3.32.37 \(\int \frac {1}{\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} (c_6+x c_7){}^2} \, dx\) [3137]

Optimal. Leaf size=884 \[ \frac {\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} \left (-c_1 c_2 c_4+c_0 c_3 c_4+c_1 c_2 \sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5-c_0 c_3 \sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5\right )}{\left (c_1 c_6-\frac {(c_0+x c_1) c_3 c_6}{c_2+x c_3}-c_0 c_7+\frac {(c_0+x c_1) c_2 c_7}{c_2+x c_3}\right ) \left (-c_3 c_4{}^2 c_6+c_1 c_5{}^2 c_6+c_2 c_4{}^2 c_7-c_0 c_5{}^2 c_7\right )}+\frac {\text {ArcTan}\left (\frac {\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} \sqrt {c_3 c_6-c_2 c_7}}{\sqrt {-c_3 c_4 c_6+c_2 c_4 c_7-c_5 \sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7}}}\right ) \left (-c_1 c_2 c_5{}^2 \sqrt {c_1 c_6-c_0 c_7}+c_0 c_3 c_5{}^2 \sqrt {c_1 c_6-c_0 c_7}+c_1 c_2 c_4 c_5 \sqrt {c_3 c_6-c_2 c_7}-c_0 c_3 c_4 c_5 \sqrt {c_3 c_6-c_2 c_7}\right )}{2 \sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7} \left (-c_3 c_4{}^2 c_6+c_1 c_5{}^2 c_6+c_2 c_4{}^2 c_7-c_0 c_5{}^2 c_7\right ) \sqrt {-c_3 c_4 c_6+c_2 c_4 c_7-c_5 \sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7}}}+\frac {\text {ArcTan}\left (\frac {\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} \sqrt {c_3 c_6-c_2 c_7}}{\sqrt {-c_3 c_4 c_6+c_2 c_4 c_7+c_5 \sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7}}}\right ) \left (-c_1 c_2 c_5{}^2 \sqrt {c_1 c_6-c_0 c_7}+c_0 c_3 c_5{}^2 \sqrt {c_1 c_6-c_0 c_7}-c_1 c_2 c_4 c_5 \sqrt {c_3 c_6-c_2 c_7}+c_0 c_3 c_4 c_5 \sqrt {c_3 c_6-c_2 c_7}\right )}{2 \sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7} \left (-c_3 c_4{}^2 c_6+c_1 c_5{}^2 c_6+c_2 c_4{}^2 c_7-c_0 c_5{}^2 c_7\right ) \sqrt {-c_3 c_4 c_6+c_2 c_4 c_7+c_5 \sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7}}} \]

[Out]

