\(\int \frac {(a+b x^2+c x^4)^2}{d+e x^2} \, dx\) [256]

Optimal result

Integrand size = 24, antiderivative size = 143 \[ \int \frac {\left (a+b x^2+c x^4\right )^2}{d+e x^2} \, dx=-\frac {(c d-b e) \left (c d^2-e (b d-2 a e)\right ) x}{e^4}+\frac {\left (c^2 d^2+b^2 e^2-2 c e (b d-a e)\right ) x^3}{3 e^3}-\frac {c (c d-2 b e) x^5}{5 e^2}+\frac {c^2 x^7}{7 e}+\frac {\left (c d^2-b d e+a e^2\right )^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} e^{9/2}} \]

[Out]

-(-b*e+c*d)*(c*d^2-e*(-2*a*e+b*d))*x/e^4+1/3*(c^2*d^2+b^2*e^2-2*c*e*(-a*e+b*d))*x^3/e^3-1/5*c*(-2*b*e+c*d)*x^5
/e^2+1/7*c^2*x^7/e+(a*e^2-b*d*e+c*d^2)^2*arctan(x*e^(1/2)/d^(1/2))/e^(9/2)/d^(1/2)

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1167, 211} \[ \int \frac {\left (a+b x^2+c x^4\right )^2}{d+e x^2} \, dx=\frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a e^2-b d e+c d^2\right )^2}{\sqrt {d} e^{9/2}}+\frac {x^3 \left (-2 c e (b d-a e)+b^2 e^2+c^2 d^2\right )}{3 e^3}-\frac {x (c d-b e) \left (c d^2-e (b d-2 a e)\right )}{e^4}-\frac {c x^5 (c d-2 b e)}{5 e^2}+\frac {c^2 x^7}{7 e} \]

[In]

Int[(a + b*x^2 + c*x^4)^2/(d + e*x^2),x]

[Out]

-(((c*d - b*e)*(c*d^2 - e*(b*d - 2*a*e))*x)/e^4) + ((c^2*d^2 + b^2*e^2 - 2*c*e*(b*d - a*e))*x^3)/(3*e^3) - (c*
(c*d - 2*b*e)*x^5)/(5*e^2) + (c^2*x^7)/(7*e) + ((c*d^2 - b*d*e + a*e^2)^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(Sqrt[d
]*e^(9/2))

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 1167

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]

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {(c d-b e) \left (c d^2-e (b d-2 a e)\right )}{e^4}+\frac {\left (c^2 d^2+b^2 e^2-2 c e (b d-a e)\right ) x^2}{e^3}-\frac {c (c d-2 b e) x^4}{e^2}+\frac {c^2 x^6}{e}+\frac {c^2 d^4-2 b c d^3 e+b^2 d^2 e^2+2 a c d^2 e^2-2 a b d e^3+a^2 e^4}{e^4 \left (d+e x^2\right )}\right ) \, dx \\ & = -\frac {(c d-b e) \left (c d^2-e (b d-2 a e)\right ) x}{e^4}+\frac {\left (c^2 d^2+b^2 e^2-2 c e (b d-a e)\right ) x^3}{3 e^3}-\frac {c (c d-2 b e) x^5}{5 e^2}+\frac {c^2 x^7}{7 e}+\frac {\left (c d^2-b d e+a e^2\right )^2 \int \frac {1}{d+e x^2} \, dx}{e^4} \\ & = -\frac {(c d-b e) \left (c d^2-e (b d-2 a e)\right ) x}{e^4}+\frac {\left (c^2 d^2+b^2 e^2-2 c e (b d-a e)\right ) x^3}{3 e^3}-\frac {c (c d-2 b e) x^5}{5 e^2}+\frac {c^2 x^7}{7 e}+\frac {\left (c d^2-b d e+a e^2\right )^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} e^{9/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.01 \[ \int \frac {\left (a+b x^2+c x^4\right )^2}{d+e x^2} \, dx=\frac {(-c d+b e) \left (c d^2-b d e+2 a e^2\right ) x}{e^4}+\frac {\left (c^2 d^2-2 b c d e+b^2 e^2+2 a c e^2\right ) x^3}{3 e^3}+\frac {c (-c d+2 b e) x^5}{5 e^2}+\frac {c^2 x^7}{7 e}+\frac {\left (c d^2-b d e+a e^2\right )^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} e^{9/2}} \]

