Integrand size = 17, antiderivative size = 430 \[ \int \frac {1}{(d+e x)^2 \left (a+c x^2\right )^4} \, dx=\frac {e \left (5 c^3 d^6+23 a c^2 d^4 e^2+47 a^2 c d^2 e^4-35 a^3 e^6\right )}{16 a^3 \left (c d^2+a e^2\right )^4 (d+e x)}+\frac {a e+c d x}{6 a \left (c d^2+a e^2\right ) (d+e x) \left (a+c x^2\right )^3}-\frac {a e \left (c d^2-7 a e^2\right )-c d \left (5 c d^2+13 a e^2\right ) x}{24 a^2 \left (c d^2+a e^2\right )^2 (d+e x) \left (a+c x^2\right )^2}-\frac {a e \left (5 c d^2-7 a e^2\right ) \left (c d^2+5 a e^2\right )-3 c d \left (5 c^2 d^4+18 a c d^2 e^2+29 a^2 e^4\right ) x}{48 a^3 \left (c d^2+a e^2\right )^3 (d+e x) \left (a+c x^2\right )}+\frac {\sqrt {c} \left (5 c^4 d^8+28 a c^3 d^6 e^2+70 a^2 c^2 d^4 e^4+140 a^3 c d^2 e^6-35 a^4 e^8\right ) \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{16 a^{7/2} \left (c d^2+a e^2\right )^5}+\frac {8 c d e^7 \log (d+e x)}{\left (c d^2+a e^2\right )^5}-\frac {4 c d e^7 \log \left (a+c x^2\right )}{\left (c d^2+a e^2\right )^5} \]
1/16*e*(-35*a^3*e^6+47*a^2*c*d^2*e^4+23*a*c^2*d^4*e^2+5*c^3*d^6)/a^3/(a*e^ 2+c*d^2)^4/(e*x+d)+1/6*(c*d*x+a*e)/a/(a*e^2+c*d^2)/(e*x+d)/(c*x^2+a)^3+1/2 4*(-a*e*(-7*a*e^2+c*d^2)+c*d*(13*a*e^2+5*c*d^2)*x)/a^2/(a*e^2+c*d^2)^2/(e* x+d)/(c*x^2+a)^2+1/48*(-a*e*(-7*a*e^2+5*c*d^2)*(5*a*e^2+c*d^2)+3*c*d*(29*a ^2*e^4+18*a*c*d^2*e^2+5*c^2*d^4)*x)/a^3/(a*e^2+c*d^2)^3/(e*x+d)/(c*x^2+a)+ 8*c*d*e^7*ln(e*x+d)/(a*e^2+c*d^2)^5-4*c*d*e^7*ln(c*x^2+a)/(a*e^2+c*d^2)^5+ 1/16*(-35*a^4*e^8+140*a^3*c*d^2*e^6+70*a^2*c^2*d^4*e^4+28*a*c^3*d^6*e^2+5* c^4*d^8)*arctan(x*c^(1/2)/a^(1/2))*c^(1/2)/a^(7/2)/(a*e^2+c*d^2)^5
Time = 0.40 (sec) , antiderivative size = 336, normalized size of antiderivative = 0.78 \[ \int \frac {1}{(d+e x)^2 \left (a+c x^2\right )^4} \, dx=\frac {-\frac {48 e^7 \left (c d^2+a e^2\right )}{d+e x}+\frac {3 c \left (c d^2+a e^2\right ) \left (5 c^3 d^6 x+23 a c^2 d^4 e^2 x+47 a^2 c d^2 e^4 x+a^3 e^5 (48 d-19 e x)\right )}{a^3 \left (a+c x^2\right )}+\frac {2 c \left (c d^2+a e^2\right )^2 \left (5 c^2 d^4 x+18 a c d^2 e^2 x+a^2 e^3 (24 d-11 e x)\right )}{a^2 \left (a+c x^2\right )^2}+\frac {8 c \left (c d^2+a e^2\right )^3 \left (c d^2 x+a e (2 d-e x)\right )}{a \left (a+c x^2\right )^3}+\frac {3 \sqrt {c} \left (5 c^4 d^8+28 a c^3 d^6 e^2+70 a^2 c^2 d^4 e^4+140 a^3 c d^2 e^6-35 a^4 e^8\right ) \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{a^{7/2}}+384 c d e^7 \log (d+e x)-192 c d e^7 \log \left (a+c x^2\right )}{48 \left (c d^2+a e^2\right )^5} \]
((-48*e^7*(c*d^2 + a*e^2))/(d + e*x) + (3*c*(c*d^2 + a*e^2)*(5*c^3*d^6*x + 23*a*c^2*d^4*e^2*x + 47*a^2*c*d^2*e^4*x + a^3*e^5*(48*d - 19*e*x)))/(a^3* (a + c*x^2)) + (2*c*(c*d^2 + a*e^2)^2*(5*c^2*d^4*x + 18*a*c*d^2*e^2*x + a^ 2*e^3*(24*d - 11*e*x)))/(a^2*(a + c*x^2)^2) + (8*c*(c*d^2 + a*e^2)^3*(c*d^ 2*x + a*e*(2*d - e*x)))/(a*(a + c*x^2)^3) + (3*Sqrt[c]*(5*c^4*d^8 + 28*a*c ^3*d^6*e^2 + 70*a^2*c^2*d^4*e^4 + 140*a^3*c*d^2*e^6 - 35*a^4*e^8)*ArcTan[( Sqrt[c]*x)/Sqrt[a]])/a^(7/2) + 384*c*d*e^7*Log[d + e*x] - 192*c*d*e^7*Log[ a + c*x^2])/(48*(c*d^2 + a*e^2)^5)
Time = 0.83 (sec) , antiderivative size = 490, normalized size of antiderivative = 1.14, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.529, Rules used = {496, 25, 686, 25, 27, 686, 27, 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.
