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

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

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Rubi [A]  time = 0.22, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {1474, 800, 634, 618, 206, 628} \begin {gather*} -\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} \end {gather*}

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 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 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 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]

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 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 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]]

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*}

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Mathematica [A]  time = 0.07, size = 126, normalized size = 0.95 \begin {gather*} \frac {\left (-a c e+b^2 e-b c d\right ) \log \left (a+b x^3+c x^6\right )+\frac {2 \left (3 a b c e-2 a c^2 d+b^3 (-e)+b^2 c d\right ) \tan ^{-1}\left (\frac {b+2 c x^3}{\sqrt {4 a c-b^2}}\right )}{\sqrt {4 a c-b^2}}+2 c x^3 (c d-b e)+c^2 e x^6}{6 c^3} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^8 \left (d+e x^3\right )}{a+b x^3+c x^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.77, size = 430, normalized size = 3.26 \begin {gather*} \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 {gather*}

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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giac [A]  time = 1.00, size = 131, normalized size = 0.99 \begin {gather*} \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 {gather*}

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)

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maple [B]  time = 0.01, size = 260, normalized size = 1.97 \begin {gather*} \frac {e \,x^{6}}{6 c}-\frac {b e \,x^{3}}{3 c^{2}}+\frac {d \,x^{3}}{3 c}+\frac {a b e \arctan \left (\frac {2 c \,x^{3}+b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, c^{2}}-\frac {2 a d \arctan \left (\frac {2 c \,x^{3}+b}{\sqrt {4 a c -b^{2}}}\right )}{3 \sqrt {4 a c -b^{2}}\, c}-\frac {b^{3} e \arctan \left (\frac {2 c \,x^{3}+b}{\sqrt {4 a c -b^{2}}}\right )}{3 \sqrt {4 a c -b^{2}}\, c^{3}}+\frac {b^{2} d \arctan \left (\frac {2 c \,x^{3}+b}{\sqrt {4 a c -b^{2}}}\right )}{3 \sqrt {4 a c -b^{2}}\, c^{2}}-\frac {a e \ln \left (c \,x^{6}+b \,x^{3}+a \right )}{6 c^{2}}+\frac {b^{2} e \ln \left (c \,x^{6}+b \,x^{3}+a \right )}{6 c^{3}}-\frac {b d \ln \left (c \,x^{6}+b \,x^{3}+a \right )}{6 c^{2}} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

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 >> 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 details)Is 4*a*c-b^2 positive or negative?

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mupad [B]  time = 2.40, size = 3586, normalized size = 27.17

