Integrand size = 22, antiderivative size = 78 \[ \int \frac {x^4}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx=\frac {x}{b d}+\frac {a^{3/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{3/2} (b c-a d)}-\frac {c^{3/2} \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{d^{3/2} (b c-a d)} \] Output:
x/b/d+a^(3/2)*arctan(b^(1/2)*x/a^(1/2))/b^(3/2)/(-a*d+b*c)-c^(3/2)*arctan( d^(1/2)*x/c^(1/2))/d^(3/2)/(-a*d+b*c)
Time = 0.06 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.95 \[ \int \frac {x^4}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx=\frac {-\frac {a x}{b}+\frac {c x}{d}+\frac {a^{3/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{3/2}}-\frac {c^{3/2} \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{d^{3/2}}}{b c-a d} \] Input:
Integrate[x^4/((a + b*x^2)*(c + d*x^2)),x]
Output:
(-((a*x)/b) + (c*x)/d + (a^(3/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(3/2) - (c ^(3/2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/d^(3/2))/(b*c - a*d)
Time = 0.23 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.14, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {381, 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.
\(\displaystyle \int \frac {x^4}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx\) |
\(\Big \downarrow \) 381 |
\(\displaystyle \frac {x}{b d}-\frac {\int \frac {(b c+a d) x^2+a c}{\left (b x^2+a\right ) \left (d x^2+c\right )}dx}{b d}\) |
\(\Big \downarrow \) 397 |
\(\displaystyle \frac {x}{b d}-\frac {\frac {b c^2 \int \frac {1}{d x^2+c}dx}{b c-a d}-\frac {a^2 d \int \frac {1}{b x^2+a}dx}{b c-a d}}{b d}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {x}{b d}-\frac {\frac {b c^{3/2} \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {d} (b c-a d)}-\frac {a^{3/2} d \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b} (b c-a d)}}{b d}\) |
Input:
Int[x^4/((a + b*x^2)*(c + d*x^2)),x]
Output:
x/(b*d) - (-((a^(3/2)*d*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[b]*(b*c - a*d)) ) + (b*c^(3/2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(Sqrt[d]*(b*c - a*d)))/(b*d)
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[e^3*(e*x)^(m - 3)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(b*d*(m + 2*(p + q) + 1))), x] - Simp[e^4/(b*d*(m + 2*(p + q) + 1)) Int[(e*x)^(m - 4)*(a + b*x^2)^p*(c + d*x^2)^q*Simp[a*c*(m - 3) + (a*d*(m + 2*q - 1) + b*c*(m + 2*p - 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, p, q }, x] && NeQ[b*c - a*d, 0] && 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]
Time = 0.57 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.94
method | result | size |
default | \(\frac {x}{b d}-\frac {a^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{b \left (a d -b c \right ) \sqrt {a b}}+\frac {c^{2} \arctan \left (\frac {x d}{\sqrt {c d}}\right )}{d \left (a d -b c \right ) \sqrt {c d}}\) | \(73\) |
