Integrand size = 32, antiderivative size = 289 \[ \int \frac {(g+h x)^2 \left (d+e x+f x^2\right )}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {2 \left (c \left (2 a e-b \left (d+\frac {a f}{c}\right )\right )-\left (2 c^2 d-b c e+b^2 f-2 a c f\right ) x\right ) (g+h x)^2}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {h \left (32 c^3 d g-15 b^3 f h-8 c^2 (2 b e g+8 a f g+b d h+4 a e h)+4 b c (6 b f g+3 b e h+13 a f h)+2 c \left (8 c^2 d-4 b c e+5 b^2 f-12 a c f\right ) h x\right ) \sqrt {a+b x+c x^2}}{4 c^3 \left (b^2-4 a c\right )}+\frac {\left (15 b^2 f h^2-12 c h (2 b f g+b e h+a f h)+8 c^2 \left (f g^2+h (2 e g+d h)\right )\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 c^{7/2}} \]
1/8*(15*b^2*f*h^2-12*c*h*(a*f*h+b*e*h+2*b*f*g)+8*c^2*(f*g^2+h*(d*h+2*e*g)) )*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(7/2)+2*(c*(2*a*e-b *(d+a*f/c))-(-2*a*c*f+b^2*f-b*c*e+2*c^2*d)*x)*(h*x+g)^2/c/(-4*a*c+b^2)/(c* x^2+b*x+a)^(1/2)+1/4*h*(32*c^3*d*g-15*b^3*f*h-8*c^2*(4*a*e*h+8*a*f*g+b*d*h +2*b*e*g)+4*b*c*(13*a*f*h+3*b*e*h+6*b*f*g)+2*c*(-12*a*c*f+5*b^2*f-4*b*c*e+ 8*c^2*d)*h*x)*(c*x^2+b*x+a)^(1/2)/c^3/(-4*a*c+b^2)
Time = 2.45 (sec) , antiderivative size = 390, normalized size of antiderivative = 1.35 \[ \int \frac {(g+h x)^2 \left (d+e x+f x^2\right )}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {-\frac {\sqrt {c} \left (15 b^4 f h^2 x+b^3 h (15 a f h+c x (-24 f g-12 e h+5 f h x))+4 b c \left (-13 a^2 f h^2+2 c^2 g (-e g x+d (g-2 h x))+a c \left (2 h (2 e g+d h+5 e h x)+f \left (2 g^2+20 g h x-5 h^2 x^2\right )\right )\right )-2 b^2 c \left (a h (12 f g+6 e h+31 f h x)+c x \left (2 h (-4 e g-2 d h+e h x)+f \left (-4 g^2+4 g h x+h^2 x^2\right )\right )\right )+8 c^2 \left (2 c^2 d g^2 x+a^2 h (8 f g+4 e h+3 f h x)+a c \left (-2 d h (2 g+h x)-2 e \left (g^2+2 g h x-h^2 x^2\right )+f x \left (-2 g^2+4 g h x+h^2 x^2\right )\right )\right )\right )}{\left (b^2-4 a c\right ) \sqrt {a+x (b+c x)}}+\left (15 b^2 f h^2-12 c h (2 b f g+b e h+a f h)+8 c^2 \left (f g^2+h (2 e g+d h)\right )\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{-\sqrt {a}+\sqrt {a+x (b+c x)}}\right )}{4 c^{7/2}} \]
(-((Sqrt[c]*(15*b^4*f*h^2*x + b^3*h*(15*a*f*h + c*x*(-24*f*g - 12*e*h + 5* f*h*x)) + 4*b*c*(-13*a^2*f*h^2 + 2*c^2*g*(-(e*g*x) + d*(g - 2*h*x)) + a*c* (2*h*(2*e*g + d*h + 5*e*h*x) + f*(2*g^2 + 20*g*h*x - 5*h^2*x^2))) - 2*b^2* c*(a*h*(12*f*g + 6*e*h + 31*f*h*x) + c*x*(2*h*(-4*e*g - 2*d*h + e*h*x) + f *(-4*g^2 + 4*g*h*x + h^2*x^2))) + 8*c^2*(2*c^2*d*g^2*x + a^2*h*(8*f*g + 4* e*h + 3*f*h*x) + a*c*(-2*d*h*(2*g + h*x) - 2*e*(g^2 + 2*g*h*x - h^2*x^2) + f*x*(-2*g^2 + 4*g*h*x + h^2*x^2)))))/((b^2 - 4*a*c)*Sqrt[a + x*(b + c*x)] )) + (15*b^2*f*h^2 - 12*c*h*(2*b*f*g + b*e*h + a*f*h) + 8*c^2*(f*g^2 + h*( 2*e*g + d*h)))*ArcTanh[(Sqrt[c]*x)/(-Sqrt[a] + Sqrt[a + x*(b + c*x)])])/(4 *c^(7/2))
Time = 0.60 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {2175, 27, 1225, 1092, 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.
