Integrand size = 19, antiderivative size = 155 \[ \int \frac {\left (a+c x^4\right )^2}{\left (d+e x^2\right )^3} \, dx=-\frac {3 c^2 d x}{e^4}+\frac {c^2 x^3}{3 e^3}+\frac {\left (c d^2+a e^2\right )^2 x}{4 d e^4 \left (d+e x^2\right )^2}-\frac {\left (13 c d^2-3 a e^2\right ) \left (c d^2+a e^2\right ) x}{8 d^2 e^4 \left (d+e x^2\right )}+\frac {\left (35 c^2 d^4+6 a c d^2 e^2+3 a^2 e^4\right ) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{5/2} e^{9/2}} \] Output:
-3*c^2*d*x/e^4+1/3*c^2*x^3/e^3+1/4*(a*e^2+c*d^2)^2*x/d/e^4/(e*x^2+d)^2-1/8 *(-3*a*e^2+13*c*d^2)*(a*e^2+c*d^2)*x/d^2/e^4/(e*x^2+d)+1/8*(3*a^2*e^4+6*a* c*d^2*e^2+35*c^2*d^4)*arctan(e^(1/2)*x/d^(1/2))/d^(5/2)/e^(9/2)
Time = 0.08 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.99 \[ \int \frac {\left (a+c x^4\right )^2}{\left (d+e x^2\right )^3} \, dx=\frac {x \left (3 a^2 e^4 \left (5 d+3 e x^2\right )-6 a c d^2 e^2 \left (3 d+5 e x^2\right )-c^2 d^2 \left (105 d^3+175 d^2 e x^2+56 d e^2 x^4-8 e^3 x^6\right )\right )}{24 d^2 e^4 \left (d+e x^2\right )^2}+\frac {\left (35 c^2 d^4+6 a c d^2 e^2+3 a^2 e^4\right ) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{5/2} e^{9/2}} \] Input:
Integrate[(a + c*x^4)^2/(d + e*x^2)^3,x]
Output:
(x*(3*a^2*e^4*(5*d + 3*e*x^2) - 6*a*c*d^2*e^2*(3*d + 5*e*x^2) - c^2*d^2*(1 05*d^3 + 175*d^2*e*x^2 + 56*d*e^2*x^4 - 8*e^3*x^6)))/(24*d^2*e^4*(d + e*x^ 2)^2) + ((35*c^2*d^4 + 6*a*c*d^2*e^2 + 3*a^2*e^4)*ArcTan[(Sqrt[e]*x)/Sqrt[ d]])/(8*d^(5/2)*e^(9/2))
Time = 0.94 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.12, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {1472, 25, 2345, 25, 1467, 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.
| \(\displaystyle \int \frac {\left (a+c x^4\right )^2}{\left (d+e x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 1472 |
| \(\displaystyle \frac {x \left (a e^2+c d^2\right )^2}{4 d e^4 \left (d+e x^2\right )^2}-\frac {\int -\frac {\frac {4 c^2 d x^6}{e}-\frac {4 c^2 d^2 x^4}{e^2}+\frac {4 c d \left (c d^2+2 a e^2\right ) x^2}{e^3}+3 a^2-\frac {2 a c d^2}{e^2}-\frac {c^2 d^4}{e^4}}{\left (e x^2+d\right )^2}dx}{4 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\frac {4 c^2 d x^6}{e}-\frac {4 c^2 d^2 x^4}{e^2}+\frac {4 c d \left (c d^2+2 a e^2\right ) x^2}{e^3}+3 a^2-\frac {2 a c d^2}{e^2}-\frac {c^2 d^4}{e^4}}{\left (e x^2+d\right )^2}dx}{4 d}+\frac {x \left (a e^2+c d^2\right )^2}{4 d e^4 \left (d+e x^2\right )^2}\) |
