3.5.85 \(\int \frac {c+d x+e x^2+f x^3}{a-b x^4} \, dx\) [485]

Optimal. Leaf size=133 \[ \frac {\left (\sqrt {b} c-\sqrt {a} e\right ) \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{3/4}}+\frac {\left (\sqrt {b} c+\sqrt {a} e\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{3/4}}+\frac {d \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \sqrt {b}}-\frac {f \log \left (a-b x^4\right )}{4 b} \]

[Out]

-1/4*f*ln(-b*x^4+a)/b+1/2*d*arctanh(x^2*b^(1/2)/a^(1/2))/a^(1/2)/b^(1/2)+1/2*arctan(b^(1/4)*x/a^(1/4))*(-e*a^(
1/2)+c*b^(1/2))/a^(3/4)/b^(3/4)+1/2*arctanh(b^(1/4)*x/a^(1/4))*(e*a^(1/2)+c*b^(1/2))/a^(3/4)/b^(3/4)

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Rubi [A]
time = 0.08, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {1890, 1181, 211, 214, 1262, 649, 266} \begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\sqrt {b} c-\sqrt {a} e\right )}{2 a^{3/4} b^{3/4}}+\frac {\left (\sqrt {a} e+\sqrt {b} c\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{3/4}}+\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 {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 211

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

Rule 214

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

Rule 266

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 649

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 1181

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 1262

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 1890

Int[(Pq_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = Sum[x^ii*((Coeff[Pq, x, ii] + Coeff[Pq, x, n/2 + ii
]*x^(n/2))/(a + b*x^n)), {ii, 0, n/2 - 1}]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ
[n/2, 0] && Expon[Pq, x] < n

Rubi steps

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

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Mathematica [A]
time = 0.04, size = 214, normalized size = 1.61 \begin {gather*} \frac {\left (\sqrt [4]{a} \sqrt {b} c-a^{3/4} e\right ) \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a b^{3/4}}-\frac {\left (\sqrt [4]{a} \sqrt {b} c+\sqrt {a} \sqrt [4]{b} d+a^{3/4} e\right ) \log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right )}{4 a b^{3/4}}-\frac {\left (-\sqrt [4]{a} \sqrt {b} c+\sqrt {a} \sqrt [4]{b} d-a^{3/4} e\right ) \log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right )}{4 a b^{3/4}}+\frac {d \log \left (\sqrt {a}+\sqrt {b} x^2\right )}{4 \sqrt {a} \sqrt {b}}-\frac {f \log \left (a-b x^4\right )}{4 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.36, size = 154, normalized size = 1.16

method	result	size
risch	\(-\frac {\munderset {\textit {\_R} =\RootOf \left (b \,\textit {\_Z}^{4}-a \right )}{\sum }\frac {\left (\textit {\_R}^{3} f +\textit {\_R}^{2} e +\textit {\_R} d +c \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}}{4 b}\)	\(44\)
default	\(\frac {c \left (\frac {a}{b}\right )^{\frac {1}{4}} \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )\right )}{4 a}+\frac {d \ln \left (\frac {a +x^{2} \sqrt {a b}}{a -x^{2} \sqrt {a b}}\right )}{4 \sqrt {a b}}-\frac {e \left (2 \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )-\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )\right )}{4 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}-\frac {f \ln \left (-b \,x^{4}+a \right )}{4 b}\)	\(154\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.50, size = 176, normalized size = 1.32 \begin {gather*} \frac {{\left (\sqrt {b} c - \sqrt {a} e\right )} \arctan \left (\frac {\sqrt {b} x}{\sqrt {\sqrt {a} \sqrt {b}}}\right )}{2 \, \sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {{\left (\sqrt {b} d - \sqrt {a} f\right )} \log \left (\sqrt {b} x^{2} + \sqrt {a}\right )}{4 \, \sqrt {a} b} - \frac {{\left (\sqrt {b} d + \sqrt {a} f\right )} \log \left (\sqrt {b} x^{2} - \sqrt {a}\right )}{4 \, \sqrt {a} b} - \frac {{\left (\sqrt {b} c + \sqrt {a} e\right )} \log \left (\frac {\sqrt {b} x - \sqrt {\sqrt {a} \sqrt {b}}}{\sqrt {b} x + \sqrt {\sqrt {a} \sqrt {b}}}\right )}{4 \, \sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(sqrt(b)*c - sqrt(a)*e)*arctan(sqrt(b)*x/sqrt(sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))*sqrt(b)) +
1/4*(sqrt(b)*d - sqrt(a)*f)*log(sqrt(b)*x^2 + sqrt(a))/(sqrt(a)*b) - 1/4*(sqrt(b)*d + sqrt(a)*f)*log(sqrt(b)*x
^2 - sqrt(a))/(sqrt(a)*b) - 1/4*(sqrt(b)*c + sqrt(a)*e)*log((sqrt(b)*x - sqrt(sqrt(a)*sqrt(b)))/(sqrt(b)*x + s
qrt(sqrt(a)*sqrt(b))))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))*sqrt(b))

