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

Optimal. Leaf size=1178 \[ -\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 ) \left (-8 c_2 c_3 c_5{}^2 c_6 c_1{}^{5/2}+3 c_2{}^2 c_5{}^2 c_7 c_1{}^{5/2}-8 c_2 c_3{}^{3/2} c_4 c_5 c_6 c_1{}^2+6 c_2{}^2 \sqrt {c_3} c_4 c_5 c_7 c_1{}^2+8 c_0 c_3{}^2 c_5{}^2 c_6 c_1{}^{3/2}+2 c_0 c_2 c_3 c_5{}^2 c_7 c_1{}^{3/2}+8 c_0 c_3{}^{5/2} c_4 c_5 c_6 c_1-4 c_0 c_2 c_3{}^{3/2} c_4 c_5 c_7 c_1-5 c_0{}^2 c_3{}^2 c_5{}^2 c_7 \sqrt {c_1}-2 c_0{}^2 c_3{}^{5/2} c_4 c_5 c_7\right )}{16 c_1{}^{3/2} c_3{}^{7/4} \left (\sqrt {c_3} c_4+\sqrt {c_1} c_5\right ){}^{5/2}}-\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 ) \sqrt {\sqrt {c_3} c_4-\sqrt {c_1} c_5} \left (8 c_2 c_3 c_5{}^2 c_6 c_1{}^{5/2}-3 c_2{}^2 c_5{}^2 c_7 c_1{}^{5/2}-8 c_2 c_3{}^{3/2} c_4 c_5 c_6 c_1{}^2+6 c_2{}^2 \sqrt {c_3} c_4 c_5 c_7 c_1{}^2-8 c_0 c_3{}^2 c_5{}^2 c_6 c_1{}^{3/2}-2 c_0 c_2 c_3 c_5{}^2 c_7 c_1{}^{3/2}+8 c_0 c_3{}^{5/2} c_4 c_5 c_6 c_1-4 c_0 c_2 c_3{}^{3/2} c_4 c_5 c_7 c_1+5 c_0{}^2 c_3{}^2 c_5{}^2 c_7 \sqrt {c_1}-2 c_0{}^2 c_3{}^{5/2} c_4 c_5 c_7\right )}{16 c_1{}^{3/2} c_3{}^{7/4} \left (\sqrt {c_1} c_5-\sqrt {c_3} c_4\right ){}^3}+\frac {\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} \left (4 x c_3{}^3 c_6 c_4{}^3+4 c_2 c_3{}^2 c_6 c_4{}^3+2 x^2 c_3{}^3 c_7 c_4{}^3-2 c_2{}^2 c_3 c_7 c_4{}^3-4 x c_1 c_3{}^2 c_5{}^2 c_6 c_4-4 c_1 c_2 c_3 c_5{}^2 c_6 c_4-c_1 c_2{}^2 c_5{}^2 c_7 c_4+3 x c_0 c_3{}^2 c_5{}^2 c_7 c_4-2 x^2 c_1 c_3{}^2 c_5{}^2 c_7 c_4+3 c_0 c_2 c_3 c_5{}^2 c_7 c_4-3 x c_1 c_2 c_3 c_5{}^2 c_7 c_4\right )}{4 c_3 \left (c_3 c_4{}^2-c_1 c_5{}^2\right ){}^2}+\frac {\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} \left (-8 x c_1 c_4{}^2 c_5 c_6 c_3{}^3-x c_0 c_4{}^2 c_5 c_7 c_3{}^3-4 x^2 c_1 c_4{}^2 c_5 c_7 c_3{}^3+8 x c_1{}^2 c_5{}^3 c_6 c_3{}^2-8 c_1 c_2 c_4{}^2 c_5 c_6 c_3{}^2+4 x^2 c_1{}^2 c_5{}^3 c_7 c_3{}^2-5 x c_0 c_1 c_5{}^3 c_7 c_3{}^2-c_0 c_2 c_4{}^2 c_5 c_7 c_3{}^2+x c_1 c_2 c_4{}^2 c_5 c_7 c_3{}^2+8 c_1{}^2 c_2 c_5{}^3 c_6 c_3+5 x c_1{}^2 c_2 c_5{}^3 c_7 c_3-5 c_0 c_1 c_2 c_5{}^3 c_7 c_3+5 c_1 c_2{}^2 c_4{}^2 c_5 c_7 c_3+c_1{}^2 c_2{}^2 c_5{}^3 c_7\right )}{8 c_1 c_3 \left (c_3 c_4{}^2-c_1 c_5{}^2\right ){}^2} \]

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Rubi [A]  time = 2.53, antiderivative size = 750, normalized size of antiderivative = 0.64, number of steps used = 7, number of rules used = 5, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.128, Rules used = {1649, 823, 827, 1166, 208} \begin {gather*} \frac {c_7 (c_1 c_2-c_0 c_3){}^2 \left (c_4-c_5 \sqrt {\frac {c_1 x+c_0}{c_3 x+c_2}}\right ) \sqrt {c_5 \sqrt {\frac {c_1 x+c_0}{c_3 x+c_2}}+c_4}}{2 c_3 \left (c_3 c_4{}^2-c_1 c_5{}^2\right ) \left (c_1-\frac {c_3 (c_1 x+c_0)}{c_3 x+c_2}\right ){}^2}-\frac {(c_3 x+c_2) \sqrt {c_5 \sqrt {\frac {c_1 x+c_0}{c_3 x+c_2}}+c_4} \left (c_3 c_5 \left (c_0 c_3 c_7 c_4{}^2-\frac {c_1{}^2 c_5{}^2 (8 c_3 c_6-3 c_2 c_7)}{c_3}+c_1 \left (8 c_3 c_4{}^2 c_6-\left (9 c_2 c_4{}^2-5 c_0 c_5{}^2\right ) c_7\right )\right ) \sqrt {\frac {c_1 x+c_0}{c_3 x+c_2}}-2 c_1 c_4 \left (4 c_3{}^2 c_6 c_4{}^2+c_1 c_2 c_5{}^2 c_7-c_3 \left (4 c_1 c_6 c_5{}^2+\left (4 c_2 c_4{}^2-3 c_0 c_5{}^2\right ) c_7\right )\right )\right )}{8 c_1 c_3 \left (c_3 c_4{}^2-c_1 c_5{}^2\right ){}^2}-\frac {c_5 \left (-c_5 (8 c_3 c_6-3 c_2 c_7) c_1{}^{3/2}-2 \sqrt {c_3} c_4 (4 c_3 c_6-3 c_2 c_7) c_1+5 c_0 c_3 c_5 c_7 \sqrt {c_1}+2 c_0 c_3{}^{3/2} c_4 c_7\right ) (c_1 c_2-c_0 c_3) \tanh ^{-1}\left (\frac {\sqrt [4]{c_3} \sqrt {c_5 \sqrt {\frac {c_1 x+c_0}{c_3 x+c_2}}+c_4}}{\sqrt {\sqrt {c_3} c_4+\sqrt {c_1} c_5}}\right )}{16 c_1{}^{3/2} c_3{}^{7/4} \left (\sqrt {c_3} c_4+\sqrt {c_1} c_5\right ){}^{5/2}}+\frac {c_5 \left (c_5 (8 c_3 c_6-3 c_2 c_7) c_1{}^{3/2}-2 \sqrt {c_3} c_4 (4 c_3 c_6-3 c_2 c_7) c_1-5 c_0 c_3 c_5 c_7 \sqrt {c_1}+2 c_0 c_3{}^{3/2} c_4 c_7\right ) (c_1 c_2-c_0 c_3) \tanh ^{-1}\left (\frac {\sqrt [4]{c_3} \sqrt {c_5 \sqrt {\frac {c_1 x+c_0}{c_3 x+c_2}}+c_4}}{\sqrt {\sqrt {c_3} c_4-\sqrt {c_1} c_5}}\right )}{16 c_1{}^{3/2} c_3{}^{7/4} \left (\sqrt {c_3} c_4-\sqrt {c_1} c_5\right ){}^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(C[6] + x*C[7])/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])^2*(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*C[3]*(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)) - (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]*(2*C[0]*C[3]^(3/2)*C[4]*C[7] + 5*C[0]*Sqrt[C[1]]*C[3]*C[5]*C[7
] - 2*C[1]*Sqrt[C[3]]*C[4]*(4*C[3]*C[6] - 3*C[2]*C[7]) - C[1]^(3/2)*C[5]*(8*C[3]*C[6] - 3*C[2]*C[7])))/(16*C[1
]^(3/2)*C[3]^(7/4)*(Sqrt[C[3]]*C[4] + Sqrt[C[1]]*C[5])^(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]*(2*C[0]*
C[3]^(3/2)*C[4]*C[7] - 5*C[0]*Sqrt[C[1]]*C[3]*C[5]*C[7] - 2*C[1]*Sqrt[C[3]]*C[4]*(4*C[3]*C[6] - 3*C[2]*C[7]) +
C[1]^(3/2)*C[5]*(8*C[3]*C[6] - 3*C[2]*C[7])))/(16*C[1]^(3/2)*C[3]^(7/4)*(Sqrt[C[3]]*C[4] - Sqrt[C[1]]*C[5])^(
5/2)) - ((C[2] + x*C[3])*Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]]*(C[3]*Sqrt[(C[0] + x*C[1])/(C
[2] + x*C[3])]*C[5]*(C[0]*C[3]*C[4]^2*C[7] - (C[1]^2*C[5]^2*(8*C[3]*C[6] - 3*C[2]*C[7]))/C[3] + C[1]*(8*C[3]*C
[4]^2*C[6] - (9*C[2]*C[4]^2 - 5*C[0]*C[5]^2)*C[7])) - 2*C[1]*C[4]*(4*C[3]^2*C[4]^2*C[6] + C[1]*C[2]*C[5]^2*C[7
] - C[3]*(4*C[1]*C[5]^2*C[6] + (4*C[2]*C[4]^2 - 3*C[0]*C[5]^2)*C[7]))))/(8*C[1]*C[3]*(C[3]*C[4]^2 - C[1]*C[5]^
2)^2)

