Integrand size = 23, antiderivative size = 221 \[ \int \frac {\sqrt {x} (A+B x)}{a+b x+c x^2} \, dx=\frac {2 B \sqrt {x}}{c}-\frac {\sqrt {2} \left (b B-A c-\frac {b^2 B-A b c-2 a B c}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{c^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2} \left (b B-A c+\frac {b^2 B-A b c-2 a B c}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{c^{3/2} \sqrt {b+\sqrt {b^2-4 a c}}} \]
2*B*x^(1/2)/c-arctan(2^(1/2)*c^(1/2)*x^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2)) *2^(1/2)*(B*b-A*c+(A*b*c+2*B*a*c-B*b^2)/(-4*a*c+b^2)^(1/2))/c^(3/2)/(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)*(B*b-A*c+(-A*b*c-2*B*a*c+B*b^2)/(-4*a*c+b^2)^(1/2))/c^(3 /2)/(b+(-4*a*c+b^2)^(1/2))^(1/2)
Time = 0.49 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.19 \[ \int \frac {\sqrt {x} (A+B x)}{a+b x+c x^2} \, dx=\frac {2 B \sqrt {x}}{c}-\frac {\sqrt {2} \left (-b^2 B+A b c+2 a B c+b B \sqrt {b^2-4 a c}-A c \sqrt {b^2-4 a c}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{c^{3/2} \sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2} \left (b^2 B-A b c-2 a B c+b B \sqrt {b^2-4 a c}-A c \sqrt {b^2-4 a c}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{c^{3/2} \sqrt {b^2-4 a c} \sqrt {b+\sqrt {b^2-4 a c}}} \]
(2*B*Sqrt[x])/c - (Sqrt[2]*(-(b^2*B) + A*b*c + 2*a*B*c + b*B*Sqrt[b^2 - 4* a*c] - A*c*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b - Sq rt[b^2 - 4*a*c]]])/(c^(3/2)*Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (Sqrt[2]*(b^2*B - A*b*c - 2*a*B*c + b*B*Sqrt[b^2 - 4*a*c] - A*c*Sqrt[b^ 2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b + Sqrt[b^2 - 4*a*c]]]) /(c^(3/2)*Sqrt[b^2 - 4*a*c]*Sqrt[b + Sqrt[b^2 - 4*a*c]])
Time = 0.45 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.02, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {1196, 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.
| \(\displaystyle \int \frac {\sqrt {x} (A+B x)}{a+b x+c x^2} \, dx\) |
| \(\Big \downarrow \) 1196 |
| \(\displaystyle \frac {\int -\frac {a B+(b B-A c) x}{\sqrt {x} \left (c x^2+b x+a\right )}dx}{c}+\frac {2 B \sqrt {x}}{c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 B \sqrt {x}}{c}-\frac {\int \frac {a B+(b B-A c) x}{\sqrt {x} \left (c x^2+b x+a\right )}dx}{c}\) |
| \(\Big \downarrow \) 1197 |
| \(\displaystyle \frac {2 B \sqrt {x}}{c}-\frac {2 \int \frac {a B+(b B-A c) x}{c x^2+b x+a}d\sqrt {x}}{c}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {2 B \sqrt {x}}{c}-\frac {2 \left (\frac {1}{2} \left (-\frac {-2 a B c-A b c+b^2 B}{\sqrt {b^2-4 a c}}-A c+b B\right ) \int \frac {1}{\frac {1}{2} \left (b-\sqrt {b^2-4 a c}\right )+c x}d\sqrt {x}+\frac {1}{2} \left (\frac {-2 a B c-A b c+b^2 B}{\sqrt {b^2-4 a c}}-A c+b B\right ) \int \frac {1}{\frac {1}{2} \left (b+\sqrt {b^2-4 a c}\right )+c x}d\sqrt {x}\right )}{c}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {2 B \sqrt {x}}{c}-\frac {2 \left (\frac {\left (-\frac {-2 a B c-A b c+b^2 B}{\sqrt {b^2-4 a c}}-A c+b B\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} \sqrt {c} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\left (\frac {-2 a B c-A b c+b^2 B}{\sqrt {b^2-4 a c}}-A c+b B\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {2} \sqrt {c} \sqrt {\sqrt {b^2-4 a c}+b}}\right )}{c}\) |
(2*B*Sqrt[x])/c - (2*(((b*B - A*c - (b^2*B - A*b*c - 2*a*B*c)/Sqrt[b^2 - 4 *a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqr t[2]*Sqrt[c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + ((b*B - A*c + (b^2*B - A*b*c - 2*a*B*c)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b + Sqr t[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[c]*Sqrt[b + Sqrt[b^2 - 4*a*c]])))/c
3.11.11.3.1 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[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[g*((d + e*x)^m/(c*m)), x] + Simp[1/c Int [(d + e*x)^(m - 1)*(Simp[c*d*f - a*e*g + (g*c*d - b*e*g + c*e*f)*x, x]/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && FractionQ[m] & & GtQ[m, 0]
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_)^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 = 0.38 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.99
| method | result | size |
| derivativedivides | \(\frac {2 B \sqrt {x}}{c}-\frac {\left (A c \sqrt {-4 a c +b^{2}}-A b c -B b \sqrt {-4 a c +b^{2}}-2 B a c +B \,b^{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}+\frac {\left (A c \sqrt {-4 a c +b^{2}}+A b c -B b \sqrt {-4 a c +b^{2}}+2 B a c -B \,b^{2}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\) | \(218\) |
| default | \(\frac {2 B \sqrt {x}}{c}-\frac {\left (A c \sqrt {-4 a c +b^{2}}-A b c -B b \sqrt {-4 a c +b^{2}}-2 B a c +B \,b^{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}+\frac {\left (A c \sqrt {-4 a c +b^{2}}+A b c -B b \sqrt {-4 a c +b^{2}}+2 B a c -B \,b^{2}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\) | \(218\) |
| risch | \(\frac {2 B \sqrt {x}}{c}-\frac {\left (A c \sqrt {-4 a c +b^{2}}-A b c -B b \sqrt {-4 a c +b^{2}}-2 B a c +B \,b^{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}+\frac {\left (A c \sqrt {-4 a c +b^{2}}+A b c -B b \sqrt {-4 a c +b^{2}}+2 B a c -B \,b^{2}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\) | \(218\) |
2*B*x^(1/2)/c-(A*c*(-4*a*c+b^2)^(1/2)-A*b*c-B*b*(-4*a*c+b^2)^(1/2)-2*B*a*c +B*b^2)/c/(-4*a*c+b^2)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arc tanh(c*x^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))+(A*c*(-4*a*c+b^2 )^(1/2)+A*b*c-B*b*(-4*a*c+b^2)^(1/2)+2*B*a*c-B*b^2)/c/(-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))
Leaf count of result is larger than twice the leaf count of optimal. 2642 vs. \(2 (179) = 358\).
