3.171 \(\int \frac{c+d x+e x^2+f x^3+g x^4}{a-b x^4} \, dx\)

Optimal. Leaf size=148 \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-\sqrt{a} \sqrt{b} e+a g+b c\right )}{2 a^{3/4} b^{5/4}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\sqrt{a} \sqrt{b} e+a g+b c\right )}{2 a^{3/4} b^{5/4}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} \sqrt{b}}-\frac{f \log \left (a-b x^4\right )}{4 b}-\frac{g x}{b} \]

[Out]

-((g*x)/b) + ((b*c - Sqrt[a]*Sqrt[b]*e + a*g)*ArcTan[(b^(1/4)*x)/a^(1/4)])/(2*a^(3/4)*b^(5/4)) + ((b*c + Sqrt[
a]*Sqrt[b]*e + a*g)*ArcTanh[(b^(1/4)*x)/a^(1/4)])/(2*a^(3/4)*b^(5/4)) + (d*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a]])/(2*
Sqrt[a]*Sqrt[b]) - (f*Log[a - b*x^4])/(4*b)

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Rubi [A]  time = 0.203053, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.258, Rules used = {1885, 1248, 635, 208, 260, 1887, 1167, 205} \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-\sqrt{a} \sqrt{b} e+a g+b c\right )}{2 a^{3/4} b^{5/4}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\sqrt{a} \sqrt{b} e+a g+b c\right )}{2 a^{3/4} b^{5/4}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} \sqrt{b}}-\frac{f \log \left (a-b x^4\right )}{4 b}-\frac{g x}{b} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x + e*x^2 + f*x^3 + g*x^4)/(a - b*x^4),x]

[Out]

-((g*x)/b) + ((b*c - Sqrt[a]*Sqrt[b]*e + a*g)*ArcTan[(b^(1/4)*x)/a^(1/4)])/(2*a^(3/4)*b^(5/4)) + ((b*c + Sqrt[
a]*Sqrt[b]*e + a*g)*ArcTanh[(b^(1/4)*x)/a^(1/4)])/(2*a^(3/4)*b^(5/4)) + (d*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a]])/(2*
Sqrt[a]*Sqrt[b]) - (f*Log[a - b*x^4])/(4*b)

Rule 1885

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], j, k}, Int[Sum[x^j*Sum[Coeff[P
q, x, j + (k*n)/2]*x^((k*n)/2), {k, 0, (2*(q - j))/n + 1}]*(a + b*x^n)^p, {j, 0, n/2 - 1}], x]] /; FreeQ[{a, b
, p}, x] && PolyQ[Pq, x] && IGtQ[n/2, 0] &&  !PolyQ[Pq, x^(n/2)]

Rule 1248

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[(d + e*x)^q
*(a + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, c, d, e, p, q}, x]

Rule 635

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[-(a*c)]

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 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 1887

Int[(Pq_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[Pq/(a + b*x^n), x], x] /; FreeQ[{a, b}, x
] && PolyQ[Pq, x] && IntegerQ[n]

Rule 1167

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-(a*c), 2]}, Dist[e/2 + (c*d)/(2*q)
, Int[1/(-q + c*x^2), x], x] + Dist[e/2 - (c*d)/(2*q), Int[1/(q + c*x^2), x], x]] /; FreeQ[{a, c, d, e}, x] &&
NeQ[c*d^2 - a*e^2, 0] && PosQ[-(a*c)]