| \(\displaystyle \int \frac {(g+h x)^2 \left (d+e x+f x^2\right )}{\left (a+b x+c x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 2175 |
| \(\displaystyle \frac {2 (g+h x)^2 \left (c \left (2 a e-b \left (\frac {a f}{c}+d\right )\right )-x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {2 \int -\frac {(g+h x) \left (f g b^2+4 (c d+a f) h b-4 a c (f g+2 e h)+c \left (\frac {5 f b^2}{c}+8 c d-4 (b e+3 a f)\right ) h x\right )}{2 c \sqrt {c x^2+b x+a}}dx}{b^2-4 a c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(g+h x) \left (f g b^2+4 (c d+a f) h b-4 a c (f g+2 e h)+\left (5 f b^2-4 c e b+8 c^2 d-12 a c f\right ) h x\right )}{\sqrt {c x^2+b x+a}}dx}{c \left (b^2-4 a c\right )}+\frac {2 (g+h x)^2 \left (c \left (2 a e-b \left (\frac {a f}{c}+d\right )\right )-x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\) |
| \(\Big \downarrow \) 1225 |
| \(\displaystyle \frac {\frac {\left (b^2-4 a c\right ) \left (-12 c h (a f h+b e h+2 b f g)+15 b^2 f h^2+8 c^2 \left (h (d h+2 e g)+f g^2\right )\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{8 c^2}+\frac {h \sqrt {a+b x+c x^2} \left (2 c h x \left (-12 a c f+5 b^2 f-4 b c e+8 c^2 d\right )-8 c^2 (4 a e h+8 a f g+b d h+2 b e g)+4 b c (13 a f h+3 b e h+6 b f g)-15 b^3 f h+32 c^3 d g\right )}{4 c^2}}{c \left (b^2-4 a c\right )}+\frac {2 (g+h x)^2 \left (c \left (2 a e-b \left (\frac {a f}{c}+d\right )\right )-x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {\frac {\left (b^2-4 a c\right ) \left (-12 c h (a f h+b e h+2 b f g)+15 b^2 f h^2+8 c^2 \left (h (d h+2 e g)+f g^2\right )\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{4 c^2}+\frac {h \sqrt {a+b x+c x^2} \left (2 c h x \left (-12 a c f+5 b^2 f-4 b c e+8 c^2 d\right )-8 c^2 (4 a e h+8 a f g+b d h+2 b e g)+4 b c (13 a f h+3 b e h+6 b f g)-15 b^3 f h+32 c^3 d g\right )}{4 c^2}}{c \left (b^2-4 a c\right )}+\frac {2 (g+h x)^2 \left (c \left (2 a e-b \left (\frac {a f}{c}+d\right )\right )-x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-12 c h (a f h+b e h+2 b f g)+15 b^2 f h^2+8 c^2 \left (h (d h+2 e g)+f g^2\right )\right )}{8 c^{5/2}}+\frac {h \sqrt {a+b x+c x^2} \left (2 c h x \left (-12 a c f+5 b^2 f-4 b c e+8 c^2 d\right )-8 c^2 (4 a e h+8 a f g+b d h+2 b e g)+4 b c (13 a f h+3 b e h+6 b f g)-15 b^3 f h+32 c^3 d g\right )}{4 c^2}}{c \left (b^2-4 a c\right )}+\frac {2 (g+h x)^2 \left (c \left (2 a e-b \left (\frac {a f}{c}+d\right )\right )-x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\) |
