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

Optimal. Leaf size=51 \[ \frac{8 d^{5/2} \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}-7 d^2 x-\frac{4}{3} d e x^3-\frac{1}{5} e^2 x^5 \]

[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]

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Rubi [A]  time = 0.0410136, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {1150, 390, 208} \[ \frac{8 d^{5/2} \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}-7 d^2 x-\frac{4}{3} d e x^3-\frac{1}{5} e^2 x^5 \]

Antiderivative was successfully verified.

[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 1150

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

Rule 390

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 208

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

Rubi steps

\begin{align*} \int \frac{\left (d+e x^2\right )^4}{d^2-e^2 x^4} \, dx &=\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]  time = 0.023367, size = 51, normalized size = 1. \[ \frac{8 d^{5/2} \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}-7 d^2 x-\frac{4}{3} d e x^3-\frac{1}{5} e^2 x^5 \]

Antiderivative was successfully verified.

[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]

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Maple [A]  time = 0.006, size = 42, normalized size = 0.8 \begin{align*} -{\frac{{e}^{2}{x}^{5}}{5}}-{\frac{4\,de{x}^{3}}{3}}-7\,{d}^{2}x+8\,{\frac{{d}^{3}}{\sqrt{de}}{\it Artanh} \left ({\frac{ex}{\sqrt{de}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.84779, size = 255, normalized size = 5. \begin{align*} \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 ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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]

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Sympy [A]  time = 0.475067, size = 75, normalized size = 1.47 \begin{align*} - 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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [B]  time = 1.18287, size = 194, normalized size = 3.8 \begin{align*} 4 \,{\left ({\left (d^{2}\right )}^{\frac{1}{4}} d^{2} e^{\frac{11}{2}} -{\left (d^{2}\right )}^{\frac{1}{4}} d{\left | d \right |} e^{\frac{11}{2}}\right )} \arctan \left (\frac{x e^{\frac{1}{2}}}{{\left (d^{2}\right )}^{\frac{1}{4}}}\right ) e^{\left (-6\right )} + 2 \,{\left ({\left (d^{2}\right )}^{\frac{1}{4}} d^{2} e^{\frac{15}{2}} +{\left (d^{2}\right )}^{\frac{3}{4}} d e^{\frac{15}{2}}\right )} e^{\left (-8\right )} \log \left ({\left |{\left (d^{2}\right )}^{\frac{1}{4}} e^{\left (-\frac{1}{2}\right )} + x \right |}\right ) - 2 \,{\left ({\left (d^{2}\right )}^{\frac{1}{4}} d^{2} e^{\frac{11}{2}} +{\left (d^{2}\right )}^{\frac{1}{4}} d{\left | d \right |} e^{\frac{11}{2}}\right )} e^{\left (-6\right )} \log \left ({\left | -{\left (d^{2}\right )}^{\frac{1}{4}} e^{\left (-\frac{1}{2}\right )} + x \right |}\right ) - \frac{1}{15} \,{\left (3 \, x^{5} e^{12} + 20 \, d x^{3} e^{11} + 105 \, d^{2} x e^{10}\right )} e^{\left (-10\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*((d^2)^(1/4)*d^2*e^(11/2) - (d^2)^(1/4)*d*abs(d)*e^(11/2))*arctan(x*e^(1/2)/(d^2)^(1/4))*e^(-6) + 2*((d^2)^(
1/4)*d^2*e^(15/2) + (d^2)^(3/4)*d*e^(15/2))*e^(-8)*log(abs((d^2)^(1/4)*e^(-1/2) + x)) - 2*((d^2)^(1/4)*d^2*e^(
11/2) + (d^2)^(1/4)*d*abs(d)*e^(11/2))*e^(-6)*log(abs(-(d^2)^(1/4)*e^(-1/2) + x)) - 1/15*(3*x^5*e^12 + 20*d*x^
3*e^11 + 105*d^2*x*e^10)*e^(-10)