Time = 1.72 (sec) , antiderivative size = 2642, normalized size of antiderivative = 11.95 \[ \int \frac {\sqrt {x} (A+B x)}{a+b x+c x^2} \, dx=\text {Too large to display} \]
1/2*(sqrt(2)*c*sqrt(-(B^2*b^3 + (4*A*B*a + A^2*b)*c^2 - (3*B^2*a*b + 2*A*B *b^2)*c + (b^2*c^3 - 4*a*c^4)*sqrt((B^4*b^4 + A^4*c^4 - 2*(A^2*B^2*a + 2*A ^3*B*b)*c^3 + (B^4*a^2 + 4*A*B^3*a*b + 6*A^2*B^2*b^2)*c^2 - 2*(B^4*a*b^2 + 2*A*B^3*b^3)*c)/(b^2*c^6 - 4*a*c^7)))/(b^2*c^3 - 4*a*c^4))*log(sqrt(2)*(B ^3*b^4 - 4*A^2*B*a*c^3 + (4*B^3*a^2 + 8*A*B^2*a*b + A^2*B*b^2)*c^2 - (5*B^ 3*a*b^2 + 2*A*B^2*b^3)*c - (B*b^3*c^3 + 8*A*a*c^5 - 2*(2*B*a*b + A*b^2)*c^ 4)*sqrt((B^4*b^4 + A^4*c^4 - 2*(A^2*B^2*a + 2*A^3*B*b)*c^3 + (B^4*a^2 + 4* A*B^3*a*b + 6*A^2*B^2*b^2)*c^2 - 2*(B^4*a*b^2 + 2*A*B^3*b^3)*c)/(b^2*c^6 - 4*a*c^7)))*sqrt(-(B^2*b^3 + (4*A*B*a + A^2*b)*c^2 - (3*B^2*a*b + 2*A*B*b^ 2)*c + (b^2*c^3 - 4*a*c^4)*sqrt((B^4*b^4 + A^4*c^4 - 2*(A^2*B^2*a + 2*A^3* B*b)*c^3 + (B^4*a^2 + 4*A*B^3*a*b + 6*A^2*B^2*b^2)*c^2 - 2*(B^4*a*b^2 + 2* A*B^3*b^3)*c)/(b^2*c^6 - 4*a*c^7)))/(b^2*c^3 - 4*a*c^4)) + 4*(B^4*a*b^2 - A*B^3*b^3 - 3*A^3*B*b*c^2 + A^4*c^3 - (B^4*a^2 + A*B^3*a*b - 3*A^2*B^2*b^2 )*c)*sqrt(x)) - sqrt(2)*c*sqrt(-(B^2*b^3 + (4*A*B*a + A^2*b)*c^2 - (3*B^2* a*b + 2*A*B*b^2)*c + (b^2*c^3 - 4*a*c^4)*sqrt((B^4*b^4 + A^4*c^4 - 2*(A^2* B^2*a + 2*A^3*B*b)*c^3 + (B^4*a^2 + 4*A*B^3*a*b + 6*A^2*B^2*b^2)*c^2 - 2*( B^4*a*b^2 + 2*A*B^3*b^3)*c)/(b^2*c^6 - 4*a*c^7)))/(b^2*c^3 - 4*a*c^4))*log (-sqrt(2)*(B^3*b^4 - 4*A^2*B*a*c^3 + (4*B^3*a^2 + 8*A*B^2*a*b + A^2*B*b^2) *c^2 - (5*B^3*a*b^2 + 2*A*B^2*b^3)*c - (B*b^3*c^3 + 8*A*a*c^5 - 2*(2*B*a*b + A*b^2)*c^4)*sqrt((B^4*b^4 + A^4*c^4 - 2*(A^2*B^2*a + 2*A^3*B*b)*c^3 ...
Leaf count of result is larger than twice the leaf count of optimal. 13942 vs. \(2 (202) = 404\).
