_0 c_5{}^2\right ) c_6 c_7+\left (96 c_2{}^2 c_4{}^4-142 c_0 c_2 c_4{}^2 c_5{}^2+61 c_0{}^2 c_5{}^4\right ) c_7{}^2\right )\right )\right )}{96 c_1{}^2 c_3{}^2 \left (c_3 c_4{}^2-c_1 c_5{}^2\right ){}^3}-\frac {\left ((c_1 c_2-c_0 c_3) c_5 \left (4 c_0{}^2 c_3{}^3 c_4{}^2 c_7{}^2-14 c_0{}^2 \sqrt {c_1} c_3{}^{5/2} c_4 c_5 c_7{}^2+28 c_0 c_1{}^{3/2} c_3{}^{3/2} c_4 c_5 c_7 (2 c_3 c_6-c_2 c_7)-c_0 c_1 c_3{}^2 c_7 \left (16 c_3 c_4{}^2 c_6-\left (8 c_2 c_4{}^2+15 c_0 c_5{}^2\right ) c_7\right )+c_1{}^3 c_5{}^2 \left (32 c_3{}^2 c_6{}^2-24 c_2 c_3 c_6 c_7+7 c_2{}^2 c_7{}^2\right )-2 c_1{}^{5/2} \sqrt {c_3} c_4 c_5 \left (32 c_3{}^2 c_6{}^2-36 c_2 c_3 c_6 c_7+11 c_2{}^2 c_7{}^2\right )+2 c_1{}^2 c_3 \left (16 c_3{}^2 c_4{}^2 c_6{}^2-4 c_3 \left (6 c_2 c_4{}^2+5 c_0 c_5{}^2\right ) c_6 c_7+5 c_2 \left (2 c_2 c_4{}^2+c_0 c_5{}^2\right ) c_7{}^2\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-x^2 c_3+c_3 c_4-\sqrt {c_1} \sqrt {c_3} c_5} \, dx,x,\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}\right )}{64 c_1{}^{5/2} c_3{}^2 \left (\sqrt {c_3} c_4-\sqrt {c_1} c_5\right ){}^3}+\frac {\left ((c_1 c_2-c_0 c_3) c_5 \left (4 c_0{}^2 c_3{}^3 c_4{}^2 c_7{}^2+14 c_0{}^2 \sqrt {c_1} c_3{}^{5/2} c_4 c_5 c_7{}^2-28 c_0 c_1{}^{3/2} c_3{}^{3/2} c_4 c_5 c_7 (2 c_3 c_6-c_2 c_7)-c_0 c_1 c_3{}^2 c_7 \left (16 c_3 c_4{}^2 c_6-\left (8 c_2 c_4{}^2+15 c_0 c_5{}^2\right ) c_7\right )+c_1{}^3 c_5{}^2 \left (32 c_3{}^2 c_6{}^2-24 c_2 c_3 c_6 c_7+7 c_2{}^2 c_7{}^2\right )+2 c_1{}^{5/2} \sqrt {c_3} c_4 c_5 \left (32 c_3{}^2 c_6{}^2-36 c_2 c_3 c_6 c_7+11 c_2{}^2 c_7{}^2\right )+2 c_1{}^2 c_3 \left (16 c_3{}^2 c_4{}^2 c_6{}^2-4 c_3 \left (6 c_2 c_4{}^2+5 c_0 c_5{}^2\right ) c_6 c_7+5 c_2 \left (2 c_2 c_4{}^2+c_0 c_5{}^2\right ) c_7{}^2\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-x^2 c_3+c_3 c_4+\sqrt {c_1} \sqrt {c_3} c_5} \, dx,x,\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}\right )}{64 c_1{}^{5/2} c_3{}^2 \left (\sqrt {c_3} c_4+\sqrt {c_1} c_5\right ){}^3}\\ &=\frac {(c_1 c_2-c_0 c_3){}^3 \left (c_4-\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5\right ) \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} c_7{}^2}{3 c_3{}^2 \left (c_1-\frac {(c_0+x c_1) c_3}{c_2+x c_3}\right ){}^3 \left (c_3 c_4{}^2-c_1 c_5{}^2\right )}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{c_3} \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}{\sqrt {\sqrt {c_3} c_4-\sqrt {c_1} c_5}}\right ) (c_1 c_2-c_0 c_3) c_5 \left (4 c_0{}^2 c_3{}^3 c_4{}^2 c_7{}^2-14 c_0{}^2 \sqrt {c_1} c_3{}^{5/2} c_4 c_5 c_7{}^2+28 c_0 c_1{}^{3/2} c_3{}^{3/2} c_4 c_5 c_7 (2 c_3 c_6-c_2 c_7)-c_0 c_1 c_3{}^2 c_7 \left (16 c_3 c_4{}^2 c_6-\left (8 c_2 c_4{}^2+15 c_0 c_5{}^2\right ) c_7\right )+c_1{}^3 c_5{}^2 \left (32 c_3{}^2 c_6{}^2-24 c_2 c_3 c_6 c_7+7 c_2{}^2 c_7{}^2\right )-2 c_1{}^{5/2} \sqrt {c_3} c_4 c_5 \left (32 c_3{}^2 c_6{}^2-36 c_2 c_3 c_6 c_7+11 c_2{}^2 c_7{}^2\right )+2 c_1{}^2 c_3 \left (16 c_3{}^2 c_4{}^2 c_6{}^2-4 c_3 \left (6 c_2 c_4{}^2+5 c_0 c_5{}^2\right ) c_6 c_7+5 c_2 \left (2 c_2 c_4{}^2+c_0 c_5{}^2\right ) c_7{}^2\right )\right )}{64 c_1{}^{5/2} c_3{}^{11/4} \left (\sqrt {c_3} c_4-\sqrt {c_1} c_5\right ){}^{7/2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{c_3} \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}{\sqrt {\sqrt {c_3} c_4+\sqrt {c_1} c_5}}\right ) (c_1 c_2-c_0 c_3) c_5 \left (4 c_0{}^2 c_3{}^3 c_4{}^2 c_7{}^2+14 c_0{}^2 \sqrt {c_1} c_3{}^{5/2} c_4 c_5 c_7{}^2-28 c_0 c_1{}^{3/2} c_3{}^{3/2} c_4 c_5 c_7 (2 c_3 c_6-c_2 c_7)-c_0 c_1 c_3{}^2 c_7 \left (16 c_3 c_4{}^2 c_6-\left (8 c_2 c_4{}^2+15 c_0 c_5{}^2\right ) c_7\right )+c_1{}^3 c_5{}^2 \left (32 c_3{}^2 c_6{}^2-24 c_2 c_3 c_6 c_7+7 c_2{}^2 c_7{}^2\right )+2 c_1{}^{5/2} \sqrt {c_3} c_4 c_5 \left (32 c_3{}^2 c_6{}^2-36 c_2 c_3 c_6 c_7+11 c_2{}^2 c_7{}^2\right )+2 c_1{}^2 c_3 \left (16 c_3{}^2 c_4{}^2 c_6{}^2-4 c_3 \left (6 c_2 c_4{}^2+5 c_0 c_5{}^2\right ) c_6 c_7+5 c_2 \left (2 c_2 c_4{}^2+c_0 c_5{}^2\right ) c_7{}^2\right )\right )}{64 c_1{}^{5/2} c_3{}^{11/4} \left (\sqrt {c_3} c_4+\sqrt {c_1} c_5\right ){}^{7/2}}-\frac {(c_1 c_2-c_0 c_3){}^2 \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} c_7 \left (\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5 \left (c_0 c_3{}^2 c_4{}^2 c_7-3 c_1{}^2 c_5{}^2 (8 c_3 c_6-5 c_2 c_7)+c_1 c_3 \left (24 c_3 c_4{}^2 c_6-\left (25 c_2 c_4{}^2-9 c_0 c_5{}^2\right ) c_7\right )\right )-2 c_1 c_4 \left (12 c_3{}^2 c_4{}^2 c_6+7 c_1 c_2 c_5{}^2 c_7-c_3 \left (12 c_1 c_5{}^2 c_6+\left (12 c_2 c_4{}^2-5 c_0 c_5{}^2\right ) c_7\right )\right )\right )}{24 c_1 c_3{}^2 \left (c_1-\frac {(c_0+x c_1) c_3}{c_2+x c_3}\right ){}^2 \left (c_3 c_4{}^2-c_1 c_5{}^2\right ){}^2}+\frac {(c_2+x c_3) \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} \left (3 \sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5 \left (2 c_0{}^2 c_3{}^4 c_4{}^4 c_7{}^2-c_0 c_1 c_3{}^3 c_4{}^2 c_7 \left (8 c_3 c_4{}^2 c_6-\left (4 c_2 c_4{}^2-7 c_0 c_5{}^2\right ) c_7\right )-c_1{}^4 c_5{}^4 \left (32 c_3{}^2 c_6{}^2-24 c_2 c_3 c_6 c_7+7 c_2{}^2 c_7{}^2\right )+c_1{}^3 c_3 c_5{}^2 \left (64 c_3{}^2 c_4{}^2 c_6{}^2-8 c_3 \left (12 c_2 c_4{}^2-5 c_0 c_5{}^2\right ) c_6 c_7+5 c_2 \left (5 c_2 c_4{}^2-2 c_0 c_5{}^2\right ) c_7{}^2\right )-c_1{}^2 c_3{}^2 \left (32 c_3{}^2 c_4{}^4 c_6{}^2-8 c_3 c_4{}^2 \left (9 c_2 c_4{}^2-4 c_0 c_5{}^2\right ) c_6 c_7+\left (38 c_2{}^2 c_4{}^4-46 c_0 c_2 c_4{}^2 c_5{}^2+15 c_0{}^2 c_5{}^4\right ) c_7{}^2\right )\right )-c_1 c_4 \left (c_0{}^2 c_3{}^3 c_4{}^2 c_5{}^2 c_7{}^2-c_1{}^3 c_5{}^4 \left (96 c_3{}^2 c_6{}^2-48 c_2 c_3 c_6 c_7+13 c_2{}^2 c_7{}^2\right )+c_1{}^2 c_3 c_5{}^2 \left (192 c_3{}^2 c_4{}^2 c_6{}^2-48 c_3 \left (5 c_2 c_4{}^2-3 c_0 c_5{}^2\right ) c_6 c_7+c_2 \left (49 c_2 c_4{}^2-22 c_0 c_5{}^2\right ) c_7{}^2\right )-c_1 c_3{}^2 \left (96 c_3{}^2 c_4{}^4 c_6{}^2-48 c_3 c_4{}^2 \left (4 c_2 c_4{}^2-3 c_0 c_5{}^2\right ) c_6 c_7+\left (96 c_2{}^2 c_4{}^4-142 c_0 c_2 c_4{}^2 c_5{}^2+61 c_0{}^2 c_5{}^4\right ) c_7{}^2\right )\right )\right )}{96 c_1{}^2 c_3{}^2 \left (c_3 c_4{}^2-c_1 c_5{}^2\right ){}^3}\\ \end {align*}

