Integrand size = 30, antiderivative size = 202 \[ \int \frac {x^4 \left (c+d x^2+e x^4+f x^6\right )}{\left (a+b x^2\right )^2} \, dx=\frac {\left (3 b^3 c-5 a b^2 d+7 a^2 b e-9 a^3 f\right ) x}{2 b^5}-\frac {\left (3 b^3 c-5 a b^2 d+7 a^2 b e-9 a^3 f\right ) x^3}{6 a b^4}+\frac {(b e-2 a f) x^5}{5 b^3}+\frac {f x^7}{7 b^2}+\frac {\left (c-\frac {a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x^5}{2 a \left (a+b x^2\right )}-\frac {\sqrt {a} \left (3 b^3 c-5 a b^2 d+7 a^2 b e-9 a^3 f\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{11/2}} \]
1/2*(-9*a^3*f+7*a^2*b*e-5*a*b^2*d+3*b^3*c)*x/b^5-1/6*(-9*a^3*f+7*a^2*b*e-5 *a*b^2*d+3*b^3*c)*x^3/a/b^4+1/5*(-2*a*f+b*e)*x^5/b^3+1/7*f*x^7/b^2+1/2*(c- a*(a^2*f-a*b*e+b^2*d)/b^3)*x^5/a/(b*x^2+a)-1/2*(-9*a^3*f+7*a^2*b*e-5*a*b^2 *d+3*b^3*c)*arctan(x*b^(1/2)/a^(1/2))*a^(1/2)/b^(11/2)
Time = 0.07 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.93 \[ \int \frac {x^4 \left (c+d x^2+e x^4+f x^6\right )}{\left (a+b x^2\right )^2} \, dx=\frac {\left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right ) x}{b^5}+\frac {\left (b^2 d-2 a b e+3 a^2 f\right ) x^3}{3 b^4}+\frac {(b e-2 a f) x^5}{5 b^3}+\frac {f x^7}{7 b^2}+\frac {\left (a b^3 c-a^2 b^2 d+a^3 b e-a^4 f\right ) x}{2 b^5 \left (a+b x^2\right )}+\frac {\sqrt {a} \left (-3 b^3 c+5 a b^2 d-7 a^2 b e+9 a^3 f\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{11/2}} \]
((b^3*c - 2*a*b^2*d + 3*a^2*b*e - 4*a^3*f)*x)/b^5 + ((b^2*d - 2*a*b*e + 3* a^2*f)*x^3)/(3*b^4) + ((b*e - 2*a*f)*x^5)/(5*b^3) + (f*x^7)/(7*b^2) + ((a* b^3*c - a^2*b^2*d + a^3*b*e - a^4*f)*x)/(2*b^5*(a + b*x^2)) + (Sqrt[a]*(-3 *b^3*c + 5*a*b^2*d - 7*a^2*b*e + 9*a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2* b^(11/2))
Time = 0.47 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.03, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2335, 9, 1584, 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.
