\(\int \frac {(a+c x^4)^2}{d+e x^2} \, dx\) [132]

Optimal result

Integrand size = 19, antiderivative size = 108 \[ \int \frac {\left (a+c x^4\right )^2}{d+e x^2} \, dx=-\frac {c d \left (c d^2+2 a e^2\right ) x}{e^4}+\frac {c \left (c d^2+2 a e^2\right ) x^3}{3 e^3}-\frac {c^2 d x^5}{5 e^2}+\frac {c^2 x^7}{7 e}+\frac {\left (c d^2+a e^2\right )^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} e^{9/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {1168, 211} \[ \int \frac {\left (a+c x^4\right )^2}{d+e x^2} \, dx=\frac {\left (a e^2+c d^2\right )^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} e^{9/2}}-\frac {c d x \left (2 a e^2+c d^2\right )}{e^4}+\frac {c x^3 \left (2 a e^2+c d^2\right )}{3 e^3}-\frac {c^2 d x^5}{5 e^2}+\frac {c^2 x^7}{7 e} \]

[In]

Int[(a + c*x^4)^2/(d + e*x^2),x]

[Out]

-((c*d*(c*d^2 + 2*a*e^2)*x)/e^4) + (c*(c*d^2 + 2*a*e^2)*x^3)/(3*e^3) - (c^2*d*x^5)/(5*e^2) + (c^2*x^7)/(7*e) +
((c*d^2 + 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 1168

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^q*(a
+ c*x^4)^p, x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {c d \left (c d^2+2 a e^2\right )}{e^4}+\frac {c \left (c d^2+2 a e^2\right ) x^2}{e^3}-\frac {c^2 d x^4}{e^2}+\frac {c^2 x^6}{e}+\frac {c^2 d^4+2 a c d^2 e^2+a^2 e^4}{e^4 \left (d+e x^2\right )}\right ) \, dx \\ & = -\frac {c d \left (c d^2+2 a e^2\right ) x}{e^4}+\frac {c \left (c d^2+2 a e^2\right ) x^3}{3 e^3}-\frac {c^2 d x^5}{5 e^2}+\frac {c^2 x^7}{7 e}+\frac {\left (c d^2+a e^2\right )^2 \int \frac {1}{d+e x^2} \, dx}{e^4} \\ & = -\frac {c d \left (c d^2+2 a e^2\right ) x}{e^4}+\frac {c \left (c d^2+2 a e^2\right ) x^3}{3 e^3}-\frac {c^2 d x^5}{5 e^2}+\frac {c^2 x^7}{7 e}+\frac {\left (c d^2+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 = 97, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+c x^4\right )^2}{d+e x^2} \, dx=\frac {c x \left (70 a e^2 \left (-3 d+e x^2\right )+c \left (-105 d^3+35 d^2 e x^2-21 d e^2 x^4+15 e^3 x^6\right )\right )}{105 e^4}+\frac {\left (c d^2+a e^2\right )^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} e^{9/2}} \]

[In]

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

[Out]

(c*x*(70*a*e^2*(-3*d + e*x^2) + c*(-105*d^3 + 35*d^2*e*x^2 - 21*d*e^2*x^4 + 15*e^3*x^6)))/(105*e^4) + ((c*d^2
+ a*e^2)^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(Sqrt[d]*e^(9/2))

Maple [A] (verified)

Time = 0.28 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.96

method	result	size
default	\(-\frac {c \left (-\frac {c \,x^{7} e^{3}}{7}+\frac {c d \,x^{5} e^{2}}{5}-\frac {\left (2 a \,e^{2}+c \,d^{2}\right ) x^{3} e}{3}+d \left (2 a \,e^{2}+c \,d^{2}\right ) x \right )}{e^{4}}+\frac {\left (a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \arctan \left (\frac {e x}{\sqrt {e d}}\right )}{e^{4} \sqrt {e d}}\)	\(104\)
risch	\(\frac {c^{2} x^{7}}{7 e}-\frac {c^{2} d \,x^{5}}{5 e^{2}}+\frac {2 c a \,x^{3}}{3 e}+\frac {c^{2} d^{2} x^{3}}{3 e^{3}}-\frac {2 c a d x}{e^{2}}-\frac {c^{2} d^{3} x}{e^{4}}-\frac {\ln \left (e x +\sqrt {-e d}\right ) a^{2}}{2 \sqrt {-e d}}-\frac {\ln \left (e x +\sqrt {-e d}\right ) a c \,d^{2}}{e^{2} \sqrt {-e d}}-\frac {\ln \left (e x +\sqrt {-e d}\right ) c^{2} d^{4}}{2 e^{4} \sqrt {-e d}}+\frac {\ln \left (-e x +\sqrt {-e d}\right ) a^{2}}{2 \sqrt {-e d}}+\frac {\ln \left (-e x +\sqrt {-e d}\right ) a c \,d^{2}}{e^{2} \sqrt {-e d}}+\frac {\ln \left (-e x +\sqrt {-e d}\right ) c^{2} d^{4}}{2 e^{4} \sqrt {-e d}}\)	\(226\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 268, normalized size of antiderivative = 2.48 \[ \int \frac {\left (a+c x^4\right )^2}{d+e x^2} \, dx=\left [\frac {30 \, c^{2} d e^{4} x^{7} - 42 \, c^{2} d^{2} e^{3} x^{5} + 70 \, {\left (c^{2} d^{3} e^{2} + 2 \, a c d e^{4}\right )} x^{3} - 105 \, {\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt {-d e} \log \left (\frac {e x^{2} - 2 \, \sqrt {-d e} x - d}{e x^{2} + d}\right ) - 210 \, {\left (c^{2} d^{4} e + 2 \, a c d^{2} e^{3}\right )} x}{210 \, d e^{5}}, \frac {15 \, c^{2} d e^{4} x^{7} - 21 \, c^{2} d^{2} e^{3} x^{5} + 35 \, {\left (c^{2} d^{3} e^{2} + 2 \, a c d e^{4}\right )} x^{3} + 105 \, {\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt {d e} \arctan \left (\frac {\sqrt {d e} x}{d}\right ) - 105 \, {\left (c^{2} d^{4} e + 2 \, a c d^{2} e^{3}\right )} x}{105 \, d e^{5}}\right ] \]