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}{\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 )}{\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 )\\ &=\frac {(c_1 c_2-c_0 c_3){}^2 \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 c_3 \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 )}+\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}{2 c_3}+\frac {1}{2} x c_1 \left (8 c_3 c_4{}^2 c_6-8 c_1 c_5{}^2 c_6+\frac {3 c_1 c_2 c_5{}^2 c_7}{c_3}-\left (8 c_2 c_4{}^2-5 c_0 c_5{}^2\right ) c_7\right )}{\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 )}{2 c_1 \left (c_3 c_4{}^2-c_1 c_5{}^2\right )}\\ &=\frac {(c_1 c_2-c_0 c_3){}^2 \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 c_3 \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 )}-\frac {(c_2+x c_3) \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} \left (c_3 \sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5 \left (c_0 c_3 c_4{}^2 c_7-\frac {c_1{}^2 c_5{}^2 (8 c_3 c_6-3 c_2 c_7)}{c_3}+c_1 \left (8 c_3 c_4{}^2 c_6-\left (9 c_2 c_4{}^2-5 c_0 c_5{}^2\right ) c_7\right )\right )-2 c_1 c_4 \left (4 c_3{}^2 c_4{}^2 c_6+c_1 c_2 c_5{}^2 c_7-c_3 \left (4 c_1 c_5{}^2 c_6+\left (4 c_2 c_4{}^2-3 c_0 c_5{}^2\right ) c_7\right )\right )\right )}{8 c_1 c_3 \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 {1}{4} x c_1 c_5{}^2 \left (c_0 c_3{}^2 c_4{}^2 c_7-c_1{}^2 c_5{}^2 (8 c_3 c_6-3 c_2 c_7)+c_1 c_3 \left (8 c_3 c_4{}^2 c_6-\left (9 c_2 c_4{}^2-5 c_0 c_5{}^2\right ) c_7\right )\right )+\frac {1}{2} c_1 c_3 c_4 c_5 \left (4 c_1{}^2 c_5{}^2 c_6+c_0 c_3 c_4{}^2 c_7-c_1 \left (4 c_3 c_4{}^2 c_6-\left (3 c_2 c_4{}^2-4 c_0 c_5{}^2\right ) c_7\right )\right )}{\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 )}{4 c_1{}^2 c_3 \left (c_3 c_4{}^2-c_1 c_5{}^2\right ){}^2}\\ &=\frac {(c_1 c_2-c_0 c_3){}^2 \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 c_3 \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 )}-\frac {(c_2+x c_3) \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} \left (c_3 \sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5 \left (c_0 c_3 c_4{}^2 c_7-\frac {c_1{}^2 c_5{}^2 (8 c_3 c_6-3 c_2 c_7)}{c_3}+c_1 \left (8 c_3 c_4{}^2 c_6-\left (9 c_2 c_4{}^2-5 c_0 c_5{}^2\right ) c_7\right )\right )-2 c_1 c_4 \left (4 c_3{}^2 c_4{}^2 c_6+c_1 c_2 c_5{}^2 c_7-c_3 \left (4 c_1 c_5{}^2 c_6+\left (4 c_2 c_4{}^2-3 c_0 c_5{}^2\right ) c_7\right )\right )\right )}{8 c_1 c_3 \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 {1}{4} x^2 c_1 c_5{}^2 \left (c_0 c_3{}^2 c_4{}^2 c_7-c_1{}^2 c_5{}^2 (8 c_3 c_6-3 c_2 c_7)+c_1 c_3 \left (8 c_3 c_4{}^2 c_6-\left (9 c_2 c_4{}^2-5 c_0 c_5{}^2\right ) c_7\right )\right )-\frac {1}{4} c_1 c_4 c_5{}^2 \left (c_0 c_3{}^2 c_4{}^2 c_7-c_1{}^2 c_5{}^2 (8 c_3 c_6-3 c_2 c_7)+c_1 c_3 \left (8 c_3 