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

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

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Rubi [A]  time = 0.03, 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 = {1150, 390, 208} \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 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]

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 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]

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 = 0.02, size = 38, normalized size = 1.00 \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]

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d+e x^2\right )^3}{d^2-e^2 x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.87, size = 90, normalized size = 2.37 \begin {gather*} \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 {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*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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giac [B]  time = 0.23, size = 123, normalized size = 3.24 \begin {gather*} 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 {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]

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)

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maple [A]  time = 0.00, size = 31, normalized size = 0.82 \begin {gather*} -\frac {e \,x^{3}}{3}+\frac {4 d^{2} \arctanh \left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {d e}}-3 d x \end {gather*}

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(1/(d*e)^(1/2)*e*x)

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maxima [A]  time = 2.45, size = 45, normalized size = 1.18 \begin {gather*} -\frac {1}{3} \, e x^{3} - \frac {2 \, d^{2} \log \left (\frac {e x - \sqrt {d e}}{e x + \sqrt {d e}}\right )}{\sqrt {d e}} - 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*e*x^3 - 2*d^2*log((e*x - sqrt(d*e))/(e*x + sqrt(d*e)))/sqrt(d*e) - 3*d*x

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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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sympy [A]  time = 0.20, 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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