| \(\displaystyle \int \frac {1}{\left (a+c x^2\right )^4 (d+e x)^2} \, dx\) |
| \(\Big \downarrow \) 496 |
| \(\displaystyle \frac {a e+c d x}{6 a \left (a+c x^2\right )^3 (d+e x) \left (a e^2+c d^2\right )}-\frac {\int -\frac {5 c d^2+6 c e x d+7 a e^2}{(d+e x)^2 \left (c x^2+a\right )^3}dx}{6 a \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {5 c d^2+6 c e x d+7 a e^2}{(d+e x)^2 \left (c x^2+a\right )^3}dx}{6 a \left (a e^2+c d^2\right )}+\frac {a e+c d x}{6 a \left (a+c x^2\right )^3 (d+e x) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 686 |
| \(\displaystyle \frac {-\frac {\int -\frac {c \left (15 c^2 d^4+34 a c e^2 d^2+4 c e \left (5 c d^2+13 a e^2\right ) x d+35 a^2 e^4\right )}{(d+e x)^2 \left (c x^2+a\right )^2}dx}{4 a c \left (a e^2+c d^2\right )}-\frac {a e \left (c d^2-7 a e^2\right )-c d x \left (13 a e^2+5 c d^2\right )}{4 a \left (a+c x^2\right )^2 (d+e x) \left (a e^2+c d^2\right )}}{6 a \left (a e^2+c d^2\right )}+\frac {a e+c d x}{6 a \left (a+c x^2\right )^3 (d+e x) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {c \left (15 c^2 d^4+34 a c e^2 d^2+4 c e \left (5 c d^2+13 a e^2\right ) x d+35 a^2 e^4\right )}{(d+e x)^2 \left (c x^2+a\right )^2}dx}{4 a c \left (a e^2+c d^2\right )}-\frac {a e \left (c d^2-7 a e^2\right )-c d x \left (13 a e^2+5 c d^2\right )}{4 a \left (a+c x^2\right )^2 (d+e x) \left (a e^2+c d^2\right )}}{6 a \left (a e^2+c d^2\right )}+\frac {a e+c d x}{6 a \left (a+c x^2\right )^3 (d+e x) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {15 c^2 d^4+34 a c e^2 d^2+4 c e \left (5 c d^2+13 a e^2\right ) x d+35 a^2 e^4}{(d+e x)^2 \left (c x^2+a\right )^2}dx}{4 a \left (a e^2+c d^2\right )}-\frac {a e \left (c d^2-7 a e^2\right )-c d x \left (13 a e^2+5 c d^2\right )}{4 a \left (a+c x^2\right )^2 (d+e x) \left (a e^2+c d^2\right )}}{6 a \left (a e^2+c d^2\right )}+\frac {a e+c d x}{6 a \left (a+c x^2\right )^3 (d+e x) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 686 |
| \(\displaystyle \frac {\frac {-\frac {\int -\frac {3 c \left (5 c^3 d^6+13 a c^2 e^2 d^4+11 a^2 c e^4 d^2+2 c e \left (5 c^2 d^4+18 a c e^2 d^2+29 a^2 e^4\right ) x d+35 a^3 e^6\right )}{(d+e x)^2 \left (c x^2+a\right )}dx}{2 a c \left (a e^2+c d^2\right )}-\frac {a e \left (5 c d^2-7 a e^2\right ) \left (5 a e^2+c d^2\right )-3 c d x \left (29 a^2 e^4+18 a c d^2 e^2+5 c^2 d^4\right )}{2 a \left (a+c x^2\right ) (d+e x) \left (a e^2+c d^2\right )}}{4 a \left (a e^2+c d^2\right )}-\frac {a e \left (c d^2-7 a e^2\right )-c d x \left (13 a e^2+5 c d^2\right )}{4 a \left (a+c x^2\right )^2 (d+e x) \left (a e^2+c d^2\right )}}{6 a \left (a e^2+c d^2\right )}+\frac {a e+c d x}{6 a \left (a+c x^2\right )^3 (d+e x) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {3 \int \frac {5 c^3 d^6+13 a c^2 e^2 d^4+11 a^2 c e^4 d^2+2 c e \left (5 c^2 d^4+18 a c e^2 d^2+29 a^2 e^4\right ) x d+35 a^3 e^6}{(d+e x)^2 \left (c x^2+a\right )}dx}{2 a \left (a e^2+c d^2\right )}-\frac {a e \left (5 c d^2-7 a e^2\right ) \left (5 a e^2+c d^2\right )-3 c d x \left (29 a^2 e^4+18 a c d^2 e^2+5 c^2 d^4\right )}{2 a \left (a+c x^2\right ) (d+e x) \left (a e^2+c d^2\right )}}{4 a \left (a e^2+c d^2\right )}-\frac {a e \left (c d^2-7 a e^2\right )-c d x \left (13 a e^2+5 c d^2\right )}{4 a \left (a+c x^2\right )^2 (d+e x) \left (a e^2+c d^2\right )}}{6 a \left (a e^2+c d^2\right )}+\frac {a e+c d x}{6 a \left (a+c x^2\right )^3 (d+e x) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 657 |
