Integrand size = 22, antiderivative size = 307 \[ \int \frac {A+B x}{(d+e x) \left (a+c x^2\right )^3} \, dx=-\frac {a (B d-A e)-(A c d+a B e) x}{4 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^2}-\frac {4 a^2 e^2 (B d-A e)+\left (a B e \left (c d^2-3 a e^2\right )-A c d \left (3 c d^2+7 a e^2\right )\right ) x}{8 a^2 \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )}-\frac {\left (a B e \left (c^2 d^4+6 a c d^2 e^2-3 a^2 e^4\right )-A c d \left (3 c^2 d^4+10 a c d^2 e^2+15 a^2 e^4\right )\right ) \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {c} \left (c d^2+a e^2\right )^3}-\frac {e^4 (B d-A e) \log (d+e x)}{\left (c d^2+a e^2\right )^3}+\frac {e^4 (B d-A e) \log \left (a+c x^2\right )}{2 \left (c d^2+a e^2\right )^3} \]
1/4*(-a*(-A*e+B*d)+(A*c*d+B*a*e)*x)/a/(a*e^2+c*d^2)/(c*x^2+a)^2+1/8*(-4*a^ 2*e^2*(-A*e+B*d)-(a*B*e*(-3*a*e^2+c*d^2)-A*c*d*(7*a*e^2+3*c*d^2))*x)/a^2/( a*e^2+c*d^2)^2/(c*x^2+a)-e^4*(-A*e+B*d)*ln(e*x+d)/(a*e^2+c*d^2)^3+1/2*e^4* (-A*e+B*d)*ln(c*x^2+a)/(a*e^2+c*d^2)^3-1/8*(a*B*e*(-3*a^2*e^4+6*a*c*d^2*e^ 2+c^2*d^4)-A*c*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)/(a*e^2+c*d^2)^3/c^(1/2)
Time = 0.19 (sec) , antiderivative size = 263, normalized size of antiderivative = 0.86 \[ \int \frac {A+B x}{(d+e x) \left (a+c x^2\right )^3} \, dx=\frac {\frac {2 \left (c d^2+a e^2\right )^2 (A c d x+a (-B d+A e+B e x))}{a \left (a+c x^2\right )^2}+\frac {\left (c d^2+a e^2\right ) \left (3 A c^2 d^3 x+a c d e (-B d+7 A e) x+a^2 e^2 (-4 B d+4 A e+3 B e x)\right )}{a^2 \left (a+c x^2\right )}+\frac {\left (a B e \left (-c^2 d^4-6 a c d^2 e^2+3 a^2 e^4\right )+A c d \left (3 c^2 d^4+10 a c d^2 e^2+15 a^2 e^4\right )\right ) \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{a^{5/2} \sqrt {c}}+8 e^4 (-B d+A e) \log (d+e x)+4 e^4 (B d-A e) \log \left (a+c x^2\right )}{8 \left (c d^2+a e^2\right )^3} \]
((2*(c*d^2 + a*e^2)^2*(A*c*d*x + a*(-(B*d) + A*e + B*e*x)))/(a*(a + c*x^2) ^2) + ((c*d^2 + a*e^2)*(3*A*c^2*d^3*x + a*c*d*e*(-(B*d) + 7*A*e)*x + a^2*e ^2*(-4*B*d + 4*A*e + 3*B*e*x)))/(a^2*(a + c*x^2)) + ((a*B*e*(-(c^2*d^4) - 6*a*c*d^2*e^2 + 3*a^2*e^4) + A*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)*Sqrt[c]) + 8*e^4*(-(B*d) + A*e)* Log[d + e*x] + 4*e^4*(B*d - A*e)*Log[a + c*x^2])/(8*(c*d^2 + a*e^2)^3)
Time = 0.71 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.14, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {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.