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Mathematica [B]  time = 18.38, size = 8468, normalized size = 2.54 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(C[6] + x*C[7])^2/Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]],x]

[Out]

Result too large to show

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IntegrateAlgebraic [B]  time = 20.10, size = 9401, normalized size = 2.82 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(C[6] + x*C[7])^2/Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]],x]

[Out]

Result too large to show

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((_C7*x+_C6)^2/(_C4+((_C1*x+_C0)/(_C3*x+_C2))^(1/2)*_C5)^(1/2),x, algorithm="fricas")

[Out]

Timed out

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((_C7*x+_C6)^2/(_C4+((_C1*x+_C0)/(_C3*x+_C2))^(1/2)*_C5)^(1/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(const ge
n & e,const index_m & i,const vecteur & l) Error: Bad Argument ValueWarning, integration of abs or sign assume
s constant sign by intervals (correct if the argument is real):Check [abs(t_nostep*1_C3+1_C2)]Warning, need to
choose a branch for the root of a polynomial with parameters. This might be wrong.Non regular value [0,0,0,0,
0,0] was discarded and replaced randomly by 0=[-39,85,33,-4,-70,15]Warning, need to choose a branch for the ro
ot of a polynomial with parameters. This might be wrong.Non regular value [0,0,0,0,0,0] was discarded and repl
aced randomly by 0=[-82,36,86,-68,-66,-39]Warning, need to choose a branch for the root of a polynomial with p
arameters. This might be wrong.Non regular value [0,0,0,0,0,0] was discarded and replaced randomly by 0=[-86,2
1,48,-16,13,80]Warning, need to choose a branch for the root of a polynomial with parameters. This might be wr
ong.Non regular value [0,0,0,0,0,0] was discarded and replaced randomly by 0=[-3,-76,-17,63,68,98]Warning, nee
d to choose a branch for the root of a polynomial with parameters. This might be wrong.Non regular value [0,0,
0,0,0,0] was discarded and replaced randomly by 0=[66,23,-29,45,75,-8]Warning, need to choose a branch for the
root of a polynomial with parameters. This might be wrong.Non regular value [0,0,0,0,0,0] was discarded and r
eplaced randomly by 0=[80,-46,-16,-32,-64,-40]Evaluation time: 11.4Done

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maple [F]  time = 0.08, size = 0, normalized size = 0.00 \[\int \frac {\left (\textit {\_C7} x +\textit {\_C6} \right )^{2}}{\sqrt {\textit {\_C4} +\sqrt {\frac {\textit {\_C1} x +\textit {\_C0}}{\textit {\_C3} x +\textit {\_C2}}}\, \textit {\_C5}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((_C7*x+_C6)^2/(_C4+((_C1*x+_C0)/(_C3*x+_C2))^(1/2)*_C5)^(1/2),x)

[Out]

int((_C7*x+_C6)^2/(_C4+((_C1*x+_C0)/(_C3*x+_C2))^(1/2)*_C5)^(1/2),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (\_{C_{7}} x + \_{C_{6}}\right )}^{2}}{\sqrt {\_{C_{5}} \sqrt {\frac {\_{C_{1}} x + \_{C_{0}}}{\_{C_{3}} x + \_{C_{2}}}} + \_{C_{4}}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((_C7*x+_C6)^2/(_C4+((_C1*x+_C0)/(_C3*x+_C2))^(1/2)*_C5)^(1/2),x, algorithm="maxima")

[Out]

integrate((_C7*x + _C6)^2/sqrt(_C5*sqrt((_C1*x + _C0)/(_C3*x + _C2)) + _C4), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (_{\mathrm {C6}}+_{\mathrm {C7}}\,x\right )}^2}{\sqrt {_{\mathrm {C4}}+_{\mathrm {C5}}\,\sqrt {\frac {_{\mathrm {C0}}+_{\mathrm {C1}}\,x}{_{\mathrm {C2}}+_{\mathrm {C3}}\,x}}}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((_C6 + _C7*x)^2/(_C4 + _C5*((_C0 + _C1*x)/(_C2 + _C3*x))^(1/2))^(1/2),x)

[Out]

int((_C6 + _C7*x)^2/(_C4 + _C5*((_C0 + _C1*x)/(_C2 + _C3*x))^(1/2))^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (_C6 + _C7 x\right )^{2}}{\sqrt {_C4 + _C5 \sqrt {\frac {_C0}{_C2 + _C3 x} + \frac {_C1 x}{_C2 + _C3 x}}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((_C7*x+_C6)**2/(_C4+((_C1*x+_C0)/(_C3*x+_C2))**(1/2)*_C5)**(1/2),x)

[Out]

Integral((_C6 + _C7*x)**2/sqrt(_C4 + _C5*sqrt(_C0/(_C2 + _C3*x) + _C1*x/(_C2 + _C3*x))), x)

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