| \(\displaystyle \int \frac {x^4 \left (c+d x^2+e x^4+f x^6\right )}{\left (a+b x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 2335 |
| \(\displaystyle \frac {x^5 \left (c-\frac {a \left (a^2 f-a b e+b^2 d\right )}{b^3}\right )}{2 a \left (a+b x^2\right )}-\frac {\int \frac {x^3 \left (-2 a f x^5-2 a \left (e-\frac {a f}{b}\right ) x^3+\left (-\frac {5 f a^3}{b^2}+\frac {5 e a^2}{b}-5 d a+3 b c\right ) x\right )}{b x^2+a}dx}{2 a b}\) |
| \(\Big \downarrow \) 9 |
| \(\displaystyle \frac {x^5 \left (c-\frac {a \left (a^2 f-a b e+b^2 d\right )}{b^3}\right )}{2 a \left (a+b x^2\right )}-\frac {\int \frac {x^4 \left (-2 a f x^4-2 a \left (e-\frac {a f}{b}\right ) x^2+3 b c-5 a d+\frac {5 a^2 e}{b}-\frac {5 a^3 f}{b^2}\right )}{b x^2+a}dx}{2 a b}\) |
| \(\Big \downarrow \) 1584 |
| \(\displaystyle \frac {x^5 \left (c-\frac {a \left (a^2 f-a b e+b^2 d\right )}{b^3}\right )}{2 a \left (a+b x^2\right )}-\frac {\int \left (-\frac {2 a f x^6}{b}-\frac {2 a (b e-2 a f) x^4}{b^2}+\frac {\left (-9 f a^3+7 b e a^2-5 b^2 d a+3 b^3 c\right ) x^2}{b^3}-\frac {a \left (-9 f a^3+7 b e a^2-5 b^2 d a+3 b^3 c\right )}{b^4}+\frac {-9 f a^5+7 b e a^4-5 b^2 d a^3+3 b^3 c a^2}{b^4 \left (b x^2+a\right )}\right )dx}{2 a b}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {x^5 \left (c-\frac {a \left (a^2 f-a b e+b^2 d\right )}{b^3}\right )}{2 a \left (a+b x^2\right )}-\frac {\frac {x^3 \left (-9 a^3 f+7 a^2 b e-5 a b^2 d+3 b^3 c\right )}{3 b^3}-\frac {a x \left (-9 a^3 f+7 a^2 b e-5 a b^2 d+3 b^3 c\right )}{b^4}+\frac {a^{3/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (-9 a^3 f+7 a^2 b e-5 a b^2 d+3 b^3 c\right )}{b^{9/2}}-\frac {2 a x^5 (b e-2 a f)}{5 b^2}-\frac {2 a f x^7}{7 b}}{2 a b}\) |
((c - (a*(b^2*d - a*b*e + a^2*f))/b^3)*x^5)/(2*a*(a + b*x^2)) - (-((a*(3*b ^3*c - 5*a*b^2*d + 7*a^2*b*e - 9*a^3*f)*x)/b^4) + ((3*b^3*c - 5*a*b^2*d + 7*a^2*b*e - 9*a^3*f)*x^3)/(3*b^3) - (2*a*(b*e - 2*a*f)*x^5)/(5*b^2) - (2*a *f*x^7)/(7*b) + (a^(3/2)*(3*b^3*c - 5*a*b^2*d + 7*a^2*b*e - 9*a^3*f)*ArcTa n[(Sqrt[b]*x)/Sqrt[a]])/b^(9/2))/(2*a*b)
3.2.25.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[1/e^(p*r) Int[u*(e*x)^(m + p*r)*ExpandToSum[Px/x^r, x]^p, x], x] /; IGtQ[r, 0]] /; FreeQ[{e, m}, x] && PolyQ[Px, x] && IntegerQ[p] && !MonomialQ[Px, x]
Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + ( c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q* (a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && NeQ[ b^2 - 4*a*c, 0] && IGtQ[p, 0] && IGtQ[q, -2]
