Integrand size = 22, antiderivative size = 140 \[ \int \frac {x^4 \left (c+d x^2\right )^3}{a+b x^2} \, dx=-\frac {a (b c-a d)^3 x}{b^5}+\frac {(b c-a d)^3 x^3}{3 b^4}+\frac {d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^5}{5 b^3}+\frac {d^2 (3 b c-a d) x^7}{7 b^2}+\frac {d^3 x^9}{9 b}+\frac {a^{3/2} (b c-a d)^3 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{11/2}} \] Output:
-a*(-a*d+b*c)^3*x/b^5+1/3*(-a*d+b*c)^3*x^3/b^4+1/5*d*(a^2*d^2-3*a*b*c*d+3* b^2*c^2)*x^5/b^3+1/7*d^2*(-a*d+3*b*c)*x^7/b^2+1/9*d^3*x^9/b+a^(3/2)*(-a*d+ b*c)^3*arctan(b^(1/2)*x/a^(1/2))/b^(11/2)
Time = 0.03 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00 \[ \int \frac {x^4 \left (c+d x^2\right )^3}{a+b x^2} \, dx=\frac {a (-b c+a d)^3 x}{b^5}+\frac {(b c-a d)^3 x^3}{3 b^4}+\frac {d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^5}{5 b^3}+\frac {d^2 (3 b c-a d) x^7}{7 b^2}+\frac {d^3 x^9}{9 b}-\frac {a^{3/2} (-b c+a d)^3 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{11/2}} \] Input:
Integrate[(x^4*(c + d*x^2)^3)/(a + b*x^2),x]
Output:
(a*(-(b*c) + a*d)^3*x)/b^5 + ((b*c - a*d)^3*x^3)/(3*b^4) + (d*(3*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*x^5)/(5*b^3) + (d^2*(3*b*c - a*d)*x^7)/(7*b^2) + (d^ 3*x^9)/(9*b) - (a^(3/2)*(-(b*c) + a*d)^3*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(1 1/2)
Time = 0.30 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {364, 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.
| \(\displaystyle \int \frac {x^4 \left (c+d x^2\right )^3}{a+b x^2} \, dx\) |
| \(\Big \downarrow \) 364 |
| \(\displaystyle \int \left (\frac {d x^4 \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{b^3}+\frac {a^5 \left (-d^3\right )+3 a^4 b c d^2-3 a^3 b^2 c^2 d+a^2 b^3 c^3}{b^5 \left (a+b x^2\right )}-\frac {a (b c-a d)^3}{b^5}+\frac {x^2 (b c-a d)^3}{b^4}+\frac {d^2 x^6 (3 b c-a d)}{b^2}+\frac {d^3 x^8}{b}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^{3/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) (b c-a d)^3}{b^{11/2}}+\frac {d x^5 \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{5 b^3}-\frac {a x (b c-a d)^3}{b^5}+\frac {x^3 (b c-a d)^3}{3 b^4}+\frac {d^2 x^7 (3 b c-a d)}{7 b^2}+\frac {d^3 x^9}{9 b}\) |
Input:
Int[(x^4*(c + d*x^2)^3)/(a + b*x^2),x]
Output:
-((a*(b*c - a*d)^3*x)/b^5) + ((b*c - a*d)^3*x^3)/(3*b^4) + (d*(3*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*x^5)/(5*b^3) + (d^2*(3*b*c - a*d)*x^7)/(7*b^2) + (d^ 3*x^9)/(9*b) + (a^(3/2)*(b*c - a*d)^3*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(11/2 )
Int[(((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_))/((c_) + (d_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*((a + b*x^2)^p/(c + d*x^2)), x], x ] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && (In tegerQ[m] || IGtQ[2*(m + 1), 0] || !RationalQ[m])
