\(\int \frac {(d+e x^2)^4}{d^2-e^2 x^4} \, dx\) [189]

Optimal result

Integrand size = 24, antiderivative size = 51 \[ \int \frac {\left (d+e x^2\right )^4}{d^2-e^2 x^4} \, dx=-7 d^2 x-\frac {4}{3} d e x^3-\frac {e^2 x^5}{5}+\frac {8 d^{5/2} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}} \]

[Out]

-7*d^2*x-4/3*d*e*x^3-1/5*e^2*x^5+8*d^(5/2)*arctanh(x*e^(1/2)/d^(1/2))/e^(1/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1164, 398, 214} \[ \int \frac {\left (d+e x^2\right )^4}{d^2-e^2 x^4} \, dx=\frac {8 d^{5/2} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}-7 d^2 x-\frac {4}{3} d e x^3-\frac {1}{5} e^2 x^5 \]

[In]

Int[(d + e*x^2)^4/(d^2 - e^2*x^4),x]

[Out]

-7*d^2*x - (4*d*e*x^3)/3 - (e^2*x^5)/5 + (8*d^(5/2)*ArcTanh[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[e]

Rule 214

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

Rule 398

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Int[PolynomialDivide[(a + b*x^n)
^p, (c + d*x^n)^(-q), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IGtQ[p, 0] && ILt
Q[q, 0] && GeQ[p, -q]

Rule 1164

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {\left (d+e x^2\right )^3}{d-e x^2} \, dx \\ & = \int \left (-7 d^2-4 d e x^2-e^2 x^4+\frac {8 d^3}{d-e x^2}\right ) \, dx \\ & = -7 d^2 x-\frac {4}{3} d e x^3-\frac {e^2 x^5}{5}+\left (8 d^3\right ) \int \frac {1}{d-e x^2} \, dx \\ & = -7 d^2 x-\frac {4}{3} d e x^3-\frac {e^2 x^5}{5}+\frac {8 d^{5/2} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00 \[ \int \frac {\left (d+e x^2\right )^4}{d^2-e^2 x^4} \, dx=-7 d^2 x-\frac {4}{3} d e x^3-\frac {e^2 x^5}{5}+\frac {8 d^{5/2} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}} \]

[In]

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

[Out]

-7*d^2*x - (4*d*e*x^3)/3 - (e^2*x^5)/5 + (8*d^(5/2)*ArcTanh[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[e]

Maple [A] (verified)

Time = 0.25 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.82

method	result	size
default	\(-\frac {e^{2} x^{5}}{5}-\frac {4 d e \,x^{3}}{3}-7 d^{2} x +\frac {8 d^{3} \operatorname {arctanh}\left (\frac {e x}{\sqrt {e d}}\right )}{\sqrt {e d}}\)	\(42\)
risch	\(-\frac {e^{2} x^{5}}{5}-\frac {4 d e \,x^{3}}{3}-7 d^{2} x -\frac {4 \sqrt {e d}\, d^{2} \ln \left (\sqrt {e d}\, x -d \right )}{e}+\frac {4 \sqrt {e d}\, d^{2} \ln \left (-\sqrt {e d}\, x -d \right )}{e}\)	\(74\)

[In]

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

[Out]

-1/5*e^2*x^5-4/3*d*e*x^3-7*d^2*x+8*d^3/(e*d)^(1/2)*arctanh(e*x/(e*d)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 116, normalized size of antiderivative = 2.27 \[ \int \frac {\left (d+e x^2\right )^4}{d^2-e^2 x^4} \, dx=\left [-\frac {1}{5} \, e^{2} x^{5} - \frac {4}{3} \, d e x^{3} + 4 \, d^{2} \sqrt {\frac {d}{e}} \log \left (\frac {e x^{2} + 2 \, e x \sqrt {\frac {d}{e}} + d}{e x^{2} - d}\right ) - 7 \, d^{2} x, -\frac {1}{5} \, e^{2} x^{5} - \frac {4}{3} \, d e x^{3} - 8 \, d^{2} \sqrt {-\frac {d}{e}} \arctan \left (\frac {e x \sqrt {-\frac {d}{e}}}{d}\right ) - 7 \, d^{2} x\right ] \]

[In]

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

[Out]

[-1/5*e^2*x^5 - 4/3*d*e*x^3 + 4*d^2*sqrt(d/e)*log((e*x^2 + 2*e*x*sqrt(d/e) + d)/(e*x^2 - d)) - 7*d^2*x, -1/5*e
^2*x^5 - 4/3*d*e*x^3 - 8*d^2*sqrt(-d/e)*arctan(e*x*sqrt(-d/e)/d) - 7*d^2*x]

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.47 \[ \int \frac {\left (d+e x^2\right )^4}{d^2-e^2 x^4} \, dx=- 7 d^{2} x - \frac {4 d e x^{3}}{3} - \frac {e^{2} x^{5}}{5} - 4 \sqrt {\frac {d^{5}}{e}} \log {\left (x - \frac {\sqrt {\frac {d^{5}}{e}}}{d^{2}} \right )} + 4 \sqrt {\frac {d^{5}}{e}} \log {\left (x + \frac {\sqrt {\frac {d^{5}}{e}}}{d^{2}} \right )} \]

[In]

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

[Out]

-7*d**2*x - 4*d*e*x**3/3 - e**2*x**5/5 - 4*sqrt(d**5/e)*log(x - sqrt(d**5/e)/d**2) + 4*sqrt(d**5/e)*log(x + sq
rt(d**5/e)/d**2)

Maxima [F(-2)]

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

[In]

integrate((e*x^2+d)^4/(-e^2*x^4+d^2),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 = 0.26 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.06 \[ \int \frac {\left (d+e x^2\right )^4}{d^2-e^2 x^4} \, dx=-\frac {8 \, d^{3} \arctan \left (\frac {e x}{\sqrt {-d e}}\right )}{\sqrt {-d e}} - \frac {3 \, e^{7} x^{5} + 20 \, d e^{6} x^{3} + 105 \, d^{2} e^{5} x}{15 \, e^{5}} \]

[In]

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

[Out]

-8*d^3*arctan(e*x/sqrt(-d*e))/sqrt(-d*e) - 1/15*(3*e^7*x^5 + 20*d*e^6*x^3 + 105*d^2*e^5*x)/e^5

Mupad [B] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.82 \[ \int \frac {\left (d+e x^2\right )^4}{d^2-e^2 x^4} \, dx=-7\,d^2\,x-\frac {e^2\,x^5}{5}-\frac {4\,d\,e\,x^3}{3}-\frac {d^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {e}\,x\,1{}\mathrm {i}}{\sqrt {d}}\right )\,8{}\mathrm {i}}{\sqrt {e}} \]

[In]

int((d + e*x^2)^4/(d^2 - e^2*x^4),x)

[Out]

- 7*d^2*x - (e^2*x^5)/5 - (d^(5/2)*atan((e^(1/2)*x*1i)/d^(1/2))*8i)/e^(1/2) - (4*d*e*x^3)/3