risch | \(\frac {x}{b d}+\frac {\sqrt {-c d}\, c \ln \left (\left (-\left (-c d \right )^{\frac {3}{2}} a \,b^{3} c^{2} d -\left (-c d \right )^{\frac {3}{2}} b^{4} c^{3}-a^{4} \sqrt {-c d}\, d^{5}-b^{4} c^{4} \sqrt {-c d}\, d \right ) x +a^{4} c \,d^{5}-a \,b^{3} c^{4} d^{2}\right )}{2 d^{2} \left (a d -b c \right )}-\frac {\sqrt {-c d}\, c \ln \left (\left (\left (-c d \right )^{\frac {3}{2}} a \,b^{3} c^{2} d +\left (-c d \right )^{\frac {3}{2}} b^{4} c^{3}+a^{4} \sqrt {-c d}\, d^{5}+b^{4} c^{4} \sqrt {-c d}\, d \right ) x +a^{4} c \,d^{5}-a \,b^{3} c^{4} d^{2}\right )}{2 d^{2} \left (a d -b c \right )}+\frac {\sqrt {-a b}\, a \ln \left (\left (-\left (-a b \right )^{\frac {3}{2}} a^{3} d^{4}-\left (-a b \right )^{\frac {3}{2}} a^{2} b c \,d^{3}-a^{4} \sqrt {-a b}\, d^{4} b -b^{5} c^{4} \sqrt {-a b}\right ) x +a^{4} b^{2} c \,d^{3}-a \,b^{5} c^{4}\right )}{2 b^{2} \left (a d -b c \right )}-\frac {\sqrt {-a b}\, a \ln \left (\left (\left (-a b \right )^{\frac {3}{2}} a^{3} d^{4}+\left (-a b \right )^{\frac {3}{2}} a^{2} b c \,d^{3}+a^{4} \sqrt {-a b}\, d^{4} b +b^{5} c^{4} \sqrt {-a b}\right ) x +a^{4} b^{2} c \,d^{3}-a \,b^{5} c^{4}\right )}{2 b^{2} \left (a d -b c \right )}\) | \(426\) |
Input:
int(x^4/(b*x^2+a)/(d*x^2+c),x,method=_RETURNVERBOSE)
Output:
x/b/d-1/b*a^2/(a*d-b*c)/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))+1/d*c^2/(a*d-b *c)/(c*d)^(1/2)*arctan(x*d/(c*d)^(1/2))
Time = 0.10 (sec) , antiderivative size = 391, normalized size of antiderivative = 5.01 \[ \int \frac {x^4}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx=\left [-\frac {a d \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} - 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right ) + b c \sqrt {-\frac {c}{d}} \log \left (\frac {d x^{2} + 2 \, d x \sqrt {-\frac {c}{d}} - c}{d x^{2} + c}\right ) - 2 \, {\left (b c - a d\right )} x}{2 \, {\left (b^{2} c d - a b d^{2}\right )}}, \frac {2 \, a d \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) - b c \sqrt {-\frac {c}{d}} \log \left (\frac {d x^{2} + 2 \, d x \sqrt {-\frac {c}{d}} - c}{d x^{2} + c}\right ) + 2 \, {\left (b c - a d\right )} x}{2 \, {\left (b^{2} c d - a b d^{2}\right )}}, -\frac {2 \, b c \sqrt {\frac {c}{d}} \arctan \left (\frac {d x \sqrt {\frac {c}{d}}}{c}\right ) + a d \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} - 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right ) - 2 \, {\left (b c - a d\right )} x}{2 \, {\left (b^{2} c d - a b d^{2}\right )}}, \frac {a d \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) - b c \sqrt {\frac {c}{d}} \arctan \left (\frac {d x \sqrt {\frac {c}{d}}}{c}\right ) + {\left (b c - a d\right )} x}{b^{2} c d - a b d^{2}}\right ] \] Input:
integrate(x^4/(b*x^2+a)/(d*x^2+c),x, algorithm="fricas")
Output:
[-1/2*(a*d*sqrt(-a/b)*log((b*x^2 - 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a)) + b* c*sqrt(-c/d)*log((d*x^2 + 2*d*x*sqrt(-c/d) - c)/(d*x^2 + c)) - 2*(b*c - a* d)*x)/(b^2*c*d - a*b*d^2), 1/2*(2*a*d*sqrt(a/b)*arctan(b*x*sqrt(a/b)/a) - b*c*sqrt(-c/d)*log((d*x^2 + 2*d*x*sqrt(-c/d) - c)/(d*x^2 + c)) + 2*(b*c - a*d)*x)/(b^2*c*d - a*b*d^2), -1/2*(2*b*c*sqrt(c/d)*arctan(d*x*sqrt(c/d)/c) + a*d*sqrt(-a/b)*log((b*x^2 - 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a)) - 2*(b*c - a*d)*x)/(b^2*c*d - a*b*d^2), (a*d*sqrt(a/b)*arctan(b*x*sqrt(a/b)/a) - b *c*sqrt(c/d)*arctan(d*x*sqrt(c/d)/c) + (b*c - a*d)*x)/(b^2*c*d - a*b*d^2)]
Leaf count of result is larger than twice the leaf count of optimal. 921 vs. \(2 (65) = 130\).