| \(\displaystyle \frac {\frac {\frac {3 \int \left (\frac {128 a^3 c d e^8}{\left (c d^2+a e^2\right )^2 (d+e x)}+\frac {\left (-5 c^3 d^6-23 a c^2 e^2 d^4-47 a^2 c e^4 d^2+35 a^3 e^6\right ) e^2}{\left (c d^2+a e^2\right ) (d+e x)^2}+\frac {c \left (5 c^4 d^8+28 a c^3 e^2 d^6+70 a^2 c^2 e^4 d^4+140 a^3 c e^6 d^2-128 a^3 c e^7 x d-35 a^4 e^8\right )}{\left (c d^2+a e^2\right )^2 \left (c x^2+a\right )}\right )dx}{2 a \left (a e^2+c d^2\right )}-\frac {a e \left (5 c d^2-7 a e^2\right ) \left (5 a e^2+c d^2\right )-3 c d x \left (29 a^2 e^4+18 a c d^2 e^2+5 c^2 d^4\right )}{2 a \left (a+c x^2\right ) (d+e x) \left (a e^2+c d^2\right )}}{4 a \left (a e^2+c d^2\right )}-\frac {a e \left (c d^2-7 a e^2\right )-c d x \left (13 a e^2+5 c d^2\right )}{4 a \left (a+c x^2\right )^2 (d+e x) \left (a e^2+c d^2\right )}}{6 a \left (a e^2+c d^2\right )}+\frac {a e+c d x}{6 a \left (a+c x^2\right )^3 (d+e x) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-\frac {64 a^3 c d e^7 \log \left (a+c x^2\right )}{\left (a e^2+c d^2\right )^2}+\frac {128 a^3 c d e^7 \log (d+e x)}{\left (a e^2+c d^2\right )^2}+\frac {e \left (-35 a^3 e^6+47 a^2 c d^2 e^4+23 a c^2 d^4 e^2+5 c^3 d^6\right )}{(d+e x) \left (a e^2+c d^2\right )}+\frac {\sqrt {c} \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (-35 a^4 e^8+140 a^3 c d^2 e^6+70 a^2 c^2 d^4 e^4+28 a c^3 d^6 e^2+5 c^4 d^8\right )}{\sqrt {a} \left (a e^2+c d^2\right )^2}\right )}{2 a \left (a e^2+c d^2\right )}-\frac {a e \left (5 c d^2-7 a e^2\right ) \left (5 a e^2+c d^2\right )-3 c d x \left (29 a^2 e^4+18 a c d^2 e^2+5 c^2 d^4\right )}{2 a \left (a+c x^2\right ) (d+e x) \left (a e^2+c d^2\right )}}{4 a \left (a e^2+c d^2\right )}-\frac {a e \left (c d^2-7 a e^2\right )-c d x \left (13 a e^2+5 c d^2\right )}{4 a \left (a+c x^2\right )^2 (d+e x) \left (a e^2+c d^2\right )}}{6 a \left (a e^2+c d^2\right )}+\frac {a e+c d x}{6 a \left (a+c x^2\right )^3 (d+e x) \left (a e^2+c d^2\right )}\) |
(a*e + c*d*x)/(6*a*(c*d^2 + a*e^2)*(d + e*x)*(a + c*x^2)^3) + (-1/4*(a*e*( c*d^2 - 7*a*e^2) - c*d*(5*c*d^2 + 13*a*e^2)*x)/(a*(c*d^2 + a*e^2)*(d + e*x )*(a + c*x^2)^2) + (-1/2*(a*e*(5*c*d^2 - 7*a*e^2)*(c*d^2 + 5*a*e^2) - 3*c* d*(5*c^2*d^4 + 18*a*c*d^2*e^2 + 29*a^2*e^4)*x)/(a*(c*d^2 + a*e^2)*(d + e*x )*(a + c*x^2)) + (3*((e*(5*c^3*d^6 + 23*a*c^2*d^4*e^2 + 47*a^2*c*d^2*e^4 - 35*a^3*e^6))/((c*d^2 + a*e^2)*(d + e*x)) + (Sqrt[c]*(5*c^4*d^8 + 28*a*c^3 *d^6*e^2 + 70*a^2*c^2*d^4*e^4 + 140*a^3*c*d^2*e^6 - 35*a^4*e^8)*ArcTan[(Sq rt[c]*x)/Sqrt[a]])/(Sqrt[a]*(c*d^2 + a*e^2)^2) + (128*a^3*c*d*e^7*Log[d + e*x])/(c*d^2 + a*e^2)^2 - (64*a^3*c*d*e^7*Log[a + c*x^2])/(c*d^2 + a*e^2)^ 2))/(2*a*(c*d^2 + a*e^2)))/(4*a*(c*d^2 + a*e^2)))/(6*a*(c*d^2 + a*e^2))
3.6.24.3.1 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[((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*(p + 1)*(b*c^2 + a*d^2))), x] + Simp[1/(2*a*(p + 1)*(b*c^2 + a*d^2)) Int[(c + d*x)^n*(a + b*x^2)^(p + 1)*Simp[b*c^2*(2*p + 3) + a*d^2*(n + 2*p + 3) + b*c*d*(n + 2 *p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[p, -1] && IntQuad raticQ[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))*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[ 1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)) Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Sim p[f*(c^2*d^2*(2*p + 3) + a*c*e^2*(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && LtQ [p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Time = 2.38 (sec) , antiderivative size = 473, normalized size of antiderivative = 1.10
| method | result | size |
