3.5.86 \(\int \frac {x^3 (c+d x+e x^2+f x^3)}{a-b x^4} \, dx\) [486]

Optimal. Leaf size=162 \[ -\frac {d x}{b}-\frac {e x^2}{2 b}-\frac {f x^3}{3 b}+\frac {\sqrt [4]{a} \left (\sqrt {b} d-\sqrt {a} f\right ) \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 b^{7/4}}+\frac {\sqrt [4]{a} \left (\sqrt {b} d+\sqrt {a} f\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 b^{7/4}}+\frac {\sqrt {a} e \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{2 b^{3/2}}-\frac {c \log \left (a-b x^4\right )}{4 b} \]

[Out]

-d*x/b-1/2*e*x^2/b-1/3*f*x^3/b-1/4*c*ln(-b*x^4+a)/b+1/2*e*arctanh(x^2*b^(1/2)/a^(1/2))*a^(1/2)/b^(3/2)+1/2*a^(
1/4)*arctan(b^(1/4)*x/a^(1/4))*(-f*a^(1/2)+d*b^(1/2))/b^(7/4)+1/2*a^(1/4)*arctanh(b^(1/4)*x/a^(1/4))*(f*a^(1/2
)+d*b^(1/2))/b^(7/4)

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Rubi [A]
time = 0.13, antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {1845, 1266, 788, 649, 214, 266, 1294, 1181, 211} \begin {gather*} \frac {\sqrt [4]{a} \text {ArcTan}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\sqrt {b} d-\sqrt {a} f\right )}{2 b^{7/4}}+\frac {\sqrt [4]{a} \left (\sqrt {a} f+\sqrt {b} d\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 b^{7/4}}+\frac {\sqrt {a} e \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{2 b^{3/2}}-\frac {c \log \left (a-b x^4\right )}{4 b}-\frac {d x}{b}-\frac {e x^2}{2 b}-\frac {f x^3}{3 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((d*x)/b) - (e*x^2)/(2*b) - (f*x^3)/(3*b) + (a^(1/4)*(Sqrt[b]*d - Sqrt[a]*f)*ArcTan[(b^(1/4)*x)/a^(1/4)])/(2*
b^(7/4)) + (a^(1/4)*(Sqrt[b]*d + Sqrt[a]*f)*ArcTanh[(b^(1/4)*x)/a^(1/4)])/(2*b^(7/4)) + (Sqrt[a]*e*ArcTanh[(Sq
rt[b]*x^2)/Sqrt[a]])/(2*b^(3/2)) - (c*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 788

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

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 1266

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

Rule 1294

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[e*f*(f*x)^(m - 1)*(
(a + c*x^4)^(p + 1)/(c*(m + 4*p + 3))), x] - Dist[f^2/(c*(m + 4*p + 3)), Int[(f*x)^(m - 2)*(a + c*x^4)^p*(a*e*
(m - 1) - c*d*(m + 4*p + 3)*x^2), x], x] /; FreeQ[{a, c, d, e, f, p}, x] && GtQ[m, 1] && NeQ[m + 4*p + 3, 0] &
& IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])

