Integrand size = 19, antiderivative size = 131 \[ \int \frac {\left (a+c x^4\right )^2}{\left (d+e x^2\right )^2} \, dx=\frac {c \left (3 c d^2+2 a e^2\right ) x}{e^4}-\frac {2 c^2 d x^3}{3 e^3}+\frac {c^2 x^5}{5 e^2}+\frac {\left (c d^2+a e^2\right )^2 x}{2 d e^4 \left (d+e x^2\right )}-\frac {\left (7 c d^2-a e^2\right ) \left (c d^2+a e^2\right ) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} e^{9/2}} \] Output:
c*(2*a*e^2+3*c*d^2)*x/e^4-2/3*c^2*d*x^3/e^3+1/5*c^2*x^5/e^2+1/2*(a*e^2+c*d ^2)^2*x/d/e^4/(e*x^2+d)-1/2*(-a*e^2+7*c*d^2)*(a*e^2+c*d^2)*arctan(e^(1/2)* x/d^(1/2))/d^(3/2)/e^(9/2)
Time = 0.08 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.02 \[ \int \frac {\left (a+c x^4\right )^2}{\left (d+e x^2\right )^2} \, dx=\frac {c \left (3 c d^2+2 a e^2\right ) x}{e^4}-\frac {2 c^2 d x^3}{3 e^3}+\frac {c^2 x^5}{5 e^2}+\frac {\left (c d^2+a e^2\right )^2 x}{2 d e^4 \left (d+e x^2\right )}-\frac {\left (7 c^2 d^4+6 a c d^2 e^2-a^2 e^4\right ) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} e^{9/2}} \] Input:
Integrate[(a + c*x^4)^2/(d + e*x^2)^2,x]
Output:
(c*(3*c*d^2 + 2*a*e^2)*x)/e^4 - (2*c^2*d*x^3)/(3*e^3) + (c^2*x^5)/(5*e^2) + ((c*d^2 + a*e^2)^2*x)/(2*d*e^4*(d + e*x^2)) - ((7*c^2*d^4 + 6*a*c*d^2*e^ 2 - a^2*e^4)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*d^(3/2)*e^(9/2))
Time = 0.78 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.08, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {1472, 25, 2341, 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.
\(\displaystyle \int \frac {\left (a+c x^4\right )^2}{\left (d+e x^2\right )^2} \, dx\) |
\(\Big \downarrow \) 1472 |
\(\displaystyle \frac {x \left (a e^2+c d^2\right )^2}{2 d e^4 \left (d+e x^2\right )}-\frac {\int -\frac {\frac {2 c^2 d x^6}{e}-\frac {2 c^2 d^2 x^4}{e^2}+\frac {2 c d \left (c d^2+2 a e^2\right ) x^2}{e^3}+a^2-\frac {2 a c d^2}{e^2}-\frac {c^2 d^4}{e^4}}{e x^2+d}dx}{2 d}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \frac {\frac {2 c^2 d x^6}{e}-\frac {2 c^2 d^2 x^4}{e^2}+\frac {2 c d \left (c d^2+2 a e^2\right ) x^2}{e^3}+a^2-\frac {2 a c d^2}{e^2}-\frac {c^2 d^4}{e^4}}{e x^2+d}dx}{2 d}+\frac {x \left (a e^2+c d^2\right )^2}{2 d e^4 \left (d+e x^2\right )}\) |
\(\Big \downarrow \) 2341 |