[In]

integrate((c*x^4+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*x^5 + 70*(c^2*d^3*e^2 + 2*a*c*d*e^4)*x^3 - 105*(c^2*d^4 + 2*a*c*d^2*
e^2 + a^2*e^4)*sqrt(-d*e)*log((e*x^2 - 2*sqrt(-d*e)*x - d)/(e*x^2 + d)) - 210*(c^2*d^4*e + 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*x^5 + 35*(c^2*d^3*e^2 + 2*a*c*d*e^4)*x^3 + 105*(c^2*d^4 + 2*a
*c*d^2*e^2 + a^2*e^4)*sqrt(d*e)*arctan(sqrt(d*e)*x/d) - 105*(c^2*d^4*e + 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. 236 vs. \(2 (100) = 200\).

Time = 0.23 (sec) , antiderivative size = 236, normalized size of antiderivative = 2.19 \[ \int \frac {\left (a+c x^4\right )^2}{d+e x^2} \, dx=- \frac {c^{2} d x^{5}}{5 e^{2}} + \frac {c^{2} x^{7}}{7 e} + x^{3} \cdot \left (\frac {2 a c}{3 e} + \frac {c^{2} d^{2}}{3 e^{3}}\right ) + x \left (- \frac {2 a c d}{e^{2}} - \frac {c^{2} d^{3}}{e^{4}}\right ) - \frac {\sqrt {- \frac {1}{d e^{9}}} \left (a e^{2} + c d^{2}\right )^{2} \log {\left (- \frac {d e^{4} \sqrt {- \frac {1}{d e^{9}}} \left (a e^{2} + c d^{2}\right )^{2}}{a^{2} e^{4} + 2 a c d^{2} e^{2} + c^{2} d^{4}} + x \right )}}{2} + \frac {\sqrt {- \frac {1}{d e^{9}}} \left (a e^{2} + c d^{2}\right )^{2} \log {\left (\frac {d e^{4} \sqrt {- \frac {1}{d e^{9}}} \left (a e^{2} + c d^{2}\right )^{2}}{a^{2} e^{4} + 2 a c d^{2} e^{2} + c^{2} d^{4}} + x \right )}}{2} \]

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

integrate((c*x^4+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 = 118, normalized size of antiderivative = 1.09 \[ \int \frac {\left (a+c x^4\right )^2}{d+e x^2} \, dx=\frac {{\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {d e} e^{4}} + \frac {15 \, c^{2} e^{6} x^{7} - 21 \, c^{2} d e^{5} x^{5} + 35 \, c^{2} d^{2} e^{4} x^{3} + 70 \, a c e^{6} x^{3} - 105 \, c^{2} d^{3} e^{3} x - 210 \, a c d e^{5} x}{105 \, e^{7}} \]

[In]

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

[Out]

(c^2*d^4 + 2*a*c*d^2*e^2 + 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 + 35*c^2*d^2*e^4*x^3 + 70*a*c*e^6*x^3 - 105*c^2*d^3*e^3*x - 210*a*c*d*e^5*x)/e^7

Mupad [B] (verification not implemented)

Time = 13.69 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.31 \[ \int \frac {\left (a+c x^4\right )^2}{d+e x^2} \, dx=x^3\,\left (\frac {c^2\,d^2}{3\,e^3}+\frac {2\,a\,c}{3\,e}\right )+\frac {c^2\,x^7}{7\,e}-\frac {c^2\,d\,x^5}{5\,e^2}+\frac {\mathrm {atan}\left (\frac {\sqrt {e}\,x\,{\left (c\,d^2+a\,e^2\right )}^2}{\sqrt {d}\,\left (a^2\,e^4+2\,a\,c\,d^2\,e^2+c^2\,d^4\right )}\right )\,{\left (c\,d^2+a\,e^2\right )}^2}{\sqrt {d}\,e^{9/2}}-\frac {d\,x\,\left (\frac {c^2\,d^2}{e^3}+\frac {2\,a\,c}{e}\right )}{e} \]

[In]

int((a + c*x^4)^2/(d + e*x^2),x)

[Out]

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