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

Optimal. Leaf size=242 \[ -\frac {x \left (7 c d^2-e (5 a e+b d)\right )}{18 d^2 e^2 \left (d+e x^3\right )}+\frac {x \left (a e^2-b d e+c d^2\right )}{6 d e^2 \left (d+e x^3\right )^2}-\frac {\log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right ) \left (e (5 a e+b d)+2 c d^2\right )}{54 d^{8/3} e^{7/3}}+\frac {\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \left (e (5 a e+b d)+2 c d^2\right )}{27 d^{8/3} e^{7/3}}-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt {3} \sqrt [3]{d}}\right ) \left (e (5 a e+b d)+2 c d^2\right )}{9 \sqrt {3} d^{8/3} e^{7/3}} \]

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Rubi [A]  time = 0.26, antiderivative size = 242, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {1409, 385, 200, 31, 634, 617, 204, 628} \begin {gather*} -\frac {x \left (7 c d^2-e (5 a e+b d)\right )}{18 d^2 e^2 \left (d+e x^3\right )}+\frac {x \left (a e^2-b d e+c d^2\right )}{6 d e^2 \left (d+e x^3\right )^2}-\frac {\log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right ) \left (e (5 a e+b d)+2 c d^2\right )}{54 d^{8/3} e^{7/3}}+\frac {\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \left (e (5 a e+b d)+2 c d^2\right )}{27 d^{8/3} e^{7/3}}-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt {3} \sqrt [3]{d}}\right ) \left (e (5 a e+b d)+2 c d^2\right )}{9 \sqrt {3} d^{8/3} e^{7/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((c*d^2 - b*d*e + a*e^2)*x)/(6*d*e^2*(d + e*x^3)^2) - ((7*c*d^2 - e*(b*d + 5*a*e))*x)/(18*d^2*e^2*(d + e*x^3))
- ((2*c*d^2 + e*(b*d + 5*a*e))*ArcTan[(d^(1/3) - 2*e^(1/3)*x)/(Sqrt[3]*d^(1/3))])/(9*Sqrt[3]*d^(8/3)*e^(7/3))
+ ((2*c*d^2 + e*(b*d + 5*a*e))*Log[d^(1/3) + e^(1/3)*x])/(27*d^(8/3)*e^(7/3)) - ((2*c*d^2 + e*(b*d + 5*a*e))*
Log[d^(2/3) - d^(1/3)*e^(1/3)*x + e^(2/3)*x^2])/(54*d^(8/3)*e^(7/3))

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 200

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[1/(3*Rt[a, 3]^2), Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x]
/; FreeQ[{a, b}, x]

Rule 204

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

Rule 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> -Simp[((b*c - a*d)*x*(a + b*x^n)^(p +
1))/(a*b*n*(p + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /
; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/n + p, 0])

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
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 1409

Int[((d_) + (e_.)*(x_)^(n_))^(q_)*((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x_Symbol] :> -Simp[((c*d^2 - b*
d*e + a*e^2)*x*(d + e*x^n)^(q + 1))/(d*e^2*n*(q + 1)), x] + Dist[1/(n*(q + 1)*d*e^2), Int[(d + e*x^n)^(q + 1)*
Simp[c*d^2 - b*d*e + a*e^2*(n*(q + 1) + 1) + c*d*e*n*(q + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, n}, x] &
& EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[q, -1]

