Integrand size = 23, antiderivative size = 199 \[ \int \frac {A+B x}{x^{3/2} \left (a+b x+c x^2\right )} \, dx=-\frac {2 A}{a \sqrt {x}}-\frac {\sqrt {2} \sqrt {c} \left (A+\frac {A b-2 a B}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{a \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2} \sqrt {c} \left (A-\frac {A b-2 a B}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{a \sqrt {b+\sqrt {b^2-4 a c}}} \]
-2*A/a/x^(1/2)-arctan(2^(1/2)*c^(1/2)*x^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2) )*2^(1/2)*c^(1/2)*(A+(A*b-2*B*a)/(-4*a*c+b^2)^(1/2))/a/(b-(-4*a*c+b^2)^(1/ 2))^(1/2)-arctan(2^(1/2)*c^(1/2)*x^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2))*2^( 1/2)*c^(1/2)*(A+(-A*b+2*B*a)/(-4*a*c+b^2)^(1/2))/a/(b+(-4*a*c+b^2)^(1/2))^ (1/2)
Time = 0.53 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.08 \[ \int \frac {A+B x}{x^{3/2} \left (a+b x+c x^2\right )} \, dx=-\frac {\frac {2 A}{\sqrt {x}}+\frac {\sqrt {2} \sqrt {c} \left (-2 a B+A \left (b+\sqrt {b^2-4 a c}\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {2} \sqrt {c} \left (2 a B+A \left (-b+\sqrt {b^2-4 a c}\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b+\sqrt {b^2-4 a c}}}}{a} \]
-(((2*A)/Sqrt[x] + (Sqrt[2]*Sqrt[c]*(-2*a*B + A*(b + Sqrt[b^2 - 4*a*c]))*A rcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*Sqrt[c]*(2*a*B + A*(-b + Sq rt[b^2 - 4*a*c]))*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b + Sqrt[b^2 - 4*a *c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/a)
Time = 0.42 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.99, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {1198, 25, 1197, 1480, 218}
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.
| \(\displaystyle \int \frac {A+B x}{x^{3/2} \left (a+b x+c x^2\right )} \, dx\) |
| \(\Big \downarrow \) 1198 |
| \(\displaystyle \frac {\int -\frac {A b-a B+A c x}{\sqrt {x} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 A}{a \sqrt {x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int \frac {A b-a B+A c x}{\sqrt {x} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 A}{a \sqrt {x}}\) |
| \(\Big \downarrow \) 1197 |
| \(\displaystyle -\frac {2 \int \frac {A b-a B+A c x}{c x^2+b x+a}d\sqrt {x}}{a}-\frac {2 A}{a \sqrt {x}}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle -\frac {2 \left (\frac {1}{2} c \left (\frac {A b-2 a B}{\sqrt {b^2-4 a c}}+A\right ) \int \frac {1}{\frac {1}{2} \left (b-\sqrt {b^2-4 a c}\right )+c x}d\sqrt {x}+\frac {1}{2} c \left (A-\frac {A b-2 a B}{\sqrt {b^2-4 a c}}\right ) \int \frac {1}{\frac {1}{2} \left (b+\sqrt {b^2-4 a c}\right )+c x}d\sqrt {x}\right )}{a}-\frac {2 A}{a \sqrt {x}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {2 \left (\frac {\sqrt {c} \left (\frac {A b-2 a B}{\sqrt {b^2-4 a c}}+A\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {c} \left (A-\frac {A b-2 a B}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {2} \sqrt {\sqrt {b^2-4 a c}+b}}\right )}{a}-\frac {2 A}{a \sqrt {x}}\) |
