### 3.9 $$\int \frac{x^8 (d+e x^3)}{a+b x^3+c x^6} \, dx$$

Optimal. Leaf size=132 $-\frac{\left (a c e+b^2 (-e)+b c d\right ) \log \left (a+b x^3+c x^6\right )}{6 c^3}-\frac{\left (3 a b c e-2 a c^2 d+b^2 c d+b^3 (-e)\right ) \tanh ^{-1}\left (\frac{b+2 c x^3}{\sqrt{b^2-4 a c}}\right )}{3 c^3 \sqrt{b^2-4 a c}}+\frac{x^3 (c d-b e)}{3 c^2}+\frac{e x^6}{6 c}$

[Out]

((c*d - b*e)*x^3)/(3*c^2) + (e*x^6)/(6*c) - ((b^2*c*d - 2*a*c^2*d - b^3*e + 3*a*b*c*e)*ArcTanh[(b + 2*c*x^3)/S
qrt[b^2 - 4*a*c]])/(3*c^3*Sqrt[b^2 - 4*a*c]) - ((b*c*d - b^2*e + a*c*e)*Log[a + b*x^3 + c*x^6])/(6*c^3)

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Rubi [A]  time = 0.217983, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 25, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.24, Rules used = {1474, 800, 634, 618, 206, 628} $-\frac{\left (a c e+b^2 (-e)+b c d\right ) \log \left (a+b x^3+c x^6\right )}{6 c^3}-\frac{\left (3 a b c e-2 a c^2 d+b^2 c d+b^3 (-e)\right ) \tanh ^{-1}\left (\frac{b+2 c x^3}{\sqrt{b^2-4 a c}}\right )}{3 c^3 \sqrt{b^2-4 a c}}+\frac{x^3 (c d-b e)}{3 c^2}+\frac{e x^6}{6 c}$

Antiderivative was successfully veriﬁed.

[In]

Int[(x^8*(d + e*x^3))/(a + b*x^3 + c*x^6),x]

[Out]

((c*d - b*e)*x^3)/(3*c^2) + (e*x^6)/(6*c) - ((b^2*c*d - 2*a*c^2*d - b^3*e + 3*a*b*c*e)*ArcTanh[(b + 2*c*x^3)/S
qrt[b^2 - 4*a*c]])/(3*c^3*Sqrt[b^2 - 4*a*c]) - ((b*c*d - b^2*e + a*c*e)*Log[a + b*x^3 + c*x^6])/(6*c^3)

Rule 1474

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :>
Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(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] && IntegerQ[Simplify[(m + 1)/n]]

Rule 800

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[((d + e*x)^m*(f + g*x))/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 -
4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[m]

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^8 \left (d+e x^3\right )}{a+b x^3+c x^6} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^2 (d+e x)}{a+b x+c x^2} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{c d-b e}{c^2}+\frac{e x}{c}-\frac{a (c d-b e)+\left (b c d-b^2 e+a c e\right ) x}{c^2 \left (a+b x+c x^2\right )}\right ) \, dx,x,x^3\right )\\ &=\frac{(c d-b e) x^3}{3 c^2}+\frac{e x^6}{6 c}-\frac{\operatorname{Subst}\left (\int \frac{a (c d-b e)+\left (b c d-b^2 e+a c e\right ) x}{a+b x+c x^2} \, dx,x,x^3\right )}{3 c^2}\\ &=\frac{(c d-b e) x^3}{3 c^2}+\frac{e x^6}{6 c}-\frac{\left (b c d-b^2 e+a c e\right ) \operatorname{Subst}\left (\int \frac{b+2 c x}{a+b x+c x^2} \, dx,x,x^3\right )}{6 c^3}+\frac{\left (b^2 c d-2 a c^2 d-b^3 e+3 a b c e\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x+c x^2} \, dx,x,x^3\right )}{6 c^3}\\ &=\frac{(c d-b e) x^3}{3 c^2}+\frac{e x^6}{6 c}-\frac{\left (b c d-b^2 e+a c e\right ) \log \left (a+b x^3+c x^6\right )}{6 c^3}-\frac{\left (b^2 c d-2 a c^2 d-b^3 e+3 a b c e\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^3\right )}{3 c^3}\\ &=\frac{(c d-b e) x^3}{3 c^2}+\frac{e x^6}{6 c}-\frac{\left (b^2 c d-2 a c^2 d-b^3 e+3 a b c e\right ) \tanh ^{-1}\left (\frac{b+2 c x^3}{\sqrt{b^2-4 a c}}\right )}{3 c^3 \sqrt{b^2-4 a c}}-\frac{\left (b c d-b^2 e+a c e\right ) \log \left (a+b x^3+c x^6\right )}{6 c^3}\\ \end{align*}

Mathematica [A]  time = 0.0654022, size = 126, normalized size = 0.95 $\frac{\frac{2 \left (3 a b c e-2 a c^2 d+b^2 c d+b^3 (-e)\right ) \tan ^{-1}\left (\frac{b+2 c x^3}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}+\left (-a c e+b^2 e-b c d\right ) \log \left (a+b x^3+c x^6\right )+2 c x^3 (c d-b e)+c^2 e x^6}{6 c^3}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(x^8*(d + e*x^3))/(a + b*x^3 + c*x^6),x]

