3.32.50 \(\int \frac {c_8+x c_9}{\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} (c_6+x c_7)} \, dx\)

Optimal. Leaf size=1716 \[ \frac {2 \tan ^{-1}\left (\frac {\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} \sqrt {c_3 c_6-c_2 c_7}}{\sqrt {-\sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7} c_5-c_3 c_4 c_6+c_2 c_4 c_7}}\right ) \sqrt {c_3 c_6-c_2 c_7} c_8}{c_7 \sqrt {-\sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7} c_5-c_3 c_4 c_6+c_2 c_4 c_7}}+\frac {2 \tan ^{-1}\left (\frac {\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} \sqrt {c_3 c_6-c_2 c_7}}{\sqrt {\sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7} c_5-c_3 c_4 c_6+c_2 c_4 c_7}}\right ) \sqrt {c_3 c_6-c_2 c_7} c_8}{c_7 \sqrt {\sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7} c_5-c_3 c_4 c_6+c_2 c_4 c_7}}+\frac {2 \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 [4]{c_3} c_8}{\sqrt {\sqrt {c_3} c_4-\sqrt {c_1} c_5} c_7}+\frac {2 \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 [4]{c_3} c_8}{\sqrt {\sqrt {c_3} c_4+\sqrt {c_1} c_5} c_7}-\frac {2 \tan ^{-1}\left (\frac {\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} \sqrt {c_2 c_7-c_3 c_6}}{\sqrt {-i \sqrt {c_1 c_6-c_0 c_7} \sqrt {c_2 c_7-c_3 c_6} c_5+c_3 c_4 c_6-c_2 c_4 c_7}}\right ) c_6 \sqrt {c_2 c_7-c_3 c_6} c_9}{c_7{}^2 \sqrt {-i \sqrt {c_1 c_6-c_0 c_7} \sqrt {c_2 c_7-c_3 c_6} c_5+c_3 c_4 c_6-c_2 c_4 c_7}}-\frac {2 \tan ^{-1}\left (\frac {\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} \sqrt {c_2 c_7-c_3 c_6}}{\sqrt {i \sqrt {c_1 c_6-c_0 c_7} \sqrt {c_2 c_7-c_3 c_6} c_5+c_3 c_4 c_6-c_2 c_4 c_7}}\right ) c_6 \sqrt {c_2 c_7-c_3 c_6} c_9}{c_7{}^2 \sqrt {i \sqrt {c_1 c_6-c_0 c_7} \sqrt {c_2 c_7-c_3 c_6} c_5+c_3 c_4 c_6-c_2 c_4 c_7}}+\frac {\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} \left (-c_1 c_2 c_4 c_9+c_0 c_3 c_4 c_9+c_1 c_2 \sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5 c_9-c_0 c_3 \sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5 c_9\right )}{\left (c_1-\frac {(c_0+x c_1) c_3}{c_2+x c_3}\right ) \left (c_1 c_5{}^2-c_3 c_4{}^2\right ) c_7}+\frac {\tan ^{-1}\left (\frac {\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}}{\sqrt {c_3} c_4-\sqrt {c_1} c_5}\right ) \left (4 \sqrt {c_1} c_4 \sqrt {-\sqrt {c_3} \left (\sqrt {c_3} c_4-\sqrt {c_1} c_5\right )} c_6 c_9 c_3{}^{3/2}-4 c_1 c_5 \sqrt {-\sqrt {c_3} \left (\sqrt {c_3} c_4-\sqrt {c_1} c_5\right )} c_6 c_9 c_3-c_0 c_5 \sqrt {-\sqrt {c_3} \left (\sqrt {c_3} c_4-\sqrt {c_1} c_5\right )} c_7 c_9 c_3+c_1 c_2 c_5 \sqrt {-\sqrt {c_3} \left (\sqrt {c_3} c_4-\sqrt {c_1} c_5\right )} c_7 c_9\right )}{2 \sqrt {c_1} c_3 \left (\sqrt {c_3} c_4-\sqrt {c_1} c_5\right ){}^2 c_7{}^2}+\frac {\tan ^{-1}\left (\frac {\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}}{\sqrt {c_3} c_4+\sqrt {c_1} c_5}\right ) \left (4 \sqrt {c_1} c_4 \sqrt {-\sqrt {c_3} \left (\sqrt {c_3} c_4+\sqrt {c_1} c_5\right )} c_6 c_9 c_3{}^{3/2}+4 c_1 c_5 \sqrt {-\sqrt {c_3} \left (\sqrt {c_3} c_4+\sqrt {c_1} c_5\right )} c_6 c_9 c_3+c_0 c_5 \sqrt {-\sqrt {c_3} \left (\sqrt {c_3} c_4+\sqrt {c_1} c_5\right )} c_7 c_9 c_3-c_1 c_2 c_5 \sqrt {-\sqrt {c_3} \left (\sqrt {c_3} c_4+\sqrt {c_1} c_5\right )} c_7 c_9\right )}{2 \sqrt {c_1} c_3 \left (\sqrt {c_3} c_4+\sqrt {c_1} c_5\right ){}^2 c_7{}^2} \]