| \(\displaystyle \int \frac {A+B x}{\left (a+c x^2\right )^3 (d+e x)} \, dx\) |
| \(\Big \downarrow \) 686 |
| \(\displaystyle -\frac {\int -\frac {c \left (3 A c d^2-a B e d+4 a A e^2+3 e (A c d+a B e) x\right )}{(d+e x) \left (c x^2+a\right )^2}dx}{4 a c \left (a e^2+c d^2\right )}-\frac {a (B d-A e)-x (a B e+A c d)}{4 a \left (a+c x^2\right )^2 \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {c \left (3 A c d^2-a B e d+4 a A e^2+3 e (A c d+a B e) x\right )}{(d+e x) \left (c x^2+a\right )^2}dx}{4 a c \left (a e^2+c d^2\right )}-\frac {a (B d-A e)-x (a B e+A c d)}{4 a \left (a+c x^2\right )^2 \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {3 A c d^2-a B e d+4 a A e^2+3 e (A c d+a B e) x}{(d+e x) \left (c x^2+a\right )^2}dx}{4 a \left (a e^2+c d^2\right )}-\frac {a (B d-A e)-x (a B e+A c d)}{4 a \left (a+c x^2\right )^2 \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 686 |
| \(\displaystyle \frac {-\frac {\int \frac {c \left (a B d e \left (c d^2+5 a e^2\right )-A \left (3 c^2 d^4+7 a c e^2 d^2+8 a^2 e^4\right )+e \left (a B e \left (c d^2-3 a e^2\right )-A c d \left (3 c d^2+7 a e^2\right )\right ) x\right )}{(d+e x) \left (c x^2+a\right )}dx}{2 a c \left (a e^2+c d^2\right )}-\frac {4 a^2 e^2 (B d-A e)+x \left (a B e \left (c d^2-3 a e^2\right )-A c d \left (7 a e^2+3 c d^2\right )\right )}{2 a \left (a+c x^2\right ) \left (a e^2+c d^2\right )}}{4 a \left (a e^2+c d^2\right )}-\frac {a (B d-A e)-x (a B e+A c d)}{4 a \left (a+c x^2\right )^2 \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\int \frac {a B d e \left (c d^2+5 a e^2\right )-A \left (3 c^2 d^4+7 a c e^2 d^2+8 a^2 e^4\right )+e \left (a B e \left (c d^2-3 a e^2\right )-A c d \left (3 c d^2+7 a e^2\right )\right ) x}{(d+e x) \left (c x^2+a\right )}dx}{2 a \left (a e^2+c d^2\right )}-\frac {4 a^2 e^2 (B d-A e)+x \left (a B e \left (c d^2-3 a e^2\right )-A c d \left (7 a e^2+3 c d^2\right )\right )}{2 a \left (a+c x^2\right ) \left (a e^2+c d^2\right )}}{4 a \left (a e^2+c d^2\right )}-\frac {a (B d-A e)-x (a B e+A c d)}{4 a \left (a+c x^2\right )^2 \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 657 |
| \(\displaystyle \frac {-\frac {\int \left (\frac {-8 a^2 c (B d-A e) x e^4+a B \left (c^2 d^4+6 a c e^2 d^2-3 a^2 e^4\right ) e-A c d \left (3 c^2 d^4+10 a c e^2 d^2+15 a^2 e^4\right )}{\left (c d^2+a e^2\right ) \left (c x^2+a\right )}-\frac {8 a^2 e^5 (A e-B d)}{\left (c d^2+a e^2\right ) (d+e x)}\right )dx}{2 a \left (a e^2+c d^2\right )}-\frac {4 a^2 e^2 (B d-A e)+x \left (a B e \left (c d^2-3 a e^2\right )-A c d \left (7 a e^2+3 c d^2\right )\right )}{2 a \left (a+c x^2\right ) \left (a e^2+c d^2\right )}}{4 a \left (a e^2+c d^2\right )}-\frac {a (B d-A e)-x (a B e+A c d)}{4 a \left (a+c x^2\right )^2 \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {\frac {\arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (a B e \left (-3 a^2 e^4+6 a c d^2 e^2+c^2 d^4\right )-A c d \left (15 a^2 e^4+10 a c d^2 e^2+3 c^2 d^4\right )\right )}{\sqrt {a} \sqrt {c} \left (a e^2+c d^2\right )}-\frac {4 a^2 e^4 \log \left (a+c x^2\right ) (B d-A e)}{a e^2+c d^2}+\frac {8 a^2 e^4 (B d-A e) \log (d+e x)}{a e^2+c d^2}}{2 a \left (a e^2+c d^2\right )}-\frac {4 a^2 e^2 (B d-A e)+x \left (a B e \left (c d^2-3 a e^2\right )-A c d \left (7 a e^2+3 c d^2\right )\right )}{2 a \left (a+c x^2\right ) \left (a e^2+c d^2\right )}}{4 a \left (a e^2+c d^2\right )}-\frac {a (B d-A e)-x (a B e+A c d)}{4 a \left (a+c x^2\right )^2 \left (a e^2+c d^2\right )}\) |