Time = 6.35 (sec) , antiderivative size = 13942, normalized size of antiderivative = 63.09 \[ \int \frac {\sqrt {x} (A+B x)}{a+b x+c x^2} \, dx=\text {Too large to display} \]
Piecewise((-A*a*log(sqrt(x) - sqrt(-a/b))/(b**2*sqrt(-a/b)) + A*a*log(sqrt (x) + sqrt(-a/b))/(b**2*sqrt(-a/b)) + 2*A*sqrt(x)/b + B*a**2*log(sqrt(x) - sqrt(-a/b))/(b**3*sqrt(-a/b)) - B*a**2*log(sqrt(x) + sqrt(-a/b))/(b**3*sq rt(-a/b)) - 2*B*a*sqrt(x)/b**2 + 2*B*x**(3/2)/(3*b), Eq(c, 0)), (A*log(sqr t(x) - sqrt(-b/c))/(c*sqrt(-b/c)) - A*log(sqrt(x) + sqrt(-b/c))/(c*sqrt(-b /c)) - B*b*log(sqrt(x) - sqrt(-b/c))/(c**2*sqrt(-b/c)) + B*b*log(sqrt(x) + sqrt(-b/c))/(c**2*sqrt(-b/c)) + 2*B*sqrt(x)/c, Eq(a, 0)), (2*sqrt(2)*A*b* c*log(sqrt(x) - sqrt(2)*sqrt(-b/c)/2)/(4*b*c**2*sqrt(-b/c) + 8*c**3*x*sqrt (-b/c)) - 2*sqrt(2)*A*b*c*log(sqrt(x) + sqrt(2)*sqrt(-b/c)/2)/(4*b*c**2*sq rt(-b/c) + 8*c**3*x*sqrt(-b/c)) - 8*A*c**2*sqrt(x)*sqrt(-b/c)/(4*b*c**2*sq rt(-b/c) + 8*c**3*x*sqrt(-b/c)) + 4*sqrt(2)*A*c**2*x*log(sqrt(x) - sqrt(2) *sqrt(-b/c)/2)/(4*b*c**2*sqrt(-b/c) + 8*c**3*x*sqrt(-b/c)) - 4*sqrt(2)*A*c **2*x*log(sqrt(x) + sqrt(2)*sqrt(-b/c)/2)/(4*b*c**2*sqrt(-b/c) + 8*c**3*x* sqrt(-b/c)) - 3*sqrt(2)*B*b**2*log(sqrt(x) - sqrt(2)*sqrt(-b/c)/2)/(4*b*c* *2*sqrt(-b/c) + 8*c**3*x*sqrt(-b/c)) + 3*sqrt(2)*B*b**2*log(sqrt(x) + sqrt (2)*sqrt(-b/c)/2)/(4*b*c**2*sqrt(-b/c) + 8*c**3*x*sqrt(-b/c)) + 12*B*b*c*s qrt(x)*sqrt(-b/c)/(4*b*c**2*sqrt(-b/c) + 8*c**3*x*sqrt(-b/c)) - 6*sqrt(2)* B*b*c*x*log(sqrt(x) - sqrt(2)*sqrt(-b/c)/2)/(4*b*c**2*sqrt(-b/c) + 8*c**3* x*sqrt(-b/c)) + 6*sqrt(2)*B*b*c*x*log(sqrt(x) + sqrt(2)*sqrt(-b/c)/2)/(4*b *c**2*sqrt(-b/c) + 8*c**3*x*sqrt(-b/c)) + 16*B*c**2*x**(3/2)*sqrt(-b/c)...
\[ \int \frac {\sqrt {x} (A+B x)}{a+b x+c x^2} \, dx=\int { \frac {{\left (B x + A\right )} \sqrt {x}}{c x^{2} + b x + a} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 3186 vs. \(2 (179) = 358\).
Time = 0.78 (sec) , antiderivative size = 3186, normalized size of antiderivative = 14.42 \[ \int \frac {\sqrt {x} (A+B x)}{a+b x+c x^2} \, dx=\text {Too large to display} \]
2*B*sqrt(x)/c + 1/4*((2*b^4*c^3 - 16*a*b^2*c^4 + 32*a^2*c^5 - sqrt(2)*sqrt (b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^4*c + 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c^2 + 2*sqrt(2)*sqrt(b^2 - 4 *a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^3*c^2 - 16*sqrt(2)*sqrt(b^2 - 4*a* c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*c^3 - 8*sqrt(2)*sqrt(b^2 - 4*a*c)*s qrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^3 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b* c + sqrt(b^2 - 4*a*c)*c)*b^2*c^3 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*c^4 - 2*(b^2 - 4*a*c)*b^2*c^3 + 8*(b^2 - 4*a*c)*a*c ^4)*A*c^2 - (2*b^5*c^2 - 16*a*b^3*c^3 + 32*a^2*b*c^4 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^5 + 8*sqrt(2)*sqrt(b^2 - 4*a*c)*s qrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^3*c + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt( b*c + sqrt(b^2 - 4*a*c)*c)*b^4*c - 16*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b*c^2 - 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + s qrt(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^2 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^3 - 2*(b^2 - 4*a*c)*b^3*c^2 + 8*(b^2 - 4*a*c)*a*b*c^3)*B *c^2 - 2*(sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^4*c^2 - 8*sqrt(2)*sq rt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^3 - 2*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^3*c^3 - 2*a*b^4*c^3 + 16*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a *c)*c)*a^3*c^4 + 8*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b*c^4 + ...