Rule 1845

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

Rubi steps

\begin {align*} \int \frac {x^3 \left (c+d x+e x^2+f x^3\right )}{a-b x^4} \, dx &=\int \left (\frac {x^3 \left (c+e x^2\right )}{a-b x^4}+\frac {x^4 \left (d+f x^2\right )}{a-b x^4}\right ) \, dx\\ &=\int \frac {x^3 \left (c+e x^2\right )}{a-b x^4} \, dx+\int \frac {x^4 \left (d+f x^2\right )}{a-b x^4} \, dx\\ &=-\frac {f x^3}{3 b}+\frac {1}{2} \text {Subst}\left (\int \frac {x (c+e x)}{a-b x^2} \, dx,x,x^2\right )+\frac {\int \frac {x^2 \left (3 a f+3 b d x^2\right )}{a-b x^4} \, dx}{3 b}\\ &=-\frac {d x}{b}-\frac {e x^2}{2 b}-\frac {f x^3}{3 b}+\frac {\int \frac {3 a b d+3 a b f x^2}{a-b x^4} \, dx}{3 b^2}-\frac {\text {Subst}\left (\int \frac {-a e-b c x}{a-b x^2} \, dx,x,x^2\right )}{2 b}\\ &=-\frac {d x}{b}-\frac {e x^2}{2 b}-\frac {f x^3}{3 b}+\frac {1}{2} c \text {Subst}\left (\int \frac {x}{a-b x^2} \, dx,x,x^2\right )+\frac {(a e) \text {Subst}\left (\int \frac {1}{a-b x^2} \, dx,x,x^2\right )}{2 b}-\frac {\left (\sqrt {a} \left (\sqrt {b} d-\sqrt {a} f\right )\right ) \int \frac {1}{-\sqrt {a} \sqrt {b}-b x^2} \, dx}{2 b}+\frac {\left (\sqrt {a} \left (\sqrt {b} d+\sqrt {a} f\right )\right ) \int \frac {1}{\sqrt {a} \sqrt {b}-b x^2} \, dx}{2 b}\\ &=-\frac {d x}{b}-\frac {e x^2}{2 b}-\frac {f x^3}{3 b}+\frac {\sqrt [4]{a} \left (\sqrt {b} d-\sqrt {a} f\right ) \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 b^{7/4}}+\frac {\sqrt [4]{a} \left (\sqrt {b} d+\sqrt {a} f\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 b^{7/4}}+\frac {\sqrt {a} e \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{2 b^{3/2}}-\frac {c \log \left (a-b x^4\right )}{4 b}\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 221, normalized size = 1.36 \begin {gather*} \frac {-12 b^{3/4} d x-6 b^{3/4} e x^2-4 b^{3/4} f x^3+6 \left (\sqrt [4]{a} \sqrt {b} d-a^{3/4} f\right ) \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )-3 \left (\sqrt [4]{a} \sqrt {b} d+\sqrt {a} \sqrt [4]{b} e+a^{3/4} f\right ) \log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right )+3 \left (\sqrt [4]{a} \sqrt {b} d-\sqrt {a} \sqrt [4]{b} e+a^{3/4} f\right ) \log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right )+3 \sqrt {a} \sqrt [4]{b} e \log \left (\sqrt {a}+\sqrt {b} x^2\right )-3 b^{3/4} c \log \left (a-b x^4\right )}{12 b^{7/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-12*b^(3/4)*d*x - 6*b^(3/4)*e*x^2 - 4*b^(3/4)*f*x^3 + 6*(a^(1/4)*Sqrt[b]*d - a^(3/4)*f)*ArcTan[(b^(1/4)*x)/a^
(1/4)] - 3*(a^(1/4)*Sqrt[b]*d + Sqrt[a]*b^(1/4)*e + a^(3/4)*f)*Log[a^(1/4) - b^(1/4)*x] + 3*(a^(1/4)*Sqrt[b]*d
- Sqrt[a]*b^(1/4)*e + a^(3/4)*f)*Log[a^(1/4) + b^(1/4)*x] + 3*Sqrt[a]*b^(1/4)*e*Log[Sqrt[a] + Sqrt[b]*x^2] -
3*b^(3/4)*c*Log[a - b*x^4])/(12*b^(7/4))

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Maple [A]
time = 0.35, size = 176, normalized size = 1.09