\(\displaystyle \frac {\int \left (\frac {2 c^2 d x^4}{e^2}-\frac {4 c^2 d^2 x^2}{e^3}+\frac {2 c d \left (3 c d^2+2 a e^2\right )}{e^4}+\frac {-7 c^2 d^4-6 a c e^2 d^2+a^2 e^4}{e^4 \left (e x^2+d\right )}\right )dx}{2 d}+\frac {x \left (a e^2+c d^2\right )^2}{2 d e^4 \left (d+e x^2\right )}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {-\frac {\left (7 c d^2-a e^2\right ) \left (a e^2+c d^2\right ) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} e^{9/2}}+\frac {2 c d x \left (2 a e^2+3 c d^2\right )}{e^4}-\frac {4 c^2 d^2 x^3}{3 e^3}+\frac {2 c^2 d x^5}{5 e^2}}{2 d}+\frac {x \left (a e^2+c d^2\right )^2}{2 d e^4 \left (d+e x^2\right )}\) |
Input:
Int[(a + c*x^4)^2/(d + e*x^2)^2,x]
Output:
((c*d^2 + a*e^2)^2*x)/(2*d*e^4*(d + e*x^2)) + ((2*c*d*(3*c*d^2 + 2*a*e^2)* x)/e^4 - (4*c^2*d^2*x^3)/(3*e^3) + (2*c^2*d*x^5)/(5*e^2) - ((7*c*d^2 - a*e ^2)*(c*d^2 + a*e^2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(Sqrt[d]*e^(9/2)))/(2*d)
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] :> Int[ExpandIntegrand[Pq* (a + b*x^2)^p, x], x] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
Time = 0.12 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.98
method | result | size |
default | \(\frac {c \left (\frac {1}{5} c \,x^{5} e^{2}-\frac {2}{3} c d \,x^{3} e +2 x a \,e^{2}+3 x c \,d^{2}\right )}{e^{4}}+\frac {\frac {\left (a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) x}{2 d \left (e \,x^{2}+d \right )}+\frac {\left (a^{2} e^{4}-6 a c \,d^{2} e^{2}-7 c^{2} d^{4}\right ) \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 d \sqrt {d e}}}{e^{4}}\) | \(129\) |
risch | \(\frac {c^{2} x^{5}}{5 e^{2}}-\frac {2 c^{2} d \,x^{3}}{3 e^{3}}+\frac {2 c x a}{e^{2}}+\frac {3 c^{2} x \,d^{2}}{e^{4}}+\frac {\left (a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) x}{2 d \,e^{4} \left (e \,x^{2}+d \right )}-\frac {\ln \left (e x +\sqrt {-d e}\right ) a^{2}}{4 \sqrt {-d e}\, d}+\frac {3 d \ln \left (e x +\sqrt {-d e}\right ) a c}{2 e^{2} \sqrt {-d e}}+\frac {7 d^{3} \ln \left (e x +\sqrt {-d e}\right ) c^{2}}{4 e^{4} \sqrt {-d e}}+\frac {\ln \left (-e x +\sqrt {-d e}\right ) a^{2}}{4 \sqrt {-d e}\, d}-\frac {3 d \ln \left (-e x +\sqrt {-d e}\right ) a c}{2 e^{2} \sqrt {-d e}}-\frac {7 d^{3} \ln \left (-e x +\sqrt {-d e}\right ) c^{2}}{4 e^{4} \sqrt {-d e}}\) | \(247\) |
