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-4/3*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)

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Rubi [A]  time = 0.04, antiderivative size = 51, 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} \[ \frac {8 d^{5/2} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}-7 d^2 x-\frac {4}{3} d e x^3-\frac {1}{5} e^2 x^5 \]

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 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 )^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*}

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Mathematica [A]  time = 0.02, size = 51, normalized size = 1.00 \[ \frac {8 d^{5/2} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}-7 d^2 x-\frac {4}{3} d e x^3-\frac {1}{5} e^2 x^5 \]

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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fricas [A]  time = 0.82, size = 116, normalized size = 2.27 \[ \left [-\frac {1}{5} \, e^{2} x^{5} - \frac {4}{3} \, d e x^{3} + 4 \, d^{2} \sqrt {\frac {d}{e}} \log \left (\frac {e x^{2} + 2 \, e x \sqrt {\frac {d}{e}} + d}{e x^{2} - d}\right ) - 7 \, d^{2} x, -\frac {1}{5} \, e^{2} x^{5} - \frac {4}{3} \, d e x^{3} - 8 \, d^{2} \sqrt {-\frac {d}{e}} \arctan \left (\frac {e x \sqrt {-\frac {d}{e}}}{d}\right ) - 7 \, d^{2} x\right ] \]

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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giac [B]  time = 0.21, size = 144, normalized size = 2.82 \[ 4 \, {\left ({\left (d^{2}\right )}^{\frac {1}{4}} d^{2} e^{\frac {11}{2}} - {\left (d^{2}\right )}^{\frac {1}{4}} d {\left | d \right |} e^{\frac {11}{2}}\right )} \arctan \left (\frac {x e^{\frac {1}{2}}}{{\left (d^{2}\right )}^{\frac {1}{4}}}\right ) e^{\left (-6\right )} + 2 \, {\left ({\left (d^{2}\right )}^{\frac {1}{4}} d^{2} e^{\frac {15}{2}} + {\left (d^{2}\right )}^{\frac {3}{4}} d e^{\frac {15}{2}}\right )} e^{\left (-8\right )} \log \left ({\left | {\left (d^{2}\right )}^{\frac {1}{4}} e^{\left (-\frac {1}{2}\right )} + x \right |}\right ) - 2 \, {\left ({\left (d^{2}\right )}^{\frac {1}{4}} d^{2} e^{\frac {11}{2}} + {\left (d^{2}\right )}^{\frac {1}{4}} d {\left | d \right |} e^{\frac {11}{2}}\right )} e^{\left (-6\right )} \log \left ({\left | -{\left (d^{2}\right )}^{\frac {1}{4}} e^{\left (-\frac {1}{2}\right )} + x \right |}\right ) - \frac {1}{15} \, {\left (3 \, x^{5} e^{12} + 20 \, d x^{3} e^{11} + 105 \, d^{2} x e^{10}\right )} e^{\left (-10\right )} \]

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)

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maple [A]  time = 0.00, size = 42, normalized size = 0.82 \[ -\frac {e^{2} x^{5}}{5}-\frac {4 d e \,x^{3}}{3}+\frac {8 d^{3} \arctanh \left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {d e}}-7 d^{2} x \]

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

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maxima [A]  time = 2.25, size = 56, normalized size = 1.10 \[ -\frac {1}{5} \, e^{2} x^{5} - \frac {4}{3} \, d e x^{3} - \frac {4 \, d^{3} \log \left (\frac {e x - \sqrt {d e}}{e x + \sqrt {d e}}\right )}{\sqrt {d e}} - 7 \, d^{2} x \]

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]

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

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mupad [B]  time = 0.09, size = 42, normalized size = 0.82 \[ -7\,d^2\,x-\frac {e^2\,x^5}{5}-\frac {4\,d\,e\,x^3}{3}-\frac {d^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {e}\,x\,1{}\mathrm {i}}{\sqrt {d}}\right )\,8{}\mathrm {i}}{\sqrt {e}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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sympy [A]  time = 0.24, size = 75, normalized size = 1.47 \[ - 7 d^{2} x - \frac {4 d e x^{3}}{3} - \frac {e^{2} x^{5}}{5} - 4 \sqrt {\frac {d^{5}}{e}} \log {\left (x - \frac {\sqrt {\frac {d^{5}}{e}}}{d^{2}} \right )} + 4 \sqrt {\frac {d^{5}}{e}} \log {\left (x + \frac {\sqrt {\frac {d^{5}}{e}}}{d^{2}} \right )} \]

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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