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

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

[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))

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Rubi [A]  time = 0.262446, antiderivative size = 242, normalized size of antiderivative = 1., 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} \[ -\frac{x \left (7 c d^2-e (5 a e+b d)\right )}{18 d^2 e^2 \left (d+e x^3\right )}+\frac{x \left (a e^2-b d e+c d^2\right )}{6 d e^2 \left (d+e x^3\right )^2}-\frac{\log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right ) \left (e (5 a e+b d)+2 c d^2\right )}{54 d^{8/3} e^{7/3}}+\frac{\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \left (e (5 a e+b d)+2 c d^2\right )}{27 d^{8/3} e^{7/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt{3} \sqrt [3]{d}}\right ) \left (e (5 a e+b d)+2 c d^2\right )}{9 \sqrt{3} d^{8/3} e^{7/3}} \]

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 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]

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 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 31

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

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 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 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 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{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*}

Mathematica [A]  time = 0.271054, size = 209, normalized size = 0.86 \[ \frac{-\log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right ) \left (e (5 a e+b d)+2 c d^2\right )-\frac{3 d^{2/3} \sqrt [3]{e} x \left (c d^2 \left (4 d+7 e x^3\right )-e \left (a e \left (8 d+5 e x^3\right )+b d \left (e x^3-2 d\right )\right )\right )}{\left (d+e x^3\right )^2}+2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \left (e (5 a e+b d)+2 c d^2\right )-2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{e} x}{\sqrt [3]{d}}}{\sqrt{3}}\right ) \left (e (5 a e+b d)+2 c d^2\right )}{54 d^{8/3} e^{7/3}} \]

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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Maple [A]  time = 0.01, size = 362, normalized size = 1.5 \begin{align*}{\frac{1}{ \left ( e{x}^{3}+d \right ) ^{2}} \left ({\frac{ \left ( 5\,a{e}^{2}+bde-7\,c{d}^{2} \right ){x}^{4}}{18\,{d}^{2}e}}+{\frac{ \left ( 4\,a{e}^{2}-bde-2\,c{d}^{2} \right ) x}{9\,d{e}^{2}}} \right ) }+{\frac{5\,a}{27\,{d}^{2}e}\ln \left ( x+\sqrt [3]{{\frac{d}{e}}} \right ) \left ({\frac{d}{e}} \right ) ^{-{\frac{2}{3}}}}+{\frac{b}{27\,d{e}^{2}}\ln \left ( x+\sqrt [3]{{\frac{d}{e}}} \right ) \left ({\frac{d}{e}} \right ) ^{-{\frac{2}{3}}}}+{\frac{2\,c}{27\,{e}^{3}}\ln \left ( x+\sqrt [3]{{\frac{d}{e}}} \right ) \left ({\frac{d}{e}} \right ) ^{-{\frac{2}{3}}}}-{\frac{5\,a}{54\,{d}^{2}e}\ln \left ({x}^{2}-\sqrt [3]{{\frac{d}{e}}}x+ \left ({\frac{d}{e}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{d}{e}} \right ) ^{-{\frac{2}{3}}}}-{\frac{b}{54\,d{e}^{2}}\ln \left ({x}^{2}-\sqrt [3]{{\frac{d}{e}}}x+ \left ({\frac{d}{e}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{d}{e}} \right ) ^{-{\frac{2}{3}}}}-{\frac{c}{27\,{e}^{3}}\ln \left ({x}^{2}-\sqrt [3]{{\frac{d}{e}}}x+ \left ({\frac{d}{e}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{d}{e}} \right ) ^{-{\frac{2}{3}}}}+{\frac{5\,\sqrt{3}a}{27\,{d}^{2}e}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{d}{e}}}}}}-1 \right ) } \right ) \left ({\frac{d}{e}} \right ) ^{-{\frac{2}{3}}}}+{\frac{\sqrt{3}b}{27\,d{e}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{d}{e}}}}}}-1 \right ) } \right ) \left ({\frac{d}{e}} \right ) ^{-{\frac{2}{3}}}}+{\frac{2\,\sqrt{3}c}{27\,{e}^{3}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{d}{e}}}}}}-1 \right ) } \right ) \left ({\frac{d}{e}} \right ) ^{-{\frac{2}{3}}}} \end{align*}

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.39834, size = 2067, normalized size = 8.54 \begin{align*} \text{result too large to display} \end{align*}

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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Sympy [A]  time = 7.55052, size = 246, normalized size = 1.02 \begin{align*} \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{align*}

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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Giac [A]  time = 1.14512, size = 340, normalized size = 1.4 \begin{align*} \frac{\sqrt{3}{\left (2 \, \left (-d e^{2}\right )^{\frac{1}{3}} c d^{2} + \left (-d e^{2}\right )^{\frac{1}{3}} b d e + 5 \, \left (-d e^{2}\right )^{\frac{1}{3}} 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 (-3\right )}}{27 \, d^{3}} - \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 (2 \, \left (-d e^{2}\right )^{\frac{1}{3}} c d^{2} + \left (-d e^{2}\right )^{\frac{1}{3}} b d e + 5 \, \left (-d e^{2}\right )^{\frac{1}{3}} a e^{2}\right )} e^{\left (-3\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 \, 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{align*}

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