### 3.51 $$\int \frac{d+e x^4}{x^4 (a+b x^4+c x^8)} \, dx$$

Optimal. Leaf size=394 $\frac{c^{3/4} \left (d-\frac{b d-2 a e}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{2 \sqrt [4]{2} a \left (-\sqrt{b^2-4 a c}-b\right )^{3/4}}+\frac{c^{3/4} \left (\frac{b d-2 a e}{\sqrt{b^2-4 a c}}+d\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{2 \sqrt [4]{2} a \left (\sqrt{b^2-4 a c}-b\right )^{3/4}}+\frac{c^{3/4} \left (d-\frac{b d-2 a e}{\sqrt{b^2-4 a c}}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{2 \sqrt [4]{2} a \left (-\sqrt{b^2-4 a c}-b\right )^{3/4}}+\frac{c^{3/4} \left (\frac{b d-2 a e}{\sqrt{b^2-4 a c}}+d\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{2 \sqrt [4]{2} a \left (\sqrt{b^2-4 a c}-b\right )^{3/4}}-\frac{d}{3 a x^3}$

[Out]

-d/(3*a*x^3) + (c^(3/4)*(d - (b*d - 2*a*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(2^(1/4)*c^(1/4)*x)/(-b - Sqrt[b^2 - 4*a*
c])^(1/4)])/(2*2^(1/4)*a*(-b - Sqrt[b^2 - 4*a*c])^(3/4)) + (c^(3/4)*(d + (b*d - 2*a*e)/Sqrt[b^2 - 4*a*c])*ArcT
an[(2^(1/4)*c^(1/4)*x)/(-b + Sqrt[b^2 - 4*a*c])^(1/4)])/(2*2^(1/4)*a*(-b + Sqrt[b^2 - 4*a*c])^(3/4)) + (c^(3/4
)*(d - (b*d - 2*a*e)/Sqrt[b^2 - 4*a*c])*ArcTanh[(2^(1/4)*c^(1/4)*x)/(-b - Sqrt[b^2 - 4*a*c])^(1/4)])/(2*2^(1/4
)*a*(-b - Sqrt[b^2 - 4*a*c])^(3/4)) + (c^(3/4)*(d + (b*d - 2*a*e)/Sqrt[b^2 - 4*a*c])*ArcTanh[(2^(1/4)*c^(1/4)*
x)/(-b + Sqrt[b^2 - 4*a*c])^(1/4)])/(2*2^(1/4)*a*(-b + Sqrt[b^2 - 4*a*c])^(3/4))

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Rubi [A]  time = 0.625247, antiderivative size = 394, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 25, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.2, Rules used = {1504, 1422, 212, 208, 205} $\frac{c^{3/4} \left (d-\frac{b d-2 a e}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{2 \sqrt [4]{2} a \left (-\sqrt{b^2-4 a c}-b\right )^{3/4}}+\frac{c^{3/4} \left (\frac{b d-2 a e}{\sqrt{b^2-4 a c}}+d\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{2 \sqrt [4]{2} a \left (\sqrt{b^2-4 a c}-b\right )^{3/4}}+\frac{c^{3/4} \left (d-\frac{b d-2 a e}{\sqrt{b^2-4 a c}}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{2 \sqrt [4]{2} a \left (-\sqrt{b^2-4 a c}-b\right )^{3/4}}+\frac{c^{3/4} \left (\frac{b d-2 a e}{\sqrt{b^2-4 a c}}+d\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{2 \sqrt [4]{2} a \left (\sqrt{b^2-4 a c}-b\right )^{3/4}}-\frac{d}{3 a x^3}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x^4)/(x^4*(a + b*x^4 + c*x^8)),x]

[Out]

