3.1.0 ∫ x5(d+ex3) _ a+bx3+cx6 dx [10]

Integrand size = 25, antiderivative size = 97 \[ \int \frac {x^5 \left (d+e x^3\right )}{a+b x^3+c x^6} \, dx=\frac {e x^3}{3 c}+\frac {\left (b c d-b^2 e+2 a c e\right ) \text {arctanh}\left (\frac {b+2 c x^3}{\sqrt {b^2-4 a c}}\right )}{3 c^2 \sqrt {b^2-4 a c}}+\frac {(c d-b e) \log \left (a+b x^3+c x^6\right )}{6 c^2} \]

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

Rubi [A] (veriﬁed)

Time = 0.08 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {1488, 787, 648, 632, 212, 642} \[ \int \frac {x^5 \left (d+e x^3\right )}{a+b x^3+c x^6} \, dx=\frac {\text {arctanh}\left (\frac {b+2 c x^3}{\sqrt {b^2-4 a c}}\right ) \left (2 a c e+b^2 (-e)+b c d\right )}{3 c^2 \sqrt {b^2-4 a c}}+\frac {(c d-b e) \log \left (a+b x^3+c x^6\right )}{6 c^2}+\frac {e x^3}{3 c} \]

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

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]

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]

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +

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] &

Int[(((d_.) + (e_.)*(x_))*((f_) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[e*g*(x/c

), x] + Dist[1/c, Int[(c*d*f - a*e*g + (c*e*f + c*d*g - b*e*g)*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c,

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]]

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

Mathematica [A] (veriﬁed)

Time = 0.08 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.96 \[ \int \frac {x^5 \left (d+e x^3\right )}{a+b x^3+c x^6} \, dx=\frac {2 c e x^3+\frac {2 \left (-b c d+b^2 e-2 a c e\right ) \arctan \left (\frac {b+2 c x^3}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}+(c d-b e) \log \left (a+b x^3+c x^6\right )}{6 c^2} \]

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

Maple [A] (veriﬁed)

Time = 0.16 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.01


method	result	size

default	\(\frac {e \,x^{3}}{3 c}+\frac {\frac {\left (-b e +c d \right ) \ln \left (c \,x^{6}+b \,x^{3}+a \right )}{2 c}+\frac {2 \left (-a e -\frac {\left (-b e +c d \right ) b}{2 c}\right ) \arctan \left (\frac {2 c \,x^{3}+b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{3 c}\)	\(98\)
risch	\(\text {Expression too large to display}\)	\(1400\)

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

Fricas [A] (veriﬁcation not implemented)

Time = 0.36 (sec) , antiderivative size = 305, normalized size of antiderivative = 3.14 \[ \int \frac {x^5 \left (d+e x^3\right )}{a+b x^3+c x^6} \, dx=\left [\frac {2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} e x^{3} + {\left (b c d - {\left (b^{2} - 2 \, a c\right )} e\right )} \sqrt {b^{2} - 4 \, a c} \log \left (\frac {2 \, c^{2} x^{6} + 2 \, b c x^{3} + b^{2} - 2 \, a c + {\left (2 \, c x^{3} + b\right )} \sqrt {b^{2} - 4 \, a c}}{c x^{6} + b x^{3} + a}\right ) + {\left ({\left (b^{2} c - 4 \, a c^{2}\right )} d - {\left (b^{3} - 4 \, a b c\right )} e\right )} \log \left (c x^{6} + b x^{3} + a\right )}{6 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )}}, \frac {2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} e x^{3} + 2 \, {\left (b c d - {\left (b^{2} - 2 \, a c\right )} e\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {{\left (2 \, c x^{3} + b\right )} \sqrt {-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) + {\left ({\left (b^{2} c - 4 \, a c^{2}\right )} d - {\left (b^{3} - 4 \, a b c\right )} e\right )} \log \left (c x^{6} + b x^{3} + a\right )}{6 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )}}\right ] \]

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

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

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

Sympy [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 434 vs. \(2 (94) = 188\).

