\(\int \frac {a b c-b^2 x+a^2 x^2}{\sqrt {c+b x+a x^2} (c+b x^2)} \, dx\) [2931]

Optimal result

Integrand size = 42, antiderivative size = 343 \[ \int \frac {a b c-b^2 x+a^2 x^2}{\sqrt {c+b x+a x^2} \left (c+b x^2\right )} \, dx=-\frac {a^{3/2} \log \left (b+2 a x-2 \sqrt {a} \sqrt {c+b x+a x^2}\right )}{b}+\frac {\text {RootSum}\left [b^2 c+b c^2-4 \sqrt {a} b c \text {$\#$1}+4 a c \text {$\#$1}^2-2 b c \text {$\#$1}^2+b \text {$\#$1}^4\&,\frac {-a^2 b c \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right )+b^3 c \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right )+a b^3 c \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right )+2 a^{5/2} c \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}-2 a^{3/2} b^2 c \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}-b^3 \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-\sqrt {a} b c+2 a c \text {$\#$1}-b c \text {$\#$1}+b \text {$\#$1}^3}\&\right ]}{2 b} \]

[Out]

Unintegrable

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1002\) vs. \(2(343)=686\).

Time = 2.88 (sec) , antiderivative size = 1002, normalized size of antiderivative = 2.92, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1092, 635, 212, 1050, 1044, 211} \[ \int \frac {a b c-b^2 x+a^2 x^2}{\sqrt {c+b x+a x^2} \left (c+b x^2\right )} \, dx=\frac {\text {arctanh}\left (\frac {b+2 a x}{2 \sqrt {a} \sqrt {a x^2+b x+c}}\right ) a^{3/2}}{b}+\frac {\sqrt {\sqrt {c} a^2+(1-b) b \sqrt {c} a-b \left (b \sqrt {c}-\sqrt {b^3+c b^2-2 a c b+a^2 c}\right )} \sqrt {b^4-a c b^3+\left (a^2 c-a \sqrt {c} \sqrt {b^3+c b^2-2 a c b+a^2 c}\right ) b^2+a^2 c b-a^3 c+a^2 \sqrt {c} \sqrt {b^3+c b^2-2 a c b+a^2 c}} \arctan \left (\frac {b \sqrt {c} \left (\sqrt {c} a^2+(1-b) b \sqrt {c} a-b^2 \sqrt {c}+b \sqrt {b^3+c b^2-2 a c b+a^2 c}\right )-\left (b^4-a \left (a-b^2\right ) \left (a c-b c-\sqrt {b^3+c b^2-2 a c b+a^2 c} \sqrt {c}\right )\right ) x}{\sqrt {2} \sqrt [4]{c} \sqrt {b^4-a c b^3+a^2 c b^2+a^2 c b-a^3 c+a \left (a-b^2\right ) \sqrt {c} \sqrt {b^3+c b^2-2 a c b+a^2 c}} \sqrt {\sqrt {c} a^2+(1-b) b \sqrt {c} a-b \left (b \sqrt {c}-\sqrt {b^3+c b^2-2 a c b+a^2 c}\right )} \sqrt {a x^2+b x+c}}\right )}{\sqrt {2} b \sqrt [4]{c} \sqrt {b^3+c b^2-2 a c b+a^2 c}}-\frac {\sqrt {\sqrt {c} a^2+(1-b) b \sqrt {c} a-b \left (\sqrt {c} b+\sqrt {b^3+c b^2-2 a c b+a^2 c}\right )} \sqrt {b^4-a c b^3+\left (c a^2+\sqrt {c} \sqrt {b^3+c b^2-2 a c b+a^2 c} a\right ) b^2+a^2 c b-a^2 \sqrt {c} \left (\sqrt {c} a+\sqrt {b^3+c b^2-2 a c b+a^2 c}\right )} \arctan \left (\frac {b \sqrt {c} \left (\sqrt {c} a^2+(1-b) b \sqrt {c} a-b \left (\sqrt {c} b+\sqrt {b^3+c b^2-2 a c b+a^2 c}\right )\right )-\left (b^4-a \left (a-b^2\right ) \left (a c-b c+\sqrt {b^3+c b^2-2 a c b+a^2 c} \sqrt {c}\right )\right ) x}{\sqrt {2} \sqrt [4]{c} \sqrt {b^4-a c b^3+a^2 c b^2+a^2 c b-a^3 c-a \left (a-b^2\right ) \sqrt {c} \sqrt {b^3+c b^2-2 a c b+a^2 c}} \sqrt {\sqrt {c} a^2+(1-b) b \sqrt {c} a-b \left (\sqrt {c} b+\sqrt {b^3+c b^2-2 a c b+a^2 c}\right )} \sqrt {a x^2+b x+c}}\right )}{\sqrt {2} b \sqrt [4]{c} \sqrt {b^3+c b^2-2 a c b+a^2 c}} \]