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Fricas [C] Result contains complex when optimal does not.
time = 5.02, size = 241149, normalized size = 1813.15 \begin {gather*} \text {too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/48*(2*(2*(1/4)^(2/3)*(-I*sqrt(3) + 1)*((sqrt(2)*a*b*sqrt(1/(a*b))*sqrt((2*a*b*c*e*sqrt(1/(a*b)) + b*c^2 + a
*e^2)/(a^2*b^2*sqrt(1/(a*b)))) - sqrt(2)*(3*a*f*sqrt(1/(a*b)) - d))^2/(a*b) + 3*sqrt(2)*(2*sqrt(2)*a^2*b*f*sqr
t(1/(a*b))*sqrt((2*a*b*c*e*sqrt(1/(a*b)) + b*c^2 + a*e^2)/(a^2*b^2*sqrt(1/(a*b)))) - 2*sqrt(2)*a*b*d*sqrt((2*a
*b*c*e*sqrt(1/(a*b)) + b*c^2 + a*e^2)/(a^2*b^2*sqrt(1/(a*b)))) + 2*sqrt(2)*(b*c*e*sqrt(1/(a*b)) + d*f)*a - sqr
t(2)*(3*a^2*f^2*sqrt(1/(a*b)) + b*c^2 - (b*d^2*sqrt(1/(a*b)) - e^2)*a))/(a^2*b^2*sqrt(1/(a*b))))/(9*(2*sqrt(2)
*a^2*b*f*sqrt(1/(a*b))*sqrt((2*a*b*c*e*sqrt(1/(a*b)) + b*c^2 + a*e^2)/(a^2*b^2*sqrt(1/(a*b)))) - 2*sqrt(2)*a*b
*d*sqrt((2*a*b*c*e*sqrt(1/(a*b)) + b*c^2 + a*e^2)/(a^2*b^2*sqrt(1/(a*b)))) + 2*sqrt(2)*(b*c*e*sqrt(1/(a*b)) +
d*f)*a - sqrt(2)*(3*a^2*f^2*sqrt(1/(a*b)) + b*c^2 - (b*d^2*sqrt(1/(a*b)) - e^2)*a))*(sqrt(2)*a*b*sqrt(1/(a*b))
*sqrt((2*a*b*c*e*sqrt(1/(a*b)) + b*c^2 + a*e^2)/(a^2*b^2*sqrt(1/(a*b)))) - sqrt(2)*(3*a*f*sqrt(1/(a*b)) - d))/
(a^2*b^2)  ...

