Integrand size = 27, antiderivative size = 671 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{\left (d+e x+f x^2\right )^3} \, dx=-\frac {(e+2 f x) \left (a+b x+c x^2\right )^{3/2}}{2 \left (e^2-4 d f\right ) \left (d+e x+f x^2\right )^2}+\frac {3 \left (4 c d e+4 a e f-b \left (e^2+4 d f\right )+2 \left (c e^2-2 b e f+4 a f^2\right ) x\right ) \sqrt {a+b x+c x^2}}{4 \left (e^2-4 d f\right )^2 \left (d+e x+f x^2\right )}-\frac {3 \left (2 (2 c d-b e+2 a f) (c e-b f) \left (e-\sqrt {e^2-4 d f}\right )-f \left (4 b e (c d+3 a f)-b^2 \left (e^2+4 d f\right )-4 a \left (c e^2+4 a f^2\right )\right )\right ) \text {arctanh}\left (\frac {4 a f-b \left (e-\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e-\sqrt {e^2-4 d f}\right )\right ) x}{2 \sqrt {2} \sqrt {c e^2-2 c d f-b e f+2 a f^2-(c e-b f) \sqrt {e^2-4 d f}} \sqrt {a+b x+c x^2}}\right )}{4 \sqrt {2} \left (e^2-4 d f\right )^{5/2} \sqrt {c e^2-2 c d f-b e f+2 a f^2-(c e-b f) \sqrt {e^2-4 d f}}}+\frac {3 \left (2 (2 c d-b e+2 a f) (c e-b f) \left (e+\sqrt {e^2-4 d f}\right )-f \left (4 b e (c d+3 a f)-b^2 \left (e^2+4 d f\right )-4 a \left (c e^2+4 a f^2\right )\right )\right ) \text {arctanh}\left (\frac {4 a f-b \left (e+\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e+\sqrt {e^2-4 d f}\right )\right ) x}{2 \sqrt {2} \sqrt {c e^2-2 c d f-b e f+2 a f^2+(c e-b f) \sqrt {e^2-4 d f}} \sqrt {a+b x+c x^2}}\right )}{4 \sqrt {2} \left (e^2-4 d f\right )^{5/2} \sqrt {c e^2-2 c d f-b e f+2 a f^2+(c e-b f) \sqrt {e^2-4 d f}}} \]
-1/2*(2*f*x+e)*(c*x^2+b*x+a)^(3/2)/(-4*d*f+e^2)/(f*x^2+e*x+d)^2+3/4*(4*c*d *e+4*a*e*f-b*(4*d*f+e^2)+2*(4*a*f^2-2*b*e*f+c*e^2)*x)*(c*x^2+b*x+a)^(1/2)/ (-4*d*f+e^2)^2/(f*x^2+e*x+d)-3/8*arctanh(1/4*(4*a*f+2*x*(b*f-c*(e-(-4*d*f+ e^2)^(1/2)))-b*(e-(-4*d*f+e^2)^(1/2)))*2^(1/2)/(c*x^2+b*x+a)^(1/2)/(c*e^2- 2*c*d*f-b*e*f+2*a*f^2-(-b*f+c*e)*(-4*d*f+e^2)^(1/2))^(1/2))*(-f*(4*b*e*(3* a*f+c*d)-b^2*(4*d*f+e^2)-4*a*(4*a*f^2+c*e^2))+2*(2*a*f-b*e+2*c*d)*(-b*f+c* e)*(e-(-4*d*f+e^2)^(1/2)))/(-4*d*f+e^2)^(5/2)*2^(1/2)/(c*e^2-2*c*d*f-b*e*f +2*a*f^2-(-b*f+c*e)*(-4*d*f+e^2)^(1/2))^(1/2)+3/8*arctanh(1/4*(4*a*f-b*(e+ (-4*d*f+e^2)^(1/2))+2*x*(b*f-c*(e+(-4*d*f+e^2)^(1/2))))*2^(1/2)/(c*x^2+b*x +a)^(1/2)/(c*e^2-2*c*d*f-b*e*f+2*a*f^2+(-b*f+c*e)*(-4*d*f+e^2)^(1/2))^(1/2 ))*(-f*(4*b*e*(3*a*f+c*d)-b^2*(4*d*f+e^2)-4*a*(4*a*f^2+c*e^2))+2*(2*a*f-b* e+2*c*d)*(-b*f+c*e)*(e+(-4*d*f+e^2)^(1/2)))/(-4*d*f+e^2)^(5/2)*2^(1/2)/(c* e^2-2*c*d*f-b*e*f+2*a*f^2+(-b*f+c*e)*(-4*d*f+e^2)^(1/2))^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(4727\) vs. \(2(671)=1342\).
