3.2.40 \(\int \frac {1}{(d+e x^2)^{3/2} (d^2-e^2 x^4)} \, dx\)

Optimal. Leaf size=80 \[ \frac {7 x}{12 d^3 \sqrt {d+e x^2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d+e x^2}}\right )}{4 \sqrt {2} d^3 \sqrt {e}}+\frac {x}{6 d^2 \left (d+e x^2\right )^{3/2}} \]

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Rubi [A]  time = 0.07, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {1150, 414, 527, 12, 377, 208} \begin {gather*} \frac {7 x}{12 d^3 \sqrt {d+e x^2}}+\frac {x}{6 d^2 \left (d+e x^2\right )^{3/2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d+e x^2}}\right )}{4 \sqrt {2} d^3 \sqrt {e}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x/(6*d^2*(d + e*x^2)^(3/2)) + (7*x)/(12*d^3*Sqrt[d + e*x^2]) + ArcTanh[(Sqrt[2]*Sqrt[e]*x)/Sqrt[d + e*x^2]]/(4
*Sqrt[2]*d^3*Sqrt[e])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 377

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 414

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(b*x*(a + b*x^n)^(p + 1)*(
c + d*x^n)^(q + 1))/(a*n*(p + 1)*(b*c - a*d)), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1)*
(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d,
n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomial
Q[a, b, c, d, n, p, q, x]

Rule 527

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> -Simp[
((b*e - a*f)*x*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*n*(b*c - a*d)*(p + 1)), x] + Dist[1/(a*n*(b*c - a*d
)*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)
*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]

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 {1}{\left (d+e x^2\right )^{3/2} \left (d^2-e^2 x^4\right )} \, dx &=\int \frac {1}{\left (d-e x^2\right ) \left (d+e x^2\right )^{5/2}} \, dx\\ &=\frac {x}{6 d^2 \left (d+e x^2\right )^{3/2}}-\frac {\int \frac {-5 d e+2 e^2 x^2}{\left (d-e x^2\right ) \left (d+e x^2\right )^{3/2}} \, dx}{6 d^2 e}\\ &=\frac {x}{6 d^2 \left (d+e x^2\right )^{3/2}}+\frac {7 x}{12 d^3 \sqrt {d+e x^2}}+\frac {\int \frac {3 d^2 e^2}{\left (d-e x^2\right ) \sqrt {d+e x^2}} \, dx}{12 d^4 e^2}\\ &=\frac {x}{6 d^2 \left (d+e x^2\right )^{3/2}}+\frac {7 x}{12 d^3 \sqrt {d+e x^2}}+\frac {\int \frac {1}{\left (d-e x^2\right ) \sqrt {d+e x^2}} \, dx}{4 d^2}\\ &=\frac {x}{6 d^2 \left (d+e x^2\right )^{3/2}}+\frac {7 x}{12 d^3 \sqrt {d+e x^2}}+\frac {\operatorname {Subst}\left (\int \frac {1}{d-2 d e x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )}{4 d^2}\\ &=\frac {x}{6 d^2 \left (d+e x^2\right )^{3/2}}+\frac {7 x}{12 d^3 \sqrt {d+e x^2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d+e x^2}}\right )}{4 \sqrt {2} d^3 \sqrt {e}}\\ \end {align*}

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Mathematica [C]  time = 3.34, size = 345, normalized size = 4.31 \begin {gather*} \frac {\frac {384 e^4 x^8 \left (d+e x^2\right )^2 \, _3F_2\left (2,2,2;1,\frac {9}{2};-\frac {2 e x^2}{d-e x^2}\right )}{e x^2-d}+\frac {384 e^4 x^8 \left (4 d^2+7 d e x^2+3 e^2 x^4\right ) \, _2F_1\left (2,2;\frac {9}{2};-\frac {2 e x^2}{d-e x^2}\right )}{e x^2-d}+\frac {35 \sqrt {2} \sqrt {\frac {e x^2}{e x^2-d}} \left (-15 d^3-5 d^2 e x^2+12 d e^2 x^4+8 e^3 x^6\right ) \left (\sqrt {2} \sqrt {\frac {e x^2}{e x^2-d}} \sqrt {\frac {d+e x^2}{d-e x^2}} \left (-3 d^2-2 d e x^2+5 e^2 x^4\right )+3 \left (d+e x^2\right )^2 \sin ^{-1}\left (\sqrt {2} \sqrt {\frac {e x^2}{e x^2-d}}\right )\right )}{\sqrt {\frac {d+e x^2}{d-e x^2}}}}{2520 d^5 e^3 x^5 \sqrt {d+e x^2} \left (1-\frac {e^2 x^4}{d^2}\right )} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((35*Sqrt[2]*Sqrt[(e*x^2)/(-d + e*x^2)]*(-15*d^3 - 5*d^2*e*x^2 + 12*d*e^2*x^4 + 8*e^3*x^6)*(Sqrt[2]*Sqrt[(e*x^
2)/(-d + e*x^2)]*Sqrt[(d + e*x^2)/(d - e*x^2)]*(-3*d^2 - 2*d*e*x^2 + 5*e^2*x^4) + 3*(d + e*x^2)^2*ArcSin[Sqrt[
2]*Sqrt[(e*x^2)/(-d + e*x^2)]]))/Sqrt[(d + e*x^2)/(d - e*x^2)] + (384*e^4*x^8*(4*d^2 + 7*d*e*x^2 + 3*e^2*x^4)*
Hypergeometric2F1[2, 2, 9/2, (-2*e*x^2)/(d - e*x^2)])/(-d + e*x^2) + (384*e^4*x^8*(d + e*x^2)^2*Hypergeometric
PFQ[{2, 2, 2}, {1, 9/2}, (-2*e*x^2)/(d - e*x^2)])/(-d + e*x^2))/(2520*d^5*e^3*x^5*Sqrt[d + e*x^2]*(1 - (e^2*x^
4)/d^2))

