Integrand size = 27, antiderivative size = 234 \[ \int \frac {(d+e x)^4 \left (A+B x+C x^2\right )}{\left (a+c x^2\right )^4} \, dx=-\frac {(a B-(A c-a C) x) (d+e x)^4}{6 a c \left (a+c x^2\right )^3}-\frac {(d+e x)^3 (a (A c+5 a C) e-c (5 A c d+a C d+4 a B e) x)}{24 a^2 c^2 \left (a+c x^2\right )^2}-\frac {\left (a (A c+5 a C) e^2+c d (5 A c d+a C d+4 a B e)\right ) (a e-c d x) (d+e x)}{16 a^3 c^3 \left (a+c x^2\right )}+\frac {\left (c d^2+a e^2\right ) \left (a (A c+5 a C) e^2+c d (5 A c d+a C d+4 a B e)\right ) \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{16 a^{7/2} c^{7/2}} \]
-1/6*(a*B-(A*c-C*a)*x)*(e*x+d)^4/a/c/(c*x^2+a)^3-1/24*(e*x+d)^3*(a*(A*c+5* C*a)*e-c*(5*A*c*d+4*B*a*e+C*a*d)*x)/a^2/c^2/(c*x^2+a)^2-1/16*(a*(A*c+5*C*a )*e^2+c*d*(5*A*c*d+4*B*a*e+C*a*d))*(-c*d*x+a*e)*(e*x+d)/a^3/c^3/(c*x^2+a)+ 1/16*(a*e^2+c*d^2)*(a*(A*c+5*C*a)*e^2+c*d*(5*A*c*d+4*B*a*e+C*a*d))*arctan( x*c^(1/2)/a^(1/2))/a^(7/2)/c^(7/2)
Time = 0.18 (sec) , antiderivative size = 437, normalized size of antiderivative = 1.87 \[ \int \frac {(d+e x)^4 \left (A+B x+C x^2\right )}{\left (a+c x^2\right )^4} \, dx=\frac {5 A c^3 d^4 x+a c^2 d^2 \left (C d^2+4 B d e+6 A e^2\right ) x+a^2 c e^2 \left (6 C d^2+e (4 B d+A e)\right ) x-a^3 e^3 (32 C d+8 B e+11 C e x)}{16 a^3 c^3 \left (a+c x^2\right )}+\frac {A c^3 d^4 x-a^3 e^3 (4 C d+B e+C e x)-a c^2 d^2 \left (4 A d e+C d^2 x+6 A e^2 x+B d (d+4 e x)\right )+a^2 c e \left (2 C d^2 (2 d+3 e x)+e (A e (4 d+e x)+2 B d (3 d+2 e x))\right )}{6 a c^3 \left (a+c x^2\right )^3}+\frac {5 A c^3 d^4 x+a c^2 d^2 \left (C d^2+4 B d e+6 A e^2\right ) x+a^3 e^3 (48 C d+12 B e+13 C e x)-a^2 c e \left (6 C d^2 (4 d+7 e x)+e (4 B d (9 d+7 e x)+A e (24 d+7 e x))\right )}{24 a^2 c^3 \left (a+c x^2\right )^2}+\frac {\left (c d^2+a e^2\right ) \left (A c \left (5 c d^2+a e^2\right )+a \left (5 a C e^2+c d (C d+4 B e)\right )\right ) \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{16 a^{7/2} c^{7/2}} \]
(5*A*c^3*d^4*x + a*c^2*d^2*(C*d^2 + 4*B*d*e + 6*A*e^2)*x + a^2*c*e^2*(6*C* d^2 + e*(4*B*d + A*e))*x - a^3*e^3*(32*C*d + 8*B*e + 11*C*e*x))/(16*a^3*c^ 3*(a + c*x^2)) + (A*c^3*d^4*x - a^3*e^3*(4*C*d + B*e + C*e*x) - a*c^2*d^2* (4*A*d*e + C*d^2*x + 6*A*e^2*x + B*d*(d + 4*e*x)) + a^2*c*e*(2*C*d^2*(2*d + 3*e*x) + e*(A*e*(4*d + e*x) + 2*B*d*(3*d + 2*e*x))))/(6*a*c^3*(a + c*x^2 )^3) + (5*A*c^3*d^4*x + a*c^2*d^2*(C*d^2 + 4*B*d*e + 6*A*e^2)*x + a^3*e^3* (48*C*d + 12*B*e + 13*C*e*x) - a^2*c*e*(6*C*d^2*(4*d + 7*e*x) + e*(4*B*d*( 9*d + 7*e*x) + A*e*(24*d + 7*e*x))))/(24*a^2*c^3*(a + c*x^2)^2) + ((c*d^2 + a*e^2)*(A*c*(5*c*d^2 + a*e^2) + a*(5*a*C*e^2 + c*d*(C*d + 4*B*e)))*ArcTa n[(Sqrt[c]*x)/Sqrt[a]])/(16*a^(7/2)*c^(7/2))
Time = 0.42 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.96, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2176, 25, 678, 487, 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.