c_4{}^2 c_6-\left (9 c_2 c_4{}^2-5 c_0 c_5{}^2\right ) c_7\right )\right )+\frac {1}{2} c_1 c_3 c_4 c_5{}^2 \left (4 c_1{}^2 c_5{}^2 c_6+c_0 c_3 c_4{}^2 c_7-c_1 \left (4 c_3 c_4{}^2 c_6-\left (3 c_2 c_4{}^2-4 c_0 c_5{}^2\right ) c_7\right )\right )}{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 )}{2 c_1{}^2 c_3 \left (c_3 c_4{}^2-c_1 c_5{}^2\right ){}^2}\\ &=\frac {(c_1 c_2-c_0 c_3){}^2 \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 c_3 \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 )}-\frac {(c_2+x c_3) \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} \left (c_3 \sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5 \left (c_0 c_3 c_4{}^2 c_7-\frac {c_1{}^2 c_5{}^2 (8 c_3 c_6-3 c_2 c_7)}{c_3}+c_1 \left (8 c_3 c_4{}^2 c_6-\left (9 c_2 c_4{}^2-5 c_0 c_5{}^2\right ) c_7\right )\right )-2 c_1 c_4 \left (4 c_3{}^2 c_4{}^2 c_6+c_1 c_2 c_5{}^2 c_7-c_3 \left (4 c_1 c_5{}^2 c_6+\left (4 c_2 c_4{}^2-3 c_0 c_5{}^2\right ) c_7\right )\right )\right )}{8 c_1 c_3 \left (c_3 c_4{}^2-c_1 c_5{}^2\right ){}^2}+\frac {\left ((c_1 c_2-c_0 c_3) c_5 \left (2 c_0 c_3{}^{3/2} c_4 c_7+5 c_0 \sqrt {c_1} c_3 c_5 c_7-2 c_1 \sqrt {c_3} c_4 (4 c_3 c_6-3 c_2 c_7)-c_1{}^{3/2} c_5 (8 c_3 c_6-3 c_2 c_7)\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 )}{16 c_1{}^{3/2} c_3 \left (\sqrt {c_3} c_4+\sqrt {c_1} c_5\right ){}^2}-\frac {\left ((c_1 c_2-c_0 c_3) c_5 \left (2 c_0 c_3{}^{3/2} c_4 c_7-5 c_0 \sqrt {c_1} c_3 c_5 c_7-2 c_1 \sqrt {c_3} c_4 (4 c_3 c_6-3 c_2 c_7)+c_1{}^{3/2} c_5 (8 c_3 c_6-3 c_2 c_7)\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 )}{16 c_1{}^{3/2} c_3 \left (\sqrt {c_3} c_4-\sqrt {c_1} c_5\right ){}^2}\\ &=\frac {(c_1 c_2-c_0 c_3){}^2 \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 c_3 \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 )}-\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 (2 c_0 c_3{}^{3/2} c_4 c_7+5 c_0 \sqrt {c_1} c_3 c_5 c_7-2 c_1 \sqrt {c_3} c_4 (4 c_3 c_6-3 c_2 c_7)-c_1{}^{3/2} c_5 (8 c_3 c_6-3 c_2 c_7)\right )}{16 c_1{}^{3/2} c_3{}^{7/4} \left (\sqrt {c_3} c_4+\sqrt {c_1} c_5\right ){}^{5/2}}+\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 (2 c_0 c_3{}^{3/2} c_4 c_7-5 c_0 \sqrt {c_1} c_3 c_5 c_7-2 c_1 \sqrt {c_3} c_4 (4 c_3 c_6-3 c_2 c_7)+c_1{}^{3/2} c_5 (8 c_3 c_6-3 c_2 c_7)\right )}{16 c_1{}^{3/2} c_3{}^{7/4} \left (\sqrt {c_3} c_4-\sqrt {c_1} c_5\right ){}^{5/2}}-\frac {(c_2+x c_3) \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} \left (c_3 \sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5 \left (c_0 c_3 c_4{}^2 c_7-\frac {c_1{}^2 c_5{}^2 (8 c_3 c_6-3 c_2 c_7)}{c_3}+c_1 \left (8 c_3 c_4{}^2 c_6-\left (9 c_2 c_4{}^2-5 c_0 c_5{}^2\right ) c_7\right )\right )-2 c_1 c_4 \left (4 c_3{}^2 c_4{}^2 c_6+c_1 c_2 c_5{}^2 c_7-c_3 \left (4 c_1 c_5{}^2 c_6+\left (4 c_2 c_4{}^2-3 c_0 c_5{}^2\right ) c_7\right )\right )\right )}{8 c_1 c_3 \left (c_3 c_4{}^2-c_1 c_5{}^2\right ){}^2}\\ \end {align*}