Rubi steps

\begin {align*} \int \frac {a+b x^3+c x^6}{\left (d+e x^3\right )^3} \, dx &=\frac {\left (c d^2-b d e+a e^2\right ) x}{6 d e^2 \left (d+e x^3\right )^2}-\frac {\int \frac {c d^2-e (b d+5 a e)-6 c d e x^3}{\left (d+e x^3\right )^2} \, dx}{6 d e^2}\\ &=\frac {\left (c d^2-b d e+a e^2\right ) x}{6 d e^2 \left (d+e x^3\right )^2}-\frac {\left (7 c d^2-e (b d+5 a e)\right ) x}{18 d^2 e^2 \left (d+e x^3\right )}+\frac {\left (2 c d^2+e (b d+5 a e)\right ) \int \frac {1}{d+e x^3} \, dx}{9 d^2 e^2}\\ &=\frac {\left (c d^2-b d e+a e^2\right ) x}{6 d e^2 \left (d+e x^3\right )^2}-\frac {\left (7 c d^2-e (b d+5 a e)\right ) x}{18 d^2 e^2 \left (d+e x^3\right )}+\frac {\left (2 c d^2+e (b d+5 a e)\right ) \int \frac {1}{\sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{27 d^{8/3} e^2}+\frac {\left (2 c d^2+e (b d+5 a e)\right ) \int \frac {2 \sqrt [3]{d}-\sqrt [3]{e} x}{d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2} \, dx}{27 d^{8/3} e^2}\\ &=\frac {\left (c d^2-b d e+a e^2\right ) x}{6 d e^2 \left (d+e x^3\right )^2}-\frac {\left (7 c d^2-e (b d+5 a e)\right ) x}{18 d^2 e^2 \left (d+e x^3\right )}+\frac {\left (2 c d^2+e (b d+5 a e)\right ) \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{27 d^{8/3} e^{7/3}}-\frac {\left (2 c d^2+e (b d+5 a e)\right ) \int \frac {-\sqrt [3]{d} \sqrt [3]{e}+2 e^{2/3} x}{d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2} \, dx}{54 d^{8/3} e^{7/3}}+\frac {\left (2 c d^2+e (b d+5 a e)\right ) \int \frac {1}{d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2} \, dx}{18 d^{7/3} e^2}\\ &=\frac {\left (c d^2-b d e+a e^2\right ) x}{6 d e^2 \left (d+e x^3\right )^2}-\frac {\left (7 c d^2-e (b d+5 a e)\right ) x}{18 d^2 e^2 \left (d+e x^3\right )}+\frac {\left (2 c d^2+e (b d+5 a e)\right ) \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{27 d^{8/3} e^{7/3}}-\frac {\left (2 c d^2+e (b d+5 a e)\right ) \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{54 d^{8/3} e^{7/3}}+\frac {\left (2 c d^2+e (b d+5 a e)\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{8/3} e^{7/3}}\\ &=\frac {\left (c d^2-b d e+a e^2\right ) x}{6 d e^2 \left (d+e x^3\right )^2}-\frac {\left (7 c d^2-e (b d+5 a e)\right ) x}{18 d^2 e^2 \left (d+e x^3\right )}-\frac {\left (2 c d^2+e (b d+5 a e)\right ) \tan ^{-1}\left (\frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt {3} \sqrt [3]{d}}\right )}{9 \sqrt {3} d^{8/3} e^{7/3}}+\frac {\left (2 c d^2+e (b d+5 a e)\right ) \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{27 d^{8/3} e^{7/3}}-\frac {\left (2 c d^2+e (b d+5 a e)\right ) \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{54 d^{8/3} e^{7/3}}\\ \end {align*}

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Mathematica [A]  time = 0.27, size = 209, normalized size = 0.86 \begin {gather*} \frac {2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \left (e (5 a e+b d)+2 c d^2\right )-2 \sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{e} x}{\sqrt [3]{d}}}{\sqrt {3}}\right ) \left (e (5 a e+b d)+2 c d^2\right )-\log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right ) \left (e (5 a e+b d)+2 c d^2\right )-\frac {3 d^{2/3} \sqrt [3]{e} x \left (c d^2 \left (4 d+7 e x^3\right )-e \left (a e \left (8 d+5 e x^3\right )+b d \left (e x^3-2 d\right )\right )\right )}{\left (d+e x^3\right )^2}}{54 d^{8/3} e^{7/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-3*d^(2/3)*e^(1/3)*x*(c*d^2*(4*d + 7*e*x^3) - e*(b*d*(-2*d + e*x^3) + a*e*(8*d + 5*e*x^3))))/(d + e*x^3)^2 -
2*Sqrt[3]*(2*c*d^2 + e*(b*d + 5*a*e))*ArcTan[(1 - (2*e^(1/3)*x)/d^(1/3))/Sqrt[3]] + 2*(2*c*d^2 + e*(b*d + 5*a
*e))*Log[d^(1/3) + e^(1/3)*x] - (2*c*d^2 + e*(b*d + 5*a*e))*Log[d^(2/3) - d^(1/3)*e^(1/3)*x + e^(2/3)*x^2])/(5
4*d^(8/3)*e^(7/3))