-d/(3*a*x^3) + (c^(3/4)*(d - (b*d - 2*a*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(2^(1/4)*c^(1/4)*x)/(-b - Sqrt[b^2 - 4*a*
c])^(1/4)])/(2*2^(1/4)*a*(-b - Sqrt[b^2 - 4*a*c])^(3/4)) + (c^(3/4)*(d + (b*d - 2*a*e)/Sqrt[b^2 - 4*a*c])*ArcT
an[(2^(1/4)*c^(1/4)*x)/(-b + Sqrt[b^2 - 4*a*c])^(1/4)])/(2*2^(1/4)*a*(-b + Sqrt[b^2 - 4*a*c])^(3/4)) + (c^(3/4
)*(d - (b*d - 2*a*e)/Sqrt[b^2 - 4*a*c])*ArcTanh[(2^(1/4)*c^(1/4)*x)/(-b - Sqrt[b^2 - 4*a*c])^(1/4)])/(2*2^(1/4
)*a*(-b - Sqrt[b^2 - 4*a*c])^(3/4)) + (c^(3/4)*(d + (b*d - 2*a*e)/Sqrt[b^2 - 4*a*c])*ArcTanh[(2^(1/4)*c^(1/4)*
x)/(-b + Sqrt[b^2 - 4*a*c])^(1/4)])/(2*2^(1/4)*a*(-b + Sqrt[b^2 - 4*a*c])^(3/4))

Rule 1504

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))*((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(p_), x_Symbol] :>
Simp[(d*(f*x)^(m + 1)*(a + b*x^n + c*x^(2*n))^(p + 1))/(a*f*(m + 1)), x] + Dist[1/(a*f^n*(m + 1)), Int[(f*x)^
(m + n)*(a + b*x^n + c*x^(2*n))^p*Simp[a*e*(m + 1) - b*d*(m + n*(p + 1) + 1) - c*d*(m + 2*n*(p + 1) + 1)*x^n,
x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && LtQ[m, -
1] && IntegerQ[p]

Rule 1422

Int[((d_) + (e_.)*(x_)^(n_))/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*
c, 2]}, Dist[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^n), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), In
t[1/(b/2 + q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && NeQ
[c*d^2 - b*d*e + a*e^2, 0] && (PosQ[b^2 - 4*a*c] ||  !IGtQ[n/2, 0])

Rule 212

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b), 2]
]}, Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&
!GtQ[a/b, 0]

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 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{d+e x^4}{x^4 \left (a+b x^4+c x^8\right )} \, dx &=-\frac{d}{3 a x^3}-\frac{\int \frac{3 (b d-a e)+3 c d x^4}{a+b x^4+c x^8} \, dx}{3 a}\\ &=-\frac{d}{3 a x^3}-\frac{\left (c \left (d-\frac{b d-2 a e}{\sqrt{b^2-4 a c}}\right )\right ) \int \frac{1}{\frac{b}{2}+\frac{1}{2} \sqrt{b^2-4 a c}+c x^4} \, dx}{2 a}-\frac{\left (c \left (d+\frac{b d-2 a e}{\sqrt{b^2-4 a c}}\right )\right ) \int \frac{1}{\frac{b}{2}-\frac{1}{2} \sqrt{b^2-4 a c}+c x^4} \, dx}{2 a}\\ &=-\frac{d}{3 a x^3}+\frac{\left (c \left (d-\frac{b d-2 a e}{\sqrt{b^2-4 a c}}\right )\right ) \int \frac{1}{\sqrt{-b-\sqrt{b^2-4 a c}}-\sqrt{2} \sqrt{c} x^2} \, dx}{2 a \sqrt{-b-\sqrt{b^2-4 a c}}}+\frac{\left (c \left (d-\frac{b d-2 a e}{\sqrt{b^2-4 a c}}\right )\right ) \int \frac{1}{\sqrt{-b-\sqrt{b^2-4 a c}}+\sqrt{2} \sqrt{c} x^2} \, dx}{2 a \sqrt{-b-\sqrt{b^2-4 a c}}}+\frac{\left (c \left (d+\frac{b d-2 a e}{\sqrt{b^2-4 a c}}\right )\right ) \int \frac{1}{\sqrt{-b+\sqrt{b^2-4 a c}}-\sqrt{2} \sqrt{c} x^2} \, dx}{2 a \sqrt{-b+\sqrt{b^2-4 a c}}}+\frac{\left (c \left (d+\frac{b d-2 a e}{\sqrt{b^2-4 a c}}\right )\right ) \int \frac{1}{\sqrt{-b+\sqrt{b^2-4 a c}}+\sqrt{2} \sqrt{c} x^2} \, dx}{2 a \sqrt{-b+\sqrt{b^2-4 a c}}}\\ &=-\frac{d}{3 a x^3}+\frac{c^{3/4} \left (d-\frac{b d-2 a e}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-b-\sqrt{b^2-4 a c}}}\right )}{2 \sqrt [4]{2} a \left (-b-\sqrt{b^2-4 a c}\right )^{3/4}}+\frac{c^{3/4} \left (d+\frac{b d-2 a e}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-b+\sqrt{b^2-4 a c}}}\right )}{2 \sqrt [4]{2} a \left (-b+\sqrt{b^2-4 a c}\right )^{3/4}}+\frac{c^{3/4} \left (d-\frac{b d-2 a e}{\sqrt{b^2-4 a c}}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-b-\sqrt{b^2-4 a c}}}\right )}{2 \sqrt [4]{2} a \left (-b-\sqrt{b^2-4 a c}\right )^{3/4}}+\frac{c^{3/4} \left (d+\frac{b d-2 a e}{\sqrt{b^2-4 a c}}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-b+\sqrt{b^2-4 a c}}}\right )}{2 \sqrt [4]{2} a \left (-b+\sqrt{b^2-4 a c}\right )^{3/4}}\\ \end{align*}