| \(\displaystyle \int \frac {(d+e x)^4 \left (A+B x+C x^2\right )}{\left (a+c x^2\right )^4} \, dx\) |
| \(\Big \downarrow \) 2176 |
| \(\displaystyle -\frac {\int -\frac {(d+e x)^3 (5 A c d+a C d+4 a B e+(A c+5 a C) e x)}{\left (c x^2+a\right )^3}dx}{6 a c}-\frac {(d+e x)^4 (a B-x (A c-a C))}{6 a c \left (a+c x^2\right )^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {(d+e x)^3 (5 A c d+a C d+4 a B e+(A c+5 a C) e x)}{\left (c x^2+a\right )^3}dx}{6 a c}-\frac {(d+e x)^4 (a B-x (A c-a C))}{6 a c \left (a+c x^2\right )^3}\) |
| \(\Big \downarrow \) 678 |
| \(\displaystyle \frac {\frac {3 \left (c d (4 a B e+a C d+5 A c d)+a e^2 (5 a C+A c)\right ) \int \frac {(d+e x)^2}{\left (c x^2+a\right )^2}dx}{4 a c}-\frac {(d+e x)^3 (a e (5 a C+A c)-c x (4 a B e+a C d+5 A c d))}{4 a c \left (a+c x^2\right )^2}}{6 a c}-\frac {(d+e x)^4 (a B-x (A c-a C))}{6 a c \left (a+c x^2\right )^3}\) |
| \(\Big \downarrow \) 487 |
| \(\displaystyle \frac {\frac {3 \left (c d (4 a B e+a C d+5 A c d)+a e^2 (5 a C+A c)\right ) \left (\frac {\left (a e^2+c d^2\right ) \int \frac {1}{c x^2+a}dx}{2 a c}-\frac {(d+e x) (a e-c d x)}{2 a c \left (a+c x^2\right )}\right )}{4 a c}-\frac {(d+e x)^3 (a e (5 a C+A c)-c x (4 a B e+a C d+5 A c d))}{4 a c \left (a+c x^2\right )^2}}{6 a c}-\frac {(d+e x)^4 (a B-x (A c-a C))}{6 a c \left (a+c x^2\right )^3}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {3 \left (\frac {\arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (a e^2+c d^2\right )}{2 a^{3/2} c^{3/2}}-\frac {(d+e x) (a e-c d x)}{2 a c \left (a+c x^2\right )}\right ) \left (c d (4 a B e+a C d+5 A c d)+a e^2 (5 a C+A c)\right )}{4 a c}-\frac {(d+e x)^3 (a e (5 a C+A c)-c x (4 a B e+a C d+5 A c d))}{4 a c \left (a+c x^2\right )^2}}{6 a c}-\frac {(d+e x)^4 (a B-x (A c-a C))}{6 a c \left (a+c x^2\right )^3}\) |
-1/6*((a*B - (A*c - a*C)*x)*(d + e*x)^4)/(a*c*(a + c*x^2)^3) + (-1/4*((d + e*x)^3*(a*(A*c + 5*a*C)*e - c*(5*A*c*d + a*C*d + 4*a*B*e)*x))/(a*c*(a + c *x^2)^2) + (3*(a*(A*c + 5*a*C)*e^2 + c*d*(5*A*c*d + a*C*d + 4*a*B*e))*(-1/ 2*((a*e - c*d*x)*(d + e*x))/(a*c*(a + c*x^2)) + ((c*d^2 + a*e^2)*ArcTan[(S qrt[c]*x)/Sqrt[a]])/(2*a^(3/2)*c^(3/2))))/(4*a*c))/(6*a*c)
3.1.64.3.1 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[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ (c + d*x)^(n - 1)*(a*d - b*c*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[(2*p + 3)*((b*c^2 + a*d^2)/(2*a*b*(p + 1))) Int[(c + d*x)^(n - 2)*( a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[n + 2*p + 2, 0] && LtQ[p, -1]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _), x_Symbol] :> Simp[(d + e*x)^m*(a + c*x^2)^(p + 1)*((a*g - c*f*x)/(2*a*c *(p + 1))), x] - Simp[m*((c*d*f + a*e*g)/(2*a*c*(p + 1))) Int[(d + e*x)^( m - 1)*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && EqQ[S implify[m + 2*p + 3], 0] && LtQ[p, -1]
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : > With[{Qx = PolynomialQuotient[Pq, a + b*x^2, x], R = Coeff[PolynomialRema inder[Pq, a + b*x^2, x], x, 0], S = Coeff[PolynomialRemainder[Pq, a + b*x^2 , x], x, 1]}, Simp[(d + e*x)^m*(a + b*x^2)^(p + 1)*((a*S - b*R*x)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*b*(p + 1)) Int[(d + e*x)^(m - 1)*(a + b*x^2)^(p + 1)*ExpandToSum[2*a*b*(p + 1)*(d + e*x)*Qx - a*e*S*m + b*d*R*(2*p + 3) + b*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, d, e}, x] && PolyQ[Pq, x ] && NeQ[b*d^2 + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 0] && !(IGtQ[m, 0] && R ationalQ[a, b, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
Leaf count of result is larger than twice the leaf count of optimal. \(543\) vs. \(2(218)=436\).
