Integrand size = 17, antiderivative size = 198 \[ \int \frac {(d+e x)^5}{\left (a+c x^2\right )^3} \, dx=-\frac {d e^2 \left (3 c d^2+7 a e^2\right ) x}{8 a^2 c^2}-\frac {(a e-c d x) (d+e x)^4}{4 a c \left (a+c x^2\right )^2}-\frac {(d+e x)^2 \left (2 a e \left (c d^2+2 a e^2\right )-c d \left (3 c d^2+5 a e^2\right ) x\right )}{8 a^2 c^2 \left (a+c x^2\right )}+\frac {d \left (3 c^2 d^4+10 a c d^2 e^2+15 a^2 e^4\right ) \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{8 a^{5/2} c^{5/2}}+\frac {e^5 \log \left (a+c x^2\right )}{2 c^3} \]
-1/8*d*e^2*(7*a*e^2+3*c*d^2)*x/a^2/c^2-1/4*(-c*d*x+a*e)*(e*x+d)^4/a/c/(c*x ^2+a)^2-1/8*(e*x+d)^2*(2*a*e*(2*a*e^2+c*d^2)-c*d*(5*a*e^2+3*c*d^2)*x)/a^2/ c^2/(c*x^2+a)+1/8*d*(15*a^2*e^4+10*a*c*d^2*e^2+3*c^2*d^4)*arctan(x*c^(1/2) /a^(1/2))/a^(5/2)/c^(5/2)+1/2*e^5*ln(c*x^2+a)/c^3
Time = 0.15 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.01 \[ \int \frac {(d+e x)^5}{\left (a+c x^2\right )^3} \, dx=\frac {-\frac {2 \left (a^3 e^5-c^3 d^5 x-5 a^2 c d e^3 (2 d+e x)+5 a c^2 d^3 e (d+2 e x)\right )}{a \left (a+c x^2\right )^2}+\frac {8 a^3 e^5+3 c^3 d^5 x+10 a c^2 d^3 e^2 x-5 a^2 c d e^3 (8 d+5 e x)}{a^2 \left (a+c x^2\right )}+\frac {\sqrt {c} d \left (3 c^2 d^4+10 a c d^2 e^2+15 a^2 e^4\right ) \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{a^{5/2}}+4 e^5 \log \left (a+c x^2\right )}{8 c^3} \]
((-2*(a^3*e^5 - c^3*d^5*x - 5*a^2*c*d*e^3*(2*d + e*x) + 5*a*c^2*d^3*e*(d + 2*e*x)))/(a*(a + c*x^2)^2) + (8*a^3*e^5 + 3*c^3*d^5*x + 10*a*c^2*d^3*e^2* x - 5*a^2*c*d*e^3*(8*d + 5*e*x))/(a^2*(a + c*x^2)) + (Sqrt[c]*d*(3*c^2*d^4 + 10*a*c*d^2*e^2 + 15*a^2*e^4)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/a^(5/2) + 4*e ^5*Log[a + c*x^2])/(8*c^3)
Time = 0.40 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.06, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {495, 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.
