Integrand size = 22, antiderivative size = 216 \[ \int \frac {(A+B x) (d+e x)^4}{\left (a+c x^2\right )^3} \, dx=-\frac {(d+e x)^3 (a (B d+A e)-(A c d-a B e) x)}{4 a c \left (a+c x^2\right )^2}-\frac {(d+e x) \left (a e \left (8 a B d e+3 A \left (c d^2+a e^2\right )\right )+\left (4 a^2 B e^3-c d \left (3 A c d^2+a e (4 B d+3 A e)\right )\right ) x\right )}{8 a^2 c^2 \left (a+c x^2\right )}+\frac {\left (3 A \left (c d^2+a e^2\right )^2+4 a B d e \left (c d^2+3 a e^2\right )\right ) \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{8 a^{5/2} c^{5/2}}+\frac {B e^4 \log \left (a+c x^2\right )}{2 c^3} \]
-1/4*(e*x+d)^3*(a*(A*e+B*d)-(A*c*d-B*a*e)*x)/a/c/(c*x^2+a)^2-1/8*(e*x+d)*( a*e*(8*B*a*d*e+3*A*(a*e^2+c*d^2))+(4*a^2*B*e^3-c*d*(3*A*c*d^2+a*e*(3*A*e+4 *B*d)))*x)/a^2/c^2/(c*x^2+a)+1/8*(3*A*(a*e^2+c*d^2)^2+4*a*B*d*e*(3*a*e^2+c *d^2))*arctan(x*c^(1/2)/a^(1/2))/a^(5/2)/c^(5/2)+1/2*B*e^4*ln(c*x^2+a)/c^3
Time = 0.13 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.22 \[ \int \frac {(A+B x) (d+e x)^4}{\left (a+c x^2\right )^3} \, dx=\frac {\frac {-2 a^3 B e^4+2 A c^3 d^4 x+2 a^2 c e^2 (A e (4 d+e x)+2 B d (3 d+2 e x))-2 a c^2 d^2 (2 A e (2 d+3 e x)+B d (d+4 e x))}{a \left (a+c x^2\right )^2}+\frac {8 a^3 B e^4+3 A c^3 d^4 x+2 a c^2 d^2 e (2 B d+3 A e) x-a^2 c e^2 (4 B d (6 d+5 e x)+A e (16 d+5 e x))}{a^2 \left (a+c x^2\right )}+\frac {\sqrt {c} \left (3 A \left (c d^2+a e^2\right )^2+4 a B d e \left (c d^2+3 a e^2\right )\right ) \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{a^{5/2}}+4 B e^4 \log \left (a+c x^2\right )}{8 c^3} \]
((-2*a^3*B*e^4 + 2*A*c^3*d^4*x + 2*a^2*c*e^2*(A*e*(4*d + e*x) + 2*B*d*(3*d + 2*e*x)) - 2*a*c^2*d^2*(2*A*e*(2*d + 3*e*x) + B*d*(d + 4*e*x)))/(a*(a + c*x^2)^2) + (8*a^3*B*e^4 + 3*A*c^3*d^4*x + 2*a*c^2*d^2*e*(2*B*d + 3*A*e)*x - a^2*c*e^2*(4*B*d*(6*d + 5*e*x) + A*e*(16*d + 5*e*x)))/(a^2*(a + c*x^2)) + (Sqrt[c]*(3*A*(c*d^2 + a*e^2)^2 + 4*a*B*d*e*(c*d^2 + 3*a*e^2))*ArcTan[( Sqrt[c]*x)/Sqrt[a]])/a^(5/2) + 4*B*e^4*Log[a + c*x^2])/(8*c^3)
Time = 0.39 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.09, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {684, 684, 452, 218, 240}
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.