Time = 0.58 (sec) , antiderivative size = 544, normalized size of antiderivative = 2.32
| method | result | size |
| default | \(\frac {\frac {\left (A \,a^{2} c \,e^{4}+6 A a \,c^{2} d^{2} e^{2}+5 A \,d^{4} c^{3}+4 B \,a^{2} c d \,e^{3}+4 B a \,c^{2} d^{3} e -11 C \,a^{3} e^{4}+6 C \,a^{2} c \,d^{2} e^{2}+C a \,c^{2} d^{4}\right ) x^{5}}{16 a^{3} c}-\frac {e^{3} \left (B e +4 C d \right ) x^{4}}{2 c}-\frac {\left (A \,a^{2} c \,e^{4}-6 A a \,c^{2} d^{2} e^{2}-5 A \,d^{4} c^{3}+4 B \,a^{2} c d \,e^{3}-4 B a \,c^{2} d^{3} e +5 C \,a^{3} e^{4}+6 C \,a^{2} c \,d^{2} e^{2}-C a \,c^{2} d^{4}\right ) x^{3}}{6 a^{2} c^{2}}-\frac {e \left (2 A c d \,e^{2}+B \,e^{3} a +3 B c \,d^{2} e +4 C a d \,e^{2}+2 C c \,d^{3}\right ) x^{2}}{2 c^{2}}-\frac {\left (A \,a^{2} c \,e^{4}+6 A a \,c^{2} d^{2} e^{2}-11 A \,d^{4} c^{3}+4 B \,a^{2} c d \,e^{3}+4 B a \,c^{2} d^{3} e +5 C \,a^{3} e^{4}+6 C \,a^{2} c \,d^{2} e^{2}+C a \,c^{2} d^{4}\right ) x}{16 a \,c^{3}}-\frac {2 A a c d \,e^{3}+4 A \,c^{2} d^{3} e +B \,e^{4} a^{2}+3 B a c \,d^{2} e^{2}+B \,c^{2} d^{4}+4 C \,a^{2} d \,e^{3}+2 C a c \,d^{3} e}{6 c^{3}}}{\left (c \,x^{2}+a \right )^{3}}+\frac {\left (A \,a^{2} c \,e^{4}+6 A a \,c^{2} d^{2} e^{2}+5 A \,d^{4} c^{3}+4 B \,a^{2} c d \,e^{3}+4 B a \,c^{2} d^{3} e +5 C \,a^{3} e^{4}+6 C \,a^{2} c \,d^{2} e^{2}+C a \,c^{2} d^{4}\right ) \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{16 a^{3} c^{3} \sqrt {a c}}\) | \(544\) |
| risch | \(\frac {\frac {\left (A \,a^{2} c \,e^{4}+6 A a \,c^{2} d^{2} e^{2}+5 A \,d^{4} c^{3}+4 B \,a^{2} c d \,e^{3}+4 B a \,c^{2} d^{3} e -11 C \,a^{3} e^{4}+6 C \,a^{2} c \,d^{2} e^{2}+C a \,c^{2} d^{4}\right ) x^{5}}{16 a^{3} c}-\frac {e^{3} \left (B e +4 C d \right ) x^{4}}{2 c}-\frac {\left (A \,a^{2} c \,e^{4}-6 A a \,c^{2} d^{2} e^{2}-5 A \,d^{4} c^{3}+4 B \,a^{2} c d \,e^{3}-4 B a \,c^{2} d^{3} e +5 C \,a^{3} e^{4}+6 C \,a^{2} c \,d^{2} e^{2}-C a \,c^{2} d^{4}\right ) x^{3}}{6 a^{2} c^{2}}-\frac {e \left (2 A c d \,e^{2}+B \,e^{3} a +3 B c \,d^{2} e +4 C a d \,e^{2}+2 C c \,d^{3}\right ) x^{2}}{2 c^{2}}-\frac {\left (A \,a^{2} c \,e^{4}+6 A a \,c^{2} d^{2} e^{2}-11 A \,d^{4} c^{3}+4 B \,a^{2} c d \,e^{3}+4 B a \,c^{2} d^{3} e +5 C \,a^{3} e^{4}+6 C \,a^{2} c \,d^{2} e^{2}+C a \,c^{2} d^{4}\right ) x}{16 a \,c^{3}}-\frac {2 A a c d \,e^{3}+4 A \,c^{2} d^{3} e +B \,e^{4} a^{2}+3 B a c \,d^{2} e^{2}+B \,c^{2} d^{4}+4 C \,a^{2} d \,e^{3}+2 C a c \,d^{3} e}{6 c^{3}}}{\left (c \,x^{2}+a \right )^{3}}-\frac {\ln \left (c x +\sqrt {-a c}\right ) A \,e^{4}}{32 \sqrt {-a c}\, c^{2} a}-\frac {3 \ln \left (c x +\sqrt {-a c}\right ) A \,d^{2} e^{2}}{16 \sqrt {-a c}\, c \,a^{2}}-\frac {5 \ln \left (c x +\sqrt {-a c}\right ) A \,d^{4}}{32 \sqrt {-a c}\, a^{3}}-\frac {\ln \left (c x +\sqrt {-a c}\right ) B d \,e^{3}}{8 \sqrt {-a c}\, c^{2} a}-\frac {\ln \left (c x +\sqrt {-a c}\right ) B \,d^{3} e}{8 \sqrt {-a c}\, c \,a^{2}}-\frac {5 \ln \left (c x +\sqrt {-a c}\right ) C \,e^{4}}{32 \sqrt {-a c}\, c^{3}}-\frac {3 \ln \left (c x +\sqrt {-a c}\right ) C \,d^{2} e^{2}}{16 \sqrt {-a c}\, c^{2} a}-\frac {\ln \left (c x +\sqrt {-a c}\right ) C \,d^{4}}{32 \sqrt {-a c}\, c \,a^{2}}+\frac {\ln \left (-c x +\sqrt {-a c}\right ) A \,e^{4}}{32 \sqrt {-a c}\, c^{2} a}+\frac {3 \ln \left (-c x +\sqrt {-a c}\right ) A \,d^{2} e^{2}}{16 \sqrt {-a c}\, c \,a^{2}}+\frac {5 \ln \left (-c x +\sqrt {-a c}\right ) A \,d^{4}}{32 \sqrt {-a c}\, a^{3}}+\frac {\ln \left (-c x +\sqrt {-a c}\right ) B d \,e^{3}}{8 \sqrt {-a c}\, c^{2} a}+\frac {\ln \left (-c x +\sqrt {-a c}\right ) B \,d^{3} e}{8 \sqrt {-a c}\, c \,a^{2}}+\frac {5 \ln \left (-c x +\sqrt {-a c}\right ) C \,e^{4}}{32 \sqrt {-a c}\, c^{3}}+\frac {3 \ln \left (-c x +\sqrt {-a c}\right ) C \,d^{2} e^{2}}{16 \sqrt {-a c}\, c^{2} a}+\frac {\ln \left (-c x +\sqrt {-a c}\right ) C \,d^{4}}{32 \sqrt {-a c}\, c \,a^{2}}\) | \(913\) |
(1/16*(A*a^2*c*e^4+6*A*a*c^2*d^2*e^2+5*A*c^3*d^4+4*B*a^2*c*d*e^3+4*B*a*c^2 *d^3*e-11*C*a^3*e^4+6*C*a^2*c*d^2*e^2+C*a*c^2*d^4)/a^3/c*x^5-1/2*e^3*(B*e+ 4*C*d)/c*x^4-1/6*(A*a^2*c*e^4-6*A*a*c^2*d^2*e^2-5*A*c^3*d^4+4*B*a^2*c*d*e^ 3-4*B*a*c^2*d^3*e+5*C*a^3*e^4+6*C*a^2*c*d^2*e^2-C*a*c^2*d^4)/a^2/c^2*x^3-1 /2*e*(2*A*c*d*e^2+B*a*e^3+3*B*c*d^2*e+4*C*a*d*e^2+2*C*c*d^3)/c^2*x^2-1/16* (A*a^2*c*e^4+6*A*a*c^2*d^2*e^2-11*A*c^3*d^4+4*B*a^2*c*d*e^3+4*B*a*c^2*d^3* e+5*C*a^3*e^4+6*C*a^2*c*d^2*e^2+C*a*c^2*d^4)/a/c^3*x-1/6*(2*A*a*c*d*e^3+4* A*c^2*d^3*e+B*a^2*e^4+3*B*a*c*d^2*e^2+B*c^2*d^4+4*C*a^2*d*e^3+2*C*a*c*d^3* e)/c^3)/(c*x^2+a)^3+1/16*(A*a^2*c*e^4+6*A*a*c^2*d^2*e^2+5*A*c^3*d^4+4*B*a^ 2*c*d*e^3+4*B*a*c^2*d^3*e+5*C*a^3*e^4+6*C*a^2*c*d^2*e^2+C*a*c^2*d^4)/a^3/c ^3/(a*c)^(1/2)*arctan(c*x/(a*c)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 922 vs. \(2 (217) = 434\).