| \(\displaystyle \int \frac {(d+e x)^5}{\left (a+c x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 495 |
| \(\displaystyle \frac {\int \frac {(d+e x)^3 \left (3 c d^2-c e x d+4 a e^2\right )}{\left (c x^2+a\right )^2}dx}{4 a c}-\frac {(d+e x)^4 (a e-c d x)}{4 a c \left (a+c x^2\right )^2}\) |
| \(\Big \downarrow \) 684 |
| \(\displaystyle \frac {\frac {\int \frac {(d+e x) \left (3 c^2 d^4+7 a c e^2 d^2-c e \left (3 c d^2+7 a e^2\right ) x d+8 a^2 e^4\right )}{c x^2+a}dx}{2 a c}-\frac {(d+e x)^2 \left (2 a e \left (2 a e^2+c d^2\right )-c d x \left (5 a e^2+3 c d^2\right )\right )}{2 a c \left (a+c x^2\right )}}{4 a c}-\frac {(d+e x)^4 (a e-c d x)}{4 a c \left (a+c x^2\right )^2}\) |
| \(\Big \downarrow \) 657 |
| \(\displaystyle \frac {\frac {\int \left (-7 a d e^4-3 c d^3 e^2+\frac {3 c^2 d^5+10 a c e^2 d^3+15 a^2 e^4 d+8 a^2 e^5 x}{c x^2+a}\right )dx}{2 a c}-\frac {(d+e x)^2 \left (2 a e \left (2 a e^2+c d^2\right )-c d x \left (5 a e^2+3 c d^2\right )\right )}{2 a c \left (a+c x^2\right )}}{4 a c}-\frac {(d+e x)^4 (a e-c d x)}{4 a c \left (a+c x^2\right )^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {\frac {d \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (15 a^2 e^4+10 a c d^2 e^2+3 c^2 d^4\right )}{\sqrt {a} \sqrt {c}}+\frac {4 a^2 e^5 \log \left (a+c x^2\right )}{c}-d e^2 x \left (7 a e^2+3 c d^2\right )}{2 a c}-\frac {(d+e x)^2 \left (2 a e \left (2 a e^2+c d^2\right )-c d x \left (5 a e^2+3 c d^2\right )\right )}{2 a c \left (a+c x^2\right )}}{4 a c}-\frac {(d+e x)^4 (a e-c d x)}{4 a c \left (a+c x^2\right )^2}\) |
-1/4*((a*e - c*d*x)*(d + e*x)^4)/(a*c*(a + c*x^2)^2) + (-1/2*((d + e*x)^2* (2*a*e*(c*d^2 + 2*a*e^2) - c*d*(3*c*d^2 + 5*a*e^2)*x))/(a*c*(a + c*x^2)) + (-(d*e^2*(3*c*d^2 + 7*a*e^2)*x) + (d*(3*c^2*d^4 + 10*a*c*d^2*e^2 + 15*a^2 *e^4)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(Sqrt[a]*Sqrt[c]) + (4*a^2*e^5*Log[a + c*x^2])/c)/(2*a*c))/(4*a*c)
3.6.12.3.1 Defintions of rubi rules used
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ (a*d - b*c*x)*(c + d*x)^(n - 1)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] - Simp[1/(2*a*b*(p + 1)) Int[(c + d*x)^(n - 2)*(a + b*x^2)^(p + 1)*Simp[a* d^2*(n - 1) - b*c^2*(2*p + 3) - b*c*d*(n + 2*p + 2)*x, x], x], x] /; FreeQ[ {a, b, c, d}, x] && LtQ[p, -1] && GtQ[n, 1] && IntQuadraticQ[a, 0, b, c, d, n, p, x]
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 = 2.20 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.09
| method | result | size |
