Integrand size = 19, antiderivative size = 130 \[ \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^3} \, dx=\frac {d^3 x}{b^3}+\frac {(b c-a d)^3 x}{4 a b^3 \left (a+b x^2\right )^2}+\frac {3 (b c-a d)^2 (b c+3 a d) x}{8 a^2 b^3 \left (a+b x^2\right )}+\frac {3 (b c-a d) \left (4 a^2 d^2+(b c+a d)^2\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{5/2} b^{7/2}} \] Output:
d^3*x/b^3+1/4*(-a*d+b*c)^3*x/a/b^3/(b*x^2+a)^2+3/8*(-a*d+b*c)^2*(3*a*d+b*c )*x/a^2/b^3/(b*x^2+a)+3/8*(-a*d+b*c)*(4*a^2*d^2+(a*d+b*c)^2)*arctan(b^(1/2 )*x/a^(1/2))/a^(5/2)/b^(7/2)
Time = 0.05 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.07 \[ \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^3} \, dx=\frac {d^3 x}{b^3}+\frac {(b c-a d)^3 x}{4 a b^3 \left (a+b x^2\right )^2}+\frac {3 (b c-a d)^2 (b c+3 a d) x}{8 a^2 b^3 \left (a+b x^2\right )}+\frac {3 \left (b^3 c^3+a b^2 c^2 d+3 a^2 b c d^2-5 a^3 d^3\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{5/2} b^{7/2}} \] Input:
Integrate[(c + d*x^2)^3/(a + b*x^2)^3,x]
Output:
(d^3*x)/b^3 + ((b*c - a*d)^3*x)/(4*a*b^3*(a + b*x^2)^2) + (3*(b*c - a*d)^2 *(b*c + 3*a*d)*x)/(8*a^2*b^3*(a + b*x^2)) + (3*(b^3*c^3 + a*b^2*c^2*d + 3* a^2*b*c*d^2 - 5*a^3*d^3)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(5/2)*b^(7/2))
Time = 0.32 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {300, 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.
| \(\displaystyle \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 300 |
| \(\displaystyle \int \left (\frac {-a^3 d^3+3 b^2 d^2 x^4 (b c-a d)+3 b d x^2 (b c-a d) (a d+b c)+b^3 c^3}{b^3 \left (a+b x^2\right )^3}+\frac {d^3}{b^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 x (b c-a d)^2 (3 a d+b c)}{8 a^2 b^3 \left (a+b x^2\right )}+\frac {3 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) (b c-a d) \left (4 a^2 d^2+(a d+b c)^2\right )}{8 a^{5/2} b^{7/2}}+\frac {x (b c-a d)^3}{4 a b^3 \left (a+b x^2\right )^2}+\frac {d^3 x}{b^3}\) |
Input:
Int[(c + d*x^2)^3/(a + b*x^2)^3,x]
Output:
(d^3*x)/b^3 + ((b*c - a*d)^3*x)/(4*a*b^3*(a + b*x^2)^2) + (3*(b*c - a*d)^2 *(b*c + 3*a*d)*x)/(8*a^2*b^3*(a + b*x^2)) + (3*(b*c - a*d)*(4*a^2*d^2 + (b *c + a*d)^2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(5/2)*b^(7/2))
