3.31.100 \(\int \frac {1}{\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} (c_6+x c_7)} \, dx\) [3100]

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

[Out]

2*arctanh(_C3^(1/4)*(_C4+((_C1*x+_C0)/(_C3*x+_C2))^(1/2)*_C5)^(1/2)/(_C3^(1/2)*_C4-_C1^(1/2)*_C5)^(1/2))*_C3^(
1/4)/(_C3^(1/2)*_C4-_C1^(1/2)*_C5)^(1/2)/_C7+2*arctanh(_C3^(1/4)*(_C4+((_C1*x+_C0)/(_C3*x+_C2))^(1/2)*_C5)^(1/
2)/(_C3^(1/2)*_C4+_C1^(1/2)*_C5)^(1/2))*_C3^(1/4)/(_C3^(1/2)*_C4+_C1^(1/2)*_C5)^(1/2)/_C7+2*arctan((_C4+((_C1*
x+_C0)/(_C3*x+_C2))^(1/2)*_C5)^(1/2)*(-_C2*_C7+_C3*_C6)^(1/2)/(-_C3*_C4*_C6+_C2*_C4*_C7-_C5*(-_C0*_C7+_C1*_C6)
^(1/2)*(-_C2*_C7+_C3*_C6)^(1/2))^(1/2))*(-_C2*_C7+_C3*_C6)^(1/2)/_C7/(-_C3*_C4*_C6+_C2*_C4*_C7-_C5*(-_C0*_C7+_
C1*_C6)^(1/2)*(-_C2*_C7+_C3*_C6)^(1/2))^(1/2)+2*arctan((_C4+((_C1*x+_C0)/(_C3*x+_C2))^(1/2)*_C5)^(1/2)*(-_C2*_
C7+_C3*_C6)^(1/2)/(-_C3*_C4*_C6+_C2*_C4*_C7+_C5*(-_C0*_C7+_C1*_C6)^(1/2)*(-_C2*_C7+_C3*_C6)^(1/2))^(1/2))*(-_C
2*_C7+_C3*_C6)^(1/2)/_C7/(-_C3*_C4*_C6+_C2*_C4*_C7+_C5*(-_C0*_C7+_C1*_C6)^(1/2)*(-_C2*_C7+_C3*_C6)^(1/2))^(1/2
)

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Rubi [A]
time = 4.39, 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 = {6873, 6860, 1180, 214} \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 214

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

Rule 1180

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 6860

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 6873

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)) \text {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)) \text {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)) \text {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)) \text {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) \text {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)) \text {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) \text {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) \text {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)) \text {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)) \text {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 = 56.01, size = 968, normalized size = 1.73 \begin {gather*} \frac {2 (c_1 c_2-c_0 c_3) \left (\frac {\text {ArcTan}\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 ) \sqrt {c_3} \left (\sqrt {c_3} c_4+\sqrt {c_1} c_5\right )}{\sqrt {-c_3 c_4-\sqrt {c_1} \sqrt {c_3} c_5} \sqrt {c_3 c_4{}^2-c_1 c_5{}^2}}+\frac {\text {ArcTan}\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 ) \sqrt {c_3} \left (\sqrt {c_3} c_4-\sqrt {c_1} c_5\right )}{\sqrt {-c_3 c_4+\sqrt {c_1} \sqrt {c_3} c_5} \sqrt {c_3 c_4{}^2-c_1 c_5{}^2}}-\frac {\text {ArcTan}\left (\frac {\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 {\frac {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}}}\right ) \sqrt {c_3 c_6-c_2 c_7} \left (c_1 c_5 c_6-c_0 c_5 c_7+c_4 \sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7}\right )}{\sqrt {c_1 c_6-c_0 c_7} \sqrt {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} \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 {\text {ArcTan}\left (\frac {\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 {\frac {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}}}\right ) \sqrt {c_3 c_6-c_2 c_7} \left (-c_1 c_5 c_6+c_0 c_5 c_7+c_4 \sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7}\right )}{\sqrt {c_1 c_6-c_0 c_7} \sqrt {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} \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 )}{(-c_1 c_2+c_0 c_3) 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*(C[1]*C[2] - C[0]*C[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]])]*Sqrt[C[3]]*(Sqrt[C[3]]*C[4] + Sqrt[C[1]]*C[5]
))/(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]])]*Sqrt[C[3]]*(Sqrt[C[3]]*C[4] - Sqrt[C[1]]*C[5]))/(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[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]]])]*Sqrt[C[3]*C[6] - C[2]*C[7]]*(C[1]*C[5]*C[6]
- C[0]*C[5]*C[7] + C[4]*Sqrt[C[1]*C[6] - C[0]*C[7]]*Sqrt[C[3]*C[6] - C[2]*C[7]]))/(Sqrt[C[1]*C[6] - C[0]*C[7]
]*Sqrt[C[3]*C[4]^2*C[6] - C[1]*C[5]^2*C[6] + (-(C[2]*C[4]^2) + 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[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]]*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[1]*C[5]*C[6]) + C[0]*C[5]*C[7] + C[4]*Sqrt[C[1]*C[6] - C[0]*C[7]]*Sqrt[C[3]*C
[6] - C[2]*C[7]]))/(Sqrt[C[1]*C[6] - C[0]*C[7]]*Sqrt[C[3]*C[4]^2*C[6] - C[1]*C[5]^2*C[6] + (-(C[2]*C[4]^2) + 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]]])))/((-(C[1]*C[2]) + C[0]*C[3])*C[7])