| default | \(\frac {-\frac {d \left (25 a^{2} e^{4}-10 a c \,d^{2} e^{2}-3 c^{2} d^{4}\right ) x^{3}}{8 a^{2} c}+\frac {e^{3} \left (e^{2} a -5 c \,d^{2}\right ) x^{2}}{c^{2}}-\frac {5 d \left (3 a^{2} e^{4}+2 a c \,d^{2} e^{2}-c^{2} d^{4}\right ) x}{8 c^{2} a}+\frac {e \left (3 a^{2} e^{4}-10 a c \,d^{2} e^{2}-5 c^{2} d^{4}\right )}{4 c^{3}}}{\left (c \,x^{2}+a \right )^{2}}+\frac {\frac {4 a^{2} e^{5} \ln \left (c \,x^{2}+a \right )}{c}+\frac {\left (15 a^{2} d \,e^{4}+10 a c \,d^{3} e^{2}+3 c^{2} d^{5}\right ) \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}}}{8 a^{2} c^{2}}\) | \(215\) |
| risch | \(\frac {-\frac {d \left (25 a^{2} e^{4}-10 a c \,d^{2} e^{2}-3 c^{2} d^{4}\right ) x^{3}}{8 a^{2} c}+\frac {e^{3} \left (e^{2} a -5 c \,d^{2}\right ) x^{2}}{c^{2}}-\frac {5 d \left (3 a^{2} e^{4}+2 a c \,d^{2} e^{2}-c^{2} d^{4}\right ) x}{8 c^{2} a}+\frac {e \left (3 a^{2} e^{4}-10 a c \,d^{2} e^{2}-5 c^{2} d^{4}\right )}{4 c^{3}}}{\left (c \,x^{2}+a \right )^{2}}+\frac {\ln \left (15 d \,e^{4} a^{3}+10 d^{3} e^{2} a^{2} c +3 d^{5} c^{2} a -\sqrt {-a c \,d^{2} \left (15 a^{2} e^{4}+10 a c \,d^{2} e^{2}+3 c^{2} d^{4}\right )^{2}}\, x \right ) e^{5}}{2 c^{3}}+\frac {\ln \left (15 d \,e^{4} a^{3}+10 d^{3} e^{2} a^{2} c +3 d^{5} c^{2} a -\sqrt {-a c \,d^{2} \left (15 a^{2} e^{4}+10 a c \,d^{2} e^{2}+3 c^{2} d^{4}\right )^{2}}\, x \right ) \sqrt {-a c \,d^{2} \left (15 a^{2} e^{4}+10 a c \,d^{2} e^{2}+3 c^{2} d^{4}\right )^{2}}}{16 a^{3} c^{3}}+\frac {\ln \left (15 d \,e^{4} a^{3}+10 d^{3} e^{2} a^{2} c +3 d^{5} c^{2} a +\sqrt {-a c \,d^{2} \left (15 a^{2} e^{4}+10 a c \,d^{2} e^{2}+3 c^{2} d^{4}\right )^{2}}\, x \right ) e^{5}}{2 c^{3}}-\frac {\ln \left (15 d \,e^{4} a^{3}+10 d^{3} e^{2} a^{2} c +3 d^{5} c^{2} a +\sqrt {-a c \,d^{2} \left (15 a^{2} e^{4}+10 a c \,d^{2} e^{2}+3 c^{2} d^{4}\right )^{2}}\, x \right ) \sqrt {-a c \,d^{2} \left (15 a^{2} e^{4}+10 a c \,d^{2} e^{2}+3 c^{2} d^{4}\right )^{2}}}{16 c^{3} a^{3}}\) | \(542\) |
(-1/8*d*(25*a^2*e^4-10*a*c*d^2*e^2-3*c^2*d^4)/a^2/c*x^3+e^3*(a*e^2-5*c*d^2 )/c^2*x^2-5/8*d*(3*a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/c^2/a*x+1/4*e*(3*a^2*e^4 -10*a*c*d^2*e^2-5*c^2*d^4)/c^3)/(c*x^2+a)^2+1/8/a^2/c^2*(4*a^2*e^5/c*ln(c* x^2+a)+(15*a^2*d*e^4+10*a*c*d^3*e^2+3*c^2*d^5)/(a*c)^(1/2)*arctan(c*x/(a*c )^(1/2)))
Time = 0.28 (sec) , antiderivative size = 707, normalized size of antiderivative = 3.57 \[ \int \frac {(d+e x)^5}{\left (a+c x^2\right )^3} \, dx=\left [-\frac {20 \, a^{3} c^{2} d^{4} e + 40 \, a^{4} c d^{2} e^{3} - 12 \, a^{5} e^{5} - 2 \, {\left (3 \, a c^{4} d^{5} + 10 \, a^{2} c^{3} d^{3} e^{2} - 25 \, a^{3} c^{2} d e^{4}\right )} x^{3} + 16 \, {\left (5 \, a^{3} c^{2} d^{2} e^{3} - a^{4} c e^{5}\right )} x^{2} + {\left (3 \, a^{2} c^{2} d^{5} + 10 \, a^{3} c d^{3} e^{2} + 15 \, a^{4} d e^{4} + {\left (3 \, c^{4} d^{5} + 10 \, a