Integrand size = 31, antiderivative size = 463 \[ \int \frac {1}{\sqrt {c+b x+a x^2} \left (c^3+a^3 b^3 x^3\right )} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {c}}{\sqrt {-b^2+a b^2+c}}+\frac {a b x}{\sqrt {c} \sqrt {-b^2+a b^2+c}}-\frac {\sqrt {a} b \sqrt {c+b x+a x^2}}{\sqrt {c} \sqrt {-b^2+a b^2+c}}\right )}{3 \sqrt {a} c^{5/2} \sqrt {-b^2+a b^2+c}}-\frac {\text {RootSum}\left [b^2 c^2+a b^2 c^2+a^2 b^2 c^2-4 \sqrt {a} b c^2 \text {$\#$1}-2 a^{3/2} b c^2 \text {$\#$1}-a b^2 c \text {$\#$1}^2-2 a^2 b^2 c \text {$\#$1}^2+4 a c^2 \text {$\#$1}^2+2 a^{3/2} b c \text {$\#$1}^3+a^2 b^2 \text {$\#$1}^4\&,\frac {2 b c \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right )+a b c \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right )-4 \sqrt {a} c \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}-a b \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{2 b c^2+a b c^2+\sqrt {a} b^2 c \text {$\#$1}+2 a^{3/2} b^2 c \text {$\#$1}-4 \sqrt {a} c^2 \text {$\#$1}-3 a b c \text {$\#$1}^2-2 a^{3/2} b^2 \text {$\#$1}^3}\&\right ]}{3 \sqrt {a} c^2} \]
Time = 0.87 (sec) , antiderivative size = 441, normalized size of antiderivative = 0.95 \[ \int \frac {1}{\sqrt {c+b x+a x^2} \left (c^3+a^3 b^3 x^3\right )} \, dx=\frac {-\frac {2 \text {arctanh}\left (\frac {\sqrt {a} \sqrt {(-1+a) b^2+c} x}{c+a b x-\sqrt {c} \sqrt {c+x (b+a x)}}\right )}{\sqrt {a} \sqrt {(-1+a) b^2+c}}-\text {RootSum}\left [a^2 b^4+a^2 b^2 c+a^2 c^2-4 a^2 b^3 \sqrt {c} \text {$\#$1}-2 a^2 b c^{3/2} \text {$\#$1}-a b^2 c \text {$\#$1}^2+4 a^2 b^2 c \text {$\#$1}^2-2 a c^2 \text {$\#$1}^2+2 a b c^{3/2} \text {$\#$1}^3+c^2 \text {$\#$1}^4\&,\frac {-a b^2 \log (x)-2 a c \log (x)+a b^2 \log \left (-\sqrt {c}+\sqrt {c+b x+a x^2}-x \text {$\#$1}\right )+2 a c \log \left (-\sqrt {c}+\sqrt {c+b x+a x^2}-x \text {$\#$1}\right )+2 a b \sqrt {c} \log (x) \text {$\#$1}-2 a b \sqrt {c} \log \left (-\sqrt {c}+\sqrt {c+b x+a x^2}-x \text {$\#$1}\right ) \text {$\#$1}+2 c \log (x) \text {$\#$1}^2-2 c \log \left (-\sqrt {c}+\sqrt {c+b x+a x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2}{2 a^2 b^3+a^2 b c+a b^2 \sqrt {c} \text {$\#$1}-4 a^2 b^2 \sqrt {c} \text {$\#$1}+2 a c^{3/2} \text {$\#$1}-3 a b c \text {$\#$1}^2-2 c^{3/2} \text {$\#$1}^3}\&\right ]}{3 c^{5/2}} \]
