Integrand size = 29, antiderivative size = 240 \[ \int \frac {(f+g x)^2}{(d+e x) \left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {2 \left (b^2 e f^2+2 a \left (a e g^2-c f (e f-2 d g)\right )-b \left (c d f^2+a g (2 e f+d g)\right )-\left (2 c^2 d f^2+b (b d-a e) g^2+c (2 a g (2 e f-d g)-b f (e f+2 d g))\right ) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt {a+b x+c x^2}}+\frac {(e f-d g)^2 \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{\left (c d^2-b d e+a e^2\right )^{3/2}} \]
(-d*g+e*f)^2*arctanh(1/2*(b*d-2*a*e+(-b*e+2*c*d)*x)/(a*e^2-b*d*e+c*d^2)^(1 /2)/(c*x^2+b*x+a)^(1/2))/(a*e^2-b*d*e+c*d^2)^(3/2)+2*(b^2*e*f^2+2*a*(a*e*g ^2-c*f*(-2*d*g+e*f))-b*(c*d*f^2+a*g*(d*g+2*e*f))-(2*c^2*d*f^2+b*(-a*e+b*d) *g^2+c*(2*a*g*(-d*g+2*e*f)-b*f*(2*d*g+e*f)))*x)/(-4*a*c+b^2)/(a*e^2-b*d*e+ c*d^2)/(c*x^2+b*x+a)^(1/2)
Time = 1.15 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.02 \[ \int \frac {(f+g x)^2}{(d+e x) \left (a+b x+c x^2\right )^{3/2}} \, dx=2 \left (\frac {-2 a^2 e g^2+2 c^2 d f^2 x-2 a c d g (2 f+g x)+2 a c e f (f+2 g x)+a b g (2 e f+d g-e g x)+b^2 \left (-e f^2+d g^2 x\right )+b c f (-e f x+d (f-2 g x))}{\left (b^2-4 a c\right ) \left (-c d^2+e (b d-a e)\right ) \sqrt {a+x (b+c x)}}+\frac {\sqrt {-c d^2+b d e-a e^2} (e f-d g)^2 \arctan \left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+x (b+c x)}}{\sqrt {-c d^2+e (b d-a e)}}\right )}{\left (c d^2+e (-b d+a e)\right )^2}\right ) \]
2*((-2*a^2*e*g^2 + 2*c^2*d*f^2*x - 2*a*c*d*g*(2*f + g*x) + 2*a*c*e*f*(f + 2*g*x) + a*b*g*(2*e*f + d*g - e*g*x) + b^2*(-(e*f^2) + d*g^2*x) + b*c*f*(- (e*f*x) + d*(f - 2*g*x)))/((b^2 - 4*a*c)*(-(c*d^2) + e*(b*d - a*e))*Sqrt[a + x*(b + c*x)]) + (Sqrt[-(c*d^2) + b*d*e - a*e^2]*(e*f - d*g)^2*ArcTan[(S qrt[c]*(d + e*x) - e*Sqrt[a + x*(b + c*x)])/Sqrt[-(c*d^2) + e*(b*d - a*e)] ])/(c*d^2 + e*(-(b*d) + a*e))^2)
Time = 0.41 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {1264, 27, 1154, 219}
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.