[In]

Integrate[(a + b*x^2 + c*x^4)^2/(d + e*x^2),x]

[Out]

((-(c*d) + b*e)*(c*d^2 - b*d*e + 2*a*e^2)*x)/e^4 + ((c^2*d^2 - 2*b*c*d*e + b^2*e^2 + 2*a*c*e^2)*x^3)/(3*e^3) +
(c*(-(c*d) + 2*b*e)*x^5)/(5*e^2) + (c^2*x^7)/(7*e) + ((c*d^2 - b*d*e + a*e^2)^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/
(Sqrt[d]*e^(9/2))

Maple [A] (verified)

Time = 0.58 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.24

method	result	size
default	\(\frac {\frac {c^{2} x^{7} e^{3}}{7}+\frac {\left (\left (b e -c d \right ) c \,e^{2}+e^{3} b c \right ) x^{5}}{5}+\frac {\left (\left (b e -c d \right ) b \,e^{2}+e c \left (2 a \,e^{2}-b d e +c \,d^{2}\right )\right ) x^{3}}{3}+\left (b e -c d \right ) \left (2 a \,e^{2}-b d e +c \,d^{2}\right ) x}{e^{4}}+\frac {\left (a^{2} e^{4}-2 a b d \,e^{3}+2 a c \,d^{2} e^{2}+b^{2} d^{2} e^{2}-2 b c \,d^{3} e +c^{2} d^{4}\right ) \arctan \left (\frac {e x}{\sqrt {e d}}\right )}{e^{4} \sqrt {e d}}\)	\(178\)
risch	\(\frac {c^{2} x^{7}}{7 e}+\frac {2 x^{5} b c}{5 e}-\frac {c^{2} d \,x^{5}}{5 e^{2}}+\frac {x^{3} b^{2}}{3 e}-\frac {2 x^{3} d b c}{3 e^{2}}+\frac {2 c a \,x^{3}}{3 e}+\frac {c^{2} d^{2} x^{3}}{3 e^{3}}+\frac {2 a b x}{e}-\frac {2 c a d x}{e^{2}}-\frac {b^{2} d x}{e^{2}}+\frac {2 b c \,d^{2} x}{e^{3}}-\frac {c^{2} d^{3} x}{e^{4}}-\frac {\ln \left (e x +\sqrt {-e d}\right ) a^{2}}{2 \sqrt {-e d}}+\frac {\ln \left (e x +\sqrt {-e d}\right ) a b d}{e \sqrt {-e d}}-\frac {\ln \left (e x +\sqrt {-e d}\right ) a c \,d^{2}}{e^{2} \sqrt {-e d}}-\frac {\ln \left (e x +\sqrt {-e d}\right ) b^{2} d^{2}}{2 e^{2} \sqrt {-e d}}+\frac {\ln \left (e x +\sqrt {-e d}\right ) b c \,d^{3}}{e^{3} \sqrt {-e d}}-\frac {\ln \left (e x +\sqrt {-e d}\right ) c^{2} d^{4}}{2 e^{4} \sqrt {-e d}}+\frac {\ln \left (-e x +\sqrt {-e d}\right ) a^{2}}{2 \sqrt {-e d}}-\frac {\ln \left (-e x +\sqrt {-e d}\right ) a b d}{e \sqrt {-e d}}+\frac {\ln \left (-e x +\sqrt {-e d}\right ) a c \,d^{2}}{e^{2} \sqrt {-e d}}+\frac {\ln \left (-e x +\sqrt {-e d}\right ) b^{2} d^{2}}{2 e^{2} \sqrt {-e d}}-\frac {\ln \left (-e x +\sqrt {-e d}\right ) b c \,d^{3}}{e^{3} \sqrt {-e d}}+\frac {\ln \left (-e x +\sqrt {-e d}\right ) c^{2} d^{4}}{2 e^{4} \sqrt {-e d}}\)	\(448\)