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^3*(d/(3*c) - (b*e)/(3*c^2)) + (e*x^6)/(6*c) - (log(a + b*x^3 + c*x^6)*(3*b^4*e + 12*a^2*c^2*e - 3*b^3*c*d +
12*a*b*c^2*d - 15*a*b^2*c*e))/(2*(36*a*c^4 - 9*b^2*c^3)) - (atan((4*c^6*(4*a*c - b^2)^(3/2)*(x^3*((b*((b^5*c^3
*d^3 - b^8*e^3 - 2*a*b^3*c^4*d^3 + a^2*b*c^5*d^3 + a^3*c^5*d^2*e - 3*b^6*c^2*d^2*e - 8*a^2*b^4*c^2*e^3 + 4*a^3
*b^2*c^3*e^3 + 5*a*b^6*c*e^3 + 3*b^7*c*d*e^2 + 9*a*b^4*c^3*d^2*e - 12*a*b^5*c^2*d*e^2 - 4*a^3*b*c^4*d*e^2 - 7*
a^2*b^2*c^4*d^2*e + 14*a^2*b^3*c^3*d*e^2)/c^6 - (((6*a^2*c^7*d^2 + 12*b^4*c^5*d^2 + 12*b^6*c^3*e^2 - 18*a*b^2*
c^6*d^2 - 42*a*b^4*c^4*e^2 + 36*a^2*b^2*c^5*e^2 - 24*b^5*c^4*d*e + 60*a*b^3*c^5*d*e - 30*a^2*b*c^6*d*e)/c^6 -
(((45*b^3*c^7*d - 45*b^4*c^6*e - 36*a*b*c^8*d + 81*a*b^2*c^7*e)/c^6 - (27*b^2*c^3*(3*b^4*e + 12*a^2*c^2*e - 3*
b^3*c*d + 12*a*b*c^2*d - 15*a*b^2*c*e))/(36*a*c^4 - 9*b^2*c^3))*(3*b^4*e + 12*a^2*c^2*e - 3*b^3*c*d + 12*a*b*c
^2*d - 15*a*b^2*c*e))/(2*(36*a*c^4 - 9*b^2*c^3)))*(3*b^4*e + 12*a^2*c^2*e - 3*b^3*c*d + 12*a*b*c^2*d - 15*a*b^
2*c*e))/(2*(36*a*c^4 - 9*b^2*c^3)) - (((((45*b^3*c^7*d - 45*b^4*c^6*e - 36*a*b*c^8*d + 81*a*b^2*c^7*e)/c^6 - (
27*b^2*c^3*(3*b^4*e + 12*a^2*c^2*e - 3*b^3*c*d + 12*a*b*c^2*d - 15*a*b^2*c*e))/(36*a*c^4 - 9*b^2*c^3))*(b^3*e
+ 2*a*c^2*d - b^2*c*d - 3*a*b*c*e))/(6*c^3*(4*a*c - b^2)^(1/2)) - (9*b^2*(b^3*e + 2*a*c^2*d - b^2*c*d - 3*a*b*
c*e)*(3*b^4*e + 12*a^2*c^2*e - 3*b^3*c*d + 12*a*b*c^2*d - 15*a*b^2*c*e))/(2*(4*a*c - b^2)^(1/2)*(36*a*c^4 - 9*
b^2*c^3)))*(b^3*e + 2*a*c^2*d - b^2*c*d - 3*a*b*c*e))/(6*c^3*(4*a*c - b^2)^(1/2)) + (3*b^2*(b^3*e + 2*a*c^2*d
- b^2*c*d - 3*a*b*c*e)^2*(3*b^4*e + 12*a^2*c^2*e - 3*b^3*c*d + 12*a*b*c^2*d - 15*a*b^2*c*e))/(4*c^3*(4*a*c - b
^2)*(36*a*c^4 - 9*b^2*c^3))))/(4*a^2*c) - ((2*a*c - b^2)*((((((45*b^3*c^7*d - 45*b^4*c^6*e - 36*a*b*c^8*d + 81
*a*b^2*c^7*e)/c^6 - (27*b^2*c^3*(3*b^4*e + 12*a^2*c^2*e - 3*b^3*c*d + 12*a*b*c^2*d - 15*a*b^2*c*e))/(36*a*c^4
- 9*b^2*c^3))*(b^3*e + 2*a*c^2*d - b^2*c*d - 3*a*b*c*e))/(6*c^3*(4*a*c - b^2)^(1/2)) - (9*b^2*(b^3*e + 2*a*c^2
*d - b^2*c*d - 3*a*b*c*e)*(3*b^4*e + 12*a^2*c^2*e - 3*b^3*c*d + 12*a*b*c^2*d - 15*a*b^2*c*e))/(2*(4*a*c - b^2)