| \(\Big \downarrow \) 2345 |
| \(\displaystyle \frac {\frac {x \left (3 a^2-\frac {10 a c d^2}{e^2}-\frac {13 c^2 d^4}{e^4}\right )}{2 d \left (d+e x^2\right )}-\frac {\int -\frac {\frac {11 c^2 d^4}{e^4}-\frac {16 c^2 x^2 d^3}{e^3}+\frac {8 c^2 x^4 d^2}{e^2}+\frac {6 a c d^2}{e^2}+3 a^2}{e x^2+d}dx}{2 d}}{4 d}+\frac {x \left (a e^2+c d^2\right )^2}{4 d e^4 \left (d+e x^2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {\frac {11 c^2 d^4}{e^4}-\frac {16 c^2 x^2 d^3}{e^3}+\frac {8 c^2 x^4 d^2}{e^2}+\frac {6 a c d^2}{e^2}+3 a^2}{e x^2+d}dx}{2 d}+\frac {x \left (3 a^2-\frac {10 a c d^2}{e^2}-\frac {13 c^2 d^4}{e^4}\right )}{2 d \left (d+e x^2\right )}}{4 d}+\frac {x \left (a e^2+c d^2\right )^2}{4 d e^4 \left (d+e x^2\right )^2}\) |
| \(\Big \downarrow \) 1467 |
| \(\displaystyle \frac {\frac {\int \left (-\frac {24 c^2 d^3}{e^4}+\frac {8 c^2 x^2 d^2}{e^3}+\frac {35 c^2 d^4+6 a c e^2 d^2+3 a^2 e^4}{e^4 \left (e x^2+d\right )}\right )dx}{2 d}+\frac {x \left (3 a^2-\frac {10 a c d^2}{e^2}-\frac {13 c^2 d^4}{e^4}\right )}{2 d \left (d+e x^2\right )}}{4 d}+\frac {x \left (a e^2+c d^2\right )^2}{4 d e^4 \left (d+e x^2\right )^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {\frac {\left (3 a^2 e^4+6 a c d^2 e^2+35 c^2 d^4\right ) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} e^{9/2}}-\frac {24 c^2 d^3 x}{e^4}+\frac {8 c^2 d^2 x^3}{3 e^3}}{2 d}+\frac {x \left (3 a^2-\frac {10 a c d^2}{e^2}-\frac {13 c^2 d^4}{e^4}\right )}{2 d \left (d+e x^2\right )}}{4 d}+\frac {x \left (a e^2+c d^2\right )^2}{4 d e^4 \left (d+e x^2\right )^2}\) |
Input:
Int[(a + c*x^4)^2/(d + e*x^2)^3,x]
Output:
((c*d^2 + a*e^2)^2*x)/(4*d*e^4*(d + e*x^2)^2) + (((3*a^2 - (13*c^2*d^4)/e^ 4 - (10*a*c*d^2)/e^2)*x)/(2*d*(d + e*x^2)) + ((-24*c^2*d^3*x)/e^4 + (8*c^2 *d^2*x^3)/(3*e^3) + ((35*c^2*d^4 + 6*a*c*d^2*e^2 + 3*a^2*e^4)*ArcTan[(Sqrt [e]*x)/Sqrt[d]])/(Sqrt[d]*e^(9/2)))/(2*d))/(4*d)
Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]
Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Wi th[{Qx = PolynomialQuotient[(a + c*x^4)^p, d + e*x^2, x], R = Coeff[Polynom ialRemainder[(a + c*x^4)^p, d + e*x^2, x], x, 0]}, Simp[(-R)*x*((d + e*x^2) ^(q + 1)/(2*d*(q + 1))), x] + Simp[1/(2*d*(q + 1)) Int[(d + e*x^2)^(q + 1 )*ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), x], x], x]] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -1]
Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuot ient[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[(a*g - b *f*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1)) In t[(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]
Time = 0.13 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.98
| method | result | size |
| default | \(-\frac {c^{2} \left (-\frac {1}{3} e \,x^{3}+3 d x \right )}{e^{4}}+\frac {\frac {\frac {e \left (3 a^{2} e^{4}-10 a c \,d^{2} e^{2}-13 c^{2} d^{4}\right ) x^{3}}{8 d^{2}}+\frac {\left (5 a^{2} e^{4}-6 a c \,d^{2} e^{2}-11 c^{2} d^{4}\right ) x}{8 d}}{\left (e \,x^{2}+d \right )^{2}}+\frac {\left (3 a^{2} e^{4}+6 a c \,d^{2} e^{2}+35 c^{2} d^{4}\right ) \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{8 d^{2} \sqrt {d e}}}{e^{4}}\) | \(152\) |
| risch | \(\frac {c^{2} x^{3}}{3 e^{3}}-\frac {3 c^{2} d x}{e^{4}}+\frac {\frac {e \left (3 a^{2} e^{4}-10 a c \,d^{2} e^{2}-13 c^{2} d^{4}\right ) x^{3}}{8 d^{2}}+\frac {\left (5 a^{2} e^{4}-6 a c \,d^{2} e^{2}-11 c^{2} d^{4}\right ) x}{8 d}}{e^{4} \left (e \,x^{2}+d \right )^{2}}-\frac {3 \ln \left (e x +\sqrt {-d e}\right ) a^{2}}{16 \sqrt {-d e}\, d^{2}}-\frac {3 \ln \left (e x +\sqrt {-d e}\right ) a c}{8 e^{2} \sqrt {-d e}}-\frac {35 d^{2} \ln \left (e x +\sqrt {-d e}\right ) c^{2}}{16 e^{4} \sqrt {-d e}}+\frac {3 \ln \left (-e x +\sqrt {-d e}\right ) a^{2}}{16 \sqrt {-d e}\, d^{2}}+\frac {3 \ln \left (-e x +\sqrt {-d e}\right ) a c}{8 e^{2} \sqrt {-d e}}+\frac {35 d^{2} \ln \left (-e x +\sqrt {-d e}\right ) c^{2}}{16 e^{4} \sqrt {-d e}}\) | \(263\) |
Input:
int((c*x^4+a)^2/(e*x^2+d)^3,x,method=_RETURNVERBOSE)
Output:
-c^2/e^4*(-1/3*e*x^3+3*d*x)+1/e^4*((1/8*e*(3*a^2*e^4-10*a*c*d^2*e^2-13*c^2 *d^4)/d^2*x^3+1/8*(5*a^2*e^4-6*a*c*d^2*e^2-11*c^2*d^4)/d*x)/(e*x^2+d)^2+1/ 8*(3*a^2*e^4+6*a*c*d^2*e^2+35*c^2*d^4)/d^2/(d*e)^(1/2)*arctan(e*x/(d*e)^(1 /2)))
Time = 0.08 (sec) , antiderivative size = 516, normalized size of antiderivative = 3.33 \[ \int \frac {\left (a+c x^4\right )^2}{\left (d+e x^2\right )^3} \, dx=\left [\frac {16 \, c^{2} d^{3} e^{4} x^{7} - 112 \, c^{2} d^{4} e^{3} x^{5} - 2 \, {\left (175 \, c^{2} d^{5} e^{2} + 30 \, a c d^{3} e^{4} - 9 \, a^{2} d e^{6}\right )} x^{3} - 3 \, {\left (35 \, c^{2} d^{6} + 6 \, a c d^{4} e^{2} + 3 \, a^{2} d^{2} e^{4} + {\left (35 \, c^{2} d^{4} e^{2} + 6 \, a c d^{2} e^{4} + 3 \, a^{2} e^{6}\right )} x^{4} + 2 \, {\left (35 \, c^{2} d^{5} e + 6 \, a c d^{3} e^{3} + 