3.424 \(\int \frac {x (c+d x+e x^2+f x^3+g x^4+h x^5)}{(a+b x^3)^3} \, dx\)

Optimal. Leaf size=323 \[ \frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (b^{2/3} (a f+2 b c)-a^{2/3} (2 a h+b e)\right )}{54 a^{7/3} b^{7/3}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (b^{2/3} (a f+2 b c)-a^{2/3} (2 a h+b e)\right )}{27 a^{7/3} b^{7/3}}-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (a^{2/3} b e+2 a^{5/3} h+a b^{2/3} f+2 b^{5/3} c\right )}{9 \sqrt {3} a^{7/3} b^{7/3}}+\frac {x \left (2 b x (a f+2 b c)+3 b x^2 (a g+b d)+a (b e-7 a h)\right )}{18 a^2 b^2 \left (a+b x^3\right )}-\frac {x \left (-b x (b c-a f)-b x^2 (b d-a g)+a (b e-a h)\right )}{6 a b^2 \left (a+b x^3\right )^2} \]

[Out]

-1/6*x*(a*(-a*h+b*e)-b*(-a*f+b*c)*x-b*(-a*g+b*d)*x^2)/a/b^2/(b*x^3+a)^2+1/18*x*(a*(-7*a*h+b*e)+2*b*(a*f+2*b*c)
*x+3*b*(a*g+b*d)*x^2)/a^2/b^2/(b*x^3+a)-1/27*(b^(2/3)*(a*f+2*b*c)-a^(2/3)*(2*a*h+b*e))*ln(a^(1/3)+b^(1/3)*x)/a
^(7/3)/b^(7/3)+1/54*(b^(2/3)*(a*f+2*b*c)-a^(2/3)*(2*a*h+b*e))*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(7/3
)/b^(7/3)-1/27*(2*b^(5/3)*c+a^(2/3)*b*e+a*b^(2/3)*f+2*a^(5/3)*h)*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)/a^(1/3)*3^(1
/2))/a^(7/3)/b^(7/3)*3^(1/2)

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Rubi [A]  time = 0.48, antiderivative size = 323, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1828, 1858, 1860, 31, 634, 617, 204, 628} \[ \frac {x \left (2 b x (a f+2 b c)+3 b x^2 (a g+b d)+a (b e-7 a h)\right )}{18 a^2 b^2 \left (a+b x^3\right )}+\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (b^{2/3} (a f+2 b c)-a^{2/3} (2 a h+b e)\right )}{54 a^{7/3} b^{7/3}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (b^{2/3} (a f+2 b c)-a^{2/3} (2 a h+b e)\right )}{27 a^{7/3} b^{7/3}}-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (a^{2/3} b e+2 a^{5/3} h+a b^{2/3} f+2 b^{5/3} c\right )}{9 \sqrt {3} a^{7/3} b^{7/3}}-\frac {x \left (-b x (b c-a f)-b x^2 (b d-a g)+a (b e-a h)\right )}{6 a b^2 \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[(x*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5))/(a + b*x^3)^3,x]

[Out]

-(x*(a*(b*e - a*h) - b*(b*c - a*f)*x - b*(b*d - a*g)*x^2))/(6*a*b^2*(a + b*x^3)^2) + (x*(a*(b*e - 7*a*h) + 2*b
*(2*b*c + a*f)*x + 3*b*(b*d + a*g)*x^2))/(18*a^2*b^2*(a + b*x^3)) - ((2*b^(5/3)*c + a^(2/3)*b*e + a*b^(2/3)*f
+ 2*a^(5/3)*h)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(7/3)*b^(7/3)) - ((b^(2/3)*(2*b
*c + a*f) - a^(2/3)*(b*e + 2*a*h))*Log[a^(1/3) + b^(1/3)*x])/(27*a^(7/3)*b^(7/3)) + ((b^(2/3)*(2*b*c + a*f) -
a^(2/3)*(b*e + 2*a*h))*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(7/3)*b^(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 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 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 1828

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = m + Expon[Pq, x]}, Module[{Q = Pol
ynomialQuotient[b^(Floor[(q - 1)/n] + 1)*x^m*Pq, a + b*x^n, x], R = PolynomialRemainder[b^(Floor[(q - 1)/n] +
1)*x^m*Pq, a + b*x^n, x]}, Dist[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)), Int[(a + b*x^n)^(p + 1)*ExpandToSum[
a*n*(p + 1)*Q + n*(p + 1)*R + D[x*R, x], x], x], x] - Simp[(x*R*(a + b*x^n)^(p + 1))/(a*n*(p + 1)*b^(Floor[(q
- 1)/n] + 1)), x]] /; GeQ[q, n]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1] && IGtQ[m, 0]

Rule 1858

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, Module[{Q = PolynomialQuotient
[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, x], R = PolynomialRemainder[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, x
]}, Dist[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)), Int[(a + b*x^n)^(p + 1)*ExpandToSum[a*n*(p + 1)*Q + n*(p +
1)*R + D[x*R, x], x], x], x] - Simp[(x*R*(a + b*x^n)^(p + 1))/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)), x]] /; G
eQ[q, n]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1]