Time = 101.95 (sec) , antiderivative size = 434, normalized size of antiderivative = 4.47 \[ \int \frac {x^5 \left (d+e x^3\right )}{a+b x^3+c x^6} \, dx=\left (- \frac {\sqrt {- 4 a c + b^{2}} \cdot \left (2 a c e - b^{2} e + b c d\right )}{6 c^{2} \cdot \left (4 a c - b^{2}\right )} - \frac {b e - c d}{6 c^{2}}\right ) \log {\left (x^{3} + \frac {- a b e - 12 a c^{2} \left (- \frac {\sqrt {- 4 a c + b^{2}} \cdot \left (2 a c e - b^{2} e + b c d\right )}{6 c^{2} \cdot \left (4 a c - b^{2}\right )} - \frac {b e - c d}{6 c^{2}}\right ) + 2 a c d + 3 b^{2} c \left (- \frac {\sqrt {- 4 a c + b^{2}} \cdot \left (2 a c e - b^{2} e + b c d\right )}{6 c^{2} \cdot \left (4 a c - b^{2}\right )} - \frac {b e - c d}{6 c^{2}}\right )}{2 a c e - b^{2} e + b c d} \right )} + \left (\frac {\sqrt {- 4 a c + b^{2}} \cdot \left (2 a c e - b^{2} e + b c d\right )}{6 c^{2} \cdot \left (4 a c - b^{2}\right )} - \frac {b e - c d}{6 c^{2}}\right ) \log {\left (x^{3} + \frac {- a b e - 12 a c^{2} \left (\frac {\sqrt {- 4 a c + b^{2}} \cdot \left (2 a c e - b^{2} e + b c d\right )}{6 c^{2} \cdot \left (4 a c - b^{2}\right )} - \frac {b e - c d}{6 c^{2}}\right ) + 2 a c d + 3 b^{2} c \left (\frac {\sqrt {- 4 a c + b^{2}} \cdot \left (2 a c e - b^{2} e + b c d\right )}{6 c^{2} \cdot \left (4 a c - b^{2}\right )} - \frac {b e - c d}{6 c^{2}}\right )}{2 a c e - b^{2} e + b c d} \right )} + \frac {e x^{3}}{3 c} \]

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

a*b*e - 12*a*c**2*(-sqrt(-4*a*c + b**2)*(2*a*c*e - b**2*e + b*c*d)/(6*c**2*(4*a*c - b**2)) - (b*e - c*d)/(6*c*

*2)) + 2*a*c*d + 3*b**2*c*(-sqrt(-4*a*c + b**2)*(2*a*c*e - b**2*e + b*c*d)/(6*c**2*(4*a*c - b**2)) - (b*e - c*

d)/(6*c**2)))/(2*a*c*e - b**2*e + b*c*d)) + (sqrt(-4*a*c + b**2)*(2*a*c*e - b**2*e + b*c*d)/(6*c**2*(4*a*c - b

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

6*c**2*(4*a*c - b**2)) - (b*e - c*d)/(6*c**2)) + 2*a*c*d + 3*b**2*c*(sqrt(-4*a*c + b**2)*(2*a*c*e - b**2*e + b

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^5 \left (d+e x^3\right )}{a+b x^3+c x^6} \, dx=\text {Exception raised: ValueError} \]

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

Giac [A] (veriﬁcation not implemented)

Time = 0.36 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.94 \[ \int \frac {x^5 \left (d+e x^3\right )}{a+b x^3+c x^6} \, dx=\frac {e x^{3}}{3 \, c} + \frac {{\left (c d - b e\right )} \log \left (c x^{6} + b x^{3} + a\right )}{6 \, c^{2}} - \frac {{\left (b c d - b^{2} e + 2 \, a c e\right )} \arctan \left (\frac {2 \, c x^{3} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{3 \, \sqrt {-b^{2} + 4 \, a c} c^{2}} \]

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

Mupad [B] (veriﬁcation not implemented)

Time = 11.31 (sec) , antiderivative size = 2624, normalized size of antiderivative = 27.05 \[ \int \frac {x^5 \left (d+e x^3\right )}{a+b x^3+c x^6} \, dx=\text {Too large to display} \]

(e*x^3)/(3*c) + (log(a + b*x^3 + c*x^6)*(3*b^3*e + 12*a*c^2*d - 3*b^2*c*d - 12*a*b*c*e))/(2*(36*a*c^3 - 9*b^2*

c^2)) + (atan((4*c^3*(4*a*c - b^2)^(3/2)*(x^3*((b*((b^2*c^3*d^3 - b^5*e^3 - a^2*b*c^2*e^3 + a^2*c^3*d*e^2 - 3*