Time = 0.57 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.65
| method | result | size |
| default | \(\frac {\frac {1}{9} b^{4} d^{3} x^{9}-\frac {1}{7} a \,b^{3} d^{3} x^{7}+\frac {3}{7} b^{4} c \,d^{2} x^{7}+\frac {1}{5} a^{2} b^{2} d^{3} x^{5}-\frac {3}{5} a \,b^{3} c \,d^{2} x^{5}+\frac {3}{5} b^{4} c^{2} d \,x^{5}-\frac {1}{3} a^{3} b \,d^{3} x^{3}+a^{2} b^{2} c \,d^{2} x^{3}-a \,b^{3} c^{2} d \,x^{3}+\frac {1}{3} b^{4} c^{3} x^{3}+a^{4} d^{3} x -3 a^{3} b c \,d^{2} x +3 a^{2} b^{2} c^{2} d x -a \,b^{3} c^{3} x}{b^{5}}-\frac {a^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{b^{5} \sqrt {a b}}\) | \(231\) |
| risch | \(\frac {d^{3} x^{9}}{9 b}-\frac {a \,d^{3} x^{7}}{7 b^{2}}+\frac {3 c \,d^{2} x^{7}}{7 b}+\frac {a^{2} d^{3} x^{5}}{5 b^{3}}-\frac {3 a c \,d^{2} x^{5}}{5 b^{2}}+\frac {3 c^{2} d \,x^{5}}{5 b}-\frac {a^{3} d^{3} x^{3}}{3 b^{4}}+\frac {a^{2} c \,d^{2} x^{3}}{b^{3}}-\frac {a \,c^{2} d \,x^{3}}{b^{2}}+\frac {c^{3} x^{3}}{3 b}+\frac {a^{4} d^{3} x}{b^{5}}-\frac {3 a^{3} c \,d^{2} x}{b^{4}}+\frac {3 a^{2} c^{2} d x}{b^{3}}-\frac {a \,c^{3} x}{b^{2}}+\frac {\sqrt {-a b}\, a^{4} \ln \left (-\sqrt {-a b}\, x -a \right ) d^{3}}{2 b^{6}}-\frac {3 \sqrt {-a b}\, a^{3} \ln \left (-\sqrt {-a b}\, x -a \right ) c \,d^{2}}{2 b^{5}}+\frac {3 \sqrt {-a b}\, a^{2} \ln \left (-\sqrt {-a b}\, x -a \right ) c^{2} d}{2 b^{4}}-\frac {\sqrt {-a b}\, a \ln \left (-\sqrt {-a b}\, x -a \right ) c^{3}}{2 b^{3}}-\frac {\sqrt {-a b}\, a^{4} \ln \left (\sqrt {-a b}\, x -a \right ) d^{3}}{2 b^{6}}+\frac {3 \sqrt {-a b}\, a^{3} \ln \left (\sqrt {-a b}\, x -a \right ) c \,d^{2}}{2 b^{5}}-\frac {3 \sqrt {-a b}\, a^{2} \ln \left (\sqrt {-a b}\, x -a \right ) c^{2} d}{2 b^{4}}+\frac {\sqrt {-a b}\, a \ln \left (\sqrt {-a b}\, x -a \right ) c^{3}}{2 b^{3}}\) | \(419\) |
Input:
int(x^4*(d*x^2+c)^3/(b*x^2+a),x,method=_RETURNVERBOSE)
Output:
1/b^5*(1/9*b^4*d^3*x^9-1/7*a*b^3*d^3*x^7+3/7*b^4*c*d^2*x^7+1/5*a^2*b^2*d^3 *x^5-3/5*a*b^3*c*d^2*x^5+3/5*b^4*c^2*d*x^5-1/3*a^3*b*d^3*x^3+a^2*b^2*c*d^2 *x^3-a*b^3*c^2*d*x^3+1/3*b^4*c^3*x^3+a^4*d^3*x-3*a^3*b*c*d^2*x+3*a^2*b^2*c ^2*d*x-a*b^3*c^3*x)-a^2*(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)/b^5/ (a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))
Time = 0.09 (sec) , antiderivative size = 468, normalized size of antiderivative = 3.34 \[ \int \frac {x^4 \left (c+d x^2\right )^3}{a+b x^2} \, dx=\left [\frac {70 \, b^{4} d^{3} x^{9} + 90 \, {\left (3 \, b^{4} c d^{2} - a b^{3} d^{3}\right )} x^{7} + 126 \, {\left (3 \, b^{4} c^{2} d - 3 \, a b^{3} c d^{2} + a^{2} b^{2} d^{3}\right )} x^{5} + 210 \, {\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x^{3} - 315 \, {\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} - 