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IntegrateAlgebraic [A]  time = 0.20, size = 94, normalized size = 1.18 \begin {gather*} \frac {\tanh ^{-1}\left (-\frac {e x^2}{\sqrt {2} d}+\frac {\sqrt {e} x \sqrt {d+e x^2}}{\sqrt {2} d}+\frac {1}{\sqrt {2}}\right )}{4 \sqrt {2} d^3 \sqrt {e}}+\frac {9 d x+7 e x^3}{12 d^3 \left (d+e x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(9*d*x + 7*e*x^3)/(12*d^3*(d + e*x^2)^(3/2)) + ArcTanh[1/Sqrt[2] - (e*x^2)/(Sqrt[2]*d) + (Sqrt[e]*x*Sqrt[d + e
*x^2])/(Sqrt[2]*d)]/(4*Sqrt[2]*d^3*Sqrt[e])

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fricas [B]  time = 1.74, size = 279, normalized size = 3.49 \begin {gather*} \left [\frac {3 \, \sqrt {2} {\left (e^{2} x^{4} + 2 \, d e x^{2} + d^{2}\right )} \sqrt {e} \log \left (\frac {17 \, e^{2} x^{4} + 14 \, d e x^{2} + 4 \, \sqrt {2} {\left (3 \, e x^{3} + d x\right )} \sqrt {e x^{2} + d} \sqrt {e} + d^{2}}{e^{2} x^{4} - 2 \, d e x^{2} + d^{2}}\right ) + 8 \, {\left (7 \, e^{2} x^{3} + 9 \, d e x\right )} \sqrt {e x^{2} + d}}{96 \, {\left (d^{3} e^{3} x^{4} + 2 \, d^{4} e^{2} x^{2} + d^{5} e\right )}}, -\frac {3 \, \sqrt {2} {\left (e^{2} x^{4} + 2 \, d e x^{2} + d^{2}\right )} \sqrt {-e} \arctan \left (\frac {\sqrt {2} {\left (3 \, e x^{2} + d\right )} \sqrt {e x^{2} + d} \sqrt {-e}}{4 \, {\left (e^{2} x^{3} + d e x\right )}}\right ) - 4 \, {\left (7 \, e^{2} x^{3} + 9 \, d e x\right )} \sqrt {e x^{2} + d}}{48 \, {\left (d^{3} e^{3} x^{4} + 2 \, d^{4} e^{2} x^{2} + d^{5} e\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/96*(3*sqrt(2)*(e^2*x^4 + 2*d*e*x^2 + d^2)*sqrt(e)*log((17*e^2*x^4 + 14*d*e*x^2 + 4*sqrt(2)*(3*e*x^3 + d*x)*
sqrt(e*x^2 + d)*sqrt(e) + d^2)/(e^2*x^4 - 2*d*e*x^2 + d^2)) + 8*(7*e^2*x^3 + 9*d*e*x)*sqrt(e*x^2 + d))/(d^3*e^
3*x^4 + 2*d^4*e^2*x^2 + d^5*e), -1/48*(3*sqrt(2)*(e^2*x^4 + 2*d*e*x^2 + d^2)*sqrt(-e)*arctan(1/4*sqrt(2)*(3*e*
x^2 + d)*sqrt(e*x^2 + d)*sqrt(-e)/(e^2*x^3 + d*e*x)) - 4*(7*e^2*x^3 + 9*d*e*x)*sqrt(e*x^2 + d))/(d^3*e^3*x^4 +
2*d^4*e^2*x^2 + d^5*e)]