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Maple [F]
time = 0.05, 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*} \text {Failed to integrate} \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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 3633 vs. \(2 (357) = 714\).
time = 173.34, size = 3633, normalized size = 6.48 \begin {gather*} \text {Too large to display} \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]

sqrt((C3*C4*C6 - C2*C4*C7 + ((C3*C4^2 - C1*C5^2)*C6*C7^2 - (C2*C4^2 - C0*C5^2)*C7^3)*sqrt((C1*C3*C5^2*C6^2 + C
0*C2*C5^2*C7^2 - (C1*C2 + C0*C3)*C5^2*C6*C7)/((C3^2*C4^4 - 2*C1*C3*C4^2*C5^2 + C1^2*C5^4)*C6^2*C7^4 - 2*(C2*C3
*C4^4 + C0*C1*C5^4 - (C1*C2 + C0*C3)*C4^2*C5^2)*C6*C7^5 + (C2^2*C4^4 - 2*C0*C2*C4^2*C5^2 + C0^2*C5^4)*C7^6)))/
((C3*C4^2 - C1*C5^2)*C6*C7^2 - (C2*C4^2 - C0*C5^2)*C7^3))*log(-4*(C3*C6 - C2*C7)*sqrt(C5*sqrt((C1*x + C0)/(C3*
x + C2)) + C4) + 4*(C3*C4*C6*C7 - C2*C4*C7^2 - ((C3*C4^2 - C1*C5^2)*C6*C7^3 - (C2*C4^2 - C0*C5^2)*C7^4)*sqrt((
C1*C3*C5^2*C6^2 + C0*C2*C5^2*C7^2 - (C1*C2 + C0*C3)*C5^2*C6*C7)/((C3^2*C4^4 - 2*C1*C3*C4^2*C5^2 + C1^2*C5^4)*C
6^2*C7^4 - 2*(C2*C3*C4^4 + C0*C1*C5^4 - (C1*C2 + C0*C3)*C4^2*C5^2)*C6*C7^5 + (C2^2*C4^4 - 2*C0*C2*C4^2*C5^2 +
C0^2*C5^4)*C7^6)))*sqrt((C3*C4*C6 - C2*C4*C7 + ((C3*C4^2 - C1*C5^2)*C6*C7^2 - (C2*C4^2 - C0*C5^2)*C7^3)*sqrt((
C1*C3*C5^2*C6^2 + C0*C2*C5^2*C7^2 - (C1*C2 + C0*C3)*C5^2*C6*C7)/((C3^2*C4^4 - 2*C1*C3*C4^2*C5^2 + C1^2*C5^4)*C
6^2*C7^4 - 2*(C2*C3*C4^4 + C0*C1*C5^4 - (C1*C2 + C0*C3)*C4^2*C5^2)*C6*C7^5 + (C2^2*C4^4 - 2*C0*C2*C4^2*C5^2 +
C0^2*C5^4)*C7^6)))/((C3*C4^2 - C1*C5^2)*C6*C7^2 - (C2*C4^2 - C0*C5^2)*C7^3))) - sqrt((C3*C4*C6 - C2*C4*C7 + ((
C3*C4^2 - C1*C5^2)*C6*C7^2 - (C2*C4^2 - C0*C5^2)*C7^3)*sqrt((C1*C3*C5^2*C6^2 + C0*C2*C5^2*C7^2 - (C1*C2 + C0*C
3)*C5^2*C6*C7)/((C3^2*C4^4 - 2*C1*C3*C4^2*C5^2 + C1^2*C5^4)*C6^2*C7^4 - 2*(C2*C3*C4^4 + C0*C1*C5^4 - (C1*C2 +
C0*C3)*C4^2*C5^2)*C6*C7^5 + (C2^2*C4^4 - 2*C0*C2*C4^2*C5^2 + C0^2*C5^4)*C7^6)))/((C3*C4^2 - C1*C5^2)*C6*C7^2 -