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Mathematica [A]  time = 9.91, size = 1140, normalized size = 0.97 \begin {gather*} -\frac {1}{16} (c_1 c_2-c_0 c_3) \left (\frac {8 (c_2+x c_3){}^2 \left (c_3 \left (c_4-3 \sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5\right ) c_4{}^2+c_1 c_5{}^2 \left (3 c_4-\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5\right )\right ) c_7 \left (c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5\right ){}^{5/2}}{(c_1 c_2-c_0 c_3) \left (c_1 c_5{}^2-c_3 c_4{}^2\right ){}^3}-\frac {2 (c_2+x c_3) \left (c_0 c_3{}^3 \sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5 c_7 c_4{}^4-c_1 c_3{}^2 \left (8 c_3 \left (c_4-\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5\right ) c_6 c_4{}^2+\left (c_2 \left (9 \sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5-8 c_4\right ) c_4{}^2+2 c_0 c_5{}^2 \left (13 c_4+4 \sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5\right )\right ) c_7\right ) c_4{}^2+c_1{}^3 c_5{}^4 \left (c_2 \left (\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5-2 c_4\right ) c_7-8 c_3 \left (c_4-\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5\right ) c_6\right )+c_1{}^2 c_3 c_5{}^2 \left (16 c_3 \left (c_4-\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5\right ) c_6 c_4{}^2+\left (2 c_2 \left (5 c_4+12 \sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5\right ) c_4{}^2+c_0 c_5{}^2 \left (10 c_4-9 \sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5\right )\right ) c_7\right )\right ) \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}{c_1 c_3 (c_1 c_2-c_0 c_3) \left (c_1 c_5{}^2-c_3 c_4{}^2\right ){}^3}+\frac {\tan ^{-1}\left (\frac {\sqrt {c_3 c_4{}^2-c_1 c_5{}^2}}{\sqrt {\sqrt {c_1} \sqrt {c_3} c_5-c_3 c_4} \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}\right ) c_5 \left (\sqrt {c_3} c_4-\sqrt {c_1} c_5\right ) \left (c_5 (8 c_3 c_6-3 c_2 c_7) c_1{}^{3/2}+2 \sqrt {c_3} c_4 (4 c_3 c_6-3 c_2 c_7) c_1-5 c_0 c_3 c_5 c_7 \sqrt {c_1}-2 c_0 c_3{}^{3/2} c_4 c_7\right )}{c_1{}^{3/2} c_3{}^{3/2} \left (\sqrt {c_3} c_4+\sqrt {c_1} c_5\right ){}^2 \sqrt {\sqrt {c_1} \sqrt {c_3} c_5-c_3 c_4} \sqrt {c_3 c_4{}^2-c_1 c_5{}^2}}+\frac {\tan ^{-1}\left (\frac {\sqrt {c_3 c_4{}^2-c_1 c_5{}^2}}{\sqrt {-c_3 c_4-\sqrt {c_1} \sqrt {c_3} c_5} \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}\right ) c_5 \left (\sqrt {c_3} c_4+\sqrt {c_1} c_5\right ) \left (c_5 (8 c_3 c_6-3 c_2 c_7) c_1{}^{3/2}+2 \sqrt {c_3} c_4 (3 c_2 c_7-4 c_3 c_6) c_1-5 c_0 c_3 c_5 c_7 \sqrt {c_1}+2 c_0 c_3{}^{3/2} c_4 c_7\right )}{c_1{}^{3/2} c_3{}^{3/2} \left (\sqrt {c_3} c_4-\sqrt {c_1} c_5\right ){}^2 \sqrt {-c_3 c_4-\sqrt {c_1} \sqrt {c_3} c_5} \sqrt {c_3 c_4{}^2-c_1 c_5{}^2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/16*((C[1]*C[2] - C[0]*C[3])*((8*(C[2] + x*C[3])^2*(C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5])^(5/2)
