Integrand size = 22, antiderivative size = 220 \[ \int \frac {(A+B x) (d+e x)^4}{\left (a+c x^2\right )^2} \, dx=-\frac {3 e^2 \left (A c d^2-4 a B d e-a A e^2\right ) x}{2 a c^2}-\frac {e^3 (A c d-2 a B e) x^2}{2 a c^2}-\frac {(d+e x)^3 (a (B d+A e)-(A c d-a B e) x)}{2 a c \left (a+c x^2\right )}+\frac {\left (4 a B d e \left (c d^2-3 a e^2\right )+A \left (c^2 d^4+6 a c d^2 e^2-3 a^2 e^4\right )\right ) \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 a^{3/2} c^{5/2}}+\frac {e^2 \left (3 B c d^2+2 A c d e-a B e^2\right ) \log \left (a+c x^2\right )}{c^3} \]
-3/2*e^2*(-A*a*e^2+A*c*d^2-4*B*a*d*e)*x/a/c^2-1/2*e^3*(A*c*d-2*B*a*e)*x^2/ a/c^2-1/2*(e*x+d)^3*(a*(A*e+B*d)-(A*c*d-B*a*e)*x)/a/c/(c*x^2+a)+1/2*(4*a*B *d*e*(-3*a*e^2+c*d^2)+A*(-3*a^2*e^4+6*a*c*d^2*e^2+c^2*d^4))*arctan(x*c^(1/ 2)/a^(1/2))/a^(3/2)/c^(5/2)+e^2*(2*A*c*d*e-B*a*e^2+3*B*c*d^2)*ln(c*x^2+a)/ c^3
Time = 0.14 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.05 \[ \int \frac {(A+B x) (d+e x)^4}{\left (a+c x^2\right )^2} \, dx=\frac {2 c e^3 (4 B d+A e) x+B c e^4 x^2+\frac {-a^3 B e^4+A c^3 d^4 x+a^2 c e^2 (A e (4 d+e x)+2 B d (3 d+2 e x))-a c^2 d^2 (2 A e (2 d+3 e x)+B d (d+4 e x))}{a \left (a+c x^2\right )}+\frac {\sqrt {c} \left (4 a B d e \left (c d^2-3 a e^2\right )+A \left (c^2 d^4+6 a c d^2 e^2-3 a^2 e^4\right )\right ) \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{a^{3/2}}+2 e^2 \left (3 B c d^2+2 A c d e-a B e^2\right ) \log \left (a+c x^2\right )}{2 c^3} \]
(2*c*e^3*(4*B*d + A*e)*x + B*c*e^4*x^2 + (-(a^3*B*e^4) + A*c^3*d^4*x + a^2 *c*e^2*(A*e*(4*d + e*x) + 2*B*d*(3*d + 2*e*x)) - a*c^2*d^2*(2*A*e*(2*d + 3 *e*x) + B*d*(d + 4*e*x)))/(a*(a + c*x^2)) + (Sqrt[c]*(4*a*B*d*e*(c*d^2 - 3 *a*e^2) + A*(c^2*d^4 + 6*a*c*d^2*e^2 - 3*a^2*e^4))*ArcTan[(Sqrt[c]*x)/Sqrt [a]])/a^(3/2) + 2*e^2*(3*B*c*d^2 + 2*A*c*d*e - a*B*e^2)*Log[a + c*x^2])/(2 *c^3)
Time = 0.43 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {684, 657, 2009}
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.