| default | \(-\frac {c \left (\frac {\frac {c^{2} \left (19 a^{4} e^{8}-28 a^{3} c \,d^{2} e^{6}-70 a^{2} c^{2} d^{4} e^{4}-28 a \,c^{3} d^{6} e^{2}-5 c^{4} d^{8}\right ) x^{5}}{16 a^{3}}+\left (-3 a d \,e^{7} c^{2}-3 d^{3} e^{5} c^{3}\right ) x^{4}+\frac {c \left (17 a^{4} e^{8}-20 a^{3} c \,d^{2} e^{6}-60 a^{2} c^{2} d^{4} e^{4}-28 a \,c^{3} d^{6} e^{2}-5 c^{4} d^{8}\right ) x^{3}}{6 a^{2}}+\left (-7 a^{2} d \,e^{7} c -8 a \,d^{3} e^{5} c^{2}-c^{3} d^{5} e^{3}\right ) x^{2}+\frac {\left (29 a^{4} e^{8}-20 a^{3} c \,d^{2} e^{6}-90 a^{2} c^{2} d^{4} e^{4}-52 a \,c^{3} d^{6} e^{2}-11 c^{4} d^{8}\right ) x}{16 a}-\frac {d e \left (13 e^{6} a^{3}+18 d^{2} e^{4} a^{2} c +6 d^{4} e^{2} c^{2} a +c^{3} d^{6}\right )}{3}}{\left (c \,x^{2}+a \right )^{3}}+\frac {64 a^{3} d \,e^{7} \ln \left (c \,x^{2}+a \right )+\frac {\left (35 a^{4} e^{8}-140 a^{3} c \,d^{2} e^{6}-70 a^{2} c^{2} d^{4} e^{4}-28 a \,c^{3} d^{6} e^{2}-5 c^{4} d^{8}\right ) \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}}}{16 a^{3}}\right )}{\left (e^{2} a +c \,d^{2}\right )^{5}}-\frac {e^{7}}{\left (e^{2} a +c \,d^{2}\right )^{4} \left (e x +d \right )}+\frac {8 c d \,e^{7} \ln \left (e x +d \right )}{\left (e^{2} a +c \,d^{2}\right )^{5}}\) | \(473\) |
| risch | \(\text {Expression too large to display}\) | \(5462\) |
-c/(a*e^2+c*d^2)^5*((1/16*c^2*(19*a^4*e^8-28*a^3*c*d^2*e^6-70*a^2*c^2*d^4* e^4-28*a*c^3*d^6*e^2-5*c^4*d^8)/a^3*x^5+(-3*a*c^2*d*e^7-3*c^3*d^3*e^5)*x^4 +1/6*c*(17*a^4*e^8-20*a^3*c*d^2*e^6-60*a^2*c^2*d^4*e^4-28*a*c^3*d^6*e^2-5* c^4*d^8)/a^2*x^3+(-7*a^2*c*d*e^7-8*a*c^2*d^3*e^5-c^3*d^5*e^3)*x^2+1/16*(29 *a^4*e^8-20*a^3*c*d^2*e^6-90*a^2*c^2*d^4*e^4-52*a*c^3*d^6*e^2-11*c^4*d^8)/ a*x-1/3*d*e*(13*a^3*e^6+18*a^2*c*d^2*e^4+6*a*c^2*d^4*e^2+c^3*d^6))/(c*x^2+ a)^3+1/16/a^3*(64*a^3*d*e^7*ln(c*x^2+a)+(35*a^4*e^8-140*a^3*c*d^2*e^6-70*a ^2*c^2*d^4*e^4-28*a*c^3*d^6*e^2-5*c^4*d^8)/(a*c)^(1/2)*arctan(c*x/(a*c)^(1 /2))))-e^7/(a*e^2+c*d^2)^4/(e*x+d)+8*c*d*e^7*ln(e*x+d)/(a*e^2+c*d^2)^5
Leaf count of result is larger than twice the leaf count of optimal. 1909 vs. \(2 (412) = 824\).
Time = 27.06 (sec) , antiderivative size = 3843, normalized size of antiderivative = 8.94 \[ \int \frac {1}{(d+e x)^2 \left (a+c x^2\right )^4} \, dx=\text {Too large to display} \]
[1/96*(32*a^3*c^4*d^8*e + 192*a^4*c^3*d^6*e^3 + 576*a^5*c^2*d^4*e^5 + 320* a^6*c*d^2*e^7 - 96*a^7*e^9 + 6*(5*c^7*d^8*e + 28*a*c^6*d^6*e^3 + 70*a^2*c^ 5*d^4*e^5 + 12*a^3*c^4*d^2*e^7 - 35*a^4*c^3*e^9)*x^6 + 6*(5*c^7*d^9 + 28*a *c^6*d^7*e^2 + 70*a^2*c^5*d^5*e^4 + 76*a^3*c^4*d^3*e^6 + 29*a^4*c^3*d*e^8) *x^5 + 16*(5*a*c^6*d^8*e + 28*a^2*c^5*d^6*e^3 + 78*a^3*c^4*d^4*e^5 + 20*a^ 4*c^3*d^2*e^7 - 35*a^5*c^2*e^9)*x^4 + 16*(5*a*c^6*d^9 + 28*a^2*c^5*d^7*e^2 + 66*a^3*c^4*d^5*e^4 + 68*a^4*c^3*d^3*e^6 + 25*a^5*c^2*d*e^8)*x^3 + 6*(11 *a^2*c^5*d^8*e + 68*a^3*c^4*d^6*e^3 + 218*a^4*c^3*d^4*e^5 + 84*a^5*c^2*d^2 *e^7 - 77*a^6*c*e^9)*x^2 - 3*(5*a^3*c^4*d^9 + 28*a^4*c^3*d^7*e^2 + 70*a^5* c^2*d^5*e^4 + 140*a^6*c*d^3*e^6 - 35*a^7*d*e^8 + (5*c^7*d^8*e + 28*a*c^6*d ^6*e^3 + 70*a^2*c^5*d^4*e^5 + 140*a^3*c^4*d^2*e^7 - 35*a^4*c^3*e^9)*x^7 + (5*c^7*d^9 + 28*a*c^6*d^7*e^2 + 70*a^2*c^5*d^5*e^4 + 140*a^3*c^4*d^3*e^6 - 35*a^4*c^3*d*e^8)*x^6 + 3*(5*a*c^6*d^8*e + 28*a^2*c^5*d^6*e^3 + 70*a^3*c^ 4*d^4*e^5 + 140*a^4*c^3*d^2*e^7 - 35*a^5*c^2*e^9)*x^5 + 3*(5*a*c^6*d^9 + 2 8*a^2*c^5*d^7*e^2 + 70*a^3*c^4*d^5*e^4 + 140*a^4*c^3*d^3*e^6 - 35*a^5*c^2* d*e^8)*x^4 + 3*(5*a^2*c^5*d^8*e + 28*a^3*c^4*d^6*e^3 + 70*a^4*c^3*d^4*e^5 + 140*a^5*c^2*d^2*e^7 - 35*a^6*c*e^9)*x^3 + 3*(5*a^2*c^5*d^9 + 28*a^3*c^4* d^7*e^2 + 70*a^4*c^3*d^5*e^4 + 140*a^5*c^2*d^3*e^6 - 35*a^6*c*d*e^8)*x^2 + (5*a^3*c^4*d^8*e + 28*a^4*c^3*d^6*e^3 + 70*a^5*c^2*d^4*e^5 + 140*a^6*c*d^ 2*e^7 - 35*a^7*e^9)*x)*sqrt(-c/a)*log((c*x^2 - 2*a*x*sqrt(-c/a) - a)/(c...