c^{3} d^{3} e^{2} + 15 \, a^{2} c^{2} d e^{4}\right )} x^{4} + 2 \, {\left (3 \, a c^{3} d^{5} + 10 \, a^{2} c^{2} d^{3} e^{2} + 15 \, a^{3} c d e^{4}\right )} x^{2}\right )} \sqrt {-a c} \log \left (\frac {c x^{2} - 2 \, \sqrt {-a c} x - a}{c x^{2} + a}\right ) - 10 \, {\left (a^{2} c^{3} d^{5} - 2 \, a^{3} c^{2} d^{3} e^{2} - 3 \, a^{4} c d e^{4}\right )} x - 8 \, {\left (a^{3} c^{2} e^{5} x^{4} + 2 \, a^{4} c e^{5} x^{2} + a^{5} e^{5}\right )} \log \left (c x^{2} + a\right )}{16 \, {\left (a^{3} c^{5} x^{4} + 2 \, a^{4} c^{4} x^{2} + a^{5} c^{3}\right )}}, -\frac {10 \, a^{3} c^{2} d^{4} e + 20 \, a^{4} c d^{2} e^{3} - 6 \, a^{5} e^{5} - {\left (3 \, a c^{4} d^{5} + 10 \, a^{2} c^{3} d^{3} e^{2} - 25 \, a^{3} c^{2} d e^{4}\right )} x^{3} + 8 \, {\left (5 \, a^{3} c^{2} d^{2} e^{3} - a^{4} c e^{5}\right )} x^{2} - {\left (3 \, a^{2} c^{2} d^{5} + 10 \, a^{3} c d^{3} e^{2} + 15 \, a^{4} d e^{4} + {\left (3 \, c^{4} d^{5} + 10 \, a c^{3} d^{3} e^{2} + 15 \, a^{2} c^{2} d e^{4}\right )} x^{4} + 2 \, {\left (3 \, a c^{3} d^{5} + 10 \, a^{2} c^{2} d^{3} e^{2} + 15 \, a^{3} c d e^{4}\right )} x^{2}\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c} x}{a}\right ) - 5 \, {\left (a^{2} c^{3} d^{5} - 2 \, a^{3} c^{2} d^{3} e^{2} - 3 \, a^{4} c d e^{4}\right )} x - 4 \, {\left (a^{3} c^{2} e^{5} x^{4} + 2 \, a^{4} c e^{5} x^{2} + a^{5} e^{5}\right )} \log \left (c x^{2} + a\right )}{8 \, {\left (a^{3} c^{5} x^{4} + 2 \, a^{4} c^{4} x^{2} + a^{5} c^{3}\right )}}\right ] \]
[-1/16*(20*a^3*c^2*d^4*e + 40*a^4*c*d^2*e^3 - 12*a^5*e^5 - 2*(3*a*c^4*d^5 + 10*a^2*c^3*d^3*e^2 - 25*a^3*c^2*d*e^4)*x^3 + 16*(5*a^3*c^2*d^2*e^3 - a^4 *c*e^5)*x^2 + (3*a^2*c^2*d^5 + 10*a^3*c*d^3*e^2 + 15*a^4*d*e^4 + (3*c^4*d^ 5 + 10*a*c^3*d^3*e^2 + 15*a^2*c^2*d*e^4)*x^4 + 2*(3*a*c^3*d^5 + 10*a^2*c^2 *d^3*e^2 + 15*a^3*c*d*e^4)*x^2)*sqrt(-a*c)*log((c*x^2 - 2*sqrt(-a*c)*x - a )/(c*x^2 + a)) - 10*(a^2*c^3*d^5 - 2*a^3*c^2*d^3*e^2 - 3*a^4*c*d*e^4)*x - 8*(a^3*c^2*e^5*x^4 + 2*a^4*c*e^5*x^2 + a^5*e^5)*log(c*x^2 + a))/(a^3*c^5*x ^4 + 2*a^4*c^4*x^2 + a^5*c^3), -1/8*(10*a^3*c^2*d^4*e + 20*a^4*c*d^2*e^3 - 6*a^5*e^5 - (3*a*c^4*d^5 + 10*a^2*c^3*d^3*e^2 - 25*a^3*c^2*d*e^4)*x^3 + 8 *(5*a^3*c^2*d^2*e^3 - a^4*c*e^5)*x^2 - (3*a^2*c^2*d^5 + 10*a^3*c*d^3*e^2 + 15*a^4*d*e^4 + (3*c^4*d^5 + 10*a*c^3*d^3*e^2 + 15*a^2*c^2*d*e^4)*x^4 + 2* (3*a*c^3*d^5 + 10*a^2*c^2*d^3*e^2 + 15*a^3*c*d*e^4)*x^2)*sqrt(a*c)*arctan( sqrt(a*c)*x/a) - 5*(a^2*c^3*d^5 - 2*a^3*c^2*d^3*e^2 - 3*a^4*c*d*e^4)*x - 4 *(a^3*c^2*e^5*x^4 + 2*a^4*c*e^5*x^2 + a^5*e^5)*log(c*x^2 + a))/(a^3*c^5*x^ 4 + 2*a^4*c^4*x^2 + a^5*c^3)]
Leaf count of result is larger than twice the leaf count of optimal. 520 vs. \(2 (189) = 378\).