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Int [PolynomialDivide[(a + b*x^2)^p, (c + d*x^2)^(-q), x], x] /; FreeQ[{a, b, c , d}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && ILtQ[q, 0] && GeQ[p, -q]
Time = 0.37 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.31
| method | result | size |
| default | \(\frac {d^{3} x}{b^{3}}-\frac {\frac {-\frac {3 b \left (3 a^{3} d^{3}-5 a^{2} b c \,d^{2}+a \,b^{2} c^{2} d +b^{3} c^{3}\right ) x^{3}}{8 a^{2}}-\frac {\left (7 a^{3} d^{3}-9 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d +5 b^{3} c^{3}\right ) x}{8 a}}{\left (b \,x^{2}+a \right )^{2}}+\frac {3 \left (5 a^{3} d^{3}-3 a^{2} b c \,d^{2}-a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 a^{2} \sqrt {a b}}}{b^{3}}\) | \(170\) |
| risch | \(\frac {d^{3} x}{b^{3}}+\frac {\frac {3 b \left (3 a^{3} d^{3}-5 a^{2} b c \,d^{2}+a \,b^{2} c^{2} d +b^{3} c^{3}\right ) x^{3}}{8 a^{2}}+\frac {\left (7 a^{3} d^{3}-9 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d +5 b^{3} c^{3}\right ) x}{8 a}}{b^{3} \left (b \,x^{2}+a \right )^{2}}-\frac {15 a \ln \left (b x -\sqrt {-a b}\right ) d^{3}}{16 b^{3} \sqrt {-a b}}+\frac {9 \ln \left (b x -\sqrt {-a b}\right ) c \,d^{2}}{16 b^{2} \sqrt {-a b}}+\frac {3 \ln \left (b x -\sqrt {-a b}\right ) c^{2} d}{16 b \sqrt {-a b}\, a}+\frac {3 \ln \left (b x -\sqrt {-a b}\right ) c^{3}}{16 \sqrt {-a b}\, a^{2}}+\frac {15 a \ln \left (-b x -\sqrt {-a b}\right ) d^{3}}{16 b^{3} \sqrt {-a b}}-\frac {9 \ln \left (-b x -\sqrt {-a b}\right ) c \,d^{2}}{16 b^{2} \sqrt {-a b}}-\frac {3 \ln \left (-b x -\sqrt {-a b}\right ) c^{2} d}{16 b \sqrt {-a b}\, a}-\frac {3 \ln \left (-b x -\sqrt {-a b}\right ) c^{3}}{16 \sqrt {-a b}\, a^{2}}\) | \(343\) |
Input:
int((d*x^2+c)^3/(b*x^2+a)^3,x,method=_RETURNVERBOSE)
Output:
d^3*x/b^3-1/b^3*((-3/8*b*(3*a^3*d^3-5*a^2*b*c*d^2+a*b^2*c^2*d+b^3*c^3)/a^2 *x^3-1/8*(7*a^3*d^3-9*a^2*b*c*d^2-3*a*b^2*c^2*d+5*b^3*c^3)/a*x)/(b*x^2+a)^ 2+3/8*(5*a^3*d^3-3*a^2*b*c*d^2-a*b^2*c^2*d-b^3*c^3)/a^2/(a*b)^(1/2)*arctan (b*x/(a*b)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 292 vs. \(2 (116) = 232\).