| \(\displaystyle \int \frac {(A+B x) (d+e x)^4}{\left (a+c x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 684 |
| \(\displaystyle \frac {\int \frac {(d+e x)^2 \left (3 A c d^2+a e (4 B d+3 A e)+4 a B e^2 x\right )}{\left (c x^2+a\right )^2}dx}{4 a c}-\frac {(d+e x)^3 (a (A e+B d)-x (A c d-a B e))}{4 a c \left (a+c x^2\right )^2}\) |
| \(\Big \downarrow \) 684 |
| \(\displaystyle \frac {\frac {\int \frac {8 a^2 B x e^4+4 a B d \left (c d^2+3 a e^2\right ) e+3 A \left (c d^2+a e^2\right )^2}{c x^2+a}dx}{2 a c}-\frac {(d+e x) \left (x \left (4 a^2 B e^3-c d \left (a e (3 A e+4 B d)+3 A c d^2\right )\right )+a e \left (3 A \left (a e^2+c d^2\right )+8 a B d e\right )\right )}{2 a c \left (a+c x^2\right )}}{4 a c}-\frac {(d+e x)^3 (a (A e+B d)-x (A c d-a B e))}{4 a c \left (a+c x^2\right )^2}\) |
| \(\Big \downarrow \) 452 |
| \(\displaystyle \frac {\frac {8 a^2 B e^4 \int \frac {x}{c x^2+a}dx+\left (3 A \left (a e^2+c d^2\right )^2+4 a B d e \left (3 a e^2+c d^2\right )\right ) \int \frac {1}{c x^2+a}dx}{2 a c}-\frac {(d+e x) \left (x \left (4 a^2 B e^3-c d \left (a e (3 A e+4 B d)+3 A c d^2\right )\right )+a e \left (3 A \left (a e^2+c d^2\right )+8 a B d e\right )\right )}{2 a c \left (a+c x^2\right )}}{4 a c}-\frac {(d+e x)^3 (a (A e+B d)-x (A c d-a B e))}{4 a c \left (a+c x^2\right )^2}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {8 a^2 B e^4 \int \frac {x}{c x^2+a}dx+\frac {\arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (3 A \left (a e^2+c d^2\right )^2+4 a B d e \left (3 a e^2+c d^2\right )\right )}{\sqrt {a} \sqrt {c}}}{2 a c}-\frac {(d+e x) \left (x \left (4 a^2 B e^3-c d \left (a e (3 A e+4 B d)+3 A c d^2\right )\right )+a e \left (3 A \left (a e^2+c d^2\right )+8 a B d e\right )\right )}{2 a c \left (a+c x^2\right )}}{4 a c}-\frac {(d+e x)^3 (a (A e+B d)-x (A c d-a B e))}{4 a c \left (a+c x^2\right )^2}\) |
| \(\Big \downarrow \) 240 |
| \(\displaystyle \frac {\frac {\frac {4 a^2 B e^4 \log \left (a+c x^2\right )}{c}+\frac {\arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (3 A \left (a e^2+c d^2\right )^2+4 a B d e \left (3 a e^2+c d^2\right )\right )}{\sqrt {a} \sqrt {c}}}{2 a c}-\frac {(d+e x) \left (x \left (4 a^2 B e^3-c d \left (a e (3 A e+4 B d)+3 A c d^2\right )\right )+a e \left (3 A \left (a e^2+c d^2\right )+8 a B d e\right )\right )}{2 a c \left (a+c x^2\right )}}{4 a c}-\frac {(d+e x)^3 (a (A e+B d)-x (A c d-a B e))}{4 a c \left (a+c x^2\right )^2}\) |
-1/4*((d + e*x)^3*(a*(B*d + A*e) - (A*c*d - a*B*e)*x))/(a*c*(a + c*x^2)^2) + (-1/2*((d + e*x)*(a*e*(8*a*B*d*e + 3*A*(c*d^2 + a*e^2)) + (4*a^2*B*e^3 - c*d*(3*A*c*d^2 + a*e*(4*B*d + 3*A*e)))*x))/(a*c*(a + c*x^2)) + (((3*A*(c *d^2 + a*e^2)^2 + 4*a*B*d*e*(c*d^2 + 3*a*e^2))*ArcTan[(Sqrt[c]*x)/Sqrt[a]] )/(Sqrt[a]*Sqrt[c]) + (4*a^2*B*e^4*Log[a + c*x^2])/c)/(2*a*c))/(4*a*c)
