3.190 \(\int \frac{(d+e x^2)^3}{d^2-e^2 x^4} \, dx\)

Optimal. Leaf size=38 \[ \frac{4 d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}-3 d x-\frac{e x^3}{3} \]

[Out]

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

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Rubi [A]  time = 0.0337036, antiderivative size = 38, 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} \[ \frac{4 d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}-3 d x-\frac{e x^3}{3} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x^2)^3/(d^2 - e^2*x^4),x]

[Out]

-3*d*x - (e*x^3)/3 + (4*d^(3/2)*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 )^3}{d^2-e^2 x^4} \, dx &=\int \frac{\left (d+e x^2\right )^2}{d-e x^2} \, dx\\ &=\int \left (-3 d-e x^2+\frac{4 d^2}{d-e x^2}\right ) \, dx\\ &=-3 d x-\frac{e x^3}{3}+\left (4 d^2\right ) \int \frac{1}{d-e x^2} \, dx\\ &=-3 d x-\frac{e x^3}{3}+\frac{4 d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}\\ \end{align*}

Mathematica [A]  time = 0.0171963, size = 38, normalized size = 1. \[ \frac{4 d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}-3 d x-\frac{e x^3}{3} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x^2)^3/(d^2 - e^2*x^4),x]

[Out]

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

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Maple [A]  time = 0.002, size = 31, normalized size = 0.8 \begin{align*} -{\frac{e{x}^{3}}{3}}-3\,dx+4\,{\frac{{d}^{2}}{\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)^3/(-e^2*x^4+d^2),x)

[Out]

-1/3*e*x^3-3*d*x+4*d^2/(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)^3/(-e^2*x^4+d^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.8496, size = 201, normalized size = 5.29 \begin{align*} \left [-\frac{1}{3} \, e x^{3} + 2 \, d \sqrt{\frac{d}{e}} \log \left (\frac{e x^{2} + 2 \, e x \sqrt{\frac{d}{e}} + d}{e x^{2} - d}\right ) - 3 \, d x, -\frac{1}{3} \, e x^{3} - 4 \, d \sqrt{-\frac{d}{e}} \arctan \left (\frac{e x \sqrt{-\frac{d}{e}}}{d}\right ) - 3 \, d x\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^3/(-e^2*x^4+d^2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.404648, size = 58, normalized size = 1.53 \begin{align*} - 3 d x - \frac{e x^{3}}{3} - 2 \sqrt{\frac{d^{3}}{e}} \log{\left (x - \frac{\sqrt{\frac{d^{3}}{e}}}{d} \right )} + 2 \sqrt{\frac{d^{3}}{e}} \log{\left (x + \frac{\sqrt{\frac{d^{3}}{e}}}{d} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x**2+d)**3/(-e**2*x**4+d**2),x)

[Out]

-3*d*x - e*x**3/3 - 2*sqrt(d**3/e)*log(x - sqrt(d**3/e)/d) + 2*sqrt(d**3/e)*log(x + sqrt(d**3/e)/d)

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Giac [B]  time = 1.1757, size = 166, normalized size = 4.37 \begin{align*} 2 \,{\left ({\left (d^{2}\right )}^{\frac{1}{4}} d e^{\frac{11}{2}} -{\left (d^{2}\right )}^{\frac{1}{4}}{\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 )} +{\left ({\left (d^{2}\right )}^{\frac{1}{4}} d e^{\frac{15}{2}} +{\left (d^{2}\right )}^{\frac{3}{4}} 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 ) -{\left ({\left (d^{2}\right )}^{\frac{1}{4}} d e^{\frac{11}{2}} +{\left (d^{2}\right )}^{\frac{1}{4}}{\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}{3} \,{\left (x^{3} e^{7} + 9 \, d x e^{6}\right )} e^{\left (-6\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^3/(-e^2*x^4+d^2),x, algorithm="giac")

[Out]

2*((d^2)^(1/4)*d*e^(11/2) - (d^2)^(1/4)*abs(d)*e^(11/2))*arctan(x*e^(1/2)/(d^2)^(1/4))*e^(-6) + ((d^2)^(1/4)*d
*e^(15/2) + (d^2)^(3/4)*e^(15/2))*e^(-8)*log(abs((d^2)^(1/4)*e^(-1/2) + x)) - ((d^2)^(1/4)*d*e^(11/2) + (d^2)^
(1/4)*abs(d)*e^(11/2))*e^(-6)*log(abs(-(d^2)^(1/4)*e^(-1/2) + x)) - 1/3*(x^3*e^7 + 9*d*x*e^6)*e^(-6)