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

12*b^2*c^4*d^2 + 12*b^4*c^2*e^2 - 18*a*b^2*c^3*e^2 - 24*b^3*c^3*d*e + 18*a*b*c^4*d*e)/c^3 - (((45*b^2*c^5*d -

45*b^3*c^4*e + 36*a*b*c^5*e)/c^3 - (27*b^2*c^3*(3*b^3*e + 12*a*c^2*d - 3*b^2*c*d - 12*a*b*c*e))/(36*a*c^3 - 9*

b^2*c^2))*(3*b^3*e + 12*a*c^2*d - 3*b^2*c*d - 12*a*b*c*e))/(2*(36*a*c^3 - 9*b^2*c^2)))*(3*b^3*e + 12*a*c^2*d -

3*b^2*c*d - 12*a*b*c*e))/(2*(36*a*c^3 - 9*b^2*c^2)) - (((((45*b^2*c^5*d - 45*b^3*c^4*e + 36*a*b*c^5*e)/c^3 -

(27*b^2*c^3*(3*b^3*e + 12*a*c^2*d - 3*b^2*c*d - 12*a*b*c*e))/(36*a*c^3 - 9*b^2*c^2))*(2*a*c*e - b^2*e + b*c*d)

)/(6*c^2*(4*a*c - b^2)^(1/2)) - (9*b^2*c*(2*a*c*e - b^2*e + b*c*d)*(3*b^3*e + 12*a*c^2*d - 3*b^2*c*d - 12*a*b*

c*e))/(2*(4*a*c - b^2)^(1/2)*(36*a*c^3 - 9*b^2*c^2)))*(2*a*c*e - b^2*e + b*c*d))/(6*c^2*(4*a*c - b^2)^(1/2)) +

(3*b^2*(2*a*c*e - b^2*e + b*c*d)^2*(3*b^3*e + 12*a*c^2*d - 3*b^2*c*d - 12*a*b*c*e))/(4*c*(4*a*c - b^2)*(36*a*

c^3 - 9*b^2*c^2))))/(4*a^2*c) + ((2*a*c - b^2)*((((((45*b^2*c^5*d - 45*b^3*c^4*e + 36*a*b*c^5*e)/c^3 - (27*b^2

*c^3*(3*b^3*e + 12*a*c^2*d - 3*b^2*c*d - 12*a*b*c*e))/(36*a*c^3 - 9*b^2*c^2))*(2*a*c*e - b^2*e + b*c*d))/(6*c^

2*(4*a*c - b^2)^(1/2)) - (9*b^2*c*(2*a*c*e - b^2*e + b*c*d)*(3*b^3*e + 12*a*c^2*d - 3*b^2*c*d - 12*a*b*c*e))/(

2*(4*a*c - b^2)^(1/2)*(36*a*c^3 - 9*b^2*c^2)))*(3*b^3*e + 12*a*c^2*d - 3*b^2*c*d - 12*a*b*c*e))/(2*(36*a*c^3 -

9*b^2*c^2)) + (b^2*(2*a*c*e - b^2*e + b*c*d)^3)/(4*c^3*(4*a*c - b^2)^(3/2)) - (((6*a^2*c^4*e^2 + 12*b^2*c^4*d

^2 + 12*b^4*c^2*e^2 - 18*a*b^2*c^3*e^2 - 24*b^3*c^3*d*e + 18*a*b*c^4*d*e)/c^3 - (((45*b^2*c^5*d - 45*b^3*c^4*e

+ 36*a*b*c^5*e)/c^3 - (27*b^2*c^3*(3*b^3*e + 12*a*c^2*d - 3*b^2*c*d - 12*a*b*c*e))/(36*a*c^3 - 9*b^2*c^2))*(3

*b^3*e + 12*a*c^2*d - 3*b^2*c*d - 12*a*b*c*e))/(2*(36*a*c^3 - 9*b^2*c^2)))*(2*a*c*e - b^2*e + b*c*d))/(6*c^2*(

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

3*d^3 + 3*a*b^3*c*d*e^2 - 3*a*b^2*c^2*d^2*e - 2*a^2*b*c^2*d*e^2)/c^3 - (((15*a*b^3*c^2*e^2 - 12*a^2*b*c^3*e^2