3.14.47.3.1 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[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x ^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
Int[((c_) + (d_.)*(x_))/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[c Int[1/ (a + b*x^2), x], x] + Simp[d Int[x/(a + b*x^2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c^2 + a*d^2, 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(d + e*x)^(m - 1)*(a + c*x^2)^(p + 1)*((a*(e*f + d*g ) - (c*d*f - a*e*g)*x)/(2*a*c*(p + 1))), x] - Simp[1/(2*a*c*(p + 1)) Int[ (d + e*x)^(m - 2)*(a + c*x^2)^(p + 1)*Simp[a*e*(e*f*(m - 1) + d*g*m) - c*d^ 2*f*(2*p + 3) + e*(a*e*g*m - c*d*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a , c, d, e, f, g}, x] && LtQ[p, -1] && GtQ[m, 1] && (EqQ[d, 0] || (EqQ[m, 2] && EqQ[p, -3] && RationalQ[a, c, d, e, f, g]) || !ILtQ[m + 2*p + 3, 0])
Time = 0.38 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.43
| method | result | size |
| default | \(\frac {-\frac {\left (5 A \,a^{2} e^{4}-6 A a c \,d^{2} e^{2}-3 d^{4} A \,c^{2}+20 B \,a^{2} d \,e^{3}-4 B a c \,d^{3} e \right ) x^{3}}{8 c \,a^{2}}-\frac {e^{2} \left (2 A c d e -B a \,e^{2}+3 B c \,d^{2}\right ) x^{2}}{c^{2}}-\frac {\left (3 A \,a^{2} e^{4}+6 A a c \,d^{2} e^{2}-5 d^{4} A \,c^{2}+12 B \,a^{2} d \,e^{3}+4 B a c \,d^{3} e \right ) x}{8 a \,c^{2}}-\frac {4 A a c d \,e^{3}+4 A \,c^{2} d^{3} e -3 B \,e^{4} a^{2}+6 B a c \,d^{2} e^{2}+B \,c^{2} d^{4}}{4 c^{3}}}{\left (c \,x^{2}+a \right )^{2}}+\frac {\frac {4 B \,e^{4} a^{2} \ln \left (c \,x^{2}+a \right )}{c}+\frac {\left (3 A \,a^{2} e^{4}+6 A a c \,d^{2} e^{2}+3 d^{4} A \,c^{2}+12 B \,a^{2} d \,e^{3}+4 B a c \,d^{3} e \right ) \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}}}{8 a^{2} c^{2}}\) | \(309\) |
| risch | \(\frac {-\frac {\left (5 A \,a^{2} e^{4}-6 A a c \,d^{2} e^{2}-3 d^{4} A \,c^{2}+20 B \,a^{2} d \,e^{3}-4 B a c \,d^{3} e \right ) x^{3}}{8 c \,a^{2}}-\frac {e^{2} \left (2 A c d e -B a \,e^{2}+3 B c \,d^{2}\right ) x^{2}}{c^{2}}-\frac {\left (3 A \,a^{2} e^{4}+6 A a c \,d^{2} e^{2}-5 d^{4} A \,c^{2}+12 B \,a^{2} d \,e^{3}+4 B a c \,d^{3} e \right ) x}{8 a \,c^{2}}-\frac {4 A a c d \,e^{3}+4 A \,c^{2} d^{3} e -3 B \,e^{4} a^{2}+6 B a c \,d^{2} e^{2}+B \,c^{2} d^{4}}{4 c^{3}}}{\left (c \,x^{2}+a \right )^{2}}+\frac {\ln \left (3 A \,a^{3} e^{4}+6 A \,a^{2} c \,d^{2} e^{2}+3 d^{4} A a \,c^{2}+12 B \,a^{3} d \,e^{3}+4 B \,a^{2} c \,d^{3} e -\sqrt {-a c \left (3 A \,a^{2} e^{4}+6 A a c \,d^{2} e^{2}+3 d^{4} A \,c^{2}+12 B \,a^{2} d \,e^{3}+4 B a c \,d^{3} e \right )^{2}}\, x \right ) B \,e^{4}}{2 c^{3}}+\frac {\ln \left (3 A \,a^{3} e^{4}+6 A \,a^{2} c \,d^{2} e^{2}+3 d^{4} A a \,c^{2}+12 B \,a^{3} d \,e^{3}+4 B \,a^{2} c \,d^{3} e -\sqrt {-a c \left (3 A \,a^{2} e^{4}+6 A a c \,d^{2} e^{2}+3 d^{4} A \,c^{2}+12 B \,a^{2} d \,e^{3}+4 B a c \,d^{3} e \right )^{2}}\, x \right ) \sqrt {-a c \left (3 A \,a^{2} e^{4}+6 A a c \,d^{2} e^{2}+3 d^{4} A \,c^{2}+12 B \,a^{2} d \,e^{3}+4 B a c \,d^{3} e \right )^{2}}}{16 a^{3} c^{3}}+\frac {\ln \left (3 A \,a^{3} e^{4}+6 A \,a^{2} c \,d^{2} e^{2}+3 d^{4} A a \,c^{2}+12 B \,a^{3} d \,e^{3}+4 B \,a^{2} c \,d^{3} e +\sqrt {-a c \left (3 A \,a^{2} e^{4}+6 A a c \,d^{2} e^{2}+3 d^{4} A \,c^{2}+12 B \,a^{2} d \,e^{3}+4 B a c \,d^{3} e \right )^{2}}\, x \right ) B \,e^{4}}{2 c^{3}}-\frac {\ln \left (3 A \,a^{3} e^{4}+6 A \,a^{2} c \,d^{2} e^{2}+3 d^{4} A a \,c^{2}+12 B \,a^{3} d \,e^{3}+4 B \,a^{2} c \,d^{3} e +\sqrt {-a c \left (3 A \,a^{2} e^{4}+6 A a c \,d^{2} e^{2}+3 d^{4} A \,c^{2}+12 B \,a^{2} d \,e^{3}+4 B a c \,d^{3} e \right )^{2}}\, x \right ) \sqrt {-a c \left (3 A \,a^{2} e^{4}+6 A a c \,d^{2} e^{2}+3 d^{4} A \,c^{2}+12 B \,a^{2} d \,e^{3}+4 B a c \,d^{3} e \right )^{2}}}{16 a^{3} c^{3}}\) | \(822\) |
(-1/8*(5*A*a^2*e^4-6*A*a*c*d^2*e^2-3*A*c^2*d^4+20*B*a^2*d*e^3-4*B*a*c*d^3* e)/c/a^2*x^3-e^2*(2*A*c*d*e-B*a*e^2+3*B*c*d^2)/c^2*x^2-1/8*(3*A*a^2*e^4+6* A*a*c*d^2*e^2-5*A*c^2*d^4+12*B*a^2*d*e^3+4*B*a*c*d^3*e)/a/c^2*x-1/4*(4*A*a *c*d*e^3+4*A*c^2*d^3*e-3*B*a^2*e^4+6*B*a*c*d^2*e^2+B*c^2*d^4)/c^3)/(c*x^2+ a)^2+1/8/a^2/c^2*(4*B*e^4*a^2/c*ln(c*x^2+a)+(3*A*a^2*e^4+6*A*a*c*d^2*e^2+3 *A*c^2*d^4+12*B*a^2*d*e^3+4*B*a*c*d^3*e)/(a*c)^(1/2)*arctan(c*x/(a*c)^(1/2 )))
Leaf count of result is larger than twice the leaf count of optimal. 518 vs. \(2 (200) = 400\).
Time = 0.44 (sec) , antiderivative size = 1055, normalized size of antiderivative = 4.88 \[ \int \frac {(A+B x) (d+e x)^4}{\left (a+c x^2\right )^3} \, dx=\left [-\frac {4 \, B a^{3} c^{2} d^{4} + 16 \, A a^{3} c^{2} d^{3} e + 24 \, B a^{4} c d^{2} e^{2} + 16 \, A a^{4} c d e^{3} - 12 \, B a^{5} e^{4} - 2 \, {\left (3 \, A a c^{4} d^{4} + 4 \, B a^{2} c^{3} d^{3} e + 6 \, A a^{2} c^{3} d^{2} e^{2} - 20 \, B a^{3} c^{2} d e^{3} - 5 \, A a^{3} c^{2} e^{4}\right )} x^{3} + 16 \, {\left (3 \, B a^{3} c^{2} d^{2} e^{2} + 2 \, A a^{3} c^{2} d e^{3} - B a^{4} c e^{4}\right )} x^{2} + {\left (3 \, A a^{2} c^{2} d^{4} + 4 \, B a^{3} c d^{3} e + 6 \, A a^{3} c d^{2} e^{2} + 12 \, B a^{4} d e^{3} + 3 \, A a^{4} e^{4} + {\left (3 \, A c^{4} d^{4} + 4 \, B a c^{3} d^{3} e + 6 \, A a c^{3} d^{2} e^{2} + 12 \, B a^{2} c^{2} d e^{3} + 3 \, A a^{2} c^{2} e^{4}\right )} x^{4} + 2 \, {\left (3 \, A a c^{3} d^{4} + 4 \, B a^{2} c^{2} d^{3} e + 6 \, A a^{2} c^{2} d^{2} e^{2} + 12 \, B a^{3} c d e^{3} + 3 \, A a^{3} c e^{4}\right )} x^{2}\right )} \sqrt {-a c} \log \left (\frac {c x^{2} - 2 \, \sqrt {-a c} x - a}{c x^{2} + a}\right ) - 2 \, {\left (5 \, A a^{2} c^{3} d^{4} - 4 \, B a^{3} c^{2} d^{3} e - 6 \, A a^{3} c^{2} d^{2} e^{2} - 12 \, B a^{4} c d e^{3} - 3 \, A a^{4} c e^{4}\right )} x - 8 \, {\left (B a^{3} c^{2} e^{4} x^{4} + 2 \, B a^{4} c e^{4} x^{2} + B a^{5} e^{4}\right )} \log \left (c x^{2} + a\right )}{16 \, {\left (a^{3} c^{5} x^{4} + 