3.31.100 \(\int \frac {1}{\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} (c_6+x c_7)} \, dx\)

Optimal. Leaf size=561 \[ \frac {2 \sqrt {c_3 c_6-c_2 c_7} \tan ^{-1}\left (\frac {\sqrt {c_3 c_6-c_2 c_7} \sqrt {c_5 \sqrt {\frac {c_1 x+c_0}{c_3 x+c_2}}+c_4}}{\sqrt {-\sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7} c_5-c_3 c_4 c_6+c_2 c_4 c_7}}\right )}{c_7 \sqrt {-\sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7} c_5-c_3 c_4 c_6+c_2 c_4 c_7}}+\frac {2 \sqrt {c_3 c_6-c_2 c_7} \tan ^{-1}\left (\frac {\sqrt {c_3 c_6-c_2 c_7} \sqrt {c_5 \sqrt {\frac {c_1 x+c_0}{c_3 x+c_2}}+c_4}}{\sqrt {\sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7} c_5-c_3 c_4 c_6+c_2 c_4 c_7}}\right )}{c_7 \sqrt {\sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7} c_5-c_3 c_4 c_6+c_2 c_4 c_7}}+\frac {2 \sqrt [4]{c_3} \tanh ^{-1}\left (\frac {\sqrt [4]{c_3} \sqrt {c_5 \sqrt {\frac {c_1 x+c_0}{c_3 x+c_2}}+c_4}}{\sqrt {\sqrt {c_3} c_4-\sqrt {c_1} c_5}}\right )}{\sqrt {\sqrt {c_3} c_4-\sqrt {c_1} c_5} c_7}+\frac {2 \sqrt [4]{c_3} \tanh ^{-1}\left (\frac {\sqrt [4]{c_3} \sqrt {c_5 \sqrt {\frac {c_1 x+c_0}{c_3 x+c_2}}+c_4}}{\sqrt {\sqrt {c_3} c_4+\sqrt {c_1} c_5}}\right )}{\sqrt {\sqrt {c_3} c_4+\sqrt {c_1} c_5} c_7} \]

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Rubi [A]  time = 5.67, antiderivative size = 561, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 4, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.098, Rules used = {6741, 6728, 1166, 208} \begin {gather*} \frac {2 \sqrt [4]{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 )}{\sqrt {\sqrt {c_3} c_4-\sqrt {c_1} c_5} c_7}-\frac {2 \sqrt {c_3 c_6-c_2 c_7} \tanh ^{-1}\left (\frac {\sqrt {c_3 c_6-c_2 c_7} \sqrt {c_5 \sqrt {\frac {c_1 x+c_0}{c_3 x+c_2}}+c_4}}{\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 )}{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 \sqrt {c_3 c_6-c_2 c_7} \tanh ^{-1}\left (\frac {\sqrt {c_3 c_6-c_2 c_7} \sqrt {c_5 \sqrt {\frac {c_1 x+c_0}{c_3 x+c_2}}+c_4}}{\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 )}{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 \sqrt [4]{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 )}{\sqrt {\sqrt {c_3} c_4+\sqrt {c_1} c_5} c_7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(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))/(Sqrt[Sqrt[C[3]]*C[4] - Sqrt[C[1]]*C[5]]*C[7]) + (2*ArcTanh[(C[3]^(1/4)*Sqrt[C[4] + Sqr
t[(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))/(Sqrt[Sqrt[C[3]
]*C[4] + Sqrt[C[1]]*C[5]]*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]*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]*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 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 {1}{\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-x^2 c_3\right ) \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 {x^2-c_4}{\left (c_1-\frac {c_3 \left (x^2-c_4\right ){}^2}{c_5{}^2}\right ) \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 {x^2-c_4}{\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 ) \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 {c_3 \left (x^2-c_4\right ) c_5{}^2}{(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}+\frac {\left (x^2-c_4\right ) c_5{}^2 (c_3 c_6-c_2 c_7)}{(c_1 c_2-c_0 c_3) c_7 \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 {(4 c_3) \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}+\frac {(4 (c_3 c_6-c_2 c_7)) \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}\\ &=\frac {(2 c_3) \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}+\frac {(2 c_3) \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}+\frac {(2 (c_3 c_6-c_2 c_7)) \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}+\frac {(2 (c_3 c_6-c_2 c_7)) \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}\\ &=\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}}{\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}}{\sqrt {\sqrt {c_3} c_4+\sqrt {c_1} c_5} 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 \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 \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*}