Time = 2.08 (sec) , antiderivative size = 520, normalized size of antiderivative = 2.63 \[ \int \frac {(d+e x)^5}{\left (a+c x^2\right )^3} \, dx=\left (\frac {e^{5}}{2 c^{3}} - \frac {d \sqrt {- a^{5} c^{7}} \cdot \left (15 a^{2} e^{4} + 10 a c d^{2} e^{2} + 3 c^{2} d^{4}\right )}{16 a^{5} c^{6}}\right ) \log {\left (x + \frac {16 a^{3} c^{3} \left (\frac {e^{5}}{2 c^{3}} - \frac {d \sqrt {- a^{5} c^{7}} \cdot \left (15 a^{2} e^{4} + 10 a c d^{2} e^{2} + 3 c^{2} d^{4}\right )}{16 a^{5} c^{6}}\right ) - 8 a^{3} e^{5}}{15 a^{2} c d e^{4} + 10 a c^{2} d^{3} e^{2} + 3 c^{3} d^{5}} \right )} + \left (\frac {e^{5}}{2 c^{3}} + \frac {d \sqrt {- a^{5} c^{7}} \cdot \left (15 a^{2} e^{4} + 10 a c d^{2} e^{2} + 3 c^{2} d^{4}\right )}{16 a^{5} c^{6}}\right ) \log {\left (x + \frac {16 a^{3} c^{3} \left (\frac {e^{5}}{2 c^{3}} + \frac {d \sqrt {- a^{5} c^{7}} \cdot \left (15 a^{2} e^{4} + 10 a c d^{2} e^{2} + 3 c^{2} d^{4}\right )}{16 a^{5} c^{6}}\right ) - 8 a^{3} e^{5}}{15 a^{2} c d e^{4} + 10 a c^{2} d^{3} e^{2} + 3 c^{3} d^{5}} \right )} + \frac {6 a^{4} e^{5} - 20 a^{3} c d^{2} e^{3} - 10 a^{2} c^{2} d^{4} e + x^{3} \left (- 25 a^{2} c^{2} d e^{4} + 10 a c^{3} d^{3} e^{2} + 3 c^{4} d^{5}\right ) + x^{2} \cdot \left (8 a^{3} c e^{5} - 40 a^{2} c^{2} d^{2} e^{3}\right ) + x \left (- 15 a^{3} c d e^{4} - 10 a^{2} c^{2} d^{3} e^{2} + 5 a c^{3} d^{5}\right )}{8 a^{4} c^{3} + 16 a^{3} c^{4} x^{2} + 8 a^{2} c^{5} x^{4}} \]
(e**5/(2*c**3) - d*sqrt(-a**5*c**7)*(15*a**2*e**4 + 10*a*c*d**2*e**2 + 3*c **2*d**4)/(16*a**5*c**6))*log(x + (16*a**3*c**3*(e**5/(2*c**3) - d*sqrt(-a **5*c**7)*(15*a**2*e**4 + 10*a*c*d**2*e**2 + 3*c**2*d**4)/(16*a**5*c**6)) - 8*a**3*e**5)/(15*a**2*c*d*e**4 + 10*a*c**2*d**3*e**2 + 3*c**3*d**5)) + ( e**5/(2*c**3) + d*sqrt(-a**5*c**7)*(15*a**2*e**4 + 10*a*c*d**2*e**2 + 3*c* *2*d**4)/(16*a**5*c**6))*log(x + (16*a**3*c**3*(e**5/(2*c**3) + d*sqrt(-a* *5*c**7)*(15*a**2*e**4 + 10*a*c*d**2*e**2 + 3*c**2*d**4)/(16*a**5*c**6)) - 8*a**3*e**5)/(15*a**2*c*d*e**4 + 10*a*c**2*d**3*e**2 + 3*c**3*d**5)) + (6 *a**4*e**5 - 20*a**3*c*d**2*e**3 - 10*a**2*c**2*d**4*e + x**3*(-25*a**2*c* *2*d*e**4 + 10*a*c**3*d**3*e**2 + 3*c**4*d**5) + x**2*(8*a**3*c*e**5 - 40* a**2*c**2*d**2*e**3) + x*(-15*a**3*c*d*e**4 - 10*a**2*c**2*d**3*e**2 + 5*a *c**3*d**5))/(8*a**4*c**3 + 16*a**3*c**4*x**2 + 8*a**2*c**5*x**4)