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 0.86, size = 941, normalized size = 3.89 \begin {gather*} \left [-\frac {3 \, {\left (7 \, c d^{4} e^{2} - b d^{3} e^{3} - 5 \, a d^{2} e^{4}\right )} x^{4} - 3 \, \sqrt {\frac {1}{3}} {\left (2 \, c d^{5} e + b d^{4} e^{2} + 5 \, a d^{3} e^{3} + {\left (2 \, c d^{3} e^{3} + b d^{2} e^{4} + 5 \, a d e^{5}\right )} x^{6} + 2 \, {\left (2 \, c d^{4} e^{2} + b d^{3} e^{3} + 5 \, a d^{2} e^{4}\right )} x^{3}\right )} \sqrt {-\frac {\left (d^{2} e\right )^{\frac {1}{3}}}{e}} \log \left (\frac {2 \, d e x^{3} - 3 \, \left (d^{2} e\right )^{\frac {1}{3}} d x - d^{2} + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, d e x^{2} + \left (d^{2} e\right )^{\frac {2}{3}} x - \left (d^{2} e\right )^{\frac {1}{3}} d\right )} \sqrt {-\frac {\left (d^{2} e\right )^{\frac {1}{3}}}{e}}}{e x^{3} + d}\right ) + {\left ({\left (2 \, c d^{2} e^{2} + b d e^{3} + 5 \, a e^{4}\right )} x^{6} + 2 \, c d^{4} + b d^{3} e + 5 \, a d^{2} e^{2} + 2 \, {\left (2 \, c d^{3} e + b d^{2} e^{2} + 5 \, a d e^{3}\right )} x^{3}\right )} \left (d^{2} e\right )^{\frac {2}{3}} \log \left (d e x^{2} - \left (d^{2} e\right )^{\frac {2}{3}} x + \left (d^{2} e\right )^{\frac {1}{3}} d\right ) - 2 \, {\left ({\left (2 \, c d^{2} e^{2} + b d e^{3} + 5 \, a e^{4}\right )} x^{6} + 2 \, c d^{4} + b d^{3} e + 5 \, a d^{2} e^{2} + 2 \, {\left (2 \, c d^{3} e + b d^{2} e^{2} + 5 \, a d e^{3}\right )} x^{3}\right )} \left (d^{2} e\right )^{\frac {2}{3}} \log \left (d e x + \left (d^{2} e\right )^{\frac {2}{3}}\right ) + 6 \, {\left (2 \, c d^{5} e + b d^{4} e^{2} - 4 \, a d^{3} e^{3}\right )} x}{54 \, {\left (d^{4} e^{5} x^{6} + 2 \, d^{5} e^{4} x^{3} + d^{6} e^{3}\right )}}, -\frac {3 \, {\left (7 \, c d^{4} e^{2} - b d^{3} e^{3} - 5 \, a d^{2} e^{4}\right )} x^{4} - 6 \, \sqrt {\frac {1}{3}} {\left (2 \, c d^{5} e + b d^{4} e^{2} + 5 \, a d^{3} e^{3} + {\left (2 \, c d^{3} e^{3} + b d^{2} e^{4} + 5 \, a d e^{5}\right )} x^{6} + 2 \, {\left (2 \, c d^{4} e^{2} + b d^{3} e^{3} + 5 \, a d^{2} e^{4}\right )} x^{3}\right )} \sqrt {\frac {\left (d^{2} e\right )^{\frac {1}{3}}}{e}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, \left (d^{2} e\right )^{\frac {2}{3}} x - \left (d^{2} e\right )^{\frac {1}{3}} d\right )} \sqrt {\frac {\left (d^{2} e\right )^{\frac {1}{3}}}{e}}}{d^{2}}\right ) + {\left ({\left (2 \, c d^{2} e^{2} + b d e^{3} + 5 \, a e^{4}\right )} x^{6} + 2 \, c d^{4} + b d^{3} e + 5 \, a d^{2} e^{2} + 2 \, {\left (2 \, c d^{3} e + b d^{2} e^{2} + 5 \, a d e^{3}\right )} x^{3}\right )} \left (d^{2} e\right )^{\frac {2}{3}} \log \left (d e x^{2} - \left (d^{2} e\right )^{\frac {2}{3}} x + \left (d^{2} e\right )^{\frac {1}{3}} d\right ) - 2 \, {\left ({\left (2 \, c d^{2} e^{2} + b d e^{3} + 5 \, a e^{4}\right )} x^{6} + 2 \, c d^{4} + b d^{3} e + 5 \, a d^{2} e^{2} + 2 \, {\left (2 \, c d^{3} e + b d^{2} e^{2} + 5 \, a d e^{3}\right )} x^{3}\right )} \left (d^{2} e\right )^{\frac {2}{3}} \log \left (d e x + \left (d^{2} e\right )^{\frac {2}{3}}\right ) + 6 \, {\left (2 \, c d^{5} e + b d^{4} e^{2} - 4 \, a d^{3} e^{3}\right )} x}{54 \, {\left (d^{4} e^{5} x^{6} + 2 \, d^{5} e^{4} x^{3} + d^{6} e^{3}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/54*(3*(7*c*d^4*e^2 - b*d^3*e^3 - 5*a*d^2*e^4)*x^4 - 3*sqrt(1/3)*(2*c*d^5*e + b*d^4*e^2 + 5*a*d^3*e^3 + (2*
c*d^3*e^3 + b*d^2*e^4 + 5*a*d*e^5)*x^6 + 2*(2*c*d^4*e^2 + b*d^3*e^3 + 5*a*d^2*e^4)*x^3)*sqrt(-(d^2*e)^(1/3)/e)
*log((2*d*e*x^3 - 3*(d^2*e)^(1/3)*d*x - d^2 + 3*sqrt(1/3)*(2*d*e*x^2 + (d^2*e)^(2/3)*x - (d^2*e)^(1/3)*d)*sqrt
(-(d^2*e)^(1/3)/e))/(e*x^3 + d)) + ((2*c*d^2*e^2 + b*d*e^3 + 5*a*e^4)*x^6 + 2*c*d^4 + b*d^3*e + 5*a*d^2*e^2 +
2*(2*c*d^3*e + b*d^2*e^2 + 5*a*d*e^3)*x^3)*(d^2*e)^(2/3)*log(d*e*x^2 - (d^2*e)^(2/3)*x + (d^2*e)^(1/3)*d) - 2*
((2*c*d^2*e^2 + b*d*e^3 + 5*a*e^4)*x^6 + 2*c*d^4 + b*d^3*e + 5*a*d^2*e^2 + 2*(2*c*d^3*e + b*d^2*e^2 + 5*a*d*e^
3)*x^3)*(d^2*e)^(2/3)*log(d*e*x + (d^2*e)^(2/3)) + 6*(2*c*d^5*e + b*d^4*e^2 - 4*a*d^3*e^3)*x)/(d^4*e^5*x^6 + 2
*d^5*e^4*x^3 + d^6*e^3), -1/54*(3*(7*c*d^4*e^2 - b*d^3*e^3 - 5*a*d^2*e^4)*x^4 - 6*sqrt(1/3)*(2*c*d^5*e + b*d^4
*e^2 + 5*a*d^3*e^3 + (2*c*d^3*e^3 + b*d^2*e^4 + 5*a*d*e^5)*x^6 + 2*(2*c*d^4*e^2 + b*d^3*e^3 + 5*a*d^2*e^4)*x^3
)*sqrt((d^2*e)^(1/3)/e)*arctan(sqrt(1/3)*(2*(d^2*e)^(2/3)*x - (d^2*e)^(1/3)*d)*sqrt((d^2*e)^(1/3)/e)/d^2) + ((
2*c*d^2*e^2 + b*d*e^3 + 5*a*e^4)*x^6 + 2*c*d^4 + b*d^3*e + 5*a*d^2*e^2 + 2*(2*c*d^3*e + b*d^2*e^2 + 5*a*d*e^3)
*x^3)*(d^2*e)^(2/3)*log(d*e*x^2 - (d^2*e)^(2/3)*x + (d^2*e)^(1/3)*d) - 2*((2*c*d^2*e^2 + b*d*e^3 + 5*a*e^4)*x^
6 + 2*c*d^4 + b*d^3*e + 5*a*d^2*e^2 + 2*(2*c*d^3*e + b*d^2*e^2 + 5*a*d*e^3)*x^3)*(d^2*e)^(2/3)*log(d*e*x + (d^
2*e)^(2/3)) + 6*(2*c*d^5*e + b*d^4*e^2 - 4*a*d^3*e^3)*x)/(d^4*e^5*x^6 + 2*d^5*e^4*x^3 + d^6*e^3)]