[In]

int((c*x^4+b*x^2+a)^2/(e*x^2+d),x,method=_RETURNVERBOSE)

[Out]

1/e^4*(1/7*c^2*x^7*e^3+1/5*((b*e-c*d)*c*e^2+e^3*b*c)*x^5+1/3*((b*e-c*d)*b*e^2+e*c*(2*a*e^2-b*d*e+c*d^2))*x^3+(
b*e-c*d)*(2*a*e^2-b*d*e+c*d^2)*x)+(a^2*e^4-2*a*b*d*e^3+2*a*c*d^2*e^2+b^2*d^2*e^2-2*b*c*d^3*e+c^2*d^4)/e^4/(e*d
)^(1/2)*arctan(e*x/(e*d)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 406, normalized size of antiderivative = 2.84 \[ \int \frac {\left (a+b x^2+c x^4\right )^2}{d+e x^2} \, dx=\left [\frac {30 \, c^{2} d e^{4} x^{7} - 42 \, {\left (c^{2} d^{2} e^{3} - 2 \, b c d e^{4}\right )} x^{5} + 70 \, {\left (c^{2} d^{3} e^{2} - 2 \, b c d^{2} e^{3} + {\left (b^{2} + 2 \, a c\right )} d e^{4}\right )} x^{3} - 105 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e - 2 \, a b d e^{3} + a^{2} e^{4} + {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2}\right )} \sqrt {-d e} \log \left (\frac {e x^{2} - 2 \, \sqrt {-d e} x - d}{e x^{2} + d}\right ) - 210 \, {\left (c^{2} d^{4} e - 2 \, b c d^{3} e^{2} - 2 \, a b d e^{4} + {\left (b^{2} + 2 \, a c\right )} d^{2} e^{3}\right )} x}{210 \, d e^{5}}, \frac {15 \, c^{2} d e^{4} x^{7} - 21 \, {\left (c^{2} d^{2} e^{3} - 2 \, b c d e^{4}\right )} x^{5} + 35 \, {\left (c^{2} d^{3} e^{2} - 2 \, b c d^{2} e^{3} + {\left (b^{2} + 2 \, a c\right )} d e^{4}\right )} x^{3} + 105 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e - 2 \, a b d e^{3} + a^{2} e^{4} + {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2}\right )} \sqrt {d e} \arctan \left (\frac {\sqrt {d e} x}{d}\right ) - 105 \, {\left (c^{2} d^{4} e - 2 \, b c d^{3} e^{2} - 2 \, a b d e^{4} + {\left (b^{2} + 2 \, a c\right )} d^{2} e^{3}\right )} x}{105 \, d e^{5}}\right ] \]

[In]

integrate((c*x^4+b*x^2+a)^2/(e*x^2+d),x, algorithm="fricas")

[Out]

[1/210*(30*c^2*d*e^4*x^7 - 42*(c^2*d^2*e^3 - 2*b*c*d*e^4)*x^5 + 70*(c^2*d^3*e^2 - 2*b*c*d^2*e^3 + (b^2 + 2*a*c
)*d*e^4)*x^3 - 105*(c^2*d^4 - 2*b*c*d^3*e - 2*a*b*d*e^3 + a^2*e^4 + (b^2 + 2*a*c)*d^2*e^2)*sqrt(-d*e)*log((e*x
^2 - 2*sqrt(-d*e)*x - d)/(e*x^2 + d)) - 210*(c^2*d^4*e - 2*b*c*d^3*e^2 - 2*a*b*d*e^4 + (b^2 + 2*a*c)*d^2*e^3)*
x)/(d*e^5), 1/105*(15*c^2*d*e^4*x^7 - 21*(c^2*d^2*e^3 - 2*b*c*d*e^4)*x^5 + 35*(c^2*d^3*e^2 - 2*b*c*d^2*e^3 + (
b^2 + 2*a*c)*d*e^4)*x^3 + 105*(c^2*d^4 - 2*b*c*d^3*e - 2*a*b*d*e^3 + a^2*e^4 + (b^2 + 2*a*c)*d^2*e^2)*sqrt(d*e
)*arctan(sqrt(d*e)*x/d) - 105*(c^2*d^4*e - 2*b*c*d^3*e^2 - 2*a*b*d*e^4 + (b^2 + 2*a*c)*d^2*e^3)*x)/(d*e^5)]

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 371 vs. \(2 (133) = 266\).