[In]

Int[(a*b*c - b^2*x + a^2*x^2)/(Sqrt[c + b*x + a*x^2]*(c + b*x^2)),x]

[Out]

(Sqrt[a^2*Sqrt[c] + a*(1 - b)*b*Sqrt[c] - b*(b*Sqrt[c] - Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c])]*Sqrt[b^4 - a^3*
c + a^2*b*c - a*b^3*c + a^2*Sqrt[c]*Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c] + b^2*(a^2*c - a*Sqrt[c]*Sqrt[b^3 + a^
2*c - 2*a*b*c + b^2*c])]*ArcTan[(b*Sqrt[c]*(a^2*Sqrt[c] + a*(1 - b)*b*Sqrt[c] - b^2*Sqrt[c] + b*Sqrt[b^3 + a^2
*c - 2*a*b*c + b^2*c]) - (b^4 - a*(a - b^2)*(a*c - b*c - Sqrt[c]*Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c]))*x)/(Sqr
t[2]*c^(1/4)*Sqrt[b^4 - a^3*c + a^2*b*c + a^2*b^2*c - a*b^3*c + a*(a - b^2)*Sqrt[c]*Sqrt[b^3 + a^2*c - 2*a*b*c
+ b^2*c]]*Sqrt[a^2*Sqrt[c] + a*(1 - b)*b*Sqrt[c] - b*(b*Sqrt[c] - Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c])]*Sqrt[
c + b*x + a*x^2])])/(Sqrt[2]*b*c^(1/4)*Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c]) - (Sqrt[a^2*Sqrt[c] + a*(1 - b)*b*
Sqrt[c] - b*(b*Sqrt[c] + Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c])]*Sqrt[b^4 + a^2*b*c - a*b^3*c - a^2*Sqrt[c]*(a*S
qrt[c] + Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c]) + b^2*(a^2*c + a*Sqrt[c]*Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c])]*A
rcTan[(b*Sqrt[c]*(a^2*Sqrt[c] + a*(1 - b)*b*Sqrt[c] - b*(b*Sqrt[c] + Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c])) - (
b^4 - a*(a - b^2)*(a*c - b*c + Sqrt[c]*Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c]))*x)/(Sqrt[2]*c^(1/4)*Sqrt[b^4 - a^
3*c + a^2*b*c + a^2*b^2*c - a*b^3*c - a*(a - b^2)*Sqrt[c]*Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c]]*Sqrt[a^2*Sqrt[c
] + a*(1 - b)*b*Sqrt[c] - b*(b*Sqrt[c] + Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c])]*Sqrt[c + b*x + a*x^2])])/(Sqrt[
2]*b*c^(1/4)*Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c]) + (a^(3/2)*ArcTanh[(b + 2*a*x)/(2*Sqrt[a]*Sqrt[c + b*x + a*x
^2])])/b

Rule 211

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

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 635

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 1044

Int[((g_) + (h_.)*(x_))/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[-2*
a*g*h, Subst[Int[1/Simp[2*a^2*g*h*c + a*e*x^2, x], x], x, Simp[a*h - g*c*x, x]/Sqrt[d + e*x + f*x^2]], x] /; F
reeQ[{a, c, d, e, f, g, h}, x] && EqQ[a*h^2*e + 2*g*h*(c*d - a*f) - g^2*c*e, 0]

Rule 1050

Int[((g_.) + (h_.)*(x_))/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> With[{q
= Rt[(c*d - a*f)^2 + a*c*e^2, 2]}, Dist[1/(2*q), Int[Simp[(-a)*h*e - g*(c*d - a*f - q) + (h*(c*d - a*f + q) -
g*c*e)*x, x]/((a + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x] - Dist[1/(2*q), Int[Simp[(-a)*h*e - g*(c*d - a*f + q
) + (h*(c*d - a*f - q) - g*c*e)*x, x]/((a + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, c, d, e, f, g,
h}, x] && NeQ[e^2 - 4*d*f, 0] && NegQ[(-a)*c]