Time = 0.09 (sec) , antiderivative size = 606, normalized size of antiderivative = 4.66 \[ \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^3} \, dx=\left [\frac {16 \, a^{3} b^{3} d^{3} x^{5} + 2 \, {\left (3 \, a b^{5} c^{3} + 3 \, a^{2} b^{4} c^{2} d - 15 \, a^{3} b^{3} c d^{2} + 25 \, a^{4} b^{2} d^{3}\right )} x^{3} + 3 \, {\left (a^{2} b^{3} c^{3} + a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - 5 \, a^{5} d^{3} + {\left (b^{5} c^{3} + a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - 5 \, a^{3} b^{2} d^{3}\right )} x^{4} + 2 \, {\left (a b^{4} c^{3} + a^{2} b^{3} c^{2} d + 3 \, a^{3} b^{2} c d^{2} - 5 \, a^{4} b d^{3}\right )} x^{2}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} + 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) + 2 \, {\left (5 \, a^{2} b^{4} c^{3} - 3 \, a^{3} b^{3} c^{2} d - 9 \, a^{4} b^{2} c d^{2} + 15 \, a^{5} b d^{3}\right )} x}{16 \, {\left (a^{3} b^{6} x^{4} + 2 \, a^{4} b^{5} x^{2} + a^{5} b^{4}\right )}}, \frac {8 \, a^{3} b^{3} d^{3} x^{5} + {\left (3 \, a b^{5} c^{3} + 3 \, a^{2} b^{4} c^{2} d - 15 \, a^{3} b^{3} c d^{2} + 25 \, a^{4} b^{2} d^{3}\right )} x^{3} + 3 \, {\left (a^{2} b^{3} c^{3} + a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - 5 \, a^{5} d^{3} + {\left (b^{5} c^{3} + a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - 5 \, a^{3} b^{2} d^{3}\right )} x^{4} + 2 \, {\left (a b^{4} c^{3} + a^{2} b^{3} c^{2} d + 3 \, a^{3} b^{2} c d^{2} - 5 \, a^{4} b d^{3}\right )} x^{2}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) + {\left (5 \, a^{2} b^{4} c^{3} - 3 \, a^{3} b^{3} c^{2} d - 9 \, a^{4} b^{2} c d^{2} + 15 \, a^{5} b d^{3}\right )} x}{8 \, {\left (a^{3} b^{6} x^{4} + 2 \, a^{4} b^{5} x^{2} + a^{5} b^{4}\right )}}\right ] \] Input:
integrate((d*x^2+c)^3/(b*x^2+a)^3,x, algorithm="fricas")
Output:
[1/16*(16*a^3*b^3*d^3*x^5 + 2*(3*a*b^5*c^3 + 3*a^2*b^4*c^2*d - 15*a^3*b^3* c*d^2 + 25*a^4*b^2*d^3)*x^3 + 3*(a^2*b^3*c^3 + a^3*b^2*c^2*d + 3*a^4*b*c*d ^2 - 5*a^5*d^3 + (b^5*c^3 + a*b^4*c^2*d + 3*a^2*b^3*c*d^2 - 5*a^3*b^2*d^3) *x^4 + 2*(a*b^4*c^3 + a^2*b^3*c^2*d + 3*a^3*b^2*c*d^2 - 5*a^4*b*d^3)*x^2)* sqrt(-a*b)*log((b*x^2 + 2*sqrt(-a*b)*x - a)/(b*x^2 + a)) + 2*(5*a^2*b^4*c^ 3 - 3*a^3*b^3*c^2*d - 9*a^4*b^2*c*d^2 + 15*a^5*b*d^3)*x)/(a^3*b^6*x^4 + 2* a^4*b^5*x^2 + a^5*b^4), 1/8*(8*a^3*b^3*d^3*x^5 + (3*a*b^5*c^3 + 3*a^2*b^4* c^2*d - 15*a^3*b^3*c*d^2 + 25*a^4*b^2*d^3)*x^3 + 3*(a^2*b^3*c^3 + a^3*b^2* c^2*d + 3*a^4*b*c*d^2 - 5*a^5*d^3 + (b^5*c^3 + a*b^4*c^2*d + 3*a^2*b^3*c*d ^2 - 5*a^3*b^2*d^3)*x^4 + 2*(a*b^4*c^3 + a^2*b^3*c^2*d + 3*a^3*b^2*c*d^2 - 5*a^4*b*d^3)*x^2)*sqrt(a*b)*arctan(sqrt(a*b)*x/a) + (5*a^2*b^4*c^3 - 3*a^ 3*b^3*c^2*d - 9*a^4*b^2*c*d^2 + 15*a^5*b*d^3)*x)/(a^3*b^6*x^4 + 2*a^4*b^5* x^2 + a^5*b^4)]
Leaf count of result is larger than twice the leaf count of optimal. 422 vs. \(2 (122) = 244\).