Rule 1860

Int[((A_) + (B_.)*(x_))/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{r = Numerator[Rt[a/b, 3]], s = Denominator[R
t[a/b, 3]]}, -Dist[(r*(B*r - A*s))/(3*a*s), Int[1/(r + s*x), x], x] + Dist[r/(3*a*s), Int[(r*(B*r + 2*A*s) + s
*(B*r - A*s)*x)/(r^2 - r*s*x + s^2*x^2), x], x]] /; FreeQ[{a, b, A, B}, x] && NeQ[a*B^3 - b*A^3, 0] && PosQ[a/
b]

Rubi steps

\begin {align*} \int \frac {x \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{\left (a+b x^3\right )^3} \, dx &=-\frac {x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{6 a b^2 \left (a+b x^3\right )^2}-\frac {\int \frac {-a (b e-a h)-2 b (2 b c+a f) x-3 b (b d+a g) x^2-6 a b h x^3}{\left (a+b x^3\right )^2} \, dx}{6 a b^2}\\ &=-\frac {x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{6 a b^2 \left (a+b x^3\right )^2}+\frac {x \left (a (b e-7 a h)+2 b (2 b c+a f) x+3 b (b d+a g) x^2\right )}{18 a^2 b^2 \left (a+b x^3\right )}+\frac {\int \frac {2 a b (b e+2 a h)+2 b^2 (2 b c+a f) x}{a+b x^3} \, dx}{18 a^2 b^3}\\ &=-\frac {x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{6 a b^2 \left (a+b x^3\right )^2}+\frac {x \left (a (b e-7 a h)+2 b (2 b c+a f) x+3 b (b d+a g) x^2\right )}{18 a^2 b^2 \left (a+b x^3\right )}+\frac {\int \frac {\sqrt [3]{a} \left (2 \sqrt [3]{a} b^2 (2 b c+a f)+4 a b^{4/3} (b e+2 a h)\right )+\sqrt [3]{b} \left (2 \sqrt [3]{a} b^2 (2 b c+a f)-2 a b^{4/3} (b e+2 a h)\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{54 a^{8/3} b^{10/3}}-\frac {\left (b^{2/3} (2 b c+a f)-a^{2/3} (b e+2 a h)\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{7/3} b^2}\\ &=-\frac {x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{6 a b^2 \left (a+b x^3\right )^2}+\frac {x \left (a (b e-7 a h)+2 b (2 b c+a f) x+3 b (b d+a g) x^2\right )}{18 a^2 b^2 \left (a+b x^3\right )}-\frac {\left (b^{2/3} (2 b c+a f)-a^{2/3} (b e+2 a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{7/3} b^{7/3}}+\frac {\left (2 b^{5/3} c+a^{2/3} b e+a b^{2/3} f+2 a^{5/3} h\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{18 a^2 b^2}+\frac {\left (b^{2/3} (2 b c+a f)-a^{2/3} (b e+2 a h)\right ) \int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{54 a^{7/3} b^{7/3}}\\ &=-\frac {x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{6 a b^2 \left (a+b x^3\right )^2}+\frac {x \left (a (b e-7 a h)+2 b (2 b c+a f) x+3 b (b d+a g) x^2\right )}{18 a^2 b^2 \left (a+b x^3\right )}-\frac {\left (b^{2/3} (2 b c+a f)-a^{2/3} (b e+2 a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{7/3} b^{7/3}}+\frac {\left (b^{2/3} (2 b c+a f)-a^{2/3} (b e+2 a h)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{7/3} b^{7/3}}+\frac {\left (2 b^{5/3} c+a^{2/3} b e+a b^{2/3} f+2 a^{5/3} h\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{9 a^{7/3} b^{7/3}}\\ &=-\frac {x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{6 a b^2 \left (a+b x^3\right )^2}+\frac {x \left (a (b e-7 a h)+2 b (2 b c+a f) x+3 b (b d+a g) x^2\right )}{18 a^2 b^2 \left (a+b x^3\right )}-\frac {\left (2 b^{5/3} c+a^{2/3} b e+a b^{2/3} f+2 a^{5/3} h\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{7/3} b^{7/3}}-\frac {\left (b^{2/3} (2 b c+a f)-a^{2/3} (b e+2 a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{7/3} b^{7/3}}+\frac {\left (b^{2/3} (2 b c+a f)-a^{2/3} (b e+2 a h)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{7/3} b^{7/3}}\\ \end {align*}