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giac [A]  time = 0.40, size = 224, normalized size = 0.93 \begin {gather*} -\frac {\sqrt {3} {\left (2 \, c d^{2} + b d e + 5 \, a e^{2}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-d e^{\left (-1\right )}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-d e^{\left (-1\right )}\right )^{\frac {1}{3}}}\right ) e^{\left (-1\right )}}{27 \, \left (-d e^{2}\right )^{\frac {2}{3}} d^{2}} - \frac {{\left (2 \, c d^{2} + b d e + 5 \, a e^{2}\right )} e^{\left (-1\right )} \log \left (x^{2} + \left (-d e^{\left (-1\right )}\right )^{\frac {1}{3}} x + \left (-d e^{\left (-1\right )}\right )^{\frac {2}{3}}\right )}{54 \, \left (-d e^{2}\right )^{\frac {2}{3}} d^{2}} - \frac {{\left (2 \, c d^{2} + b d e + 5 \, a e^{2}\right )} \left (-d e^{\left (-1\right )}\right )^{\frac {1}{3}} e^{\left (-2\right )} \log \left ({\left | x - \left (-d e^{\left (-1\right )}\right )^{\frac {1}{3}} \right |}\right )}{27 \, d^{3}} - \frac {{\left (7 \, c d^{2} x^{4} e - b d x^{4} e^{2} - 5 \, a x^{4} e^{3} + 4 \, c d^{3} x + 2 \, b d^{2} x e - 8 \, a d x e^{2}\right )} e^{\left (-2\right )}}{18 \, {\left (x^{3} e + d\right )}^{2} d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/27*sqrt(3)*(2*c*d^2 + b*d*e + 5*a*e^2)*arctan(1/3*sqrt(3)*(2*x + (-d*e^(-1))^(1/3))/(-d*e^(-1))^(1/3))*e^(-
1)/((-d*e^2)^(2/3)*d^2) - 1/54*(2*c*d^2 + b*d*e + 5*a*e^2)*e^(-1)*log(x^2 + (-d*e^(-1))^(1/3)*x + (-d*e^(-1))^
(2/3))/((-d*e^2)^(2/3)*d^2) - 1/27*(2*c*d^2 + b*d*e + 5*a*e^2)*(-d*e^(-1))^(1/3)*e^(-2)*log(abs(x - (-d*e^(-1)
)^(1/3)))/d^3 - 1/18*(7*c*d^2*x^4*e - b*d*x^4*e^2 - 5*a*x^4*e^3 + 4*c*d^3*x + 2*b*d^2*x*e - 8*a*d*x*e^2)*e^(-2
)/((x^3*e + d)^2*d^2)