(2*(c*(2*a*e - b*(d + (a*f)/c)) - (2*c^2*d - b*c*e + b^2*f - 2*a*c*f)*x)*( g + h*x)^2)/(c*(b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2]) + ((h*(32*c^3*d*g - 15 *b^3*f*h - 8*c^2*(2*b*e*g + 8*a*f*g + b*d*h + 4*a*e*h) + 4*b*c*(6*b*f*g + 3*b*e*h + 13*a*f*h) + 2*c*(8*c^2*d - 4*b*c*e + 5*b^2*f - 12*a*c*f)*h*x)*Sq rt[a + b*x + c*x^2])/(4*c^2) + ((b^2 - 4*a*c)*(15*b^2*f*h^2 - 12*c*h*(2*b* f*g + b*e*h + a*f*h) + 8*c^2*(f*g^2 + h*(2*e*g + d*h)))*ArcTanh[(b + 2*c*x )/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(8*c^(5/2)))/(c*(b^2 - 4*a*c))
3.3.34.3.1 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/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[I nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a , b, c}, x]
Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*( x_)^2)^(p_), x_Symbol] :> Simp[(-(b*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 2*c*e*g*(p + 1)*x))*((a + b*x + c*x^2)^(p + 1)/(2*c^2*(p + 1)*(2*p + 3))), x] + Simp[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(2*c^2*(2*p + 3)) Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c , d, e, f, g, p}, x] && !LeQ[p, -1]
Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p _), x_Symbol] :> With[{Qx = PolynomialQuotient[Pq, a + b*x + c*x^2, x], R = Coeff[PolynomialRemainder[Pq, a + b*x + c*x^2, x], x, 0], S = Coeff[Polyno mialRemainder[Pq, a + b*x + c*x^2, x], x, 1]}, Simp[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*((R*b - 2*a*S + (2*c*R - b*S)*x)/((p + 1)*(b^2 - 4*a*c))), x ] + Simp[1/((p + 1)*(b^2 - 4*a*c)) Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2 )^(p + 1)*ExpandToSum[(p + 1)*(b^2 - 4*a*c)*(d + e*x)*Qx + S*(2*a*e*m + b*d *(2*p + 3)) - R*(b*e*m + 2*c*d*(2*p + 3)) - e*(2*c*R - b*S)*(m + 2*p + 3)*x , x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a *c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 0] && (Inte gerQ[p] || !IntegerQ[m] || !RationalQ[a, b, c, d, e]) && !(IGtQ[m, 0] && RationalQ[a, b, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
Leaf count of result is larger than twice the leaf count of optimal. \(547\) vs. \(2(273)=546\).