________________________________________________________________________________________

Rubi [A]  time = 33.94, antiderivative size = 944, normalized size of antiderivative = 0.55, number of steps used = 15, number of rules used = 5, integrand size = 48, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.104, Rules used = {6741, 6728, 1178, 1166, 208} \begin {gather*} \frac {(c_2+x c_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_9}{\left (c_3 c_4{}^2-c_1 c_5{}^2\right ) c_7}-\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 c_9}{2 \sqrt {c_1} c_3{}^{3/4} \left (\sqrt {c_3} c_4-\sqrt {c_1} c_5\right ){}^{3/2} c_7}+\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 c_9}{2 \sqrt {c_1} c_3{}^{3/4} \left (\sqrt {c_3} c_4+\sqrt {c_1} c_5\right ){}^{3/2} c_7}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} \sqrt {c_3 c_6-c_2 c_7}}{\sqrt {-\sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7} c_5+c_3 c_4 c_6-c_2 c_4 c_7}}\right ) \sqrt {c_3 c_6-c_2 c_7} (c_7 c_8-c_6 c_9)}{c_7{}^2 \sqrt {-\sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7} c_5+c_3 c_4 c_6-c_2 c_4 c_7}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} \sqrt {c_3 c_6-c_2 c_7}}{\sqrt {\sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7} c_5+c_3 c_4 c_6-c_2 c_4 c_7}}\right ) \sqrt {c_3 c_6-c_2 c_7} (c_7 c_8-c_6 c_9)}{c_7{}^2 \sqrt {\sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7} c_5+c_3 c_4 c_6-c_2 c_4 c_7}}+\frac {2 \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 [4]{c_3} (c_7 c_8-c_6 c_9)}{\sqrt {\sqrt {c_3} c_4-\sqrt {c_1} c_5} c_7{}^2}+\frac {2 \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 [4]{c_3} (c_7 c_8-c_6 c_9)}{\sqrt {\sqrt {c_3} c_4+\sqrt {c_1} c_5} c_7{}^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 1178

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(x*(a*b*e - d*(b^2 - 2*
a*c) - c*(b*d - 2*a*e)*x^2)*(a + b*x^2 + c*x^4)^(p + 1))/(2*a*(p + 1)*(b^2 - 4*a*c)), x] + Dist[1/(2*a*(p + 1)
*(b^2 - 4*a*c)), Int[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 7)*(d*b - 2*a*e)*c*x^2, x]*(a +
b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e
^2, 0] && LtQ[p, -1] && IntegerQ[2*p]