(_C4+((_C1*x+_C0)/(_C3*x+_C2))^(1/2)*_C5)^(1/2)*(-_C1*_C2*_C4+_C0*_C3*_C4+_C1*_C2*((_C1*x+_C0)/(_C3*x+_C2))^(1
/2)*_C5-_C0*_C3*((_C1*x+_C0)/(_C3*x+_C2))^(1/2)*_C5)/(_C1*_C6-(_C1*x+_C0)*_C3*_C6/(_C3*x+_C2)-_C0*_C7+(_C1*x+_
C0)*_C2*_C7/(_C3*x+_C2))/(-_C0*_C5^2*_C7+_C1*_C5^2*_C6+_C2*_C4^2*_C7-_C3*_C4^2*_C6)+1/2*arctan((_C4+((_C1*x+_C
0)/(_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))*(-_C1*_C2*_C5^2*(-_C0*_C7+_C1*_C6)^(1/2)+_C0*_C3*_C5^2*(-_C0*_C7+_C1*_C6)^
(1/2)+_C1*_C2*_C4*_C5*(-_C2*_C7+_C3*_C6)^(1/2)-_C0*_C3*_C4*_C5*(-_C2*_C7+_C3*_C6)^(1/2))/(-_C0*_C7+_C1*_C6)^(1
/2)/(-_C2*_C7+_C3*_C6)^(1/2)/(-_C0*_C5^2*_C7+_C1*_C5^2*_C6+_C2*_C4^2*_C7-_C3*_C4^2*_C6)/(-_C3*_C4*_C6+_C2*_C4*
_C7-_C5*(-_C0*_C7+_C1*_C6)^(1/2)*(-_C2*_C7+_C3*_C6)^(1/2))^(1/2)+1/2*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*_C
6)^(1/2))^(1/2))*(-_C1*_C2*_C5^2*(-_C0*_C7+_C1*_C6)^(1/2)+_C0*_C3*_C5^2*(-_C0*_C7+_C1*_C6)^(1/2)-_C1*_C2*_C4*_
C5*(-_C2*_C7+_C3*_C6)^(1/2)+_C0*_C3*_C4*_C5*(-_C2*_C7+_C3*_C6)^(1/2))/(-_C0*_C7+_C1*_C6)^(1/2)/(-_C2*_C7+_C3*_
C6)^(1/2)/(-_C0*_C5^2*_C7+_C1*_C5^2*_C6+_C2*_C4^2*_C7-_C3*_C4^2*_C6)/(-_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 = 3.62, antiderivative size = 711, normalized size of antiderivative = 0.80, number of steps used = 6, number of rules used = 4, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.098, Rules used = {837, 841, 1180, 214} \begin {gather*} -\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_4-c_5 \sqrt {\frac {c_1 x+c_0}{c_3 x+c_2}}\right )}{\left (c_5{}^2 (c_1 c_6-c_0 c_7)-c_4{}^2 (c_3 c_6-c_2 c_7)\right ) (c_7 x+c_6)}+\frac {(c_1 c_2-c_0 c_3) c_5 \left (\sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7} c_4+c_1 c_5 c_6-c_0 c_5 c_7\right ) \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 )}{2 (c_1 c_6-c_0 c_7) \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} \left (c_5{}^2 (c_1 c_6-c_0 c_7)-c_4{}^2 (c_3 c_6-c_2 c_7)\right )}+\frac {(c_1 c_2-c_0 c_3) c_5 \left (-\sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7} c_4+c_1 c_5 c_6-c_0 c_5 c_7\right ) \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 )}{2 (c_1 c_6-c_0 c_7) \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} \left (c_5{}^2 (c_1 c_6-c_0 c_7)-c_4{}^2 (c_3 c_6-c_2 c_7)\right )} \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])^2),x]

[Out]

-(((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[6] + x*C[7])*(C[5]^2*(C[1]*C[6] - C[0]*C[7]) - C[4]^2*(C[3]*C[6] - C[2]*C[7])))) + (ArcT
anh[(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]]]]*(C[1]*C[2] - C[0]*C[3])*C[5]
*(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]]))/(2*(C[1]*C[
6] - C[0]*C[7])*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]]]*(C[5]^2*(C[1]*C[6] - C[0]*C[7]) - C[4]^2*(C[3]*C[6] - C[2]*C[7]))) + (ArcTa
nh[(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]]]]*(C[1]*C[2] - C[0]*C[3])*C[5]*
(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]]))/(2*(C[1]*C[6
] - C[0]*C[7])*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]]]*(C[5]^2*(C[1]*C[6] - C[0]*C[7]) - C[4]^2*(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 837

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] +
Dist[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 841

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 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]