Time = 0.96 (sec) , antiderivative size = 548, normalized size of antiderivative = 1.90
| method | result | size |
| risch | \(-\frac {h \left (-2 c f h x +7 b f h -4 e h c -8 c f g \right ) \sqrt {c \,x^{2}+b x +a}}{4 c^{3}}-\frac {-\frac {16 c^{3} d \,g^{2} \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\frac {8 a^{2} c f \,h^{2} \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\frac {14 a \,b^{2} f \,h^{2} \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\frac {8 a b c e \,h^{2} \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\frac {16 a b c f g h \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\left (12 a \,c^{2} f \,h^{2}-15 b^{2} c f \,h^{2}+12 b \,c^{2} e \,h^{2}+24 b \,c^{2} f g h -8 c^{3} d \,h^{2}-16 c^{3} e g h -8 c^{3} f \,g^{2}\right ) \left (-\frac {x}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}\right )+\left (-4 a b c f \,h^{2}+8 a \,c^{2} e \,h^{2}+16 a \,c^{2} f g h -7 b^{3} f \,h^{2}+4 b^{2} c e \,h^{2}+8 b^{2} c f g h -16 c^{3} d g h -8 c^{3} e \,g^{2}\right ) \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{8 c^{3}}\) | \(548\) |
| default | \(\frac {2 d \,g^{2} \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+f \,h^{2} \left (\frac {x^{3}}{2 c \sqrt {c \,x^{2}+b x +a}}-\frac {5 b \left (\frac {x^{2}}{c \sqrt {c \,x^{2}+b x +a}}-\frac {3 b \left (-\frac {x}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \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 )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}\right )}{2 c}-\frac {2 a \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 )}{c}\right )}{4 c}-\frac {3 a \left (-\frac {x}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \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 )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}\right )}{2 c}\right )+\left (e \,h^{2}+2 f g h \right ) \left (\frac {x^{2}}{c \sqrt {c \,x^{2}+b x +a}}-\frac {3 b \left (-\frac {x}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \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 )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}\right )}{2 c}-\frac {2 a \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 )}{c}\right )+\left (2 d g h +g^{2} e \right ) \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 )+\left (d \,h^{2}+2 e g h +f \,g^{2}\right ) \left (-\frac {x}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \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 )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}\right )\) | \(775\) |
-1/4*h*(-2*c*f*h*x+7*b*f*h-4*c*e*h-8*c*f*g)*(c*x^2+b*x+a)^(1/2)/c^3-1/8/c^ 3*(-16*c^3*d*g^2*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+8*a^2*c*f*h^2*( 2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)-14*a*b^2*f*h^2*(2*c*x+b)/(4*a*c-b ^2)/(c*x^2+b*x+a)^(1/2)+8*a*b*c*e*h^2*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^ (1/2)+16*a*b*c*f*g*h*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+(12*a*c^2*f *h^2-15*b^2*c*f*h^2+12*b*c^2*e*h^2+24*b*c^2*f*g*h-8*c^3*d*h^2-16*c^3*e*g*h -8*c^3*f*g^2)*(-x/c/(c*x^2+b*x+a)^(1/2)-1/2*b/c*(-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))+1/c^(3/2)*ln((1/2*b+c*x)/c^ (1/2)+(c*x^2+b*x+a)^(1/2)))+(-4*a*b*c*f*h^2+8*a*c^2*e*h^2+16*a*c^2*f*g*h-7 *b^3*f*h^2+4*b^2*c*e*h^2+8*b^2*c*f*g*h-16*c^3*d*g*h-8*c^3*e*g^2)*(-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)))
Leaf count of result is larger than twice the leaf count of optimal. 883 vs. \(2 (271) = 542\).