| \(\displaystyle \int \frac {(f+g x)^2}{(d+e x) \left (a+b x+c x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1264 |
| \(\displaystyle \frac {2 \left (-x \left (c (2 a g (2 e f-d g)-b f (2 d g+e f))+b g^2 (b d-a e)+2 c^2 d f^2\right )-b \left (a g (d g+2 e f)+c d f^2\right )+2 a \left (a e g^2-c f (e f-2 d g)\right )+b^2 e f^2\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}-\frac {2 \int -\frac {\left (b^2-4 a c\right ) (e f-d g)^2}{2 \left (c d^2-b e d+a e^2\right ) (d+e x) \sqrt {c x^2+b x+a}}dx}{b^2-4 a c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(e f-d g)^2 \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{a e^2-b d e+c d^2}+\frac {2 \left (-x \left (c (2 a g (2 e f-d g)-b f (2 d g+e f))+b g^2 (b d-a e)+2 c^2 d f^2\right )-b \left (a g (d g+2 e f)+c d f^2\right )+2 a \left (a e g^2-c f (e f-2 d g)\right )+b^2 e f^2\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {2 \left (-x \left (c (2 a g (2 e f-d g)-b f (2 d g+e f))+b g^2 (b d-a e)+2 c^2 d f^2\right )-b \left (a g (d g+2 e f)+c d f^2\right )+2 a \left (a e g^2-c f (e f-2 d g)\right )+b^2 e f^2\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}-\frac {2 (e f-d g)^2 \int \frac {1}{4 \left (c d^2-b e d+a e^2\right )-\frac {(b d-2 a e+(2 c d-b e) x)^2}{c x^2+b x+a}}d\left (-\frac {b d-2 a e+(2 c d-b e) x}{\sqrt {c x^2+b x+a}}\right )}{a e^2-b d e+c d^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {(e f-d g)^2 \text {arctanh}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{\left (a e^2-b d e+c d^2\right )^{3/2}}+\frac {2 \left (-x \left (c (2 a g (2 e f-d g)-b f (2 d g+e f))+b g^2 (b d-a e)+2 c^2 d f^2\right )-b \left (a g (d g+2 e f)+c d f^2\right )+2 a \left (a e g^2-c f (e f-2 d g)\right )+b^2 e f^2\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}\) |
(2*(b^2*e*f^2 + 2*a*(a*e*g^2 - c*f*(e*f - 2*d*g)) - b*(c*d*f^2 + a*g*(2*e* f + d*g)) - (2*c^2*d*f^2 + b*(b*d - a*e)*g^2 + c*(2*a*g*(2*e*f - d*g) - b* f*(e*f + 2*d*g)))*x))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*Sqrt[a + b*x + c*x^2]) + ((e*f - d*g)^2*ArcTanh[(b*d - 2*a*e + (2*c*d - b*e)*x)/(2*Sqrt [c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x + c*x^2])])/(c*d^2 - b*d*e + a*e^2)^( 3/2)
3.9.80.3.1 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[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
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[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_ ) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(d + e*x) ^m*(f + g*x)^n, a + b*x + c*x^2, x], R = Coeff[PolynomialRemainder[(d + e*x )^m*(f + g*x)^n, a + b*x + c*x^2, x], x, 0], S = Coeff[PolynomialRemainder[ (d + e*x)^m*(f + g*x)^n, a + b*x + c*x^2, x], x, 1]}, Simp[(b*R - 2*a*S + ( 2*c*R - b*S)*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + S imp[1/((p + 1)*(b^2 - 4*a*c)) Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*E xpandToSum[((p + 1)*(b^2 - 4*a*c)*Q)/(d + e*x)^m - ((2*p + 3)*(2*c*R - b*S) )/(d + e*x)^m, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[n, 1] && LtQ[p, -1] && ILtQ[m, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(554\) vs. \(2(230)=460\).