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){}^2} \, dx &=(2 (c_1 c_2-c_0 c_3)) \text {Subst}\left (\int \frac {x}{\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 ){}^2} \, dx,x,\sqrt {\frac {c_0+x c_1}{c_2+x c_3}}\right )\\ &=-\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_6+x c_7) \left (c_5{}^2 (c_1 c_6-c_0 c_7)-c_4{}^2 (c_3 c_6-c_2 c_7)\right )}-\frac {(c_1 c_2-c_0 c_3) \text {Subst}\left (\int \frac {-\frac {1}{2} c_4 c_5 (c_1 c_6-c_0 c_7) (c_3 c_6-c_2 c_7)+\frac {1}{2} x c_5{}^2 (c_1 c_6-c_0 c_7) (c_3 c_6-c_2 c_7)}{\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 )}{(c_1 c_6-c_0 c_7) (c_3 c_6-c_2 c_7) \left (c_5{}^2 (c_1 c_6-c_0 c_7)-c_4{}^2 (c_3 c_6-c_2 c_7)\right )}\\ &=-\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_6+x c_7) \left (c_5{}^2 (c_1 c_6-c_0 c_7)-c_4{}^2 (c_3 c_6-c_2 c_7)\right )}-\frac {(2 (c_1 c_2-c_0 c_3)) \text {Subst}\left (\int \frac {\frac {1}{2} x^2 c_5{}^2 (c_1 c_6-c_0 c_7) (c_3 c_6-c_2 c_7)-c_4 c_5{}^2 (c_1 c_6-c_0 c_7) (c_3 c_6-c_2 c_7)}{c_5{}^2 (-c_1 c_6+c_0 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)+c_4{}^2 (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_1 c_6-c_0 c_7) (c_3 c_6-c_2 c_7) \left (c_5{}^2 (c_1 c_6-c_0 c_7)-c_4{}^2 (c_3 c_6-c_2 c_7)\right )}\\ &=-\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_6+x c_7) \left (c_5{}^2 (c_1 c_6-c_0 c_7)-c_4{}^2 (c_3 c_6-c_2 c_7)\right )}-\frac {\left ((c_1 c_2-c_0 c_3) c_5 \left (c_5 \sqrt {c_1 c_6-c_0 c_7}-c_4 \sqrt {c_3 c_6-c_2 c_7}\right )\right ) \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 )}{2 \sqrt {c_1 c_6-c_0 c_7} \left (c_5{}^2 (c_1 c_6-c_0 c_7)-c_4{}^2 (c_3 c_6-c_2 c_7)\right )}-\frac {\left ((c_1 c_2-c_0 c_3) c_5 \left (c_5 \sqrt {c_1 c_6-c_0 c_7}+c_4 \sqrt {c_3 c_6-c_2 c_7}\right )\right ) \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 )}{2 \sqrt {c_1 c_6-c_0 c_7} \left (c_5{}^2 (c_1 c_6-c_0 c_7)-c_4{}^2 (c_3 c_6-c_2 c_7)\right )}\\ &=-\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_6+x c_7) \left (c_5{}^2 (c_1 c_6-c_0 c_7)-c_4{}^2 (c_3 c_6-c_2 c_7)\right )}+\frac {\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 ) (c_1 c_2-c_0 c_3) c_5 \left (c_5 \sqrt {c_1 c_6-c_0 c_7}+c_4 \sqrt {c_3 c_6-c_2 c_7}\right )}{2 \sqrt {c_1 c_6-c_0 c_7} \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}} \left (c_5{}^2 (c_1 c_6-c_0 c_7)-c_4{}^2 (c_3 c_6-c_2 c_7)\right )}+\frac {\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 ) (c_1 c_2-c_0 c_3) c_5 \left (c_5 \sqrt {c_1 c_6-c_0 c_7}-c_4 \sqrt {c_3 c_6-c_2 c_7}\right )}{2 \sqrt {c_1 c_6-c_0 c_7} \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}} \left (c_5{}^2 (c_1 c_6-c_0 c_7)-c_4{}^2 (c_3 c_6-c_2 c_7)\right )}\\ \end {align*}