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maple [A]  time = 0.01, size = 362, normalized size = 1.50 \begin {gather*} \frac {5 \sqrt {3}\, a \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {d}{e}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{27 \left (\frac {d}{e}\right )^{\frac {2}{3}} d^{2} e}+\frac {5 a \ln \left (x +\left (\frac {d}{e}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {d}{e}\right )^{\frac {2}{3}} d^{2} e}-\frac {5 a \ln \left (x^{2}-\left (\frac {d}{e}\right )^{\frac {1}{3}} x +\left (\frac {d}{e}\right )^{\frac {2}{3}}\right )}{54 \left (\frac {d}{e}\right )^{\frac {2}{3}} d^{2} e}+\frac {\sqrt {3}\, b \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {d}{e}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{27 \left (\frac {d}{e}\right )^{\frac {2}{3}} d \,e^{2}}+\frac {b \ln \left (x +\left (\frac {d}{e}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {d}{e}\right )^{\frac {2}{3}} d \,e^{2}}-\frac {b \ln \left (x^{2}-\left (\frac {d}{e}\right )^{\frac {1}{3}} x +\left (\frac {d}{e}\right )^{\frac {2}{3}}\right )}{54 \left (\frac {d}{e}\right )^{\frac {2}{3}} d \,e^{2}}+\frac {2 \sqrt {3}\, c \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {d}{e}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{27 \left (\frac {d}{e}\right )^{\frac {2}{3}} e^{3}}+\frac {2 c \ln \left (x +\left (\frac {d}{e}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {d}{e}\right )^{\frac {2}{3}} e^{3}}-\frac {c \ln \left (x^{2}-\left (\frac {d}{e}\right )^{\frac {1}{3}} x +\left (\frac {d}{e}\right )^{\frac {2}{3}}\right )}{27 \left (\frac {d}{e}\right )^{\frac {2}{3}} e^{3}}+\frac {\frac {\left (5 a \,e^{2}+d e b -7 c \,d^{2}\right ) x^{4}}{18 d^{2} e}+\frac {\left (4 a \,e^{2}-d e b -2 c \,d^{2}\right ) x}{9 d \,e^{2}}}{\left (e \,x^{3}+d \right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1/18*(5*a*e^2+b*d*e-7*c*d^2)/d^2/e*x^4+1/9*(4*a*e^2-b*d*e-2*c*d^2)/d/e^2*x)/(e*x^3+d)^2+5/27/e/d^2/(d/e)^(2/3
)*ln(x+(d/e)^(1/3))*a+1/27/e^2/d/(d/e)^(2/3)*ln(x+(d/e)^(1/3))*b+2/27/e^3/(d/e)^(2/3)*ln(x+(d/e)^(1/3))*c-5/54
/e/d^2/(d/e)^(2/3)*ln(x^2-(d/e)^(1/3)*x+(d/e)^(2/3))*a-1/54/e^2/d/(d/e)^(2/3)*ln(x^2-(d/e)^(1/3)*x+(d/e)^(2/3)
)*b-1/27/e^3/(d/e)^(2/3)*ln(x^2-(d/e)^(1/3)*x+(d/e)^(2/3))*c+5/27/e/d^2/(d/e)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)
*(2/(d/e)^(1/3)*x-1))*a+1/27/e^2/d/(d/e)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(d/e)^(1/3)*x-1))*b+2/27/e^3/(d/e
)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(d/e)^(1/3)*x-1))*c

