\(\int \frac {x^3}{a+b x^4+c x^8} \, dx\) [314]

Optimal result

Integrand size = 18, antiderivative size = 38 \[ \int \frac {x^3}{a+b x^4+c x^8} \, dx=-\frac {\text {arctanh}\left (\frac {b+2 c x^4}{\sqrt {b^2-4 a c}}\right )}{2 \sqrt {b^2-4 a c}} \]

[Out]

-1/2*arctanh((2*c*x^4+b)/(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1366, 632, 212} \[ \int \frac {x^3}{a+b x^4+c x^8} \, dx=-\frac {\text {arctanh}\left (\frac {b+2 c x^4}{\sqrt {b^2-4 a c}}\right )}{2 \sqrt {b^2-4 a c}} \]

[In]

Int[x^3/(a + b*x^4 + c*x^8),x]

[Out]

-1/2*ArcTanh[(b + 2*c*x^4)/Sqrt[b^2 - 4*a*c]]/Sqrt[b^2 - 4*a*c]

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 1366

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[(a + b*x +
c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[Simplify[m - n + 1], 0]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} \text {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,x^4\right ) \\ & = -\left (\frac {1}{2} \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^4\right )\right ) \\ & = -\frac {\tanh ^{-1}\left (\frac {b+2 c x^4}{\sqrt {b^2-4 a c}}\right )}{2 \sqrt {b^2-4 a c}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.11 \[ \int \frac {x^3}{a+b x^4+c x^8} \, dx=\frac {\arctan \left (\frac {b+2 c x^4}{\sqrt {-b^2+4 a c}}\right )}{2 \sqrt {-b^2+4 a c}} \]

[In]

Integrate[x^3/(a + b*x^4 + c*x^8),x]

[Out]

ArcTan[(b + 2*c*x^4)/Sqrt[-b^2 + 4*a*c]]/(2*Sqrt[-b^2 + 4*a*c])

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.97

method	result	size
default	\(\frac {\arctan \left (\frac {2 c \,x^{4}+b}{\sqrt {4 a c -b^{2}}}\right )}{2 \sqrt {4 a c -b^{2}}}\)	\(37\)
risch	\(-\frac {\ln \left (\left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{4}-2 a \right )}{4 \sqrt {-4 a c +b^{2}}}+\frac {\ln \left (\left (b +\sqrt {-4 a c +b^{2}}\right ) x^{4}+2 a \right )}{4 \sqrt {-4 a c +b^{2}}}\)	\(70\)

[In]

int(x^3/(c*x^8+b*x^4+a),x,method=_RETURNVERBOSE)

[Out]

1/2/(4*a*c-b^2)^(1/2)*arctan((2*c*x^4+b)/(4*a*c-b^2)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 129, normalized size of antiderivative = 3.39 \[ \int \frac {x^3}{a+b x^4+c x^8} \, dx=\left [\frac {\log \left (\frac {2 \, c^{2} x^{8} + 2 \, b c x^{4} + b^{2} - 2 \, a c - {\left (2 \, c x^{4} + b\right )} \sqrt {b^{2} - 4 \, a c}}{c x^{8} + b x^{4} + a}\right )}{4 \, \sqrt {b^{2} - 4 \, a c}}, -\frac {\sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {{\left (2 \, c x^{4} + b\right )} \sqrt {-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right )}{2 \, {\left (b^{2} - 4 \, a c\right )}}\right ] \]

[In]

integrate(x^3/(c*x^8+b*x^4+a),x, algorithm="fricas")

[Out]

[1/4*log((2*c^2*x^8 + 2*b*c*x^4 + b^2 - 2*a*c - (2*c*x^4 + b)*sqrt(b^2 - 4*a*c))/(c*x^8 + b*x^4 + a))/sqrt(b^2
- 4*a*c), -1/2*sqrt(-b^2 + 4*a*c)*arctan(-(2*c*x^4 + b)*sqrt(-b^2 + 4*a*c)/(b^2 - 4*a*c))/(b^2 - 4*a*c)]

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 131 vs. \(2 (36) = 72\).