[Out]

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

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Maple [B]  time = 0.004, size = 260, normalized size = 2. \begin{align*}{\frac{e{x}^{6}}{6\,c}}-{\frac{be{x}^{3}}{3\,{c}^{2}}}+{\frac{d{x}^{3}}{3\,c}}-{\frac{\ln \left ( c{x}^{6}+b{x}^{3}+a \right ) ae}{6\,{c}^{2}}}+{\frac{\ln \left ( c{x}^{6}+b{x}^{3}+a \right ){b}^{2}e}{6\,{c}^{3}}}-{\frac{\ln \left ( c{x}^{6}+b{x}^{3}+a \right ) bd}{6\,{c}^{2}}}+{\frac{abe}{{c}^{2}}\arctan \left ({(2\,c{x}^{3}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{2\,ad}{3\,c}\arctan \left ({(2\,c{x}^{3}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{{b}^{3}e}{3\,{c}^{3}}\arctan \left ({(2\,c{x}^{3}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{\frac{{b}^{2}d}{3\,{c}^{2}}\arctan \left ({(2\,c{x}^{3}+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^8*(e*x^3+d)/(c*x^6+b*x^3+a),x)

[Out]

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

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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^8*(e*x^3+d)/(c*x^6+b*x^3+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 3.25384, size = 902, normalized size = 6.83 \begin{align*} \left [\frac{{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} e x^{6} + 2 \,{\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d -{\left (b^{3} c - 4 \, a b c^{2}\right )} e\right )} x^{3} + \sqrt{b^{2} - 4 \, a c}{\left ({\left (b^{2} c - 2 \, a c^{2}\right )} d -{\left (b^{3} - 3 \, a b c\right )} e\right )} \log \left (\frac{2 \, c^{2} x^{6} + 2 \, b c x^{3} + b^{2} - 2 \, a c -{\left (2 \, c x^{3} + b\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{6} + b x^{3} + a}\right ) -{\left ({\left (b^{3} c - 4 \, a b c^{2}\right )} d -{\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} e\right )} \log \left (c x^{6} + b x^{3} + a\right )}{6 \,{\left (b^{2} c^{3} - 4 \, a c^{4}\right )}}, \frac{{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} e x^{6} + 2 \,{\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d -{\left (b^{3} c - 4 \, a b c^{2}\right )} e\right )} x^{3} - 2 \, \sqrt{-b^{2} + 4 \, a c}{\left ({\left (b^{2} c - 2 \, a c^{2}\right )} d -{\left (b^{3} - 3 \, a b c\right )} e\right )} \arctan \left (-\frac{{\left (2 \, c x^{3} + b\right )} \sqrt{-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) -{\left ({\left (b^{3} c - 4 \, a b c^{2}\right )} d -{\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} e\right )} \log \left (c x^{6} + b x^{3} + a\right )}{6 \,{\left (b^{2} c^{3} - 4 \, a c^{4}\right )}}\right ] \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(e*x^3+d)/(c*x^6+b*x^3+a),x, algorithm="fricas")

[Out]

[1/6*((b^2*c^2 - 4*a*c^3)*e*x^6 + 2*((b^2*c^2 - 4*a*c^3)*d - (b^3*c - 4*a*b*c^2)*e)*x^3 + sqrt(b^2 - 4*a*c)*((
b^2*c - 2*a*c^2)*d - (b^3 - 3*a*b*c)*e)*log((2*c^2*x^6 + 2*b*c*x^3 + b^2 - 2*a*c - (2*c*x^3 + b)*sqrt(b^2 - 4*
a*c))/(c*x^6 + b*x^3 + a)) - ((b^3*c - 4*a*b*c^2)*d - (b^4 - 5*a*b^2*c + 4*a^2*c^2)*e)*log(c*x^6 + b*x^3 + a))
/(b^2*c^3 - 4*a*c^4), 1/6*((b^2*c^2 - 4*a*c^3)*e*x^6 + 2*((b^2*c^2 - 4*a*c^3)*d - (b^3*c - 4*a*b*c^2)*e)*x^3 -
2*sqrt(-b^2 + 4*a*c)*((b^2*c - 2*a*c^2)*d - (b^3 - 3*a*b*c)*e)*arctan(-(2*c*x^3 + b)*sqrt(-b^2 + 4*a*c)/(b^2
- 4*a*c)) - ((b^3*c - 4*a*b*c^2)*d - (b^4 - 5*a*b^2*c + 4*a^2*c^2)*e)*log(c*x^6 + b*x^3 + a))/(b^2*c^3 - 4*a*c
^4)]