Time = 17.36 (sec) , antiderivative size = 4727, normalized size of antiderivative = 7.04 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{\left (d+e x+f x^2\right )^3} \, dx=\text {Result too large to show} \]
(-2*f^2*(a + x*(b + c*x))^(3/2))/((e^2 - 4*d*f)^(3/2)*(e - Sqrt[e^2 - 4*d* f] + 2*f*x)^2) + (6*f^2*(a + x*(b + c*x))^(3/2))/((e^2 - 4*d*f)^2*(e - Sqr t[e^2 - 4*d*f] + 2*f*x)) + (2*f^2*(a + x*(b + c*x))^(3/2))/((e^2 - 4*d*f)^ (3/2)*(e + Sqrt[e^2 - 4*d*f] + 2*f*x)^2) + (6*f^2*(a + x*(b + c*x))^(3/2)) /((e^2 - 4*d*f)^2*(e + Sqrt[e^2 - 4*d*f] + 2*f*x)) + (9*f^2*(a + x*(b + c* x))^(3/2)*(((-4*b*c*f - 2*c*(b*f + 2*c*(-e + Sqrt[e^2 - 4*d*f])) - 4*c^2*f *x)*Sqrt[a + b*x + c*x^2])/(8*c*f^2) - ((2*Sqrt[c]*(b^2*f^2 + 4*c^2*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f]) + 4*c*f*(a*f - b*(e - Sqrt[e^2 - 4*d*f])))*A rcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/f + (2*Sqrt[2]*Sqrt [c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - c*e*Sqrt[e^2 - 4*d*f] + b*f*Sqrt[e^2 - 4*d*f]]*(4*c*f*(8*a*b*f^2 - 3*b^2*f*(e - Sqrt[e^2 - 4*d*f]) - 4*a*c*f*(e - Sqrt[e^2 - 4*d*f]) + 4*b*c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])) + 4*c*( -e + Sqrt[e^2 - 4*d*f])*(b^2*f^2 + 4*c^2*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f ]) + 4*c*f*(a*f - b*(e - Sqrt[e^2 - 4*d*f]))))*ArcTanh[(-4*a*f - b*(-e + S qrt[e^2 - 4*d*f]) - (2*b*f + 2*c*(-e + Sqrt[e^2 - 4*d*f]))*x)/(2*Sqrt[2]*S qrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - c*e*Sqrt[e^2 - 4*d*f] + b*f*Sqrt[e ^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(f*(16*a*f^2 + 8*b*f*(-e + Sqrt[e^2 - 4*d*f]) + 4*c*(-e + Sqrt[e^2 - 4*d*f])^2)))/(16*c*f^2)))/((e^2 - 4*d*f)^ 2*(a + b*x + c*x^2)^(3/2)) - (3*f^2*(a + x*(b + c*x))^(3/2)*(((-4*c*f*(4*a *f - b*(e - Sqrt[e^2 - 4*d*f])) - 2*(b*f - c*(e - Sqrt[e^2 - 4*d*f]))*(...
Time = 1.13 (sec) , antiderivative size = 693, normalized size of antiderivative = 1.03, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {1302, 27, 1346, 27, 1365, 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.