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right ) - 630 \, {\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} x}{630 \, b^{5}}, \frac {35 \, b^{4} d^{3} x^{9} + 45 \, {\left (3 \, b^{4} c d^{2} - a b^{3} d^{3}\right )} x^{7} + 63 \, {\left (3 \, b^{4} c^{2} d - 3 \, a b^{3} c d^{2} + a^{2} b^{2} d^{3}\right )} x^{5} + 105 \, {\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x^{3} + 315 \, {\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) - 315 \, {\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} x}{315 \, b^{5}}\right ] \] Input:
integrate(x^4*(d*x^2+c)^3/(b*x^2+a),x, algorithm="fricas")
Output:
[1/630*(70*b^4*d^3*x^9 + 90*(3*b^4*c*d^2 - a*b^3*d^3)*x^7 + 126*(3*b^4*c^2 *d - 3*a*b^3*c*d^2 + a^2*b^2*d^3)*x^5 + 210*(b^4*c^3 - 3*a*b^3*c^2*d + 3*a ^2*b^2*c*d^2 - a^3*b*d^3)*x^3 - 315*(a*b^3*c^3 - 3*a^2*b^2*c^2*d + 3*a^3*b *c*d^2 - a^4*d^3)*sqrt(-a/b)*log((b*x^2 - 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a )) - 630*(a*b^3*c^3 - 3*a^2*b^2*c^2*d + 3*a^3*b*c*d^2 - a^4*d^3)*x)/b^5, 1 /315*(35*b^4*d^3*x^9 + 45*(3*b^4*c*d^2 - a*b^3*d^3)*x^7 + 63*(3*b^4*c^2*d - 3*a*b^3*c*d^2 + a^2*b^2*d^3)*x^5 + 105*(b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2* b^2*c*d^2 - a^3*b*d^3)*x^3 + 315*(a*b^3*c^3 - 3*a^2*b^2*c^2*d + 3*a^3*b*c* d^2 - a^4*d^3)*sqrt(a/b)*arctan(b*x*sqrt(a/b)/a) - 315*(a*b^3*c^3 - 3*a^2* b^2*c^2*d + 3*a^3*b*c*d^2 - a^4*d^3)*x)/b^5]
Leaf count of result is larger than twice the leaf count of optimal. 343 vs. \(2 (128) = 256\).
Time = 0.36 (sec) , antiderivative size = 343, normalized size of antiderivative = 2.45 \[ \int \frac {x^4 \left (c+d x^2\right )^3}{a+b x^2} \, dx=x^{7} \left (- \frac {a d^{3}}{7 b^{2}} + \frac {3 c d^{2}}{7 b}\right ) + x^{5} \left (\frac {a^{2} d^{3}}{5 b^{3}} - \frac {3 a c d^{2}}{5 b^{2}} + \frac {3 c^{2} d}{5 b}\right ) + x^{3} \left (- \frac {a^{3} d^{3}}{3 b^{4}} + \frac {a^{2} c d^{2}}{b^{3}} - \frac {a c^{2} d}{b^{2}} + \frac {c^{3}}{3 b}\right ) + x \left (\frac {a^{4} d^{3}}{b^{5}} - \frac {3 a^{3} c d^{2}}{b^{4}} + \frac {3 a^{2} c^{2} d}{b^{3}} - \frac {a c^{3}}{b^{2}}\right ) + \frac {\sqrt {- \frac {a^{3}}{b^{11}}} \left (a d - b c\right )^{3} \log {\left (- \frac {b^{5} \sqrt {- \frac {a^{3}}{b^{11}}} \left (a d - b c\right )^{3}}{a^{4} d^{3} - 3 a^{3} b c d^{2} + 3 a^{2} b^{2} c^{2} d - a b^{3} c^{3}} + x \right )}}{2} - \frac {\sqrt {- \frac {a^{3}}{b^{11}}} \left (a d - b c\right )^{3} \log {\left (\frac {b^{5} \sqrt {- \frac {a^{3}}{b^{11}}} \left (a d - b c\right )^{3}}{a^{4} d^{3} - 3 a^{3} b c d^{2} + 3 a^{2} b^{2} c^{2} d - a b^{3} c^{3}} + x \right )}}{2} + \frac {d^{3} x^{9}}{9 b} \] Input:
integrate(x**4*(d*x**2+c)**3/(b*x**2+a),x)
Output:
x**7*(-a*d**3/(7*b**2) + 3*c*d**2/(7*b)) + x**5*(a**2*d**3/(5*b**3) - 3*a* c*d**2/(5*b**2) + 3*c**2*d/(5*b)) + x**3*(-a**3*d**3/(3*b**4) + a**2*c*d** 2/b**3 - a*c**2*d/b**2 + c**3/(3*b)) + x*(a**4*d**3/b**5 - 3*a**3*c*d**2/b **4 + 3*a**2*c**2*d/b**3 - a*c**3/b**2) + sqrt(-a**3/b**11)*(a*d - b*c)**3 *log(-b**5*sqrt(-a**3/b**11)*(a*d - b*c)**3/(a**4*d**3 - 3*a**3*b*c*d**2 + 3*a**2*b**2*c**2*d - a*b**3*c**3) + x)/2 - sqrt(-a**3/b**11)*(a*d - b*c)* *3*log(b**5*sqrt(-a**3/b**11)*(a*d - b*c)**3/(a**4*d**3 - 3*a**3*b*c*d**2 + 3*a**2*b**2*c**2*d - a*b**3*c**3) + x)/2 + d**3*x**9/(9*b)
Time = 0.12 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.59 \[ \int \frac {x^4 \left (c+d x^2\right )^3}{a+b x^2} \, dx=\frac {{\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{5}} + \frac {35 \, b^{4} d^{3} x^{9} + 45 \, {\left (3 \, b^{4} c d^{2} - a b^{3} d^{3}\right )} x^{7} + 63 \, {\left (3 \, b^{4} c^{2} d - 3 \, a b^{3} c d^{2} + a^{2} b^{2} d^{3}\right )} x^{5} + 105 \, {\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x^{3} - 315 \, {\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} x}{315 \, b^{5}} \] Input:
integrate(x^4*(d*x^2+c)^3/(b*x^2+a),x, algorithm="maxima")
Output:
(a^2*b^3*c^3 - 3*a^3*b^2*c^2*d + 3*a^4*b*c*d^2 - a^5*d^3)*arctan(b*x/sqrt( a*b))/(sqrt(a*b)*b^5) + 1/315*(35*b^4*d^3*x^9 + 45*(3*b^4*c*d^2 - a*b^3*d^ 3)*x^7 + 63*(3*b^4*c^2*d - 3*a*b^3*c*d^2 + a^2*b^2*d^3)*x^5 + 105*(b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*x^3 - 315*(a*b^3*c^3 - 3*a ^2*b^2*c^2*d + 3*a^3*b*c*d^2 - a^4*d^3)*x)/b^5
Time = 0.12 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.72 \[ \int \frac {x^4 \left (c+d x^2\right )^3}{a+b x^2} \, dx=\frac {{\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{5}} + \frac {35 \, b^{8} d^{3} x^{9} + 135 \, b^{8} c d^{2} x^{7} - 45 \, a b^{7} d^{3} x^{7} + 189 \, b^{8} c^{2} d x^{5} - 189 \, a b^{7} c d^{2} x^{5} + 63 \, a^{2} b^{6} d^{3} x^{5} + 105 \, b^{8} c^{3} x^{3} - 315 \, a b^{7} c^{2} d x^{3} + 315 \, a^{2} b^{6} c d^{2} x^{3} - 105 \, a^{3} b^{5} d^{3} x^{3} - 315 \, a b^{7} c^{3} x + 945 \, a^{2} b^{6} c^{2} d x - 945 \, a^{3} b^{5} c d^{2} x + 315 \, a^{4} b^{4} d^{3} x}{315 \, b^{9}} \] Input:
integrate(x^4*(d*x^2+c)^3/(b*x^2+a),x, algorithm="giac")
Output:
(a^2*b^3*c^3 - 3*a^3*b^2*c^2*d + 3*a^4*b*c*d^2 - a^5*d^3)*arctan(b*x/sqrt( a*b))/(sqrt(a*b)*b^5) + 1/315*(35*b^8*d^3*x^9 + 135*b^8*c*d^2*x^7 - 45*a*b ^7*d^3*x^7 + 189*b^8*c^2*d*x^5 - 189*a*b^7*c*d^2*x^5 + 63*a^2*b^6*d^3*x^5 + 105*b^8*c^3*x^3 - 315*a*b^7*c^2*d*x^3 + 315*a^2*b^6*c*d^2*x^3 - 105*a^3* b^5*d^3*x^3 - 315*a*b^7*c^3*x + 945*a^2*b^6*c^2*d*x - 945*a^3*b^5*c*d^2*x + 315*a^4*b^4*d^3*x)/b^9