^(1/2)*(36*a*c^4 - 9*b^2*c^3)))*(3*b^4*e + 12*a^2*c^2*e - 3*b^3*c*d + 12*a*b*c^2*d - 15*a*b^2*c*e))/(2*(36*a*c
^4 - 9*b^2*c^3)) - (((6*a^2*c^7*d^2 + 12*b^4*c^5*d^2 + 12*b^6*c^3*e^2 - 18*a*b^2*c^6*d^2 - 42*a*b^4*c^4*e^2 +
36*a^2*b^2*c^5*e^2 - 24*b^5*c^4*d*e + 60*a*b^3*c^5*d*e - 30*a^2*b*c^6*d*e)/c^6 - (((45*b^3*c^7*d - 45*b^4*c^6*
e - 36*a*b*c^8*d + 81*a*b^2*c^7*e)/c^6 - (27*b^2*c^3*(3*b^4*e + 12*a^2*c^2*e - 3*b^3*c*d + 12*a*b*c^2*d - 15*a
*b^2*c*e))/(36*a*c^4 - 9*b^2*c^3))*(3*b^4*e + 12*a^2*c^2*e - 3*b^3*c*d + 12*a*b*c^2*d - 15*a*b^2*c*e))/(2*(36*
a*c^4 - 9*b^2*c^3)))*(b^3*e + 2*a*c^2*d - b^2*c*d - 3*a*b*c*e))/(6*c^3*(4*a*c - b^2)^(1/2)) + (b^2*(b^3*e + 2*
a*c^2*d - b^2*c*d - 3*a*b*c*e)^3)/(4*c^6*(4*a*c - b^2)^(3/2))))/(4*a^2*c*(4*a*c - b^2)^(1/2))) - (b*((a*b^7*e^
3 - a*b^4*c^3*d^3 - 4*a^2*b^5*c*e^3 - 2*a^4*b*c^3*e^3 + a^4*c^4*d*e^2 + a^2*b^2*c^4*d^3 + 5*a^3*b^3*c^2*e^3 -
3*a*b^6*c*d*e^2 + 3*a*b^5*c^2*d^2*e + 2*a^3*b*c^4*d^2*e - 6*a^2*b^3*c^3*d^2*e + 9*a^2*b^4*c^2*d*e^2 - 7*a^3*b^
2*c^3*d*e^2)/c^6 + (((15*a*b^3*c^5*d^2 - 12*a^2*b*c^6*d^2 + 15*a*b^5*c^3*e^2 + 27*a^3*b*c^5*e^2 - 42*a^2*b^3*c
^4*e^2 - 12*a^3*c^6*d*e - 30*a*b^4*c^4*d*e + 54*a^2*b^2*c^5*d*e)/c^6 + (((36*a^2*c^8*d - 72*a*b^2*c^7*d + 72*a
*b^3*c^6*e - 108*a^2*b*c^7*e)/c^6 + (54*a*b*c^3*(3*b^4*e + 12*a^2*c^2*e - 3*b^3*c*d + 12*a*b*c^2*d - 15*a*b^2*
c*e))/(36*a*c^4 - 9*b^2*c^3))*(3*b^4*e + 12*a^2*c^2*e - 3*b^3*c*d + 12*a*b*c^2*d - 15*a*b^2*c*e))/(2*(36*a*c^4
- 9*b^2*c^3)))*(3*b^4*e + 12*a^2*c^2*e - 3*b^3*c*d + 12*a*b*c^2*d - 15*a*b^2*c*e))/(2*(36*a*c^4 - 9*b^2*c^3))
- (((((36*a^2*c^8*d - 72*a*b^2*c^7*d + 72*a*b^3*c^6*e - 108*a^2*b*c^7*e)/c^6 + (54*a*b*c^3*(3*b^4*e + 12*a^2*
c^2*e - 3*b^3*c*d + 12*a*b*c^2*d - 15*a*b^2*c*e))/(36*a*c^4 - 9*b^2*c^3))*(b^3*e + 2*a*c^2*d - b^2*c*d - 3*a*b
*c*e))/(6*c^3*(4*a*c - b^2)^(1/2)) + (9*a*b*(b^3*e + 2*a*c^2*d - b^2*c*d - 3*a*b*c*e)*(3*b^4*e + 12*a^2*c^2*e
- 3*b^3*c*d + 12*a*b*c^2*d - 15*a*b^2*c*e))/((4*a*c - b^2)^(1/2)*(36*a*c^4 - 9*b^2*c^3)))*(b^3*e + 2*a*c^2*d -
b^2*c*d - 3*a*b*c*e))/(6*c^3*(4*a*c - b^2)^(1/2)) - (3*a*b*(b^3*e + 2*a*c^2*d - b^2*c*d - 3*a*b*c*e)^2*(3*b^4
*e + 12*a^2*c^2*e - 3*b^3*c*d + 12*a*b*c^2*d - 15*a*b^2*c*e))/(2*c^3*(4*a*c - b^2)*(36*a*c^4 - 9*b^2*c^3))))/(