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Unab
le to transpose Error: Bad Argument Value

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maple [B]  time = 0.03, size = 911, normalized size = 11.39 \begin {gather*} -\frac {\sqrt {2}\, e \ln \left (\frac {4 d +2 \sqrt {2}\, \sqrt {2 d +\left (x -\frac {\sqrt {d e}}{e}\right )^{2} e +2 \sqrt {d e}\, \left (x -\frac {\sqrt {d e}}{e}\right )}\, \sqrt {d}+2 \sqrt {d e}\, \left (x -\frac {\sqrt {d e}}{e}\right )}{x -\frac {\sqrt {d e}}{e}}\right )}{8 \sqrt {d e}\, \left (\sqrt {-d e}+\sqrt {d e}\right ) \left (\sqrt {-d e}-\sqrt {d e}\right ) d^{\frac {3}{2}}}+\frac {\sqrt {2}\, e \ln \left (\frac {4 d +2 \sqrt {2}\, \sqrt {2 d +\left (x +\frac {\sqrt {d e}}{e}\right )^{2} e -2 \sqrt {d e}\, \left (x +\frac {\sqrt {d e}}{e}\right )}\, \sqrt {d}-2 \sqrt {d e}\, \left (x +\frac {\sqrt {d e}}{e}\right )}{x +\frac {\sqrt {d e}}{e}}\right )}{8 \sqrt {d e}\, \left (\sqrt {-d e}+\sqrt {d e}\right ) \left (\sqrt {-d e}-\sqrt {d e}\right ) d^{\frac {3}{2}}}+\frac {e}{4 \sqrt {d e}\, \left (\sqrt {-d e}+\sqrt {d e}\right ) \left (\sqrt {-d e}-\sqrt {d e}\right ) \sqrt {2 d +\left (x -\frac {\sqrt {d e}}{e}\right )^{2} e +2 \sqrt {d e}\, \left (x -\frac {\sqrt {d e}}{e}\right )}\, d}-\frac {e}{4 \sqrt {d e}\, \left (\sqrt {-d e}+\sqrt {d e}\right ) \left (\sqrt {-d e}-\sqrt {d e}\right ) \sqrt {2 d +\left (x +\frac {\sqrt {d e}}{e}\right )^{2} e -2 \sqrt {d e}\, \left (x +\frac {\sqrt {d e}}{e}\right )}\, d}-\frac {e x}{3 \left (\sqrt {-d e}+\sqrt {d e}\right ) \left (\sqrt {-d e}-\sqrt {d e}\right ) \sqrt {\left (x -\frac {\sqrt {-d e}}{e}\right )^{2} e +2 \sqrt {-d e}\, \left (x -\frac {\sqrt {-d e}}{e}\right )}\, d^{2}}-\frac {e x}{4 \left (\sqrt {-d e}+\sqrt {d e}\right ) \left (\sqrt {-d e}-\sqrt {d e}\right ) \sqrt {2 d +\left (x -\frac {\sqrt {d e}}{e}\right )^{2} e +2 \sqrt {d e}\, \left (x -\frac {\sqrt {d e}}{e}\right )}\, d^{2}}-\frac {e x}{3 \left (\sqrt {-d e}+\sqrt {d e}\right ) \left (\sqrt {-d e}-\sqrt {d e}\right ) \sqrt {\left (x +\frac {\sqrt {-d e}}{e}\right )^{2} e -2 \sqrt {-d e}\, \left (x +\frac {\sqrt {-d e}}{e}\right )}\, d^{2}}-\frac {e x}{4 \left (\sqrt {-d e}+\sqrt {d e}\right ) \left (\sqrt {-d e}-\sqrt {d e}\right ) \sqrt {2 d +\left (x +\frac {\sqrt {d e}}{e}\right )^{2} e -2 \sqrt {d e}\, \left (x +\frac {\sqrt {d e}}{e}\right )}\, d^{2}}-\frac {1}{6 \left (\sqrt {-d e}+\sqrt {d e}\right ) \left (\sqrt {-d e}-\sqrt {d e}\right ) \left (x -\frac {\sqrt {-d e}}{e}\right ) \sqrt {\left (x -\frac {\sqrt {-d e}}{e}\right )^{2} e +2 \sqrt {-d e}\, \left (x -\frac {\sqrt {-d e}}{e}\right )}\, d}-\frac {1}{6 \left (\sqrt {-d e}+\sqrt {d e}\right ) \left (\sqrt {-d e}-\sqrt {d e}\right ) \left (x +\frac {\sqrt {-d e}}{e}\right ) \sqrt {\left (x +\frac {\sqrt {-d e}}{e}\right )^{2} e -2 \sqrt {-d e}\, \left (x +\frac {\sqrt {-d e}}{e}\right )}\, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6/((-d*e)^(1/2)+(d*e)^(1/2))/((-d*e)^(1/2)-(d*e)^(1/2))/d/(x-(-d*e)^(1/2)/e)/((x-(-d*e)^(1/2)/e)^2*e+2*(-d*
e)^(1/2)*(x-(-d*e)^(1/2)/e))^(1/2)-1/3*e/((-d*e)^(1/2)+(d*e)^(1/2))/((-d*e)^(1/2)-(d*e)^(1/2))/d^2/((x-(-d*e)^
(1/2)/e)^2*e+2*(-d*e)^(1/2)*(x-(-d*e)^(1/2)/e))^(1/2)*x+1/4*e/(d*e)^(1/2)/((-d*e)^(1/2)+(d*e)^(1/2))/((-d*e)^(
1/2)-(d*e)^(1/2))/d/(2*d+(x-(d*e)^(1/2)/e)^2*e+2*(d*e)^(1/2)*(x-(d*e)^(1/2)/e))^(1/2)-1/4*e/((-d*e)^(1/2)+(d*e
)^(1/2))/((-d*e)^(1/2)-(d*e)^(1/2))/d^2/(2*d+(x-(d*e)^(1/2)/e)^2*e+2*(d*e)^(1/2)*(x-(d*e)^(1/2)/e))^(1/2)*x-1/
8*e/(d*e)^(1/2)/((-d*e)^(1/2)+(d*e)^(1/2))/((-d*e)^(1/2)-(d*e)^(1/2))/d^(3/2)*2^(1/2)*ln((4*d+2*2^(1/2)*(2*d+(
x-(d*e)^(1/2)/e)^2*e+2*(d*e)^(1/2)*(x-(d*e)^(1/2)/e))^(1/2)*d^(1/2)+2*(d*e)^(1/2)*(x-(d*e)^(1/2)/e))/(x-(d*e)^
(1/2)/e))-1/6/((-d*e)^(1/2)+(d*e)^(1/2))/((-d*e)^(1/2)-(d*e)^(1/2))/d/(x+(-d*e)^(1/2)/e)/((x+(-d*e)^(1/2)/e)^2
*e-2*(-d*e)^(1/2)*(x+(-d*e)^(1/2)/e))^(1/2)-1/3*e/((-d*e)^(1/2)+(d*e)^(1/2))/((-d*e)^(1/2)-(d*e)^(1/2))/d^2/((
x+(-d*e)^(1/2)/e)^2*e-2*(-d*e)^(1/2)*(x+(-d*e)^(1/2)/e))^(1/2)*x-1/4*e/(d*e)^(1/2)/((-d*e)^(1/2)+(d*e)^(1/2))/
((-d*e)^(1/2)-(d*e)^(1/2))/d/(2*d+(x+(d*e)^(1/2)/e)^2*e-2*(d*e)^(1/2)*(x+(d*e)^(1/2)/e))^(1/2)-1/4*e/((-d*e)^(
1/2)+(d*e)^(1/2))/((-d*e)^(1/2)-(d*e)^(1/2))/d^2/(2*d+(x+(d*e)^(1/2)/e)^2*e-2*(d*e)^(1/2)*(x+(d*e)^(1/2)/e))^(
1/2)*x+1/8*e/(d*e)^(1/2)/((-d*e)^(1/2)+(d*e)^(1/2))/((-d*e)^(1/2)-(d*e)^(1/2))/d^(3/2)*2^(1/2)*ln((4*d+2*2^(1/
2)*(2*d+(x+(d*e)^(1/2)/e)^2*e-2*(d*e)^(1/2)*(x+(d*e)^(1/2)/e))^(1/2)*d^(1/2)-2*(d*e)^(1/2)*(x+(d*e)^(1/2)/e))/
(x+(d*e)^(1/2)/e))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\int \frac {1}{{\left (e^{2} x^{4} - d^{2}\right )} {\left (e x^{2} + d\right )}^{\frac {3}{2}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\left (d^2-e^2\,x^4\right )\,{\left (e\,x^2+d\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {1}{- d^{3} \sqrt {d + e x^{2}} - d^{2} e x^{2} \sqrt {d + e x^{2}} + d e^{2} x^{4} \sqrt {d + e x^{2}} + e^{3} x^{6} \sqrt {d + e x^{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x**2+d)**(3/2)/(-e**2*x**4+d**2),x)

[Out]

-Integral(1/(-d**3*sqrt(d + e*x**2) - d**2*e*x**2*sqrt(d + e*x**2) + d*e**2*x**4*sqrt(d + e*x**2) + e**3*x**6*
sqrt(d + e*x**2)), x)

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