3.310 \(\int \frac{x^{11}}{a+b x^4+c x^8} \, dx\)

Optimal. Leaf size=81 \[ -\frac{\left (b^2-2 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^4}{\sqrt{b^2-4 a c}}\right )}{4 c^2 \sqrt{b^2-4 a c}}-\frac{b \log \left (a+b x^4+c x^8\right )}{8 c^2}+\frac{x^4}{4 c} \]

[Out]

x^4/(4*c) - ((b^2 - 2*a*c)*ArcTanh[(b + 2*c*x^4)/Sqrt[b^2 - 4*a*c]])/(4*c^2*Sqrt[b^2 - 4*a*c]) - (b*Log[a + b*
x^4 + c*x^8])/(8*c^2)

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Rubi [A]  time = 0.0837916, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {1357, 703, 634, 618, 206, 628} \[ -\frac{\left (b^2-2 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^4}{\sqrt{b^2-4 a c}}\right )}{4 c^2 \sqrt{b^2-4 a c}}-\frac{b \log \left (a+b x^4+c x^8\right )}{8 c^2}+\frac{x^4}{4 c} \]

Antiderivative was successfully verified.

[In]

Int[x^11/(a + b*x^4 + c*x^8),x]

[Out]

x^4/(4*c) - ((b^2 - 2*a*c)*ArcTanh[(b + 2*c*x^4)/Sqrt[b^2 - 4*a*c]])/(4*c^2*Sqrt[b^2 - 4*a*c]) - (b*Log[a + b*
x^4 + c*x^8])/(8*c^2)

Rule 1357

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && NeQ[
b^2 - 4*a*c, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 703

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

Rule 634

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

Rule 618

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 206

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

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int \frac{x^{11}}{a+b x^4+c x^8} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{x^2}{a+b x+c x^2} \, dx,x,x^4\right )\\ &=\frac{x^4}{4 c}+\frac{\operatorname{Subst}\left (\int \frac{-a-b x}{a+b x+c x^2} \, dx,x,x^4\right )}{4 c}\\ &=\frac{x^4}{4 c}-\frac{b \operatorname{Subst}\left (\int \frac{b+2 c x}{a+b x+c x^2} \, dx,x,x^4\right )}{8 c^2}+\frac{\left (b^2-2 a c\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x+c x^2} \, dx,x,x^4\right )}{8 c^2}\\ &=\frac{x^4}{4 c}-\frac{b \log \left (a+b x^4+c x^8\right )}{8 c^2}-\frac{\left (b^2-2 a c\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^4\right )}{4 c^2}\\ &=\frac{x^4}{4 c}-\frac{\left (b^2-2 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^4}{\sqrt{b^2-4 a c}}\right )}{4 c^2 \sqrt{b^2-4 a c}}-\frac{b \log \left (a+b x^4+c x^8\right )}{8 c^2}\\ \end{align*}

Mathematica [A]  time = 0.05244, size = 78, normalized size = 0.96 \[ \frac{\frac{2 \left (b^2-2 a c\right ) \tan ^{-1}\left (\frac{b+2 c x^4}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}-b \log \left (a+b x^4+c x^8\right )+2 c x^4}{8 c^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^11/(a + b*x^4 + c*x^8),x]

[Out]

(2*c*x^4 + (2*(b^2 - 2*a*c)*ArcTan[(b + 2*c*x^4)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] - b*Log[a + b*x^4 + c
*x^8])/(8*c^2)

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Maple [A]  time = 0.004, size = 111, normalized size = 1.4 \begin{align*}{\frac{{x}^{4}}{4\,c}}-{\frac{b\ln \left ( c{x}^{8}+b{x}^{4}+a \right ) }{8\,{c}^{2}}}-{\frac{a}{2\,c}\arctan \left ({(2\,c{x}^{4}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{\frac{{b}^{2}}{4\,{c}^{2}}\arctan \left ({(2\,c{x}^{4}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^11/(c*x^8+b*x^4+a),x)

[Out]

1/4*x^4/c-1/8*b*ln(c*x^8+b*x^4+a)/c^2-1/2/c/(4*a*c-b^2)^(1/2)*arctan((2*c*x^4+b)/(4*a*c-b^2)^(1/2))*a+1/4/c^2/
(4*a*c-b^2)^(1/2)*arctan((2*c*x^4+b)/(4*a*c-b^2)^(1/2))*b^2

