\(\int \frac {x^4}{(a+b x^2)^2 (c+d x^2)} \, dx\) [683]

Optimal result

Integrand size = 22, antiderivative size = 109 \[ \int \frac {x^4}{\left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx=\frac {a x}{2 b (b c-a d) \left (a+b x^2\right )}-\frac {\sqrt {a} (3 b c-a d) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{3/2} (b c-a d)^2}+\frac {c^{3/2} \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {d} (b c-a d)^2} \] Output:

1/2*a*x/b/(-a*d+b*c)/(b*x^2+a)-1/2*a^(1/2)*(-a*d+3*b*c)*arctan(b^(1/2)*x/a 
^(1/2))/b^(3/2)/(-a*d+b*c)^2+c^(3/2)*arctan(d^(1/2)*x/c^(1/2))/d^(1/2)/(-a 
*d+b*c)^2
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.87 \[ \int \frac {x^4}{\left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx=\frac {\frac {a (b c-a d) x}{b \left (a+b x^2\right )}+\frac {\sqrt {a} (-3 b c+a d) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{3/2}}+\frac {2 c^{3/2} \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {d}}}{2 (b c-a d)^2} \] Input:

Integrate[x^4/((a + b*x^2)^2*(c + d*x^2)),x]
 

Output:

((a*(b*c - a*d)*x)/(b*(a + b*x^2)) + (Sqrt[a]*(-3*b*c + a*d)*ArcTan[(Sqrt[ 
b]*x)/Sqrt[a]])/b^(3/2) + (2*c^(3/2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/Sqrt[d]) 
/(2*(b*c - a*d)^2)
 

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.16, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {372, 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.

\(\displaystyle \int \frac {x^4}{\left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx\)

\(\Big \downarrow \) 372

\(\displaystyle \frac {a x}{2 b \left (a+b x^2\right ) (b c-a d)}-\frac {\int \frac {a c-(2 b c-a d) x^2}{\left (b x^2+a\right ) \left (d x^2+c\right )}dx}{2 b (b c-a d)}\)

\(\Big \downarrow \) 397

\(\displaystyle \frac {a x}{2 b \left (a+b x^2\right ) (b c-a d)}-\frac {\frac {a (3 b c-a d) \int \frac {1}{b x^2+a}dx}{b c-a d}-\frac {2 b c^2 \int \frac {1}{d x^2+c}dx}{b c-a d}}{2 b (b c-a d)}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {a x}{2 b \left (a+b x^2\right ) (b c-a d)}-\frac {\frac {\sqrt {a} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) (3 b c-a d)}{\sqrt {b} (b c-a d)}-\frac {2 b c^{3/2} \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {d} (b c-a d)}}{2 b (b c-a d)}\)

Input:

Int[x^4/((a + b*x^2)^2*(c + d*x^2)),x]
 

Output:

(a*x)/(2*b*(b*c - a*d)*(a + b*x^2)) - ((Sqrt[a]*(3*b*c - a*d)*ArcTan[(Sqrt 
[b]*x)/Sqrt[a]])/(Sqrt[b]*(b*c - a*d)) - (2*b*c^(3/2)*ArcTan[(Sqrt[d]*x)/S 
qrt[c]])/(Sqrt[d]*(b*c - a*d)))/(2*b*(b*c - a*d))
 

Defintions of rubi rules used

rule 218

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]
 

rule 372

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]
 

rule 397

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]
 

Maple [A] (verified)

Time = 0.59 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.86

method	result	size
default	\(-\frac {a \left (\frac {\left (a d -b c \right ) x}{2 b \left (b \,x^{2}+a \right )}-\frac {\left (a d -3 b c \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 b \sqrt {a b}}\right )}{\left (a d -b c \right )^{2}}+\frac {c^{2} \arctan \left (\frac {x d}{\sqrt {c d}}\right )}{\left (a d -b c \right )^{2} \sqrt {c d}}\)	\(94\)
risch	\(\text {Expression too large to display}\)	\(1177\)

Input:

int(x^4/(b*x^2+a)^2/(d*x^2+c),x,method=_RETURNVERBOSE)
 