*(C[3]*C[4]^2*(C[4] - 3*Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]) + C[1]*C[5]^2*(3*C[4] - Sqrt[(C[0] + x*C[1
])/(C[2] + x*C[3])]*C[5]))*C[7])/((C[1]*C[2] - C[0]*C[3])*(-(C[3]*C[4]^2) + C[1]*C[5]^2)^3) + (ArcTan[Sqrt[C[3
]*C[4]^2 - C[1]*C[5]^2]/(Sqrt[-(C[3]*C[4]) + Sqrt[C[1]]*Sqrt[C[3]]*C[5]]*Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2
] + x*C[3])]*C[5]])]*C[5]*(Sqrt[C[3]]*C[4] - Sqrt[C[1]]*C[5])*(-2*C[0]*C[3]^(3/2)*C[4]*C[7] - 5*C[0]*Sqrt[C[1]
]*C[3]*C[5]*C[7] + 2*C[1]*Sqrt[C[3]]*C[4]*(4*C[3]*C[6] - 3*C[2]*C[7]) + C[1]^(3/2)*C[5]*(8*C[3]*C[6] - 3*C[2]*
C[7])))/(C[1]^(3/2)*C[3]^(3/2)*(Sqrt[C[3]]*C[4] + Sqrt[C[1]]*C[5])^2*Sqrt[-(C[3]*C[4]) + Sqrt[C[1]]*Sqrt[C[3]]
*C[5]]*Sqrt[C[3]*C[4]^2 - C[1]*C[5]^2]) + (ArcTan[Sqrt[C[3]*C[4]^2 - C[1]*C[5]^2]/(Sqrt[-(C[3]*C[4]) - Sqrt[C[
1]]*Sqrt[C[3]]*C[5]]*Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]])]*C[5]*(Sqrt[C[3]]*C[4] + Sqrt[C[
1]]*C[5])*(2*C[0]*C[3]^(3/2)*C[4]*C[7] - 5*C[0]*Sqrt[C[1]]*C[3]*C[5]*C[7] + C[1]^(3/2)*C[5]*(8*C[3]*C[6] - 3*C
[2]*C[7]) + 2*C[1]*Sqrt[C[3]]*C[4]*(-4*C[3]*C[6] + 3*C[2]*C[7])))/(C[1]^(3/2)*C[3]^(3/2)*(Sqrt[C[3]]*C[4] - Sq
rt[C[1]]*C[5])^2*Sqrt[-(C[3]*C[4]) - Sqrt[C[1]]*Sqrt[C[3]]*C[5]]*Sqrt[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]]*(C[0]*C[3]^3*Sqrt[(C[0] + x*C[1])/(C[2] + x*C
[3])]*C[4]^4*C[5]*C[7] + C[1]^3*C[5]^4*(-8*C[3]*(C[4] - Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5])*C[6] + C[2
]*(-2*C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5])*C[7]) - C[1]*C[3]^2*C[4]^2*(8*C[3]*C[4]^2*(C[4] - Sqr
t[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5])*C[6] + (2*C[0]*C[5]^2*(13*C[4] + 4*Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3
])]*C[5]) + C[2]*C[4]^2*(-8*C[4] + 9*Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]))*C[7]) + C[1]^2*C[3]*C[5]^2*(
16*C[3]*C[4]^2*(C[4] - Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5])*C[6] + (C[0]*C[5]^2*(10*C[4] - 9*Sqrt[(C[0]
+ x*C[1])/(C[2] + x*C[3])]*C[5]) + 2*C[2]*C[4]^2*(5*C[4] + 12*Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]))*C[
7])))/(C[1]*C[3]*(C[1]*C[2] - C[0]*C[3])*(-(C[3]*C[4]^2) + C[1]*C[5]^2)^3)))

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IntegrateAlgebraic [A]  time = 4.91, size = 2065, normalized size = 1.75 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(ArcTan[(Sqrt[-(C[3]*C[4]) - Sqrt[C[1]]*Sqrt[C[3]]*C[5]]*Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5