(C2*C4^2 - C0*C5^2)*C7^3))*log(-4*(C3*C6 - C2*C7)*sqrt(C5*sqrt((C1*x + C0)/(C3*x + C2)) + C4) - 4*(C3*C4*C6*C
7 - C2*C4*C7^2 - ((C3*C4^2 - C1*C5^2)*C6*C7^3 - (C2*C4^2 - C0*C5^2)*C7^4)*sqrt((C1*C3*C5^2*C6^2 + C0*C2*C5^2*C
7^2 - (C1*C2 + C0*C3)*C5^2*C6*C7)/((C3^2*C4^4 - 2*C1*C3*C4^2*C5^2 + C1^2*C5^4)*C6^2*C7^4 - 2*(C2*C3*C4^4 + C0*
C1*C5^4 - (C1*C2 + C0*C3)*C4^2*C5^2)*C6*C7^5 + (C2^2*C4^4 - 2*C0*C2*C4^2*C5^2 + C0^2*C5^4)*C7^6)))*sqrt((C3*C4
*C6 - C2*C4*C7 + ((C3*C4^2 - C1*C5^2)*C6*C7^2 - (C2*C4^2 - C0*C5^2)*C7^3)*sqrt((C1*C3*C5^2*C6^2 + C0*C2*C5^2*C
7^2 - (C1*C2 + C0*C3)*C5^2*C6*C7)/((C3^2*C4^4 - 2*C1*C3*C4^2*C5^2 + C1^2*C5^4)*C6^2*C7^4 - 2*(C2*C3*C4^4 + C0*
C1*C5^4 - (C1*C2 + C0*C3)*C4^2*C5^2)*C6*C7^5 + (C2^2*C4^4 - 2*C0*C2*C4^2*C5^2 + C0^2*C5^4)*C7^6)))/((C3*C4^2 -
C1*C5^2)*C6*C7^2 - (C2*C4^2 - C0*C5^2)*C7^3))) + sqrt((C3*C4*C6 - C2*C4*C7 - ((C3*C4^2 - C1*C5^2)*C6*C7^2 - (
C2*C4^2 - C0*C5^2)*C7^3)*sqrt((C1*C3*C5^2*C6^2 + C0*C2*C5^2*C7^2 - (C1*C2 + C0*C3)*C5^2*C6*C7)/((C3^2*C4^4 - 2
*C1*C3*C4^2*C5^2 + C1^2*C5^4)*C6^2*C7^4 - 2*(C2*C3*C4^4 + C0*C1*C5^4 - (C1*C2 + C0*C3)*C4^2*C5^2)*C6*C7^5 + (C
2^2*C4^4 - 2*C0*C2*C4^2*C5^2 + C0^2*C5^4)*C7^6)))/((C3*C4^2 - C1*C5^2)*C6*C7^2 - (C2*C4^2 - C0*C5^2)*C7^3))*lo
g(-4*(C3*C6 - C2*C7)*sqrt(C5*sqrt((C1*x + C0)/(C3*x + C2)) + C4) + 4*(C3*C4*C6*C7 - C2*C4*C7^2 + ((C3*C4^2 - C
1*C5^2)*C6*C7^3 - (C2*C4^2 - C0*C5^2)*C7^4)*sqrt((C1*C3*C5^2*C6^2 + C0*C2*C5^2*C7^2 - (C1*C2 + C0*C3)*C5^2*C6*
C7)/((C3^2*C4^4 - 2*C1*C3*C4^2*C5^2 + C1^2*C5^4)*C6^2*C7^4 - 2*(C2*C3*C4^4 + C0*C1*C5^4 - (C1*C2 + C0*C3)*C4^2
*C5^2)*C6*C7^5 + (C2^2*C4^4 - 2*C0*C2*C4^2*C5^2 + C0^2*C5^4)*C7^6)))*sqrt((C3*C4*C6 - C2*C4*C7 - ((C3*C4^2 - C
1*C5^2)*C6*C7^2 - (C2*C4^2 - C0*C5^2)*C7^3)*sqrt((C1*C3*C5^2*C6^2 + C0*C2*C5^2*C7^2 - (C1*C2 + C0*C3)*C5^2*C6*
C7)/((C3^2*C4^4 - 2*C1*C3*C4^2*C5^2 + C1^2*C5^4)*C6^2*C7^4 - 2*(C2*C3*C4^4 + C0*C1*C5^4 - (C1*C2 + C0*C3)*C4^2
*C5^2)*C6*C7^5 + (C2^2*C4^4 - 2*C0*C2*C4^2*C5^2 + C0^2*C5^4)*C7^6)))/((C3*C4^2 - C1*C5^2)*C6*C7^2 - (C2*C4^2 -