method	result	size
risch	\(-\frac {f \,x^{3}}{3 b}-\frac {e \,x^{2}}{2 b}-\frac {d x}{b}+\frac {\munderset {\textit {\_R} =\RootOf \left (b \,\textit {\_Z}^{4}-a \right )}{\sum }\frac {\left (-\textit {\_R}^{3} b c -\textit {\_R}^{2} a f -\textit {\_R} a e -a d \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}}{4 b^{2}}\)	\(79\)
default	\(-\frac {\frac {1}{3} f \,x^{3}+\frac {1}{2} e \,x^{2}+d x}{b}+\frac {\frac {d \left (\frac {a}{b}\right )^{\frac {1}{4}} \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )\right )}{4}+\frac {a e \ln \left (\frac {a +x^{2} \sqrt {a b}}{a -x^{2} \sqrt {a b}}\right )}{4 \sqrt {a b}}-\frac {a f \left (2 \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )-\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )\right )}{4 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}-\frac {c \ln \left (-b \,x^{4}+a \right )}{4}}{b}\)	\(176\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.49, size = 211, normalized size = 1.30 \begin {gather*} -\frac {2 \, f x^{3} + 3 \, x^{2} e + 6 \, d x}{6 \, b} + \frac {\frac {2 \, {\left (a \sqrt {b} d - a^{\frac {3}{2}} f\right )} \arctan \left (\frac {\sqrt {b} x}{\sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {{\left (\sqrt {a} b c - a \sqrt {b} e\right )} \log \left (\sqrt {b} x^{2} + \sqrt {a}\right )}{\sqrt {a} b} - \frac {{\left (\sqrt {a} b c + a \sqrt {b} e\right )} \log \left (\sqrt {b} x^{2} - \sqrt {a}\right )}{\sqrt {a} b} - \frac {{\left (a \sqrt {b} d + a^{\frac {3}{2}} f\right )} \log \left (\frac {\sqrt {b} x - \sqrt {\sqrt {a} \sqrt {b}}}{\sqrt {b} x + \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}}}{4 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(2*f*x^3 + 3*x^2*e + 6*d*x)/b + 1/4*(2*(a*sqrt(b)*d - a^(3/2)*f)*arctan(sqrt(b)*x/sqrt(sqrt(a)*sqrt(b)))/
(sqrt(a)*sqrt(sqrt(a)*sqrt(b))*sqrt(b)) - (sqrt(a)*b*c - a*sqrt(b)*e)*log(sqrt(b)*x^2 + sqrt(a))/(sqrt(a)*b) -
(sqrt(a)*b*c + a*sqrt(b)*e)*log(sqrt(b)*x^2 - sqrt(a))/(sqrt(a)*b) - (a*sqrt(b)*d + a^(3/2)*f)*log((sqrt(b)*x
- sqrt(sqrt(a)*sqrt(b)))/(sqrt(b)*x + sqrt(sqrt(a)*sqrt(b))))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))*sqrt(b)))/b

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Fricas [C] Result contains complex when optimal does not.
time = 3.95, size = 220680, normalized size = 1362.22 \begin {gather*} \text {too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/48*(16*f*x^3 + 24*e*x^2 + 2*(2*(1/4)^(2/3)*(-I*sqrt(3) + 1)*((sqrt(2)*b^2*sqrt(a/b)*sqrt((2*a*b*d*f*sqrt(a/
b) + a*b*d^2 + a^2*f^2)/(b^4*sqrt(a/b))) - sqrt(2)*(3*b*c*sqrt(a/b) - a*e))^2/(a*b^3) + 3*sqrt(2)*(2*sqrt(2)*b
^3*c*sqrt(a/b)*sqrt((2*a*b*d*f*sqrt(a/b) + a*b*d^2 + a^2*f^2)/(b^4*sqrt(a/b))) - 2*sqrt(2)*a*b^2*e*sqrt((2*a*b
*d*f*sqrt(a/b) + a*b*d^2 + a^2*f^2)/(b^4*sqrt(a/b))) + 2*sqrt(2)*(b*d*f*sqrt(a/b) + b*c*e)*a - sqrt(2)*(3*b^2*
c^2*sqrt(a/b) + a^2*f^2 - (b*e^2*sqrt(a/b) - b*d^2)*a))/(b^4*sqrt(a/b)))/(9*(2*sqrt(2)*b^3*c*sqrt(a/b)*sqrt((2
*a*b*d*f*sqrt(a/b) + a*b*d^2 + a^2*f^2)/(b^4*sqrt(a/b))) - 2*sqrt(2)*a*b^2*e*sqrt((2*a*b*d*f*sqrt(a/b) + a*b*d
^2 + a^2*f^2)/(b^4*sqrt(a/b))) + 2*sqrt(2)*(b*d*f*sqrt(a/b) + b*c*e)*a - sqrt(2)*(3*b^2*c^2*sqrt(a/b) + a^2*f^
2 - (b*e^2*sqrt(a/b) - b*d^2)*a))*(sqrt(2)*b^2*sqrt(a/b)*sqrt((2*a*b*d*f*sqrt(a/b) + a*b*d^2 + a^2*f^2)/(b^4*s
qrt(a/b))) - sqrt(2)*(3*b*c*sqrt(a/b) - a*e))/(a*b^5) + sqrt(2)*(sqrt(2)*b^2*sqrt(a/b)*sqrt((2*a*b*d*f*sqrt(a/
b) + a*b*d ...