Time = 0.77 (sec) , antiderivative size = 555, normalized size of antiderivative = 2.31
| method | result | size |
| default | \(-\frac {g \left (\frac {2 d g \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\frac {4 e f \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-e g \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )\right )}{e^{2}}+\frac {\left (d^{2} g^{2}-2 d e f g +e^{2} f^{2}\right ) \left (\frac {e^{2}}{\left (e^{2} a -b d e +c \,d^{2}\right ) \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}-\frac {\left (b e -2 c d \right ) e \left (2 c \left (x +\frac {d}{e}\right )+\frac {b e -2 c d}{e}\right )}{\left (e^{2} a -b d e +c \,d^{2}\right ) \left (\frac {4 c \left (e^{2} a -b d e +c \,d^{2}\right )}{e^{2}}-\frac {\left (b e -2 c d \right )^{2}}{e^{2}}\right ) \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}-\frac {e^{2} \ln \left (\frac {\frac {2 e^{2} a -2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (e^{2} a -b d e +c \,d^{2}\right ) \sqrt {\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}\right )}{e^{3}}\) | \(555\) |
-g/e^2*(2*d*g*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)-4*e*f*(2*c*x+b)/(4 *a*c-b^2)/(c*x^2+b*x+a)^(1/2)-e*g*(-1/c/(c*x^2+b*x+a)^(1/2)-b/c*(2*c*x+b)/ (4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)))+(d^2*g^2-2*d*e*f*g+e^2*f^2)/e^3*(1/(a*e^ 2-b*d*e+c*d^2)*e^2/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/ e^2)^(1/2)-(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^2)*(2*c*(x+d/e)+(b*e-2*c*d)/e)/( 4*c*(a*e^2-b*d*e+c*d^2)/e^2-(b*e-2*c*d)^2/e^2)/((x+d/e)^2*c+(b*e-2*c*d)/e* (x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)-1/(a*e^2-b*d*e+c*d^2)*e^2/((a*e^2-b *d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x+d/e) +2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e ^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e)))
Leaf count of result is larger than twice the leaf count of optimal. 990 vs. \(2 (230) = 460\).
Time = 3.33 (sec) , antiderivative size = 2023, normalized size of antiderivative = 8.43 \[ \int \frac {(f+g x)^2}{(d+e x) \left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Too large to display} \]
[1/2*(((a*b^2 - 4*a^2*c)*e^2*f^2 - 2*(a*b^2 - 4*a^2*c)*d*e*f*g + (a*b^2 - 4*a^2*c)*d^2*g^2 + ((b^2*c - 4*a*c^2)*e^2*f^2 - 2*(b^2*c - 4*a*c^2)*d*e*f* g + (b^2*c - 4*a*c^2)*d^2*g^2)*x^2 + ((b^3 - 4*a*b*c)*e^2*f^2 - 2*(b^3 - 4 *a*b*c)*d*e*f*g + (b^3 - 4*a*b*c)*d^2*g^2)*x)*sqrt(c*d^2 - b*d*e + a*e^2)* log((8*a*b*d*e - 8*a^2*e^2 - (b^2 + 4*a*c)*d^2 - (8*c^2*d^2 - 8*b*c*d*e + (b^2 + 4*a*c)*e^2)*x^2 - 4*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c*x^2 + b*x + a)*(b*d - 2*a*e + (2*c*d - b*e)*x) - 2*(4*b*c*d^2 + 4*a*b*e^2 - (3*b^2 + 4 *a*c)*d*e)*x)/(e^2*x^2 + 2*d*e*x + d^2)) - 4*((b*c^2*d^3 - 2*(b^2*c - a*c^ 2)*d^2*e + (b^3 - a*b*c)*d*e^2 - (a*b^2 - 2*a^2*c)*e^3)*f^2 - 2*(2*a*c^2*d ^3 - 3*a*b*c*d^2*e - a^2*b*e^3 + (a*b^2 + 2*a^2*c)*d*e^2)*f*g + (a*b*c*d^3 + 3*a^2*b*d*e^2 - 2*a^3*e^3 - (a*b^2 + 2*a^2*c)*d^2*e)*g^2 + ((2*c^3*d^3 - 3*b*c^2*d^2*e - a*b*c*e^3 + (b^2*c + 2*a*c^2)*d*e^2)*f^2 - 2*(b*c^2*d^3 + 3*a*b*c*d*e^2 - 2*a^2*c*e^3 - (b^2*c + 2*a*c^2)*d^2*e)*f*g - (a^2*b*e^3 - (b^2*c - 2*a*c^2)*d^3 + (b^3 - a*b*c)*d^2*e - 2*(a*b^2 - a^2*c)*d*e^2)*g ^2)*x)*sqrt(c*x^2 + b*x + a))/((a*b^2*c^2 - 4*a^2*c^3)*d^4 - 2*(a*b^3*c - 4*a^2*b*c^2)*d^3*e + (a*b^4 - 2*a^2*b^2*c - 8*a^3*c^2)*d^2*e^2 - 2*(a^2*b^ 3 - 4*a^3*b*c)*d*e^3 + (a^3*b^2 - 4*a^4*c)*e^4 + ((b^2*c^3 - 4*a*c^4)*d^4 - 2*(b^3*c^2 - 4*a*b*c^3)*d^3*e + (b^4*c - 2*a*b^2*c^2 - 8*a^2*c^3)*d^2*e^ 2 - 2*(a*b^3*c - 4*a^2*b*c^2)*d*e^3 + (a^2*b^2*c - 4*a^3*c^2)*e^4)*x^2 + ( (b^3*c^2 - 4*a*b*c^3)*d^4 - 2*(b^4*c - 4*a*b^2*c^2)*d^3*e + (b^5 - 2*a*...