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ {Q = PolynomialQuotient[Pq, a + b*x^2, x], f = Coeff[PolynomialRemainder[Pq , a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[(c*x)^m*(a + b*x^2)^(p + 1)*((a*g - b*f*x)/(2*a*b*(p + 1))), x] + Simp[c/(2*a*b*(p + 1)) Int[(c*x)^(m - 1)*(a + b*x^2)^(p + 1)*ExpandToSu m[2*a*b*(p + 1)*x*Q - a*g*m + b*f*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && GtQ[m, 0]
Time = 3.44 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.90
| method | result | size |
| default | \(-\frac {-\frac {1}{7} f \,x^{7} b^{3}+\frac {2}{5} a \,b^{2} f \,x^{5}-\frac {1}{5} b^{3} e \,x^{5}-a^{2} b f \,x^{3}+\frac {2}{3} a \,b^{2} e \,x^{3}-\frac {1}{3} b^{3} d \,x^{3}+4 f \,a^{3} x -3 a^{2} b e x +2 a \,b^{2} d x -b^{3} c x}{b^{5}}+\frac {a \left (\frac {\left (-\frac {1}{2} f \,a^{3}+\frac {1}{2} a^{2} b e -\frac {1}{2} a \,b^{2} d +\frac {1}{2} b^{3} c \right ) x}{b \,x^{2}+a}+\frac {\left (9 f \,a^{3}-7 a^{2} b e +5 a \,b^{2} d -3 b^{3} c \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{b^{5}}\) | \(182\) |
| risch | \(\frac {f \,x^{7}}{7 b^{2}}-\frac {2 a f \,x^{5}}{5 b^{3}}+\frac {e \,x^{5}}{5 b^{2}}+\frac {a^{2} f \,x^{3}}{b^{4}}-\frac {2 a e \,x^{3}}{3 b^{3}}+\frac {d \,x^{3}}{3 b^{2}}-\frac {4 f \,a^{3} x}{b^{5}}+\frac {3 a^{2} e x}{b^{4}}-\frac {2 a d x}{b^{3}}+\frac {c x}{b^{2}}+\frac {\left (-\frac {1}{2} a^{4} f +\frac {1}{2} a^{3} b e -\frac {1}{2} a^{2} b^{2} d +\frac {1}{2} a \,b^{3} c \right ) x}{b^{5} \left (b \,x^{2}+a \right )}+\frac {9 \sqrt {-a b}\, \ln \left (-\sqrt {-a b}\, x +a \right ) f \,a^{3}}{4 b^{6}}-\frac {7 \sqrt {-a b}\, \ln \left (-\sqrt {-a b}\, x +a \right ) a^{2} e}{4 b^{5}}+\frac {5 \sqrt {-a b}\, \ln \left (-\sqrt {-a b}\, x +a \right ) a d}{4 b^{4}}-\frac {3 \sqrt {-a b}\, \ln \left (-\sqrt {-a b}\, x +a \right ) c}{4 b^{3}}-\frac {9 \sqrt {-a b}\, \ln \left (\sqrt {-a b}\, x +a \right ) f \,a^{3}}{4 b^{6}}+\frac {7 \sqrt {-a b}\, \ln \left (\sqrt {-a b}\, x +a \right ) a^{2} e}{4 b^{5}}-\frac {5 \sqrt {-a b}\, \ln \left (\sqrt {-a b}\, x +a \right ) a d}{4 b^{4}}+\frac {3 \sqrt {-a b}\, \ln \left (\sqrt {-a b}\, x +a \right ) c}{4 b^{3}}\) | \(340\) |
-1/b^5*(-1/7*f*x^7*b^3+2/5*a*b^2*f*x^5-1/5*b^3*e*x^5-a^2*b*f*x^3+2/3*a*b^2 *e*x^3-1/3*b^3*d*x^3+4*f*a^3*x-3*a^2*b*e*x+2*a*b^2*d*x-b^3*c*x)+a/b^5*((-1 /2*f*a^3+1/2*a^2*b*e-1/2*a*b^2*d+1/2*b^3*c)*x/(b*x^2+a)+1/2*(9*a^3*f-7*a^2 *b*e+5*a*b^2*d-3*b^3*c)/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2)))