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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(x**3*(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. 328 vs. \(2 (120) = 240\).
time = 0.49, size = 328, normalized size = 2.02 \begin {gather*} -\frac {c \log \left ({\left | b x^{4} - a \right |}\right )}{4 \, b} - \frac {\sqrt {2} {\left (\sqrt {2} \sqrt {-a b} b^{2} e - \left (-a b^{3}\right )^{\frac {1}{4}} b^{2} d - \left (-a b^{3}\right )^{\frac {3}{4}} f\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 \, b^{4}} - \frac {\sqrt {2} {\left (\sqrt {2} \sqrt {-a b} b^{2} e - \left (-a b^{3}\right )^{\frac {1}{4}} b^{2} d - \left (-a b^{3}\right )^{\frac {3}{4}} f\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 \, b^{4}} + \frac {\sqrt {2} {\left (\left (-a b^{3}\right )^{\frac {1}{4}} b^{2} d - \left (-a b^{3}\right )^{\frac {3}{4}} f\right )} \log \left (x^{2} + \sqrt {2} x \left (-\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {-\frac {a}{b}}\right )}{8 \, b^{4}} - \frac {\sqrt {2} {\left (\left (-a b^{3}\right )^{\frac {1}{4}} b^{2} d - \left (-a b^{3}\right )^{\frac {3}{4}} f\right )} \log \left (x^{2} - \sqrt {2} x \left (-\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {-\frac {a}{b}}\right )}{8 \, b^{4}} - \frac {2 \, b^{2} f x^{3} + 3 \, b^{2} x^{2} e + 6 \, b^{2} d x}{6 \, b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*c*log(abs(b*x^4 - a))/b - 1/4*sqrt(2)*(sqrt(2)*sqrt(-a*b)*b^2*e - (-a*b^3)^(1/4)*b^2*d - (-a*b^3)^(3/4)*f
)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(-a/b)^(1/4))/(-a/b)^(1/4))/b^4 - 1/4*sqrt(2)*(sqrt(2)*sqrt(-a*b)*b^2*e -
(-a*b^3)^(1/4)*b^2*d - (-a*b^3)^(3/4)*f)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(-a/b)^(1/4))/(-a/b)^(1/4))/b^4 + 1
/8*sqrt(2)*((-a*b^3)^(1/4)*b^2*d - (-a*b^3)^(3/4)*f)*log(x^2 + sqrt(2)*x*(-a/b)^(1/4) + sqrt(-a/b))/b^4 - 1/8*
sqrt(2)*((-a*b^3)^(1/4)*b^2*d - (-a*b^3)^(3/4)*f)*log(x^2 - sqrt(2)*x*(-a/b)^(1/4) + sqrt(-a/b))/b^4 - 1/6*(2*
b^2*f*x^3 + 3*b^2*x^2*e + 6*b^2*d*x)/b^3