| \(\displaystyle \int \frac {\left (a+b x+c x^2\right )^{3/2}}{\left (d+e x+f x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 1302 |
| \(\displaystyle \frac {\int \frac {3 (b e-4 a f+2 (c e-b f) x) \sqrt {c x^2+b x+a}}{2 \left (f x^2+e x+d\right )^2}dx}{2 \left (e^2-4 d f\right )}-\frac {(e+2 f x) \left (a+b x+c x^2\right )^{3/2}}{2 \left (e^2-4 d f\right ) \left (d+e x+f x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 \int \frac {(b e-4 a f+2 (c e-b f) x) \sqrt {c x^2+b x+a}}{\left (f x^2+e x+d\right )^2}dx}{4 \left (e^2-4 d f\right )}-\frac {(e+2 f x) \left (a+b x+c x^2\right )^{3/2}}{2 \left (e^2-4 d f\right ) \left (d+e x+f x^2\right )^2}\) |
| \(\Big \downarrow \) 1346 |
| \(\displaystyle \frac {3 \left (\frac {\int -\frac {-\left (\left (e^2+4 d f\right ) b^2\right )+4 e (c d+3 a f) b-4 a \left (c e^2+4 a f^2\right )+4 (2 c d-b e+2 a f) (c e-b f) x}{2 \sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{e^2-4 d f}+\frac {\sqrt {a+b x+c x^2} \left (2 x \left (4 a f^2-2 b e f+c e^2\right )+4 a e f-b \left (4 d f+e^2\right )+4 c d e\right )}{\left (e^2-4 d f\right ) \left (d+e x+f x^2\right )}\right )}{4 \left (e^2-4 d f\right )}-\frac {(e+2 f x) \left (a+b x+c x^2\right )^{3/2}}{2 \left (e^2-4 d f\right ) \left (d+e x+f x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 \left (\frac {\sqrt {a+b x+c x^2} \left (2 x \left (4 a f^2-2 b e f+c e^2\right )+4 a e f-b \left (4 d f+e^2\right )+4 c d e\right )}{\left (e^2-4 d f\right ) \left (d+e x+f x^2\right )}-\frac {\int \frac {-\left (\left (e^2+4 d f\right ) b^2\right )+4 e (c d+3 a f) b-4 a \left (c e^2+4 a f^2\right )+4 (2 c d-b e+2 a f) (c e-b f) x}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{2 \left (e^2-4 d f\right )}\right )}{4 \left (e^2-4 d f\right )}-\frac {(e+2 f x) \left (a+b x+c x^2\right )^{3/2}}{2 \left (e^2-4 d f\right ) \left (d+e x+f x^2\right )^2}\) |
| \(\Big \downarrow \) 1365 |
| \(\displaystyle \frac {3 \left (\frac {\sqrt {a+b x+c x^2} \left (2 x \left (4 a f^2-2 b e f+c e^2\right )+4 a e f-b \left (4 d f+e^2\right )+4 c d e\right )}{\left (e^2-4 d f\right ) \left (d+e x+f x^2\right )}-\frac {\frac {2 \left (2 \left (\sqrt {e^2-4 d f}+e\right ) (c e-b f) (2 a f-b e+2 c d)-f \left (4 b e (3 a f+c d)-4 a \left (4 a f^2+c e^2\right )-\left (b^2 \left (4 d f+e^2\right )\right )\right )\right ) \int \frac {1}{\left (e+2 f x+\sqrt {e^2-4 d f}\right ) \sqrt {c x^2+b x+a}}dx}{\sqrt {e^2-4 d f}}-\frac {2 \left (2 \left (e-\sqrt {e^2-4 d f}\right ) (c e-b f) (2 a f-b e+2 c d)-f \left (4 b e (3 a f+c d)-4 a \left (4 a f^2+c e^2\right )-\left (b^2 \left (4 d f+e^2\right )\right )\right )\right ) \int \frac {1}{\left (e+2 f x-\sqrt {e^2-4 d f}\right ) \sqrt {c x^2+b x+a}}dx}{\sqrt {e^2-4 d f}}}{2 \left (e^2-4 d f\right )}\right )}{4 \left (e^2-4 d f\right )}-\frac {(e+2 f x) \left (a+b x+c x^2\right )^{3/2}}{2 \left (e^2-4 d f\right ) \left (d+e x+f x^2\right )^2}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {3 \left (\frac {\sqrt {a+b x+c x^2} \left (2 x \left (4 a f^2-2 b e f+c e^2\right )+4 a e f-b \left (4 d f+e^2\right )+4 c d e\right )}{\left (e^2-4 d f\right ) \left (d+e x+f x^2\right )}-\frac {\frac {4 \left (2 \left (e-\sqrt {e^2-4 d f}\right ) (c e-b f) (2 a f-b e+2 c d)-f \left (4 b e (3 a f+c d)-4 a \left (4 a f^2+c e^2\right )-\left (b^2 \left (4 d f+e^2\right )\right )\right )\right ) \int \frac {1}{4 \left (4 a f^2-2 b \left (e-\sqrt {e^2-4 d f}\right ) f+c \left (e-\sqrt {e^2-4 d f}\right )^2\right )-\frac {\left (4 a f-b \left (e-\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e-\sqrt {e^2-4 d f}\right )\right ) x\right )^2}{c x^2+b x+a}}d\frac {4 a f-b \left (e-\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e-\sqrt {e^2-4 d f}\right )\right ) x}{\sqrt {c x^2+b x+a}}}{\sqrt {e^2-4 d f}}-\frac {4 \left (2 \left (\sqrt {e^2-4 d f}+e\right ) (c e-b f) (2 a f-b e+2 c d)-f \left (4 b e (3 a f+c d)-4 a \left (4 a f^2+c e^2\right )-\left (b^2 \left (4 d f+e^2\right )\right )\right )\right ) \int \frac {1}{4 \left (4 a f^2-2 b \left (e+\sqrt {e^2-4 d f}\right ) f+c \left (e+\sqrt {e^2-4 d f}\right )^2\right )-\frac {\left (4 a f-b \left (e+\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e+\sqrt {e^2-4 d f}\right )\right ) x\right )^2}{c x^2+b x+a}}d\frac {4 a f-b \left (e+\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e+\sqrt {e^2-4 d f}\right )\right ) x}{\sqrt {c x^2+b x+a}}}{\sqrt {e^2-4 d f}}}{2 \left (e^2-4 d f\right )}\right )}{4 \left (e^2-4 d f\right )}-\frac {(e+2 f x) \left (a+b x+c x^2\right )^{3/2}}{2 \left (e^2-4 d f\right ) \left (d+e x+f x^2\right )^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {3 \left (\frac {\sqrt {a+b x+c x^2} \left (2 x \left (4 a f^2-2 b e f+c e^2\right )+4 a e f-b \left (4 d f+e^2\right )+4 c d e\right )}{\left (e^2-4 d f\right ) \left (d+e x+f x^2\right )}-\frac {\frac {\sqrt {2} \left (2 \left (e-\sqrt {e^2-4 d f}\right ) (c e-b f) (2 a f-b e+2 c d)-f \left (4 b e (3 a f+c d)-4 a \left (4 a f^2+c e^2\right )-\left (b^2 \left (4 d f+e^2\right )\right )\right )\right ) \text {arctanh}\left (\frac {4 a f+2 x \left (b f-c \left (e-\sqrt {e^2-4 d f}\right )\right )-b \left (e-\sqrt {e^2-4 d f}\right )}{2 \sqrt {2} \sqrt {a+b x+c x^2} \sqrt {2 a f^2-\sqrt {e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right )}{\sqrt {e^2-4 d f} \sqrt {2 a f^2-\sqrt {e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}-\frac {\sqrt {2} \left (2 \left (\sqrt {e^2-4 d f}+e\right ) (c e-b f) (2 a f-b e+2 c d)-f \left (4 b e (3 a f+c d)-4 a \left (4 a f^2+c e^2\right )-\left (b^2 \left (4 d f+e^2\right )\right )\right )\right ) \text {arctanh}\left (\frac {4 a f+2 x \left (b f-c \left (\sqrt {e^2-4 d f}+e\right )\right )-b \left (\sqrt {e^2-4 d f}+e\right )}{2 \sqrt {2} \sqrt {a+b x+c x^2} \sqrt {2 a f^2+\sqrt {e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right )}{\sqrt {e^2-4 d f} \sqrt {2 a f^2+\sqrt {e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}}{2 \left (e^2-4 d f\right )}\right )}{4 \left (e^2-4 d f\right )}-\frac {(e+2 f x) \left (a+b x+c x^2\right )^{3/2}}{2 \left (e^2-4 d f\right ) \left (d+e x+f x^2\right )^2}\) |