Time = 0.29 (sec) , antiderivative size = 478, normalized size of antiderivative = 2.37 \[ \int \frac {x^4 \left (c+d x^2+e x^4+f x^6\right )}{\left (a+b x^2\right )^2} \, dx=\left [\frac {60 \, b^{4} f x^{9} + 12 \, {\left (7 \, b^{4} e - 9 \, a b^{3} f\right )} x^{7} + 28 \, {\left (5 \, b^{4} d - 7 \, a b^{3} e + 9 \, a^{2} b^{2} f\right )} x^{5} + 140 \, {\left (3 \, b^{4} c - 5 \, a b^{3} d + 7 \, a^{2} b^{2} e - 9 \, a^{3} b f\right )} x^{3} - 105 \, {\left (3 \, a b^{3} c - 5 \, a^{2} b^{2} d + 7 \, a^{3} b e - 9 \, a^{4} f + {\left (3 \, b^{4} c - 5 \, a b^{3} d + 7 \, a^{2} b^{2} e - 9 \, a^{3} b f\right )} x^{2}\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} + 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right ) + 210 \, {\left (3 \, a b^{3} c - 5 \, a^{2} b^{2} d + 7 \, a^{3} b e - 9 \, a^{4} f\right )} x}{420 \, {\left (b^{6} x^{2} + a b^{5}\right )}}, \frac {30 \, b^{4} f x^{9} + 6 \, {\left (7 \, b^{4} e - 9 \, a b^{3} f\right )} x^{7} + 14 \, {\left (5 \, b^{4} d - 7 \, a b^{3} e + 9 \, a^{2} b^{2} f\right )} x^{5} + 70 \, {\left (3 \, b^{4} c - 5 \, a b^{3} d + 7 \, a^{2} b^{2} e - 9 \, a^{3} b f\right )} x^{3} - 105 \, {\left (3 \, a b^{3} c - 5 \, a^{2} b^{2} d + 7 \, a^{3} b e - 9 \, a^{4} f + {\left (3 \, b^{4} c - 5 \, a b^{3} d + 7 \, a^{2} b^{2} e - 9 \, a^{3} b f\right )} x^{2}\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) + 105 \, {\left (3 \, a b^{3} c - 5 \, a^{2} b^{2} d + 7 \, a^{3} b e - 9 \, a^{4} f\right )} x}{210 \, {\left (b^{6} x^{2} + a b^{5}\right )}}\right ] \]
[1/420*(60*b^4*f*x^9 + 12*(7*b^4*e - 9*a*b^3*f)*x^7 + 28*(5*b^4*d - 7*a*b^ 3*e + 9*a^2*b^2*f)*x^5 + 140*(3*b^4*c - 5*a*b^3*d + 7*a^2*b^2*e - 9*a^3*b* f)*x^3 - 105*(3*a*b^3*c - 5*a^2*b^2*d + 7*a^3*b*e - 9*a^4*f + (3*b^4*c - 5 *a*b^3*d + 7*a^2*b^2*e - 9*a^3*b*f)*x^2)*sqrt(-a/b)*log((b*x^2 + 2*b*x*sqr t(-a/b) - a)/(b*x^2 + a)) + 210*(3*a*b^3*c - 5*a^2*b^2*d + 7*a^3*b*e - 9*a ^4*f)*x)/(b^6*x^2 + a*b^5), 1/210*(30*b^4*f*x^9 + 6*(7*b^4*e - 9*a*b^3*f)* x^7 + 14*(5*b^4*d - 7*a*b^3*e + 9*a^2*b^2*f)*x^5 + 70*(3*b^4*c - 5*a*b^3*d + 7*a^2*b^2*e - 9*a^3*b*f)*x^3 - 105*(3*a*b^3*c - 5*a^2*b^2*d + 7*a^3*b*e - 9*a^4*f + (3*b^4*c - 5*a*b^3*d + 7*a^2*b^2*e - 9*a^3*b*f)*x^2)*sqrt(a/b )*arctan(b*x*sqrt(a/b)/a) + 105*(3*a*b^3*c - 5*a^2*b^2*d + 7*a^3*b*e - 9*a ^4*f)*x)/(b^6*x^2 + a*b^5)]