Rule 6728

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rule 6741

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rubi steps

\begin {align*} \int \frac {c_8+x c_9}{\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} (c_6+x c_7)} \, dx &=(2 (c_1 c_2-c_0 c_3)) \operatorname {Subst}\left (\int \frac {x \left (-c_1 c_8+c_0 c_9+x^2 (c_3 c_8-c_2 c_9)\right )}{\left (c_1-x^2 c_3\right ){}^2 \sqrt {c_4+x c_5} \left (-c_1 c_6+c_0 c_7+x^2 (c_3 c_6-c_2 c_7)\right )} \, dx,x,\sqrt {\frac {c_0+x c_1}{c_2+x c_3}}\right )\\ &=\frac {(4 (c_1 c_2-c_0 c_3)) \operatorname {Subst}\left (\int \frac {\left (x^2-c_4\right ) \left (-c_1 c_8+c_0 c_9+\frac {\left (x^2-c_4\right ){}^2 (c_3 c_8-c_2 c_9)}{c_5{}^2}\right )}{\left (c_1-\frac {c_3 \left (x^2-c_4\right ){}^2}{c_5{}^2}\right ){}^2 \left (-c_1 c_6+c_0 c_7+\frac {\left (x^2-c_4\right ){}^2 (c_3 c_6-c_2 c_7)}{c_5{}^2}\right )} \, dx,x,\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}\right )}{c_5{}^2}\\ &=\frac {(4 (c_1 c_2-c_0 c_3)) \operatorname {Subst}\left (\int \frac {\left (x^2-c_4\right ) \left (c_1 c_8-\frac {c_3 c_4{}^2 c_8}{c_5{}^2}-c_0 c_9+\frac {c_2 c_4{}^2 c_9}{c_5{}^2}-\frac {x^4 (c_3 c_8-c_2 c_9)}{c_5{}^2}+\frac {2 x^2 c_4 (c_3 c_8-c_2 c_9)}{c_5{}^2}\right )}{\left (c_1-\frac {x^4 c_3}{c_5{}^2}+\frac {2 x^2 c_3 c_4}{c_5{}^2}-\frac {c_3 c_4{}^2}{c_5{}^2}\right ){}^2 \left (c_1 c_6-\frac {c_3 c_4{}^2 c_6}{c_5{}^2}-c_0 c_7+\frac {c_2 c_4{}^2 c_7}{c_5{}^2}-\frac {x^4 (c_3 c_6-c_2 c_7)}{c_5{}^2}+\frac {2 x^2 c_4 (c_3 c_6-c_2 c_7)}{c_5{}^2}\right )} \, dx,x,\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}\right )}{c_5{}^2}\\ &=\frac {(4 (c_1 c_2-c_0 c_3)) \operatorname {Subst}\left (\int \left (\frac {\left (x^2-c_4\right ) c_5{}^4 c_9}{\left (x^4 c_3-2 x^2 c_3 c_4+c_3 c_4{}^2-c_1 c_5{}^2\right ){}^2 c_7}+\frac {c_3 \left (x^2-c_4\right ) c_5{}^2 (c_7 c_8-c_6 c_9)}{(c_1 c_2-c_0 c_3) \left (-x^4 c_3+2 x^2 c_3 c_4-c_3 c_4{}^2+c_1 c_5{}^2\right ) c_7{}^2}+\frac {\left (x^2-c_4\right ) c_5{}^2 (c_3 c_6-c_2 c_7) (c_7 c_8-c_6 c_9)}{(c_1 c_2-c_0 c_3) c_7{}^2 \left (c_3 c_4{}^2 c_6-c_1 c_5{}^2 c_6-\left (c_2 c_4{}^2-c_0 c_5{}^2\right ) c_7+x^4 (c_3 c_6-c_2 c_7)-2 x^2 c_4 (c_3 c_6-c_2 c_7)\right )}\right ) \, dx,x,\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}\right )}{c_5{}^2}\\ &=\frac {\left (4 (c_1 c_2-c_0 c_3) c_5{}^2 c_9\right ) \operatorname {Subst}\left (\int \frac {x^2-c_4}{\left (x^4 c_3-2 x^2 c_3 c_4+c_3 c_4{}^2-c_1 c_5{}^2\right ){}^2} \, dx,x,\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}\right )}{c_7}+\frac {(4 c_3 (c_7 c_8-c_6 c_9)) \operatorname {Subst}\left (\int \frac {x^2-c_4}{-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 )}{c_7{}^2}+\frac {(4 (c_3 c_6-c_2 c_7) (c_7 c_8-c_6 c_9)) \operatorname {Subst}\left (\int \frac {x^2-c_4}{c_3 c_4{}^2 c_6-c_1 c_5{}^2 c_6-\left (c_2 c_4{}^2-c_0 c_5{}^2\right ) c_7+x^4 (c_3 c_6-c_2 c_7)-2 x^2 c_4 (c_3 c_6-c_2 c_7)} \, dx,x,\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}\right )}{c_7{}^2}\\ &=\frac {(c_2+x c_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_9}{\left (c_3 c_4{}^2-c_1 c_5{}^2\right ) c_7}-\frac {((c_1 c_2-c_0 c_3) c_9) \operatorname {Subst}\left (\int \frac {-2 x^2 c_1 c_3 c_5{}^2+4 c_1 c_3 c_4 c_5{}^2}{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 c_3 \left (c_3 c_4{}^2-c_1 c_5{}^2\right ) c_7}+\frac {(2 c_3 (c_7 c_8-c_6 c_9)) \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 )}{c_7{}^2}+\frac {(2 c_3 (c_7 c_8-c_6 c_9)) \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 )}{c_7{}^2}+\frac {(2 (c_3 c_6-c_2 c_7) (c_7 c_8-c_6 c_9)) \operatorname {Subst}\left (\int \frac {1}{-c_5 \sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7}+x^2 (c_3 c_6-c_2 c_7)-c_4 (c_3 c_6-c_2 c_7)} \, dx,x,\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}\right )}{c_7{}^2}+\frac {(2 (c_3 c_6-c_2 c_7) (c_7 c_8-c_6 c_9)) \operatorname {Subst}\left (\int \frac {1}{c_5 \sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7}+x^2 (c_3 c_6-c_2 c_7)-c_4 (c_3 c_6-c_2 c_7)} \, dx,x,\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}\right )}{c_7{}^2}\\ &=\frac {(c_2+x c_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_9}{\left (c_3 c_4{}^2-c_1 c_5{}^2\right ) c_7}+\frac {2 \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 [4]{c_3} (c_7 c_8-c_6 c_9)}{\sqrt {\sqrt {c_3} c_4-\sqrt {c_1} c_5} c_7{}^2}+\frac {2 \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 [4]{c_3} (c_7 c_8-c_6 c_9)}{\sqrt {\sqrt {c_3} c_4+\sqrt {c_1} c_5} c_7{}^2}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} \sqrt {c_3 c_6-c_2 c_7}}{\sqrt {c_3 c_4 c_6-c_2 c_4 c_7-c_5 \sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7}}}\right ) \sqrt {c_3 c_6-c_2 c_7} (c_7 c_8-c_6 c_9)}{c_7{}^2 \sqrt {c_3 c_4 c_6-c_2 c_4 c_7-c_5 \sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7}}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} \sqrt {c_3 c_6-c_2 c_7}}{\sqrt {c_3 c_4 c_6-c_2 c_4 c_7+c_5 \sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7}}}\right ) \sqrt {c_3 c_6-c_2 c_7} (c_7 c_8-c_6 c_9)}{c_7{}^2 \sqrt {c_3 c_4 c_6-c_2 c_4 c_7+c_5 \sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7}}}+\frac {((c_1 c_2-c_0 c_3) c_5 c_9) \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 )}{2 \sqrt {c_1} \left (\sqrt {c_3} c_4-\sqrt {c_1} c_5\right ) c_7}-\frac {((c_1 c_2-c_0 c_3) c_5 c_9) \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 )}{2 \sqrt {c_1} \left (\sqrt {c_3} c_4+\sqrt {c_1} c_5\right ) c_7}\\ &=-\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 c_9}{2 \sqrt {c_1} c_3{}^{3/4} \left (\sqrt {c_3} c_4-\sqrt {c_1} c_5\right ){}^{3/2} c_7}+\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 c_9}{2 \sqrt {c_1} c_3{}^{3/4} \left (\sqrt {c_3} c_4+\sqrt {c_1} c_5\right ){}^{3/2} c_7}+\frac {(c_2+x c_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_9}{\left (c_3 c_4{}^2-c_1 c_5{}^2\right ) c_7}+\frac {2 \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 [4]{c_3} (c_7 c_8-c_6 c_9)}{\sqrt {\sqrt {c_3} c_4-\sqrt {c_1} c_5} c_7{}^2}+\frac {2 \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 [4]{c_3} (c_7 c_8-c_6 c_9)}{\sqrt {\sqrt {c_3} c_4+\sqrt {c_1} c_5} c_7{}^2}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} \sqrt {c_3 c_6-c_2 c_7}}{\sqrt {c_3 c_4 c_6-c_2 c_4 c_7-c_5 \sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7}}}\right ) \sqrt {c_3 c_6-c_2 c_7} (c_7 c_8-c_6 c_9)}{c_7{}^2 \sqrt {c_3 c_4 c_6-c_2 c_4 c_7-c_5 \sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7}}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} \sqrt {c_3 c_6-c_2 c_7}}{\sqrt {c_3 c_4 c_6-c_2 c_4 c_7+c_5 \sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7}}}\right ) \sqrt {c_3 c_6-c_2 c_7} (c_7 c_8-c_6 c_9)}{c_7{}^2 \sqrt {c_3 c_4 c_6-c_2 c_4 c_7+c_5 \sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7}}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 16.13, size = 2281, normalized size = 1.33 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(C[8] + x*C[9])/(Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]]*(C[6] + x*C[7])),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 6.33, size = 1344, normalized size = 0.78 \begin {gather*} \frac {\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} \left (-c_1 c_2 c_4 c_9+c_0 c_3 c_4 c_9+c_1 c_2 \sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5 c_9-c_0 c_3 \sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5 c_9\right )}{\left (c_1-\frac {(c_0+x c_1) c_3}{c_2+x c_3}\right ) \left (-c_3 