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Mupad [B]
time = 4.85, size = 846, normalized size = 5.22 \begin {gather*} \left (\sum _{k=1}^4\ln \left (-\frac {a^4\,f^3-2\,a^3\,b\,c\,e\,f-a^3\,b\,d^2\,f+a^3\,b\,d\,e^2+a^2\,b^2\,c^2\,d}{b^2}-\mathrm {root}\left (256\,b^7\,z^4+256\,b^6\,c\,z^3-64\,a\,b^4\,d\,f\,z^2-32\,a\,b^4\,e^2\,z^2+96\,b^5\,c^2\,z^2-32\,a\,b^3\,c\,d\,f\,z+16\,a^2\,b^2\,e\,f^2\,z+16\,a\,b^3\,d^2\,e\,z-16\,a\,b^3\,c\,e^2\,z+16\,b^4\,c^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^2\,b\,e^4+b^3\,c^4-a\,b^2\,d^4-a^3\,f^4,z,k\right )\,\left (\mathrm {root}\left (256\,b^7\,z^4+256\,b^6\,c\,z^3-64\,a\,b^4\,d\,f\,z^2-32\,a\,b^4\,e^2\,z^2+96\,b^5\,c^2\,z^2-32\,a\,b^3\,c\,d\,f\,z+16\,a^2\,b^2\,e\,f^2\,z+16\,a\,b^3\,d^2\,e\,z-16\,a\,b^3\,c\,e^2\,z+16\,b^4\,c^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^2\,b\,e^4+b^3\,c^4-a\,b^2\,d^4-a^3\,f^4,z,k\right )\,\left (16\,a^2\,b^2\,d-16\,a^2\,b^2\,e\,x\right )+\frac {8\,a^2\,b^3\,c\,d-8\,a^3\,b^2\,e\,f}{b^2}+\frac {x\,\left (4\,a^3\,b\,f^2+4\,a^2\,b^2\,d^2-8\,c\,e\,a^2\,b^2\right )}{b}\right )-\frac {x\,\left (a^3\,c\,f^2-2\,a^3\,d\,e\,f+a^3\,e^3-b\,a^2\,c^2\,e+b\,a^2\,c\,d^2\right )}{b}\right )\,\mathrm {root}\left (256\,b^7\,z^4+256\,b^6\,c\,z^3-64\,a\,b^4\,d\,f\,z^2-32\,a\,b^4\,e^2\,z^2+96\,b^5\,c^2\,z^2-32\,a\,b^3\,c\,d\,f\,z+16\,a^2\,b^2\,e\,f^2\,z+16\,a\,b^3\,d^2\,e\,z-16\,a\,b^3\,c\,e^2\,z+16\,b^4\,c^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^2\,b\,e^4+b^3\,c^4-a\,b^2\,d^4-a^3\,f^4,z,k\right )\right )-\frac {e\,x^2}{2\,b}-\frac {f\,x^3}{3\,b}-\frac {d\,x}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

symsum(log(- (a^4*f^3 + a^2*b^2*c^2*d + a^3*b*d*e^2 - a^3*b*d^2*f - 2*a^3*b*c*e*f)/b^2 - root(256*b^7*z^4 + 25
6*b^6*c*z^3 - 64*a*b^4*d*f*z^2 - 32*a*b^4*e^2*z^2 + 96*b^5*c^2*z^2 - 32*a*b^3*c*d*f*z + 16*a^2*b^2*e*f^2*z + 1
6*a*b^3*d^2*e*z - 16*a*b^3*c*e^2*z + 16*b^4*c^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^2*b*e^4 + b^3*c^4 - a*b^2*d^4 - a^3*f^4, z, k)*(root(256*b
^7*z^4 + 256*b^6*c*z^3 - 64*a*b^4*d*f*z^2 - 32*a*b^4*e^2*z^2 + 96*b^5*c^2*z^2 - 32*a*b^3*c*d*f*z + 16*a^2*b^2*
e*f^2*z + 16*a*b^3*d^2*e*z - 16*a*b^3*c*e^2*z + 16*b^4*c^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^2*b*e^4 + b^3*c^4 - a*b^2*d^4 - a^3*f^4, z, k)*
(16*a^2*b^2*d - 16*a^2*b^2*e*x) + (8*a^2*b^3*c*d - 8*a^3*b^2*e*f)/b^2 + (x*(4*a^3*b*f^2 + 4*a^2*b^2*d^2 - 8*a^
2*b^2*c*e))/b) - (x*(a^3*e^3 + a^3*c*f^2 - 2*a^3*d*e*f + a^2*b*c*d^2 - a^2*b*c^2*e))/b)*root(256*b^7*z^4 + 256
*b^6*c*z^3 - 64*a*b^4*d*f*z^2 - 32*a*b^4*e^2*z^2 + 96*b^5*c^2*z^2 - 32*a*b^3*c*d*f*z + 16*a^2*b^2*e*f^2*z + 16
*a*b^3*d^2*e*z - 16*a*b^3*c*e^2*z + 16*b^4*c^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^2*b*e^4 + b^3*c^4 - a*b^2*d^4 - a^3*f^4, z, k), k, 1, 4) -
(e*x^2)/(2*b) - (f*x^3)/(3*b) - (d*x)/b

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