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Mathematica [A]  time = 3.35, size = 535, normalized size = 0.95 \begin {gather*} \frac {2 \left (\frac {\sqrt [4]{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 )}{\sqrt {\sqrt {c_3} c_4-\sqrt {c_1} c_5}}+\sqrt {c_3 c_6-c_2 c_7} \left (-\frac {\tanh ^{-1}\left (\frac {\sqrt {c_3 c_6-c_2 c_7} \sqrt {c_5 \sqrt {\frac {c_1 x+c_0}{c_3 x+c_2}}+c_4}}{\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 {-\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 {\tanh ^{-1}\left (\frac {\sqrt {c_3 c_6-c_2 c_7} \sqrt {c_5 \sqrt {\frac {c_1 x+c_0}{c_3 x+c_2}}+c_4}}{\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 {\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 )+\frac {\sqrt [4]{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 )}{\sqrt {\sqrt {c_3} c_4+\sqrt {c_1} c_5}}\right )}{c_7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(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))/Sqrt[Sqrt[C[3]]*C[4] - Sqrt[C[1]]*C[5]] + (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))/Sqrt[Sqrt[C[3]]*C[4] +
Sqrt[C[1]]*C[5]] + Sqrt[C[3]*C[6] - C[2]*C[7]]*(-(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]]*Sqr
t[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]]]) - 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]]]]/Sqr
t[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[7]

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IntegrateAlgebraic [A]  time = 3.15, size = 645, normalized size = 1.15 \begin {gather*} -\frac {2 \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 ) \sqrt {-\sqrt {c_3} \left (\sqrt {c_3} c_4-\sqrt {c_1} c_5\right )}}{\left (\sqrt {c_3} c_4-\sqrt {c_1} c_5\right ) c_7}-\frac {2 \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 ) \sqrt {-\sqrt {c_3} \left (\sqrt {c_3} c_4+\sqrt {c_1} c_5\right )}}{\left (\sqrt {c_3} c_4+\sqrt {c_1} c_5\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 \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 \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 {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-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])]*Sqrt[-(Sqrt[C[3]]*(Sqrt[C[3]]*C[4] - Sqrt[C[1]]*C[5]))])/((Sqrt[C[
3]]*C[4] - Sqrt[C[1]]*C[5])*C[7]) - (2*ArcTan[(Sqrt[-(C[3]*C[4]) - Sqrt[C[1]]*Sqrt[C[3]]*C[5]]*Sqrt[C[4] + Sqr
t[(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]))])/((Sqrt[C[3]]*C[4] + Sqrt[C[1]]*C[5])*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]*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] + Sqr
t[(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]*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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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(1/(_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(1/(_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=[33,-4,-70,15,2,72]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 replac
ed randomly by 0=[86,-68,-66,-39,-82,10]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=[48,-16,
13,80,82,-92]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=[-17,63,68,98,0,-13]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=[-29,45,75,-8,42,-53]Warning, need to choose a branch for the r
oot of a polynomial with parameters. This might be wrong.Non regular value [0,0,0,0,0,0] was discarded and rep
laced randomly by 0=[-16,-32,-64,-40,-89,-64]Evaluation time: 7.18Done

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maple [F]  time = 0.07, size = 0, normalized size = 0.00 \[\int \frac {1}{\sqrt {\textit {\_C4} +\sqrt {\frac {\textit {\_C1} x +\textit {\_C0}}{\textit {\_C3} x +\textit {\_C2}}}\, \textit {\_C5}}\, \left (\textit {\_C7} x +\textit {\_C6} \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {_C4 + _C5 \sqrt {\frac {_C0}{_C2 + _C3 x} + \frac {_C1 x}{_C2 + _C3 x}}} \left (_C6 + _C7 x\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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