| \(\displaystyle \int \frac {(A+B x) (d+e x)^4}{\left (a+c x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 684 |
| \(\displaystyle \frac {\int \frac {(d+e x)^2 \left (A c d^2+a e (4 B d+3 A e)-2 e (A c d-2 a B e) x\right )}{c x^2+a}dx}{2 a c}-\frac {(d+e x)^3 (a (A e+B d)-x (A c d-a B e))}{2 a c \left (a+c x^2\right )}\) |
| \(\Big \downarrow \) 657 |
| \(\displaystyle \frac {\int \left (-\frac {2 (A c d-2 a B e) x e^3}{c}-\frac {3 \left (A c d^2-4 a B e d-a A e^2\right ) e^2}{c}+\frac {4 a \left (3 B c d^2+2 A c e d-a B e^2\right ) x e^2+4 a B d \left (c d^2-3 a e^2\right ) e+A \left (c^2 d^4+6 a c e^2 d^2-3 a^2 e^4\right )}{c \left (c x^2+a\right )}\right )dx}{2 a c}-\frac {(d+e x)^3 (a (A e+B d)-x (A c d-a B e))}{2 a c \left (a+c x^2\right )}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {\arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (A \left (-3 a^2 e^4+6 a c d^2 e^2+c^2 d^4\right )+4 a B d e \left (c d^2-3 a e^2\right )\right )}{\sqrt {a} c^{3/2}}+\frac {2 a e^2 \log \left (a+c x^2\right ) \left (-a B e^2+2 A c d e+3 B c d^2\right )}{c^2}-\frac {3 e^2 x \left (-a A e^2-4 a B d e+A c d^2\right )}{c}-\frac {e^3 x^2 (A c d-2 a B e)}{c}}{2 a c}-\frac {(d+e x)^3 (a (A e+B d)-x (A c d-a B e))}{2 a c \left (a+c x^2\right )}\) |
-1/2*((d + e*x)^3*(a*(B*d + A*e) - (A*c*d - a*B*e)*x))/(a*c*(a + c*x^2)) + ((-3*e^2*(A*c*d^2 - 4*a*B*d*e - a*A*e^2)*x)/c - (e^3*(A*c*d - 2*a*B*e)*x^ 2)/c + ((4*a*B*d*e*(c*d^2 - 3*a*e^2) + A*(c^2*d^4 + 6*a*c*d^2*e^2 - 3*a^2* e^4))*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(Sqrt[a]*c^(3/2)) + (2*a*e^2*(3*B*c*d^2 + 2*A*c*d*e - a*B*e^2)*Log[a + c*x^2])/c^2)/(2*a*c)
3.14.39.3.1 Defintions of rubi rules used
Int[(((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.))/((a_) + (c_.)*( x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*((f + g*x)^n/(a + c*x^ 2)), x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && IntegersQ[n]
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 = 0.39 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.20
| method | result | size |
| default | \(\frac {e^{3} \left (\frac {1}{2} B e \,x^{2}+A e x +4 B d x \right )}{c^{2}}-\frac {\frac {-\frac {\left (A \,a^{2} e^{4}-6 A a c \,d^{2} e^{2}+d^{4} A \,c^{2}+4 B \,a^{2} d \,e^{3}-4 B a c \,d^{3} e \right ) x}{2 a}-\frac {4 A a c d \,e^{3}-4 A \,c^{2} d^{3} e -B \,e^{4} a^{2}+6 B a c \,d^{2} e^{2}-B \,c^{2} d^{4}}{2 c}}{c \,x^{2}+a}+\frac {\frac {\left (-8 A a c d \,e^{3}+4 B \,e^{4} a^{2}-12 B a c \,d^{2} e^{2}\right ) \ln \left (c \,x^{2}+a \right )}{2 c}+\frac {\left (3 A \,a^{2} e^{4}-6 A a c \,d^{2} e^{2}-d^{4} A \,c^{2}+12 B \,a^{2} d \,e^{3}-4 B a c \,d^{3} e \right ) \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}}}{2 a}}{c^{2}}\) | \(263\) |
| risch | \(\text {Expression too large to display}\) | \(1257\) |
e^3/c^2*(1/2*B*e*x^2+A*e*x+4*B*d*x)-1/c^2*((-1/2*(A*a^2*e^4-6*A*a*c*d^2*e^ 2+A*c^2*d^4+4*B*a^2*d*e^3-4*B*a*c*d^3*e)/a*x-1/2*(4*A*a*c*d*e^3-4*A*c^2*d^ 3*e-B*a^2*e^4+6*B*a*c*d^2*e^2-B*c^2*d^4)/c)/(c*x^2+a)+1/2/a*(1/2*(-8*A*a*c *d*e^3+4*B*a^2*e^4-12*B*a*c*d^2*e^2)/c*ln(c*x^2+a)+(3*A*a^2*e^4-6*A*a*c*d^ 2*e^2-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. 413 vs. \(2 (204) = 408\).
