Integrand size = 22, antiderivative size = 79 \[ \int \frac {x^5}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=-\frac {\sqrt {a} \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{2 \sqrt {b} (b c-a d)}+\frac {\sqrt {c} \arctan \left (\frac {\sqrt {d} x^2}{\sqrt {c}}\right )}{2 \sqrt {d} (b c-a d)} \] Output:
-1/2*a^(1/2)*arctan(b^(1/2)*x^2/a^(1/2))/b^(1/2)/(-a*d+b*c)+1/2*c^(1/2)*ar ctan(d^(1/2)*x^2/c^(1/2))/d^(1/2)/(-a*d+b*c)
Time = 0.03 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.84 \[ \int \frac {x^5}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=\frac {-\frac {\sqrt {a} \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {c} \arctan \left (\frac {\sqrt {d} x^2}{\sqrt {c}}\right )}{\sqrt {d}}}{2 b c-2 a d} \] Input:
Integrate[x^5/((a + b*x^4)*(c + d*x^4)),x]
Output:
(-((Sqrt[a]*ArcTan[(Sqrt[b]*x^2)/Sqrt[a]])/Sqrt[b]) + (Sqrt[c]*ArcTan[(Sqr t[d]*x^2)/Sqrt[c]])/Sqrt[d])/(2*b*c - 2*a*d)
Time = 0.35 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {965, 383, 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.
\(\displaystyle \int \frac {x^5}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx\) |
\(\Big \downarrow \) 965 |
\(\displaystyle \frac {1}{2} \int \frac {x^4}{\left (b x^4+a\right ) \left (d x^4+c\right )}dx^2\) |
\(\Big \downarrow \) 383 |
\(\displaystyle \frac {1}{2} \left (\frac {c \int \frac {1}{d x^4+c}dx^2}{b c-a d}-\frac {a \int \frac {1}{b x^4+a}dx^2}{b c-a d}\right )\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {1}{2} \left (\frac {\sqrt {c} \arctan \left (\frac {\sqrt {d} x^2}{\sqrt {c}}\right )}{\sqrt {d} (b c-a d)}-\frac {\sqrt {a} \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{\sqrt {b} (b c-a d)}\right )\) |
Input:
Int[x^5/((a + b*x^4)*(c + d*x^4)),x]
Output:
(-((Sqrt[a]*ArcTan[(Sqrt[b]*x^2)/Sqrt[a]])/(Sqrt[b]*(b*c - a*d))) + (Sqrt[ c]*ArcTan[(Sqrt[d]*x^2)/Sqrt[c]])/(Sqrt[d]*(b*c - a*d)))/2
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)*((c_) + (d_.)*(x_)^2)), x_Sym bol] :> Simp[(-a)*(e^2/(b*c - a*d)) Int[(e*x)^(m - 2)/(a + b*x^2), x], x] + Simp[c*(e^2/(b*c - a*d)) Int[(e*x)^(m - 2)/(c + d*x^2), x], x] /; Free Q[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && LeQ[2, m, 3]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p*(c + d*x^(n/k))^q, x], x, x^k], x] /; k != 1] /; Free Q[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IntegerQ[m]
Time = 0.15 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.76
method | result | size |
default | \(\frac {a \arctan \left (\frac {b \,x^{2}}{\sqrt {a b}}\right )}{2 \left (a d -c b \right ) \sqrt {a b}}-\frac {c \arctan \left (\frac {d \,x^{2}}{\sqrt {c d}}\right )}{2 \left (a d -c b \right ) \sqrt {c d}}\) | \(60\) |
risch | \(\frac {\sqrt {-c d}\, \ln \left (\left (-a^{2} d^{4}+a b c \,d^{3}\right ) x^{2}+\left (-c d \right )^{\frac {3}{2}} a b d +\left (-c d \right )^{\frac {3}{2}} b^{2} c +a^{2} d^{3} \sqrt {-c d}+\sqrt {-c d}\, b^{2} c^{2} d \right )}{4 d \left (a d -c b \right )}-\frac {\sqrt {-c d}\, \ln \left (\left (-a^{2} d^{4}+a b c \,d^{3}\right ) x^{2}-\left (-c d \right )^{\frac {3}{2}} a b d -\left (-c d \right )^{\frac {3}{2}} b^{2} c -a^{2} d^{3} \sqrt {-c d}-\sqrt {-c d}\, b^{2} c^{2} d \right )}{4 d \left (a d -c b \right )}+\frac {\sqrt {-a b}\, \ln \left (\left (-a \,b^{3} c d +b^{4} c^{2}\right ) x^{2}+\left (-a b \right )^{\frac {3}{2}} a \,d^{2}+\left (-a b \right )^{\frac {3}{2}} b c d +\sqrt {-a b}\, d^{2} a^{2} b +b^{3} c^{2} \sqrt {-a b}\right )}{4 b \left (a d -c b \right )}-\frac {\sqrt {-a b}\, \ln \left (\left (-a \,b^{3} c d +b^{4} c^{2}\right ) x^{2}-\left (-a b \right )^{\frac {3}{2}} a \,d^{2}-\left (-a b \right )^{\frac {3}{2}} b c d -\sqrt {-a b}\, d^{2} a^{2} b -b^{3} c^{2} \sqrt {-a b}\right )}{4 b \left (a d -c b \right )}\) | \(374\) |