2 \, a^{4} c^{4} x^{2} + a^{5} c^{3}\right )}}, -\frac {2 \, B a^{3} c^{2} d^{4} + 8 \, A a^{3} c^{2} d^{3} e + 12 \, B a^{4} c d^{2} e^{2} + 8 \, A a^{4} c d e^{3} - 6 \, B a^{5} e^{4} - {\left (3 \, A a c^{4} d^{4} + 4 \, B a^{2} c^{3} d^{3} e + 6 \, A a^{2} c^{3} d^{2} e^{2} - 20 \, B a^{3} c^{2} d e^{3} - 5 \, A a^{3} c^{2} e^{4}\right )} x^{3} + 8 \, {\left (3 \, B a^{3} c^{2} d^{2} e^{2} + 2 \, A a^{3} c^{2} d e^{3} - B a^{4} c e^{4}\right )} x^{2} - {\left (3 \, A a^{2} c^{2} d^{4} + 4 \, B a^{3} c d^{3} e + 6 \, A a^{3} c d^{2} e^{2} + 12 \, B a^{4} d e^{3} + 3 \, A a^{4} e^{4} + {\left (3 \, A c^{4} d^{4} + 4 \, B a c^{3} d^{3} e + 6 \, A a c^{3} d^{2} e^{2} + 12 \, B a^{2} c^{2} d e^{3} + 3 \, A a^{2} c^{2} e^{4}\right )} x^{4} + 2 \, {\left (3 \, A a c^{3} d^{4} + 4 \, B a^{2} c^{2} d^{3} e + 6 \, A a^{2} c^{2} d^{2} e^{2} + 12 \, B a^{3} c d e^{3} + 3 \, A a^{3} c e^{4}\right )} x^{2}\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c} x}{a}\right ) - {\left (5 \, A a^{2} c^{3} d^{4} - 4 \, B a^{3} c^{2} d^{3} e - 6 \, A a^{3} c^{2} d^{2} e^{2} - 12 \, B a^{4} c d e^{3} - 3 \, A a^{4} c e^{4}\right )} x - 4 \, {\left (B a^{3} c^{2} e^{4} x^{4} + 2 \, B a^{4} c e^{4} x^{2} + B a^{5} e^{4}\right )} \log \left (c x^{2} + a\right )}{8 \, {\left (a^{3} c^{5} x^{4} + 2 \, a^{4} c^{4} x^{2} + a^{5} c^{3}\right )}}\right ] \]
[-1/16*(4*B*a^3*c^2*d^4 + 16*A*a^3*c^2*d^3*e + 24*B*a^4*c*d^2*e^2 + 16*A*a ^4*c*d*e^3 - 12*B*a^5*e^4 - 2*(3*A*a*c^4*d^4 + 4*B*a^2*c^3*d^3*e + 6*A*a^2 *c^3*d^2*e^2 - 20*B*a^3*c^2*d*e^3 - 5*A*a^3*c^2*e^4)*x^3 + 16*(3*B*a^3*c^2 *d^2*e^2 + 2*A*a^3*c^2*d*e^3 - B*a^4*c*e^4)*x^2 + (3*A*a^2*c^2*d^4 + 4*B*a ^3*c*d^3*e + 6*A*a^3*c*d^2*e^2 + 12*B*a^4*d*e^3 + 3*A*a^4*e^4 + (3*A*c^4*d ^4 + 4*B*a*c^3*d^3*e + 6*A*a*c^3*d^2*e^2 + 12*B*a^2*c^2*d*e^3 + 3*A*a^2*c^ 2*e^4)*x^4 + 2*(3*A*a*c^3*d^4 + 4*B*a^2*c^2*d^3*e + 6*A*a^2*c^2*d^2*e^2 + 12*B*a^3*c*d*e^3 + 3*A*a^3*c*e^4)*x^2)*sqrt(-a*c)*log((c*x^2 - 2*sqrt(-a*c )*x - a)/(c*x^2 + a)) - 2*(5*A*a^2*c^3*d^4 - 4*B*a^3*c^2*d^3*e - 6*A*a^3*c ^2*d^2*e^2 - 12*B*a^4*c*d*e^3 - 3*A*a^4*c*e^4)*x - 8*(B*a^3*c^2*e^4*x^4 + 2*B*a^4*c*e^4*x^2 + B*a^5*e^4)*log(c*x^2 + a))/(a^3*c^5*x^4 + 2*a^4*c^4*x^ 2 + a^5*c^3), -1/8*(2*B*a^3*c^2*d^4 + 8*A*a^3*c^2*d^3*e + 12*B*a^4*c*d^2*e ^2 + 8*A*a^4*c*d*e^3 - 6*B*a^5*e^4 - (3*A*a*c^4*d^4 + 4*B*a^2*c^3*d^3*e + 6*A*a^2*c^3*d^2*e^2 - 20*B*a^3*c^2*d*e^3 - 5*A*a^3*c^2*e^4)*x^3 + 8*(3*B*a ^3*c^2*d^2*e^2 + 2*A*a^3*c^2*d*e^3 - B*a^4*c*e^4)*x^2 - (3*A*a^2*c^2*d^4 + 4*B*a^3*c*d^3*e + 6*A*a^3*c*d^2*e^2 + 12*B*a^4*d*e^3 + 3*A*a^4*e^4 + (3*A *c^4*d^4 + 4*B*a*c^3*d^3*e + 6*A*a*c^3*d^2*e^2 + 12*B*a^2*c^2*d*e^3 + 3*A* a^2*c^2*e^4)*x^4 + 2*(3*A*a*c^3*d^4 + 4*B*a^2*c^2*d^3*e + 6*A*a^2*c^2*d^2* e^2 + 12*B*a^3*c*d*e^3 + 3*A*a^3*c*e^4)*x^2)*sqrt(a*c)*arctan(sqrt(a*c)*x/ a) - (5*A*a^2*c^3*d^4 - 4*B*a^3*c^2*d^3*e - 6*A*a^3*c^2*d^2*e^2 - 12*B*...