-1/2*((e + 2*f*x)*(a + b*x + c*x^2)^(3/2))/((e^2 - 4*d*f)*(d + e*x + f*x^2 )^2) + (3*(((4*c*d*e + 4*a*e*f - b*(e^2 + 4*d*f) + 2*(c*e^2 - 2*b*e*f + 4* a*f^2)*x)*Sqrt[a + b*x + c*x^2])/((e^2 - 4*d*f)*(d + e*x + f*x^2)) - ((Sqr t[2]*(2*(2*c*d - b*e + 2*a*f)*(c*e - b*f)*(e - Sqrt[e^2 - 4*d*f]) - f*(4*b *e*(c*d + 3*a*f) - b^2*(e^2 + 4*d*f) - 4*a*(c*e^2 + 4*a*f^2)))*ArcTanh[(4* a*f - b*(e - Sqrt[e^2 - 4*d*f]) + 2*(b*f - c*(e - Sqrt[e^2 - 4*d*f]))*x)/( 2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(Sqrt[e^2 - 4*d*f]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*Sqrt[e^2 - 4*d*f]]) - (Sqrt[2]*(2*(2*c*d - b*e + 2*a*f)*(c*e - b*f)*(e + Sqrt[e^2 - 4*d*f]) - f*(4*b*e*(c*d + 3*a*f) - b^2*(e^2 + 4*d*f) - 4*a*(c*e^2 + 4*a*f^2)))*ArcTanh[(4*a*f - b*(e + Sqrt [e^2 - 4*d*f]) + 2*(b*f - c*(e + Sqrt[e^2 - 4*d*f]))*x)/(2*Sqrt[2]*Sqrt[c* e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(Sqrt[e^2 - 4*d*f]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*Sqrt[e^2 - 4*d*f]]))/(2*(e^2 - 4*d*f))))/(4*(e^2 - 4*d*f))
3.2.8.3.1 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[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_.) + (e_.)*(x_) + (f_.)*(x _)^2)^(q_), x_Symbol] :> Simp[(b + 2*c*x)*(a + b*x + c*x^2)^(p + 1)*((d + e *x + f*x^2)^q/((b^2 - 4*a*c)*(p + 1))), x] - Simp[1/((b^2 - 4*a*c)*(p + 1)) Int[(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q - 1)*Simp[2*c*d*(2*p + 3) + b*e*q + (2*b*f*q + 2*c*e*(2*p + q + 3))*x + 2*c*f*(2*p + 2*q + 3)*x^ 2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ [e^2 - 4*d*f, 0] && LtQ[p, -1] && GtQ[q, 0] && !IGtQ[q, 0]
Int[((g_.) + (h_.)*(x_))*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (e _.)*(x_) + (f_.)*(x_)^2)^(q_), x_Symbol] :> Simp[(g*b - 2*a*h - (b*h - 2*g* c)*x)*(a + b*x + c*x^2)^(p + 1)*((d + e*x + f*x^2)^q/((b^2 - 4*a*c)*(p + 1) )), x] - Simp[1/((b^2 - 4*a*c)*(p + 1)) Int[(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q - 1)*Simp[e*q*(g*b - 2*a*h) - d*(b*h - 2*g*c)*(2*p + 3) + (2*f*q*(g*b - 2*a*h) - e*(b*h - 2*g*c)*(2*p + q + 3))*x - f*(b*h - 2*g*c)* (2*p + 2*q + 3)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && Ne Q[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && LtQ[p, -1] && GtQ[q, 0]
Int[((g_.) + (h_.)*(x_))/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + ( e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Sim p[(2*c*g - h*(b - q))/q Int[1/((b - q + 2*c*x)*Sqrt[d + e*x + f*x^2]), x] , x] - Simp[(2*c*g - h*(b + q))/q Int[1/((b + q + 2*c*x)*Sqrt[d + e*x + f *x^2]), x], 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] && PosQ[b^2 - 4*a*c]
Leaf count of result is larger than twice the leaf count of optimal. \(16308\) vs. \(2(611)=1222\).
Time = 1.61 (sec) , antiderivative size = 16309, normalized size of antiderivative = 24.31
Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{\left (d+e x+f x^2\right )^3} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{\left (d+e x+f x^2\right )^3} \, dx=\text {Timed out} \]
\[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{\left (d+e x+f x^2\right )^3} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}}{{\left (f x^{2} + e x + d\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{\left (d+e x+f x^2\right )^3} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{\left (d+e x+f x^2\right )^3} \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^{3/2}}{{\left (f\,x^2+e\,x+d\right )}^3} \,d x \]