Input:
int((c*x^4+a)^2/(e*x^2+d)^2,x,method=_RETURNVERBOSE)
Output:
c/e^4*(1/5*c*x^5*e^2-2/3*c*d*x^3*e+2*x*a*e^2+3*x*c*d^2)+1/e^4*(1/2*(a^2*e^ 4+2*a*c*d^2*e^2+c^2*d^4)/d*x/(e*x^2+d)+1/2*(a^2*e^4-6*a*c*d^2*e^2-7*c^2*d^ 4)/d/(d*e)^(1/2)*arctan(e*x/(d*e)^(1/2)))
Time = 0.07 (sec) , antiderivative size = 394, normalized size of antiderivative = 3.01 \[ \int \frac {\left (a+c x^4\right )^2}{\left (d+e x^2\right )^2} \, dx=\left [\frac {12 \, c^{2} d^{2} e^{4} x^{7} - 28 \, c^{2} d^{3} e^{3} x^{5} + 20 \, {\left (7 \, c^{2} d^{4} e^{2} + 6 \, a c d^{2} e^{4}\right )} x^{3} + 15 \, {\left (7 \, c^{2} d^{5} + 6 \, a c d^{3} e^{2} - a^{2} d e^{4} + {\left (7 \, c^{2} d^{4} e + 6 \, a c d^{2} e^{3} - a^{2} e^{5}\right )} x^{2}\right )} \sqrt {-d e} \log \left (\frac {e x^{2} - 2 \, \sqrt {-d e} x - d}{e x^{2} + d}\right ) + 30 \, {\left (7 \, c^{2} d^{5} e + 6 \, a c d^{3} e^{3} + a^{2} d e^{5}\right )} x}{60 \, {\left (d^{2} e^{6} x^{2} + d^{3} e^{5}\right )}}, \frac {6 \, c^{2} d^{2} e^{4} x^{7} - 14 \, c^{2} d^{3} e^{3} x^{5} + 10 \, {\left (7 \, c^{2} d^{4} e^{2} + 6 \, a c d^{2} e^{4}\right )} x^{3} - 15 \, {\left (7 \, c^{2} d^{5} + 6 \, a c d^{3} e^{2} - a^{2} d e^{4} + {\left (7 \, c^{2} d^{4} e + 6 \, a c d^{2} e^{3} - a^{2} e^{5}\right )} x^{2}\right )} \sqrt {d e} \arctan \left (\frac {\sqrt {d e} x}{d}\right ) + 15 \, {\left (7 \, c^{2} d^{5} e + 6 \, a c d^{3} e^{3} + a^{2} d e^{5}\right )} x}{30 \, {\left (d^{2} e^{6} x^{2} + d^{3} e^{5}\right )}}\right ] \] Input:
integrate((c*x^4+a)^2/(e*x^2+d)^2,x, algorithm="fricas")
Output:
[1/60*(12*c^2*d^2*e^4*x^7 - 28*c^2*d^3*e^3*x^5 + 20*(7*c^2*d^4*e^2 + 6*a*c *d^2*e^4)*x^3 + 15*(7*c^2*d^5 + 6*a*c*d^3*e^2 - a^2*d*e^4 + (7*c^2*d^4*e + 6*a*c*d^2*e^3 - a^2*e^5)*x^2)*sqrt(-d*e)*log((e*x^2 - 2*sqrt(-d*e)*x - d) /(e*x^2 + d)) + 30*(7*c^2*d^5*e + 6*a*c*d^3*e^3 + a^2*d*e^5)*x)/(d^2*e^6*x ^2 + d^3*e^5), 1/30*(6*c^2*d^2*e^4*x^7 - 14*c^2*d^3*e^3*x^5 + 10*(7*c^2*d^ 4*e^2 + 6*a*c*d^2*e^4)*x^3 - 15*(7*c^2*d^5 + 6*a*c*d^3*e^2 - a^2*d*e^4 + ( 7*c^2*d^4*e + 6*a*c*d^2*e^3 - a^2*e^5)*x^2)*sqrt(d*e)*arctan(sqrt(d*e)*x/d ) + 15*(7*c^2*d^5*e + 6*a*c*d^3*e^3 + a^2*d*e^5)*x)/(d^2*e^6*x^2 + d^3*e^5 )]
Leaf count of result is larger than twice the leaf count of optimal. 314 vs. \(2 (122) = 244\).