Time = 0.37 (sec) , antiderivative size = 1864, normalized size of antiderivative = 7.97 \[ \int \frac {(d+e x)^4 \left (A+B x+C x^2\right )}{\left (a+c x^2\right )^4} \, dx=\text {Too large to display} \]
[-1/96*(16*B*a^4*c^3*d^4 + 48*B*a^5*c^2*d^2*e^2 + 16*B*a^6*c*e^4 - 6*(4*B* a^2*c^5*d^3*e + 4*B*a^3*c^4*d*e^3 + (C*a^2*c^5 + 5*A*a*c^6)*d^4 + 6*(C*a^3 *c^4 + A*a^2*c^5)*d^2*e^2 - (11*C*a^4*c^3 - A*a^3*c^4)*e^4)*x^5 + 32*(C*a^ 5*c^2 + 2*A*a^4*c^3)*d^3*e + 32*(2*C*a^6*c + A*a^5*c^2)*d*e^3 + 48*(4*C*a^ 4*c^3*d*e^3 + B*a^4*c^3*e^4)*x^4 - 16*(4*B*a^3*c^4*d^3*e - 4*B*a^4*c^3*d*e ^3 + (C*a^3*c^4 + 5*A*a^2*c^5)*d^4 - 6*(C*a^4*c^3 - A*a^3*c^4)*d^2*e^2 - ( 5*C*a^5*c^2 + A*a^4*c^3)*e^4)*x^3 + 48*(2*C*a^4*c^3*d^3*e + 3*B*a^4*c^3*d^ 2*e^2 + B*a^5*c^2*e^4 + 2*(2*C*a^5*c^2 + A*a^4*c^3)*d*e^3)*x^2 + 3*(4*B*a^ 4*c^2*d^3*e + 4*B*a^5*c*d*e^3 + (4*B*a*c^5*d^3*e + 4*B*a^2*c^4*d*e^3 + (C* a*c^5 + 5*A*c^6)*d^4 + 6*(C*a^2*c^4 + A*a*c^5)*d^2*e^2 + (5*C*a^3*c^3 + A* a^2*c^4)*e^4)*x^6 + (C*a^4*c^2 + 5*A*a^3*c^3)*d^4 + 6*(C*a^5*c + A*a^4*c^2 )*d^2*e^2 + (5*C*a^6 + A*a^5*c)*e^4 + 3*(4*B*a^2*c^4*d^3*e + 4*B*a^3*c^3*d *e^3 + (C*a^2*c^4 + 5*A*a*c^5)*d^4 + 6*(C*a^3*c^3 + A*a^2*c^4)*d^2*e^2 + ( 5*C*a^4*c^2 + A*a^3*c^3)*e^4)*x^4 + 3*(4*B*a^3*c^3*d^3*e + 4*B*a^4*c^2*d*e ^3 + (C*a^3*c^3 + 5*A*a^2*c^4)*d^4 + 6*(C*a^4*c^2 + A*a^3*c^3)*d^2*e^2 + ( 5*C*a^5*c + A*a^4*c^2)*e^4)*x^2)*sqrt(-a*c)*log((c*x^2 - 2*sqrt(-a*c)*x - a)/(c*x^2 + a)) + 6*(4*B*a^4*c^3*d^3*e + 4*B*a^5*c^2*d*e^3 + (C*a^4*c^3 - 11*A*a^3*c^4)*d^4 + 6*(C*a^5*c^2 + A*a^4*c^3)*d^2*e^2 + (5*C*a^6*c + A*a^5 *c^2)*e^4)*x)/(a^4*c^7*x^6 + 3*a^5*c^6*x^4 + 3*a^6*c^5*x^2 + a^7*c^4), -1/ 48*(8*B*a^4*c^3*d^4 + 24*B*a^5*c^2*d^2*e^2 + 8*B*a^6*c*e^4 - 3*(4*B*a^2...
Timed out. \[ \int \frac {(d+e x)^4 \left (A+B x+C x^2\right )}{\left (a+c x^2\right )^4} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 599 vs. \(2 (217) = 434\).
Time = 0.28 (sec) , antiderivative size = 599, normalized size of antiderivative = 2.56 \[ \int \frac {(d+e x)^4 \left (A+B x+C x^2\right )}{\left (a+c x^2\right )^4} \, dx=-\frac {8 \, B a^{3} c^{2} d^{4} + 24 \, B a^{4} c d^{2} e^{2} + 8 \, B a^{5} e^{4} - 3 \, {\left (4 \, B a c^{4} d^{3} e + 4 \, B a^{2} c^{3} d e^{3} + {\left (C a c^{4} + 5 \, A c^{5}\right )} d^{4} + 6 \, {\left (C a^{2} c^{3} + A a c^{4}\right )} d^{2} e^{2} - {\left (11 \, C a^{3} c^{2} - A a^{2} c^{3}\right )} e^{4}\right )} x^{5} + 16 \, {\left (C a^{4} c + 2 \, A a^{3} c^{2}\right )} d^{3} e + 16 \, {\left (2 \, C a^{5} + A a^{4} c\right )} d e^{3} + 24 \, {\left (4 \, C a^{3} c^{2} d