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Mathematica [A]
time = 1.77, size = 821, normalized size = 0.93 \begin {gather*} \frac {(c_1 c_2-c_0 c_3) \left (\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_1 c_6-c_0 c_7) (-c_3 c_6+c_2 c_7)}{(c_1 c_2-c_0 c_3) (c_6+x c_7)}+c_4 c_5 \sqrt {c_1 c_6-c_0 c_7} (c_3 c_6-c_2 c_7){}^{3/4} \left (-\frac {\text {ArcTan}\left (\frac {\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} \sqrt [4]{c_3 c_6-c_2 c_7}}{\sqrt {c_5 \sqrt {c_1 c_6-c_0 c_7}-c_4 \sqrt {c_3 c_6-c_2 c_7}}}\right )}{\sqrt {c_5 \sqrt {c_1 c_6-c_0 c_7}-c_4 \sqrt {c_3 c_6-c_2 c_7}}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} \sqrt [4]{c_3 c_6-c_2 c_7}}{\sqrt {c_5 \sqrt {c_1 c_6-c_0 c_7}+c_4 \sqrt {c_3 c_6-c_2 c_7}}}\right )}{\sqrt {c_5 \sqrt {c_1 c_6-c_0 c_7}+c_4 \sqrt {c_3 c_6-c_2 c_7}}}\right )-\frac {1}{2} c_5 \sqrt {c_1 c_6-c_0 c_7} \sqrt [4]{c_3 c_6-c_2 c_7} \left (\text {ArcTan}\left (\frac {\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} \sqrt [4]{c_3 c_6-c_2 c_7}}{\sqrt {c_5 \sqrt {c_1 c_6-c_0 c_7}-c_4 \sqrt {c_3 c_6-c_2 c_7}}}\right ) \sqrt {c_5 \sqrt {c_1 c_6-c_0 c_7}-c_4 \sqrt {c_3 c_6-c_2 c_7}}-\tanh ^{-1}\left (\frac {\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} \sqrt [4]{c_3 c_6-c_2 c_7}}{\sqrt {c_5 \sqrt {c_1 c_6-c_0 c_7}+c_4 \sqrt {c_3 c_6-c_2 c_7}}}\right ) \sqrt {c_5 \sqrt {c_1 c_6-c_0 c_7}+c_4 \sqrt {c_3 c_6-c_2 c_7}}\right )\right )}{(-c_1 c_6+c_0 c_7) (c_3 c_6-c_2 c_7) \left (c_5{}^2 (-c_1 c_6+c_0 c_7)+c_4{}^2 (c_3 c_6-c_2 c_7)\right )} \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])^2),x]

[Out]

((C[1]*C[2] - C[0]*C[3])*(((C[2] + x*C[3])*(C[4] - Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5])*Sqrt[C[4] + Sqr
t[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]]*(C[1]*C[6] - C[0]*C[7])*(-(C[3]*C[6]) + C[2]*C[7]))/((C[1]*C[2] - C[0
]*C[3])*(C[6] + x*C[7])) + C[4]*C[5]*Sqrt[C[1]*C[6] - C[0]*C[7]]*(C[3]*C[6] - C[2]*C[7])^(3/4)*(-(ArcTan[(Sqrt
[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]]*(C[3]*C[6] - C[2]*C[7])^(1/4))/Sqrt[C[5]*Sqrt[C[1]*C[6] -
C[0]*C[7]] - C[4]*Sqrt[C[3]*C[6] - C[2]*C[7]]]]/Sqrt[C[5]*Sqrt[C[1]*C[6] - C[0]*C[7]] - C[4]*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]]*(C[3]*C[6] - C[2]*C[7])^(1/4))
/Sqrt[C[5]*Sqrt[C[1]*C[6] - C[0]*C[7]] + C[4]*Sqrt[C[3]*C[6] - C[2]*C[7]]]]/Sqrt[C[5]*Sqrt[C[1]*C[6] - C[0]*C[
7]] + C[4]*Sqrt[C[3]*C[6] - C[2]*C[7]]]) - (C[5]*Sqrt[C[1]*C[6] - C[0]*C[7]]*(C[3]*C[6] - C[2]*C[7])^(1/4)*(Ar
cTan[(Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]]*(C[3]*C[6] - C[2]*C[7])^(1/4))/Sqrt[C[5]*Sqrt[C[
1]*C[6] - C[0]*C[7]] - C[4]*Sqrt[C[3]*C[6] - C[2]*C[7]]]]*Sqrt[C[5]*Sqrt[C[1]*C[6] - C[0]*C[7]] - C[4]*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]]*(C[3]*C[6] - C[2]*C[7
])^(1/4))/Sqrt[C[5]*Sqrt[C[1]*C[6] - C[0]*C[7]] + C[4]*Sqrt[C[3]*C[6] - C[2]*C[7]]]]*Sqrt[C[5]*Sqrt[C[1]*C[6]
- C[0]*C[7]] + C[4]*Sqrt[C[3]*C[6] - C[2]*C[7]]]))/2))/((-(C[1]*C[6]) + C[0]*C[7])*(C[3]*C[6] - C[2]*C[7])*(C[
5]^2*(-(C[1]*C[6]) + C[0]*C[7]) + C[4]^2*(C[3]*C[6] - C[2]*C[7])))

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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {1}{\sqrt {\textit {\_C4} +\sqrt {\frac {\textit {\_C1} x +\textit {\_C0}}{\textit {\_C3} x +\textit {\_C2}}}\, \textit {\_C5}}\, \left (\textit {\_C7} x +\textit {\_C6} \right )^{2}}\, dx\]