Timed out. \[ \int \frac {(A+B x) (d+e x)^4}{\left (a+c x^2\right )^3} \, dx=\text {Timed out} \]
Time = 0.33 (sec) , antiderivative size = 352, normalized size of antiderivative = 1.63 \[ \int \frac {(A+B x) (d+e x)^4}{\left (a+c x^2\right )^3} \, dx=\frac {B e^{4} \log \left (c x^{2} + a\right )}{2 \, c^{3}} - \frac {2 \, B a^{2} c^{2} d^{4} + 8 \, A a^{2} c^{2} d^{3} e + 12 \, B a^{3} c d^{2} e^{2} + 8 \, A a^{3} c d e^{3} - 6 \, B a^{4} e^{4} - {\left (3 \, A c^{4} d^{4} + 4 \, B a c^{3} d^{3} e + 6 \, A a c^{3} d^{2} e^{2} - 20 \, B a^{2} c^{2} d e^{3} - 5 \, A a^{2} c^{2} e^{4}\right )} x^{3} + 8 \, {\left (3 \, B a^{2} c^{2} d^{2} e^{2} + 2 \, A a^{2} c^{2} d e^{3} - B a^{3} c e^{4}\right )} x^{2} - {\left (5 \, A a c^{3} d^{4} - 4 \, B a^{2} c^{2} d^{3} e - 6 \, A a^{2} c^{2} d^{2} e^{2} - 12 \, B a^{3} c d e^{3} - 3 \, A a^{3} c e^{4}\right )} x}{8 \, {\left (a^{2} c^{5} x^{4} + 2 \, a^{3} c^{4} x^{2} + a^{4} c^{3}\right )}} + \frac {{\left (3 \, A c^{2} d^{4} + 4 \, B a c d^{3} e + 6 \, A a c d^{2} e^{2} + 12 \, B a^{2} d e^{3} + 3 \, A a^{2} e^{4}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 \, \sqrt {a c} a^{2} c^{2}} \]
1/2*B*e^4*log(c*x^2 + a)/c^3 - 1/8*(2*B*a^2*c^2*d^4 + 8*A*a^2*c^2*d^3*e + 12*B*a^3*c*d^2*e^2 + 8*A*a^3*c*d*e^3 - 6*B*a^4*e^4 - (3*A*c^4*d^4 + 4*B*a* c^3*d^3*e + 6*A*a*c^3*d^2*e^2 - 20*B*a^2*c^2*d*e^3 - 5*A*a^2*c^2*e^4)*x^3 + 8*(3*B*a^2*c^2*d^2*e^2 + 2*A*a^2*c^2*d*e^3 - B*a^3*c*e^4)*x^2 - (5*A*a*c ^3*d^4 - 4*B*a^2*c^2*d^3*e - 6*A*a^2*c^2*d^2*e^2 - 12*B*a^3*c*d*e^3 - 3*A* a^3*c*e^4)*x)/(a^2*c^5*x^4 + 2*a^3*c^4*x^2 + a^4*c^3) + 1/8*(3*A*c^2*d^4 + 4*B*a*c*d^3*e + 6*A*a*c*d^2*e^2 + 12*B*a^2*d*e^3 + 3*A*a^2*e^4)*arctan(c* x/sqrt(a*c))/(sqrt(a*c)*a^2*c^2)
Time = 0.27 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.50 \[ \int \frac {(A+B x) (d+e x)^4}{\left (a+c x^2\right )^3} \, dx=\frac {B e^{4} \log \left (c x^{2} + a\right )}{2 \, c^{3}} + \frac {{\left (3 \, A c^{2} d^{4} + 4 \, B a c d^{3} e + 6 \, A a c d^{2} e^{2} + 12 \, B a^{2} d e^{3} + 3 \, A a^{2} e^{4}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 \, \sqrt {a c} a^{2} c^{2}} + \frac {{\left (3 \, A c^{3} d^{4} + 