3.44 \(\int \frac{x^3 (d+e x^4)}{a+b x^4+c x^8} \, dx\)

Optimal. Leaf size=72 \[ \frac{e \log \left (a+b x^4+c x^8\right )}{8 c}-\frac{(2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x^4}{\sqrt{b^2-4 a c}}\right )}{4 c \sqrt{b^2-4 a c}} \]

[Out]

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

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Rubi [A]  time = 0.0721409, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {1468, 634, 618, 206, 628} \[ \frac{e \log \left (a+b x^4+c x^8\right )}{8 c}-\frac{(2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x^4}{\sqrt{b^2-4 a c}}\right )}{4 c \sqrt{b^2-4 a c}} \]

Antiderivative was successfully verified.

[In]

Int[(x^3*(d + e*x^4))/(a + b*x^4 + c*x^8),x]

[Out]

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

Rule 1468

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

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^3 \left (d+e x^4\right )}{a+b x^4+c x^8} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{d+e x}{a+b x+c x^2} \, dx,x,x^4\right )\\ &=\frac{e \operatorname{Subst}\left (\int \frac{b+2 c x}{a+b x+c x^2} \, dx,x,x^4\right )}{8 c}+\frac{(2 c d-b e) \operatorname{Subst}\left (\int \frac{1}{a+b x+c x^2} \, dx,x,x^4\right )}{8 c}\\ &=\frac{e \log \left (a+b x^4+c x^8\right )}{8 c}-\frac{(2 c d-b e) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^4\right )}{4 c}\\ &=-\frac{(2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x^4}{\sqrt{b^2-4 a c}}\right )}{4 c \sqrt{b^2-4 a c}}+\frac{e \log \left (a+b x^4+c x^8\right )}{8 c}\\ \end{align*}

Mathematica [A]  time = 0.0555689, size = 71, normalized size = 0.99 \[ \frac{e \log \left (a+b x^4+c x^8\right )-\frac{2 (b e-2 c d) \tan ^{-1}\left (\frac{b+2 c x^4}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}}{8 c} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^3*(d + e*x^4))/(a + b*x^4 + c*x^8),x]

[Out]

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

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Maple [A]  time = 0.003, size = 99, normalized size = 1.4 \begin{align*}{\frac{e\ln \left ( c{x}^{8}+b{x}^{4}+a \right ) }{8\,c}}+{\frac{d}{2}\arctan \left ({(2\,c{x}^{4}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{be}{4\,c}\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^3*(e*x^4+d)/(c*x^8+b*x^4+a),x)

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.22427, size = 481, normalized size = 6.68 \begin{align*} \left [\frac{{\left (b^{2} - 4 \, a c\right )} e \log \left (c x^{8} + b x^{4} + a\right ) - \sqrt{b^{2} - 4 \, a c}{\left (2 \, c d - b e\right )} \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 )}{8 \,{\left (b^{2} c - 4 \, a c^{2}\right )}}, \frac{{\left (b^{2} - 4 \, a c\right )} e \log \left (c x^{8} + b x^{4} + a\right ) - 2 \, \sqrt{-b^{2} + 4 \, a c}{\left (2 \, c d - b e\right )} \arctan \left (-\frac{{\left (2 \, c x^{4} + b\right )} \sqrt{-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right )}{8 \,{\left (b^{2} c - 4 \, a c^{2}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 7.51907, size = 287, normalized size = 3.99 \begin{align*} \left (\frac{e}{8 c} - \frac{\sqrt{- 4 a c + b^{2}} \left (b e - 2 c d\right )}{8 c \left (4 a c - b^{2}\right )}\right ) \log{\left (x^{4} + \frac{- 16 a c \left (\frac{e}{8 c} - \frac{\sqrt{- 4 a c + b^{2}} \left (b e - 2 c d\right )}{8 c \left (4 a c - b^{2}\right )}\right ) + 2 a e + 4 b^{2} \left (\frac{e}{8 c} - \frac{\sqrt{- 4 a c + b^{2}} \left (b e - 2 c d\right )}{8 c \left (4 a c - b^{2}\right )}\right ) - b d}{b e - 2 c d} \right )} + \left (\frac{e}{8 c} + \frac{\sqrt{- 4 a c + b^{2}} \left (b e - 2 c d\right )}{8 c \left (4 a c - b^{2}\right )}\right ) \log{\left (x^{4} + \frac{- 16 a c \left (\frac{e}{8 c} + \frac{\sqrt{- 4 a c + b^{2}} \left (b e - 2 c d\right )}{8 c \left (4 a c - b^{2}\right )}\right ) + 2 a e + 4 b^{2} \left (\frac{e}{8 c} + \frac{\sqrt{- 4 a c + b^{2}} \left (b e - 2 c d\right )}{8 c \left (4 a c - b^{2}\right )}\right ) - b d}{b e - 2 c d} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 6.16674, size = 95, normalized size = 1.32 \begin{align*} \frac{e \log \left (c x^{8} + b x^{4} + a\right )}{8 \, c} + \frac{{\left (2 \, c d - b e\right )} \arctan \left (\frac{2 \, c x^{4} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{4 \, \sqrt{-b^{2} + 4 \, a c} c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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