e^{3} + B a^{3} c^{2} e^{4}\right )} x^{4} - 8 \, {\left (4 \, B a^{2} c^{3} d^{3} e - 4 \, B a^{3} c^{2} d e^{3} + {\left (C a^{2} c^{3} + 5 \, A a c^{4}\right )} d^{4} - 6 \, {\left (C a^{3} c^{2} - A a^{2} c^{3}\right )} d^{2} e^{2} - {\left (5 \, C a^{4} c + A a^{3} c^{2}\right )} e^{4}\right )} x^{3} + 24 \, {\left (2 \, C a^{3} c^{2} d^{3} e + 3 \, B a^{3} c^{2} d^{2} e^{2} + B a^{4} c e^{4} + 2 \, {\left (2 \, C a^{4} c + A a^{3} c^{2}\right )} d e^{3}\right )} x^{2} + 3 \, {\left (4 \, B a^{3} c^{2} d^{3} e + 4 \, B a^{4} c d e^{3} + {\left (C a^{3} c^{2} - 11 \, A a^{2} c^{3}\right )} d^{4} + 6 \, {\left (C a^{4} c + A a^{3} c^{2}\right )} d^{2} e^{2} + {\left (5 \, C a^{5} + A a^{4} c\right )} e^{4}\right )} x}{48 \, {\left (a^{3} c^{6} x^{6} + 3 \, a^{4} c^{5} x^{4} + 3 \, a^{5} c^{4} x^{2} + a^{6} c^{3}\right )}} + \frac {{\left (4 \, B a c^{2} d^{3} e + 4 \, B a^{2} c d e^{3} + {\left (C a c^{2} + 5 \, A c^{3}\right )} d^{4} + 6 \, {\left (C a^{2} c + A a c^{2}\right )} d^{2} e^{2} + {\left (5 \, C a^{3} + A a^{2} c\right )} e^{4}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{16 \, \sqrt {a c} a^{3} c^{3}} \]
-1/48*(8*B*a^3*c^2*d^4 + 24*B*a^4*c*d^2*e^2 + 8*B*a^5*e^4 - 3*(4*B*a*c^4*d ^3*e + 4*B*a^2*c^3*d*e^3 + (C*a*c^4 + 5*A*c^5)*d^4 + 6*(C*a^2*c^3 + A*a*c^ 4)*d^2*e^2 - (11*C*a^3*c^2 - A*a^2*c^3)*e^4)*x^5 + 16*(C*a^4*c + 2*A*a^3*c ^2)*d^3*e + 16*(2*C*a^5 + A*a^4*c)*d*e^3 + 24*(4*C*a^3*c^2*d*e^3 + B*a^3*c ^2*e^4)*x^4 - 8*(4*B*a^2*c^3*d^3*e - 4*B*a^3*c^2*d*e^3 + (C*a^2*c^3 + 5*A* a*c^4)*d^4 - 6*(C*a^3*c^2 - A*a^2*c^3)*d^2*e^2 - (5*C*a^4*c + A*a^3*c^2)*e ^4)*x^3 + 24*(2*C*a^3*c^2*d^3*e + 3*B*a^3*c^2*d^2*e^2 + B*a^4*c*e^4 + 2*(2 *C*a^4*c + A*a^3*c^2)*d*e^3)*x^2 + 3*(4*B*a^3*c^2*d^3*e + 4*B*a^4*c*d*e^3 + (C*a^3*c^2 - 11*A*a^2*c^3)*d^4 + 6*(C*a^4*c + A*a^3*c^2)*d^2*e^2 + (5*C* a^5 + A*a^4*c)*e^4)*x)/(a^3*c^6*x^6 + 3*a^4*c^5*x^4 + 3*a^5*c^4*x^2 + a^6* c^3) + 1/16*(4*B*a*c^2*d^3*e + 4*B*a^2*c*d*e^3 + (C*a*c^2 + 5*A*c^3)*d^4 + 6*(C*a^2*c + A*a*c^2)*d^2*e^2 + (5*C*a^3 + A*a^2*c)*e^4)*arctan(c*x/sqrt( a*c))/(sqrt(a*c)*a^3*c^3)
Leaf count of result is larger than twice the leaf count of optimal. 659 vs. \(2 (217) = 434\).