c_4{}^2+c_1 c_5{}^2\right ) c_7}-\frac {2 \tan ^{-1}\left (\frac {\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} \sqrt {c_3 c_6-c_2 c_7}}{\sqrt {-c_3 c_4 c_6+c_2 c_4 c_7-c_5 \sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7}}}\right ) \sqrt {c_3 c_6-c_2 c_7} (-c_7 c_8+c_6 c_9)}{c_7{}^2 \sqrt {-c_3 c_4 c_6+c_2 c_4 c_7-c_5 \sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7}}}-\frac {2 \tan ^{-1}\left (\frac {\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} \sqrt {c_3 c_6-c_2 c_7}}{\sqrt {-c_3 c_4 c_6+c_2 c_4 c_7+c_5 \sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7}}}\right ) \sqrt {c_3 c_6-c_2 c_7} (-c_7 c_8+c_6 c_9)}{c_7{}^2 \sqrt {-c_3 c_4 c_6+c_2 c_4 c_7+c_5 \sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7}}}+\frac {\tan ^{-1}\left (\frac {\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}}{\sqrt {c_3} c_4-\sqrt {c_1} c_5}\right ) \left (-4 \sqrt {c_1} c_3{}^{3/2} c_4 \sqrt {-\sqrt {c_3} \left (\sqrt {c_3} c_4-\sqrt {c_1} c_5\right )} c_7 c_8+4 c_1 c_3 c_5 \sqrt {-\sqrt {c_3} \left (\sqrt {c_3} c_4-\sqrt {c_1} c_5\right )} c_7 c_8+4 \sqrt {c_1} c_3{}^{3/2} c_4 \sqrt {-\sqrt {c_3} \left (\sqrt {c_3} c_4-\sqrt {c_1} c_5\right )} c_6 c_9-4 c_1 c_3 c_5 \sqrt {-\sqrt {c_3} \left (\sqrt {c_3} c_4-\sqrt {c_1} c_5\right )} c_6 c_9+c_1 c_2 c_5 \sqrt {-\sqrt {c_3} \left (\sqrt {c_3} c_4-\sqrt {c_1} c_5\right )} c_7 c_9-c_0 c_3 c_5 \sqrt {-\sqrt {c_3} \left (\sqrt {c_3} c_4-\sqrt {c_1} c_5\right )} c_7 c_9\right )}{2 \sqrt {c_1} c_3 \left (\sqrt {c_3} c_4-\sqrt {c_1} c_5\right ){}^2 c_7{}^2}+\frac {\tan ^{-1}\left (\frac {\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}}{\sqrt {c_3} c_4+\sqrt {c_1} c_5}\right ) \left (-4 \sqrt {c_1} c_3{}^{3/2} c_4 \sqrt {-\sqrt {c_3} \left (\sqrt {c_3} c_4+\sqrt {c_1} c_5\right )} c_7 c_8-4 c_1 c_3 c_5 \sqrt {-\sqrt {c_3} \left (\sqrt {c_3} c_4+\sqrt {c_1} c_5\right )} c_7 c_8+4 \sqrt {c_1} c_3{}^{3/2} c_4 \sqrt {-\sqrt {c_3} \left (\sqrt {c_3} c_4+\sqrt {c_1} c_5\right )} c_6 c_9+4 c_1 c_3 c_5 \sqrt {-\sqrt {c_3} \left (\sqrt {c_3} c_4+\sqrt {c_1} c_5\right )} c_6 c_9-c_1 c_2 c_5 \sqrt {-\sqrt {c_3} \left (\sqrt {c_3} c_4+\sqrt {c_1} c_5\right )} c_7 c_9+c_0 c_3 c_5 \sqrt {-\sqrt {c_3} \left (\sqrt {c_3} c_4+\sqrt {c_1} c_5\right )} c_7 c_9\right )}{2 \sqrt {c_1} c_3 \left (\sqrt {c_3} c_4+\sqrt {c_1} c_5\right ){}^2 c_7{}^2} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(C[8] + x*C[9])/(Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]]*(C[6] + x*C[7])),x
]