Time = 0.35 (sec) , antiderivative size = 849, normalized size of antiderivative = 3.86 \[ \int \frac {(A+B x) (d+e x)^4}{\left (a+c x^2\right )^2} \, dx=\left [\frac {2 \, B a^{2} c^{2} e^{4} x^{4} + 2 \, B a^{3} c e^{4} x^{2} - 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} - 2 \, B a^{4} e^{4} + 4 \, {\left (4 \, B a^{2} c^{2} d e^{3} + A a^{2} c^{2} e^{4}\right )} x^{3} + {\left (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} + {\left (A c^{3} d^{4} + 4 \, B a c^{2} d^{3} e + 6 \, A a c^{2} d^{2} e^{2} - 12 \, B a^{2} c d e^{3} - 3 \, A a^{2} 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 (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 + 4 \, {\left (3 \, B a^{3} c d^{2} e^{2} + 2 \, A a^{3} c d e^{3} - B a^{4} e^{4} + {\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}\right )} \log \left (c x^{2} + a\right )}{4 \, {\left (a^{2} c^{4} x^{2} + a^{3} c^{3}\right )}}, \frac {B a^{2} c^{2} e^{4} x^{4} + B a^{3} c 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} - B a^{4} e^{4} + 2 \, {\left (4 \, B a^{2} c^{2} d e^{3} + A a^{2} c^{2} e^{4}\right )} x^{3} + {\left (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} + {\left (A c^{3} d^{4} + 4 \, B a c^{2} d^{3} e + 6 \, A a c^{2} d^{2} e^{2} - 12 \, B a^{2} c d e^{3} - 3 \, A a^{2} c e^{4}\right )} x^{2}\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c} x}{a}\right ) + {\left (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 \, {\left (3 \, B a^{3} c d^{2} e^{2} + 2 \, A a^{3} c d e^{3} - B a^{4} e^{4} + {\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}\right )} \log \left (c x^{2} + a\right )}{2 \, {\left (a^{2} c^{4} x^{2} + a^{3} c^{3}\right )}}\right ] \]
[1/4*(2*B*a^2*c^2*e^4*x^4 + 2*B*a^3*c*e^4*x^2 - 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 - 2*B*a^4*e^4 + 4*(4*B*a^ 2*c^2*d*e^3 + A*a^2*c^2*e^4)*x^3 + (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 + (A*c^3*d^4 + 4*B*a*c^2*d^3*e + 6*A*a*c^2*d^2*e^2 - 12*B*a^2*c*d*e^3 - 3*A*a^2*c*e^4)*x^2)*sqrt(-a*c)*lo g((c*x^2 + 2*sqrt(-a*c)*x - a)/(c*x^2 + a)) + 2*(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 + 4*(3* B*a^3*c*d^2*e^2 + 2*A*a^3*c*d*e^3 - B*a^4*e^4 + (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)*log(c*x^2 + a))/(a^2*c^4*x^2 + a^3*c^3) , 1/2*(B*a^2*c^2*e^4*x^4 + B*a^3*c*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 - B*a^4*e^4 + 2*(4*B*a^2*c^2*d* e^3 + A*a^2*c^2*e^4)*x^3 + (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 + (A*c^3*d^4 + 4*B*a*c^2*d^3*e + 6*A*a* c^2*d^2*e^2 - 12*B*a^2*c*d*e^3 - 3*A*a^2*c*e^4)*x^2)*sqrt(a*c)*arctan(sqrt (a*c)*x/a) + (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*(3*B*a^3*c*d^2*e^2 + 2*A*a^3*c*d*e^3 - B*a^4*e^4 + (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)* log(c*x^2 + a))/(a^2*c^4*x^2 + a^3*c^3)]
Leaf count of result is larger than twice the leaf count of optimal. 836 vs. \(2 (212) = 424\).