(-2*A)/(a*Sqrt[x]) - (2*((Sqrt[c]*(A + (A*b - 2*a*B)/Sqrt[b^2 - 4*a*c])*Ar cTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt [b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[c]*(A - (A*b - 2*a*B)/Sqrt[b^2 - 4*a*c])* ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sq rt[b + Sqrt[b^2 - 4*a*c]])))/a
3.11.13.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)), x_Symbol] :> Simp[2 Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; Fr eeQ[{a, b, c, d, e, f, g}, x]
Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(e*f - d*g)*((d + e*x)^(m + 1)/((m + 1)*(c *d^2 - b*d*e + a*e^2))), x] + Simp[1/(c*d^2 - b*d*e + a*e^2) Int[(d + e*x )^(m + 1)*(Simp[c*d*f - f*b*e + a*e*g - c*(e*f - d*g)*x, x]/(a + b*x + c*x^ 2)), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && FractionQ[m] && LtQ[m, -1 ]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q)) Int[1/( b/2 - q/2 + c*x^2), x], x] + Simp[(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]
Time = 0.39 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.88
| method | result | size |
| risch | \(-\frac {2 A}{a \sqrt {x}}-\frac {8 c \left (\frac {\left (A \sqrt {-4 a c +b^{2}}-A b +2 B a \right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}-\frac {\left (A \sqrt {-4 a c +b^{2}}+A b -2 B a \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{a}\) | \(175\) |
| derivativedivides | \(\frac {8 c \left (-\frac {\left (-A \sqrt {-4 a c +b^{2}}-A b +2 B a \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}+\frac {\left (-A \sqrt {-4 a c +b^{2}}+A b -2 B a \right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{a}-\frac {2 A}{a \sqrt {x}}\) | \(177\) |
| default | \(\frac {8 c \left (-\frac {\left (-A \sqrt {-4 a c +b^{2}}-A b +2 B a \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}+\frac {\left (-A \sqrt {-4 a c +b^{2}}+A b -2 B a \right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{a}-\frac {2 A}{a \sqrt {x}}\) | \(177\) |
-2*A/a/x^(1/2)-8/a*c*(1/8*(A*(-4*a*c+b^2)^(1/2)-A*b+2*B*a)/(-4*a*c+b^2)^(1 /2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x^(1/2)*2^(1/2)/((b+ (-4*a*c+b^2)^(1/2))*c)^(1/2))-1/8*(A*(-4*a*c+b^2)^(1/2)+A*b-2*B*a)/(-4*a*c +b^2)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(c*x^(1/2)*2^ (1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 2925 vs. \(2 (159) = 318\).
Time = 1.56 (sec) , antiderivative size = 2925, normalized size of antiderivative = 14.70 \[ \int \frac {A+B x}{x^{3/2} \left (a+b x+c x^2\right )} \, dx=\text {Too large to display} \]