Mathematica [C]  time = 0.0735016, size = 86, normalized size = 0.22 $-\frac{3 \text{RootSum}\left [\text{\#1}^4 b+\text{\#1}^8 c+a\& ,\frac{\text{\#1}^4 c d \log (x-\text{\#1})-a e \log (x-\text{\#1})+b d \log (x-\text{\#1})}{\text{\#1}^3 b+2 \text{\#1}^7 c}\& \right ]+\frac{4 d}{x^3}}{12 a}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x^4)/(x^4*(a + b*x^4 + c*x^8)),x]

[Out]

-((4*d)/x^3 + 3*RootSum[a + b*#1^4 + c*#1^8 & , (b*d*Log[x - #1] - a*e*Log[x - #1] + c*d*Log[x - #1]*#1^4)/(b*
#1^3 + 2*c*#1^7) & ])/(12*a)

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Maple [C]  time = 0.007, size = 68, normalized size = 0.2 \begin{align*} -{\frac{d}{3\,a{x}^{3}}}+{\frac{1}{4\,a}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}c+{{\it \_Z}}^{4}b+a \right ) }{\frac{ \left ( -cd{{\it \_R}}^{4}+ae-bd \right ) \ln \left ( x-{\it \_R} \right ) }{2\,{{\it \_R}}^{7}c+{{\it \_R}}^{3}b}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^4+d)/x^4/(c*x^8+b*x^4+a),x)

[Out]

-1/3*d/a/x^3+1/4/a*sum((-_R^4*c*d+a*e-b*d)/(2*_R^7*c+_R^3*b)*ln(x-_R),_R=RootOf(_Z^8*c+_Z^4*b+a))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{\mathit{sage}_{2}}{a} - \frac{d}{3 \, a x^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^4+d)/x^4/(c*x^8+b*x^4+a),x, algorithm="maxima")

[Out]

-integrate((c*d*x^4 + b*d - a*e)/(c*x^8 + b*x^4 + a), x)/a - 1/3*d/(a*x^3)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^4+d)/x^4/(c*x^8+b*x^4+a),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x**4+d)/x**4/(c*x**8+b*x**4+a),x)

[Out]

Timed out

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^4+d)/x^4/(c*x^8+b*x^4+a),x, algorithm="giac")

[Out]

Exception raised: TypeError