Time = 11.13 (sec) , antiderivative size = 6401, normalized size of antiderivative = 28.96 \[ \int \frac {\sqrt {x} (A+B x)}{a+b x+c x^2} \, dx=\text {Too large to display} \]
(2*B*x^(1/2))/c - atan(((((8*(4*B*a^2*c^3 - B*a*b^2*c^2))/c - (8*x^(1/2)*( b^3*c^3 - 4*a*b*c^4)*(-(B^2*b^5 + A^2*b^3*c^2 - A^2*c^2*(-(4*a*c - b^2)^3) ^(1/2) - B^2*b^2*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*b^4*c - 16*A*B*a^2*c^3 - 4*A^2*a*b*c^3 - 7*B^2*a*b^3*c + B^2*a*c*(-(4*a*c - b^2)^3)^(1/2) + 12*B^2 *a^2*b*c^2 + 2*A*B*b*c*(-(4*a*c - b^2)^3)^(1/2) + 12*A*B*a*b^2*c^2)/(2*(16 *a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2))/c)*(-(B^2*b^5 + A^2*b^3*c^2 - A ^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - B^2*b^2*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B *b^4*c - 16*A*B*a^2*c^3 - 4*A^2*a*b*c^3 - 7*B^2*a*b^3*c + B^2*a*c*(-(4*a*c - b^2)^3)^(1/2) + 12*B^2*a^2*b*c^2 + 2*A*B*b*c*(-(4*a*c - b^2)^3)^(1/2) + 12*A*B*a*b^2*c^2)/(2*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2) - (8*x^ (1/2)*(B^2*b^4 - 2*A^2*a*c^3 + A^2*b^2*c^2 + 2*B^2*a^2*c^2 - 2*A*B*b^3*c - 4*B^2*a*b^2*c + 6*A*B*a*b*c^2))/c)*(-(B^2*b^5 + A^2*b^3*c^2 - A^2*c^2*(-( 4*a*c - b^2)^3)^(1/2) - B^2*b^2*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*b^4*c - 1 6*A*B*a^2*c^3 - 4*A^2*a*b*c^3 - 7*B^2*a*b^3*c + B^2*a*c*(-(4*a*c - b^2)^3) ^(1/2) + 12*B^2*a^2*b*c^2 + 2*A*B*b*c*(-(4*a*c - b^2)^3)^(1/2) + 12*A*B*a* b^2*c^2)/(2*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2)*1i - (((8*(4*B*a^ 2*c^3 - B*a*b^2*c^2))/c + (8*x^(1/2)*(b^3*c^3 - 4*a*b*c^4)*(-(B^2*b^5 + A^ 2*b^3*c^2 - A^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - B^2*b^2*(-(4*a*c - b^2)^3)^ (1/2) - 2*A*B*b^4*c - 16*A*B*a^2*c^3 - 4*A^2*a*b*c^3 - 7*B^2*a*b^3*c + B^2 *a*c*(-(4*a*c - b^2)^3)^(1/2) + 12*B^2*a^2*b*c^2 + 2*A*B*b*c*(-(4*a*c -...