1/2*(sqrt(2)*a*x*sqrt(-(B^2*a^2*b - 2*A*B*a*b^2 + A^2*b^3 + (4*A*B*a^2 - 3 *A^2*a*b)*c + (a^3*b^2 - 4*a^4*c)*sqrt((B^4*a^4 - 4*A*B^3*a^3*b + 6*A^2*B^ 2*a^2*b^2 - 4*A^3*B*a*b^3 + A^4*b^4 + A^4*a^2*c^2 - 2*(A^2*B^2*a^3 - 2*A^3 *B*a^2*b + A^4*a*b^2)*c)/(a^6*b^2 - 4*a^7*c)))/(a^3*b^2 - 4*a^4*c))*log(sq rt(2)*(B^3*a^3*b^2 - 3*A*B^2*a^2*b^3 + 3*A^2*B*a*b^4 - A^3*b^5 + 4*(A^2*B* a^3 - A^3*a^2*b)*c^2 - (4*B^3*a^4 - 12*A*B^2*a^3*b + 13*A^2*B*a^2*b^2 - 5* A^3*a*b^3)*c - (B*a^4*b^3 - A*a^3*b^4 - 8*A*a^5*c^2 - 2*(2*B*a^5*b - 3*A*a ^4*b^2)*c)*sqrt((B^4*a^4 - 4*A*B^3*a^3*b + 6*A^2*B^2*a^2*b^2 - 4*A^3*B*a*b ^3 + A^4*b^4 + A^4*a^2*c^2 - 2*(A^2*B^2*a^3 - 2*A^3*B*a^2*b + A^4*a*b^2)*c )/(a^6*b^2 - 4*a^7*c)))*sqrt(-(B^2*a^2*b - 2*A*B*a*b^2 + A^2*b^3 + (4*A*B* a^2 - 3*A^2*a*b)*c + (a^3*b^2 - 4*a^4*c)*sqrt((B^4*a^4 - 4*A*B^3*a^3*b + 6 *A^2*B^2*a^2*b^2 - 4*A^3*B*a*b^3 + A^4*b^4 + A^4*a^2*c^2 - 2*(A^2*B^2*a^3 - 2*A^3*B*a^2*b + A^4*a*b^2)*c)/(a^6*b^2 - 4*a^7*c)))/(a^3*b^2 - 4*a^4*c)) + 4*(A^4*a*c^3 + (A^3*B*a*b - A^4*b^2)*c^2 - (B^4*a^3 - 3*A*B^3*a^2*b + 3 *A^2*B^2*a*b^2 - A^3*B*b^3)*c)*sqrt(x)) - sqrt(2)*a*x*sqrt(-(B^2*a^2*b - 2 *A*B*a*b^2 + A^2*b^3 + (4*A*B*a^2 - 3*A^2*a*b)*c + (a^3*b^2 - 4*a^4*c)*sqr t((B^4*a^4 - 4*A*B^3*a^3*b + 6*A^2*B^2*a^2*b^2 - 4*A^3*B*a*b^3 + A^4*b^4 + A^4*a^2*c^2 - 2*(A^2*B^2*a^3 - 2*A^3*B*a^2*b + A^4*a*b^2)*c)/(a^6*b^2 - 4 *a^7*c)))/(a^3*b^2 - 4*a^4*c))*log(-sqrt(2)*(B^3*a^3*b^2 - 3*A*B^2*a^2*b^3 + 3*A^2*B*a*b^4 - A^3*b^5 + 4*(A^2*B*a^3 - A^3*a^2*b)*c^2 - (4*B^3*a^4...
Leaf count of result is larger than twice the leaf count of optimal. 305978 vs. \(2 (180) = 360\).
Time = 75.83 (sec) , antiderivative size = 305978, normalized size of antiderivative = 1537.58 \[ \int \frac {A+B x}{x^{3/2} \left (a+b x+c x^2\right )} \, dx=\text {Too large to display} \]
Piecewise((-A*log(sqrt(x) - sqrt(-a/b))/(a*sqrt(-a/b)) + A*log(sqrt(x) + s qrt(-a/b))/(a*sqrt(-a/b)) - 2*A/(a*sqrt(x)) + B*log(sqrt(x) - sqrt(-a/b))/ (b*sqrt(-a/b)) - B*log(sqrt(x) + sqrt(-a/b))/(b*sqrt(-a/b)), Eq(c, 0)), (- 2*A/(3*b*x**(3/2)) + A*c*log(sqrt(x) - sqrt(-b/c))/(b**2*sqrt(-b/c)) - A*c *log(sqrt(x) + sqrt(-b/c))/(b**2*sqrt(-b/c)) + 2*A*c/(b**2*sqrt(x)) - B*lo g(sqrt(x) - sqrt(-b/c))/(b*sqrt(-b/c)) + B*log(sqrt(x) + sqrt(-b/c))/(b*sq rt(-b/c)) - 2*B/(b*sqrt(x)), Eq(a, 0)), (-6*sqrt(2)*A*b*c*sqrt(x)*log(sqrt (x) - sqrt(2)*sqrt(-b/c)/2)/(b**3*sqrt(x)*sqrt(-b/c) + 2*b**2*c*x**(3/2)*s qrt(-b/c)) + 6*sqrt(2)*A*b*c*sqrt(x)*log(sqrt(x) + sqrt(2)*sqrt(-b/c)/2)/( b**3*sqrt(x)*sqrt(-b/c) + 2*b**2*c*x**(3/2)*sqrt(-b/c)) - 8*A*b*c*sqrt(-b/ c)/(b**3*sqrt(x)*sqrt(-b/c) + 2*b**2*c*x**(3/2)*sqrt(-b/c)) - 12*sqrt(2)*A *c**2*x**(3/2)*log(sqrt(x) - sqrt(2)*sqrt(-b/c)/2)/(b**3*sqrt(x)*sqrt(-b/c ) + 2*b**2*c*x**(3/2)*sqrt(-b/c)) + 12*sqrt(2)*A*c**2*x**(3/2)*log(sqrt(x) + sqrt(2)*sqrt(-b/c)/2)/(b**3*sqrt(x)*sqrt(-b/c) + 2*b**2*c*x**(3/2)*sqrt (-b/c)) - 24*A*c**2*x*sqrt(-b/c)/(b**3*sqrt(x)*sqrt(-b/c) + 2*b**2*c*x**(3 /2)*sqrt(-b/c)) + sqrt(2)*B*b**2*sqrt(x)*log(sqrt(x) - sqrt(2)*sqrt(-b/c)/ 2)/(b**3*sqrt(x)*sqrt(-b/c) + 2*b**2*c*x**(3/2)*sqrt(-b/c)) - sqrt(2)*B*b* *2*sqrt(x)*log(sqrt(x) + sqrt(2)*sqrt(-b/c)/2)/(b**3*sqrt(x)*sqrt(-b/c) + 2*b**2*c*x**(3/2)*sqrt(-b/c)) + 2*sqrt(2)*B*b*c*x**(3/2)*log(sqrt(x) - sqr t(2)*sqrt(-b/c)/2)/(b**3*sqrt(x)*sqrt(-b/c) + 2*b**2*c*x**(3/2)*sqrt(-b...
\[ \int \frac {A+B x}{x^{3/2} \left (a+b x+c x^2\right )} \, dx=\int { \frac {B x + A}{{\left (c x^{2} + b x + a\right )} x^{\frac {3}{2}}} \,d x } \]
-2*(A*a/sqrt(x) - (B*a - A*b)*sqrt(x))/a^2 + integrate(-((B*a*c - A*b*c)*x ^(3/2) + (B*a*b - (b^2 - a*c)*A)*sqrt(x))/(a^2*c*x^2 + a^2*b*x + a^3), x)
Leaf count of result is larger than twice the leaf count of optimal. 2809 vs. \(2 (159) = 318\).
Time = 0.91 (sec) , antiderivative size = 2809, normalized size of antiderivative = 14.12 \[ \int \frac {A+B x}{x^{3/2} \left (a+b x+c x^2\right )} \, dx=\text {Too large to display} \]
-2*A/(a*sqrt(x)) - 1/4*((2*b^4*c^2 - 16*a*b^2*c^3 + 32*a^2*c^4 - sqrt(2)*s qrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^4 + 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c + 2*sqrt(2)*sqrt(b^2 - 4* a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^3*c - 16*sqrt(2)*sqrt(b^2 - 4*a*c)* sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*c^2 - 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt (b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^2 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^2*c^2 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqr t(b^2 - 4*a*c)*c)*a*c^3 - 2*(b^2 - 4*a*c)*b^2*c^2 + 8*(b^2 - 4*a*c)*a*c^3) *A*a^2 + 2*(sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^5 - 8*sqrt(2)*sqrt (b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^3*c - 2*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4* a*c)*c)*a*b^4*c - 2*a*b^5*c + 16*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a ^3*b*c^2 + 8*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^2 + sqrt(2) *sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^3*c^2 + 16*a^2*b^3*c^2 - 4*sqrt(2)*sq rt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b*c^3 - 32*a^3*b*c^3 + 2*(b^2 - 4*a*c)*a *b^3*c - 8*(b^2 - 4*a*c)*a^2*b*c^2)*A*abs(a) - 2*(sqrt(2)*sqrt(b*c + sqrt( b^2 - 4*a*c)*c)*a^2*b^4 - 8*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^ 2*c - 2*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^3*c - 2*a^2*b^4*c + 16*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*c^2 + 8*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b*c^2 + sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a ^2*b^2*c^2 + 16*a^3*b^2*c^2 - 4*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)...