C0*C5^2)*C7^3))) - sqrt((C3*C4*C6 - C2*C4*C7 - ((C3*C4^2 - C1*C5^2)*C6*C7^2 - (C2*C4^2 - C0*C5^2)*C7^3)*sqrt(
(C1*C3*C5^2*C6^2 + C0*C2*C5^2*C7^2 - (C1*C2 + C0*C3)*C5^2*C6*C7)/((C3^2*C4^4 - 2*C1*C3*C4^2*C5^2 + C1^2*C5^4)*
C6^2*C7^4 - 2*(C2*C3*C4^4 + C0*C1*C5^4 - (C1*C2 + C0*C3)*C4^2*C5^2)*C6*C7^5 + (C2^2*C4^4 - 2*C0*C2*C4^2*C5^2 +
C0^2*C5^4)*C7^6)))/((C3*C4^2 - C1*C5^2)*C6*C7^2 - (C2*C4^2 - C0*C5^2)*C7^3))*log(-4*(C3*C6 - C2*C7)*sqrt(C5*s
qrt((C1*x + C0)/(C3*x + C2)) + C4) - 4*(C3*C4*C6*C7 - C2*C4*C7^2 + ((C3*C4^2 - C1*C5^2)*C6*C7^3 - (C2*C4^2 - C
0*C5^2)*C7^4)*sqrt((C1*C3*C5^2*C6^2 + C0*C2*C5^2*C7^2 - (C1*C2 + C0*C3)*C5^2*C6*C7)/((C3^2*C4^4 - 2*C1*C3*C4^2
*C5^2 + C1^2*C5^4)*C6^2*C7^4 - 2*(C2*C3*C4^4 + C0*C1*C5^4 - (C1*C2 + C0*C3)*C4^2*C5^2)*C6*C7^5 + (C2^2*C4^4 -
2*C0*C2*C4^2*C5^2 + C0^2*C5^4)*C7^6)))*sqrt((C3*C4*C6 - C2*C4*C7 - ((C3*C4^2 - C1*C5^2)*C6*C7^2 - (C2*C4^2 - C
0*C5^2)*C7^3)*sqrt((C1*C3*C5^2*C6^2 + C0*C2*C5^2*C7^2 - (C1*C2 + C0*C3)*C5^2*C6*C7)/((C3^2*C4^4 - 2*C1*C3*C4^2
*C5^2 + C1^2*C5^4)*C6^2*C7^4 - 2*(C2*C3*C4^4 + C0*C1*C5^4 - (C1*C2 + C0*C3)*C4^2*C5^2)*C6*C7^5 + (C2^2*C4^4 -
2*C0*C2*C4^2*C5^2 + C0^2*C5^4)*C7^6)))/((C3*C4^2 - C1*C5^2)*C6*C7^2 - (C2*C4^2 - C0*C5^2)*C7^3))) - sqrt(((C3*
C4^2 - C1*C5^2)*C7^2*sqrt(C1*C3*C5^2/((C3^2*C4^4 - 2*C1*C3*C4^2*C5^2 + C1^2*C5^4)*C7^4)) + C3*C4)/((C3*C4^2 -
C1*C5^2)*C7^2))*log(4*sqrt(C5*sqrt((C1*x + C0)/(C3*x + C2)) + C4)*C3 + 4*((C3*C4^2 - C1*C5^2)*C7^3*sqrt(C1*C3*
C5^2/((C3^2*C4^4 - 2*C1*C3*C4^2*C5^2 + C1^2*C5^...

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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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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,sageVARx):;OUTP
UT:Warning, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Ch
eck [abs(sa

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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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