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 280 vs. \(2 (95) = 190\).
time = 0.49, size = 280, normalized size = 2.11 \begin {gather*} -\frac {\sqrt {2} {\left (b^{2} c - \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} b d + \sqrt {-a b} b 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 \, \left (-a b^{3}\right )^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (b^{2} c + \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} b d - \sqrt {-a b} b 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 \, \left (-a b^{3}\right )^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (b^{2} c - \sqrt {-a b} b e\right )} \log \left (x^{2} + \sqrt {2} x \left (-\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {-\frac {a}{b}}\right )}{8 \, \left (-a b^{3}\right )^{\frac {3}{4}}} + \frac {\sqrt {2} {\left (b^{2} c - \sqrt {-a b} b e\right )} \log \left (x^{2} - \sqrt {2} x \left (-\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {-\frac {a}{b}}\right )}{8 \, \left (-a b^{3}\right )^{\frac {3}{4}}} - \frac {f \log \left ({\left | b x^{4} - a \right |}\right )}{4 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 5.66, size = 1970, normalized size = 14.81 \begin {gather*} \sum _{k=1}^4\ln \left (-b^2\,c\,d^2+b^2\,c^2\,e-b^2\,d^3\,x-a\,b\,e^3-a\,b\,c\,f^2-{\mathrm {root}\left (256\,a^3\,b^4\,z^4+256\,a^3\,b^3\,f\,z^3-64\,a^2\,b^3\,c\,e\,z^2+96\,a^3\,b^2\,f^2\,z^2-32\,a^2\,b^3\,d^2\,z^2-32\,a^2\,b^2\,c\,e\,f\,z-16\,a^2\,b^2\,d^2\,f\,z+16\,a^2\,b^2\,d\,e^2\,z+16\,a\,b^3\,c^2\,d\,z+16\,a^3\,b\,f^3\,z+4\,a^2\,b\,d\,e^2\,f-4\,a^2\,b\,c\,e\,f^2+4\,a\,b^2\,c^2\,d\,f-4\,a\,b^2\,c\,d^2\,e-2\,a^2\,b\,d^2\,f^2+2\,a\,b^2\,c^2\,e^2+a\,b^2\,d^4+a^3\,f^4-a^2\,b\,e^4-b^3\,c^4,z,k\right )}^2\,a\,b^3\,c\,16-\mathrm {root}\left (256\,a^3\,b^4\,z^4+256\,a^3\,b^3\,f\,z^3-64\,a^2\,b^3\,c\,e\,z^2+96\,a^3\,b^2\,f^2\,z^2-32\,a^2\,b^3\,d^2\,z^2-32\,a^2\,b^2\,c\,e\,f\,z-16\,a^2\,b^2\,d^2\,f\,z+16\,a^2\,b^2\,d\,e^2\,z+16\,a\,b^3\,c^2\,d\,z+16\,a^3\,b\,f^3\,z+4\,a^2\,b\,d\,e^2\,f-4\,a^2\,b\,c\,e\,f^2+4\,a\,b^2\,c^2\,d\,f-4\,a\,b^2\,c\,d^2\,e-2\,a^2\,b\,d^2\,f^2+2\,a\,b^2\,c^2\,e^2+a\,b^2\,d^4+a^3\,f^4-a^2\,b\,e^4-b^3\,c^4,z,k\right )\,b^3\,c^2\,x\,4-b^2\,c^2\,f\,x+{\mathrm {root}\left (256\,a^3\,b^4\,z^4+256\,a^3\,b^3\,f\,z^3-64\,a^2\,b^3\,c\,e\,z^2+96\,a^3\,b^2\,f^2\,z^2-32\,a^2\,b^3\,d^2\,z^2-32\,a^2\,b^2\,c\,e\,f\,z-16\,a^2\,b^2\,d^2\,f\,z+16\,a^2\,b^2\,d\,e^2\,z+16\,a\,b^3\,c^2\,d\,z+16\,a^3\,b\,f^3\,z+4\,a^2\,b\,d\,e^2\,f-4\,a^2\,b\,c\,e\,f^2+4\,a\,b^2\,c^2\,d\,f-4\,a\,b^2\,c\,d^2\,e-2\,a^2\,b\,d^2\,f^2+2\,a\,b^2\,c^2\,e^2+a\,b^2\,d^4+a^3\,f^4-a^2\,b\,e^4-b^3\,c^4,z,k\right )}^2\,a\,b^3\,d\,x\,16-\mathrm {root}\left (256\,a^3\,b^4\,z^4+256\,a^3\,b^3\,f\,z^3-64\,a^2\,b^3\,c\,e\,z^2+96\,a^3\,b^2\,f^2\,z^2-32\,a^2\,b^3\,d^2\,z^2-32\,a^2\,b^2\,c\,e\,f\,z-16\,a^2\,b^2\,d^2\,f\,z+16\,a^2\,b^2\,d\,e^2\,z+16\,a\,b^3\,c^2\,d\,z+16\,a^3\,b\,f^3\,z+4\,a^2\,b\,d\,e^2\,f-4\,a^2\,b\,c\,e\,f^2+4\,a\,b^2\,c^2\,d\,f-4\,a\,b^2\,c\,d^2\,e-2\,a^2\,b\,d^2\,f^2+2\,a\,b^2\,c^2\,e^2+a\,b^2\,d^4+a^3\,f^4-a^2\,b\,e^4-b^3\,c^4,z,k\right )\,a\,b^2\,e^2\,x\,4+2\,a\,b\,d\,e\,f-\mathrm {root}\left (256\,a^3\,b^4\,z^4+256\,a^3\,b^3\,f\,z^3-64\,a^2\,b^3\,c\,e\,z^2+96\,a^3\,b^2\,f^2\,z^2-32\,a^2\,b^3\,d^2\,z^2-32\,a^2\,b^2\,c\,e\,f\,z-16\,a^2\,b^2\,d^2\,f\,z+16\,a^2\,b^2\,d\,e^2\,z+16\,a\,b^3\,c^2\,d\,z+16\,a^3\,b\,f^3\,z+4\,a^2\,b\,d\,e^2\,f-4\,a^2\,b\,c\,e\,f^2+4\,a\,b^2\,c^2\,d\,f-4\,a\,b^2\,c\,d^2\,e-2\,a^2\,b\,d^2\,f^2+2\,a\,b^2\,c^2\,e^2+a\,b^2\,d^4+a^3\,f^4-a^2\,b\,e^4-b^3\,c^4,z,k\right )\,a\,b^2\,c\,f\,8+\mathrm {root}\left (256\,a^3\,b^4\,z^4+256\,a^3\,b^3\,f\,z^3-64\,a^2\,b^3\,c\,e\,z^2+96\,a^3\,b^2\,f^2\,z^2-32\,a^2\,b^3\,d^2\,z^2-32\,a^2\,b^2\,c\,e\,f\,z-16\,a^2\,b^2\,d^2\,f\,z+16\,a^2\,b^2\,d\,e^2\,z+16\,a\,b^3\,c^2\,d\,z+16\,a^3\,b\,f^3\,z+4\,a^2\,b\,d\,e^2\,f-4\,a^2\,b\,c\,e\,f^2+4\,a\,b^2\,c^2\,d\,f-4\,a\,b^2\,c\,d^2\,e-2\,a^2\,b\,d^2\,f^2+2\,a\,b^2\,c^2\,e^2+a\,b^2\,d^4+a^3\,f^4-a^2\,b\,e^4-b^3\,c^4,z,k\right )\,a\,b^2\,d\,e\,8+a\,b\,d\,f^2\,x-a\,b\,e^2\,f\,x+2\,b^2\,c\,d\,e\,x+\mathrm {root}\left (256\,a^3\,b^4\,z^4+256\,a^3\,b^3\,f\,z^3-64\,a^2\,b^3\,c\,e\,z^2+96\,a^3\,b^2\,f^2\,z^2-32\,a^2\,b^3\,d^2\,z^2-32\,a^2\,b^2\,c\,e\,f\,z-16\,a^2\,b^2\,d^2\,f\,z+16\,a^2\,b^2\,d\,e^2\,z+16\,a\,b^3\,c^2\,d\,z+16\,a^3\,b\,f^3\,z+4\,a^2\,b\,d\,e^2\,f-4\,a^2\,b\,c\,e\,f^2+4\,a\,b^2\,c^2\,d\,f-4\,a\,b^2\,c\,d^2\,e-2\,a^2\,b\,d^2\,f^2+2\,a\,b^2\,c^2\,e^2+a\,b^2\,d^4+a^3\,f^4-a^2\,b\,e^4-b^3\,c^4,z,k\right )\,a\,b^2\,d\,f\,x\,8\right )\,\mathrm {root}\left (256\,a^3\,b^4\,z^4+256\,a^3\,b^3\,f\,z^3-64\,a^2\,b^3\,c\,e\,z^2+96\,a^3\,b^2\,f^2\,z^2-32\,a^2\,b^3\,d^2\,z^2-32\,a^2\,b^2\,c\,e\,f\,z-16\,a^2\,b^2\,d^2\,f\,z+16\,a^2\,b^2\,d\,e^2\,z+16\,a\,b^3\,c^2\,d\,z+16\,a^3\,b\,f^3\,z+4\,a^2\,b\,d\,e^2\,f-4\,a^2\,b\,c\,e\,f^2+4\,a\,b^2\,c^2\,d\,f-4\,a\,b^2\,c\,d^2\,e-2\,a^2\,b\,d^2\,f^2+2\,a\,b^2\,c^2\,e^2+a\,b^2\,d^4+a^3\,f^4-a^2\,b\,e^4-b^3\,c^4,z,k\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