Timed out. \[ \int \frac {1}{(d+e x)^2 \left (a+c x^2\right )^4} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 1215 vs. \(2 (412) = 824\).
Time = 0.33 (sec) , antiderivative size = 1215, normalized size of antiderivative = 2.83 \[ \int \frac {1}{(d+e x)^2 \left (a+c x^2\right )^4} \, dx=-\frac {4 \, c d e^{7} \log \left (c x^{2} + a\right )}{c^{5} d^{10} + 5 \, a c^{4} d^{8} e^{2} + 10 \, a^{2} c^{3} d^{6} e^{4} + 10 \, a^{3} c^{2} d^{4} e^{6} + 5 \, a^{4} c d^{2} e^{8} + a^{5} e^{10}} + \frac {8 \, c d e^{7} \log \left (e x + d\right )}{c^{5} d^{10} + 5 \, a c^{4} d^{8} e^{2} + 10 \, a^{2} c^{3} d^{6} e^{4} + 10 \, a^{3} c^{2} d^{4} e^{6} + 5 \, a^{4} c d^{2} e^{8} + a^{5} e^{10}} + \frac {{\left (5 \, c^{5} d^{8} + 28 \, a c^{4} d^{6} e^{2} + 70 \, a^{2} c^{3} d^{4} e^{4} + 140 \, a^{3} c^{2} d^{2} e^{6} - 35 \, a^{4} c e^{8}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{16 \, {\left (a^{3} c^{5} d^{10} + 5 \, a^{4} c^{4} d^{8} e^{2} + 10 \, a^{5} c^{3} d^{6} e^{4} + 10 \, a^{6} c^{2} d^{4} e^{6} + 5 \, a^{7} c d^{2} e^{8} + a^{8} e^{10}\right )} \sqrt {a c}} + \frac {16 \, a^{3} c^{3} d^{6} e + 80 \, a^{4} c^{2} d^{4} e^{3} + 208 \, a^{5} c d^{2} e^{5} - 48 \, a^{6} e^{7} + 3 \, {\left (5 \, c^{6} d^{6} e + 23 \, a c^{5} d^{4} e^{3} + 47 \, a^{2} c^{4} d^{2} e^{5} - 35 \, a^{3} c^{3} e^{7}\right )} x^{6} + 3 \, {\left (5 \, c^{6} d^{7} + 23 \, a c^{5} d^{5} e^{2} + 47 \, a^{2} c^{4} d^{3} e^{4} + 29 \, a^{3} c^{3} d e^{6}\right )} x^{5} + 8 \, {\left (5 \, a c^{5} d^{6} e + 23 \, a^{2} c^{4} d^{4} e^{3} + 55 \, a^{3} c^{3} d^{2} e^{5} - 35 \, a^{4} c^{2} e^{7}\right )} x^{4} + 8 \, {\left (5 \, a c^{5} d^{7} + 23 \, a^{2} c^{4} d^{5} e^{2} + 43 \, a^{3} c^{3} d^{3} e^{4} + 25 \, a^{4} c^{2} d e^{6}\right )} x^{3} + 3 \, {\left (11 \, a^{2} c^{4} d^{6} e + 57 \, a^{3} c^{3} d^{4} e^{3} + 161 \, a^{4} c^{2} d^{2} e^{5} - 77 \, a^{5} c e^{7}\right )} x^{2} + {\left (33 \, a^{2} c^{4} d^{7} + 139 \, a^{3} c^{3} d^{5} e^{2} + 227 \, a^{4} c^{2} d^{3} e^{4} + 121 \, a^{5} c d e^{6}\right )} x}{48 \, {\left (a^{6} c^{4} d^{9} + 4 \, a^{7} c^{3} d^{7} e^{2} + 6 \, a^{8} c^{2} d^{5} e^{4} + 4 \, a^{9} c d^{3} e^{6} + a^{10} d e^{8} + {\left (a^{3} c^{7} d^{8} e + 4 \, a^{4} c^{6} d^{6} e^{3} + 6 \, a^{5} c^{5} d^{4} e^{5} + 4 \, a^{6} c^{4} d^{2} e^{7} + a^{7} c^{3} e^{9}\right )} x^{7} + {\left (a^{3} c^{7} d^{9} + 4 \, a^{4} c^{6} d^{7} e^{2} + 6 \, a^{5} c^{5} d^{5} e^{4} + 4 \, a^{6} c^{4} d^{3} e^{6} + a^{7} c^{3} d e^{8}\right )} x^{6} + 3 \, {\left (a^{4} c^{6} d^{8} e + 4 \, a^{5} c^{5} d^{6} e^{3} + 6 \, a^{6} c^{4} d^{4} e^{5} + 4 \, a^{7} c^{3} d^{2} e^{7} + a^{8} c^{2} e^{9}\right )} x^{5} + 3 \, {\left (a^{4} c^{6} d^{9} + 4 \, a^{5} c^{5} d^{7} e^{2} + 6 \, a^{6} c^{4} d^{5} e^{4} + 4 \, a^{7} c^{3} d^{3} e^{6} + a^{8} c^{2} d e^{8}\right )} x^{4} + 3 \, {\left (a^{5} c^{5} d^{8} e + 4 \, a^{6} c^{4} d^{6} e^{3} + 6 \, a^{7} c^{3} d^{4} e^{5} + 4 \, a^{8} c^{2} d^{2} e^{7} + a^{9} c e^{9}\right )} x^{3} + 3 \, {\left (a^{5} c^{5} d^{9} + 4 \, a^{6} c^{4} d^{7} e^{2} + 6 \, a^{7} c^{3} d^{5} e^{4} + 4 \, a^{8} c^{2} d^{3} e^{6} + a^{9} c d e^{8}\right )} x^{2} + {\left (a^{6} c^{4} d^{8} e + 4 \, a^{7} c^{3} d^{6} e^{3} + 6 \, a^{8} c^{2} d^{4} e^{5} + 4 \, a^{9} c d^{2} e^{7} + a^{10} e^{9}\right )} x\right )}} \]
-4*c*d*e^7*log(c*x^2 + a)/(c^5*d^10 + 5*a*c^4*d^8*e^2 + 10*a^2*c^3*d^6*e^4 + 10*a^3*c^2*d^4*e^6 + 5*a^4*c*d^2*e^8 + a^5*e^10) + 8*c*d*e^7*log(e*x + d)/(c^5*d^10 + 5*a*c^4*d^8*e^2 + 10*a^2*c^3*d^6*e^4 + 10*a^3*c^2*d^4*e^6 + 5*a^4*c*d^2*e^8 + a^5*e^10) + 1/16*(5*c^5*d^8 + 28*a*c^4*d^6*e^2 + 70*a^2 *c^3*d^4*e^4 + 140*a^3*c^2*d^2*e^6 - 35*a^4*c*e^8)*arctan(c*x/sqrt(a*c))/( (a^3*c^5*d^10 + 5*a^4*c^4*d^8*e^2 + 10*a^5*c^3*d^6*e^4 + 10*a^6*c^2*d^4*e^ 6 + 5*a^7*c*d^2*e^8 + a^8*e^10)*sqrt(a*c)) + 1/48*(16*a^3*c^3*d^6*e + 80*a ^4*c^2*d^4*e^3 + 208*a^5*c*d^2*e^5 - 48*a^6*e^7 + 3*(5*c^6*d^6*e + 23*a*c^ 5*d^4*e^3 + 47*a^2*c^4*d^2*e^5 - 35*a^3*c^3*e^7)*x^6 + 3*(5*c^6*d^7 + 23*a *c^5*d^5*e^2 + 47*a^2*c^4*d^3*e^4 + 29*a^3*c^3*d*e^6)*x^5 + 8*(5*a*c^5*d^6 *e + 23*a^2*c^4*d^4*e^3 + 55*a^3*c^3*d^2*e^5 - 35*a^4*c^2*e^7)*x^4 + 8*(5* a*c^5*d^7 + 23*a^2*c^4*d^5*e^2 + 43*a^3*c^3*d^3*e^4 + 25*a^4*c^2*d*e^6)*x^ 3 + 3*(11*a^2*c^4*d^6*e + 57*a^3*c^3*d^4*e^3 + 161*a^4*c^2*d^2*e^5 - 77*a^ 5*c*e^7)*x^2 + (33*a^2*c^4*d^7 + 139*a^3*c^3*d^5*e^2 + 227*a^4*c^2*d^3*e^4 + 121*a^5*c*d*e^6)*x)/(a^6*c^4*d^9 + 4*a^7*c^3*d^7*e^2 + 6*a^8*c^2*d^5*e^ 4 + 4*a^9*c*d^3*e^6 + a^10*d*e^8 + (a^3*c^7*d^8*e + 4*a^4*c^6*d^6*e^3 + 6* a^5*c^5*d^4*e^5 + 4*a^6*c^4*d^2*e^7 + a^7*c^3*e^9)*x^7 + (a^3*c^7*d^9 + 4* a^4*c^6*d^7*e^2 + 6*a^5*c^5*d^5*e^4 + 4*a^6*c^4*d^3*e^6 + a^7*c^3*d*e^8)*x ^6 + 3*(a^4*c^6*d^8*e + 4*a^5*c^5*d^6*e^3 + 6*a^6*c^4*d^4*e^5 + 4*a^7*c^3* d^2*e^7 + a^8*c^2*e^9)*x^5 + 3*(a^4*c^6*d^9 + 4*a^5*c^5*d^7*e^2 + 6*a^6...
Leaf count of result is larger than twice the leaf count of optimal. 906 vs. \(2 (412) = 824\).
