3.2.90 \(\int \frac {(d+e x^2)^3}{d^2-e^2 x^4} \, dx\) [190]

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

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, 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 = {1164, 398, 214} \begin {gather*} \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} \end {gather*}

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 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*} \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*}

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Mathematica [A]
time = 0.01, size = 38, normalized size = 1.00 \begin {gather*} -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 {gather*}

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 = 0.12, size = 31, normalized size = 0.82

method	result	size
default	\(-\frac {e \,x^{3}}{3}-3 d x +\frac {4 d^{2} \arctanh \left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {d e}}\)	\(31\)
risch	\(-\frac {e \,x^{3}}{3}-3 d x +\frac {2 \sqrt {d e}\, d \ln \left (\sqrt {d e}\, x +d \right )}{e}-\frac {2 \sqrt {d e}\, d \ln \left (-\sqrt {d e}\, x +d \right )}{e}\)	\(55\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^2+d)^3/(-e^2*x^4+d^2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.52, size = 46, normalized size = 1.21 \begin {gather*} -\frac {1}{3} \, x^{3} e - 2 \, d^{\frac {3}{2}} e^{\left (-\frac {1}{2}\right )} \log \left (\frac {x e - \sqrt {d} e^{\frac {1}{2}}}{x e + \sqrt {d} e^{\frac {1}{2}}}\right ) - 3 \, d x \end {gather*}

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]

-1/3*x^3*e - 2*d^(3/2)*e^(-1/2)*log((x*e - sqrt(d)*e^(1/2))/(x*e + sqrt(d)*e^(1/2))) - 3*d*x

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Fricas [A]
time = 0.37, size = 87, normalized size = 2.29 \begin {gather*} \left [-\frac {1}{3} \, x^{3} e + 2 \, d^{\frac {3}{2}} e^{\left (-\frac {1}{2}\right )} \log \left (\frac {x^{2} e + 2 \, \sqrt {d} x e^{\frac {1}{2}} + d}{x^{2} e - d}\right ) - 3 \, d x, -\frac {1}{3} \, x^{3} e - 4 \, \sqrt {-d e^{\left (-1\right )}} d \arctan \left (\frac {\sqrt {-d e^{\left (-1\right )}} x e}{d}\right ) - 3 \, d x\right ] \end {gather*}

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

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Sympy [A]
time = 0.08, size = 58, normalized size = 1.53 \begin {gather*} - 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 {gather*}

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 [A]
time = 4.20, size = 42, normalized size = 1.11 \begin {gather*} -\frac {4 \, d^{2} \arctan \left (\frac {x e}{\sqrt {-d e}}\right )}{\sqrt {-d e}} - \frac {1}{3} \, {\left (x^{3} e^{4} + 9 \, d x e^{3}\right )} e^{\left (-3\right )} \end {gather*}

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]

-4*d^2*arctan(x*e/sqrt(-d*e))/sqrt(-d*e) - 1/3*(x^3*e^4 + 9*d*x*e^3)*e^(-3)

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Mupad [B]
time = 0.05, size = 28, normalized size = 0.74 \begin {gather*} \frac {4\,d^{3/2}\,\mathrm {atanh}\left (\frac {\sqrt {e}\,x}{\sqrt {d}}\right )}{\sqrt {e}}-\frac {e\,x^3}{3}-3\,d\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x^2)^3/(d^2 - e^2*x^4),x)

[Out]

(4*d^(3/2)*atanh((e^(1/2)*x)/d^(1/2)))/e^(1/2) - (e*x^3)/3 - 3*d*x

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