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{c+d x+e x^2+f x^3+g x^4}{a-b x^4} \, dx &=\int \left (\frac{x \left (d+f x^2\right )}{a-b x^4}+\frac{c+e x^2+g x^4}{a-b x^4}\right ) \, dx\\ &=\int \frac{x \left (d+f x^2\right )}{a-b x^4} \, dx+\int \frac{c+e x^2+g x^4}{a-b x^4} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{d+f x}{a-b x^2} \, dx,x,x^2\right )+\int \left (-\frac{g}{b}+\frac{b c+a g+b e x^2}{b \left (a-b x^4\right )}\right ) \, dx\\ &=-\frac{g x}{b}+\frac{\int \frac{b c+a g+b e x^2}{a-b x^4} \, dx}{b}+\frac{1}{2} d \operatorname{Subst}\left (\int \frac{1}{a-b x^2} \, dx,x,x^2\right )+\frac{1}{2} f \operatorname{Subst}\left (\int \frac{x}{a-b x^2} \, dx,x,x^2\right )\\ &=-\frac{g x}{b}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} \sqrt{b}}-\frac{f \log \left (a-b x^4\right )}{4 b}+\frac{1}{2} \left (e-\frac{b c+a g}{\sqrt{a} \sqrt{b}}\right ) \int \frac{1}{-\sqrt{a} \sqrt{b}-b x^2} \, dx+\frac{1}{2} \left (e+\frac{b c+a g}{\sqrt{a} \sqrt{b}}\right ) \int \frac{1}{\sqrt{a} \sqrt{b}-b x^2} \, dx\\ &=-\frac{g x}{b}+\frac{\left (b c-\sqrt{a} \sqrt{b} e+a g\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{5/4}}+\frac{\left (b c+\sqrt{a} \sqrt{b} e+a g\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{5/4}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} \sqrt{b}}-\frac{f \log \left (a-b x^4\right )}{4 b}\\ \end{align*}

Mathematica [A]  time = 0.0812388, size = 249, normalized size = 1.68 \[ \frac{-a^{3/4} \sqrt [4]{b} f \log \left (a-b x^4\right )-4 a^{3/4} \sqrt [4]{b} g x-\log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right ) \left (\sqrt [4]{a} b^{3/4} d+\sqrt{a} \sqrt{b} e+a g+b c\right )+\sqrt [4]{a} b^{3/4} d \log \left (\sqrt{a}+\sqrt{b} x^2\right )-\sqrt [4]{a} b^{3/4} d \log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right )+2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-\sqrt{a} \sqrt{b} e+a g+b c\right )+b c \log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right )+\sqrt{a} \sqrt{b} e \log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right )+a g \log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right )}{4 a^{3/4} b^{5/4}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4)/(a - b*x^4),x]

[Out]

(-4*a^(3/4)*b^(1/4)*g*x + 2*(b*c - Sqrt[a]*Sqrt[b]*e + a*g)*ArcTan[(b^(1/4)*x)/a^(1/4)] - (b*c + a^(1/4)*b^(3/
4)*d + Sqrt[a]*Sqrt[b]*e + a*g)*Log[a^(1/4) - b^(1/4)*x] + b*c*Log[a^(1/4) + b^(1/4)*x] - a^(1/4)*b^(3/4)*d*Lo
g[a^(1/4) + b^(1/4)*x] + Sqrt[a]*Sqrt[b]*e*Log[a^(1/4) + b^(1/4)*x] + a*g*Log[a^(1/4) + b^(1/4)*x] + a^(1/4)*b
^(3/4)*d*Log[Sqrt[a] + Sqrt[b]*x^2] - a^(3/4)*b^(1/4)*f*Log[a - b*x^4])/(4*a^(3/4)*b^(5/4))

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Maple [B]  time = 0.005, size = 244, normalized size = 1.7 \begin{align*} -{\frac{gx}{b}}+{\frac{g}{2\,b}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ) }+{\frac{c}{2\,a}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ) }+{\frac{g}{4\,b}\sqrt [4]{{\frac{a}{b}}}\ln \left ({ \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{c}{4\,a}\sqrt [4]{{\frac{a}{b}}}\ln \left ({ \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ) }-{\frac{d}{4}\ln \left ({ \left ( -a+{x}^{2}\sqrt{ab} \right ) \left ( -a-{x}^{2}\sqrt{ab} \right ) ^{-1}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{e}{2\,b}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{e}{4\,b}\ln \left ({ \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{f\ln \left ( b{x}^{4}-a \right ) }{4\,b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x^4+f*x^3+e*x^2+d*x+c)/(-b*x^4+a),x)

[Out]