Time = 0.27 (sec) , antiderivative size = 659, normalized size of antiderivative = 2.82 \[ \int \frac {(d+e x)^4 \left (A+B x+C x^2\right )}{\left (a+c x^2\right )^4} \, dx=\frac {{\left (C a c^{2} d^{4} + 5 \, A c^{3} d^{4} + 4 \, B a c^{2} d^{3} e + 6 \, C a^{2} c d^{2} e^{2} + 6 \, A a c^{2} d^{2} e^{2} + 4 \, B a^{2} c d e^{3} + 5 \, C a^{3} e^{4} + A a^{2} c e^{4}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{16 \, \sqrt {a c} a^{3} c^{3}} + \frac {3 \, C a c^{4} d^{4} x^{5} + 15 \, A c^{5} d^{4} x^{5} + 12 \, B a c^{4} d^{3} e x^{5} + 18 \, C a^{2} c^{3} d^{2} e^{2} x^{5} + 18 \, A a c^{4} d^{2} e^{2} x^{5} + 12 \, B a^{2} c^{3} d e^{3} x^{5} - 33 \, C a^{3} c^{2} e^{4} x^{5} + 3 \, A a^{2} c^{3} e^{4} x^{5} - 96 \, C a^{3} c^{2} d e^{3} x^{4} - 24 \, B a^{3} c^{2} e^{4} x^{4} + 8 \, C a^{2} c^{3} d^{4} x^{3} + 40 \, A a c^{4} d^{4} x^{3} + 32 \, B a^{2} c^{3} d^{3} e x^{3} - 48 \, C a^{3} c^{2} d^{2} e^{2} x^{3} + 48 \, A a^{2} c^{3} d^{2} e^{2} x^{3} - 32 \, B a^{3} c^{2} d e^{3} x^{3} - 40 \, C a^{4} c e^{4} x^{3} - 8 \, A a^{3} c^{2} e^{4} x^{3} - 48 \, C a^{3} c^{2} d^{3} e x^{2} - 72 \, B a^{3} c^{2} d^{2} e^{2} x^{2} - 96 \, C a^{4} c d e^{3} x^{2} - 48 \, A a^{3} c^{2} d e^{3} x^{2} - 24 \, B a^{4} c e^{4} x^{2} - 3 \, C a^{3} c^{2} d^{4} x + 33 \, A a^{2} c^{3} d^{4} x - 12 \, B a^{3} c^{2} d^{3} e x - 18 \, C a^{4} c d^{2} e^{2} x - 18 \, A a^{3} c^{2} d^{2} e^{2} x - 12 \, B a^{4} c d e^{3} x - 15 \, C a^{5} e^{4} x - 3 \, A a^{4} c e^{4} x - 8 \, B a^{3} c^{2} d^{4} - 16 \, C a^{4} c d^{3} e - 32 \, A a^{3} c^{2} d^{3} e - 24 \, B a^{4} c d^{2} e^{2} - 32 \, C a^{5} d e^{3} - 16 \, A a^{4} c d e^{3} - 8 \, B a^{5} e^{4}}{48 \, {\left (c x^{2} + a\right )}^{3} a^{3} c^{3}} \]
1/16*(C*a*c^2*d^4 + 5*A*c^3*d^4 + 4*B*a*c^2*d^3*e + 6*C*a^2*c*d^2*e^2 + 6* A*a*c^2*d^2*e^2 + 4*B*a^2*c*d*e^3 + 5*C*a^3*e^4 + A*a^2*c*e^4)*arctan(c*x/ sqrt(a*c))/(sqrt(a*c)*a^3*c^3) + 1/48*(3*C*a*c^4*d^4*x^5 + 15*A*c^5*d^4*x^ 5 + 12*B*a*c^4*d^3*e*x^5 + 18*C*a^2*c^3*d^2*e^2*x^5 + 18*A*a*c^4*d^2*e^2*x ^5 + 12*B*a^2*c^3*d*e^3*x^5 - 33*C*a^3*c^2*e^4*x^5 + 3*A*a^2*c^3*e^4*x^5 - 96*C*a^3*c^2*d*e^3*x^4 - 24*B*a^3*c^2*e^4*x^4 + 8*C*a^2*c^3*d^4*x^3 + 40* A*a*c^4*d^4*x^3 + 32*B*a^2*c^3*d^3*e*x^3 - 48*C*a^3*c^2*d^2*e^2*x^3 + 48*A *a^2*c^3*d^2*e^2*x^3 - 32*B*a^3*c^2*d*e^3*x^3 - 40*C*a^4*c*e^4*x^3 - 8*A*a ^3*c^2*e^4*x^3 - 48*C*a^3*c^2*d^3*e*x^2 - 72*B*a^3*c^2*d^2*e^2*x^2 - 96*C* a^4*c*d*e^3*x^2 - 48*A*a^3*c^2*d*e^3*x^2 - 24*B*a^4*c*e^4*x^2 - 3*C*a^3*c^ 2*d^4*x + 33*A*a^2*c^3*d^4*x - 12*B*a^3*c^2*d^3*e*x - 18*C*a^4*c*d^2*e^2*x - 18*A*a^3*c^2*d^2*e^2*x - 12*B*a^4*c*d*e^3*x - 15*C*a^5*e^4*x - 3*A*a^4* c*e^4*x - 8*B*a^3*c^2*d^4 - 16*C*a^4*c*d^3*e - 32*A*a^3*c^2*d^3*e - 24*B*a ^4*c*d^2*e^2 - 32*C*a^5*d*e^3 - 16*A*a^4*c*d*e^3 - 8*B*a^5*e^4)/((c*x^2 + a)^3*a^3*c^3)