symsum(log(b^2*c^2*e - b^2*c*d^2 - b^2*d^3*x - a*b*e^3 - a*b*c*f^2 - 16*root(256*a^3*b^4*z^4 + 256*a^3*b^3*f*z
^3 - 64*a^2*b^3*c*e*z^2 + 96*a^3*b^2*f^2*z^2 - 32*a^2*b^3*d^2*z^2 - 32*a^2*b^2*c*e*f*z - 16*a^2*b^2*d^2*f*z +
16*a^2*b^2*d*e^2*z + 16*a*b^3*c^2*d*z + 16*a^3*b*f^3*z + 4*a^2*b*d*e^2*f - 4*a^2*b*c*e*f^2 + 4*a*b^2*c^2*d*f -
4*a*b^2*c*d^2*e - 2*a^2*b*d^2*f^2 + 2*a*b^2*c^2*e^2 + a*b^2*d^4 + a^3*f^4 - a^2*b*e^4 - b^3*c^4, z, k)^2*a*b^
3*c - 4*root(256*a^3*b^4*z^4 + 256*a^3*b^3*f*z^3 - 64*a^2*b^3*c*e*z^2 + 96*a^3*b^2*f^2*z^2 - 32*a^2*b^3*d^2*z^
2 - 32*a^2*b^2*c*e*f*z - 16*a^2*b^2*d^2*f*z + 16*a^2*b^2*d*e^2*z + 16*a*b^3*c^2*d*z + 16*a^3*b*f^3*z + 4*a^2*b
*d*e^2*f - 4*a^2*b*c*e*f^2 + 4*a*b^2*c^2*d*f - 4*a*b^2*c*d^2*e - 2*a^2*b*d^2*f^2 + 2*a*b^2*c^2*e^2 + a*b^2*d^4
+ a^3*f^4 - a^2*b*e^4 - b^3*c^4, z, k)*b^3*c^2*x - b^2*c^2*f*x + 16*root(256*a^3*b^4*z^4 + 256*a^3*b^3*f*z^3
- 64*a^2*b^3*c*e*z^2 + 96*a^3*b^2*f^2*z^2 - 32*a^2*b^3*d^2*z^2 - 32*a^2*b^2*c*e*f*z - 16*a^2*b^2*d^2*f*z + 16*
a^2*b^2*d*e^2*z + 16*a*b^3*c^2*d*z + 16*a^3*b*f^3*z + 4*a^2*b*d*e^2*f - 4*a^2*b*c*e*f^2 + 4*a*b^2*c^2*d*f - 4*
a*b^2*c*d^2*e - 2*a^2*b*d^2*f^2 + 2*a*b^2*c^2*e^2 + a*b^2*d^4 + a^3*f^4 - a^2*b*e^4 - b^3*c^4, z, k)^2*a*b^3*d
*x - 4*root(256*a^3*b^4*z^4 + 256*a^3*b^3*f*z^3 - 64*a^2*b^3*c*e*z^2 + 96*a^3*b^2*f^2*z^2 - 32*a^2*b^3*d^2*z^2
- 32*a^2*b^2*c*e*f*z - 16*a^2*b^2*d^2*f*z + 16*a^2*b^2*d*e^2*z + 16*a*b^3*c^2*d*z + 16*a^3*b*f^3*z + 4*a^2*b*
d*e^2*f - 4*a^2*b*c*e*f^2 + 4*a*b^2*c^2*d*f - 4*a*b^2*c*d^2*e - 2*a^2*b*d^2*f^2 + 2*a*b^2*c^2*e^2 + a*b^2*d^4
+ a^3*f^4 - a^2*b*e^4 - b^3*c^4, z, k)*a*b^2*e^2*x + 2*a*b*d*e*f - 8*root(256*a^3*b^4*z^4 + 256*a^3*b^3*f*z^3