4 \, B a c^{2} d^{3} e + 6 \, A a c^{2} d^{2} e^{2} - 20 \, B a^{2} c d e^{3} - 5 \, A a^{2} c e^{4}\right )} x^{3} - 8 \, {\left (3 \, B a^{2} c d^{2} e^{2} + 2 \, A a^{2} c d e^{3} - B a^{3} e^{4}\right )} x^{2} + {\left (5 \, A a c^{2} d^{4} - 4 \, B a^{2} c d^{3} e - 6 \, A a^{2} c d^{2} e^{2} - 12 \, B a^{3} d e^{3} - 3 \, A a^{3} e^{4}\right )} x - \frac {2 \, {\left (B a^{2} c^{2} d^{4} + 4 \, A a^{2} c^{2} d^{3} e + 6 \, B a^{3} c d^{2} e^{2} + 4 \, A a^{3} c d e^{3} - 3 \, B a^{4} e^{4}\right )}}{c}}{8 \, {\left (c x^{2} + a\right )}^{2} a^{2} c^{2}} \]
1/2*B*e^4*log(c*x^2 + a)/c^3 + 1/8*(3*A*c^2*d^4 + 4*B*a*c*d^3*e + 6*A*a*c* d^2*e^2 + 12*B*a^2*d*e^3 + 3*A*a^2*e^4)*arctan(c*x/sqrt(a*c))/(sqrt(a*c)*a ^2*c^2) + 1/8*((3*A*c^3*d^4 + 4*B*a*c^2*d^3*e + 6*A*a*c^2*d^2*e^2 - 20*B*a ^2*c*d*e^3 - 5*A*a^2*c*e^4)*x^3 - 8*(3*B*a^2*c*d^2*e^2 + 2*A*a^2*c*d*e^3 - B*a^3*e^4)*x^2 + (5*A*a*c^2*d^4 - 4*B*a^2*c*d^3*e - 6*A*a^2*c*d^2*e^2 - 1 2*B*a^3*d*e^3 - 3*A*a^3*e^4)*x - 2*(B*a^2*c^2*d^4 + 4*A*a^2*c^2*d^3*e + 6* B*a^3*c*d^2*e^2 + 4*A*a^3*c*d*e^3 - 3*B*a^4*e^4)/c)/((c*x^2 + a)^2*a^2*c^2 )
Time = 11.66 (sec) , antiderivative size = 763, normalized size of antiderivative = 3.53 \[ \int \frac {(A+B x) (d+e x)^4}{\left (a+c x^2\right )^3} \, dx=\frac {5\,A\,d^4\,x}{8\,\left (a^3+2\,a^2\,c\,x^2+a\,c^2\,x^4\right )}-\frac {B\,d^4}{4\,\left (a^2\,c+2\,a\,c^2\,x^2+c^3\,x^4\right )}+\frac {3\,B\,a^2\,e^4}{4\,\left (a^2\,c^3+2\,a\,c^4\,x^2+c^5\,x^4\right )}-\frac {A\,d^3\,e}{a^2\,c+2\,a\,c^2\,x^2+c^3\,x^4}-\frac {5\,A\,e^4\,x^3}{8\,\left (a^2\,c+2\,a\,c^2\,x^2+c^3\,x^4\right )}+\frac {B\,e^4\,\ln \left (c\,x^2+a\right )}{2\,c^3}-\frac {A\,a\,d\,e^3}{a^2\,c^2+2\,a\,c^3\,x^2+c^4\,x^4}+\frac {3\,A\,c\,d^4\,x^3}{8\,\left (a^4+2\,a^3\,c\,x^2+a^2\,c^2\,x^4\right )}-\frac {3\,A\,a\,e^4\,x}{8\,\left (a^2\,c^2+2\,a\,c^3\,x^2+c^4\,x^4\right )}+\frac {3\,A\,d^2\,e^2\,x^3}{4\,\left (a^3+2\,a^2\,c\,x^2+a\,c^2\,x^4\right )}-\frac {3\,A\,d^2\,e^2\,x}{4\,\left (a^2\,c+2\,a\,c^2\,x^2+c^3\,x^4\right )}-\frac {2\,A\,d\,e^3\,x^2}{a^2\,c+2\,a\,c^2\,x^2+c^3\,x^4}-\frac {5\,B\,d\,e^3\,x^3}{2\,\left (a^2\,c+2\,a\,c^2\,x^2+c^3\,x^4\right )}-\frac {3\,B\,a\,d^2\,e^2}{2\,\left (a^2\,c^2+2\,a\,c^3\,x^2+c^4\,x^4\right )}+\frac {B\,a\,e^4\,x^2}{a^2\,c^2+2\,a\,c^3\,x^2+c^4\,x^4}-\frac {3\,B\,d^2\,e^2\,x^2}{a^2\,c+2\,a\,c^2\,x^2+c^3\,x^4}+\frac {3\,A\,d^4\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )}{8\,a^{5/2}\,\sqrt {c}}+\frac {3\,A\,e^4\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )}{8\,\sqrt {a}\,c^{5/2}}+\frac {B\,d^3\,e\,x^3}{2\,\left (a^3+2\,a^2\,c\,x^2+a\,c^2\,x^4\right )}-\frac {B\,d^3\,e\,x}{2\,\left (a^2\,c+2\,a\,c^2\,x^2+c^3\,x^4\right )}+\frac {3\,B\,d\,e^3\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )}{2\,\sqrt {a}\,c^{5/2}}+\frac {B\,d^3\,e\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )}{2\,a^{3/2}\,c^{3/2}}+\frac {3\,A\,d^2\,e^2\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )}{4\,a^{3/2}\,c^{3/2}}-\frac {3\,B\,a\,d\,e^3\,x}{2\,\left (a^2\,c^2+2\,a\,c^3\,x^2+c^4\,x^4\right )} \]
(5*A*d^4*x)/(8*(a^3 + 2*a^2*c*x^2 + a*c^2*x^4)) - (B*d^4)/(4*(a^2*c + c^3* x^4 + 2*a*c^2*x^2)) + (3*B*a^2*e^4)/(4*(a^2*c^3 + c^5*x^4 + 2*a*c^4*x^2)) - (A*d^3*e)/(a^2*c + c^3*x^4 + 2*a*c^2*x^2) - (5*A*e^4*x^3)/(8*(a^2*c + c^ 3*x^4 + 2*a*c^2*x^2)) + (B*e^4*log(a + c*x^2))/(2*c^3) - (A*a*d*e^3)/(a^2* c^2 + c^4*x^4 + 2*a*c^3*x^2) + (3*A*c*d^4*x^3)/(8*(a^4 + 2*a^3*c*x^2 + a^2 *c^2*x^4)) - (3*A*a*e^4*x)/(8*(a^2*c^2 + c^4*x^4 + 2*a*c^3*x^2)) + (3*A*d^ 2*e^2*x^3)/(4*(a^3 + 2*a^2*c*x^2 + a*c^2*x^4)) - (3*A*d^2*e^2*x)/(4*(a^2*c + c^3*x^4 + 2*a*c^2*x^2)) - (2*A*d*e^3*x^2)/(a^2*c + c^3*x^4 + 2*a*c^2*x^ 2) - (5*B*d*e^3*x^3)/(2*(a^2*c + c^3*x^4 + 2*a*c^2*x^2)) - (3*B*a*d^2*e^2) /(2*(a^2*c^2 + c^4*x^4 + 2*a*c^3*x^2)) + (B*a*e^4*x^2)/(a^2*c^2 + c^4*x^4 + 2*a*c^3*x^2) - (3*B*d^2*e^2*x^2)/(a^2*c + c^3*x^4 + 2*a*c^2*x^2) + (3*A* d^4*atan((c^(1/2)*x)/a^(1/2)))/(8*a^(5/2)*c^(1/2)) + (3*A*e^4*atan((c^(1/2 )*x)/a^(1/2)))/(8*a^(1/2)*c^(5/2)) + (B*d^3*e*x^3)/(2*(a^3 + 2*a^2*c*x^2 + a*c^2*x^4)) - (B*d^3*e*x)/(2*(a^2*c + c^3*x^4 + 2*a*c^2*x^2)) + (3*B*d*e^ 3*atan((c^(1/2)*x)/a^(1/2)))/(2*a^(1/2)*c^(5/2)) + (B*d^3*e*atan((c^(1/2)* x)/a^(1/2)))/(2*a^(3/2)*c^(3/2)) + (3*A*d^2*e^2*atan((c^(1/2)*x)/a^(1/2))) /(4*a^(3/2)*c^(3/2)) - (3*B*a*d*e^3*x)/(2*(a^2*c^2 + c^4*x^4 + 2*a*c^3*x^2 ))