Time = 7.78 (sec) , antiderivative size = 836, normalized size of antiderivative = 3.80 \[ \int \frac {(A+B x) (d+e x)^4}{\left (a+c x^2\right )^2} \, dx=\frac {B e^{4} x^{2}}{2 c^{2}} + x \left (\frac {A e^{4}}{c^{2}} + \frac {4 B d e^{3}}{c^{2}}\right ) + \left (- \frac {e^{2} \left (- 2 A c d e + B a e^{2} - 3 B c d^{2}\right )}{c^{3}} - \frac {\sqrt {- a^{3} c^{7}} \cdot \left (3 A a^{2} e^{4} - 6 A a c d^{2} e^{2} - A c^{2} d^{4} + 12 B a^{2} d e^{3} - 4 B a c d^{3} e\right )}{4 a^{3} c^{6}}\right ) \log {\left (x + \frac {8 A a^{2} c d e^{3} - 4 B a^{3} e^{4} + 12 B a^{2} c d^{2} e^{2} - 4 a^{2} c^{3} \left (- \frac {e^{2} \left (- 2 A c d e + B a e^{2} - 3 B c d^{2}\right )}{c^{3}} - \frac {\sqrt {- a^{3} c^{7}} \cdot \left (3 A a^{2} e^{4} - 6 A a c d^{2} e^{2} - A c^{2} d^{4} + 12 B a^{2} d e^{3} - 4 B a c d^{3} e\right )}{4 a^{3} c^{6}}\right )}{3 A a^{2} c e^{4} - 6 A a c^{2} d^{2} e^{2} - A c^{3} d^{4} + 12 B a^{2} c d e^{3} - 4 B a c^{2} d^{3} e} \right )} + \left (- \frac {e^{2} \left (- 2 A c d e + B a e^{2} - 3 B c d^{2}\right )}{c^{3}} + \frac {\sqrt {- a^{3} c^{7}} \cdot \left (3 A a^{2} e^{4} - 6 A a c d^{2} e^{2} - A c^{2} d^{4} + 12 B a^{2} d e^{3} - 4 B a c d^{3} e\right )}{4 a^{3} c^{6}}\right ) \log {\left (x + \frac {8 A a^{2} c d e^{3} - 4 B a^{3} e^{4} + 12 B a^{2} c d^{2} e^{2} - 4 a^{2} c^{3} \left (- \frac {e^{2} \left (- 2 A c d e + B a e^{2} - 3 B c d^{2}\right )}{c^{3}} + \frac {\sqrt {- a^{3} c^{7}} \cdot \left (3 A a^{2} e^{4} - 6 A a c d^{2} e^{2} - A c^{2} d^{4} + 12 B a^{2} d e^{3} - 4 B a c d^{3} e\right )}{4 a^{3} c^{6}}\right )}{3 A a^{2} c e^{4} - 6 A a c^{2} d^{2} e^{2} - A c^{3} d^{4} + 12 B a^{2} c d e^{3} - 4 B a c^{2} d^{3} e} \right )} + \frac {4 A a^{2} c d e^{3} - 4 A a c^{2} d^{3} e - B a^{3} e^{4} + 6 B a^{2} c d^{2} e^{2} - B a c^{2} d^{4} + x \left (A a^{2} c e^{4} - 6 A a c^{2} d^{2} e^{2} + A c^{3} d^{4} + 4 B a^{2} c d e^{3} - 4 B a c^{2} d^{3} e\right )}{2 a^{2} c^{3} + 2 a c^{4} x^{2}} \]