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)^2,x)

[Out]

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

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \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)^2,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 56059 vs. \(2 (579) = 1158\).
time = 47.17, size = 56059, normalized size = 63.42 \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)^2,x, algorithm="fricas")

[Out]

1/4*(((C3*C4^2 - C1*C5^2)*C6^2 - (C2*C4^2 - C0*C5^2)*C6*C7 + ((C3*C4^2 - C1*C5^2)*C6*C7 - (C2*C4^2 - C0*C5^2)*
C7^2)*x)*sqrt((((C1^2*C2^2*C3 - 2*C0*C1*C2*C3^2 + C0^2*C3^3)*C4^3*C5^2 + 3*(C1^3*C2^2 - 2*C0*C1^2*C2*C3 + C0^2
*C1*C3^2)*C4*C5^4)*C6 - ((C1^2*C2^3 - 2*C0*C1*C2^2*C3 + C0^2*C2*C3^2)*C4^3*C5^2 + 3*(C0*C1^2*C2^2 - 2*C0^2*C1*
C2*C3 + C0^3*C3^2)*C4*C5^4)*C7 + ((C1*C3^4*C4^6 - 3*C1^2*C3^3*C4^4*C5^2 + 3*C1^3*C3^2*C4^2*C5^4 - C1^4*C3*C5^6
)*C6^5 - ((4*C1*C2*C3^3 + C0*C3^4)*C4^6 - 3*(3*C1^2*C2*C3^2 + 2*C0*C1*C3^3)*C4^4*C5^2 + 3*(2*C1^3*C2*C3 + 3*C0
*C1^2*C3^2)*C4^2*C5^4 - (C1^4*C2 + 4*C0*C1^3*C3)*C5^6)*C6^4*C7 + (2*(3*C1*C2^2*C3^2 + 2*C0*C2*C3^3)*C4^6 - 3*(
3*C1^2*C2^2*C3 + 6*C0*C1*C2*C3^2 + C0^2*C3^3)*C4^4*C5^2 + 3*(C1^3*C2^2 + 6*C0*C1^2*C2*C3 + 3*C0^2*C1*C3^2)*C4^
2*C5^4 - 2*(2*C0*C1^3*C2 + 3*C0^2*C1^2*C3)*C5^6)*C6^3*C7^2 - (2*(2*C1*C2^3*C3 + 3*C0*C2^2*C3^2)*C4^6 - 3*(C1^2
*C2^3 + 6*C0*C1*C2^2*C3 + 3*C0^2*C2*C3^2)*C4^4*C5^2 + 3*(3*C0*C1^2*C2^2 + 6*C0^2*C1*C2*C3 + C0^3*C3^2)*C4^2*C5
^4 - 2*(3*C0^2*C1^2*C2 + 2*C0^3*C1*C3)*C5^6)*C6^2*C7^3 + ((C1*C2^4 + 4*C0*C2^3*C3)*C4^6 - 3*(2*C0*C1*C2^3 + 3*
C0^2*C2^2*C3)*C4^4*C5^2 + 3*(3*C0^2*C1*C2^2 + 2*C0^3*C2*C3)*C4^2*C5^4 - (4*C0^3*C1*C2 + C0^4*C3)*C5^6)*C6*C7^4
- (C0*C2^4*C4^6 - 3*C0^2*C2^3*C4^4*C5^2 + 3*C0^3*C2^2*C4^2*C5^4 - C0^4*C2*C5^6)*C7^5)*sqrt(((9*(C1^4*C2^4*C3^
2 - 4*C0*C1^3*C2^3*C3^3 + 6*C0^2*C1^2*C2^2*C3^4 - 4*C0^3*C1*C2*C3^5 + C0^4*C3^6)*C4^4*C5^6 + 6*(C1^5*C2^4*C3 -
4*C0*C1^4*C2^3*C3^2 + 6*C0^2*C1^3*C2^2*C3^3 - 4*C0^3*C1^2*C2*C3^4 + C0^4*C1*C3^5)*C4^2*C5^8 + (C1^6*C2^4 - 4*