-1/4*(a*(B*d - A*e) - (A*c*d + a*B*e)*x)/(a*(c*d^2 + a*e^2)*(a + c*x^2)^2) + (-1/2*(4*a^2*e^2*(B*d - A*e) + (a*B*e*(c*d^2 - 3*a*e^2) - A*c*d*(3*c*d^ 2 + 7*a*e^2))*x)/(a*(c*d^2 + a*e^2)*(a + c*x^2)) - (((a*B*e*(c^2*d^4 + 6*a *c*d^2*e^2 - 3*a^2*e^4) - A*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]])/(Sqrt[a]*Sqrt[c]*(c*d^2 + a*e^2)) + (8*a^2*e ^4*(B*d - A*e)*Log[d + e*x])/(c*d^2 + a*e^2) - (4*a^2*e^4*(B*d - A*e)*Log[ a + c*x^2])/(c*d^2 + a*e^2))/(2*a*(c*d^2 + a*e^2)))/(4*a*(c*d^2 + a*e^2))
3.14.52.3.1 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[(((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 = 0.52 (sec) , antiderivative size = 435, normalized size of antiderivative = 1.42
| method | result | size |
| default | \(\frac {\frac {\frac {c \left (7 A \,a^{2} c d \,e^{4}+10 A a \,c^{2} d^{3} e^{2}+3 d^{5} A \,c^{3}+3 B \,e^{5} a^{3}+2 B \,a^{2} c \,d^{2} e^{3}-B a \,c^{2} d^{4} e \right ) x^{3}}{8 a^{2}}+\left (\frac {1}{2} A a c \,e^{5}+\frac {1}{2} A \,c^{2} d^{2} e^{3}-\frac {1}{2} B a c d \,e^{4}-\frac {1}{2} B \,c^{2} d^{3} e^{2}\right ) x^{2}+\frac {\left (9 A \,a^{2} c d \,e^{4}+14 A a \,c^{2} d^{3} e^{2}+5 d^{5} A \,c^{3}+5 B \,e^{5} a^{3}+6 B \,a^{2} c \,d^{2} e^{3}+B a \,c^{2} d^{4} e \right ) x}{8 a}+\frac {3 A \,a^{2} e^{5}}{4}+A a c \,d^{2} e^{3}+\frac {A \,c^{2} d^{4} e}{4}-\frac {3 B \,a^{2} d \,e^{4}}{4}-B a c \,d^{3} e^{2}-\frac {B \,c^{2} d^{5}}{4}}{\left (c \,x^{2}+a \right )^{2}}+\frac {\frac {\left (-8 A \,a^{2} c \,e^{5}+8 B \,a^{2} c d \,e^{4}\right ) \ln \left (c \,x^{2}+a \right )}{2 c}+\frac {\left (15 A \,a^{2} c d \,e^{4}+10 A a \,c^{2} d^{3} e^{2}+3 d^{5} A \,c^{3}+3 B \,e^{5} a^{3}-6 B \,a^{2} c \,d^{2} e^{3}-B a \,c^{2} d^{4} e \right ) \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}}}{8 a^{2}}}{\left (e^{2} a +c \,d^{2}\right )^{3}}+\frac {\left (A e -B d \right ) e^{4} \ln \left (e x +d \right )}{\left (e^{2} a +c \,d^{2}\right )^{3}}\) | \(435\) |
| risch | \(\text {Expression too large to display}\) | \(11965\) |
1/(a*e^2+c*d^2)^3*((1/8*c*(7*A*a^2*c*d*e^4+10*A*a*c^2*d^3*e^2+3*A*c^3*d^5+ 3*B*a^3*e^5+2*B*a^2*c*d^2*e^3-B*a*c^2*d^4*e)/a^2*x^3+(1/2*A*a*c*e^5+1/2*A* c^2*d^2*e^3-1/2*B*a*c*d*e^4-1/2*B*c^2*d^3*e^2)*x^2+1/8*(9*A*a^2*c*d*e^4+14 *A*a*c^2*d^3*e^2+5*A*c^3*d^5+5*B*a^3*e^5+6*B*a^2*c*d^2*e^3+B*a*c^2*d^4*e)/ a*x+3/4*A*a^2*e^5+A*a*c*d^2*e^3+1/4*A*c^2*d^4*e-3/4*B*a^2*d*e^4-B*a*c*d^3* e^2-1/4*B*c^2*d^5)/(c*x^2+a)^2+1/8/a^2*(1/2*(-8*A*a^2*c*e^5+8*B*a^2*c*d*e^ 4)/c*ln(c*x^2+a)+(15*A*a^2*c*d*e^4+10*A*a*c^2*d^3*e^2+3*A*c^3*d^5+3*B*a^3* e^5-6*B*a^2*c*d^2*e^3-B*a*c^2*d^4*e)/(a*c)^(1/2)*arctan(c*x/(a*c)^(1/2)))) +(A*e-B*d)*e^4/(a*e^2+c*d^2)^3*ln(e*x+d)
Leaf count of result is larger than twice the leaf count of optimal. 889 vs. \(2 (292) = 584\).