((-2*ArcTanh[(Sqrt[a]*Sqrt[(-1 + a)*b^2 + c]*x)/(c + a*b*x - Sqrt[c]*Sqrt[ c + x*(b + a*x)])])/(Sqrt[a]*Sqrt[(-1 + a)*b^2 + c]) - RootSum[a^2*b^4 + a ^2*b^2*c + a^2*c^2 - 4*a^2*b^3*Sqrt[c]*#1 - 2*a^2*b*c^(3/2)*#1 - a*b^2*c*# 1^2 + 4*a^2*b^2*c*#1^2 - 2*a*c^2*#1^2 + 2*a*b*c^(3/2)*#1^3 + c^2*#1^4 & , (-(a*b^2*Log[x]) - 2*a*c*Log[x] + a*b^2*Log[-Sqrt[c] + Sqrt[c + b*x + a*x^ 2] - x*#1] + 2*a*c*Log[-Sqrt[c] + Sqrt[c + b*x + a*x^2] - x*#1] + 2*a*b*Sq rt[c]*Log[x]*#1 - 2*a*b*Sqrt[c]*Log[-Sqrt[c] + Sqrt[c + b*x + a*x^2] - x*# 1]*#1 + 2*c*Log[x]*#1^2 - 2*c*Log[-Sqrt[c] + Sqrt[c + b*x + a*x^2] - x*#1] *#1^2)/(2*a^2*b^3 + a^2*b*c + a*b^2*Sqrt[c]*#1 - 4*a^2*b^2*Sqrt[c]*#1 + 2* a*c^(3/2)*#1 - 3*a*b*c*#1^2 - 2*c^(3/2)*#1^3) & ])/(3*c^(5/2))
Time = 1.75 (sec) , antiderivative size = 521, normalized size of antiderivative = 1.13, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {2533, 1154, 217, 1368, 27, 1362, 221}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{\sqrt {a x^2+b x+c} \left (a^3 b^3 x^3+c^3\right )} \, dx\) |
| \(\Big \downarrow \) 2533 |
| \(\displaystyle \frac {\int \frac {2 c-a b x}{\sqrt {a x^2+b x+c} \left (c^2-a b x c+a^2 b^2 x^2\right )}dx}{3 c^2}+\frac {\int \frac {1}{(c+a b x) \sqrt {a x^2+b x+c}}dx}{3 c^2}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {\int \frac {2 c-a b x}{\sqrt {a x^2+b x+c} \left (c^2-a b x c+a^2 b^2 x^2\right )}dx}{3 c^2}-\frac {2 \int \frac {1}{-\frac {\left ((1-2 a) b c-a \left (b^2-2 c\right ) x\right )^2}{a x^2+b x+c}-4 a \left ((1-a) b^2-c\right ) c}d\left (-\frac {(1-2 a) b c-a \left (b^2-2 c\right ) x}{\sqrt {a x^2+b x+c}}\right )}{3 c^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\int \frac {2 c-a b x}{\sqrt {a x^2+b x+c} \left (c^2-a b x c+a^2 b^2 x^2\right )}dx}{3 c^2}-\frac {\arctan \left (\frac {(1-2 a) b c-a x \left (b^2-2 c\right )}{2 \sqrt {a} \sqrt {c} \sqrt {(1-a) b^2-c} \sqrt {a x^2+b x+c}}\right )}{3 \sqrt {a} c^{5/2} \sqrt {(1-a) b^2-c}}\) |
| \(\Big \downarrow \) 1368 |