Input:
int(x^5/(b*x^4+a)/(d*x^4+c),x,method=_RETURNVERBOSE)
Output:
1/2*a/(a*d-b*c)/(a*b)^(1/2)*arctan(b*x^2/(a*b)^(1/2))-1/2*c/(a*d-b*c)/(c*d )^(1/2)*arctan(d*x^2/(c*d)^(1/2))
Time = 0.14 (sec) , antiderivative size = 325, normalized size of antiderivative = 4.11 \[ \int \frac {x^5}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=\left [-\frac {\sqrt {-\frac {a}{b}} \log \left (\frac {b x^{4} + 2 \, b x^{2} \sqrt {-\frac {a}{b}} - a}{b x^{4} + a}\right ) + \sqrt {-\frac {c}{d}} \log \left (\frac {d x^{4} - 2 \, d x^{2} \sqrt {-\frac {c}{d}} - c}{d x^{4} + c}\right )}{4 \, {\left (b c - a d\right )}}, -\frac {2 \, \sqrt {\frac {a}{b}} \arctan \left (\frac {b x^{2} \sqrt {\frac {a}{b}}}{a}\right ) + \sqrt {-\frac {c}{d}} \log \left (\frac {d x^{4} - 2 \, d x^{2} \sqrt {-\frac {c}{d}} - c}{d x^{4} + c}\right )}{4 \, {\left (b c - a d\right )}}, \frac {2 \, \sqrt {\frac {c}{d}} \arctan \left (\frac {d x^{2} \sqrt {\frac {c}{d}}}{c}\right ) - \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{4} + 2 \, b x^{2} \sqrt {-\frac {a}{b}} - a}{b x^{4} + a}\right )}{4 \, {\left (b c - a d\right )}}, -\frac {\sqrt {\frac {a}{b}} \arctan \left (\frac {b x^{2} \sqrt {\frac {a}{b}}}{a}\right ) - \sqrt {\frac {c}{d}} \arctan \left (\frac {d x^{2} \sqrt {\frac {c}{d}}}{c}\right )}{2 \, {\left (b c - a d\right )}}\right ] \] Input:
integrate(x^5/(b*x^4+a)/(d*x^4+c),x, algorithm="fricas")
Output:
[-1/4*(sqrt(-a/b)*log((b*x^4 + 2*b*x^2*sqrt(-a/b) - a)/(b*x^4 + a)) + sqrt (-c/d)*log((d*x^4 - 2*d*x^2*sqrt(-c/d) - c)/(d*x^4 + c)))/(b*c - a*d), -1/ 4*(2*sqrt(a/b)*arctan(b*x^2*sqrt(a/b)/a) + sqrt(-c/d)*log((d*x^4 - 2*d*x^2 *sqrt(-c/d) - c)/(d*x^4 + c)))/(b*c - a*d), 1/4*(2*sqrt(c/d)*arctan(d*x^2* sqrt(c/d)/c) - sqrt(-a/b)*log((b*x^4 + 2*b*x^2*sqrt(-a/b) - a)/(b*x^4 + a) ))/(b*c - a*d), -1/2*(sqrt(a/b)*arctan(b*x^2*sqrt(a/b)/a) - sqrt(c/d)*arct an(d*x^2*sqrt(c/d)/c))/(b*c - a*d)]
Timed out. \[ \int \frac {x^5}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=\text {Timed out} \] Input:
integrate(x**5/(b*x**4+a)/(d*x**4+c),x)
Output:
Timed out
Time = 0.11 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.75 \[ \int \frac {x^5}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=-\frac {a \arctan \left (\frac {b x^{2}}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} {\left (b c - a d\right )}} + \frac {c \arctan \left (\frac {d x^{2}}{\sqrt {c d}}\right )}{2 \, {\left (b c - a d\right )} \sqrt {c d}} \] Input:
integrate(x^5/(b*x^4+a)/(d*x^4+c),x, algorithm="maxima")
Output:
-1/2*a*arctan(b*x^2/sqrt(a*b))/(sqrt(a*b)*(b*c - a*d)) + 1/2*c*arctan(d*x^ 2/sqrt(c*d))/((b*c - a*d)*sqrt(c*d))
Time = 0.12 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.75 \[ \int \frac {x^5}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=-\frac {a \arctan \left (\frac {b x^{2}}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} {\left (b c - a d\right )}} + \frac {c \arctan \left (\frac {d x^{2}}{\sqrt {c d}}\right )}{2 \, {\left (b c - a d\right )} \sqrt {c d}} \] Input:
integrate(x^5/(b*x^4+a)/(d*x^4+c),x, algorithm="giac")
Output:
-1/2*a*arctan(b*x^2/sqrt(a*b))/(sqrt(a*b)*(b*c - a*d)) + 1/2*c*arctan(d*x^ 2/sqrt(c*d))/((b*c - a*d)*sqrt(c*d))
Time = 4.51 (sec) , antiderivative size = 379, normalized size of antiderivative = 4.80 \[ \int \frac {x^5}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=\frac {\ln \left (d^2\,{\left (-a\,b\right )}^{5/2}+b^4\,c^2\,\sqrt {-a\,b}-b^5\,c^2\,x^2+2\,b^2\,c\,d\,{\left (-a\,b\right )}^{3/2}-a^2\,b^3\,d^2\,x^2+2\,a\,b^4\,c\,d\,x^2\right )\,\sqrt {-a\,b}}{4\,b^2\,c-4\,a\,b\,d}-\frac {\ln \left (d^2\,{\left (-a\,b\right )}^{5/2}+b^4\,c^2\,\sqrt {-a\,b}+b^5\,c^2\,x^2+2\,b^2\,c\,d\,{\left (-a\,b\right )}^{3/2}+a^2\,b^3\,d^2\,x^2-2\,a\,b^4\,c\,d\,x^2\right )\,\sqrt {-a\,b}}{4\,\left (b^2\,c-a\,b\,d\right )}-\frac {\ln \left (b^2\,{\left (-c\,d\right )}^{5/2}+a^2\,d^4\,\sqrt {-c\,d}+a^2\,d^5\,x^2+2\,a\,b\,d^2\,{\left (-c\,d\right )}^{3/2}+b^2\,c^2\,d^3\,x^2-2\,a\,b\,c\,d^4\,x^2\right )\,\sqrt {-c\,d}}{4\,\left (a\,d^2-b\,c\,d\right )}+\frac {\ln \left (b^2\,{\left (-c\,d\right )}^{5/2}+a^2\,d^4\,\sqrt {-c\,d}-a^2\,d^5\,x^2+2\,a\,b\,d^2\,{\left (-c\,d\right )}^{3/2}-b^2\,c^2\,d^3\,x^2+2\,a\,b\,c\,d^4\,x^2\right )\,\sqrt {-c\,d}}{4\,a\,d^2-4\,b\,c\,d} \] Input:
int(x^5/((a + b*x^4)*(c + d*x^4)),x)
Output:
(log(d^2*(-a*b)^(5/2) + b^4*c^2*(-a*b)^(1/2) - b^5*c^2*x^2 + 2*b^2*c*d*(-a *b)^(3/2) - a^2*b^3*d^2*x^2 + 2*a*b^4*c*d*x^2)*(-a*b)^(1/2))/(4*b^2*c - 4* a*b*d) - (log(d^2*(-a*b)^(5/2) + b^4*c^2*(-a*b)^(1/2) + b^5*c^2*x^2 + 2*b^ 2*c*d*(-a*b)^(3/2) + a^2*b^3*d^2*x^2 - 2*a*b^4*c*d*x^2)*(-a*b)^(1/2))/(4*( b^2*c - a*b*d)) - (log(b^2*(-c*d)^(5/2) + a^2*d^4*(-c*d)^(1/2) + a^2*d^5*x ^2 + 2*a*b*d^2*(-c*d)^(3/2) + b^2*c^2*d^3*x^2 - 2*a*b*c*d^4*x^2)*(-c*d)^(1 /2))/(4*(a*d^2 - b*c*d)) + (log(b^2*(-c*d)^(5/2) + a^2*d^4*(-c*d)^(1/2) - a^2*d^5*x^2 + 2*a*b*d^2*(-c*d)^(3/2) - b^2*c^2*d^3*x^2 + 2*a*b*c*d^4*x^2)* (-c*d)^(1/2))/(4*a*d^2 - 4*b*c*d)
Time = 0.25 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.94 \[ \int \frac {x^5}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=\frac {-\sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 \sqrt {b}\, x}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) d -\sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 \sqrt {b}\, x}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) d +\sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}-2 \sqrt {d}\, x}{d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}}\right ) b +\sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}+2 \sqrt {d}\, x}{d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}}\right ) b}{2 b d \left (a d -b c \right )} \] Input:
int(x^5/(b*x^4+a)/(d*x^4+c),x)
Output:
( - sqrt(b)*sqrt(a)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(b)*x)/(b**(1/ 4)*a**(1/4)*sqrt(2)))*d - sqrt(b)*sqrt(a)*atan((b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(b)*x)/(b**(1/4)*a**(1/4)*sqrt(2)))*d + sqrt(d)*sqrt(c)*atan((d**( 1/4)*c**(1/4)*sqrt(2) - 2*sqrt(d)*x)/(d**(1/4)*c**(1/4)*sqrt(2)))*b + sqrt (d)*sqrt(c)*atan((d**(1/4)*c**(1/4)*sqrt(2) + 2*sqrt(d)*x)/(d**(1/4)*c**(1 /4)*sqrt(2)))*b)/(2*b*d*(a*d - b*c))