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maxima [A]  time = 1.69, size = 240, normalized size = 0.99 \begin {gather*} -\frac {{\left (7 \, c d^{2} e - b d e^{2} - 5 \, a e^{3}\right )} x^{4} + 2 \, {\left (2 \, c d^{3} + b d^{2} e - 4 \, a d e^{2}\right )} x}{18 \, {\left (d^{2} e^{4} x^{6} + 2 \, d^{3} e^{3} x^{3} + d^{4} e^{2}\right )}} + \frac {\sqrt {3} {\left (2 \, c d^{2} + b d e + 5 \, a e^{2}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {d}{e}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {d}{e}\right )^{\frac {1}{3}}}\right )}{27 \, d^{2} e^{3} \left (\frac {d}{e}\right )^{\frac {2}{3}}} - \frac {{\left (2 \, c d^{2} + b d e + 5 \, a e^{2}\right )} \log \left (x^{2} - x \left (\frac {d}{e}\right )^{\frac {1}{3}} + \left (\frac {d}{e}\right )^{\frac {2}{3}}\right )}{54 \, d^{2} e^{3} \left (\frac {d}{e}\right )^{\frac {2}{3}}} + \frac {{\left (2 \, c d^{2} + b d e + 5 \, a e^{2}\right )} \log \left (x + \left (\frac {d}{e}\right )^{\frac {1}{3}}\right )}{27 \, d^{2} e^{3} \left (\frac {d}{e}\right )^{\frac {2}{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/18*((7*c*d^2*e - b*d*e^2 - 5*a*e^3)*x^4 + 2*(2*c*d^3 + b*d^2*e - 4*a*d*e^2)*x)/(d^2*e^4*x^6 + 2*d^3*e^3*x^3
+ d^4*e^2) + 1/27*sqrt(3)*(2*c*d^2 + b*d*e + 5*a*e^2)*arctan(1/3*sqrt(3)*(2*x - (d/e)^(1/3))/(d/e)^(1/3))/(d^
2*e^3*(d/e)^(2/3)) - 1/54*(2*c*d^2 + b*d*e + 5*a*e^2)*log(x^2 - x*(d/e)^(1/3) + (d/e)^(2/3))/(d^2*e^3*(d/e)^(2
/3)) + 1/27*(2*c*d^2 + b*d*e + 5*a*e^2)*log(x + (d/e)^(1/3))/(d^2*e^3*(d/e)^(2/3))