Time = 0.06 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.86 \[ \int \frac {x^4 \left (c+d x^2\right )^3}{a+b x^2} \, dx=x^3\,\left (\frac {c^3}{3\,b}-\frac {a\,\left (\frac {3\,c^2\,d}{b}+\frac {a\,\left (\frac {a\,d^3}{b^2}-\frac {3\,c\,d^2}{b}\right )}{b}\right )}{3\,b}\right )-x^7\,\left (\frac {a\,d^3}{7\,b^2}-\frac {3\,c\,d^2}{7\,b}\right )+x^5\,\left (\frac {3\,c^2\,d}{5\,b}+\frac {a\,\left (\frac {a\,d^3}{b^2}-\frac {3\,c\,d^2}{b}\right )}{5\,b}\right )+\frac {d^3\,x^9}{9\,b}-\frac {a^{3/2}\,\mathrm {atan}\left (\frac {a^{3/2}\,\sqrt {b}\,x\,{\left (a\,d-b\,c\right )}^3}{a^5\,d^3-3\,a^4\,b\,c\,d^2+3\,a^3\,b^2\,c^2\,d-a^2\,b^3\,c^3}\right )\,{\left (a\,d-b\,c\right )}^3}{b^{11/2}}-\frac {a\,x\,\left (\frac {c^3}{b}-\frac {a\,\left (\frac {3\,c^2\,d}{b}+\frac {a\,\left (\frac {a\,d^3}{b^2}-\frac {3\,c\,d^2}{b}\right )}{b}\right )}{b}\right )}{b} \] Input:
int((x^4*(c + d*x^2)^3)/(a + b*x^2),x)
Output:
x^3*(c^3/(3*b) - (a*((3*c^2*d)/b + (a*((a*d^3)/b^2 - (3*c*d^2)/b))/b))/(3* b)) - x^7*((a*d^3)/(7*b^2) - (3*c*d^2)/(7*b)) + x^5*((3*c^2*d)/(5*b) + (a* ((a*d^3)/b^2 - (3*c*d^2)/b))/(5*b)) + (d^3*x^9)/(9*b) - (a^(3/2)*atan((a^( 3/2)*b^(1/2)*x*(a*d - b*c)^3)/(a^5*d^3 - a^2*b^3*c^3 + 3*a^3*b^2*c^2*d - 3 *a^4*b*c*d^2))*(a*d - b*c)^3)/b^(11/2) - (a*x*(c^3/b - (a*((3*c^2*d)/b + ( a*((a*d^3)/b^2 - (3*c*d^2)/b))/b))/b))/b
Time = 0.21 (sec) , antiderivative size = 282, normalized size of antiderivative = 2.01 \[ \int \frac {x^4 \left (c+d x^2\right )^3}{a+b x^2} \, dx=\frac {-315 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{4} d^{3}+945 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{3} b c \,d^{2}-945 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{2} b^{2} c^{2} d +315 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a \,b^{3} c^{3}+315 a^{4} b \,d^{3} x -945 a^{3} b^{2} c \,d^{2} x -105 a^{3} b^{2} d^{3} x^{3}+945 a^{2} b^{3} c^{2} d x +315 a^{2} b^{3} c \,d^{2} x^{3}+63 a^{2} b^{3} d^{3} x^{5}-315 a \,b^{4} c^{3} x -315 a \,b^{4} c^{2} d \,x^{3}-189 a \,b^{4} c \,d^{2} x^{5}-45 a \,b^{4} d^{3} x^{7}+105 b^{5} c^{3} x^{3}+189 b^{5} c^{2} d \,x^{5}+135 b^{5} c \,d^{2} x^{7}+35 b^{5} d^{3} x^{9}}{315 b^{6}} \] Input:
int(x^4*(d*x^2+c)^3/(b*x^2+a),x)
Output:
( - 315*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**4*d**3 + 945*sqrt (b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**3*b*c*d**2 - 945*sqrt(b)*sqrt (a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**2*b**2*c**2*d + 315*sqrt(b)*sqrt(a)*a tan((b*x)/(sqrt(b)*sqrt(a)))*a*b**3*c**3 + 315*a**4*b*d**3*x - 945*a**3*b* *2*c*d**2*x - 105*a**3*b**2*d**3*x**3 + 945*a**2*b**3*c**2*d*x + 315*a**2* b**3*c*d**2*x**3 + 63*a**2*b**3*d**3*x**5 - 315*a*b**4*c**3*x - 315*a*b**4 *c**2*d*x**3 - 189*a*b**4*c*d**2*x**5 - 45*a*b**4*d**3*x**7 + 105*b**5*c** 3*x**3 + 189*b**5*c**2*d*x**5 + 135*b**5*c*d**2*x**7 + 35*b**5*d**3*x**9)/ (315*b**6)