Time = 0.28 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.19 \[ \int \frac {(d+e x)^5}{\left (a+c x^2\right )^3} \, dx=\frac {e^{5} \log \left (c x^{2} + a\right )}{2 \, c^{3}} - \frac {10 \, a^{2} c^{2} d^{4} e + 20 \, a^{3} c d^{2} e^{3} - 6 \, a^{4} e^{5} - {\left (3 \, c^{4} d^{5} + 10 \, a c^{3} d^{3} e^{2} - 25 \, a^{2} c^{2} d e^{4}\right )} x^{3} + 8 \, {\left (5 \, a^{2} c^{2} d^{2} e^{3} - a^{3} c e^{5}\right )} x^{2} - 5 \, {\left (a c^{3} d^{5} - 2 \, a^{2} c^{2} d^{3} e^{2} - 3 \, a^{3} c d e^{4}\right )} x}{8 \, {\left (a^{2} c^{5} x^{4} + 2 \, a^{3} c^{4} x^{2} + a^{4} c^{3}\right )}} + \frac {{\left (3 \, c^{2} d^{5} + 10 \, a c d^{3} e^{2} + 15 \, a^{2} d e^{4}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 \, \sqrt {a c} a^{2} c^{2}} \]
1/2*e^5*log(c*x^2 + a)/c^3 - 1/8*(10*a^2*c^2*d^4*e + 20*a^3*c*d^2*e^3 - 6* a^4*e^5 - (3*c^4*d^5 + 10*a*c^3*d^3*e^2 - 25*a^2*c^2*d*e^4)*x^3 + 8*(5*a^2 *c^2*d^2*e^3 - a^3*c*e^5)*x^2 - 5*(a*c^3*d^5 - 2*a^2*c^2*d^3*e^2 - 3*a^3*c *d*e^4)*x)/(a^2*c^5*x^4 + 2*a^3*c^4*x^2 + a^4*c^3) + 1/8*(3*c^2*d^5 + 10*a *c*d^3*e^2 + 15*a^2*d*e^4)*arctan(c*x/sqrt(a*c))/(sqrt(a*c)*a^2*c^2)
Time = 0.38 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.10 \[ \int \frac {(d+e x)^5}{\left (a+c x^2\right )^3} \, dx=\frac {e^{5} \log \left (c x^{2} + a\right )}{2 \, c^{3}} + \frac {{\left (3 \, c^{2} d^{5} + 10 \, a c d^{3} e^{2} + 15 \, a^{2} d e^{4}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 \, \sqrt {a c} a^{2} c^{2}} + \frac {{\left (3 \, c^{3} d^{5} + 10 \, a c^{2} d^{3} e^{2} - 25 \, a^{2} c d e^{4}\right )} x^{3} - 8 \, {\left (5 \, a^{2} c d^{2} e^{3} - a^{3} e^{5}\right )} x^{2} + 5 \, {\left (a c^{2} d^{5} - 2 \, a^{2} c d^{3} e^{2} - 3 \, a^{3} d e^{4}\right )} x - \frac {2 \, {\left (5 \, a^{2} c^{2} d^{4} e + 10 \, a^{3} c d^{2} e^{3} - 3 \, a^{4} e^{5}\right )}}{c}}{8 \, {\left (c x^{2} + a\right )}^{2} a^{2} c^{2}} \]
1/2*e^5*log(c*x^2 + a)/c^3 + 1/8*(3*c^2*d^5 + 10*a*c*d^3*e^2 + 15*a^2*d*e^ 4)*arctan(c*x/sqrt(a*c))/(sqrt(a*c)*a^2*c^2) + 1/8*((3*c^3*d^5 + 10*a*c^2* d^3*e^2 - 25*a^2*c*d*e^4)*x^3 - 8*(5*a^2*c*d^2*e^3 - a^3*e^5)*x^2 + 5*(a*c ^2*d^5 - 2*a^2*c*d^3*e^2 - 3*a^3*d*e^4)*x - 2*(5*a^2*c^2*d^4*e + 10*a^3*c* d^2*e^3 - 3*a^4*e^5)/c)/((c*x^2 + a)^2*a^2*c^2)
