Integrand size = 22, antiderivative size = 162 \[ \int \frac {x^4}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\frac {(b c+a d) x}{2 b (b c-a d)^2 \left (c+d x^2\right )}+\frac {a x}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac {\sqrt {a} (3 b c+a d) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 \sqrt {b} (b c-a d)^3}+\frac {\sqrt {c} (b c+3 a d) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 \sqrt {d} (b c-a d)^3} \]
1/2*(a*d+b*c)*x/b/(-a*d+b*c)^2/(d*x^2+c)+1/2*a*x/b/(-a*d+b*c)/(b*x^2+a)/(d *x^2+c)-1/2*(a*d+3*b*c)*arctan(x*b^(1/2)/a^(1/2))*a^(1/2)/(-a*d+b*c)^3/b^( 1/2)+1/2*(3*a*d+b*c)*arctan(x*d^(1/2)/c^(1/2))*c^(1/2)/(-a*d+b*c)^3/d^(1/2 )
Time = 0.14 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.84 \[ \int \frac {x^4}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\frac {1}{2} \left (\frac {\sqrt {a} (3 b c+a d) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b} (-b c+a d)^3}+\frac {\frac {(b c-a d) x \left (2 a c+b c x^2+a d x^2\right )}{\left (a+b x^2\right ) \left (c+d x^2\right )}+\frac {\sqrt {c} (b c+3 a d) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {d}}}{(b c-a d)^3}\right ) \]
((Sqrt[a]*(3*b*c + a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[b]*(-(b*c) + a* d)^3) + (((b*c - a*d)*x*(2*a*c + b*c*x^2 + a*d*x^2))/((a + b*x^2)*(c + d*x ^2)) + (Sqrt[c]*(b*c + 3*a*d)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/Sqrt[d])/(b*c - a*d)^3)/2
Time = 0.32 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.13, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {372, 402, 27, 397, 218}
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.
\(\displaystyle \int \frac {x^4}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx\) |
\(\Big \downarrow \) 372 |
\(\displaystyle \frac {a x}{2 b \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}-\frac {\int \frac {a c-(2 b c+a d) x^2}{\left (b x^2+a\right ) \left (d x^2+c\right )^2}dx}{2 b (b c-a d)}\) |
\(\Big \downarrow \) 402 |
\(\displaystyle \frac {a x}{2 b \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}-\frac {\frac {\int \frac {2 b c \left (2 a c-(b c+a d) x^2\right )}{\left (b x^2+a\right ) \left (d x^2+c\right )}dx}{2 c (b c-a d)}-\frac {x (a d+b c)}{\left (c+d x^2\right ) (b c-a d)}}{2 b (b c-a d)}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {a x}{2 b \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}-\frac {\frac {b \int \frac {2 a c-(b c+a d) x^2}{\left (b x^2+a\right ) \left (d x^2+c\right )}dx}{b c-a d}-\frac {x (a d+b c)}{\left (c+d x^2\right ) (b c-a d)}}{2 b (b c-a d)}\) |
\(\Big \downarrow \) 397 |
\(\displaystyle \frac {a x}{2 b \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}-\frac {\frac {b \left (\frac {a (a d+3 b c) \int \frac {1}{b x^2+a}dx}{b c-a d}-\frac {c (3 a d+b c) \int \frac {1}{d x^2+c}dx}{b c-a d}\right )}{b c-a d}-\frac {x (a d+b c)}{\left (c+d x^2\right ) (b c-a d)}}{2 b (b c-a d)}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {a x}{2 b \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}-\frac {\frac {b \left (\frac {\sqrt {a} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) (a d+3 b c)}{\sqrt {b} (b c-a d)}-\frac {\sqrt {c} (3 a d+b c) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {d} (b c-a d)}\right )}{b c-a d}-\frac {x (a d+b c)}{\left (c+d x^2\right ) (b c-a d)}}{2 b (b c-a d)}\) |
(a*x)/(2*b*(b*c - a*d)*(a + b*x^2)*(c + d*x^2)) - (-(((b*c + a*d)*x)/((b*c - a*d)*(c + d*x^2))) + (b*((Sqrt[a]*(3*b*c + a*d)*ArcTan[(Sqrt[b]*x)/Sqrt [a]])/(Sqrt[b]*(b*c - a*d)) - (Sqrt[c]*(b*c + 3*a*d)*ArcTan[(Sqrt[d]*x)/Sq rt[c]])/(Sqrt[d]*(b*c - a*d))))/(b*c - a*d))/(2*b*(b*c - a*d))