Time = 11.22 (sec) , antiderivative size = 6367, normalized size of antiderivative = 31.99 \[ \int \frac {A+B x}{x^{3/2} \left (a+b x+c x^2\right )} \, dx=\text {Too large to display} \]
- atan((((-(A^2*b^5 + B^2*a^2*b^3 + A^2*b^2*(-(4*a*c - b^2)^3)^(1/2) + B^2 *a^2*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*a*b^4 - 16*A*B*a^3*c^2 - 7*A^2*a*b^3 *c - A^2*a*c*(-(4*a*c - b^2)^3)^(1/2) - 4*B^2*a^3*b*c + 12*A^2*a^2*b*c^2 - 2*A*B*a*b*(-(4*a*c - b^2)^3)^(1/2) + 12*A*B*a^2*b^2*c)/(2*(a^3*b^4 + 16*a ^5*c^2 - 8*a^4*b^2*c)))^(1/2)*(x^(1/2)*(32*a^6*b*c^3 - 8*a^5*b^3*c^2)*(-(A ^2*b^5 + B^2*a^2*b^3 + A^2*b^2*(-(4*a*c - b^2)^3)^(1/2) + B^2*a^2*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*a*b^4 - 16*A*B*a^3*c^2 - 7*A^2*a*b^3*c - A^2*a*c* (-(4*a*c - b^2)^3)^(1/2) - 4*B^2*a^3*b*c + 12*A^2*a^2*b*c^2 - 2*A*B*a*b*(- (4*a*c - b^2)^3)^(1/2) + 12*A*B*a^2*b^2*c)/(2*(a^3*b^4 + 16*a^5*c^2 - 8*a^ 4*b^2*c)))^(1/2) - 32*B*a^6*c^3 + 32*A*a^5*b*c^3 - 8*A*a^4*b^3*c^2 + 8*B*a ^5*b^2*c^2) + x^(1/2)*(16*A^2*a^4*c^4 - 16*B^2*a^5*c^3 - 8*A^2*a^3*b^2*c^3 + 16*A*B*a^4*b*c^3))*(-(A^2*b^5 + B^2*a^2*b^3 + A^2*b^2*(-(4*a*c - b^2)^3 )^(1/2) + B^2*a^2*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*a*b^4 - 16*A*B*a^3*c^2 - 7*A^2*a*b^3*c - A^2*a*c*(-(4*a*c - b^2)^3)^(1/2) - 4*B^2*a^3*b*c + 12*A^ 2*a^2*b*c^2 - 2*A*B*a*b*(-(4*a*c - b^2)^3)^(1/2) + 12*A*B*a^2*b^2*c)/(2*(a ^3*b^4 + 16*a^5*c^2 - 8*a^4*b^2*c)))^(1/2)*1i + ((-(A^2*b^5 + B^2*a^2*b^3 + A^2*b^2*(-(4*a*c - b^2)^3)^(1/2) + B^2*a^2*(-(4*a*c - b^2)^3)^(1/2) - 2* A*B*a*b^4 - 16*A*B*a^3*c^2 - 7*A^2*a*b^3*c - A^2*a*c*(-(4*a*c - b^2)^3)^(1 /2) - 4*B^2*a^3*b*c + 12*A^2*a^2*b*c^2 - 2*A*B*a*b*(-(4*a*c - b^2)^3)^(1/2 ) + 12*A*B*a^2*b^2*c)/(2*(a^3*b^4 + 16*a^5*c^2 - 8*a^4*b^2*c)))^(1/2)*(...