Time = 92.99 (sec) , antiderivative size = 1797, normalized size of antiderivative = 5.85 \[ \int \frac {A+B x}{(d+e x) \left (a+c x^2\right )^3} \, dx=\text {Too large to display} \]
[-1/16*(4*B*a^3*c^3*d^5 - 4*A*a^3*c^3*d^4*e + 16*B*a^4*c^2*d^3*e^2 - 16*A* a^4*c^2*d^2*e^3 + 12*B*a^5*c*d*e^4 - 12*A*a^5*c*e^5 - 2*(3*A*a*c^5*d^5 - B *a^2*c^4*d^4*e + 10*A*a^2*c^4*d^3*e^2 + 2*B*a^3*c^3*d^2*e^3 + 7*A*a^3*c^3* d*e^4 + 3*B*a^4*c^2*e^5)*x^3 + 8*(B*a^3*c^3*d^3*e^2 - A*a^3*c^3*d^2*e^3 + B*a^4*c^2*d*e^4 - A*a^4*c^2*e^5)*x^2 + (3*A*a^2*c^3*d^5 - B*a^3*c^2*d^4*e + 10*A*a^3*c^2*d^3*e^2 - 6*B*a^4*c*d^2*e^3 + 15*A*a^4*c*d*e^4 + 3*B*a^5*e^ 5 + (3*A*c^5*d^5 - B*a*c^4*d^4*e + 10*A*a*c^4*d^3*e^2 - 6*B*a^2*c^3*d^2*e^ 3 + 15*A*a^2*c^3*d*e^4 + 3*B*a^3*c^2*e^5)*x^4 + 2*(3*A*a*c^4*d^5 - B*a^2*c ^3*d^4*e + 10*A*a^2*c^3*d^3*e^2 - 6*B*a^3*c^2*d^2*e^3 + 15*A*a^3*c^2*d*e^4 + 3*B*a^4*c*e^5)*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^4*d^5 + B*a^3*c^3*d^4*e + 14*A*a^3*c^3*d^3*e^2 + 6*B* a^4*c^2*d^2*e^3 + 9*A*a^4*c^2*d*e^4 + 5*B*a^5*c*e^5)*x - 8*(B*a^5*c*d*e^4 - A*a^5*c*e^5 + (B*a^3*c^3*d*e^4 - A*a^3*c^3*e^5)*x^4 + 2*(B*a^4*c^2*d*e^4 - A*a^4*c^2*e^5)*x^2)*log(c*x^2 + a) + 16*(B*a^5*c*d*e^4 - A*a^5*c*e^5 + (B*a^3*c^3*d*e^4 - A*a^3*c^3*e^5)*x^4 + 2*(B*a^4*c^2*d*e^4 - A*a^4*c^2*e^5 )*x^2)*log(e*x + d))/(a^5*c^4*d^6 + 3*a^6*c^3*d^4*e^2 + 3*a^7*c^2*d^2*e^4 + a^8*c*e^6 + (a^3*c^6*d^6 + 3*a^4*c^5*d^4*e^2 + 3*a^5*c^4*d^2*e^4 + a^6*c ^3*e^6)*x^4 + 2*(a^4*c^5*d^6 + 3*a^5*c^4*d^4*e^2 + 3*a^6*c^3*d^2*e^4 + a^7 *c^2*e^6)*x^2), -1/8*(2*B*a^3*c^3*d^5 - 2*A*a^3*c^3*d^4*e + 8*B*a^4*c^2*d^ 3*e^2 - 8*A*a^4*c^2*d^2*e^3 + 6*B*a^5*c*d*e^4 - 6*A*a^5*c*e^5 - (3*A*a*...