Time = 7.60 (sec) , antiderivative size = 921, normalized size of antiderivative = 11.81 \[ \int \frac {x^4}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx =\text {Too large to display} \] Input:
integrate(x**4/(b*x**2+a)/(d*x**2+c),x)
Output:
-sqrt(-a**3/b**3)*log(x + (-a**4*d**4*sqrt(-a**3/b**3)/(a*d - b*c) - a**3* b**3*d**6*(-a**3/b**3)**(3/2)/(a*d - b*c)**3 + a**2*b**4*c*d**5*(-a**3/b** 3)**(3/2)/(a*d - b*c)**3 + a*b**5*c**2*d**4*(-a**3/b**3)**(3/2)/(a*d - b*c )**3 - b**6*c**3*d**3*(-a**3/b**3)**(3/2)/(a*d - b*c)**3 - b**4*c**4*sqrt( -a**3/b**3)/(a*d - b*c))/(a**3*c*d**2 + a**2*b*c**2*d + a*b**2*c**3))/(2*( a*d - b*c)) + sqrt(-a**3/b**3)*log(x + (a**4*d**4*sqrt(-a**3/b**3)/(a*d - b*c) + a**3*b**3*d**6*(-a**3/b**3)**(3/2)/(a*d - b*c)**3 - a**2*b**4*c*d** 5*(-a**3/b**3)**(3/2)/(a*d - b*c)**3 - a*b**5*c**2*d**4*(-a**3/b**3)**(3/2 )/(a*d - b*c)**3 + b**6*c**3*d**3*(-a**3/b**3)**(3/2)/(a*d - b*c)**3 + b** 4*c**4*sqrt(-a**3/b**3)/(a*d - b*c))/(a**3*c*d**2 + a**2*b*c**2*d + a*b**2 *c**3))/(2*(a*d - b*c)) - sqrt(-c**3/d**3)*log(x + (-a**4*d**4*sqrt(-c**3/ d**3)/(a*d - b*c) - a**3*b**3*d**6*(-c**3/d**3)**(3/2)/(a*d - b*c)**3 + a* *2*b**4*c*d**5*(-c**3/d**3)**(3/2)/(a*d - b*c)**3 + a*b**5*c**2*d**4*(-c** 3/d**3)**(3/2)/(a*d - b*c)**3 - b**6*c**3*d**3*(-c**3/d**3)**(3/2)/(a*d - b*c)**3 - b**4*c**4*sqrt(-c**3/d**3)/(a*d - b*c))/(a**3*c*d**2 + a**2*b*c* *2*d + a*b**2*c**3))/(2*(a*d - b*c)) + sqrt(-c**3/d**3)*log(x + (a**4*d**4 *sqrt(-c**3/d**3)/(a*d - b*c) + a**3*b**3*d**6*(-c**3/d**3)**(3/2)/(a*d - b*c)**3 - a**2*b**4*c*d**5*(-c**3/d**3)**(3/2)/(a*d - b*c)**3 - a*b**5*c** 2*d**4*(-c**3/d**3)**(3/2)/(a*d - b*c)**3 + b**6*c**3*d**3*(-c**3/d**3)**( 3/2)/(a*d - b*c)**3 + b**4*c**4*sqrt(-c**3/d**3)/(a*d - b*c))/(a**3*c*d...