4*a^2*c) + ((2*a*c - b^2)*((((((36*a^2*c^8*d - 72*a*b^2*c^7*d + 72*a*b^3*c^6*e - 108*a^2*b*c^7*e)/c^6 + (54*a*
b*c^3*(3*b^4*e + 12*a^2*c^2*e - 3*b^3*c*d + 12*a*b*c^2*d - 15*a*b^2*c*e))/(36*a*c^4 - 9*b^2*c^3))*(b^3*e + 2*a
*c^2*d - b^2*c*d - 3*a*b*c*e))/(6*c^3*(4*a*c - b^2)^(1/2)) + (9*a*b*(b^3*e + 2*a*c^2*d - b^2*c*d - 3*a*b*c*e)*
(3*b^4*e + 12*a^2*c^2*e - 3*b^3*c*d + 12*a*b*c^2*d - 15*a*b^2*c*e))/((4*a*c - b^2)^(1/2)*(36*a*c^4 - 9*b^2*c^3
)))*(3*b^4*e + 12*a^2*c^2*e - 3*b^3*c*d + 12*a*b*c^2*d - 15*a*b^2*c*e))/(2*(36*a*c^4 - 9*b^2*c^3)) + (((15*a*b
^3*c^5*d^2 - 12*a^2*b*c^6*d^2 + 15*a*b^5*c^3*e^2 + 27*a^3*b*c^5*e^2 - 42*a^2*b^3*c^4*e^2 - 12*a^3*c^6*d*e - 30
*a*b^4*c^4*d*e + 54*a^2*b^2*c^5*d*e)/c^6 + (((36*a^2*c^8*d - 72*a*b^2*c^7*d + 72*a*b^3*c^6*e - 108*a^2*b*c^7*e
)/c^6 + (54*a*b*c^3*(3*b^4*e + 12*a^2*c^2*e - 3*b^3*c*d + 12*a*b*c^2*d - 15*a*b^2*c*e))/(36*a*c^4 - 9*b^2*c^3)
)*(3*b^4*e + 12*a^2*c^2*e - 3*b^3*c*d + 12*a*b*c^2*d - 15*a*b^2*c*e))/(2*(36*a*c^4 - 9*b^2*c^3)))*(b^3*e + 2*a
*c^2*d - b^2*c*d - 3*a*b*c*e))/(6*c^3*(4*a*c - b^2)^(1/2)) - (a*b*(b^3*e + 2*a*c^2*d - b^2*c*d - 3*a*b*c*e)^3)
/(2*c^6*(4*a*c - b^2)^(3/2))))/(4*a^2*c*(4*a*c - b^2)^(1/2))))/(b^9*e^3 + 8*a^3*c^6*d^3 - b^6*c^3*d^3 + 6*a*b^
4*c^4*d^3 + 3*b^7*c^2*d^2*e - 12*a^2*b^2*c^5*d^3 + 27*a^2*b^5*c^2*e^3 - 27*a^3*b^3*c^3*e^3 - 9*a*b^7*c*e^3 - 3
*b^8*c*d*e^2 - 21*a*b^5*c^3*d^2*e + 24*a*b^6*c^2*d*e^2 - 36*a^3*b*c^5*d^2*e + 48*a^2*b^3*c^4*d^2*e - 63*a^2*b^
4*c^3*d*e^2 + 54*a^3*b^2*c^4*d*e^2))*(b^3*e + 2*a*c^2*d - b^2*c*d - 3*a*b*c*e))/(3*c^3*(4*a*c - b^2)^(1/2))

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sympy [B]  time = 55.47, size = 620, normalized size = 4.70 \begin {gather*} x^{3} \left (- \frac {b e}{3 c^{2}} + \frac {d}{3 c}\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 )} + \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} \end {gather*}

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]

x**3*(-b*e/(3*c**2) + d/(3*c)) + (-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*(-s
qrt(-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)) + (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)

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