[Out]

(Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]]*(-(C[1]*C[2]*C[4]*C[9]) + C[0]*C[3]*C[4]*C[9] + C[1]*
C[2]*Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]*C[9] - C[0]*C[3]*Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]*C[
9]))/((C[1] - ((C[0] + x*C[1])*C[3])/(C[2] + x*C[3]))*(-(C[3]*C[4]^2) + C[1]*C[5]^2)*C[7]) - (2*ArcTan[(Sqrt[C
[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]]*Sqrt[C[3]*C[6] - C[2]*C[7]])/Sqrt[-(C[3]*C[4]*C[6]) + C[2]*C
[4]*C[7] - C[5]*Sqrt[C[1]*C[6] - C[0]*C[7]]*Sqrt[C[3]*C[6] - C[2]*C[7]]]]*Sqrt[C[3]*C[6] - C[2]*C[7]]*(-(C[7]*
C[8]) + C[6]*C[9]))/(C[7]^2*Sqrt[-(C[3]*C[4]*C[6]) + C[2]*C[4]*C[7] - C[5]*Sqrt[C[1]*C[6] - C[0]*C[7]]*Sqrt[C[
3]*C[6] - C[2]*C[7]]]) - (2*ArcTan[(Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]]*Sqrt[C[3]*C[6] - C
[2]*C[7]])/Sqrt[-(C[3]*C[4]*C[6]) + C[2]*C[4]*C[7] + C[5]*Sqrt[C[1]*C[6] - C[0]*C[7]]*Sqrt[C[3]*C[6] - C[2]*C[
7]]]]*Sqrt[C[3]*C[6] - C[2]*C[7]]*(-(C[7]*C[8]) + C[6]*C[9]))/(C[7]^2*Sqrt[-(C[3]*C[4]*C[6]) + C[2]*C[4]*C[7]
+ C[5]*Sqrt[C[1]*C[6] - C[0]*C[7]]*Sqrt[C[3]*C[6] - C[2]*C[7]]]) + (ArcTan[(Sqrt[-(C[3]*C[4]) + Sqrt[C[1]]*Sqr
t[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])]*(-
4*Sqrt[C[1]]*C[3]^(3/2)*C[4]*Sqrt[-(Sqrt[C[3]]*(Sqrt[C[3]]*C[4] - Sqrt[C[1]]*C[5]))]*C[7]*C[8] + 4*C[1]*C[3]*C
[5]*Sqrt[-(Sqrt[C[3]]*(Sqrt[C[3]]*C[4] - Sqrt[C[1]]*C[5]))]*C[7]*C[8] + 4*Sqrt[C[1]]*C[3]^(3/2)*C[4]*Sqrt[-(Sq
rt[C[3]]*(Sqrt[C[3]]*C[4] - Sqrt[C[1]]*C[5]))]*C[6]*C[9] - 4*C[1]*C[3]*C[5]*Sqrt[-(Sqrt[C[3]]*(Sqrt[C[3]]*C[4]
- Sqrt[C[1]]*C[5]))]*C[6]*C[9] + C[1]*C[2]*C[5]*Sqrt[-(Sqrt[C[3]]*(Sqrt[C[3]]*C[4] - Sqrt[C[1]]*C[5]))]*C[7]*
C[9] - C[0]*C[3]*C[5]*Sqrt[-(Sqrt[C[3]]*(Sqrt[C[3]]*C[4] - Sqrt[C[1]]*C[5]))]*C[7]*C[9]))/(2*Sqrt[C[1]]*C[3]*(
Sqrt[C[3]]*C[4] - Sqrt[C[1]]*C[5])^2*C[7]^2) + (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])]*(-4*Sqrt[C[1]]*C[3]^(3
/2)*C[4]*Sqrt[-(Sqrt[C[3]]*(Sqrt[C[3]]*C[4] + Sqrt[C[1]]*C[5]))]*C[7]*C[8] - 4*C[1]*C[3]*C[5]*Sqrt[-(Sqrt[C[3]
]*(Sqrt[C[3]]*C[4] + Sqrt[C[1]]*C[5]))]*C[7]*C[8] + 4*Sqrt[C[1]]*C[3]^(3/2)*C[4]*Sqrt[-(Sqrt[C[3]]*(Sqrt[C[3]]
*C[4] + Sqrt[C[1]]*C[5]))]*C[6]*C[9] + 4*C[1]*C[3]*C[5]*Sqrt[-(Sqrt[C[3]]*(Sqrt[C[3]]*C[4] + Sqrt[C[1]]*C[5]))
]*C[6]*C[9] - C[1]*C[2]*C[5]*Sqrt[-(Sqrt[C[3]]*(Sqrt[C[3]]*C[4] + Sqrt[C[1]]*C[5]))]*C[7]*C[9] + C[0]*C[3]*C[5
]*Sqrt[-(Sqrt[C[3]]*(Sqrt[C[3]]*C[4] + Sqrt[C[1]]*C[5]))]*C[7]*C[9]))/(2*Sqrt[C[1]]*C[3]*(Sqrt[C[3]]*C[4] + Sq
rt[C[1]]*C[5])^2*C[7]^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((_C9*x+_C8)/(_C4+((_C1*x+_C0)/(_C3*x+_C2))^(1/2)*_C5)^(1/2)/(_C7*x+_C6),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((_C9*x+_C8)/(_C4+((_C1*x+_C0)/(_C3*x+_C2))^(1/2)*_C5)^(1/2)/(_C7*x+_C6),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=[-70,15,2,72,-32,97]Warning, need to choose a branch for the roo
t of a polynomial with parameters. This might be wrong.Non regular value [0,0,0,0,0,0] was discarded and repla
ced randomly by 0=[-66,-39,-82,10,16,91]Warning, need to choose a branch for the root of a polynomial with par
ameters. This might be wrong.Non regular value [0,0,0,0,0,0] was discarded and replaced randomly by 0=[13,80,8
2,-92,-95,41]Warning, need to choose a branch for the root of a polynomial with parameters. This might be wron
g.Non regular value [0,0,0,0,0,0] was discarded and replaced randomly by 0=[51,55,80,34,31,-65]Warning, need t
o 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=[-46,-16,-32,-64,-40,-89]Warning, need to choose a branch for t
he 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=[68,-24,87,84,-84,46]Warning, need to choose a branch for the root of a polynomial wit
h parameters. This might be wrong.Non regular value [0,0,0,0,0,0] was discarded and replaced randomly by 0=[69
,-62,15,83,-10,18]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=[-3,-5,78,-5,-95,-95]Warning,
integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [abs(t_n
ostep*1_C3+1_C2)]Evaluation time: 13.68Done