-g*x/b+1/2/b*(1/b*a)^(1/4)*arctan(x/(1/b*a)^(1/4))*g+1/2*c*(1/b*a)^(1/4)/a*arctan(x/(1/b*a)^(1/4))+1/4/b*(1/b*
a)^(1/4)*ln((x+(1/b*a)^(1/4))/(x-(1/b*a)^(1/4)))*g+1/4*c*(1/b*a)^(1/4)/a*ln((x+(1/b*a)^(1/4))/(x-(1/b*a)^(1/4)
))-1/4*d/(a*b)^(1/2)*ln((-a+x^2*(a*b)^(1/2))/(-a-x^2*(a*b)^(1/2)))-1/2*e/b/(1/b*a)^(1/4)*arctan(x/(1/b*a)^(1/4
))+1/4*e/b/(1/b*a)^(1/4)*ln((x+(1/b*a)^(1/4))/(x-(1/b*a)^(1/4)))-1/4/b*f*ln(b*x^4-a)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x^4+f*x^3+e*x^2+d*x+c)/(-b*x^4+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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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((g*x^4+f*x^3+e*x^2+d*x+c)/(-b*x^4+a),x, algorithm="fricas")

[Out]

Timed out

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Sympy [B]  time = 156.777, size = 2394, normalized size = 16.18 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x**4+f*x**3+e*x**2+d*x+c)/(-b*x**4+a),x)

[Out]