Timed out. \[ \int \frac {A+B x}{(d+e x) \left (a+c x^2\right )^3} \, dx=\text {Timed out} \]
Time = 0.30 (sec) , antiderivative size = 525, normalized size of antiderivative = 1.71 \[ \int \frac {A+B x}{(d+e x) \left (a+c x^2\right )^3} \, dx=\frac {{\left (B d e^{4} - A e^{5}\right )} \log \left (c x^{2} + a\right )}{2 \, {\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )}} - \frac {{\left (B d e^{4} - A e^{5}\right )} \log \left (e x + d\right )}{c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}} + \frac {{\left (3 \, A c^{3} d^{5} - B a c^{2} d^{4} e + 10 \, A a c^{2} d^{3} e^{2} - 6 \, B a^{2} c d^{2} e^{3} + 15 \, A a^{2} c d e^{4} + 3 \, B a^{3} e^{5}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 \, {\left (a^{2} c^{3} d^{6} + 3 \, a^{3} c^{2} d^{4} e^{2} + 3 \, a^{4} c d^{2} e^{4} + a^{5} e^{6}\right )} \sqrt {a c}} - \frac {2 \, B a^{2} c d^{3} - 2 \, A a^{2} c d^{2} e + 6 \, B a^{3} d e^{2} - 6 \, A a^{3} e^{3} - {\left (3 \, A c^{3} d^{3} - B a c^{2} d^{2} e + 7 \, A a c^{2} d e^{2} + 3 \, B a^{2} c e^{3}\right )} x^{3} + 4 \, {\left (B a^{2} c d e^{2} - A a^{2} c e^{3}\right )} x^{2} - {\left (5 \, A a c^{2} d^{3} + B a^{2} c d^{2} e + 9 \, A a^{2} c d e^{2} + 5 \, B a^{3} e^{3}\right )} x}{8 \, {\left (a^{4} c^{2} d^{4} + 2 \, a^{5} c d^{2} e^{2} + a^{6} e^{4} + {\left (a^{2} c^{4} d^{4} + 2 \, a^{3} c^{3} d^{2} e^{2} + a^{4} c^{2} e^{4}\right )} x^{4} + 2 \, {\left (a^{3} c^{3} d^{4} + 2 \, a^{4} c^{2} d^{2} e^{2} + a^{5} c e^{4}\right )} x^{2}\right )}} \]
1/2*(B*d*e^4 - A*e^5)*log(c*x^2 + a)/(c^3*d^6 + 3*a*c^2*d^4*e^2 + 3*a^2*c* d^2*e^4 + a^3*e^6) - (B*d*e^4 - A*e^5)*log(e*x + d)/(c^3*d^6 + 3*a*c^2*d^4 *e^2 + 3*a^2*c*d^2*e^4 + a^3*e^6) + 1/8*(3*A*c^3*d^5 - B*a*c^2*d^4*e + 10* A*a*c^2*d^3*e^2 - 6*B*a^2*c*d^2*e^3 + 15*A*a^2*c*d*e^4 + 3*B*a^3*e^5)*arct an(c*x/sqrt(a*c))/((a^2*c^3*d^6 + 3*a^3*c^2*d^4*e^2 + 3*a^4*c*d^2*e^4 + a^ 5*e^6)*sqrt(a*c)) - 1/8*(2*B*a^2*c*d^3 - 2*A*a^2*c*d^2*e + 6*B*a^3*d*e^2 - 6*A*a^3*e^3 - (3*A*c^3*d^3 - B*a*c^2*d^2*e + 7*A*a*c^2*d*e^2 + 3*B*a^2*c* e^3)*x^3 + 4*(B*a^2*c*d*e^2 - A*a^2*c*e^3)*x^2 - (5*A*a*c^2*d^3 + B*a^2*c* d^2*e + 9*A*a^2*c*d*e^2 + 5*B*a^3*e^3)*x)/(a^4*c^2*d^4 + 2*a^5*c*d^2*e^2 + a^6*e^4 + (a^2*c^4*d^4 + 2*a^3*c^3*d^2*e^2 + a^4*c^2*e^4)*x^4 + 2*(a^3*c^ 3*d^4 + 2*a^4*c^2*d^2*e^2 + a^5*c*e^4)*x^2)
