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

Optimal. Leaf size=884 \[ \frac {\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} \left (-c_1 c_2 c_4+c_0 c_3 c_4+c_1 c_2 \sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5-c_0 c_3 \sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5\right )}{\left (c_1 c_6-\frac {(c_0+x c_1) c_3 c_6}{c_2+x c_3}-c_0 c_7+\frac {(c_0+x c_1) c_2 c_7}{c_2+x c_3}\right ) \left (-c_3 c_6 c_4{}^2+c_2 c_7 c_4{}^2+c_1 c_5{}^2 c_6-c_0 c_5{}^2 c_7\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} \sqrt {c_3 c_6-c_2 c_7}}{\sqrt {-\sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7} c_5-c_3 c_4 c_6+c_2 c_4 c_7}}\right ) \left (-c_1 c_2 \sqrt {c_1 c_6-c_0 c_7} c_5{}^2+c_0 c_3 \sqrt {c_1 c_6-c_0 c_7} c_5{}^2+c_1 c_2 c_4 \sqrt {c_3 c_6-c_2 c_7} c_5-c_0 c_3 c_4 \sqrt {c_3 c_6-c_2 c_7} c_5\right )}{2 \sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7} \left (-c_3 c_6 c_4{}^2+c_2 c_7 c_4{}^2+c_1 c_5{}^2 c_6-c_0 c_5{}^2 c_7\right ) \sqrt {-\sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7} c_5-c_3 c_4 c_6+c_2 c_4 c_7}}+\frac {\tan ^{-1}\left (\frac {\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} \sqrt {c_3 c_6-c_2 c_7}}{\sqrt {\sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7} c_5-c_3 c_4 c_6+c_2 c_4 c_7}}\right ) \left (-c_1 c_2 \sqrt {c_1 c_6-c_0 c_7} c_5{}^2+c_0 c_3 \sqrt {c_1 c_6-c_0 c_7} c_5{}^2-c_1 c_2 c_4 \sqrt {c_3 c_6-c_2 c_7} c_5+c_0 c_3 c_4 \sqrt {c_3 c_6-c_2 c_7} c_5\right )}{2 \sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7} \left (-c_3 c_6 c_4{}^2+c_2 c_7 c_4{}^2+c_1 c_5{}^2 c_6-c_0 c_5{}^2 c_7\right ) \sqrt {\sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7} c_5-c_3 c_4 c_6+c_2 c_4 c_7}} \]

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Rubi [A]  time = 4.64, 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 = {823, 827, 1166, 208} \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 208

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

Rule 823

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(
m + 1)*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*(a + c*x^2)^(p + 1))/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), x] + Di
st[1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Simp[f*(c^2*d^2*(2*p + 3) + a*c*e^2*
(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x]
&& NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 827

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2, Subst[Int[(e*f
- d*g + g*x^2)/(c*d^2 + a*e^2 - 2*c*d*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e, f, g}, x]
&& NeQ[c*d^2 + a*e^2, 0]

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

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)) \operatorname {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) \operatorname {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)) \operatorname {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 ) \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 )}{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 ) \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 )}{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*}

________________________________________________________________________________________

Mathematica [A]  time = 2.59, 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_2 c_7-c_3 c_6)}{(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 {\tan ^{-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}}}-\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 {\sqrt {c_3 c_6-c_2 c_7} c_4+c_5 \sqrt {c_1 c_6-c_0 c_7}}}\right )}{\sqrt {\sqrt {c_3 c_6-c_2 c_7} c_4+c_5 \sqrt {c_1 c_6-c_0 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 (\tan ^{-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}}-\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 {\sqrt {c_3 c_6-c_2 c_7} c_4+c_5 \sqrt {c_1 c_6-c_0 c_7}}}\right ) \sqrt {\sqrt {c_3 c_6-c_2 c_7} c_4+c_5 \sqrt {c_1 c_6-c_0 c_7}}\right )\right )}{(c_0 c_7-c_1 c_6) (c_3 c_6-c_2 c_7) \left ((c_3 c_6-c_2 c_7) c_4{}^2+c_5{}^2 (c_0 c_7-c_1 c_6)\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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IntegrateAlgebraic [A]  time = 3.13, size = 884, normalized size = 1.00 \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_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 {\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 ) \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 {\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 ) \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}}} \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])^2),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[0]*C[3]*C[4] + C[1]*C[2]*Sqrt[
(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5] - C[0]*C[3]*Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]))/((-(C[1]*C[6])
+ ((C[0] + x*C[1])*C[3]*C[6])/(C[2] + x*C[3]) + C[0]*C[7] - ((C[0] + x*C[1])*C[2]*C[7])/(C[2] + x*C[3]))*(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])) + (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]]]]*(-(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]]))/(2*Sqrt[C[1]*C[6] - C[0]*C[7]]*Sqrt[C[3]*C[6] - C[2]*C[7]]*(-(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[(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[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]]))/(2*Sqrt[C[1]*C[6] - C[0]*C[7]]*Sqrt[C[3]*C[6] - C[2]*C[7]]*(-(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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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)^2,x, algorithm="fricas")

[Out]

Timed out

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giac [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)^2,x, algorithm="giac")

[Out]

Timed out

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maple [F]  time = 0.07, size = 0, normalized size = 0.00 \[\int \frac {1}{\sqrt {\textit {\_C4} +\sqrt {\frac {\textit {\_C1} x +\textit {\_C0}}{\textit {\_C3} x +\textit {\_C2}}}\, \textit {\_C5}}\, \left (\textit {\_C7} x +\textit {\_C6} \right )^{2}}\, dx\]

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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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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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(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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