-RootSum(256*_t**4*a**3*b**5 - 256*_t**3*a**3*b**4*f + _t**2*(-64*a**3*b**3*e*g + 96*a**3*b**3*f**2 - 64*a**2*
b**4*c*e - 32*a**2*b**4*d**2) + _t*(-16*a**3*b**2*d*g**2 + 32*a**3*b**2*e*f*g - 16*a**3*b**2*f**3 - 32*a**2*b*
*3*c*d*g + 32*a**2*b**3*c*e*f + 16*a**2*b**3*d**2*f - 16*a**2*b**3*d*e**2 - 16*a*b**4*c**2*d) - a**4*g**4 - 4*
a**3*b*c*g**3 + 4*a**3*b*d*f*g**2 + 2*a**3*b*e**2*g**2 - 4*a**3*b*e*f**2*g + a**3*b*f**4 - 6*a**2*b**2*c**2*g*
*2 + 8*a**2*b**2*c*d*f*g + 4*a**2*b**2*c*e**2*g - 4*a**2*b**2*c*e*f**2 - 4*a**2*b**2*d**2*e*g - 2*a**2*b**2*d*
*2*f**2 + 4*a**2*b**2*d*e**2*f - a**2*b**2*e**4 - 4*a*b**3*c**3*g + 4*a*b**3*c**2*d*f + 2*a*b**3*c**2*e**2 - 4
*a*b**3*c*d**2*e + a*b**3*d**4 - b**4*c**4, Lambda(_t, _t*log(x + (64*_t**3*a**5*b**4*e*g**2 + 128*_t**3*a**4*
b**5*c*e*g - 128*_t**3*a**4*b**5*d**2*g + 64*_t**3*a**4*b**5*e**3 + 64*_t**3*a**3*b**6*c**2*e - 128*_t**3*a**3
*b**6*c*d**2 + 16*_t**2*a**5*b**3*d*g**3 - 48*_t**2*a**5*b**3*e*f*g**2 + 48*_t**2*a**4*b**4*c*d*g**2 - 96*_t**
2*a**4*b**4*c*e*f*g + 96*_t**2*a**4*b**4*d**2*f*g - 48*_t**2*a**4*b**4*d*e**2*g - 48*_t**2*a**4*b**4*e**3*f +
48*_t**2*a**3*b**5*c**2*d*g - 48*_t**2*a**3*b**5*c**2*e*f + 96*_t**2*a**3*b**5*c*d**2*f - 48*_t**2*a**3*b**5*c
*d*e**2 + 32*_t**2*a**3*b**5*d**3*e + 16*_t**2*a**2*b**6*c**3*d - 4*_t*a**6*b*g**5 - 20*_t*a**5*b**2*c*g**4 -
8*_t*a**5*b**2*d*f*g**3 - 16*_t*a**5*b**2*e**2*g**3 + 12*_t*a**5*b**2*e*f**2*g**2 - 40*_t*a**4*b**3*c**2*g**3
- 24*_t*a**4*b**3*c*d*f*g**2 - 48*_t*a**4*b**3*c*e**2*g**2 + 24*_t*a**4*b**3*c*e*f**2*g + 36*_t*a**4*b**3*d**2
*e*g**2 - 24*_t*a**4*b**3*d**2*f**2*g + 24*_t*a**4*b**3*d*e**2*f*g - 12*_t*a**4*b**3*e**4*g + 12*_t*a**4*b**3*
e**3*f**2 - 40*_t*a**3*b**4*c**3*g**2 - 24*_t*a**3*b**4*c**2*d*f*g - 48*_t*a**3*b**4*c**2*e**2*g + 12*_t*a**3*
b**4*c**2*e*f**2 + 72*_t*a**3*b**4*c*d**2*e*g - 24*_t*a**3*b**4*c*d**2*f**2 + 24*_t*a**3*b**4*c*d*e**2*f - 12*
_t*a**3*b**4*c*e**4 + 8*_t*a**3*b**4*d**4*g - 16*_t*a**3*b**4*d**3*e*f - 12*_t*a**3*b**4*d**2*e**3 - 20*_t*a**
2*b**5*c**4*g - 8*_t*a**2*b**5*c**3*d*f - 16*_t*a**2*b**5*c**3*e**2 + 36*_t*a**2*b**5*c**2*d**2*e + 8*_t*a**2*
b**5*c*d**4 - 4*_t*a*b**6*c**5 + a**6*f*g**5 + 5*a**5*b*c*f*g**4 - 5*a**5*b*d*e*g**4 + a**5*b*d*f**2*g**3 + 4*
a**5*b*e**2*f*g**3 - a**5*b*e*f**3*g**2 + 10*a**4*b**2*c**2*f*g**3 - 20*a**4*b**2*c*d*e*g**3 + 3*a**4*b**2*c*d
*f**2*g**2 + 12*a**4*b**2*c*e**2*f*g**2 - 2*a**4*b**2*c*e*f**3*g + 5*a**4*b**2*d**3*g**3 - 9*a**4*b**2*d**2*e*
f*g**2 + 2*a**4*b**2*d**2*f**3*g - 3*a**4*b**2*d*e**2*f**2*g + 3*a**4*b**2*e**4*f*g - a**4*b**2*e**3*f**3 + 10
*a**3*b**3*c**3*f*g**2 - 30*a**3*b**3*c**2*d*e*g**2 + 3*a**3*b**3*c**2*d*f**2*g + 12*a**3*b**3*c**2*e**2*f*g -
a**3*b**3*c**2*e*f**3 + 15*a**3*b**3*c*d**3*g**2 - 18*a**3*b**3*c*d**2*e*f*g + 2*a**3*b**3*c*d**2*f**3 - 3*a*
*3*b**3*c*d*e**2*f**2 + 3*a**3*b**3*c*e**4*f - 2*a**3*b**3*d**4*f*g + 5*a**3*b**3*d**3*e**2*g + 2*a**3*b**3*d*
*3*e*f**2 + 3*a**3*b**3*d**2*e**3*f - 3*a**3*b**3*d*e**5 + 5*a**2*b**4*c**4*f*g - 20*a**2*b**4*c**3*d*e*g + a*
*2*b**4*c**3*d*f**2 + 4*a**2*b**4*c**3*e**2*f + 15*a**2*b**4*c**2*d**3*g - 9*a**2*b**4*c**2*d**2*e*f - 2*a**2*
b**4*c*d**4*f + 5*a**2*b**4*c*d**3*e**2 - 2*a**2*b**4*d**5*e + a*b**5*c**5*f - 5*a*b**5*c**4*d*e + 5*a*b**5*c*
*3*d**3)/(a**6*g**6 + 6*a**5*b*c*g**5 + a**5*b*e**2*g**4 + 15*a**4*b**2*c**2*g**4 + 4*a**4*b**2*c*e**2*g**3 -
8*a**4*b**2*d**2*e*g**3 - a**4*b**2*e**4*g**2 + 20*a**3*b**3*c**3*g**3 + 6*a**3*b**3*c**2*e**2*g**2 - 24*a**3*
b**3*c*d**2*e*g**2 - 2*a**3*b**3*c*e**4*g + 4*a**3*b**3*d**4*g**2 + 8*a**3*b**3*d**2*e**3*g - a**3*b**3*e**6 +
15*a**2*b**4*c**4*g**2 + 4*a**2*b**4*c**3*e**2*g - 24*a**2*b**4*c**2*d**2*e*g - a**2*b**4*c**2*e**4 + 8*a**2*
b**4*c*d**4*g + 8*a**2*b**4*c*d**2*e**3 - 4*a**2*b**4*d**4*e**2 + 6*a*b**5*c**5*g + a*b**5*c**4*e**2 - 8*a*b**
5*c**3*d**2*e + 4*a*b**5*c**2*d**4 + b**6*c**6)))) - g*x/b