Time = 0.49 (sec) , antiderivative size = 371, normalized size of antiderivative = 2.59 \[ \int \frac {\left (a+b x^2+c x^4\right )^2}{d+e x^2} \, dx=\frac {c^{2} x^{7}}{7 e} + x^{5} \cdot \left (\frac {2 b c}{5 e} - \frac {c^{2} d}{5 e^{2}}\right ) + x^{3} \cdot \left (\frac {2 a c}{3 e} + \frac {b^{2}}{3 e} - \frac {2 b c d}{3 e^{2}} + \frac {c^{2} d^{2}}{3 e^{3}}\right ) + x \left (\frac {2 a b}{e} - \frac {2 a c d}{e^{2}} - \frac {b^{2} d}{e^{2}} + \frac {2 b c d^{2}}{e^{3}} - \frac {c^{2} d^{3}}{e^{4}}\right ) - \frac {\sqrt {- \frac {1}{d e^{9}}} \left (a e^{2} - b d e + c d^{2}\right )^{2} \log {\left (- \frac {d e^{4} \sqrt {- \frac {1}{d e^{9}}} \left (a e^{2} - b d e + c d^{2}\right )^{2}}{a^{2} e^{4} - 2 a b d e^{3} + 2 a c d^{2} e^{2} + b^{2} d^{2} e^{2} - 2 b c d^{3} e + c^{2} d^{4}} + x \right )}}{2} + \frac {\sqrt {- \frac {1}{d e^{9}}} \left (a e^{2} - b d e + c d^{2}\right )^{2} \log {\left (\frac {d e^{4} \sqrt {- \frac {1}{d e^{9}}} \left (a e^{2} - b d e + c d^{2}\right )^{2}}{a^{2} e^{4} - 2 a b d e^{3} + 2 a c d^{2} e^{2} + b^{2} d^{2} e^{2} - 2 b c d^{3} e + c^{2} d^{4}} + x \right )}}{2} \]

[In]

integrate((c*x**4+b*x**2+a)**2/(e*x**2+d),x)

[Out]

c**2*x**7/(7*e) + x**5*(2*b*c/(5*e) - c**2*d/(5*e**2)) + x**3*(2*a*c/(3*e) + b**2/(3*e) - 2*b*c*d/(3*e**2) + c
**2*d**2/(3*e**3)) + x*(2*a*b/e - 2*a*c*d/e**2 - b**2*d/e**2 + 2*b*c*d**2/e**3 - c**2*d**3/e**4) - sqrt(-1/(d*
e**9))*(a*e**2 - b*d*e + c*d**2)**2*log(-d*e**4*sqrt(-1/(d*e**9))*(a*e**2 - b*d*e + c*d**2)**2/(a**2*e**4 - 2*
a*b*d*e**3 + 2*a*c*d**2*e**2 + b**2*d**2*e**2 - 2*b*c*d**3*e + c**2*d**4) + x)/2 + sqrt(-1/(d*e**9))*(a*e**2 -
b*d*e + c*d**2)**2*log(d*e**4*sqrt(-1/(d*e**9))*(a*e**2 - b*d*e + c*d**2)**2/(a**2*e**4 - 2*a*b*d*e**3 + 2*a*
c*d**2*e**2 + b**2*d**2*e**2 - 2*b*c*d**3*e + c**2*d**4) + x)/2