| \(\displaystyle \frac {\frac {\int \frac {a c \left (c \left ((1-a) b^2+2 \left (c+\sqrt {\left (a^2+a+1\right ) b^4+(c-a c) b^2+c^2}\right )\right )-a b \left ((a+2) b^2+c+\sqrt {\left (a^2+a+1\right ) b^4+(c-a c) b^2+c^2}\right ) x\right )}{\sqrt {a x^2+b x+c} \left (c^2-a b x c+a^2 b^2 x^2\right )}dx}{2 a c \sqrt {\left (a^2+a+1\right ) b^4+b^2 (c-a c)+c^2}}-\frac {\int \frac {a c \left (c \left ((1-a) b^2+2 c-2 \sqrt {\left (a^2+a+1\right ) b^4+(c-a c) b^2+c^2}\right )-a b \left ((a+2) b^2+c-\sqrt {\left (a^2+a+1\right ) b^4+(c-a c) b^2+c^2}\right ) x\right )}{\sqrt {a x^2+b x+c} \left (c^2-a b x c+a^2 b^2 x^2\right )}dx}{2 a c \sqrt {\left (a^2+a+1\right ) b^4+b^2 (c-a c)+c^2}}}{3 c^2}-\frac {\arctan \left (\frac {(1-2 a) b c-a x \left (b^2-2 c\right )}{2 \sqrt {a} \sqrt {c} \sqrt {(1-a) b^2-c} \sqrt {a x^2+b x+c}}\right )}{3 \sqrt {a} c^{5/2} \sqrt {(1-a) b^2-c}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {c \left ((1-a) b^2+2 \left (c+\sqrt {\left (a^2+a+1\right ) b^4+(c-a c) b^2+c^2}\right )\right )-a b \left ((a+2) b^2+c+\sqrt {\left (a^2+a+1\right ) b^4+(c-a c) b^2+c^2}\right ) x}{\sqrt {a x^2+b x+c} \left (c^2-a b x c+a^2 b^2 x^2\right )}dx}{2 \sqrt {\left (a^2+a+1\right ) b^4+b^2 (c-a c)+c^2}}-\frac {\int \frac {c \left ((1-a) b^2+2 c-2 \sqrt {\left (a^2+a+1\right ) b^4+(c-a c) b^2+c^2}\right )-a b \left ((a+2) b^2+c-\sqrt {\left (a^2+a+1\right ) b^4+(c-a c) b^2+c^2}\right ) x}{\sqrt {a x^2+b x+c} \left (c^2-a b x c+a^2 b^2 x^2\right )}dx}{2 \sqrt {\left (a^2+a+1\right ) b^4+b^2 (c-a c)+c^2}}}{3 c^2}-\frac {\arctan \left (\frac {(1-2 a) b c-a x \left (b^2-2 c\right )}{2 \sqrt {a} \sqrt {c} \sqrt {(1-a) b^2-c} \sqrt {a x^2+b x+c}}\right )}{3 \sqrt {a} c^{5/2} \sqrt {(1-a) b^2-c}}\) |
| \(\Big \downarrow \) 1362 |
| \(\displaystyle \frac {\frac {3 a (a+1) b^3 c^3 \left (-2 \sqrt {\left (a^2+a+1\right ) b^4+b^2 (c-a c)+c^2}+(1-a) b^2+2 c\right ) \int \frac {1}{\frac {9 a^2 (a+1) b^5 c^4 \left ((a+1) b c-a \left (a b^2-c+\sqrt {\left (a^2+a+1\right ) b^4+(c-a c) b^2+c^2}\right ) x\right )^2}{a x^2+b x+c}-9 a^3 (a+1) b^5 c^5 \left ((1-a) b^2+2 c-2 \sqrt {\left (a^2+a+1\right ) b^4+(c-a c) b^2+c^2}\right )}d\frac {3 a b^2 c \left ((a+1) b c-a \left (a b^2-c+\sqrt {\left (a^2+a+1\right ) b^4+(c-a c) b^2+c^2}\right ) x\right )}{\sqrt {a x^2+b x+c}}}{\sqrt {\left (a^2+a+1\right ) b^4+b^2 (c-a c)+c^2}}-\frac {3 a (a+1) b^3 c^3 \left (2 \left (\sqrt {\left (a^2+a+1\right ) b^4+b^2 (c-a c)+c^2}+c\right )+(1-a) b^2\right ) \int \frac {1}{\frac {9 a^2 (a+1) b^5 c^4 \left ((a+1) b c-a \left (a b^2-c-\sqrt {\left (a^2+a+1\right ) b^4+(c-a