Time = 0.44 (sec) , antiderivative size = 131, normalized size of antiderivative = 3.45 \[ \int \frac {x^3}{a+b x^4+c x^8} \, dx=- \frac {\sqrt {- \frac {1}{4 a c - b^{2}}} \log {\left (x^{4} + \frac {- 4 a c \sqrt {- \frac {1}{4 a c - b^{2}}} + b^{2} \sqrt {- \frac {1}{4 a c - b^{2}}} + b}{2 c} \right )}}{4} + \frac {\sqrt {- \frac {1}{4 a c - b^{2}}} \log {\left (x^{4} + \frac {4 a c \sqrt {- \frac {1}{4 a c - b^{2}}} - b^{2} \sqrt {- \frac {1}{4 a c - b^{2}}} + b}{2 c} \right )}}{4} \]

[In]

integrate(x**3/(c*x**8+b*x**4+a),x)

[Out]

-sqrt(-1/(4*a*c - b**2))*log(x**4 + (-4*a*c*sqrt(-1/(4*a*c - b**2)) + b**2*sqrt(-1/(4*a*c - b**2)) + b)/(2*c))
/4 + sqrt(-1/(4*a*c - b**2))*log(x**4 + (4*a*c*sqrt(-1/(4*a*c - b**2)) - b**2*sqrt(-1/(4*a*c - b**2)) + b)/(2*
c))/4

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^3}{a+b x^4+c x^8} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(x^3/(c*x^8+b*x^4+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more deta

Giac [A] (verification not implemented)

none

Time = 1.66 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.95 \[ \int \frac {x^3}{a+b x^4+c x^8} \, dx=\frac {\arctan \left (\frac {2 \, c x^{4} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{2 \, \sqrt {-b^{2} + 4 \, a c}} \]

[In]

integrate(x^3/(c*x^8+b*x^4+a),x, algorithm="giac")

[Out]

1/2*arctan((2*c*x^4 + b)/sqrt(-b^2 + 4*a*c))/sqrt(-b^2 + 4*a*c)

Mupad [B] (verification not implemented)

Time = 8.21 (sec) , antiderivative size = 260, normalized size of antiderivative = 6.84 \[ \int \frac {x^3}{a+b x^4+c x^8} \, dx=-\frac {\mathrm {atan}\left (\frac {{\left (4\,a\,c-b^2\right )}^2\,\left (\frac {\left (\frac {4\,a\,c^4}{4\,a\,c-b^2}-\frac {4\,a\,b^2\,c^4}{{\left (4\,a\,c-b^2\right )}^2}\right )\,\left (b^3-3\,a\,b\,c\right )}{8\,a^3\,c^2\,\sqrt {4\,a\,c-b^2}}-x^4\,\left (\frac {\left (\frac {2\,c^4}{\sqrt {4\,a\,c-b^2}}-\frac {6\,b^2\,c^4}{{\left (4\,a\,c-b^2\right )}^{3/2}}\right )\,\left (a\,c-b^2\right )}{8\,a^3\,c^2}-\frac {\left (b^3-3\,a\,b\,c\right )\,\left (\frac {6\,b\,c^4}{4\,a\,c-b^2}-\frac {2\,b^3\,c^4}{{\left (4\,a\,c-b^2\right )}^2}\right )}{8\,a^3\,c^2\,\sqrt {4\,a\,c-b^2}}\right )+\frac {b\,c^2\,\left (a\,c-b^2\right )}{a^2\,{\left (4\,a\,c-b^2\right )}^{3/2}}\right )}{2\,c^4}\right )}{2\,\sqrt {4\,a\,c-b^2}} \]

[In]

int(x^3/(a + b*x^4 + c*x^8),x)

[Out]

-atan(((4*a*c - b^2)^2*((((4*a*c^4)/(4*a*c - b^2) - (4*a*b^2*c^4)/(4*a*c - b^2)^2)*(b^3 - 3*a*b*c))/(8*a^3*c^2
*(4*a*c - b^2)^(1/2)) - x^4*((((2*c^4)/(4*a*c - b^2)^(1/2) - (6*b^2*c^4)/(4*a*c - b^2)^(3/2))*(a*c - b^2))/(8*
a^3*c^2) - ((b^3 - 3*a*b*c)*((6*b*c^4)/(4*a*c - b^2) - (2*b^3*c^4)/(4*a*c - b^2)^2))/(8*a^3*c^2*(4*a*c - b^2)^
(1/2))) + (b*c^2*(a*c - b^2))/(a^2*(4*a*c - b^2)^(3/2))))/(2*c^4))/(2*(4*a*c - b^2)^(1/2))