Time = 9.77 (sec) , antiderivative size = 467, normalized size of antiderivative = 2.36 \[ \int \frac {(d+e x)^5}{\left (a+c x^2\right )^3} \, dx=\frac {e^5\,\ln \left (c\,x^2+a\right )}{2\,c^3}-\frac {5\,d^4\,e}{4\,\left (a^2\,c+2\,a\,c^2\,x^2+c^3\,x^4\right )}+\frac {5\,d^5\,x}{8\,\left (a^3+2\,a^2\,c\,x^2+a\,c^2\,x^4\right )}+\frac {3\,a^2\,e^5}{4\,\left (a^2\,c^3+2\,a\,c^4\,x^2+c^5\,x^4\right )}-\frac {5\,a\,d^2\,e^3}{2\,\left (a^2\,c^2+2\,a\,c^3\,x^2+c^4\,x^4\right )}+\frac {a\,e^5\,x^2}{a^2\,c^2+2\,a\,c^3\,x^2+c^4\,x^4}-\frac {5\,d^2\,e^3\,x^2}{a^2\,c+2\,a\,c^2\,x^2+c^3\,x^4}+\frac {3\,d^5\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )}{8\,a^{5/2}\,\sqrt {c}}+\frac {3\,c\,d^5\,x^3}{8\,\left (a^4+2\,a^3\,c\,x^2+a^2\,c^2\,x^4\right )}+\frac {5\,d^3\,e^2\,x^3}{4\,\left (a^3+2\,a^2\,c\,x^2+a\,c^2\,x^4\right )}-\frac {5\,d^3\,e^2\,x}{4\,\left (a^2\,c+2\,a\,c^2\,x^2+c^3\,x^4\right )}-\frac {25\,d\,e^4\,x^3}{8\,\left (a^2\,c+2\,a\,c^2\,x^2+c^3\,x^4\right )}+\frac {5\,d^3\,e^2\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )}{4\,a^{3/2}\,c^{3/2}}-\frac {15\,a\,d\,e^4\,x}{8\,\left (a^2\,c^2+2\,a\,c^3\,x^2+c^4\,x^4\right )}+\frac {15\,d\,e^4\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )}{8\,\sqrt {a}\,c^{5/2}} \]
(e^5*log(a + c*x^2))/(2*c^3) - (5*d^4*e)/(4*(a^2*c + c^3*x^4 + 2*a*c^2*x^2 )) + (5*d^5*x)/(8*(a^3 + 2*a^2*c*x^2 + a*c^2*x^4)) + (3*a^2*e^5)/(4*(a^2*c ^3 + c^5*x^4 + 2*a*c^4*x^2)) - (5*a*d^2*e^3)/(2*(a^2*c^2 + c^4*x^4 + 2*a*c ^3*x^2)) + (a*e^5*x^2)/(a^2*c^2 + c^4*x^4 + 2*a*c^3*x^2) - (5*d^2*e^3*x^2) /(a^2*c + c^3*x^4 + 2*a*c^2*x^2) + (3*d^5*atan((c^(1/2)*x)/a^(1/2)))/(8*a^ (5/2)*c^(1/2)) + (3*c*d^5*x^3)/(8*(a^4 + 2*a^3*c*x^2 + a^2*c^2*x^4)) + (5* d^3*e^2*x^3)/(4*(a^3 + 2*a^2*c*x^2 + a*c^2*x^4)) - (5*d^3*e^2*x)/(4*(a^2*c + c^3*x^4 + 2*a*c^2*x^2)) - (25*d*e^4*x^3)/(8*(a^2*c + c^3*x^4 + 2*a*c^2* x^2)) + (5*d^3*e^2*atan((c^(1/2)*x)/a^(1/2)))/(4*a^(3/2)*c^(3/2)) - (15*a* d*e^4*x)/(8*(a^2*c^2 + c^4*x^4 + 2*a*c^3*x^2)) + (15*d*e^4*atan((c^(1/2)*x )/a^(1/2)))/(8*a^(1/2)*c^(5/2))