+ 15*a*b*c^4*d^2 + 12*a^2*c^4*d*e - 30*a*b^2*c^3*d*e)/c^3 - (((36*a^2*c^5*e + 72*a*b*c^5*d - 72*a*b^2*c^4*e)/c

^3 - (54*a*b*c^3*(3*b^3*e + 12*a*c^2*d - 3*b^2*c*d - 12*a*b*c*e))/(36*a*c^3 - 9*b^2*c^2))*(3*b^3*e + 12*a*c^2*

d - 3*b^2*c*d - 12*a*b*c*e))/(2*(36*a*c^3 - 9*b^2*c^2)))*(3*b^3*e + 12*a*c^2*d - 3*b^2*c*d - 12*a*b*c*e))/(2*(

36*a*c^3 - 9*b^2*c^2)) - (((((36*a^2*c^5*e + 72*a*b*c^5*d - 72*a*b^2*c^4*e)/c^3 - (54*a*b*c^3*(3*b^3*e + 12*a*

c^2*d - 3*b^2*c*d - 12*a*b*c*e))/(36*a*c^3 - 9*b^2*c^2))*(2*a*c*e - b^2*e + b*c*d))/(6*c^2*(4*a*c - b^2)^(1/2)

) - (9*a*b*c*(2*a*c*e - b^2*e + b*c*d)*(3*b^3*e + 12*a*c^2*d - 3*b^2*c*d - 12*a*b*c*e))/((4*a*c - b^2)^(1/2)*(

36*a*c^3 - 9*b^2*c^2)))*(2*a*c*e - b^2*e + b*c*d))/(6*c^2*(4*a*c - b^2)^(1/2)) + (3*a*b*(2*a*c*e - b^2*e + b*c

*d)^2*(3*b^3*e + 12*a*c^2*d - 3*b^2*c*d - 12*a*b*c*e))/(2*c*(4*a*c - b^2)*(36*a*c^3 - 9*b^2*c^2))))/(4*a^2*c)

+ ((2*a*c - b^2)*((((((36*a^2*c^5*e + 72*a*b*c^5*d - 72*a*b^2*c^4*e)/c^3 - (54*a*b*c^3*(3*b^3*e + 12*a*c^2*d -

3*b^2*c*d - 12*a*b*c*e))/(36*a*c^3 - 9*b^2*c^2))*(2*a*c*e - b^2*e + b*c*d))/(6*c^2*(4*a*c - b^2)^(1/2)) - (9*

a*b*c*(2*a*c*e - b^2*e + b*c*d)*(3*b^3*e + 12*a*c^2*d - 3*b^2*c*d - 12*a*b*c*e))/((4*a*c - b^2)^(1/2)*(36*a*c^

3 - 9*b^2*c^2)))*(3*b^3*e + 12*a*c^2*d - 3*b^2*c*d - 12*a*b*c*e))/(2*(36*a*c^3 - 9*b^2*c^2)) - (((15*a*b^3*c^2

*e^2 - 12*a^2*b*c^3*e^2 + 15*a*b*c^4*d^2 + 12*a^2*c^4*d*e - 30*a*b^2*c^3*d*e)/c^3 - (((36*a^2*c^5*e + 72*a*b*c

^5*d - 72*a*b^2*c^4*e)/c^3 - (54*a*b*c^3*(3*b^3*e + 12*a*c^2*d - 3*b^2*c*d - 12*a*b*c*e))/(36*a*c^3 - 9*b^2*c^

2))*(3*b^3*e + 12*a*c^2*d - 3*b^2*c*d - 12*a*b*c*e))/(2*(36*a*c^3 - 9*b^2*c^2)))*(2*a*c*e - b^2*e + b*c*d))/(6

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

b^2)^(1/2))))/(8*a^3*c^3*e^3 - b^6*e^3 + b^3*c^3*d^3 - 3*b^4*c^2*d^2*e - 12*a^2*b^2*c^2*e^3 + 6*a*b^4*c*e^3 +

3*b^5*c*d*e^2 + 6*a*b^2*c^3*d^2*e - 12*a*b^3*c^2*d*e^2 + 12*a^2*b*c^3*d*e^2))*(2*a*c*e - b^2*e + b*c*d))/(3*c^

\(\int \frac {x^5 (d+e x^3)}{a+b x^3+c x^6} \, dx\) [10]

Optimal result