Rule 1092

Int[((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Sym
bol] :> Dist[C/c, Int[1/Sqrt[d + e*x + f*x^2], x], x] + Dist[1/c, Int[(A*c - a*C + B*c*x)/((a + c*x^2)*Sqrt[d
+ e*x + f*x^2]), x], x] /; FreeQ[{a, c, d, e, f, A, B, C}, x] && NeQ[e^2 - 4*d*f, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {-a^2 c+a b^2 c-b^3 x}{\sqrt {c+b x+a x^2} \left (c+b x^2\right )} \, dx}{b}+\frac {a^2 \int \frac {1}{\sqrt {c+b x+a x^2}} \, dx}{b} \\ & = \frac {\left (2 a^2\right ) \text {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {b+2 a x}{\sqrt {c+b x+a x^2}}\right )}{b}+\frac {\int \frac {c \left (b^4-a \left (a-b^2\right ) \left (a c-b c+\sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )\right )+b^2 \sqrt {c} \left (a^2 \sqrt {c}+a (1-b) b \sqrt {c}-b^2 \sqrt {c}-b \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right ) x}{\sqrt {c+b x+a x^2} \left (c+b x^2\right )} \, dx}{2 b \sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}}-\frac {\int \frac {c \left (b^4-a \left (a-b^2\right ) \left (a c-b c-\sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )\right )+b^2 \sqrt {c} \left (a^2 \sqrt {c}+a (1-b) b \sqrt {c}-b^2 \sqrt {c}+b \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right ) x}{\sqrt {c+b x+a x^2} \left (c+b x^2\right )} \, dx}{2 b \sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}} \\ & = \frac {a^{3/2} \text {arctanh}\left (\frac {b+2 a x}{2 \sqrt {a} \sqrt {c+b x+a x^2}}\right )}{b}+\frac {\left (b c^2 \left (a^2 \sqrt {c}+a (1-b) b \sqrt {c}-b \left (b \sqrt {c}-\sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )\right ) \left (b^4-a \left (a-b^2\right ) \left (a c-b c-\sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )\right )\right ) \text {Subst}\left (\int \frac {1}{2 b^3 c^{7/2} \left (a^2 \sqrt {c}+a (1-b) b \sqrt {c}-b \left (b \sqrt {c}-\sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )\right ) \left (b^4-a \left (a-b^2\right ) \left (a c-b c-\sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )\right )+b c x^2} \, dx,x,\frac {b^2 c^{3/2} \left (a^2 \sqrt {c}+a (1-b) b \sqrt {c}-b^2 \sqrt {c}+b \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )-b c \left (b^4-a \left (a-b^2\right ) \left (a c-b c-\sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )\right ) x}{\sqrt {c+b x+a x^2}}\right )}{\sqrt {b^3+a^2 c-2 a b c+b^2 c}}-\frac {\left (b c^2 \left (a^2 \sqrt {c}+a (1-b) b \sqrt {c}-b \left (b \sqrt {c}+\sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )\right ) \left (b^4-a \left (a-b^2\right ) \left (a c-b c+\sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )\right )\right ) \text {Subst}\left (\int \frac {1}{2 b^3 c^{7/2} \left (a^2 \sqrt {c}+a (1-b) b \sqrt {c}-b \left (b \sqrt {c}+\sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )\right ) \left (b^4-a \left (a-b^2\right ) \left (a c-b c+\sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )\right )+b c x^2} \, dx,x,\frac {b^2 c^{3/2} \left (a^2 \sqrt {c}+a (1-b) b \sqrt {c}-b \left (b \sqrt {c}+\sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )\right )-b c \left (b^4-a \left (a-b^2\right ) \left (a c-b c+\sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )\right ) x}{\sqrt {c+b x+a x^2}}\right )}{\sqrt {b^3+a^2 c-2 a b c+b^2 c}} \\ & = \frac {\sqrt {a^2 \sqrt {c}+a (1-b) b \sqrt {c}-b \left (b \sqrt {c}-\sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )} \sqrt {b^4-a^3 c+a^2 b c-a b^3 c+a^2 \sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}+b^2 \left (a^2 c-a \sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )} \arctan \left (\frac {b \sqrt {c} \left (a^2 \sqrt {c}+a (1-b) b \sqrt {c}-b^2 \sqrt {c}+b \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )-\left (b^4-a \left (a-b^2\right ) \left (a c-b c-\sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )\right ) x}{\sqrt {2} \sqrt [4]{c} \sqrt {b^4-a^3 c+a^2 b c+a^2 b^2 c-a b^3 c+a \left (a-b^2\right ) \sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}} \sqrt {a^2 \sqrt {c}+a (1-b) b \sqrt {c}-b \left (b \sqrt {c}-\sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )} \sqrt {c+b x+a x^2}}\right )}{\sqrt {2} b \sqrt [4]{c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}}-\frac {\sqrt {a^2 \sqrt {c}+a (1-b) b \sqrt {c}-b \left (b \sqrt {c}+\sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )} \sqrt {b^4+a^2 b c-a b^3 c-a^2 \sqrt {c} \left (a \sqrt {c}+\sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )+b^2 \left (a^2 c+a \sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )} \arctan \left (\frac {b \sqrt {c} \left (a^2 \sqrt {c}+a (1-b) b \sqrt {c}-b \left (b \sqrt {c}+\sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )\right )-\left (b^4-a \left (a-b^2\right ) \left (a c-b c+\sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )\right ) x}{\sqrt {2} \sqrt [4]{c} \sqrt {b^4-a^3 c+a^2 b c+a^2 b^2 c-a b^3 c-a \left (a-b^2\right ) \sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}} \sqrt {a^2 \sqrt {c}+a (1-b) b \sqrt {c}-b \left (b \sqrt {c}+\sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )} \sqrt {c+b x+a x^2}}\right )}{\sqrt {2} b \sqrt [4]{c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}}+\frac {a^{3/2} \text {arctanh}\left (\frac {b+2 a x}{2 \sqrt {a} \sqrt {c+b x+a x^2}}\right )}{b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.56 (sec) , antiderivative size = 417, normalized size of antiderivative = 1.22 \[ \int \frac {a b c-b^2 x+a^2 x^2}{\sqrt {c+b x+a x^2} \left (c+b x^2\right )} \, dx=-\frac {4 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} x}{\sqrt {c}-\sqrt {c+x (b+a x)}}\right )+\frac {\text {RootSum}\left [b^3+a^2 c-4 b^2 \sqrt {c} \text {$\#$1}-2 a c \text {$\#$1}^2+4 b c \text {$\#$1}^2+c \text {$\#$1}^4\&,\frac {-b^4 \log (x)+a^3 c \log (x)-a^2 b^2 c \log (x)+b^4 \log \left (-\sqrt {c}+\sqrt {c+b x+a x^2}-x \text {$\#$1}\right )-a^3 c \log \left (-\sqrt {c}+\sqrt {c+b x+a x^2}-x \text {$\#$1}\right )+a^2 b^2 c \log \left (-\sqrt {c}+\sqrt {c+b x+a x^2}-x \text {$\#$1}\right )+2 b^3 \sqrt {c} \log (x) \text {$\#$1}-2 b^3 \sqrt {c} \log \left (-\sqrt {c}+\sqrt {c+b x+a x^2}-x \text {$\#$1}\right ) \text {$\#$1}-a^2 c \log (x) \text {$\#$1}^2+a b^2 c \log (x) \text {$\#$1}^2+a^2 c \log \left (-\sqrt {c}+\sqrt {c+b x+a x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2-a b^2 c \log \left (-\sqrt {c}+\sqrt {c+b x+a x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2}{b^2+a \sqrt {c} \text {$\#$1}-2 b \sqrt {c} \text {$\#$1}-\sqrt {c} \text {$\#$1}^3}\&\right ]}{\sqrt {c}}}{2 b} \]