Time = 1.08 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.27 \[ \int \frac {x^4 \left (c+d x^2+e x^4+f x^6\right )}{\left (a+b x^2\right )^2} \, dx=x^{5} \left (- \frac {2 a f}{5 b^{3}} + \frac {e}{5 b^{2}}\right ) + x^{3} \left (\frac {a^{2} f}{b^{4}} - \frac {2 a e}{3 b^{3}} + \frac {d}{3 b^{2}}\right ) + x \left (- \frac {4 a^{3} f}{b^{5}} + \frac {3 a^{2} e}{b^{4}} - \frac {2 a d}{b^{3}} + \frac {c}{b^{2}}\right ) + \frac {x \left (- a^{4} f + a^{3} b e - a^{2} b^{2} d + a b^{3} c\right )}{2 a b^{5} + 2 b^{6} x^{2}} - \frac {\sqrt {- \frac {a}{b^{11}}} \cdot \left (9 a^{3} f - 7 a^{2} b e + 5 a b^{2} d - 3 b^{3} c\right ) \log {\left (- b^{5} \sqrt {- \frac {a}{b^{11}}} + x \right )}}{4} + \frac {\sqrt {- \frac {a}{b^{11}}} \cdot \left (9 a^{3} f - 7 a^{2} b e + 5 a b^{2} d - 3 b^{3} c\right ) \log {\left (b^{5} \sqrt {- \frac {a}{b^{11}}} + x \right )}}{4} + \frac {f x^{7}}{7 b^{2}} \]
x**5*(-2*a*f/(5*b**3) + e/(5*b**2)) + x**3*(a**2*f/b**4 - 2*a*e/(3*b**3) + d/(3*b**2)) + x*(-4*a**3*f/b**5 + 3*a**2*e/b**4 - 2*a*d/b**3 + c/b**2) + x*(-a**4*f + a**3*b*e - a**2*b**2*d + a*b**3*c)/(2*a*b**5 + 2*b**6*x**2) - sqrt(-a/b**11)*(9*a**3*f - 7*a**2*b*e + 5*a*b**2*d - 3*b**3*c)*log(-b**5* sqrt(-a/b**11) + x)/4 + sqrt(-a/b**11)*(9*a**3*f - 7*a**2*b*e + 5*a*b**2*d - 3*b**3*c)*log(b**5*sqrt(-a/b**11) + x)/4 + f*x**7/(7*b**2)
Time = 0.29 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.91 \[ \int \frac {x^4 \left (c+d x^2+e x^4+f x^6\right )}{\left (a+b x^2\right )^2} \, dx=\frac {{\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} x}{2 \, {\left (b^{6} x^{2} + a b^{5}\right )}} - \frac {{\left (3 \, a b^{3} c - 5 \, a^{2} b^{2} d + 7 \, a^{3} b e - 9 \, a^{4} f\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} b^{5}} + \frac {15 \, b^{3} f x^{7} + 21 \, {\left (b^{3} e - 2 \, a b^{2} f\right )} x^{5} + 35 \, {\left (b^{3} d - 2 \, a b^{2} e + 3 \, a^{2} b f\right )} x^{3} + 105 \, {\left (b^{3} c - 2 \, a b^{2} d + 3 \, a^{2} b e - 4 \, a^{3} f\right )} x}{105 \, b^{5}} \]
1/2*(a*b^3*c - a^2*b^2*d + a^3*b*e - a^4*f)*x/(b^6*x^2 + a*b^5) - 1/2*(3*a *b^3*c - 5*a^2*b^2*d + 7*a^3*b*e - 9*a^4*f)*arctan(b*x/sqrt(a*b))/(sqrt(a* b)*b^5) + 1/105*(15*b^3*f*x^7 + 21*(b^3*e - 2*a*b^2*f)*x^5 + 35*(b^3*d - 2 *a*b^2*e + 3*a^2*b*f)*x^3 + 105*(b^3*c - 2*a*b^2*d + 3*a^2*b*e - 4*a^3*f)* x)/b^5