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (a+b x^2+c x^4\right )^2}{d+e x^2} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((c*x^4+b*x^2+a)^2/(e*x^2+d),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.43 \[ \int \frac {\left (a+b x^2+c x^4\right )^2}{d+e x^2} \, dx=\frac {{\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2} + 2 \, a c d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {d e} e^{4}} + \frac {15 \, c^{2} e^{6} x^{7} - 21 \, c^{2} d e^{5} x^{5} + 42 \, b c e^{6} x^{5} + 35 \, c^{2} d^{2} e^{4} x^{3} - 70 \, b c d e^{5} x^{3} + 35 \, b^{2} e^{6} x^{3} + 70 \, a c e^{6} x^{3} - 105 \, c^{2} d^{3} e^{3} x + 210 \, b c d^{2} e^{4} x - 105 \, b^{2} d e^{5} x - 210 \, a c d e^{5} x + 210 \, a b e^{6} x}{105 \, e^{7}} \]

[In]

integrate((c*x^4+b*x^2+a)^2/(e*x^2+d),x, algorithm="giac")

[Out]

(c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2 + 2*a*c*d^2*e^2 - 2*a*b*d*e^3 + a^2*e^4)*arctan(e*x/sqrt(d*e))/(sqrt(d*e)
*e^4) + 1/105*(15*c^2*e^6*x^7 - 21*c^2*d*e^5*x^5 + 42*b*c*e^6*x^5 + 35*c^2*d^2*e^4*x^3 - 70*b*c*d*e^5*x^3 + 35
*b^2*e^6*x^3 + 70*a*c*e^6*x^3 - 105*c^2*d^3*e^3*x + 210*b*c*d^2*e^4*x - 105*b^2*d*e^5*x - 210*a*c*d*e^5*x + 21
0*a*b*e^6*x)/e^7

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.60 \[ \int \frac {\left (a+b x^2+c x^4\right )^2}{d+e x^2} \, dx=x^3\,\left (\frac {b^2+2\,a\,c}{3\,e}+\frac {d\,\left (\frac {c^2\,d}{e^2}-\frac {2\,b\,c}{e}\right )}{3\,e}\right )-x\,\left (\frac {d\,\left (\frac {b^2+2\,a\,c}{e}+\frac {d\,\left (\frac {c^2\,d}{e^2}-\frac {2\,b\,c}{e}\right )}{e}\right )}{e}-\frac {2\,a\,b}{e}\right )-x^5\,\left (\frac {c^2\,d}{5\,e^2}-\frac {2\,b\,c}{5\,e}\right )+\frac {c^2\,x^7}{7\,e}+\frac {\mathrm {atan}\left (\frac {\sqrt {e}\,x\,{\left (c\,d^2-b\,d\,e+a\,e^2\right )}^2}{\sqrt {d}\,\left (a^2\,e^4-2\,a\,b\,d\,e^3+2\,a\,c\,d^2\,e^2+b^2\,d^2\,e^2-2\,b\,c\,d^3\,e+c^2\,d^4\right )}\right )\,{\left (c\,d^2-b\,d\,e+a\,e^2\right )}^2}{\sqrt {d}\,e^{9/2}} \]

[In]

int((a + b*x^2 + c*x^4)^2/(d + e*x^2),x)

[Out]

x^3*((2*a*c + b^2)/(3*e) + (d*((c^2*d)/e^2 - (2*b*c)/e))/(3*e)) - x*((d*((2*a*c + b^2)/e + (d*((c^2*d)/e^2 - (
2*b*c)/e))/e))/e - (2*a*b)/e) - x^5*((c^2*d)/(5*e^2) - (2*b*c)/(5*e)) + (c^2*x^7)/(7*e) + (atan((e^(1/2)*x*(a*
e^2 + c*d^2 - b*d*e)^2)/(d^(1/2)*(a^2*e^4 + c^2*d^4 + b^2*d^2*e^2 - 2*a*b*d*e^3 - 2*b*c*d^3*e + 2*a*c*d^2*e^2)
))*(a*e^2 + c*d^2 - b*d*e)^2)/(d^(1/2)*e^(9/2))