Time = 0.12 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.92 \[ \int \frac {x^4}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx=\frac {a^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{{\left (b^{2} c - a b d\right )} \sqrt {a b}} - \frac {c^{2} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{{\left (b c d - a d^{2}\right )} \sqrt {c d}} + \frac {x}{b d} \] Input:
integrate(x^4/(b*x^2+a)/(d*x^2+c),x, algorithm="maxima")
Output:
a^2*arctan(b*x/sqrt(a*b))/((b^2*c - a*b*d)*sqrt(a*b)) - c^2*arctan(d*x/sqr t(c*d))/((b*c*d - a*d^2)*sqrt(c*d)) + x/(b*d)
Time = 0.13 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.92 \[ \int \frac {x^4}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx=\frac {a^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{{\left (b^{2} c - a b d\right )} \sqrt {a b}} - \frac {c^{2} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{{\left (b c d - a d^{2}\right )} \sqrt {c d}} + \frac {x}{b d} \] Input:
integrate(x^4/(b*x^2+a)/(d*x^2+c),x, algorithm="giac")
Output:
a^2*arctan(b*x/sqrt(a*b))/((b^2*c - a*b*d)*sqrt(a*b)) - c^2*arctan(d*x/sqr t(c*d))/((b*c*d - a*d^2)*sqrt(c*d)) + x/(b*d)
Time = 0.68 (sec) , antiderivative size = 343, normalized size of antiderivative = 4.40 \[ \int \frac {x^4}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx=\frac {\ln \left (a^5\,b^4\,d^3-a^2\,b^7\,c^3+d^3\,x\,{\left (-a^3\,b^3\right )}^{3/2}+b^6\,c^3\,x\,\sqrt {-a^3\,b^3}\right )\,\sqrt {-a^3\,b^3}}{2\,b^4\,c-2\,a\,b^3\,d}-\frac {\ln \left (a^2\,b^7\,c^3-a^5\,b^4\,d^3+d^3\,x\,{\left (-a^3\,b^3\right )}^{3/2}+b^6\,c^3\,x\,\sqrt {-a^3\,b^3}\right )\,\sqrt {-a^3\,b^3}}{2\,\left (b^4\,c-a\,b^3\,d\right )}+\frac {x}{b\,d}-\frac {\ln \left (a^3\,c^2\,d^7-b^3\,c^5\,d^4+b^3\,x\,{\left (-c^3\,d^3\right )}^{3/2}+a^3\,d^6\,x\,\sqrt {-c^3\,d^3}\right )\,\sqrt {-c^3\,d^3}}{2\,\left (a\,d^4-b\,c\,d^3\right )}+\frac {\ln \left (b^3\,c^5\,d^4-a^3\,c^2\,d^7+b^3\,x\,{\left (-c^3\,d^3\right )}^{3/2}+a^3\,d^6\,x\,\sqrt {-c^3\,d^3}\right )\,\sqrt {-c^3\,d^3}}{2\,a\,d^4-2\,b\,c\,d^3} \] Input:
int(x^4/((a + b*x^2)*(c + d*x^2)),x)
Output:
(log(a^5*b^4*d^3 - a^2*b^7*c^3 + d^3*x*(-a^3*b^3)^(3/2) + b^6*c^3*x*(-a^3* b^3)^(1/2))*(-a^3*b^3)^(1/2))/(2*b^4*c - 2*a*b^3*d) - (log(a^2*b^7*c^3 - a ^5*b^4*d^3 + d^3*x*(-a^3*b^3)^(3/2) + b^6*c^3*x*(-a^3*b^3)^(1/2))*(-a^3*b^ 3)^(1/2))/(2*(b^4*c - a*b^3*d)) + x/(b*d) - (log(a^3*c^2*d^7 - b^3*c^5*d^4 + b^3*x*(-c^3*d^3)^(3/2) + a^3*d^6*x*(-c^3*d^3)^(1/2))*(-c^3*d^3)^(1/2))/ (2*(a*d^4 - b*c*d^3)) + (log(b^3*c^5*d^4 - a^3*c^2*d^7 + b^3*x*(-c^3*d^3)^ (3/2) + a^3*d^6*x*(-c^3*d^3)^(1/2))*(-c^3*d^3)^(1/2))/(2*a*d^4 - 2*b*c*d^3 )
Time = 0.25 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.97 \[ \int \frac {x^4}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx=\frac {-\sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a \,d^{2}+\sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {d x}{\sqrt {d}\, \sqrt {c}}\right ) b^{2} c +a b \,d^{2} x -b^{2} c d x}{b^{2} d^{2} \left (a d -b c \right )} \] Input:
int(x^4/(b*x^2+a)/(d*x^2+c),x)
Output:
( - sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a*d**2 + sqrt(d)*sqrt(c) *atan((d*x)/(sqrt(d)*sqrt(c)))*b**2*c + a*b*d**2*x - b**2*c*d*x)/(b**2*d** 2*(a*d - b*c))