Time = 0.90 (sec) , antiderivative size = 422, normalized size of antiderivative = 3.25 \[ \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^3} \, dx=\frac {3 \sqrt {- \frac {1}{a^{5} b^{7}}} \left (a d - b c\right ) \left (5 a^{2} d^{2} + 2 a b c d + b^{2} c^{2}\right ) \log {\left (- \frac {3 a^{3} b^{3} \sqrt {- \frac {1}{a^{5} b^{7}}} \left (a d - b c\right ) \left (5 a^{2} d^{2} + 2 a b c d + b^{2} c^{2}\right )}{15 a^{3} d^{3} - 9 a^{2} b c d^{2} - 3 a b^{2} c^{2} d - 3 b^{3} c^{3}} + x \right )}}{16} - \frac {3 \sqrt {- \frac {1}{a^{5} b^{7}}} \left (a d - b c\right ) \left (5 a^{2} d^{2} + 2 a b c d + b^{2} c^{2}\right ) \log {\left (\frac {3 a^{3} b^{3} \sqrt {- \frac {1}{a^{5} b^{7}}} \left (a d - b c\right ) \left (5 a^{2} d^{2} + 2 a b c d + b^{2} c^{2}\right )}{15 a^{3} d^{3} - 9 a^{2} b c d^{2} - 3 a b^{2} c^{2} d - 3 b^{3} c^{3}} + x \right )}}{16} + \frac {x^{3} \cdot \left (9 a^{3} b d^{3} - 15 a^{2} b^{2} c d^{2} + 3 a b^{3} c^{2} d + 3 b^{4} c^{3}\right ) + x \left (7 a^{4} d^{3} - 9 a^{3} b c d^{2} - 3 a^{2} b^{2} c^{2} d + 5 a b^{3} c^{3}\right )}{8 a^{4} b^{3} + 16 a^{3} b^{4} x^{2} + 8 a^{2} b^{5} x^{4}} + \frac {d^{3} x}{b^{3}} \] Input:
integrate((d*x**2+c)**3/(b*x**2+a)**3,x)
Output:
3*sqrt(-1/(a**5*b**7))*(a*d - b*c)*(5*a**2*d**2 + 2*a*b*c*d + b**2*c**2)*l og(-3*a**3*b**3*sqrt(-1/(a**5*b**7))*(a*d - b*c)*(5*a**2*d**2 + 2*a*b*c*d + b**2*c**2)/(15*a**3*d**3 - 9*a**2*b*c*d**2 - 3*a*b**2*c**2*d - 3*b**3*c* *3) + x)/16 - 3*sqrt(-1/(a**5*b**7))*(a*d - b*c)*(5*a**2*d**2 + 2*a*b*c*d + b**2*c**2)*log(3*a**3*b**3*sqrt(-1/(a**5*b**7))*(a*d - b*c)*(5*a**2*d**2 + 2*a*b*c*d + b**2*c**2)/(15*a**3*d**3 - 9*a**2*b*c*d**2 - 3*a*b**2*c**2* d - 3*b**3*c**3) + x)/16 + (x**3*(9*a**3*b*d**3 - 15*a**2*b**2*c*d**2 + 3* a*b**3*c**2*d + 3*b**4*c**3) + x*(7*a**4*d**3 - 9*a**3*b*c*d**2 - 3*a**2*b **2*c**2*d + 5*a*b**3*c**3))/(8*a**4*b**3 + 16*a**3*b**4*x**2 + 8*a**2*b** 5*x**4) + d**3*x/b**3
Time = 0.11 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.42 \[ \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^3} \, dx=\frac {d^{3} x}{b^{3}} + \frac {3 \, {\left (b^{4} c^{3} + a b^{3} c^{2} d - 5 \, a^{2} b^{2} c d^{2} + 3 \, a^{3} b d^{3}\right )} x^{3} + {\left (5 \, a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d - 9 \, a^{3} b c d^{2} + 7 \, a^{4} d^{3}\right )} x}{8 \, {\left (a^{2} b^{5} x^{4} + 2 \, a^{3} b^{4} x^{2} + a^{4} b^{3}\right )}} + \frac {3 \, {\left (b^{3} c^{3} + a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{2} b^{3}} \] Input:
integrate((d*x^2+c)^3/(b*x^2+a)^3,x, algorithm="maxima")
Output:
d^3*x/b^3 + 1/8*(3*(b^4*c^3 + a*b^3*c^2*d - 5*a^2*b^2*c*d^2 + 3*a^3*b*d^3) *x^3 + (5*a*b^3*c^3 - 3*a^2*b^2*c^2*d - 9*a^3*b*c*d^2 + 7*a^4*d^3)*x)/(a^2 *b^5*x^4 + 2*a^3*b^4*x^2 + a^4*b^3) + 3/8*(b^3*c^3 + a*b^2*c^2*d + 3*a^2*b *c*d^2 - 5*a^3*d^3)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^2*b^3)
Time = 0.12 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.37 \[ \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^3} \, dx=\frac {d^{3} x}{b^{3}} + \frac {3 \, {\left (b^{3} c^{3} + a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{2} b^{3}} + \frac {3 \, b^{4} c^{3} x^{3} + 3 \, a b^{3} c^{2} d x^{3} - 15 \, a^{2} b^{2} c d^{2} x^{3} + 9 \, a^{3} b d^{3} x^{3} + 5 \, a b^{3} c^{3} x - 3 \, a^{2} b^{2} c^{2} d x - 9 \, a^{3} b c d^{2} x + 7 \, a^{4} d^{3} x}{8 \, {\left (b x^{2} + a\right )}^{2} a^{2} b^{3}} \] Input:
integrate((d*x^2+c)^3/(b*x^2+a)^3,x, algorithm="giac")
Output:
d^3*x/b^3 + 3/8*(b^3*c^3 + a*b^2*c^2*d + 3*a^2*b*c*d^2 - 5*a^3*d^3)*arctan (b*x/sqrt(a*b))/(sqrt(a*b)*a^2*b^3) + 1/8*(3*b^4*c^3*x^3 + 3*a*b^3*c^2*d*x ^3 - 15*a^2*b^2*c*d^2*x^3 + 9*a^3*b*d^3*x^3 + 5*a*b^3*c^3*x - 3*a^2*b^2*c^ 2*d*x - 9*a^3*b*c*d^2*x + 7*a^4*d^3*x)/((b*x^2 + a)^2*a^2*b^3)
Time = 0.36 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.85 \[ \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^3} \, dx=\frac {\frac {x\,\left (7\,a^3\,d^3-9\,a^2\,b\,c\,d^2-3\,a\,b^2\,c^2\,d+5\,b^3\,c^3\right )}{8\,a}+\frac {3\,x^3\,\left (3\,a^3\,b\,d^3-5\,a^2\,b^2\,c\,d^2+a\,b^3\,c^2\,d+b^4\,c^3\right )}{8\,a^2}}{a^2\,b^3+2\,a\,b^4\,x^2+b^5\,x^4}+\frac {d^3\,x}{b^3}+\frac {3\,\mathrm {atan}\left (\frac {\sqrt {b}\,x\,\left (a\,d-b\,c\right )\,\left (5\,a^2\,d^2+2\,a\,b\,c\,d+b^2\,c^2\right )}{\sqrt {a}\,\left (-5\,a^3\,d^3+3\,a^2\,b\,c\,d^2+a\,b^2\,c^2\,d+b^3\,c^3\right )}\right )\,\left (a\,d-b\,c\right )\,\left (5\,a^2\,d^2+2\,a\,b\,c\,d+b^2\,c^2\right )}{8\,a^{5/2}\,b^{7/2}} \] Input:
int((c + d*x^2)^3/(a + b*x^2)^3,x)
Output:
((x*(7*a^3*d^3 + 5*b^3*c^3 - 3*a*b^2*c^2*d - 9*a^2*b*c*d^2))/(8*a) + (3*x^ 3*(b^4*c^3 + 3*a^3*b*d^3 - 5*a^2*b^2*c*d^2 + a*b^3*c^2*d))/(8*a^2))/(a^2*b ^3 + b^5*x^4 + 2*a*b^4*x^2) + (d^3*x)/b^3 + (3*atan((b^(1/2)*x*(a*d - b*c) *(5*a^2*d^2 + b^2*c^2 + 2*a*b*c*d))/(a^(1/2)*(b^3*c^3 - 5*a^3*d^3 + a*b^2* c^2*d + 3*a^2*b*c*d^2)))*(a*d - b*c)*(5*a^2*d^2 + b^2*c^2 + 2*a*b*c*d))/(8 *a^(5/2)*b^(7/2))