C0*C1^5*C2^3*C3 + 6*C0^2*C1^4*C2^2*C3^2 - 4*C0^3*C1^3*C2*C3^3 + C0^4*C1^2*C3^4)*C5^10)*C6^2 - 2*(9*(C1^4*C2^5*
C3 - 4*C0*C1^3*C2^4*C3^2 + 6*C0^2*C1^2*C2^3*C3^3 - 4*C0^3*C1*C2^2*C3^4 + C0^4*C2*C3^5)*C4^4*C5^6 + 3*(C1^5*C2^
5 - 3*C0*C1^4*C2^4*C3 + 2*C0^2*C1^3*C2^3*C3^2 + 2*C0^3*C1^2*C2^2*C3^3 - 3*C0^4*C1*C2*C3^4 + C0^5*C3^5)*C4^2*C5
^8 + (C0*C1^5*C2^4 - 4*C0^2*C1^4*C2^3*C3 + 6*C0^3*C1^3*C2^2*C3^2 - 4*C0^4*C1^2*C2*C3^3 + C0^5*C1*C3^4)*C5^10)*
C6*C7 + (9*(C1^4*C2^6 - 4*C0*C1^3*C2^5*C3 + 6*C0^2*C1^2*C2^4*C3^2 - 4*C0^3*C1*C2^3*C3^3 + C0^4*C2^2*C3^4)*C4^4
*C5^6 + 6*(C0*C1^4*C2^5 - 4*C0^2*C1^3*C2^4*C3 + 6*C0^3*C1^2*C2^3*C3^2 - 4*C0^4*C1*C2^2*C3^3 + C0^5*C2*C3^4)*C4
^2*C5^8 + (C0^2*C1^4*C2^4 - 4*C0^3*C1^3*C2^3*C3 + 6*C0^4*C1^2*C2^2*C3^2 - 4*C0^5*C1*C2*C3^3 + C0^6*C3^4)*C5^10
)*C7^2)/((C1*C3^9*C4^12 - 6*C1^2*C3^8*C4^10*C5^2 + 15*C1^3*C3^7*C4^8*C5^4 - 20*C1^4*C3^6*C4^6*C5^6 + 15*C1^5*C
3^5*C4^4*C5^8 - 6*C1^6*C3^4*C4^2*C5^10 + C1^7*C3^3*C5^12)*C6^10 - ((9*C1*C2*C3^8 + C0*C3^9)*C4^12 - 12*(4*C1^2
*C2*C3^7 + C0*C1*C3^8)*C4^10*C5^2 + 15*(7*C1^3*C2*C3^6 + 3*C0*C1^2*C3^7)*C4^8*C5^4 - 40*(3*C1^4*C2*C3^5 + 2*C0
*C1^3*C3^6)*C4^6*C5^6 + 75*(C1^5*C2*C3^4 + C0*C1^4*C3^5)*C4^4*C5^8 - 12*(2*C1^6*C2*C3^3 + 3*C0*C1^5*C3^4)*C4^2
*C5^10 + (3*C1^7*C2*C3^2 + 7*C0*C1^6*C3^3)*C5^12)*C6^9*C7 + 3*(3*(4*C1*C2^2*C3^7 + C0*C2*C3^8)*C4^12 - 2*(28*C
1^2*C2^2*C3^6 + 16*C0*C1*C2*C3^7 + C0^2*C3^8)*C4^10*C5^2 + 15*(7*C1^3*C2^2*C3^5 + 7*C0*C1^2*C2*C3^6 + C0^2*C1*
C3^7)*C4^8*C5^4 - 20*(5*C1^4*C2^2*C3^4 + 8*C0*C1^3*C2*C3^5 + 2*C0^2*C1^2*C3^6)*C4^6*C5^6 + 25*(2*C1^5*C2^2*C3^
3 + 5*C0*C1^4*C2*C3^4 + 2*C0^2*C1^3*C3^5)*C4^4*C5^8 - 6*(2*C1^6*C2^2*C3^2 + 8*C0*C1^5*C2*C3^3 + 5*C0^2*C1^4*C3
^4)*C4^2*C5^10 + (C1^7*C2^2*C3 + 7*C0*C1^6*C2*C3^2 + 7*C0^2*C1^5*C3^3)*C5^12)*C6^8*C7^2 - (12*(7*C1*C2^3*C3^6
+ 3*C0*C2^2*C3^7)*C4^12 - 48*(7*C1^2*C2^3*C3^5 + 7*C0*C1*C2^2*C3^6 + C0^2*C2*C3^7)*C4^10*C5^2 + 15*(35*C1^3*C2
^3*C3^4 + 63*C0*C1^2*C2^2*C3^5 + 21*C0^2*C1*C2*C3^6 + C0^3*C3^7)*C4^8*C5^4 - 80*(5*C1^4*C2^3*C3^3 + 15*C0*C1^3