B*e**4*x**2/(2*c**2) + x*(A*e**4/c**2 + 4*B*d*e**3/c**2) + (-e**2*(-2*A*c* d*e + B*a*e**2 - 3*B*c*d**2)/c**3 - sqrt(-a**3*c**7)*(3*A*a**2*e**4 - 6*A* a*c*d**2*e**2 - A*c**2*d**4 + 12*B*a**2*d*e**3 - 4*B*a*c*d**3*e)/(4*a**3*c **6))*log(x + (8*A*a**2*c*d*e**3 - 4*B*a**3*e**4 + 12*B*a**2*c*d**2*e**2 - 4*a**2*c**3*(-e**2*(-2*A*c*d*e + B*a*e**2 - 3*B*c*d**2)/c**3 - sqrt(-a**3 *c**7)*(3*A*a**2*e**4 - 6*A*a*c*d**2*e**2 - A*c**2*d**4 + 12*B*a**2*d*e**3 - 4*B*a*c*d**3*e)/(4*a**3*c**6)))/(3*A*a**2*c*e**4 - 6*A*a*c**2*d**2*e**2 - A*c**3*d**4 + 12*B*a**2*c*d*e**3 - 4*B*a*c**2*d**3*e)) + (-e**2*(-2*A*c *d*e + B*a*e**2 - 3*B*c*d**2)/c**3 + sqrt(-a**3*c**7)*(3*A*a**2*e**4 - 6*A *a*c*d**2*e**2 - A*c**2*d**4 + 12*B*a**2*d*e**3 - 4*B*a*c*d**3*e)/(4*a**3* c**6))*log(x + (8*A*a**2*c*d*e**3 - 4*B*a**3*e**4 + 12*B*a**2*c*d**2*e**2 - 4*a**2*c**3*(-e**2*(-2*A*c*d*e + B*a*e**2 - 3*B*c*d**2)/c**3 + sqrt(-a** 3*c**7)*(3*A*a**2*e**4 - 6*A*a*c*d**2*e**2 - A*c**2*d**4 + 12*B*a**2*d*e** 3 - 4*B*a*c*d**3*e)/(4*a**3*c**6)))/(3*A*a**2*c*e**4 - 6*A*a*c**2*d**2*e** 2 - A*c**3*d**4 + 12*B*a**2*c*d*e**3 - 4*B*a*c**2*d**3*e)) + (4*A*a**2*c*d *e**3 - 4*A*a*c**2*d**3*e - B*a**3*e**4 + 6*B*a**2*c*d**2*e**2 - B*a*c**2* d**4 + x*(A*a**2*c*e**4 - 6*A*a*c**2*d**2*e**2 + A*c**3*d**4 + 4*B*a**2*c* d*e**3 - 4*B*a*c**2*d**3*e))/(2*a**2*c**3 + 2*a*c**4*x**2)
Time = 0.28 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.22 \[ \int \frac {(A+B x) (d+e x)^4}{\left (a+c x^2\right )^2} \, dx=-\frac {B a c^{2} d^{4} + 4 \, A a c^{2} d^{3} e - 6 \, B a^{2} c d^{2} e^{2} - 4 \, A a^{2} c d e^{3} + B a^{3} e^{4} - {\left (A c^{3} d^{4} - 4 \, B a c^{2} d^{3} e - 6 \, A a c^{2} d^{2} e^{2} + 4 \, B a^{2} c d e^{3} + A a^{2} c e^{4}\right )} x}{2 \, {\left (a c^{4} x^{2} + a^{2} c^{3}\right )}} + \frac {B e^{4} x^{2} + 2 \, {\left (4 \, B d e^{3} + A e^{4}\right )} x}{2 \, c^{2}} + \frac {{\left (3 \, B c d^{2} e^{2} + 2 \, A c d e^{3} - B a e^{4}\right )} \log \left (c x^{2} + a\right )}{c^{3}} + \frac {{\left (A c^{2} d^{4} + 4 \, B a c d^{3} e + 6 \, A a c d^{2} e^{2} - 12 \, B a^{2} d e^{3} - 3 \, A a^{2} e^{4}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \, \sqrt {a c} a c^{2}} \]
-1/2*(B*a*c^2*d^4 + 4*A*a*c^2*d^3*e - 6*B*a^2*c*d^2*e^2 - 4*A*a^2*c*d*e^3 + B*a^3*e^4 - (A*c^3*d^4 - 4*B*a*c^2*d^3*e - 6*A*a*c^2*d^2*e^2 + 4*B*a^2*c *d*e^3 + A*a^2*c*e^4)*x)/(a*c^4*x^2 + a^2*c^3) + 1/2*(B*e^4*x^2 + 2*(4*B*d *e^3 + A*e^4)*x)/c^2 + (3*B*c*d^2*e^2 + 2*A*c*d*e^3 - B*a*e^4)*log(c*x^2 + a)/c^3 + 1/2*(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*c^2)