Time = 0.24 (sec) , antiderivative size = 487, normalized size of antiderivative = 3.75 \[ \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^3} \, dx=\frac {-15 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{5} d^{3}+9 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{4} b c \,d^{2}-30 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{4} b \,d^{3} x^{2}+3 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{3} b^{2} c^{2} d +18 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{3} b^{2} c \,d^{2} x^{2}-15 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{3} b^{2} d^{3} x^{4}+3 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{2} b^{3} c^{3}+6 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{2} b^{3} c^{2} d \,x^{2}+9 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{2} b^{3} c \,d^{2} x^{4}+6 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a \,b^{4} c^{3} x^{2}+3 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a \,b^{4} c^{2} d \,x^{4}+3 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) b^{5} c^{3} x^{4}+15 a^{5} b \,d^{3} x -9 a^{4} b^{2} c \,d^{2} x +25 a^{4} b^{2} d^{3} x^{3}-3 a^{3} b^{3} c^{2} d x -15 a^{3} b^{3} c \,d^{2} x^{3}+8 a^{3} b^{3} d^{3} x^{5}+5 a^{2} b^{4} c^{3} x +3 a^{2} b^{4} c^{2} d \,x^{3}+3 a \,b^{5} c^{3} x^{3}}{8 a^{3} b^{4} \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )} \] Input:
int((d*x^2+c)^3/(b*x^2+a)^3,x)
Output:
( - 15*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**5*d**3 + 9*sqrt(b) *sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**4*b*c*d**2 - 30*sqrt(b)*sqrt(a)* atan((b*x)/(sqrt(b)*sqrt(a)))*a**4*b*d**3*x**2 + 3*sqrt(b)*sqrt(a)*atan((b *x)/(sqrt(b)*sqrt(a)))*a**3*b**2*c**2*d + 18*sqrt(b)*sqrt(a)*atan((b*x)/(s qrt(b)*sqrt(a)))*a**3*b**2*c*d**2*x**2 - 15*sqrt(b)*sqrt(a)*atan((b*x)/(sq rt(b)*sqrt(a)))*a**3*b**2*d**3*x**4 + 3*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b )*sqrt(a)))*a**2*b**3*c**3 + 6*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a) ))*a**2*b**3*c**2*d*x**2 + 9*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a))) *a**2*b**3*c*d**2*x**4 + 6*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a *b**4*c**3*x**2 + 3*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a*b**4*c **2*d*x**4 + 3*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b**5*c**3*x** 4 + 15*a**5*b*d**3*x - 9*a**4*b**2*c*d**2*x + 25*a**4*b**2*d**3*x**3 - 3*a **3*b**3*c**2*d*x - 15*a**3*b**3*c*d**2*x**3 + 8*a**3*b**3*d**3*x**5 + 5*a **2*b**4*c**3*x + 3*a**2*b**4*c**2*d*x**3 + 3*a*b**5*c**3*x**3)/(8*a**3*b* *4*(a**2 + 2*a*b*x**2 + b**2*x**4))