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Giac [B]  time = 1.08823, size = 539, normalized size = 3.64 \begin{align*} -\frac{g x}{b} - \frac{f \log \left ({\left | b x^{4} - a \right |}\right )}{4 \, b} + \frac{\sqrt{2}{\left (\sqrt{2} \sqrt{-a b} a b^{4} d + \left (-a b^{3}\right )^{\frac{1}{4}} a b^{4} c + \left (-a b^{3}\right )^{\frac{1}{4}} a^{2} b^{3} g + \left (-a b^{3}\right )^{\frac{3}{4}} a b^{2} e\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (-\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (-\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{4 \, a^{2} b^{5}} + \frac{\sqrt{2}{\left (\sqrt{2} \sqrt{-a b} a b^{4} d + \left (-a b^{3}\right )^{\frac{1}{4}} a b^{4} c + \left (-a b^{3}\right )^{\frac{1}{4}} a^{2} b^{3} g + \left (-a b^{3}\right )^{\frac{3}{4}} a b^{2} e\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (-\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (-\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{4 \, a^{2} b^{5}} + \frac{\sqrt{2}{\left (\left (-a b^{3}\right )^{\frac{1}{4}} a b^{4} c + \left (-a b^{3}\right )^{\frac{1}{4}} a^{2} b^{3} g - \left (-a b^{3}\right )^{\frac{3}{4}} a b^{2} e\right )} \log \left (x^{2} + \sqrt{2} x \left (-\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{-\frac{a}{b}}\right )}{8 \, a^{2} b^{5}} - \frac{\sqrt{2}{\left (\left (-a b^{3}\right )^{\frac{1}{4}} a b^{4} c + \left (-a b^{3}\right )^{\frac{1}{4}} a^{2} b^{3} g - \left (-a b^{3}\right )^{\frac{3}{4}} a b^{2} e\right )} \log \left (x^{2} - \sqrt{2} x \left (-\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{-\frac{a}{b}}\right )}{8 \, a^{2} b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x^4+f*x^3+e*x^2+d*x+c)/(-b*x^4+a),x, algorithm="giac")

[Out]

-g*x/b - 1/4*f*log(abs(b*x^4 - a))/b + 1/4*sqrt(2)*(sqrt(2)*sqrt(-a*b)*a*b^4*d + (-a*b^3)^(1/4)*a*b^4*c + (-a*
b^3)^(1/4)*a^2*b^3*g + (-a*b^3)^(3/4)*a*b^2*e)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(-a/b)^(1/4))/(-a/b)^(1/4))/(
a^2*b^5) + 1/4*sqrt(2)*(sqrt(2)*sqrt(-a*b)*a*b^4*d + (-a*b^3)^(1/4)*a*b^4*c + (-a*b^3)^(1/4)*a^2*b^3*g + (-a*b
^3)^(3/4)*a*b^2*e)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(-a/b)^(1/4))/(-a/b)^(1/4))/(a^2*b^5) + 1/8*sqrt(2)*((-a*
b^3)^(1/4)*a*b^4*c + (-a*b^3)^(1/4)*a^2*b^3*g - (-a*b^3)^(3/4)*a*b^2*e)*log(x^2 + sqrt(2)*x*(-a/b)^(1/4) + sqr
t(-a/b))/(a^2*b^5) - 1/8*sqrt(2)*((-a*b^3)^(1/4)*a*b^4*c + (-a*b^3)^(1/4)*a^2*b^3*g - (-a*b^3)^(3/4)*a*b^2*e)*
log(x^2 - sqrt(2)*x*(-a/b)^(1/4) + sqrt(-a/b))/(a^2*b^5)