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Sympy [B]  time = 15.0021, size = 619, normalized size = 4.69 \begin{align*} \left (- \frac{\sqrt{- 4 a c + b^{2}} \left (3 a b c e - 2 a c^{2} d - b^{3} e + b^{2} c d\right )}{6 c^{3} \left (4 a c - b^{2}\right )} - \frac{a c e - b^{2} e + b c d}{6 c^{3}}\right ) \log{\left (x^{3} + \frac{2 a^{2} c e - a b^{2} e + a b c d + 12 a c^{3} \left (- \frac{\sqrt{- 4 a c + b^{2}} \left (3 a b c e - 2 a c^{2} d - b^{3} e + b^{2} c d\right )}{6 c^{3} \left (4 a c - b^{2}\right )} - \frac{a c e - b^{2} e + b c d}{6 c^{3}}\right ) - 3 b^{2} c^{2} \left (- \frac{\sqrt{- 4 a c + b^{2}} \left (3 a b c e - 2 a c^{2} d - b^{3} e + b^{2} c d\right )}{6 c^{3} \left (4 a c - b^{2}\right )} - \frac{a c e - b^{2} e + b c d}{6 c^{3}}\right )}{3 a b c e - 2 a c^{2} d - b^{3} e + b^{2} c d} \right )} + \left (\frac{\sqrt{- 4 a c + b^{2}} \left (3 a b c e - 2 a c^{2} d - b^{3} e + b^{2} c d\right )}{6 c^{3} \left (4 a c - b^{2}\right )} - \frac{a c e - b^{2} e + b c d}{6 c^{3}}\right ) \log{\left (x^{3} + \frac{2 a^{2} c e - a b^{2} e + a b c d + 12 a c^{3} \left (\frac{\sqrt{- 4 a c + b^{2}} \left (3 a b c e - 2 a c^{2} d - b^{3} e + b^{2} c d\right )}{6 c^{3} \left (4 a c - b^{2}\right )} - \frac{a c e - b^{2} e + b c d}{6 c^{3}}\right ) - 3 b^{2} c^{2} \left (\frac{\sqrt{- 4 a c + b^{2}} \left (3 a b c e - 2 a c^{2} d - b^{3} e + b^{2} c d\right )}{6 c^{3} \left (4 a c - b^{2}\right )} - \frac{a c e - b^{2} e + b c d}{6 c^{3}}\right )}{3 a b c e - 2 a c^{2} d - b^{3} e + b^{2} c d} \right )} + \frac{e x^{6}}{6 c} - \frac{x^{3} \left (b e - c d\right )}{3 c^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**8*(e*x**3+d)/(c*x**6+b*x**3+a),x)

[Out]

(-sqrt(-4*a*c + b**2)*(3*a*b*c*e - 2*a*c**2*d - b**3*e + b**2*c*d)/(6*c**3*(4*a*c - b**2)) - (a*c*e - b**2*e +
b*c*d)/(6*c**3))*log(x**3 + (2*a**2*c*e - a*b**2*e + a*b*c*d + 12*a*c**3*(-sqrt(-4*a*c + b**2)*(3*a*b*c*e - 2
*a*c**2*d - b**3*e + b**2*c*d)/(6*c**3*(4*a*c - b**2)) - (a*c*e - b**2*e + b*c*d)/(6*c**3)) - 3*b**2*c**2*(-sq
rt(-4*a*c + b**2)*(3*a*b*c*e - 2*a*c**2*d - b**3*e + b**2*c*d)/(6*c**3*(4*a*c - b**2)) - (a*c*e - b**2*e + b*c
*d)/(6*c**3)))/(3*a*b*c*e - 2*a*c**2*d - b**3*e + b**2*c*d)) + (sqrt(-4*a*c + b**2)*(3*a*b*c*e - 2*a*c**2*d -
b**3*e + b**2*c*d)/(6*c**3*(4*a*c - b**2)) - (a*c*e - b**2*e + b*c*d)/(6*c**3))*log(x**3 + (2*a**2*c*e - a*b**
2*e + a*b*c*d + 12*a*c**3*(sqrt(-4*a*c + b**2)*(3*a*b*c*e - 2*a*c**2*d - b**3*e + b**2*c*d)/(6*c**3*(4*a*c - b
**2)) - (a*c*e - b**2*e + b*c*d)/(6*c**3)) - 3*b**2*c**2*(sqrt(-4*a*c + b**2)*(3*a*b*c*e - 2*a*c**2*d - b**3*e
+ b**2*c*d)/(6*c**3*(4*a*c - b**2)) - (a*c*e - b**2*e + b*c*d)/(6*c**3)))/(3*a*b*c*e - 2*a*c**2*d - b**3*e +
b**2*c*d)) + e*x**6/(6*c) - x**3*(b*e - c*d)/(3*c**2)

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Giac [A]  time = 1.36096, size = 177, normalized size = 1.34 \begin{align*} \frac{c x^{6} e + 2 \, c d x^{3} - 2 \, b x^{3} e}{6 \, c^{2}} - \frac{{\left (b c d - b^{2} e + a c e\right )} \log \left (c x^{6} + b x^{3} + a\right )}{6 \, c^{3}} + \frac{{\left (b^{2} c d - 2 \, a c^{2} d - b^{3} e + 3 \, a b c e\right )} \arctan \left (\frac{2 \, c x^{3} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{3 \, \sqrt{-b^{2} + 4 \, a c} c^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(e*x^3+d)/(c*x^6+b*x^3+a),x, algorithm="giac")

[Out]

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