Time = 0.43 (sec) , antiderivative size = 314, normalized size of antiderivative = 2.40 \[ \int \frac {\left (a+c x^4\right )^2}{\left (d+e x^2\right )^2} \, dx=- \frac {2 c^{2} d x^{3}}{3 e^{3}} + \frac {c^{2} x^{5}}{5 e^{2}} + x \left (\frac {2 a c}{e^{2}} + \frac {3 c^{2} d^{2}}{e^{4}}\right ) + \frac {x \left (a^{2} e^{4} + 2 a c d^{2} e^{2} + c^{2} d^{4}\right )}{2 d^{2} e^{4} + 2 d e^{5} x^{2}} - \frac {\sqrt {- \frac {1}{d^{3} e^{9}}} \left (a e^{2} - 7 c d^{2}\right ) \left (a e^{2} + c d^{2}\right ) \log {\left (- \frac {d^{2} e^{4} \sqrt {- \frac {1}{d^{3} e^{9}}} \left (a e^{2} - 7 c d^{2}\right ) \left (a e^{2} + c d^{2}\right )}{a^{2} e^{4} - 6 a c d^{2} e^{2} - 7 c^{2} d^{4}} + x \right )}}{4} + \frac {\sqrt {- \frac {1}{d^{3} e^{9}}} \left (a e^{2} - 7 c d^{2}\right ) \left (a e^{2} + c d^{2}\right ) \log {\left (\frac {d^{2} e^{4} \sqrt {- \frac {1}{d^{3} e^{9}}} \left (a e^{2} - 7 c d^{2}\right ) \left (a e^{2} + c d^{2}\right )}{a^{2} e^{4} - 6 a c d^{2} e^{2} - 7 c^{2} d^{4}} + x \right )}}{4} \] Input:
integrate((c*x**4+a)**2/(e*x**2+d)**2,x)
Output:
-2*c**2*d*x**3/(3*e**3) + c**2*x**5/(5*e**2) + x*(2*a*c/e**2 + 3*c**2*d**2 /e**4) + x*(a**2*e**4 + 2*a*c*d**2*e**2 + c**2*d**4)/(2*d**2*e**4 + 2*d*e* *5*x**2) - sqrt(-1/(d**3*e**9))*(a*e**2 - 7*c*d**2)*(a*e**2 + c*d**2)*log( -d**2*e**4*sqrt(-1/(d**3*e**9))*(a*e**2 - 7*c*d**2)*(a*e**2 + c*d**2)/(a** 2*e**4 - 6*a*c*d**2*e**2 - 7*c**2*d**4) + x)/4 + sqrt(-1/(d**3*e**9))*(a*e **2 - 7*c*d**2)*(a*e**2 + c*d**2)*log(d**2*e**4*sqrt(-1/(d**3*e**9))*(a*e* *2 - 7*c*d**2)*(a*e**2 + c*d**2)/(a**2*e**4 - 6*a*c*d**2*e**2 - 7*c**2*d** 4) + x)/4
Exception generated. \[ \int \frac {\left (a+c x^4\right )^2}{\left (d+e x^2\right )^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((c*x^4+a)^2/(e*x^2+d)^2,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 = 0.15 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.10 \[ \int \frac {\left (a+c x^4\right )^2}{\left (d+e x^2\right )^2} \, dx=-\frac {{\left (7 \, c^{2} d^{4} + 6 \, a c d^{2} e^{2} - a^{2} e^{4}\right )} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 \, \sqrt {d e} d e^{4}} + \frac {c^{2} d^{4} x + 2 \, a c d^{2} e^{2} x + a^{2} e^{4} x}{2 \, {\left (e x^{2} + d\right )} d e^{4}} + \frac {3 \, c^{2} e^{8} x^{5} - 10 \, c^{2} d e^{7} x^{3} + 45 \, c^{2} d^{2} e^{6} x + 30 \, a c e^{8} x}{15 \, e^{10}} \] Input:
integrate((c*x^4+a)^2/(e*x^2+d)^2,x, algorithm="giac")
Output:
-1/2*(7*c^2*d^4 + 6*a*c*d^2*e^2 - a^2*e^4)*arctan(e*x/sqrt(d*e))/(sqrt(d*e )*d*e^4) + 1/2*(c^2*d^4*x + 2*a*c*d^2*e^2*x + a^2*e^4*x)/((e*x^2 + d)*d*e^ 4) + 1/15*(3*c^2*e^8*x^5 - 10*c^2*d*e^7*x^3 + 45*c^2*d^2*e^6*x + 30*a*c*e^ 8*x)/e^10