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maple [F]  time = 0.07, size = 0, normalized size = 0.00 \[\int \frac {\textit {\_C9} x +\textit {\_C8}}{\sqrt {\textit {\_C4} +\sqrt {\frac {\textit {\_C1} x +\textit {\_C0}}{\textit {\_C3} x +\textit {\_C2}}}\, \textit {\_C5}}\, \left (\textit {\_C7} x +\textit {\_C6} \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((_C9*x+_C8)/(_C4+((_C1*x+_C0)/(_C3*x+_C2))^(1/2)*_C5)^(1/2)/(_C7*x+_C6),x)

[Out]

int((_C9*x+_C8)/(_C4+((_C1*x+_C0)/(_C3*x+_C2))^(1/2)*_C5)^(1/2)/(_C7*x+_C6),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\_{C_{9}} x + \_{C_{8}}}{{\left (\_{C_{7}} x + \_{C_{6}}\right )} \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((_C9*x+_C8)/(_C4+((_C1*x+_C0)/(_C3*x+_C2))^(1/2)*_C5)^(1/2)/(_C7*x+_C6),x, algorithm="maxima")

[Out]

integrate((_C9*x + _C8)/((_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 {C8}}+_{\mathrm {C9}}\,x}{\sqrt {_{\mathrm {C4}}+_{\mathrm {C5}}\,\sqrt {\frac {_{\mathrm {C0}}+_{\mathrm {C1}}\,x}{_{\mathrm {C2}}+_{\mathrm {C3}}\,x}}}\,\left (_{\mathrm {C6}}+_{\mathrm {C7}}\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((_C8 + _C9*x)/((_C4 + _C5*((_C0 + _C1*x)/(_C2 + _C3*x))^(1/2))^(1/2)*(_C6 + _C7*x)),x)

[Out]

int((_C8 + _C9*x)/((_C4 + _C5*((_C0 + _C1*x)/(_C2 + _C3*x))^(1/2))^(1/2)*(_C6 + _C7*x)), x)

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sympy [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((_C9*x+_C8)/(_C4+((_C1*x+_C0)/(_C3*x+_C2))**(1/2)*_C5)**(1/2)/(_C7*x+_C6),x)

[Out]

Timed out

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