c) b^2+c^2}\right ) x\right )^2}{a x^2+b x+c}-9 a^3 (a+1) b^5 c^5 \left ((1-a) b^2+2 \left (c+\sqrt {\left (a^2+a+1\right ) b^4+(c-a c) b^2+c^2}\right )\right )}d\frac {3 a b^2 c \left ((a+1) b c-a \left (a b^2-c-\sqrt {\left (a^2+a+1\right ) b^4+(c-a c) b^2+c^2}\right ) x\right )}{\sqrt {a x^2+b x+c}}}{\sqrt {\left (a^2+a+1\right ) b^4+b^2 (c-a c)+c^2}}}{3 c^2}-\frac {\arctan \left (\frac {(1-2 a) b c-a x \left (b^2-2 c\right )}{2 \sqrt {a} \sqrt {c} \sqrt {(1-a) b^2-c} \sqrt {a x^2+b x+c}}\right )}{3 \sqrt {a} c^{5/2} \sqrt {(1-a) b^2-c}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {\sqrt {2 \left (\sqrt {\left (a^2+a+1\right ) b^4+b^2 (c-a c)+c^2}+c\right )+(1-a) b^2} \text {arctanh}\left (\frac {(a+1) b c-a x \left (-\sqrt {\left (a^2+a+1\right ) b^4+b^2 (c-a c)+c^2}+a b^2-c\right )}{\sqrt {a} \sqrt {c} \sqrt {2 \left (\sqrt {\left (a^2+a+1\right ) b^4+b^2 (c-a c)+c^2}+c\right )+(1-a) b^2} \sqrt {a x^2+b x+c}}\right )}{\sqrt {a} \sqrt {c} \sqrt {\left (a^2+a+1\right ) b^4+b^2 (c-a c)+c^2}}-\frac {\sqrt {-2 \sqrt {\left (a^2+a+1\right ) b^4+b^2 (c-a c)+c^2}+(1-a) b^2+2 c} \text {arctanh}\left (\frac {(a+1) b c-a x \left (\sqrt {\left (a^2+a+1\right ) b^4+b^2 (c-a c)+c^2}+a b^2-c\right )}{\sqrt {a} \sqrt {c} \sqrt {-2 \sqrt {\left (a^2+a+1\right ) b^4+b^2 (c-a c)+c^2}+(1-a) b^2+2 c} \sqrt {a x^2+b x+c}}\right )}{\sqrt {a} \sqrt {c} \sqrt {\left (a^2+a+1\right ) b^4+b^2 (c-a c)+c^2}}}{3 c^2}-\frac {\arctan \left (\frac {(1-2 a) b c-a x \left (b^2-2 c\right )}{2 \sqrt {a} \sqrt {c} \sqrt {(1-a) b^2-c} \sqrt {a x^2+b x+c}}\right )}{3 \sqrt {a} c^{5/2} \sqrt {(1-a) b^2-c}}\) |
-1/3*ArcTan[((1 - 2*a)*b*c - a*(b^2 - 2*c)*x)/(2*Sqrt[a]*Sqrt[(1 - a)*b^2 - c]*Sqrt[c]*Sqrt[c + b*x + a*x^2])]/(Sqrt[a]*Sqrt[(1 - a)*b^2 - c]*c^(5/2 )) + ((Sqrt[(1 - a)*b^2 + 2*(c + Sqrt[(1 + a + a^2)*b^4 + c^2 + b^2*(c - a *c)])]*ArcTanh[((1 + a)*b*c - a*(a*b^2 - c - Sqrt[(1 + a + a^2)*b^4 + c^2 + b^2*(c - a*c)])*x)/(Sqrt[a]*Sqrt[c]*Sqrt[(1 - a)*b^2 + 2*(c + Sqrt[(1 + a + a^2)*b^4 + c^2 + b^2*(c - a*c)])]*Sqrt[c + b*x + a*x^2])])/(Sqrt[a]*Sq rt[c]*Sqrt[(1 + a + a^2)*b^4 + c^2 + b^2*(c - a*c)]) - (Sqrt[(1 - a)*b^2 + 2*c - 2*Sqrt[(1 + a + a^2)*b^4 + c^2 + b^2*(c - a*c)]]*ArcTanh[((1 + a)*b *c - a*(a*b^2 - c + Sqrt[(1 + a + a^2)*b^4 + c^2 + b^2*(c - a*c)])*x)/(Sqr t[a]*Sqrt[c]*Sqrt[(1 - a)*b^2 + 2*c - 2*Sqrt[(1 + a + a^2)*b^4 + c^2 + b^2 *(c - a*c)]]*Sqrt[c + b*x + a*x^2])])/(Sqrt[a]*Sqrt[c]*Sqrt[(1 + a + a^2)* b^4 + c^2 + b^2*(c - a*c)]))/(3*c^2)