- 64*a^2*b^3*c*e*z^2 + 96*a^3*b^2*f^2*z^2 - 32*a^2*b^3*d^2*z^2 - 32*a^2*b^2*c*e*f*z - 16*a^2*b^2*d^2*f*z + 16*
a^2*b^2*d*e^2*z + 16*a*b^3*c^2*d*z + 16*a^3*b*f^3*z + 4*a^2*b*d*e^2*f - 4*a^2*b*c*e*f^2 + 4*a*b^2*c^2*d*f - 4*
a*b^2*c*d^2*e - 2*a^2*b*d^2*f^2 + 2*a*b^2*c^2*e^2 + a*b^2*d^4 + a^3*f^4 - a^2*b*e^4 - b^3*c^4, z, k)*a*b^2*c*f
+ 8*root(256*a^3*b^4*z^4 + 256*a^3*b^3*f*z^3 - 64*a^2*b^3*c*e*z^2 + 96*a^3*b^2*f^2*z^2 - 32*a^2*b^3*d^2*z^2 -
32*a^2*b^2*c*e*f*z - 16*a^2*b^2*d^2*f*z + 16*a^2*b^2*d*e^2*z + 16*a*b^3*c^2*d*z + 16*a^3*b*f^3*z + 4*a^2*b*d*
e^2*f - 4*a^2*b*c*e*f^2 + 4*a*b^2*c^2*d*f - 4*a*b^2*c*d^2*e - 2*a^2*b*d^2*f^2 + 2*a*b^2*c^2*e^2 + a*b^2*d^4 +
a^3*f^4 - a^2*b*e^4 - b^3*c^4, z, k)*a*b^2*d*e + a*b*d*f^2*x - a*b*e^2*f*x + 2*b^2*c*d*e*x + 8*root(256*a^3*b^
4*z^4 + 256*a^3*b^3*f*z^3 - 64*a^2*b^3*c*e*z^2 + 96*a^3*b^2*f^2*z^2 - 32*a^2*b^3*d^2*z^2 - 32*a^2*b^2*c*e*f*z
- 16*a^2*b^2*d^2*f*z + 16*a^2*b^2*d*e^2*z + 16*a*b^3*c^2*d*z + 16*a^3*b*f^3*z + 4*a^2*b*d*e^2*f - 4*a^2*b*c*e*
f^2 + 4*a*b^2*c^2*d*f - 4*a*b^2*c*d^2*e - 2*a^2*b*d^2*f^2 + 2*a*b^2*c^2*e^2 + a*b^2*d^4 + a^3*f^4 - a^2*b*e^4
- b^3*c^4, z, k)*a*b^2*d*f*x)*root(256*a^3*b^4*z^4 + 256*a^3*b^3*f*z^3 - 64*a^2*b^3*c*e*z^2 + 96*a^3*b^2*f^2*z
^2 - 32*a^2*b^3*d^2*z^2 - 32*a^2*b^2*c*e*f*z - 16*a^2*b^2*d^2*f*z + 16*a^2*b^2*d*e^2*z + 16*a*b^3*c^2*d*z + 16
*a^3*b*f^3*z + 4*a^2*b*d*e^2*f - 4*a^2*b*c*e*f^2 + 4*a*b^2*c^2*d*f - 4*a*b^2*c*d^2*e - 2*a^2*b*d^2*f^2 + 2*a*b
^2*c^2*e^2 + a*b^2*d^4 + a^3*f^4 - a^2*b*e^4 - b^3*c^4, z, k), k, 1, 4)

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