3 \, a^{2} d e^{5}\right )} x^{2}\right )} \sqrt {-d e} \log \left (\frac {e x^{2} - 2 \, \sqrt {-d e} x - d}{e x^{2} + d}\right ) - 6 \, {\left (35 \, c^{2} d^{6} e + 6 \, a c d^{4} e^{3} - 5 \, a^{2} d^{2} e^{5}\right )} x}{48 \, {\left (d^{3} e^{7} x^{4} + 2 \, d^{4} e^{6} x^{2} + d^{5} e^{5}\right )}}, \frac {8 \, c^{2} d^{3} e^{4} x^{7} - 56 \, c^{2} d^{4} e^{3} x^{5} - {\left (175 \, c^{2} d^{5} e^{2} + 30 \, a c d^{3} e^{4} - 9 \, a^{2} d e^{6}\right )} x^{3} + 3 \, {\left (35 \, c^{2} d^{6} + 6 \, a c d^{4} e^{2} + 3 \, a^{2} d^{2} e^{4} + {\left (35 \, c^{2} d^{4} e^{2} + 6 \, a c d^{2} e^{4} + 3 \, a^{2} e^{6}\right )} x^{4} + 2 \, {\left (35 \, c^{2} d^{5} e + 6 \, a c d^{3} e^{3} + 3 \, a^{2} d e^{5}\right )} x^{2}\right )} \sqrt {d e} \arctan \left (\frac {\sqrt {d e} x}{d}\right ) - 3 \, {\left (35 \, c^{2} d^{6} e + 6 \, a c d^{4} e^{3} - 5 \, a^{2} d^{2} e^{5}\right )} x}{24 \, {\left (d^{3} e^{7} x^{4} + 2 \, d^{4} e^{6} x^{2} + d^{5} e^{5}\right )}}\right ] \] Input:
integrate((c*x^4+a)^2/(e*x^2+d)^3,x, algorithm="fricas")
Output:
[1/48*(16*c^2*d^3*e^4*x^7 - 112*c^2*d^4*e^3*x^5 - 2*(175*c^2*d^5*e^2 + 30* a*c*d^3*e^4 - 9*a^2*d*e^6)*x^3 - 3*(35*c^2*d^6 + 6*a*c*d^4*e^2 + 3*a^2*d^2 *e^4 + (35*c^2*d^4*e^2 + 6*a*c*d^2*e^4 + 3*a^2*e^6)*x^4 + 2*(35*c^2*d^5*e + 6*a*c*d^3*e^3 + 3*a^2*d*e^5)*x^2)*sqrt(-d*e)*log((e*x^2 - 2*sqrt(-d*e)*x - d)/(e*x^2 + d)) - 6*(35*c^2*d^6*e + 6*a*c*d^4*e^3 - 5*a^2*d^2*e^5)*x)/( d^3*e^7*x^4 + 2*d^4*e^6*x^2 + d^5*e^5), 1/24*(8*c^2*d^3*e^4*x^7 - 56*c^2*d ^4*e^3*x^5 - (175*c^2*d^5*e^2 + 30*a*c*d^3*e^4 - 9*a^2*d*e^6)*x^3 + 3*(35* c^2*d^6 + 6*a*c*d^4*e^2 + 3*a^2*d^2*e^4 + (35*c^2*d^4*e^2 + 6*a*c*d^2*e^4 + 3*a^2*e^6)*x^4 + 2*(35*c^2*d^5*e + 6*a*c*d^3*e^3 + 3*a^2*d*e^5)*x^2)*sqr t(d*e)*arctan(sqrt(d*e)*x/d) - 3*(35*c^2*d^6*e + 6*a*c*d^4*e^3 - 5*a^2*d^2 *e^5)*x)/(d^3*e^7*x^4 + 2*d^4*e^6*x^2 + d^5*e^5)]
Time = 0.75 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.66 \[ \int \frac {\left (a+c x^4\right )^2}{\left (d+e x^2\right )^3} \, dx=- \frac {3 c^{2} d x}{e^{4}} + \frac {c^{2} x^{3}}{3 e^{3}} - \frac {\sqrt {- \frac {1}{d^{5} e^{9}}} \cdot \left (3 a^{2} e^{4} + 6 a c d^{2} e^{2} + 35 c^{2} d^{4}\right ) \log {\left (- d^{3} e^{4} \sqrt {- \frac {1}{d^{5} e^{9}}} + x \right )}}{16} + \frac {\sqrt {- \frac {1}{d^{5} e^{9}}} \cdot \left (3 a^{2} e^{4} + 6 a c d^{2} e^{2} + 35 c^{2} d^{4}\right ) \log {\left (d^{3} e^{4} \sqrt {- \frac {1}{d^{5} e^{9}}} + x \right )}}{16} + \frac {x^{3} \cdot \left (3 a^{2} e^{5} - 10 a c d^{2} e^{3} - 13 c^{2} d^{4} e\right ) + x \left (5 a^{2} d e^{4} - 6 a c d^{3} e^{2} - 11 c^{2} d^{5}\right )}{8 d^{4} e^{4} + 16 d^{3} e^{5} x^{2} + 8 d^{2} e^{6} x^{4}} \] Input:
integrate((c*x**4+a)**2/(e*x**2+d)**3,x)
Output:
-3*c**2*d*x/e**4 + c**2*x**3/(3*e**3) - sqrt(-1/(d**5*e**9))*(3*a**2*e**4 + 6*a*c*d**2*e**2 + 35*c**2*d**4)*log(-d**3*e**4*sqrt(-1/(d**5*e**9)) + x) /16 + sqrt(-1/(d**5*e**9))*(3*a**2*e**4 + 6*a*c*d**2*e**2 + 35*c**2*d**4)* log(d**3*e**4*sqrt(-1/(d**5*e**9)) + x)/16 + (x**3*(3*a**2*e**5 - 10*a*c*d **2*e**3 - 13*c**2*d**4*e) + x*(5*a**2*d*e**4 - 6*a*c*d**3*e**2 - 11*c**2* d**5))/(8*d**4*e**4 + 16*d**3*e**5*x**2 + 8*d**2*e**6*x**4)
Exception generated. \[ \int \frac {\left (a+c x^4\right )^2}{\left (d+e x^2\right )^3} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((c*x^4+a)^2/(e*x^2+d)^3,x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
Time = 0.11 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.03 \[ \int \frac {\left (a+c x^4\right )^2}{\left (d+e x^2\right )^3} \, dx=\frac {{\left (35 \, c^{2} d^{4} + 6 \, a c d^{2} e^{2} + 3 \, a^{2} e^{4}\right )} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{8 \, \sqrt {d e} d^{2} e^{4}} - \frac {13 \, c^{2} d^{4} e x^{3} + 10 \, a c d^{2} e^{3} x^{3} - 3 \, a^{2} e^{5} x^{3} + 11 \, c^{2} d^{5} x + 6 \, a c d^{3} e^{2} x - 5 \, a^{2} d e^{4} x}{8 \, {\left (e x^{2} + d\right )}^{2} d^{2} e^{4}} + \frac {c^{2} e^{6} x^{3} - 9 \, c^{2} d e^{5} x}{3 \, e^{9}} \] Input:
integrate((c*x^4+a)^2/(e*x^2+d)^3,x, algorithm="giac")
Output:
1/8*(35*c^2*d^4 + 6*a*c*d^2*e^2 + 3*a^2*e^4)*arctan(e*x/sqrt(d*e))/(sqrt(d *e)*d^2*e^4) - 1/8*(13*c^2*d^4*e*x^3 + 10*a*c*d^2*e^3*x^3 - 3*a^2*e^5*x^3 + 11*c^2*d^5*x + 6*a*c*d^3*e^2*x - 5*a^2*d*e^4*x)/((e*x^2 + d)^2*d^2*e^4) + 1/3*(c^2*e^6*x^3 - 9*c^2*d*e^5*x)/e^9
Time = 0.10 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.06 \[ \int \frac {\left (a+c x^4\right )^2}{\left (d+e x^2\right )^3} \, dx=\frac {c^2\,x^3}{3\,e^3}-\frac {\frac {x^3\,\left (-3\,a^2\,e^5+10\,a\,c\,d^2\,e^3+13\,c^2\,d^4\,e\right )}{8\,d^2}+\frac {x\,\left (-5\,a^2\,e^4+6\,a\,c\,d^2\,e^2+11\,c^2\,d^4\right )}{8\,d}}{d^2\,e^4+2\,d\,e^5\,x^2+e^6\,x^4}+\frac {\mathrm {atan}\left (\frac {\sqrt {e}\,x}{\sqrt {d}}\right )\,\left (3\,a^2\,e^4+6\,a\,c\,d^2\,e^2+35\,c^2\,d^4\right )}{8\,d^{5/2}\,e^{9/2}}-\frac {3\,c^2\,d\,x}{e^4} \] Input:
int((a + c*x^4)^2/(d + e*x^2)^3,x)
Output:
(c^2*x^3)/(3*e^3) - ((x^3*(13*c^2*d^4*e - 3*a^2*e^5 + 10*a*c*d^2*e^3))/(8* d^2) + (x*(11*c^2*d^4 - 5*a^2*e^4 + 6*a*c*d^2*e^2))/(8*d))/(d^2*e^4 + e^6* x^4 + 2*d*e^5*x^2) + (atan((e^(1/2)*x)/d^(1/2))*(3*a^2*e^4 + 35*c^2*d^4 + 6*a*c*d^2*e^2))/(8*d^(5/2)*e^(9/2)) - (3*c^2*d*x)/e^4
Time = 0.18 (sec) , antiderivative size = 377, normalized size of antiderivative = 2.43 \[ \int \frac {\left (a+c x^4\right )^2}{\left (d+e x^2\right )^3} \, dx=\frac {9 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) a^{2} d^{2} e^{4}+18 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) a^{2} d \,e^{5} x^{2}+9 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) a^{2} e^{6} x^{4}+18 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) a c \,d^{4} e^{2}+36 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) a c \,d^{3} e^{3} x^{2}+18 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) a c \,d^{2} e^{4} x^{4}+105 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) c^{2} d^{6}+210 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) c^{2} d^{5} e \,x^{2}+105 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) c^{2} d^{4} e^{2} x^{4}+15 a^{2} d^{2} e^{5} x +9 a^{2} d \,e^{6} x^{3}-18 a c \,d^{4} e^{3} x -30 a c \,d^{3} e^{4} x^{3}-105 c^{2} d^{6} e x -175 c^{2} d^{5} e^{2} x^{3}-56 c^{2} d^{4} e^{3} x^{5}+8 c^{2} d^{3} e^{4} x^{7}}{24 d^{3} e^{5} \left (e^{2} x^{4}+2 d e \,x^{2}+d^{2}\right )} \] Input:
int((c*x^4+a)^2/(e*x^2+d)^3,x)
Output:
(9*sqrt(e)*sqrt(d)*atan((e*x)/(sqrt(e)*sqrt(d)))*a**2*d**2*e**4 + 18*sqrt( e)*sqrt(d)*atan((e*x)/(sqrt(e)*sqrt(d)))*a**2*d*e**5*x**2 + 9*sqrt(e)*sqrt (d)*atan((e*x)/(sqrt(e)*sqrt(d)))*a**2*e**6*x**4 + 18*sqrt(e)*sqrt(d)*atan ((e*x)/(sqrt(e)*sqrt(d)))*a*c*d**4*e**2 + 36*sqrt(e)*sqrt(d)*atan((e*x)/(s qrt(e)*sqrt(d)))*a*c*d**3*e**3*x**2 + 18*sqrt(e)*sqrt(d)*atan((e*x)/(sqrt( e)*sqrt(d)))*a*c*d**2*e**4*x**4 + 105*sqrt(e)*sqrt(d)*atan((e*x)/(sqrt(e)* sqrt(d)))*c**2*d**6 + 210*sqrt(e)*sqrt(d)*atan((e*x)/(sqrt(e)*sqrt(d)))*c* *2*d**5*e*x**2 + 105*sqrt(e)*sqrt(d)*atan((e*x)/(sqrt(e)*sqrt(d)))*c**2*d* *4*e**2*x**4 + 15*a**2*d**2*e**5*x + 9*a**2*d*e**6*x**3 - 18*a*c*d**4*e**3 *x - 30*a*c*d**3*e**4*x**3 - 105*c**2*d**6*e*x - 175*c**2*d**5*e**2*x**3 - 56*c**2*d**4*e**3*x**5 + 8*c**2*d**3*e**4*x**7)/(24*d**3*e**5*(d**2 + 2*d *e*x**2 + e**2*x**4))