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mupad [B]  time = 0.29, size = 221, normalized size = 0.91 \begin {gather*} \frac {\ln \left (e^{1/3}\,x+d^{1/3}\right )\,\left (2\,c\,d^2+b\,d\,e+5\,a\,e^2\right )}{27\,d^{8/3}\,e^{7/3}}-\frac {\frac {x\,\left (2\,c\,d^2+b\,d\,e-4\,a\,e^2\right )}{9\,d\,e^2}-\frac {x^4\,\left (-7\,c\,d^2+b\,d\,e+5\,a\,e^2\right )}{18\,d^2\,e}}{d^2+2\,d\,e\,x^3+e^2\,x^6}+\frac {\ln \left (2\,e^{1/3}\,x-d^{1/3}+\sqrt {3}\,d^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (2\,c\,d^2+b\,d\,e+5\,a\,e^2\right )}{27\,d^{8/3}\,e^{7/3}}-\frac {\ln \left (d^{1/3}-2\,e^{1/3}\,x+\sqrt {3}\,d^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (2\,c\,d^2+b\,d\,e+5\,a\,e^2\right )}{27\,d^{8/3}\,e^{7/3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(log(e^(1/3)*x + d^(1/3))*(5*a*e^2 + 2*c*d^2 + b*d*e))/(27*d^(8/3)*e^(7/3)) - ((x*(2*c*d^2 - 4*a*e^2 + b*d*e))
/(9*d*e^2) - (x^4*(5*a*e^2 - 7*c*d^2 + b*d*e))/(18*d^2*e))/(d^2 + e^2*x^6 + 2*d*e*x^3) + (log(3^(1/2)*d^(1/3)*
1i + 2*e^(1/3)*x - d^(1/3))*((3^(1/2)*1i)/2 - 1/2)*(5*a*e^2 + 2*c*d^2 + b*d*e))/(27*d^(8/3)*e^(7/3)) - (log(3^
(1/2)*d^(1/3)*1i - 2*e^(1/3)*x + d^(1/3))*((3^(1/2)*1i)/2 + 1/2)*(5*a*e^2 + 2*c*d^2 + b*d*e))/(27*d^(8/3)*e^(7
/3))

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sympy [A]  time = 5.23, size = 246, normalized size = 1.02 \begin {gather*} \frac {x^{4} \left (5 a e^{3} + b d e^{2} - 7 c d^{2} e\right ) + x \left (8 a d e^{2} - 2 b d^{2} e - 4 c d^{3}\right )}{18 d^{4} e^{2} + 36 d^{3} e^{3} x^{3} + 18 d^{2} e^{4} x^{6}} + \operatorname {RootSum} {\left (19683 t^{3} d^{8} e^{7} - 125 a^{3} e^{6} - 75 a^{2} b d e^{5} - 150 a^{2} c d^{2} e^{4} - 15 a b^{2} d^{2} e^{4} - 60 a b c d^{3} e^{3} - 60 a c^{2} d^{4} e^{2} - b^{3} d^{3} e^{3} - 6 b^{2} c d^{4} e^{2} - 12 b c^{2} d^{5} e - 8 c^{3} d^{6}, \left (t \mapsto t \log {\left (\frac {27 t d^{3} e^{2}}{5 a e^{2} + b d e + 2 c d^{2}} + x \right )} \right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x**4*(5*a*e**3 + b*d*e**2 - 7*c*d**2*e) + x*(8*a*d*e**2 - 2*b*d**2*e - 4*c*d**3))/(18*d**4*e**2 + 36*d**3*e**
3*x**3 + 18*d**2*e**4*x**6) + RootSum(19683*_t**3*d**8*e**7 - 125*a**3*e**6 - 75*a**2*b*d*e**5 - 150*a**2*c*d*
*2*e**4 - 15*a*b**2*d**2*e**4 - 60*a*b*c*d**3*e**3 - 60*a*c**2*d**4*e**2 - b**3*d**3*e**3 - 6*b**2*c*d**4*e**2
- 12*b*c**2*d**5*e - 8*c**3*d**6, Lambda(_t, _t*log(27*_t*d**3*e**2/(5*a*e**2 + b*d*e + 2*c*d**2) + x)))

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