]])/(Sqrt[C[3]]*C[4] + Sqrt[C[1]]*C[5])]*Sqrt[-(Sqrt[C[3]]*(Sqrt[C[3]]*C[4] + Sqrt[C[1]]*C[5]))]*(-8*C[1]^2*C[
2]*C[3]^(3/2)*C[4]*C[5]*C[6] + 8*C[0]*C[1]*C[3]^(5/2)*C[4]*C[5]*C[6] - 8*C[1]^(5/2)*C[2]*C[3]*C[5]^2*C[6] + 8*
C[0]*C[1]^(3/2)*C[3]^2*C[5]^2*C[6] + 6*C[1]^2*C[2]^2*Sqrt[C[3]]*C[4]*C[5]*C[7] - 4*C[0]*C[1]*C[2]*C[3]^(3/2)*C
[4]*C[5]*C[7] - 2*C[0]^2*C[3]^(5/2)*C[4]*C[5]*C[7] + 3*C[1]^(5/2)*C[2]^2*C[5]^2*C[7] + 2*C[0]*C[1]^(3/2)*C[2]*
C[3]*C[5]^2*C[7] - 5*C[0]^2*Sqrt[C[1]]*C[3]^2*C[5]^2*C[7]))/(16*C[1]^(3/2)*C[3]^2*(Sqrt[C[3]]*C[4] + Sqrt[C[1]
]*C[5])^3) + (ArcTan[(Sqrt[-(C[3]*C[4]) + Sqrt[C[1]]*Sqrt[C[3]]*C[5]]*Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] +
x*C[3])]*C[5]])/(Sqrt[C[3]]*C[4] - Sqrt[C[1]]*C[5])]*Sqrt[-(Sqrt[C[3]]*(Sqrt[C[3]]*C[4] - Sqrt[C[1]]*C[5]))]*
(-8*C[1]^2*C[2]*C[3]^(3/2)*C[4]*C[5]*C[6] + 8*C[0]*C[1]*C[3]^(5/2)*C[4]*C[5]*C[6] + 8*C[1]^(5/2)*C[2]*C[3]*C[5
]^2*C[6] - 8*C[0]*C[1]^(3/2)*C[3]^2*C[5]^2*C[6] + 6*C[1]^2*C[2]^2*Sqrt[C[3]]*C[4]*C[5]*C[7] - 4*C[0]*C[1]*C[2]
*C[3]^(3/2)*C[4]*C[5]*C[7] - 2*C[0]^2*C[3]^(5/2)*C[4]*C[5]*C[7] - 3*C[1]^(5/2)*C[2]^2*C[5]^2*C[7] - 2*C[0]*C[1
]^(3/2)*C[2]*C[3]*C[5]^2*C[7] + 5*C[0]^2*Sqrt[C[1]]*C[3]^2*C[5]^2*C[7]))/(16*C[1]^(3/2)*C[3]^2*(-(Sqrt[C[3]]*C
[4]) + Sqrt[C[1]]*C[5])^3) + (Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]]*(8*C[1]^3*C[2]*C[3]^2*C[
4]^3*C[6] - 8*C[0]*C[1]^2*C[3]^3*C[4]^3*C[6] - (8*C[1]^2*(C[0] + x*C[1])*C[2]*C[3]^3*C[4]^3*C[6])/(C[2] + x*C[
3]) + (8*C[0]*C[1]*(C[0] + x*C[1])*C[3]^4*C[4]^3*C[6])/(C[2] + x*C[3]) - 8*C[1]^3*C[2]*C[3]^2*Sqrt[(C[0] + x*C
[1])/(C[2] + x*C[3])]*C[4]^2*C[5]*C[6] + 8*C[0]*C[1]^2*C[3]^3*Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[4]^2*C[5
]*C[6] + 8*C[1]^2*C[2]*C[3]^3*((C[0] + x*C[1])/(C[2] + x*C[3]))^(3/2)*C[4]^2*C[5]*C[6] - 8*C[0]*C[1]*C[3]^4*((
C[0] + x*C[1])/(C[2] + x*C[3]))^(3/2)*C[4]^2*C[5]*C[6] - 8*C[1]^4*C[2]*C[3]*C[4]*C[5]^2*C[6] + 8*C[0]*C[1]^3*C
[3]^2*C[4]*C[5]^2*C[6] + (8*C[1]^3*(C[0] + x*C[1])*C[2]*C[3]^2*C[4]*C[5]^2*C[6])/(C[2] + x*C[3]) - (8*C[0]*C[1
]^2*(C[0] + x*C[1])*C[3]^3*C[4]*C[5]^2*C[6])/(C[2] + x*C[3]) + 8*C[1]^4*C[2]*C[3]*Sqrt[(C[0] + x*C[1])/(C[2] +
x*C[3])]*C[5]^3*C[6] - 8*C[0]*C[1]^3*C[3]^2*Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]^3*C[6] - 8*C[1]^3*C[2]
*C[3]^2*((C[0] + x*C[1])/(C[2] + x*C[3]))^(3/2)*C[5]^3*C[6] + 8*C[0]*C[1]^2*C[3]^3*((C[0] + x*C[1])/(C[2] + x*
C[3]))^(3/2)*C[5]^3*C[6] - 4*C[1]^3*C[2]^2*C[3]*C[4]^3*C[7] + 4*C[0]^2*C[1]*C[3]^3*C[4]^3*C[7] + (8*C[1]^2*(C[
0] + x*C[1])*C[2]^2*C[3]^2*C[4]^3*C[7])/(C[2] + x*C[3]) - (8*C[0]*C[1]*(C[0] + x*C[1])*C[2]*C[3]^3*C[4]^3*C[7]