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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(x^11/(c*x^8+b*x^4+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.90724, size = 556, normalized size = 6.86 \begin{align*} \left [\frac{2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x^{4} -{\left (b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c} \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 ) -{\left (b^{3} - 4 \, a b c\right )} \log \left (c x^{8} + b x^{4} + a\right )}{8 \,{\left (b^{2} c^{2} - 4 \, a c^{3}\right )}}, \frac{2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x^{4} - 2 \,{\left (b^{2} - 2 \, a c\right )} \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 ) -{\left (b^{3} - 4 \, a b c\right )} \log \left (c x^{8} + b x^{4} + a\right )}{8 \,{\left (b^{2} c^{2} - 4 \, a c^{3}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11/(c*x^8+b*x^4+a),x, algorithm="fricas")

[Out]

[1/8*(2*(b^2*c - 4*a*c^2)*x^4 - (b^2 - 2*a*c)*sqrt(b^2 - 4*a*c)*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)) - (b^3 - 4*a*b*c)*log(c*x^8 + b*x^4 + a))/(b^2*c^2 - 4*a*c^
3), 1/8*(2*(b^2*c - 4*a*c^2)*x^4 - 2*(b^2 - 2*a*c)*sqrt(-b^2 + 4*a*c)*arctan(-(2*c*x^4 + b)*sqrt(-b^2 + 4*a*c)
/(b^2 - 4*a*c)) - (b^3 - 4*a*b*c)*log(c*x^8 + b*x^4 + a))/(b^2*c^2 - 4*a*c^3)]

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Sympy [B]  time = 2.90928, size = 316, normalized size = 3.9 \begin{align*} \left (- \frac{b}{8 c^{2}} - \frac{\sqrt{- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{8 c^{2} \left (4 a c - b^{2}\right )}\right ) \log{\left (x^{4} + \frac{- a b - 16 a c^{2} \left (- \frac{b}{8 c^{2}} - \frac{\sqrt{- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{8 c^{2} \left (4 a c - b^{2}\right )}\right ) + 4 b^{2} c \left (- \frac{b}{8 c^{2}} - \frac{\sqrt{- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{8 c^{2} \left (4 a c - b^{2}\right )}\right )}{2 a c - b^{2}} \right )} + \left (- \frac{b}{8 c^{2}} + \frac{\sqrt{- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{8 c^{2} \left (4 a c - b^{2}\right )}\right ) \log{\left (x^{4} + \frac{- a b - 16 a c^{2} \left (- \frac{b}{8 c^{2}} + \frac{\sqrt{- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{8 c^{2} \left (4 a c - b^{2}\right )}\right ) + 4 b^{2} c \left (- \frac{b}{8 c^{2}} + \frac{\sqrt{- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{8 c^{2} \left (4 a c - b^{2}\right )}\right )}{2 a c - b^{2}} \right )} + \frac{x^{4}}{4 c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**11/(c*x**8+b*x**4+a),x)

[Out]

(-b/(8*c**2) - sqrt(-4*a*c + b**2)*(2*a*c - b**2)/(8*c**2*(4*a*c - b**2)))*log(x**4 + (-a*b - 16*a*c**2*(-b/(8
*c**2) - sqrt(-4*a*c + b**2)*(2*a*c - b**2)/(8*c**2*(4*a*c - b**2))) + 4*b**2*c*(-b/(8*c**2) - sqrt(-4*a*c + b
**2)*(2*a*c - b**2)/(8*c**2*(4*a*c - b**2))))/(2*a*c - b**2)) + (-b/(8*c**2) + sqrt(-4*a*c + b**2)*(2*a*c - b*
*2)/(8*c**2*(4*a*c - b**2)))*log(x**4 + (-a*b - 16*a*c**2*(-b/(8*c**2) + sqrt(-4*a*c + b**2)*(2*a*c - b**2)/(8
*c**2*(4*a*c - b**2))) + 4*b**2*c*(-b/(8*c**2) + sqrt(-4*a*c + b**2)*(2*a*c - b**2)/(8*c**2*(4*a*c - b**2))))/
(2*a*c - b**2)) + x**4/(4*c)

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Giac [A]  time = 7.44507, size = 101, normalized size = 1.25 \begin{align*} \frac{x^{4}}{4 \, c} - \frac{b \log \left (c x^{8} + b x^{4} + a\right )}{8 \, c^{2}} + \frac{{\left (b^{2} - 2 \, a c\right )} \arctan \left (\frac{2 \, c x^{4} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{4 \, \sqrt{-b^{2} + 4 \, a c} c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11/(c*x^8+b*x^4+a),x, algorithm="giac")

[Out]

1/4*x^4/c - 1/8*b*log(c*x^8 + b*x^4 + a)/c^2 + 1/4*(b^2 - 2*a*c)*arctan((2*c*x^4 + b)/sqrt(-b^2 + 4*a*c))/(sqr
t(-b^2 + 4*a*c)*c^2)