Time = 0.26 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.24 \[ \int \frac {(A+B x) (d+e x)^4}{\left (a+c x^2\right )^2} \, dx=\frac {{\left (3 \, B c d^{2} e^{2} + 2 \, A c d e^{3} - B a e^{4}\right )} \log \left (c x^{2} + a\right )}{c^{3}} + \frac {{\left (A c^{2} d^{4} + 4 \, B a c d^{3} e + 6 \, A a c d^{2} e^{2} - 12 \, B a^{2} d e^{3} - 3 \, A a^{2} e^{4}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \, \sqrt {a c} a c^{2}} + \frac {B c^{2} e^{4} x^{2} + 8 \, B c^{2} d e^{3} x + 2 \, A c^{2} e^{4} x}{2 \, c^{4}} - \frac {B a c^{2} d^{4} + 4 \, A a c^{2} d^{3} e - 6 \, B a^{2} c d^{2} e^{2} - 4 \, A a^{2} c d e^{3} + B a^{3} e^{4} - {\left (A c^{3} d^{4} - 4 \, B a c^{2} d^{3} e - 6 \, A a c^{2} d^{2} e^{2} + 4 \, B a^{2} c d e^{3} + A a^{2} c e^{4}\right )} x}{2 \, {\left (c x^{2} + a\right )} a c^{3}} \]
(3*B*c*d^2*e^2 + 2*A*c*d*e^3 - B*a*e^4)*log(c*x^2 + a)/c^3 + 1/2*(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*c^2) + 1/2*(B*c^2*e^4*x^2 + 8*B*c^2*d*e^3*x + 2*A*c^2*e^4*x)/c^4 - 1/2*(B*a*c^2*d^4 + 4*A*a*c^2*d^3*e - 6*B*a^2*c*d^2*e ^2 - 4*A*a^2*c*d*e^3 + B*a^3*e^4 - (A*c^3*d^4 - 4*B*a*c^2*d^3*e - 6*A*a*c^ 2*d^2*e^2 + 4*B*a^2*c*d*e^3 + A*a^2*c*e^4)*x)/((c*x^2 + a)*a*c^3)
Time = 10.79 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.25 \[ \int \frac {(A+B x) (d+e x)^4}{\left (a+c x^2\right )^2} \, dx=\frac {x\,\left (A\,e^4+4\,B\,d\,e^3\right )}{c^2}-\frac {\frac {B\,a^2\,e^4-6\,B\,a\,c\,d^2\,e^2-4\,A\,a\,c\,d\,e^3+B\,c^2\,d^4+4\,A\,c^2\,d^3\,e}{2\,c}-\frac {x\,\left (4\,B\,a^2\,d\,e^3+A\,a^2\,e^4-4\,B\,a\,c\,d^3\,e-6\,A\,a\,c\,d^2\,e^2+A\,c^2\,d^4\right )}{2\,a}}{c^3\,x^2+a\,c^2}+\frac {\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )\,\left (-12\,B\,a^2\,d\,e^3-3\,A\,a^2\,e^4+4\,B\,a\,c\,d^3\,e+6\,A\,a\,c\,d^2\,e^2+A\,c^2\,d^4\right )}{2\,a^{3/2}\,c^{5/2}}+\frac {\ln \left (c\,x^2+a\right )\,\left (-32\,B\,a^4\,c^3\,e^4+96\,B\,a^3\,c^4\,d^2\,e^2+64\,A\,a^3\,c^4\,d\,e^3\right )}{32\,a^3\,c^6}+\frac {B\,e^4\,x^2}{2\,c^2} \]
(x*(A*e^4 + 4*B*d*e^3))/c^2 - ((B*a^2*e^4 + B*c^2*d^4 + 4*A*c^2*d^3*e - 4* A*a*c*d*e^3 - 6*B*a*c*d^2*e^2)/(2*c) - (x*(A*a^2*e^4 + A*c^2*d^4 + 4*B*a^2 *d*e^3 - 4*B*a*c*d^3*e - 6*A*a*c*d^2*e^2))/(2*a))/(a*c^2 + c^3*x^2) + (ata n((c^(1/2)*x)/a^(1/2))*(A*c^2*d^4 - 3*A*a^2*e^4 - 12*B*a^2*d*e^3 + 4*B*a*c *d^3*e + 6*A*a*c*d^2*e^2))/(2*a^(3/2)*c^(5/2)) + (log(a + c*x^2)*(64*A*a^3 *c^4*d*e^3 - 32*B*a^4*c^3*e^4 + 96*B*a^3*c^4*d^2*e^2))/(32*a^3*c^6) + (B*e ^4*x^2)/(2*c^2)