*C2^2*C3^4 + 9*C0^2*C1^2*C2*C3^5 + C0^3*C1*C3^6)*C4^6*C5^6 + 150*(C1^5*C2^3*C3^2 + 5*C0*C1^4*C2^2*C3^3 + 5*C0^
2*C1^3*C2*C3^4 + C0^3*C1^2*C3^5)*C4^4*C5^8 - 24*(C1^6*C2^3*C3 + 9*C0*C1^5*C2^2*C3^2 + 15*C0^2*C1^4*C2*C3^3 + 5
*C0^3*C1^3*C3^4)*C4^2*C5^10 + (C1^7*C2^3 + 21*C0*C1^6*C2^2*C3 + 63*C0^2*C1^5*C2*C3^2 + 35*C0^3*C1^4*C3^3)*C5^1
2)*C6^7*C7^3 + (42*(3*C1*C2^4*C3^5 + 2*C0*C2^3*C3^6)*C4^12 - 84*(5*C1^2*C2^4*C3^4 + 8*C0*C1*C2^3*C3^5 + 2*C0^2
*C2^2*C3^6)*C4^10*C5^2 + 105*(5*C1^3*C2^4*C3^3 + 15*C0*C1^2*C2^3*C3^4 + 9*C0^2*C1*C2^2*C3^5 + C0^3*C2*C3^6)*C4
^8*C5^4 - 20*(15*C1^4*C2^4*C3^2 + 80*C0*C1^3*C2^3*C3^3 + 90*C0^2*C1^2*C2^2*C3^4 + 24*C0^3*C1*C2*C3^5 + C0^4*C3
^6)*C4^6*C5^6 + 75*(C1^5*C2^4*C3 + 10*C0*C1^4*C2^3*C3^2 + 20*C0^2*C1^3*C2^2*C3^3 + 10*C0^3*C1^2*C2*C3^4 + C0^4
*C1*C3^5)*C4^4*C5^8 - 6*(C1^6*C2^4 + 24*C0*C1^5*C2^3*C3 + 90*C0^2*C1^4*C2^2*C3^2 + 80*C0^3*C1^3*C2*C3^3 + 15*C
0^4*C1^2*C3^4)*C4^2*C5^10 + 7*(C0*C1^6*C2^3 + 9*C0^2*C1^5*C2^2*C3 + 15*C0^3*C1^4*C2*C3^2 + 5*C0^4*C1^3*C3^3)*C
5^12)*C6^6*C7^4 - 3*(42*(C1*C2^5*C3^4 + C0*C2^4*C3^5)*C4^12 - 56*(2*C1^2*C2^5*C3^3 + 5*C0*C1*C2^4*C3^4 + 2*C0^
2*C2^3*C3^5)*C4^10*C5^2 + 105*(C1^3*C2^5*C3^2 + 5*C0*C1^2*C2^4*C3^3 + 5*C0^2*C1*C2^3*C3^4 + C0^3*C2^2*C3^5)*C4
^8*C5^4 - 40*(C1^4*C2^5*C3 + 10*C0*C1^3*C2^4*C3^2 + 20*C0^2*C1^2*C2^3*C3^3 + 10*C0^3*C1*C2^2*C3^4 + C0^4*C2*C3
^5)*C4^6*C5^6 + 5*(C1^5*C2^5 + 25*C0*C1^4*C2^4*C3 + 100*C0^2*C1^3*C2^3*C3^2 + 100*C0^3*C1^2*C2^2*C3^3 + 25*C0^
4*C1*C2*C3^4 + C0^5*C3^5)*C4^4*C5^8 - 12*(C0*C1...

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Sympy [F(-1)] Timed out
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)**2,x)

[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)^2,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 )}^2} \,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)^2),x)

[Out]

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

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