[In]

Integrate[(a*b*c - b^2*x + a^2*x^2)/(Sqrt[c + b*x + a*x^2]*(c + b*x^2)),x]

[Out]

-1/2*(4*a^(3/2)*ArcTanh[(Sqrt[a]*x)/(Sqrt[c] - Sqrt[c + x*(b + a*x)])] + RootSum[b^3 + a^2*c - 4*b^2*Sqrt[c]*#
1 - 2*a*c*#1^2 + 4*b*c*#1^2 + c*#1^4 & , (-(b^4*Log[x]) + a^3*c*Log[x] - a^2*b^2*c*Log[x] + b^4*Log[-Sqrt[c] +
Sqrt[c + b*x + a*x^2] - x*#1] - a^3*c*Log[-Sqrt[c] + Sqrt[c + b*x + a*x^2] - x*#1] + a^2*b^2*c*Log[-Sqrt[c] +
Sqrt[c + b*x + a*x^2] - x*#1] + 2*b^3*Sqrt[c]*Log[x]*#1 - 2*b^3*Sqrt[c]*Log[-Sqrt[c] + Sqrt[c + b*x + a*x^2]
- x*#1]*#1 - a^2*c*Log[x]*#1^2 + a*b^2*c*Log[x]*#1^2 + a^2*c*Log[-Sqrt[c] + Sqrt[c + b*x + a*x^2] - x*#1]*#1^2
- a*b^2*c*Log[-Sqrt[c] + Sqrt[c + b*x + a*x^2] - x*#1]*#1^2)/(b^2 + a*Sqrt[c]*#1 - 2*b*Sqrt[c]*#1 - Sqrt[c]*#
1^3) & ]/Sqrt[c])/b

Maple [N/A] (verified)

Time = 0.32 (sec) , antiderivative size = 520, normalized size of antiderivative = 1.52

method	result	size
default	\(\frac {a^{\frac {3}{2}} \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x +c}\right )}{b}-\frac {\left (-a \,b^{2} c -\sqrt {-b c}\, b^{2}+a^{2} c \right ) \ln \left (\frac {-\frac {2 \left (\sqrt {-b c}\, b +a c -b c \right )}{b}-\frac {\sqrt {-b c}\, \left (\sqrt {-b c}\, b +2 a c \right ) \left (x +\frac {\sqrt {-b c}}{b}\right )}{b c}+2 \sqrt {-\frac {\sqrt {-b c}\, b +a c -b c}{b}}\, \sqrt {\left (x +\frac {\sqrt {-b c}}{b}\right )^{2} a -\frac {\sqrt {-b c}\, \left (\sqrt {-b c}\, b +2 a c \right ) \left (x +\frac {\sqrt {-b c}}{b}\right )}{b c}-\frac {\sqrt {-b c}\, b +a c -b c}{b}}}{x +\frac {\sqrt {-b c}}{b}}\right )}{2 \sqrt {-b c}\, b \sqrt {-\frac {\sqrt {-b c}\, b +a c -b c}{b}}}-\frac {\left (a \,b^{2} c -\sqrt {-b c}\, b^{2}-a^{2} c \right ) \ln \left (\frac {-\frac {2 \left (-\sqrt {-b c}\, b +a c -b c \right )}{b}+\frac {\sqrt {-b c}\, \left (-\sqrt {-b c}\, b +2 a c \right ) \left (x -\frac {\sqrt {-b c}}{b}\right )}{b c}+2 \sqrt {-\frac {-\sqrt {-b c}\, b +a c -b c}{b}}\, \sqrt {\left (x -\frac {\sqrt {-b c}}{b}\right )^{2} a +\frac {\sqrt {-b c}\, \left (-\sqrt {-b c}\, b +2 a c \right ) \left (x -\frac {\sqrt {-b c}}{b}\right )}{b c}-\frac {-\sqrt {-b c}\, b +a c -b c}{b}}}{x -\frac {\sqrt {-b c}}{b}}\right )}{2 \sqrt {-b c}\, b \sqrt {-\frac {-\sqrt {-b c}\, b +a c -b c}{b}}}\)	\(520\)

[In]

int((a^2*x^2+a*b*c-b^2*x)/(a*x^2+b*x+c)^(1/2)/(b*x^2+c),x,method=_RETURNVERBOSE)