3.4.1.3.1 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[((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[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ ), x_Symbol] :> Simp[(-a)*e^3*(e*x)^(m - 3)*(a + b*x^2)^(p + 1)*((c + d*x^2 )^(q + 1)/(2*b*(b*c - a*d)*(p + 1))), x] + Simp[e^4/(2*b*(b*c - a*d)*(p + 1 )) Int[(e*x)^(m - 4)*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[a*c*(m - 3) + (a*d*(m + 2*q - 1) + 2*b*c*(p + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[m, 3] && IntBinomialQ[a , b, c, d, e, m, 2, p, q, x]
Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*((c_) + (d_.)*(x_)^2)), x_ Symbol] :> Simp[(b*e - a*f)/(b*c - a*d) Int[1/(a + b*x^2), x], x] - Simp[ (d*e - c*f)/(b*c - a*d) Int[1/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d, e , f}, x]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x _)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ (q + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) *(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, q}, x] && LtQ[p, -1]
Time = 2.78 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.72
method | result | size |
default | \(\frac {a \left (\frac {\left (\frac {a d}{2}-\frac {b c}{2}\right ) x}{b \,x^{2}+a}+\frac {\left (a d +3 b c \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{\left (a d -b c \right )^{3}}-\frac {c \left (\frac {\left (-\frac {a d}{2}+\frac {b c}{2}\right ) x}{d \,x^{2}+c}+\frac {\left (3 a d +b c \right ) \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{2 \sqrt {c d}}\right )}{\left (a d -b c \right )^{3}}\) | \(117\) |
risch | \(\text {Expression too large to display}\) | \(1816\) |
a/(a*d-b*c)^3*((1/2*a*d-1/2*b*c)*x/(b*x^2+a)+1/2*(a*d+3*b*c)/(a*b)^(1/2)*a rctan(b*x/(a*b)^(1/2)))-c/(a*d-b*c)^3*((-1/2*a*d+1/2*b*c)*x/(d*x^2+c)+1/2* (3*a*d+b*c)/(c*d)^(1/2)*arctan(d*x/(c*d)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 331 vs. \(2 (138) = 276\).
Time = 0.36 (sec) , antiderivative size = 1407, normalized size of antiderivative = 8.69 \[ \int \frac {x^4}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\text {Too large to display} \]
[1/4*(2*(b^2*c^2 - a^2*d^2)*x^3 - ((3*b^2*c*d + a*b*d^2)*x^4 + 3*a*b*c^2 + a^2*c*d + (3*b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^2)*sqrt(-a/b)*log((b*x^2 + 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a)) - ((b^2*c*d + 3*a*b*d^2)*x^4 + a*b*c^2 + 3*a^2*c*d + (b^2*c^2 + 4*a*b*c*d + 3*a^2*d^2)*x^2)*sqrt(-c/d)*log((d*x^2 - 2*d*x*sqrt(-c/d) - c)/(d*x^2 + c)) + 4*(a*b*c^2 - a^2*c*d)*x)/(a*b^3*c^ 4 - 3*a^2*b^2*c^3*d + 3*a^3*b*c^2*d^2 - a^4*c*d^3 + (b^4*c^3*d - 3*a*b^3*c ^2*d^2 + 3*a^2*b^2*c*d^3 - a^3*b*d^4)*x^4 + (b^4*c^4 - 2*a*b^3*c^3*d + 2*a ^3*b*c*d^3 - a^4*d^4)*x^2), 1/4*(2*(b^2*c^2 - a^2*d^2)*x^3 - 2*((3*b^2*c*d + a*b*d^2)*x^4 + 3*a*b*c^2 + a^2*c*d + (3*b^2*c^2 + 4*a*b*c*d + a^2*d^2)* x^2)*sqrt(a/b)*arctan(b*x*sqrt(a/b)/a) - ((b^2*c*d + 3*a*b*d^2)*x^4 + a*b* c^2 + 3*a^2*c*d + (b^2*c^2 + 4*a*b*c*d + 3*a^2*d^2)*x^2)*sqrt(-c/d)*log((d *x^2 - 2*d*x*sqrt(-c/d) - c)/(d*x^2 + c)) + 4*(a*b*c^2 - a^2*c*d)*x)/(a*b^ 3*c^4 - 3*a^2*b^2*c^3*d + 3*a^3*b*c^2*d^2 - a^4*c*d^3 + (b^4*c^3*d - 3*a*b ^3*c^2*d^2 + 3*a^2*b^2*c*d^3 - a^3*b*d^4)*x^4 + (b^4*c^4 - 2*a*b^3*c^3*d + 2*a^3*b*c*d^3 - a^4*d^4)*x^2), 1/4*(2*(b^2*c^2 - a^2*d^2)*x^3 + 2*((b^2*c *d + 3*a*b*d^2)*x^4 + a*b*c^2 + 3*a^2*c*d + (b^2*c^2 + 4*a*b*c*d + 3*a^2*d ^2)*x^2)*sqrt(c/d)*arctan(d*x*sqrt(c/d)/c) - ((3*b^2*c*d + a*b*d^2)*x^4 + 3*a*b*c^2 + a^2*c*d + (3*b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^2)*sqrt(-a/b)*lo g((b*x^2 + 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a)) + 4*(a*b*c^2 - a^2*c*d)*x)/( a*b^3*c^4 - 3*a^2*b^2*c^3*d + 3*a^3*b*c^2*d^2 - a^4*c*d^3 + (b^4*c^3*d ...