3.31.52.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
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]
Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym bol] :> Simp[-2 Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c , d, e}, x]
Int[((g_.) + (h_.)*(x_))/(((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[-2*g*(g*b - 2*a*h) Subst[I nt[1/Simp[g*(g*b - 2*a*h)*(b^2 - 4*a*c) - (b*d - a*e)*x^2, x], x], x, Simp[ g*b - 2*a*h - (b*h - 2*g*c)*x, x]/Sqrt[d + e*x + f*x^2]], x] /; FreeQ[{a, b , c, d, e, f, g, h}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && Ne Q[b*d - a*e, 0] && EqQ[h^2*(b*d - a*e) - 2*g*h*(c*d - a*f) + g^2*(c*e - b*f ), 0]
Int[((g_.) + (h_.)*(x_))/(((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[(c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f), 2]}, Simp[1/(2*q) Int[Simp[h*(b*d - a*e) - g*(c*d - a*f - q) - (g*(c*e - b*f) - h*(c*d - a*f + q))*x, x]/((a + b*x + c*x^2)*Sqr t[d + e*x + f*x^2]), x], x] - Simp[1/(2*q) Int[Simp[h*(b*d - a*e) - g*(c* d - a*f + q) - (g*(c*e - b*f) - h*(c*d - a*f - q))*x, x]/((a + b*x + c*x^2) *Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && N eQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && NeQ[b*d - a*e, 0] && NegQ[b^2 - 4*a*c]
Int[1/(Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]*((a_) + (b_.)*(x_)^3)), x_Sy mbol] :> With[{r = Numerator[Rt[a/b, 3]], s = Denominator[Rt[a/b, 3]]}, Sim p[r/(3*a) Int[1/((r + s*x)*Sqrt[d + e*x + f*x^2]), x], x] + Simp[r/(3*a) Int[(2*r - s*x)/((r^2 - r*s*x + s^2*x^2)*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, b, d, e, f}, x] && PosQ[a/b]
Time = 0.58 (sec) , antiderivative size = 1195, normalized size of antiderivative = 2.58
4*a*b/(3*a*b*c+(-3*a^2*b^2*c^2)^(1/2))/(-3*a*b*c+(-3*a^2*b^2*c^2)^(1/2))/( c*(a*b^2-b^2+c)/a/b^2)^(1/2)*ln((2*c*(a*b^2-b^2+c)/a/b^2+(b^2-2*c)/b*(x+c/ a/b)+2*(c*(a*b^2-b^2+c)/a/b^2)^(1/2)*(a*(x+c/a/b)^2+(b^2-2*c)/b*(x+c/a/b)+ c*(a*b^2-b^2+c)/a/b^2)^(1/2))/(x+c/a/b))-2*a*b/(-3*a*b*c+(-3*a^2*b^2*c^2)^ (1/2))/(-3*a^2*b^2*c^2)^(1/2)*2^(1/2)/(1/a^2/b^3*(2*a^2*b^3*c+a*b^3*c-a*b* c^2-(-3*a^2*b^2*c^2)^(1/2)*b^2-(-3*a^2*b^2*c^2)^(1/2)*c))^(1/2)*ln((1/a^2/ b^3*(2*a^2*b^3*c+a*b^3*c-a*b*c^2-(-3*a^2*b^2*c^2)^(1/2)*b^2-(-3*a^2*b^2*c^ 2)^(1/2)*c)+(a*b^3+a*b*c-(-3*a^2*b^2*c^2)^(1/2))/a/b^2*(x+1/2*(-a*b*c+(-3* a^2*b^2*c^2)^(1/2))/a^2/b^2)+1/2*2^(1/2)*(1/a^2/b^3*(2*a^2*b^3*c+a*b^3*c-a *b*c^2-(-3*a^2*b^2*c^2)^(1/2)*b^2-(-3*a^2*b^2*c^2)^(1/2)*c))^(1/2)*(4*a*(x +1/2*(-a*b*c+(-3*a^2*b^2*c^2)^(1/2))/a^2/b^2)^2+4*(a*b^3+a*b*c-(-3*a^2*b^2 *c^2)^(1/2))/a/b^2*(x+1/2*(-a*b*c+(-3*a^2*b^2*c^2)^(1/2))/a^2/b^2)+2/a^2/b ^3*(2*a^2*b^3*c+a*b^3*c-a*b*c^2-(-3*a^2*b^2*c^2)^(1/2)*b^2-(-3*a^2*b^2*c^2 )^(1/2)*c))^(1/2))/(x+1/2*(-a*b*c+(-3*a^2*b^2*c^2)^(1/2))/a^2/b^2))-2*a*b/ (3*a*b*c+(-3*a^2*b^2*c^2)^(1/2))/(-3*a^2*b^2*c^2)^(1/2)*2^(1/2)/(1/a^2/b^3 *(2*a^2*b^3*c+a*b^3*c-a*b*c^2+(-3*a^2*b^2*c^2)^(1/2)*b^2+(-3*a^2*b^2*c^2)^ (1/2)*c))^(1/2)*ln((1/a^2/b^3*(2*a^2*b^3*c+a*b^3*c-a*b*c^2+(-3*a^2*b^2*c^2 )^(1/2)*b^2+(-3*a^2*b^2*c^2)^(1/2)*c)+(a*b^3+a*b*c+(-3*a^2*b^2*c^2)^(1/2)) /a/b^2*(x-1/2*(a*b*c+(-3*a^2*b^2*c^2)^(1/2))/a^2/b^2)+1/2*2^(1/2)*(1/a^2/b ^3*(2*a^2*b^3*c+a*b^3*c-a*b*c^2+(-3*a^2*b^2*c^2)^(1/2)*b^2+(-3*a^2*b^2*...
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 8.24 (sec) , antiderivative size = 7282, normalized size of antiderivative = 15.73 \[ \int \frac {1}{\sqrt {c+b x+a x^2} \left (c^3+a^3 b^3 x^3\right )} \, dx=\text {Too large to display} \]
Not integrable
Time = 3.41 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.09 \[ \int \frac {1}{\sqrt {c+b x+a x^2} \left (c^3+a^3 b^3 x^3\right )} \, dx=\int \frac {1}{\left (a b x + c\right ) \sqrt {a x^{2} + b x + c} \left (a^{2} b^{2} x^{2} - a b c x + c^{2}\right )}\, dx \]
Not integrable
Time = 0.25 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.07 \[ \int \frac {1}{\sqrt {c+b x+a x^2} \left (c^3+a^3 b^3 x^3\right )} \, dx=\int { \frac {1}{{\left (a^{3} b^{3} x^{3} + c^{3}\right )} \sqrt {a x^{2} + b x + c}} \,d x } \]
Exception generated. \[ \int \frac {1}{\sqrt {c+b x+a x^2} \left (c^3+a^3 b^3 x^3\right )} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Not integrable
Time = 7.14 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.07 \[ \int \frac {1}{\sqrt {c+b x+a x^2} \left (c^3+a^3 b^3 x^3\right )} \, dx=\int \frac {1}{\left (a^3\,b^3\,x^3+c^3\right )\,\sqrt {a\,x^2+b\,x+c}} \,d x \]