\[ \int \frac {(f+g x)^2}{(d+e x) \left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {\left (f + g x\right )^{2}}{\left (d + e x\right ) \left (a + b x + c x^{2}\right )^{\frac {3}{2}}}\, dx \]
Exception generated. \[ \int \frac {(f+g x)^2}{(d+e x) \left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume((b/e-(2*c*d)/e^2)^2>0)', see `as sume?` for
Leaf count of result is larger than twice the leaf count of optimal. 773 vs. \(2 (230) = 460\).
Time = 0.29 (sec) , antiderivative size = 773, normalized size of antiderivative = 3.22 \[ \int \frac {(f+g x)^2}{(d+e x) \left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {2 \, {\left (\frac {{\left (2 \, c^{3} d^{3} f^{2} - 3 \, b c^{2} d^{2} e f^{2} + b^{2} c d e^{2} f^{2} + 2 \, a c^{2} d e^{2} f^{2} - a b c e^{3} f^{2} - 2 \, b c^{2} d^{3} f g + 2 \, b^{2} c d^{2} e f g + 4 \, a c^{2} d^{2} e f g - 6 \, a b c d e^{2} f g + 4 \, a^{2} c e^{3} f g + b^{2} c d^{3} g^{2} - 2 \, a c^{2} d^{3} g^{2} - b^{3} d^{2} e g^{2} + a b c d^{2} e g^{2} + 2 \, a b^{2} d e^{2} g^{2} - 2 \, a^{2} c d e^{2} g^{2} - a^{2} b e^{3} g^{2}\right )} x}{b^{2} c^{2} d^{4} - 4 \, a c^{3} d^{4} - 2 \, b^{3} c d^{3} e + 8 \, a b c^{2} d^{3} e + b^{4} d^{2} e^{2} - 2 \, a b^{2} c d^{2} e^{2} - 8 \, a^{2} c^{2} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + 8 \, a^{2} b c d e^{3} + a^{2} b^{2} e^{4} - 4 \, a^{3} c e^{4}} + \frac {b c^{2} d^{3} f^{2} - 2 \, b^{2} c d^{2} e f^{2} + 2 \, a c^{2} d^{2} e f^{2} + b^{3} d e^{2} f^{2} - a b c d e^{2} f^{2} - a b^{2} e^{3} f^{2} + 2 \, a^{2} c e^{3} f^{2} - 4 \, a c^{2} d^{3} f g + 6 \, a b c d^{2} e f g - 2 \, a b^{2} d e^{2} f g - 4 \, a^{2} c d e^{2} f g + 2 \, a^{2} b e^{3} f g + a b c d^{3} g^{2} - a b^{2} d^{2} e g^{2} - 2 \, a^{2} c d^{2} e g^{2} + 3 \, a^{2} b d e^{2} g^{2} - 2 \, a^{3} e^{3} g^{2}}{b^{2} c^{2} d^{4} - 4 \, a c^{3} d^{4} - 2 \, b^{3} c d^{3} e + 8 \, a b c^{2} d^{3} e + b^{4} d^{2} e^{2} - 2 \, a b^{2} c d^{2} e^{2} - 8 \, a^{2} c^{2} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + 8 \, a^{2} b c d e^{3} + a^{2} b^{2} e^{4} - 4 \, a^{3} c e^{4}}\right )}}{\sqrt {c x^{2} + b x + a}} + \frac {2 \, {\left (e^{2} f^{2} - 2 \, d e f g + d^{2} g^{2}\right )} \arctan \left (-\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} + b d e - a e^{2}}}\right )}{{\left (c d^{2} - b d e + a e^{2}\right )} \sqrt {-c d^{2} + b d e - a e^{2}}} \]
-2*((2*c^3*d^3*f^2 - 3*b*c^2*d^2*e*f^2 + b^2*c*d*e^2*f^2 + 2*a*c^2*d*e^2*f ^2 - a*b*c*e^3*f^2 - 2*b*c^2*d^3*f*g + 2*b^2*c*d^2*e*f*g + 4*a*c^2*d^2*e*f *g - 6*a*b*c*d*e^2*f*g + 4*a^2*c*e^3*f*g + b^2*c*d^3*g^2 - 2*a*c^2*d^3*g^2 - b^3*d^2*e*g^2 + a*b*c*d^2*e*g^2 + 2*a*b^2*d*e^2*g^2 - 2*a^2*c*d*e^2*g^2 - a^2*b*e^3*g^2)*x/(b^2*c^2*d^4 - 4*a*c^3*d^4 - 2*b^3*c*d^3*e + 8*a*b*c^2 *d^3*e + b^4*d^2*e^2 - 2*a*b^2*c*d^2*e^2 - 8*a^2*c^2*d^2*e^2 - 2*a*b^3*d*e ^3 + 8*a^2*b*c*d*e^3 + a^2*b^2*e^4 - 4*a^3*c*e^4) + (b*c^2*d^3*f^2 - 2*b^2 *c*d^2*e*f^2 + 2*a*c^2*d^2*e*f^2 + b^3*d*e^2*f^2 - a*b*c*d*e^2*f^2 - a*b^2 *e^3*f^2 + 2*a^2*c*e^3*f^2 - 4*a*c^2*d^3*f*g + 6*a*b*c*d^2*e*f*g - 2*a*b^2 *d*e^2*f*g - 4*a^2*c*d*e^2*f*g + 2*a^2*b*e^3*f*g + a*b*c*d^3*g^2 - a*b^2*d ^2*e*g^2 - 2*a^2*c*d^2*e*g^2 + 3*a^2*b*d*e^2*g^2 - 2*a^3*e^3*g^2)/(b^2*c^2 *d^4 - 4*a*c^3*d^4 - 2*b^3*c*d^3*e + 8*a*b*c^2*d^3*e + b^4*d^2*e^2 - 2*a*b ^2*c*d^2*e^2 - 8*a^2*c^2*d^2*e^2 - 2*a*b^3*d*e^3 + 8*a^2*b*c*d*e^3 + a^2*b ^2*e^4 - 4*a^3*c*e^4))/sqrt(c*x^2 + b*x + a) + 2*(e^2*f^2 - 2*d*e*f*g + d^ 2*g^2)*arctan(-((sqrt(c)*x - sqrt(c*x^2 + b*x + a))*e + sqrt(c)*d)/sqrt(-c *d^2 + b*d*e - a*e^2))/((c*d^2 - b*d*e + a*e^2)*sqrt(-c*d^2 + b*d*e - a*e^ 2))
Timed out. \[ \int \frac {(f+g x)^2}{(d+e x) \left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {{\left (f+g\,x\right )}^2}{\left (d+e\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{3/2}} \,d x \]