Timed out. \[ \int \frac {x^4}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\text {Timed out} \]
Time = 0.28 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.54 \[ \int \frac {x^4}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=-\frac {{\left (3 \, a b c + a^{2} d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {a b}} + \frac {{\left (b c^{2} + 3 \, a c d\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{2 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {c d}} + \frac {{\left (b c + a d\right )} x^{3} + 2 \, a c x}{2 \, {\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} + {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{4} + {\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x^{2}\right )}} \]
-1/2*(3*a*b*c + a^2*d)*arctan(b*x/sqrt(a*b))/((b^3*c^3 - 3*a*b^2*c^2*d + 3 *a^2*b*c*d^2 - a^3*d^3)*sqrt(a*b)) + 1/2*(b*c^2 + 3*a*c*d)*arctan(d*x/sqrt (c*d))/((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*sqrt(c*d)) + 1 /2*((b*c + a*d)*x^3 + 2*a*c*x)/(a*b^2*c^3 - 2*a^2*b*c^2*d + a^3*c*d^2 + (b ^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*x^4 + (b^3*c^3 - a*b^2*c^2*d - a^2*b *c*d^2 + a^3*d^3)*x^2)
Time = 0.27 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.22 \[ \int \frac {x^4}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=-\frac {{\left (3 \, a b c + a^{2} d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {a b}} + \frac {{\left (b c^{2} + 3 \, a c d\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{2 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {c d}} + \frac {b c x^{3} + a d x^{3} + 2 \, a c x}{2 \, {\left (b d x^{4} + b c x^{2} + a d x^{2} + a c\right )} {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}} \]
-1/2*(3*a*b*c + a^2*d)*arctan(b*x/sqrt(a*b))/((b^3*c^3 - 3*a*b^2*c^2*d + 3 *a^2*b*c*d^2 - a^3*d^3)*sqrt(a*b)) + 1/2*(b*c^2 + 3*a*c*d)*arctan(d*x/sqrt (c*d))/((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*sqrt(c*d)) + 1 /2*(b*c*x^3 + a*d*x^3 + 2*a*c*x)/((b*d*x^4 + b*c*x^2 + a*d*x^2 + a*c)*(b^2 *c^2 - 2*a*b*c*d + a^2*d^2))
Time = 6.60 (sec) , antiderivative size = 5395, normalized size of antiderivative = 33.30 \[ \int \frac {x^4}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\text {Too large to display} \]
((x^3*(a*d + b*c))/(2*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) + (a*c*x)/(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))/(a*c + x^2*(a*d + b*c) + b*d*x^4) - (atan((((-c*d) ^(1/2)*((x*(a^4*b*d^5 + b^5*c^4*d + 6*a*b^4*c^3*d^2 + 6*a^3*b^2*c*d^4 + 18 *a^2*b^3*c^2*d^3))/(2*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3 *d - 4*a^3*b*c*d^3)) - ((-c*d)^(1/2)*((4*a*b^8*c^7*d^2 + 4*a^7*b^2*c*d^8 - 24*a^2*b^7*c^6*d^3 + 60*a^3*b^6*c^5*d^4 - 80*a^4*b^5*c^4*d^5 + 60*a^5*b^4 *c^3*d^6 - 24*a^6*b^3*c^2*d^7)/(a^6*d^6 + b^6*c^6 + 15*a^2*b^4*c^4*d^2 - 2 0*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a*b^5*c^5*d - 6*a^5*b*c*d^5) - (x*(-c*d)^(1/2)*(3*a*d + b*c)*(16*a^7*b^2*d^9 + 16*b^9*c^7*d^2 - 80*a*b^8* c^6*d^3 - 80*a^6*b^3*c*d^8 + 144*a^2*b^7*c^5*d^4 - 80*a^3*b^6*c^4*d^5 - 80 *a^4*b^5*c^3*d^6 + 144*a^5*b^4*c^2*d^7))/(8*(a^3*d^4 - b^3*c^3*d + 3*a*b^2 *c^2*d^2 - 3*a^2*b*c*d^3)*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3 *c^3*d - 4*a^3*b*c*d^3)))*(3*a*d + b*c))/(4*(a^3*d^4 - b^3*c^3*d + 3*a*b^2 *c^2*d^2 - 3*a^2*b*c*d^3)))*(3*a*d + b*c)*1i)/(4*(a^3*d^4 - b^3*c^3*d + 3* a*b^2*c^2*d^2 - 3*a^2*b*c*d^3)) + ((-c*d)^(1/2)*((x*(a^4*b*d^5 + b^5*c^4*d + 6*a*b^4*c^3*d^2 + 6*a^3*b^2*c*d^4 + 18*a^2*b^3*c^2*d^3))/(2*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3)) + ((-c*d)^(1 /2)*((4*a*b^8*c^7*d^2 + 4*a^7*b^2*c*d^8 - 24*a^2*b^7*c^6*d^3 + 60*a^3*b^6* c^5*d^4 - 80*a^4*b^5*c^4*d^5 + 60*a^5*b^4*c^3*d^6 - 24*a^6*b^3*c^2*d^7)/(a ^6*d^6 + b^6*c^6 + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2...