Time = 0.30 (sec) , antiderivative size = 906, normalized size of antiderivative = 2.11 \[ \int \frac {1}{(d+e x)^2 \left (a+c x^2\right )^4} \, dx=-\frac {e^{15}}{{\left (c^{4} d^{8} e^{8} + 4 \, a c^{3} d^{6} e^{10} + 6 \, a^{2} c^{2} d^{4} e^{12} + 4 \, a^{3} c d^{2} e^{14} + a^{4} e^{16}\right )} {\left (e x + d\right )}} - \frac {4 \, c d e^{7} \log \left (c - \frac {2 \, c d}{e x + d} + \frac {c d^{2}}{{\left (e x + d\right )}^{2}} + \frac {a e^{2}}{{\left (e x + d\right )}^{2}}\right )}{c^{5} d^{10} + 5 \, a c^{4} d^{8} e^{2} + 10 \, a^{2} c^{3} d^{6} e^{4} + 10 \, a^{3} c^{2} d^{4} e^{6} + 5 \, a^{4} c d^{2} e^{8} + a^{5} e^{10}} + \frac {{\left (5 \, c^{5} d^{8} e^{2} + 28 \, a c^{4} d^{6} e^{4} + 70 \, a^{2} c^{3} d^{4} e^{6} + 140 \, a^{3} c^{2} d^{2} e^{8} - 35 \, a^{4} c e^{10}\right )} \arctan \left (\frac {c d - \frac {c d^{2}}{e x + d} - \frac {a e^{2}}{e x + d}}{\sqrt {a c} e}\right )}{16 \, {\left (a^{3} c^{5} d^{10} + 5 \, a^{4} c^{4} d^{8} e^{2} + 10 \, a^{5} c^{3} d^{6} e^{4} + 10 \, a^{6} c^{2} d^{4} e^{6} + 5 \, a^{7} c d^{2} e^{8} + a^{8} e^{10}\right )} \sqrt {a c} e^{2}} + \frac {15 \, c^{7} d^{7} e + 79 \, a c^{6} d^{5} e^{3} + 185 \, a^{2} c^{5} d^{3} e^{5} - 295 \, a^{3} c^{4} d e^{7} - \frac {3 \, {\left (25 \, c^{7} d^{8} e^{2} + 130 \, a c^{6} d^{6} e^{4} + 300 \, a^{2} c^{5} d^{4} e^{6} - 618 \, a^{3} c^{4} d^{2} e^{8} + 19 \, a^{4} c^{3} e^{10}\right )}}{{\left (e x + d\right )} e} + \frac {6 \, {\left (25 \, c^{7} d^{9} e^{3} + 135 \, a c^{6} d^{7} e^{5} + 327 \, a^{2} c^{5} d^{5} e^{7} - 691 \, a^{3} c^{4} d^{3} e^{9} - 76 \, a^{4} c^{3} d e^{11}\right )}}{{\left (e x + d\right )}^{2} e^{2}} - \frac {2 \, {\left (75 \, c^{7} d^{10} e^{4} + 440 \, a c^{6} d^{8} e^{6} + 1162 \, a^{2} c^{5} d^{6} e^{8} - 2212 \, a^{3} c^{4} d^{4} e^{10} - 1277 \, a^{4} c^{3} d^{2} e^{12} + 68 \, a^{5} c^{2} e^{14}\right )}}{{\left (e x + d\right )}^{3} e^{3}} + \frac {3 \, {\left (25 \, c^{7} d^{11} e^{5} + 165 \, a c^{6} d^{9} e^{7} + 490 \, a^{2} c^{5} d^{7} e^{9} - 742 \, a^{3} c^{4} d^{5} e^{11} - 1139 \, a^{4} c^{3} d^{3} e^{13} - 47 \, a^{5} c^{2} d e^{15}\right )}}{{\left (e x + d\right )}^{4} e^{4}} - \frac {3 \, {\left (5 \, c^{7} d^{12} e^{6} + 38 \, a c^{6} d^{10} e^{8} + 131 \, a^{2} c^{5} d^{8} e^{10} - 140 \, a^{3} c^{4} d^{6} e^{12} - 517 \, a^{4} c^{3} d^{4} e^{14} - 250 \, a^{5} c^{2} d^{2} e^{16} + 29 \, a^{6} c e^{18}\right )}}{{\left (e x + d\right )}^{5} e^{5}}}{48 \, {\left (c d^{2} + a e^{2}\right )}^{5} a^{3} {\left (c - \frac {2 \, c d}{e x + d} + \frac {c d^{2}}{{\left (e x + d\right )}^{2}} + \frac {a e^{2}}{{\left (e x + d\right )}^{2}}\right )}^{3}} \]