Time = 5.98 (sec) , antiderivative size = 1769, normalized size of antiderivative = 6.12 \[ \int \frac {(g+h x)^2 \left (d+e x+f x^2\right )}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Too large to display} \]
[-1/16*((8*(a*b^2*c^2 - 4*a^2*c^3)*f*g^2 + 8*(2*(a*b^2*c^2 - 4*a^2*c^3)*e - 3*(a*b^3*c - 4*a^2*b*c^2)*f)*g*h + (8*(a*b^2*c^2 - 4*a^2*c^3)*d - 12*(a* b^3*c - 4*a^2*b*c^2)*e + 3*(5*a*b^4 - 24*a^2*b^2*c + 16*a^3*c^2)*f)*h^2 + (8*(b^2*c^3 - 4*a*c^4)*f*g^2 + 8*(2*(b^2*c^3 - 4*a*c^4)*e - 3*(b^3*c^2 - 4 *a*b*c^3)*f)*g*h + (8*(b^2*c^3 - 4*a*c^4)*d - 12*(b^3*c^2 - 4*a*b*c^3)*e + 3*(5*b^4*c - 24*a*b^2*c^2 + 16*a^2*c^3)*f)*h^2)*x^2 + (8*(b^3*c^2 - 4*a*b *c^3)*f*g^2 + 8*(2*(b^3*c^2 - 4*a*b*c^3)*e - 3*(b^4*c - 4*a*b^2*c^2)*f)*g* h + (8*(b^3*c^2 - 4*a*b*c^3)*d - 12*(b^4*c - 4*a*b^2*c^2)*e + 3*(5*b^5 - 2 4*a*b^3*c + 16*a^2*b*c^2)*f)*h^2)*x)*sqrt(c)*log(-8*c^2*x^2 - 8*b*c*x - b^ 2 + 4*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(c) - 4*a*c) - 4*(2*(b^2*c^3 - 4*a*c^4)*f*h^2*x^3 - 8*(b*c^4*d - 2*a*c^4*e + a*b*c^3*f)*g^2 + 8*(4*a*c^4 *d - 2*a*b*c^3*e + (3*a*b^2*c^2 - 8*a^2*c^3)*f)*g*h - (8*a*b*c^3*d - 4*(3* a*b^2*c^2 - 8*a^2*c^3)*e + (15*a*b^3*c - 52*a^2*b*c^2)*f)*h^2 + (8*(b^2*c^ 3 - 4*a*c^4)*f*g*h + (4*(b^2*c^3 - 4*a*c^4)*e - 5*(b^3*c^2 - 4*a*b*c^3)*f) *h^2)*x^2 - (8*(2*c^5*d - b*c^4*e + (b^2*c^3 - 2*a*c^4)*f)*g^2 - 8*(2*b*c^ 4*d - 2*(b^2*c^3 - 2*a*c^4)*e + (3*b^3*c^2 - 10*a*b*c^3)*f)*g*h + (8*(b^2* c^3 - 2*a*c^4)*d - 4*(3*b^3*c^2 - 10*a*b*c^3)*e + (15*b^4*c - 62*a*b^2*c^2 + 24*a^2*c^3)*f)*h^2)*x)*sqrt(c*x^2 + b*x + a))/(a*b^2*c^4 - 4*a^2*c^5 + (b^2*c^5 - 4*a*c^6)*x^2 + (b^3*c^4 - 4*a*b*c^5)*x), -1/8*((8*(a*b^2*c^2 - 4*a^2*c^3)*f*g^2 + 8*(2*(a*b^2*c^2 - 4*a^2*c^3)*e - 3*(a*b^3*c - 4*a^2*...
\[ \int \frac {(g+h x)^2 \left (d+e x+f x^2\right )}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {\left (g + h x\right )^{2} \left (d + e x + f x^{2}\right )}{\left (a + b x + c x^{2}\right )^{\frac {3}{2}}}\, dx \]
Exception generated. \[ \int \frac {(g+h x)^2 \left (d+e x+f x^2\right )}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for more deta
Leaf count of result is larger than twice the leaf count of optimal. 565 vs. \(2 (271) = 542\).