Time = 0.28 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.97 \[ \int \frac {x^4 \left (c+d x^2+e x^4+f x^6\right )}{\left (a+b x^2\right )^2} \, dx=-\frac {{\left (3 \, a b^{3} c - 5 \, a^{2} b^{2} d + 7 \, a^{3} b e - 9 \, a^{4} f\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} b^{5}} + \frac {a b^{3} c x - a^{2} b^{2} d x + a^{3} b e x - a^{4} f x}{2 \, {\left (b x^{2} + a\right )} b^{5}} + \frac {15 \, b^{12} f x^{7} + 21 \, b^{12} e x^{5} - 42 \, a b^{11} f x^{5} + 35 \, b^{12} d x^{3} - 70 \, a b^{11} e x^{3} + 105 \, a^{2} b^{10} f x^{3} + 105 \, b^{12} c x - 210 \, a b^{11} d x + 315 \, a^{2} b^{10} e x - 420 \, a^{3} b^{9} f x}{105 \, b^{14}} \]
-1/2*(3*a*b^3*c - 5*a^2*b^2*d + 7*a^3*b*e - 9*a^4*f)*arctan(b*x/sqrt(a*b)) /(sqrt(a*b)*b^5) + 1/2*(a*b^3*c*x - a^2*b^2*d*x + a^3*b*e*x - a^4*f*x)/((b *x^2 + a)*b^5) + 1/105*(15*b^12*f*x^7 + 21*b^12*e*x^5 - 42*a*b^11*f*x^5 + 35*b^12*d*x^3 - 70*a*b^11*e*x^3 + 105*a^2*b^10*f*x^3 + 105*b^12*c*x - 210* a*b^11*d*x + 315*a^2*b^10*e*x - 420*a^3*b^9*f*x)/b^14
Time = 5.78 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.43 \[ \int \frac {x^4 \left (c+d x^2+e x^4+f x^6\right )}{\left (a+b x^2\right )^2} \, dx=x^5\,\left (\frac {e}{5\,b^2}-\frac {2\,a\,f}{5\,b^3}\right )+x\,\left (\frac {c}{b^2}-\frac {a^2\,\left (\frac {e}{b^2}-\frac {2\,a\,f}{b^3}\right )}{b^2}+\frac {2\,a\,\left (\frac {a^2\,f}{b^4}-\frac {d}{b^2}+\frac {2\,a\,\left (\frac {e}{b^2}-\frac {2\,a\,f}{b^3}\right )}{b}\right )}{b}\right )-x^3\,\left (\frac {a^2\,f}{3\,b^4}-\frac {d}{3\,b^2}+\frac {2\,a\,\left (\frac {e}{b^2}-\frac {2\,a\,f}{b^3}\right )}{3\,b}\right )-\frac {x\,\left (\frac {f\,a^4}{2}-\frac {e\,a^3\,b}{2}+\frac {d\,a^2\,b^2}{2}-\frac {c\,a\,b^3}{2}\right )}{b^6\,x^2+a\,b^5}+\frac {f\,x^7}{7\,b^2}+\frac {\sqrt {a}\,\mathrm {atan}\left (\frac {\sqrt {a}\,\sqrt {b}\,x\,\left (-9\,f\,a^3+7\,e\,a^2\,b-5\,d\,a\,b^2+3\,c\,b^3\right )}{9\,f\,a^4-7\,e\,a^3\,b+5\,d\,a^2\,b^2-3\,c\,a\,b^3}\right )\,\left (-9\,f\,a^3+7\,e\,a^2\,b-5\,d\,a\,b^2+3\,c\,b^3\right )}{2\,b^{11/2}} \]
x^5*(e/(5*b^2) - (2*a*f)/(5*b^3)) + x*(c/b^2 - (a^2*(e/b^2 - (2*a*f)/b^3)) /b^2 + (2*a*((a^2*f)/b^4 - d/b^2 + (2*a*(e/b^2 - (2*a*f)/b^3))/b))/b) - x^ 3*((a^2*f)/(3*b^4) - d/(3*b^2) + (2*a*(e/b^2 - (2*a*f)/b^3))/(3*b)) - (x*( (a^4*f)/2 + (a^2*b^2*d)/2 - (a*b^3*c)/2 - (a^3*b*e)/2))/(a*b^5 + b^6*x^2) + (f*x^7)/(7*b^2) + (a^(1/2)*atan((a^(1/2)*b^(1/2)*x*(3*b^3*c - 9*a^3*f - 5*a*b^2*d + 7*a^2*b*e))/(9*a^4*f + 5*a^2*b^2*d - 3*a*b^3*c - 7*a^3*b*e))*( 3*b^3*c - 9*a^3*f - 5*a*b^2*d + 7*a^2*b*e))/(2*b^(11/2))