Output:

-a/(a*d-b*c)^2*(1/2*(a*d-b*c)/b*x/(b*x^2+a)-1/2*(a*d-3*b*c)/b/(a*b)^(1/2)* 
arctan(b*x/(a*b)^(1/2)))+c^2/(a*d-b*c)^2/(c*d)^(1/2)*arctan(x*d/(c*d)^(1/2 
))
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 726, normalized size of antiderivative = 6.66 \[ \int \frac {x^4}{\left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx=\left [-\frac {{\left (3 \, a b c - a^{2} d + {\left (3 \, b^{2} c - a b d\right )} x^{2}\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} + 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right ) - 2 \, {\left (b^{2} c x^{2} + a b c\right )} \sqrt {-\frac {c}{d}} \log \left (\frac {d x^{2} + 2 \, d x \sqrt {-\frac {c}{d}} - c}{d x^{2} + c}\right ) - 2 \, {\left (a b c - a^{2} d\right )} x}{4 \, {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2} + {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{2}\right )}}, -\frac {{\left (3 \, a b c - a^{2} d + {\left (3 \, b^{2} c - a b d\right )} x^{2}\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) - {\left (b^{2} c x^{2} + a b c\right )} \sqrt {-\frac {c}{d}} \log \left (\frac {d x^{2} + 2 \, d x \sqrt {-\frac {c}{d}} - c}{d x^{2} + c}\right ) - {\left (a b c - a^{2} d\right )} x}{2 \, {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2} + {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{2}\right )}}, \frac {4 \, {\left (b^{2} c x^{2} + a b c\right )} \sqrt {\frac {c}{d}} \arctan \left (\frac {d x \sqrt {\frac {c}{d}}}{c}\right ) - {\left (3 \, a b c - a^{2} d + {\left (3 \, b^{2} c - a b d\right )} x^{2}\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} + 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right ) + 2 \, {\left (a b c - a^{2} d\right )} x}{4 \, {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2} + {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{2}\right )}}, -\frac {{\left (3 \, a b c - a^{2} d + {\left (3 \, b^{2} c - a b d\right )} x^{2}\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) - 2 \, {\left (b^{2} c x^{2} + a b c\right )} \sqrt {\frac {c}{d}} \arctan \left (\frac {d x \sqrt {\frac {c}{d}}}{c}\right ) - {\left (a b c - a^{2} d\right )} x}{2 \, {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2} + {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{2}\right )}}\right ] \] Input:

integrate(x^4/(b*x^2+a)^2/(d*x^2+c),x, algorithm="fricas")
 

Output:

[-1/4*((3*a*b*c - a^2*d + (3*b^2*c - a*b*d)*x^2)*sqrt(-a/b)*log((b*x^2 + 2 
*b*x*sqrt(-a/b) - a)/(b*x^2 + a)) - 2*(b^2*c*x^2 + a*b*c)*sqrt(-c/d)*log(( 
d*x^2 + 2*d*x*sqrt(-c/d) - c)/(d*x^2 + c)) - 2*(a*b*c - a^2*d)*x)/(a*b^3*c 
^2 - 2*a^2*b^2*c*d + a^3*b*d^2 + (b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2)*x^2 
), -1/2*((3*a*b*c - a^2*d + (3*b^2*c - a*b*d)*x^2)*sqrt(a/b)*arctan(b*x*sq 
rt(a/b)/a) - (b^2*c*x^2 + a*b*c)*sqrt(-c/d)*log((d*x^2 + 2*d*x*sqrt(-c/d) 
- c)/(d*x^2 + c)) - (a*b*c - a^2*d)*x)/(a*b^3*c^2 - 2*a^2*b^2*c*d + a^3*b* 
d^2 + (b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2)*x^2), 1/4*(4*(b^2*c*x^2 + a*b* 
c)*sqrt(c/d)*arctan(d*x*sqrt(c/d)/c) - (3*a*b*c - a^2*d + (3*b^2*c - a*b*d 
)*x^2)*sqrt(-a/b)*log((b*x^2 + 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a)) + 2*(a*b 
*c - a^2*d)*x)/(a*b^3*c^2 - 2*a^2*b^2*c*d + a^3*b*d^2 + (b^4*c^2 - 2*a*b^3 
*c*d + a^2*b^2*d^2)*x^2), -1/2*((3*a*b*c - a^2*d + (3*b^2*c - a*b*d)*x^2)* 
sqrt(a/b)*arctan(b*x*sqrt(a/b)/a) - 2*(b^2*c*x^2 + a*b*c)*sqrt(c/d)*arctan 
(d*x*sqrt(c/d)/c) - (a*b*c - a^2*d)*x)/(a*b^3*c^2 - 2*a^2*b^2*c*d + a^3*b* 
d^2 + (b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2)*x^2)]
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^4}{\left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx=\text {Timed out} \] Input:

integrate(x**4/(b*x**2+a)**2/(d*x**2+c),x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.22 \[ \int \frac {x^4}{\left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx=\frac {c^{2} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {c d}} + \frac {a x}{2 \, {\left (a b^{2} c - a^{2} b d + {\left (b^{3} c - a b^{2} d\right )} x^{2}\right )}} - \frac {{\left (3 \, a b c - a^{2} d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} \sqrt {a b}} \] Input:

integrate(x^4/(b*x^2+a)^2/(d*x^2+c),x, algorithm="maxima")
 

Output:

c^2*arctan(d*x/sqrt(c*d))/((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt(c*d)) + 1/ 
2*a*x/(a*b^2*c - a^2*b*d + (b^3*c - a*b^2*d)*x^2) - 1/2*(3*a*b*c - a^2*d)* 
arctan(b*x/sqrt(a*b))/((b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)*sqrt(a*b))
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.12 \[ \int \frac {x^4}{\left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx=\frac {c^{2} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {c d}} - \frac {{\left (3 \, a b c - a^{2} d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} \sqrt {a b}} + \frac {a x}{2 \, {\left (b^{2} c - a b d\right )} {\left (b x^{2} + a\right )}} \] Input:

integrate(x^4/(b*x^2+a)^2/(d*x^2+c),x, algorithm="giac")
 

Output:

c^2*arctan(d*x/sqrt(c*d))/((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt(c*d)) - 1/ 
2*(3*a*b*c - a^2*d)*arctan(b*x/sqrt(a*b))/((b^3*c^2 - 2*a*b^2*c*d + a^2*b* 
d^2)*sqrt(a*b)) + 1/2*a*x/((b^2*c - a*b*d)*(b*x^2 + a))
 

Mupad [B] (verification not implemented)

Time = 1.45 (sec) , antiderivative size = 3558, normalized size of antiderivative = 32.64 \[ \int \frac {x^4}{\left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx=\text {Too large to display} \] Input:

int(x^4/((a + b*x^2)^2*(c + d*x^2)),x)
 

Output:

(atan((((((-c^3*d)^(1/2)*((2*a*b^6*c^5*d^2 + 2*a^5*b^2*c*d^6 - 8*a^2*b^5*c 
^4*d^3 + 12*a^3*b^4*c^3*d^4 - 8*a^4*b^3*c^2*d^5)/(2*(b^4*c^3 - a^3*b*d^3 + 
 3*a^2*b^2*c*d^2 - 3*a*b^3*c^2*d)) - (x*(-c^3*d)^(1/2)*(16*a^5*b^3*d^7 + 1 
6*b^8*c^5*d^2 - 48*a*b^7*c^4*d^3 - 48*a^4*b^4*c*d^6 + 32*a^2*b^6*c^3*d^4 + 
 32*a^3*b^5*c^2*d^5))/(8*(b^3*c^2 + a^2*b*d^2 - 2*a*b^2*c*d)*(a^2*d^3 + b^ 
2*c^2*d - 2*a*b*c*d^2))))/(2*(a^2*d^3 + b^2*c^2*d - 2*a*b*c*d^2)) - (x*(a^ 
4*d^5 + 4*b^4*c^4*d + 9*a^2*b^2*c^2*d^3 - 6*a^3*b*c*d^4))/(4*(b^3*c^2 + a^ 
2*b*d^2 - 2*a*b^2*c*d)))*(-c^3*d)^(1/2)*1i)/(a^2*d^3 + b^2*c^2*d - 2*a*b*c 
*d^2) - ((((-c^3*d)^(1/2)*((2*a*b^6*c^5*d^2 + 2*a^5*b^2*c*d^6 - 8*a^2*b^5* 
c^4*d^3 + 12*a^3*b^4*c^3*d^4 - 8*a^4*b^3*c^2*d^5)/(2*(b^4*c^3 - a^3*b*d^3 
+ 3*a^2*b^2*c*d^2 - 3*a*b^3*c^2*d)) + (x*(-c^3*d)^(1/2)*(16*a^5*b^3*d^7 + 
16*b^8*c^5*d^2 - 48*a*b^7*c^4*d^3 - 48*a^4*b^4*c*d^6 + 32*a^2*b^6*c^3*d^4 
+ 32*a^3*b^5*c^2*d^5))/(8*(b^3*c^2 + a^2*b*d^2 - 2*a*b^2*c*d)*(a^2*d^3 + b 
^2*c^2*d - 2*a*b*c*d^2))))/(2*(a^2*d^3 + b^2*c^2*d - 2*a*b*c*d^2)) + (x*(a 
^4*d^5 + 4*b^4*c^4*d + 9*a^2*b^2*c^2*d^3 - 6*a^3*b*c*d^4))/(4*(b^3*c^2 + a 
^2*b*d^2 - 2*a*b^2*c*d)))*(-c^3*d)^(1/2)*1i)/(a^2*d^3 + b^2*c^2*d - 2*a*b* 
c*d^2))/(((a^3*c^2*d^3)/2 - (5*a^2*b*c^3*d^2)/2 + 3*a*b^2*c^4*d)/(b^4*c^3 
- a^3*b*d^3 + 3*a^2*b^2*c*d^2 - 3*a*b^3*c^2*d) + ((((-c^3*d)^(1/2)*((2*a*b 
^6*c^5*d^2 + 2*a^5*b^2*c*d^6 - 8*a^2*b^5*c^4*d^3 + 12*a^3*b^4*c^3*d^4 - 8* 
a^4*b^3*c^2*d^5)/(2*(b^4*c^3 - a^3*b*d^3 + 3*a^2*b^2*c*d^2 - 3*a*b^3*c^...
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 229, normalized size of antiderivative = 2.10 \[ \int \frac {x^4}{\left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx=\frac {\sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{2} d^{2}-3 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a b c d +\sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a b \,d^{2} x^{2}-3 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) b^{2} c d \,x^{2}+2 \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {d x}{\sqrt {d}\, \sqrt {c}}\right ) a \,b^{2} c +2 \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {d x}{\sqrt {d}\, \sqrt {c}}\right ) b^{3} c \,x^{2}-a^{2} b \,d^{2} x +a \,b^{2} c d x}{2 b^{2} d \left (a^{2} b \,d^{2} x^{2}-2 a \,b^{2} c d \,x^{2}+b^{3} c^{2} x^{2}+a^{3} d^{2}-2 a^{2} b c d +a \,b^{2} c^{2}\right )} \] Input:

int(x^4/(b*x^2+a)^2/(d*x^2+c),x)
 

Output:

(sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**2*d**2 - 3*sqrt(b)*sqrt( 
a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a*b*c*d + sqrt(b)*sqrt(a)*atan((b*x)/(sqr 
t(b)*sqrt(a)))*a*b*d**2*x**2 - 3*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt( 
a)))*b**2*c*d*x**2 + 2*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a*b** 
2*c + 2*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*b**3*c*x**2 - a**2*b 
*d**2*x + a*b**2*c*d*x)/(2*b**2*d*(a**3*d**2 - 2*a**2*b*c*d + a**2*b*d**2* 
x**2 + a*b**2*c**2 - 2*a*b**2*c*d*x**2 + b**3*c**2*x**2))