-e^15/((c^4*d^8*e^8 + 4*a*c^3*d^6*e^10 + 6*a^2*c^2*d^4*e^12 + 4*a^3*c*d^2* e^14 + a^4*e^16)*(e*x + d)) - 4*c*d*e^7*log(c - 2*c*d/(e*x + d) + c*d^2/(e *x + d)^2 + a*e^2/(e*x + d)^2)/(c^5*d^10 + 5*a*c^4*d^8*e^2 + 10*a^2*c^3*d^ 6*e^4 + 10*a^3*c^2*d^4*e^6 + 5*a^4*c*d^2*e^8 + a^5*e^10) + 1/16*(5*c^5*d^8 *e^2 + 28*a*c^4*d^6*e^4 + 70*a^2*c^3*d^4*e^6 + 140*a^3*c^2*d^2*e^8 - 35*a^ 4*c*e^10)*arctan((c*d - c*d^2/(e*x + d) - a*e^2/(e*x + d))/(sqrt(a*c)*e))/ ((a^3*c^5*d^10 + 5*a^4*c^4*d^8*e^2 + 10*a^5*c^3*d^6*e^4 + 10*a^6*c^2*d^4*e ^6 + 5*a^7*c*d^2*e^8 + a^8*e^10)*sqrt(a*c)*e^2) + 1/48*(15*c^7*d^7*e + 79* a*c^6*d^5*e^3 + 185*a^2*c^5*d^3*e^5 - 295*a^3*c^4*d*e^7 - 3*(25*c^7*d^8*e^ 2 + 130*a*c^6*d^6*e^4 + 300*a^2*c^5*d^4*e^6 - 618*a^3*c^4*d^2*e^8 + 19*a^4 *c^3*e^10)/((e*x + d)*e) + 6*(25*c^7*d^9*e^3 + 135*a*c^6*d^7*e^5 + 327*a^2 *c^5*d^5*e^7 - 691*a^3*c^4*d^3*e^9 - 76*a^4*c^3*d*e^11)/((e*x + d)^2*e^2) - 2*(75*c^7*d^10*e^4 + 440*a*c^6*d^8*e^6 + 1162*a^2*c^5*d^6*e^8 - 2212*a^3 *c^4*d^4*e^10 - 1277*a^4*c^3*d^2*e^12 + 68*a^5*c^2*e^14)/((e*x + d)^3*e^3) + 3*(25*c^7*d^11*e^5 + 165*a*c^6*d^9*e^7 + 490*a^2*c^5*d^7*e^9 - 742*a^3* c^4*d^5*e^11 - 1139*a^4*c^3*d^3*e^13 - 47*a^5*c^2*d*e^15)/((e*x + d)^4*e^4 ) - 3*(5*c^7*d^12*e^6 + 38*a*c^6*d^10*e^8 + 131*a^2*c^5*d^8*e^10 - 140*a^3 *c^4*d^6*e^12 - 517*a^4*c^3*d^4*e^14 - 250*a^5*c^2*d^2*e^16 + 29*a^6*c*e^1 8)/((e*x + d)^5*e^5))/((c*d^2 + a*e^2)^5*a^3*(c - 2*c*d/(e*x + d) + c*d^2/ (e*x + d)^2 + a*e^2/(e*x + d)^2)^3)
Time = 11.45 (sec) , antiderivative size = 1876, normalized size of antiderivative = 4.36 \[ \int \frac {1}{(d+e x)^2 \left (a+c x^2\right )^4} \, dx=\text {Too large to display} \]
((c^3*d^6*e - 3*a^3*e^7 + 5*a*c^2*d^4*e^3 + 13*a^2*c*d^2*e^5)/(3*(a*e^2 + c*d^2)*(a^3*e^6 + c^3*d^6 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4)) + (x*(33*c ^3*d^5 + 106*a*c^2*d^3*e^2 + 121*a^2*c*d*e^4))/(48*a*(a^3*e^6 + c^3*d^6 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4)) + (x^6*(5*c^6*d^6*e - 35*a^3*c^3*e^7 + 23*a*c^5*d^4*e^3 + 47*a^2*c^4*d^2*e^5))/(16*a^3*(a^4*e^8 + c^4*d^8 + 4*a* c^3*d^6*e^2 + 4*a^3*c*d^2*e^6 + 6*a^2*c^2*d^4*e^4)) + (x^3*(5*c^4*d^5 + 18 *a*c^3*d^3*e^2 + 25*a^2*c^2*d*e^4))/(6*a^2*(a^3*e^6 + c^3*d^6 + 3*a*c^2*d^ 4*e^2 + 3*a^2*c*d^2*e^4)) + (x^5*(5*c^5*d^5 + 18*a*c^4*d^3*e^2 + 29*a^2*c^ 3*d*e^4))/(16*a^3*(a^3*e^6 + c^3*d^6 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4)) + (x^4*(5*c^5*d^6*e - 35*a^3*c^2*e^7 + 23*a*c^4*d^4*e^3 + 55*a^2*c^3*d^2* e^5))/(6*a^2*(a*e^2 + c*d^2)*(a^3*e^6 + c^3*d^6 + 3*a*c^2*d^4*e^2 + 3*a^2* c*d^2*e^4)) + (x^2*(11*c^4*d^6*e - 77*a^3*c*e^7 + 57*a*c^3*d^4*e^3 + 161*a ^2*c^2*d^2*e^5))/(16*a*(a*e^2 + c*d^2)*(a^3*e^6 + c^3*d^6 + 3*a*c^2*d^4*e^ 2 + 3*a^2*c*d^2*e^4)))/(a^3*d + c^3*d*x^6 + c^3*e*x^7 + a^3*e*x + 3*a^2*c* d*x^2 + 3*a*c^2*d*x^4 + 3*a^2*c*e*x^3 + 3*a*c^2*e*x^5) - (log(25*c^9*d^20* (-a^7*c)^(3/2) - 1225*a^17*e^20*(-a^7*c)^(1/2) + 25*a^10*c^11*d^20*x - 291 237*a*d^4*e^16*(-a^7*c)^(5/2) - 184696*c*d^6*e^14*(-a^7*c)^(5/2) + 140106* a^9*d^2*e^18*(-a^7*c)^(3/2) + 1225*a^20*c*e^20*x + 2069*a^2*c^7*d^16*e^4*( -a^7*c)^(3/2) + 8568*a^3*c^6*d^14*e^6*(-a^7*c)^(3/2) + 24514*a^4*c^5*d^12* e^8*(-a^7*c)^(3/2) + 47740*a^5*c^4*d^10*e^10*(-a^7*c)^(3/2) + 62370*a^6...