Time = 16.94 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.40 \[ \int \frac {\left (a+c x^4\right )^2}{\left (d+e x^2\right )^2} \, dx=x\,\left (\frac {3\,c^2\,d^2}{e^4}+\frac {2\,a\,c}{e^2}\right )+\frac {c^2\,x^5}{5\,e^2}-\frac {2\,c^2\,d\,x^3}{3\,e^3}+\frac {x\,\left (a^2\,e^4+2\,a\,c\,d^2\,e^2+c^2\,d^4\right )}{2\,d\,\left (e^5\,x^2+d\,e^4\right )}-\frac {\mathrm {atan}\left (\frac {\sqrt {e}\,x\,\left (c\,d^2+a\,e^2\right )\,\left (a\,e^2-7\,c\,d^2\right )}{\sqrt {d}\,\left (-a^2\,e^4+6\,a\,c\,d^2\,e^2+7\,c^2\,d^4\right )}\right )\,\left (c\,d^2+a\,e^2\right )\,\left (a\,e^2-7\,c\,d^2\right )}{2\,d^{3/2}\,e^{9/2}} \] Input:
int((a + c*x^4)^2/(d + e*x^2)^2,x)
Output:
x*((3*c^2*d^2)/e^4 + (2*a*c)/e^2) + (c^2*x^5)/(5*e^2) - (2*c^2*d*x^3)/(3*e ^3) + (x*(a^2*e^4 + c^2*d^4 + 2*a*c*d^2*e^2))/(2*d*(d*e^4 + e^5*x^2)) - (a tan((e^(1/2)*x*(a*e^2 + c*d^2)*(a*e^2 - 7*c*d^2))/(d^(1/2)*(7*c^2*d^4 - a^ 2*e^4 + 6*a*c*d^2*e^2)))*(a*e^2 + c*d^2)*(a*e^2 - 7*c*d^2))/(2*d^(3/2)*e^( 9/2))
Time = 0.17 (sec) , antiderivative size = 263, normalized size of antiderivative = 2.01 \[ \int \frac {\left (a+c x^4\right )^2}{\left (d+e x^2\right )^2} \, dx=\frac {15 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) a^{2} d \,e^{4}+15 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) a^{2} e^{5} x^{2}-90 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) a c \,d^{3} e^{2}-90 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) a c \,d^{2} e^{3} x^{2}-105 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) c^{2} d^{5}-105 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) c^{2} d^{4} e \,x^{2}+15 a^{2} d \,e^{5} x +90 a c \,d^{3} e^{3} x +60 a c \,d^{2} e^{4} x^{3}+105 c^{2} d^{5} e x +70 c^{2} d^{4} e^{2} x^{3}-14 c^{2} d^{3} e^{3} x^{5}+6 c^{2} d^{2} e^{4} x^{7}}{30 d^{2} e^{5} \left (e \,x^{2}+d \right )} \] Input:
int((c*x^4+a)^2/(e*x^2+d)^2,x)
Output:
(15*sqrt(e)*sqrt(d)*atan((e*x)/(sqrt(e)*sqrt(d)))*a**2*d*e**4 + 15*sqrt(e) *sqrt(d)*atan((e*x)/(sqrt(e)*sqrt(d)))*a**2*e**5*x**2 - 90*sqrt(e)*sqrt(d) *atan((e*x)/(sqrt(e)*sqrt(d)))*a*c*d**3*e**2 - 90*sqrt(e)*sqrt(d)*atan((e* x)/(sqrt(e)*sqrt(d)))*a*c*d**2*e**3*x**2 - 105*sqrt(e)*sqrt(d)*atan((e*x)/ (sqrt(e)*sqrt(d)))*c**2*d**5 - 105*sqrt(e)*sqrt(d)*atan((e*x)/(sqrt(e)*sqr t(d)))*c**2*d**4*e*x**2 + 15*a**2*d*e**5*x + 90*a*c*d**3*e**3*x + 60*a*c*d **2*e**4*x**3 + 105*c**2*d**5*e*x + 70*c**2*d**4*e**2*x**3 - 14*c**2*d**3* e**3*x**5 + 6*c**2*d**2*e**4*x**7)/(30*d**2*e**5*(d + e*x**2))