Time = 0.29 (sec) , antiderivative size = 565, normalized size of antiderivative = 1.96 \[ \int \frac {(g+h x)^2 \left (d+e x+f x^2\right )}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {{\left ({\left (\frac {2 \, {\left (b^{2} c^{2} f h^{2} - 4 \, a c^{3} f h^{2}\right )} x}{b^{2} c^{3} - 4 \, a c^{4}} + \frac {8 \, b^{2} c^{2} f g h - 32 \, a c^{3} f g h + 4 \, b^{2} c^{2} e h^{2} - 16 \, a c^{3} e h^{2} - 5 \, b^{3} c f h^{2} + 20 \, a b c^{2} f h^{2}}{b^{2} c^{3} - 4 \, a c^{4}}\right )} x - \frac {16 \, c^{4} d g^{2} - 8 \, b c^{3} e g^{2} + 8 \, b^{2} c^{2} f g^{2} - 16 \, a c^{3} f g^{2} - 16 \, b c^{3} d g h + 16 \, b^{2} c^{2} e g h - 32 \, a c^{3} e g h - 24 \, b^{3} c f g h + 80 \, a b c^{2} f g h + 8 \, b^{2} c^{2} d h^{2} - 16 \, a c^{3} d h^{2} - 12 \, b^{3} c e h^{2} + 40 \, a b c^{2} e h^{2} + 15 \, b^{4} f h^{2} - 62 \, a b^{2} c f h^{2} + 24 \, a^{2} c^{2} f h^{2}}{b^{2} c^{3} - 4 \, a c^{4}}\right )} x - \frac {8 \, b c^{3} d g^{2} - 16 \, a c^{3} e g^{2} + 8 \, a b c^{2} f g^{2} - 32 \, a c^{3} d g h + 16 \, a b c^{2} e g h - 24 \, a b^{2} c f g h + 64 \, a^{2} c^{2} f g h + 8 \, a b c^{2} d h^{2} - 12 \, a b^{2} c e h^{2} + 32 \, a^{2} c^{2} e h^{2} + 15 \, a b^{3} f h^{2} - 52 \, a^{2} b c f h^{2}}{b^{2} c^{3} - 4 \, a c^{4}}}{4 \, \sqrt {c x^{2} + b x + a}} - \frac {{\left (8 \, c^{2} f g^{2} + 16 \, c^{2} e g h - 24 \, b c f g h + 8 \, c^{2} d h^{2} - 12 \, b c e h^{2} + 15 \, b^{2} f h^{2} - 12 \, a c f h^{2}\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{8 \, c^{\frac {7}{2}}} \]
1/4*(((2*(b^2*c^2*f*h^2 - 4*a*c^3*f*h^2)*x/(b^2*c^3 - 4*a*c^4) + (8*b^2*c^ 2*f*g*h - 32*a*c^3*f*g*h + 4*b^2*c^2*e*h^2 - 16*a*c^3*e*h^2 - 5*b^3*c*f*h^ 2 + 20*a*b*c^2*f*h^2)/(b^2*c^3 - 4*a*c^4))*x - (16*c^4*d*g^2 - 8*b*c^3*e*g ^2 + 8*b^2*c^2*f*g^2 - 16*a*c^3*f*g^2 - 16*b*c^3*d*g*h + 16*b^2*c^2*e*g*h - 32*a*c^3*e*g*h - 24*b^3*c*f*g*h + 80*a*b*c^2*f*g*h + 8*b^2*c^2*d*h^2 - 1 6*a*c^3*d*h^2 - 12*b^3*c*e*h^2 + 40*a*b*c^2*e*h^2 + 15*b^4*f*h^2 - 62*a*b^ 2*c*f*h^2 + 24*a^2*c^2*f*h^2)/(b^2*c^3 - 4*a*c^4))*x - (8*b*c^3*d*g^2 - 16 *a*c^3*e*g^2 + 8*a*b*c^2*f*g^2 - 32*a*c^3*d*g*h + 16*a*b*c^2*e*g*h - 24*a* b^2*c*f*g*h + 64*a^2*c^2*f*g*h + 8*a*b*c^2*d*h^2 - 12*a*b^2*c*e*h^2 + 32*a ^2*c^2*e*h^2 + 15*a*b^3*f*h^2 - 52*a^2*b*c*f*h^2)/(b^2*c^3 - 4*a*c^4))/sqr t(c*x^2 + b*x + a) - 1/8*(8*c^2*f*g^2 + 16*c^2*e*g*h - 24*b*c*f*g*h + 8*c^ 2*d*h^2 - 12*b*c*e*h^2 + 15*b^2*f*h^2 - 12*a*c*f*h^2)*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) + b))/c^(7/2)
Timed out. \[ \int \frac {(g+h x)^2 \left (d+e x+f x^2\right )}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {{\left (g+h\,x\right )}^2\,\left (f\,x^2+e\,x+d\right )}{{\left (c\,x^2+b\,x+a\right )}^{3/2}} \,d x \]