[Out]

a^(3/2)/b*ln((1/2*b+a*x)/a^(1/2)+(a*x^2+b*x+c)^(1/2))-1/2*(-a*b^2*c-(-b*c)^(1/2)*b^2+a^2*c)/(-b*c)^(1/2)/b/(-(
(-b*c)^(1/2)*b+a*c-b*c)/b)^(1/2)*ln((-2*((-b*c)^(1/2)*b+a*c-b*c)/b-(-b*c)^(1/2)/b/c*((-b*c)^(1/2)*b+2*a*c)*(x+
(-b*c)^(1/2)/b)+2*(-((-b*c)^(1/2)*b+a*c-b*c)/b)^(1/2)*((x+(-b*c)^(1/2)/b)^2*a-(-b*c)^(1/2)/b/c*((-b*c)^(1/2)*b
+2*a*c)*(x+(-b*c)^(1/2)/b)-((-b*c)^(1/2)*b+a*c-b*c)/b)^(1/2))/(x+(-b*c)^(1/2)/b))-1/2*(a*b^2*c-(-b*c)^(1/2)*b^
2-a^2*c)/(-b*c)^(1/2)/b/(-(-(-b*c)^(1/2)*b+a*c-b*c)/b)^(1/2)*ln((-2*(-(-b*c)^(1/2)*b+a*c-b*c)/b+(-b*c)^(1/2)/b
/c*(-(-b*c)^(1/2)*b+2*a*c)*(x-(-b*c)^(1/2)/b)+2*(-(-(-b*c)^(1/2)*b+a*c-b*c)/b)^(1/2)*((x-(-b*c)^(1/2)/b)^2*a+(
-b*c)^(1/2)/b/c*(-(-b*c)^(1/2)*b+2*a*c)*(x-(-b*c)^(1/2)/b)-(-(-b*c)^(1/2)*b+a*c-b*c)/b)^(1/2))/(x-(-b*c)^(1/2)
/b))

Fricas [F(-1)]

Timed out. \[ \int \frac {a b c-b^2 x+a^2 x^2}{\sqrt {c+b x+a x^2} \left (c+b x^2\right )} \, dx=\text {Timed out} \]

[In]

integrate((a^2*x^2+a*b*c-b^2*x)/(a*x^2+b*x+c)^(1/2)/(b*x^2+c),x, algorithm="fricas")

[Out]

Timed out

Sympy [N/A]

Not integrable

Time = 2.73 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.11 \[ \int \frac {a b c-b^2 x+a^2 x^2}{\sqrt {c+b x+a x^2} \left (c+b x^2\right )} \, dx=\int \frac {a^{2} x^{2} + a b c - b^{2} x}{\left (b x^{2} + c\right ) \sqrt {a x^{2} + b x + c}}\, dx \]

[In]

integrate((a**2*x**2+a*b*c-b**2*x)/(a*x**2+b*x+c)**(1/2)/(b*x**2+c),x)

[Out]

Integral((a**2*x**2 + a*b*c - b**2*x)/((b*x**2 + c)*sqrt(a*x**2 + b*x + c)), x)

Maxima [N/A]

Not integrable

Time = 0.23 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.12 \[ \int \frac {a b c-b^2 x+a^2 x^2}{\sqrt {c+b x+a x^2} \left (c+b x^2\right )} \, dx=\int { \frac {a^{2} x^{2} + a b c - b^{2} x}{\sqrt {a x^{2} + b x + c} {\left (b x^{2} + c\right )}} \,d x } \]

[In]

integrate((a^2*x^2+a*b*c-b^2*x)/(a*x^2+b*x+c)^(1/2)/(b*x^2+c),x, algorithm="maxima")

[Out]

integrate((a^2*x^2 + a*b*c - b^2*x)/(sqrt(a*x^2 + b*x + c)*(b*x^2 + c)), x)

Giac [F(-2)]

Exception generated. \[ \int \frac {a b c-b^2 x+a^2 x^2}{\sqrt {c+b x+a x^2} \left (c+b x^2\right )} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((a^2*x^2+a*b*c-b^2*x)/(a*x^2+b*x+c)^(1/2)/(b*x^2+c),x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Error: Bad Argument Type

Mupad [N/A]

Not integrable

Time = 7.39 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.12 \[ \int \frac {a b c-b^2 x+a^2 x^2}{\sqrt {c+b x+a x^2} \left (c+b x^2\right )} \, dx=\int \frac {a^2\,x^2+c\,a\,b-b^2\,x}{\left (b\,x^2+c\right )\,\sqrt {a\,x^2+b\,x+c}} \,d x \]

[In]

int((a^2*x^2 - b^2*x + a*b*c)/((c + b*x^2)*(c + b*x + a*x^2)^(1/2)),x)

[Out]

int((a^2*x^2 - b^2*x + a*b*c)/((c + b*x^2)*(c + b*x + a*x^2)^(1/2)), x)