)/(C[2] + x*C[3]) + 5*C[1]^3*C[2]^2*C[3]*Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[4]^2*C[5]*C[7] - 2*C[0]*C[1]^
2*C[2]*C[3]^2*Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[4]^2*C[5]*C[7] - 3*C[0]^2*C[1]*C[3]^3*Sqrt[(C[0] + x*C[1
])/(C[2] + x*C[3])]*C[4]^2*C[5]*C[7] - 9*C[1]^2*C[2]^2*C[3]^2*((C[0] + x*C[1])/(C[2] + x*C[3]))^(3/2)*C[4]^2*C
[5]*C[7] + 10*C[0]*C[1]*C[2]*C[3]^3*((C[0] + x*C[1])/(C[2] + x*C[3]))^(3/2)*C[4]^2*C[5]*C[7] - C[0]^2*C[3]^4*(
(C[0] + x*C[1])/(C[2] + x*C[3]))^(3/2)*C[4]^2*C[5]*C[7] - 2*C[1]^4*C[2]^2*C[4]*C[5]^2*C[7] + 12*C[0]*C[1]^3*C[
2]*C[3]*C[4]*C[5]^2*C[7] - 10*C[0]^2*C[1]^2*C[3]^2*C[4]*C[5]^2*C[7] - (2*C[1]^3*(C[0] + x*C[1])*C[2]^2*C[3]*C[
4]*C[5]^2*C[7])/(C[2] + x*C[3]) - (4*C[0]*C[1]^2*(C[0] + x*C[1])*C[2]*C[3]^2*C[4]*C[5]^2*C[7])/(C[2] + x*C[3])
+ (6*C[0]^2*C[1]*(C[0] + x*C[1])*C[3]^3*C[4]*C[5]^2*C[7])/(C[2] + x*C[3]) + C[1]^4*C[2]^2*Sqrt[(C[0] + x*C[1]
)/(C[2] + x*C[3])]*C[5]^3*C[7] - 10*C[0]*C[1]^3*C[2]*C[3]*Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]^3*C[7] +
9*C[0]^2*C[1]^2*C[3]^2*Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]^3*C[7] + 3*C[1]^3*C[2]^2*C[3]*((C[0] + x*C[1
])/(C[2] + x*C[3]))^(3/2)*C[5]^3*C[7] + 2*C[0]*C[1]^2*C[2]*C[3]^2*((C[0] + x*C[1])/(C[2] + x*C[3]))^(3/2)*C[5]
^3*C[7] - 5*C[0]^2*C[1]*C[3]^3*((C[0] + x*C[1])/(C[2] + x*C[3]))^(3/2)*C[5]^3*C[7]))/(8*C[1]*C[3]*(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)

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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)/(_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)/(_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: 11.29Done

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maple [F]  time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {\textit {\_C7} x +\textit {\_C6}}{\sqrt {\textit {\_C4} +\sqrt {\frac {\textit {\_C1} x +\textit {\_C0}}{\textit {\_C3} x +\textit {\_C2}}}\, \textit {\_C5}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((_C7*x+_C6)/(_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 {\_{C_{7}} x + \_{C_{6}}}{\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)/(_C4+((_C1*x+_C0)/(_C3*x+_C2))^(1/2)*_C5)^(1/2),x, algorithm="maxima")

[Out]

integrate((_C7*x + _C6)/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 {_{\mathrm {C6}}+_{\mathrm {C7}}\,x}{\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)/(_C4 + _C5*((_C0 + _C1*x)/(_C2 + _C3*x))^(1/2))^(1/2),x)

[Out]

int((_C6 + _C7*x)/(_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 {_C6 + _C7 x}{\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)/(_C4+((_C1*x+_C0)/(_C3*x+_C2))**(1/2)*_C5)**(1/2),x)

[Out]

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

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