Time = 0.26 (sec) , antiderivative size = 562, normalized size of antiderivative = 1.83 \[ \int \frac {A+B x}{(d+e x) \left (a+c x^2\right )^3} \, dx=\frac {{\left (B d e^{4} - A e^{5}\right )} \log \left (c x^{2} + a\right )}{2 \, {\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )}} - \frac {{\left (B d e^{5} - A e^{6}\right )} \log \left ({\left | e x + d \right |}\right )}{c^{3} d^{6} e + 3 \, a c^{2} d^{4} e^{3} + 3 \, a^{2} c d^{2} e^{5} + a^{3} e^{7}} + \frac {{\left (3 \, A c^{3} d^{5} - B a c^{2} d^{4} e + 10 \, A a c^{2} d^{3} e^{2} - 6 \, B a^{2} c d^{2} e^{3} + 15 \, A a^{2} c d e^{4} + 3 \, B a^{3} e^{5}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 \, {\left (a^{2} c^{3} d^{6} + 3 \, a^{3} c^{2} d^{4} e^{2} + 3 \, a^{4} c d^{2} e^{4} + a^{5} e^{6}\right )} \sqrt {a c}} - \frac {2 \, B a^{2} c^{2} d^{5} - 2 \, A a^{2} c^{2} d^{4} e + 8 \, B a^{3} c d^{3} e^{2} - 8 \, A a^{3} c d^{2} e^{3} + 6 \, B a^{4} d e^{4} - 6 \, A a^{4} e^{5} - {\left (3 \, A c^{4} d^{5} - B a c^{3} d^{4} e + 10 \, A a c^{3} d^{3} e^{2} + 2 \, B a^{2} c^{2} d^{2} e^{3} + 7 \, A a^{2} c^{2} d e^{4} + 3 \, B a^{3} c e^{5}\right )} x^{3} + 4 \, {\left (B a^{2} c^{2} d^{3} e^{2} - A a^{2} c^{2} d^{2} e^{3} + B a^{3} c d e^{4} - A a^{3} c e^{5}\right )} x^{2} - {\left (5 \, A a c^{3} d^{5} + B a^{2} c^{2} d^{4} e + 14 \, A a^{2} c^{2} d^{3} e^{2} + 6 \, B a^{3} c d^{2} e^{3} + 9 \, A a^{3} c d e^{4} + 5 \, B a^{4} e^{5}\right )} x}{8 \, {\left (c d^{2} + a e^{2}\right )}^{3} {\left (c x^{2} + a\right )}^{2} a^{2}} \]
1/2*(B*d*e^4 - A*e^5)*log(c*x^2 + a)/(c^3*d^6 + 3*a*c^2*d^4*e^2 + 3*a^2*c* d^2*e^4 + a^3*e^6) - (B*d*e^5 - A*e^6)*log(abs(e*x + d))/(c^3*d^6*e + 3*a* c^2*d^4*e^3 + 3*a^2*c*d^2*e^5 + a^3*e^7) + 1/8*(3*A*c^3*d^5 - B*a*c^2*d^4* e + 10*A*a*c^2*d^3*e^2 - 6*B*a^2*c*d^2*e^3 + 15*A*a^2*c*d*e^4 + 3*B*a^3*e^ 5)*arctan(c*x/sqrt(a*c))/((a^2*c^3*d^6 + 3*a^3*c^2*d^4*e^2 + 3*a^4*c*d^2*e ^4 + a^5*e^6)*sqrt(a*c)) - 1/8*(2*B*a^2*c^2*d^5 - 2*A*a^2*c^2*d^4*e + 8*B* a^3*c*d^3*e^2 - 8*A*a^3*c*d^2*e^3 + 6*B*a^4*d*e^4 - 6*A*a^4*e^5 - (3*A*c^4 *d^5 - B*a*c^3*d^4*e + 10*A*a*c^3*d^3*e^2 + 2*B*a^2*c^2*d^2*e^3 + 7*A*a^2* c^2*d*e^4 + 3*B*a^3*c*e^5)*x^3 + 4*(B*a^2*c^2*d^3*e^2 - A*a^2*c^2*d^2*e^3 + B*a^3*c*d*e^4 - A*a^3*c*e^5)*x^2 - (5*A*a*c^3*d^5 + B*a^2*c^2*d^4*e + 14 *A*a^2*c^2*d^3*e^2 + 6*B*a^3*c*d^2*e^3 + 9*A*a^3*c*d*e^4 + 5*B*a^4*e^5)*x) /((c*d^2 + a*e^2)^3*(c*x^2 + a)^2*a^2)
Time = 13.66 (sec) , antiderivative size = 2415, normalized size of antiderivative = 7.87 \[ \int \frac {A+B x}{(d+e x) \left (a+c x^2\right )^3} \, dx=\text {Too large to display} \]
((3*A*a*e^3 - B*c*d^3 - 3*B*a*d*e^2 + A*c*d^2*e)/(4*(a^2*e^4 + c^2*d^4 + 2 *a*c*d^2*e^2)) + (x^2*(A*c*e^3 - B*c*d*e^2))/(2*(a^2*e^4 + c^2*d^4 + 2*a*c *d^2*e^2)) + (x^3*(3*A*c^3*d^3 + 3*B*a^2*c*e^3 + 7*A*a*c^2*d*e^2 - B*a*c^2 *d^2*e))/(8*a^2*(a^2*e^4 + c^2*d^4 + 2*a*c*d^2*e^2)) + (x*(5*A*c^2*d^3 + 5 *B*a^2*e^3 + 9*A*a*c*d*e^2 + B*a*c*d^2*e))/(8*a*(a^2*e^4 + c^2*d^4 + 2*a*c *d^2*e^2)))/(a^2 + c^2*x^4 + 2*a*c*x^2) - (log(576*A^2*a^7*e^14*(-a^5*c)^( 3/2) + 9*A^2*c^7*d^14*(-a^5*c)^(3/2) - 9*B^2*a^13*e^14*(-a^5*c)^(1/2) + 55 8*B^2*a^7*d^2*e^12*(-a^5*c)^(3/2) + 9*B^2*a^15*c*e^14*x + 9*A^2*a^7*c^9*d^ 14*x + 576*A^2*a^14*c^2*e^14*x - 1377*A^2*a*d^2*e^12*(-a^5*c)^(5/2) - 1119 *B^2*a*d^4*e^10*(-a^5*c)^(5/2) - 1326*A^2*c*d^4*e^10*(-a^5*c)^(5/2) - 612* B^2*c*d^6*e^8*(-a^5*c)^(5/2) + 78*A^2*a^8*c^8*d^12*e^2*x + 319*A^2*a^9*c^7 *d^10*e^4*x + 740*A^2*a^10*c^6*d^8*e^6*x + 1015*A^2*a^11*c^5*d^6*e^8*x + 1 326*A^2*a^12*c^4*d^4*e^10*x + 1377*A^2*a^13*c^3*d^2*e^12*x + B^2*a^9*c^7*d ^12*e^2*x + 14*B^2*a^10*c^6*d^10*e^4*x + 55*B^2*a^11*c^5*d^8*e^6*x + 612*B ^2*a^12*c^4*d^6*e^8*x + 1119*B^2*a^13*c^3*d^4*e^10*x + 558*B^2*a^14*c^2*d^ 2*e^12*x + 78*A^2*a*c^6*d^12*e^2*(-a^5*c)^(3/2) + 2244*A*B*a*d^3*e^11*(-a^ 5*c)^(5/2) - 1062*A*B*a^7*d*e^13*(-a^5*c)^(3/2) + 1434*A*B*c*d^5*e^9*(-a^5 *c)^(5/2) + 319*A^2*a^2*c^5*d^10*e^4*(-a^5*c)^(3/2) + 740*A^2*a^3*c^4*d^8* e^6*(-a^5*c)^(3/2) + 1015*A^2*a^4*c^3*d^6*e^8*(-a^5*c)^(3/2) + B^2*a^2*c^5 *d^12*e^2*(-a^5*c)^(3/2) + 14*B^2*a^3*c^4*d^10*e^4*(-a^5*c)^(3/2) + 55*...