Time = 13.86 (sec) , antiderivative size = 669, normalized size of antiderivative = 2.86 \[ \int \frac {(d+e x)^4 \left (A+B x+C x^2\right )}{\left (a+c x^2\right )^4} \, dx=\frac {\mathrm {atan}\left (\frac {\sqrt {c}\,x\,\left (c\,d^2+a\,e^2\right )\,\left (5\,C\,a^2\,e^2+C\,a\,c\,d^2+4\,B\,a\,c\,d\,e+A\,a\,c\,e^2+5\,A\,c^2\,d^2\right )}{\sqrt {a}\,\left (5\,C\,a^3\,e^4+6\,C\,a^2\,c\,d^2\,e^2+4\,B\,a^2\,c\,d\,e^3+A\,a^2\,c\,e^4+C\,a\,c^2\,d^4+4\,B\,a\,c^2\,d^3\,e+6\,A\,a\,c^2\,d^2\,e^2+5\,A\,c^3\,d^4\right )}\right )\,\left (c\,d^2+a\,e^2\right )\,\left (5\,C\,a^2\,e^2+C\,a\,c\,d^2+4\,B\,a\,c\,d\,e+A\,a\,c\,e^2+5\,A\,c^2\,d^2\right )}{16\,a^{7/2}\,c^{7/2}}-\frac {\frac {4\,C\,a^2\,d\,e^3+B\,a^2\,e^4+2\,C\,a\,c\,d^3\,e+3\,B\,a\,c\,d^2\,e^2+2\,A\,a\,c\,d\,e^3+B\,c^2\,d^4+4\,A\,c^2\,d^3\,e}{6\,c^3}+\frac {x^2\,\left (B\,a\,e^4+2\,A\,c\,d\,e^3+4\,C\,a\,d\,e^3+2\,C\,c\,d^3\,e+3\,B\,c\,d^2\,e^2\right )}{2\,c^2}+\frac {x^4\,\left (B\,e^4+4\,C\,d\,e^3\right )}{2\,c}+\frac {x\,\left (5\,C\,a^3\,e^4+6\,C\,a^2\,c\,d^2\,e^2+4\,B\,a^2\,c\,d\,e^3+A\,a^2\,c\,e^4+C\,a\,c^2\,d^4+4\,B\,a\,c^2\,d^3\,e+6\,A\,a\,c^2\,d^2\,e^2-11\,A\,c^3\,d^4\right )}{16\,a\,c^3}-\frac {x^3\,\left (-5\,C\,a^3\,e^4-6\,C\,a^2\,c\,d^2\,e^2-4\,B\,a^2\,c\,d\,e^3-A\,a^2\,c\,e^4+C\,a\,c^2\,d^4+4\,B\,a\,c^2\,d^3\,e+6\,A\,a\,c^2\,d^2\,e^2+5\,A\,c^3\,d^4\right )}{6\,a^2\,c^2}-\frac {x^5\,\left (-11\,C\,a^3\,e^4+6\,C\,a^2\,c\,d^2\,e^2+4\,B\,a^2\,c\,d\,e^3+A\,a^2\,c\,e^4+C\,a\,c^2\,d^4+4\,B\,a\,c^2\,d^3\,e+6\,A\,a\,c^2\,d^2\,e^2+5\,A\,c^3\,d^4\right )}{16\,a^3\,c}}{a^3+3\,a^2\,c\,x^2+3\,a\,c^2\,x^4+c^3\,x^6} \]
(atan((c^(1/2)*x*(a*e^2 + c*d^2)*(5*A*c^2*d^2 + 5*C*a^2*e^2 + A*a*c*e^2 + C*a*c*d^2 + 4*B*a*c*d*e))/(a^(1/2)*(5*A*c^3*d^4 + 5*C*a^3*e^4 + A*a^2*c*e^ 4 + C*a*c^2*d^4 + 6*A*a*c^2*d^2*e^2 + 6*C*a^2*c*d^2*e^2 + 4*B*a*c^2*d^3*e + 4*B*a^2*c*d*e^3)))*(a*e^2 + c*d^2)*(5*A*c^2*d^2 + 5*C*a^2*e^2 + A*a*c*e^ 2 + C*a*c*d^2 + 4*B*a*c*d*e))/(16*a^(7/2)*c^(7/2)) - ((B*a^2*e^4 + B*c^2*d ^4 + 4*A*c^2*d^3*e + 4*C*a^2*d*e^3 + 2*A*a*c*d*e^3 + 2*C*a*c*d^3*e + 3*B*a *c*d^2*e^2)/(6*c^3) + (x^2*(B*a*e^4 + 2*A*c*d*e^3 + 4*C*a*d*e^3 + 2*C*c*d^ 3*e + 3*B*c*d^2*e^2))/(2*c^2) + (x^4*(B*e^4 + 4*C*d*e^3))/(2*c) + (x*(5*C* a^3*e^4 - 11*A*c^3*d^4 + A*a^2*c*e^4 + C*a*c^2*d^4 + 6*A*a*c^2*d^2*e^2 + 6 *C*a^2*c*d^2*e^2 + 4*B*a*c^2*d^3*e + 4*B*a^2*c*d*e^3))/(16*a*c^3) - (x^3*( 5*A*c^3*d^4 - 5*C*a^3*e^4 - A*a^2*c*e^4 + C*a*c^2*d^4 + 6*A*a*c^2*d^2*e^2 - 6*C*a^2*c*d^2*e^2 + 4*B*a*c^2*d^3*e - 4*B*a^2*c*d*e^3))/(6*a^2*c^2) - (x ^5*(5*A*c^3*d^4 - 11*C*a^3*e^4 + A*a^2*c*e^4 + C*a*c^2*d^4 + 6*A*a*c^2*d^2 *e^2 + 6*C*a^2*c*d^2*e^2 + 4*B*a*c^2*d^3*e + 4*B*a^2*c*d*e^3))/(16*a^3*c)) /(a^3 + c^3*x^6 + 3*a^2*c*x^2 + 3*a*c^2*x^4)