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Mathematica [A]  time = 0.36, size = 297, normalized size = 0.92 \[ \frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-a^{2/3} b e-2 a^{5/3} h+a b^{2/3} f+2 b^{5/3} c\right )+2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^{2/3} b e+2 a^{5/3} h-a b^{2/3} f-2 b^{5/3} c\right )-2 \sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right ) \left (a^{2/3} b e+2 a^{5/3} h+a b^{2/3} f+2 b^{5/3} c\right )-\frac {3 \sqrt [3]{a} \sqrt [3]{b} \left (a^2 (6 g+7 h x)-a b x (e+2 f x)-4 b^2 c x^2\right )}{a+b x^3}+\frac {9 a^{4/3} \sqrt [3]{b} \left (a^2 (g+h x)-a b (d+x (e+f x))+b^2 c x^2\right )}{\left (a+b x^3\right )^2}}{54 a^{7/3} b^{7/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5))/(a + b*x^3)^3,x]

[Out]

((-3*a^(1/3)*b^(1/3)*(-4*b^2*c*x^2 - a*b*x*(e + 2*f*x) + a^2*(6*g + 7*h*x)))/(a + b*x^3) + (9*a^(4/3)*b^(1/3)*
(b^2*c*x^2 + a^2*(g + h*x) - a*b*(d + x*(e + f*x))))/(a + b*x^3)^2 - 2*Sqrt[3]*(2*b^(5/3)*c + a^(2/3)*b*e + a*
b^(2/3)*f + 2*a^(5/3)*h)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 2*(-2*b^(5/3)*c + a^(2/3)*b*e - a*b^(2/
3)*f + 2*a^(5/3)*h)*Log[a^(1/3) + b^(1/3)*x] + (2*b^(5/3)*c - a^(2/3)*b*e + a*b^(2/3)*f - 2*a^(5/3)*h)*Log[a^(
2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(7/3)*b^(7/3))

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fricas [C]  time = 5.63, size = 7190, normalized size = 22.26 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^3+a)^3,x, algorithm="fricas")

[Out]

-1/108*(36*a^2*b*g*x^3 - 12*(2*b^3*c + a*b^2*f)*x^5 - 6*(a*b^2*e - 7*a^2*b*h)*x^4 + 18*a^2*b*d + 18*a^3*g - 6*
(7*a*b^2*c - a^2*b*f)*x^2 + 2*(a^2*b^4*x^6 + 2*a^3*b^3*x^3 + a^4*b^2)*((1/2)^(1/3)*(I*sqrt(3) + 1)*((8*b^5*c^3
+ a^2*b^3*e^3 + 12*a*b^4*c^2*f + 6*a^2*b^3*c*f^2 + a^3*b^2*f^3 + 6*a^3*b^2*e^2*h + 12*a^4*b*e*h^2 + 8*a^5*h^3
)/(a^7*b^7) - (8*b^5*c^3 + 12*a*b^4*c^2*f - 12*a^4*b*e*h^2 - 8*a^5*h^3 + (f^3 - 6*e^2*h)*a^3*b^2 - (e^3 - 6*c*
f^2)*a^2*b^3)/(a^7*b^7))^(1/3) - 2*(1/2)^(2/3)*(2*b^2*c*e + 2*a^2*f*h + (e*f + 4*c*h)*a*b)*(-I*sqrt(3) + 1)/(a
^4*b^4*((8*b^5*c^3 + a^2*b^3*e^3 + 12*a*b^4*c^2*f + 6*a^2*b^3*c*f^2 + a^3*b^2*f^3 + 6*a^3*b^2*e^2*h + 12*a^4*b
*e*h^2 + 8*a^5*h^3)/(a^7*b^7) - (8*b^5*c^3 + 12*a*b^4*c^2*f - 12*a^4*b*e*h^2 - 8*a^5*h^3 + (f^3 - 6*e^2*h)*a^3
*b^2 - (e^3 - 6*c*f^2)*a^2*b^3)/(a^7*b^7))^(1/3)))*log(8*a*b^4*c^2*e + 8*a^2*b^3*c*e*f + 2*a^3*b^2*e*f^2 + 1/4
*(2*a^5*b^6*c + a^6*b^5*f)*((1/2)^(1/3)*(I*sqrt(3) + 1)*((8*b^5*c^3 + a^2*b^3*e^3 + 12*a*b^4*c^2*f + 6*a^2*b^3
*c*f^2 + a^3*b^2*f^3 + 6*a^3*b^2*e^2*h + 12*a^4*b*e*h^2 + 8*a^5*h^3)/(a^7*b^7) - (8*b^5*c^3 + 12*a*b^4*c^2*f -
12*a^4*b*e*h^2 - 8*a^5*h^3 + (f^3 - 6*e^2*h)*a^3*b^2 - (e^3 - 6*c*f^2)*a^2*b^3)/(a^7*b^7))^(1/3) - 2*(1/2)^(2
/3)*(2*b^2*c*e + 2*a^2*f*h + (e*f + 4*c*h)*a*b)*(-I*sqrt(3) + 1)/(a^4*b^4*((8*b^5*c^3 + a^2*b^3*e^3 + 12*a*b^4
*c^2*f + 6*a^2*b^3*c*f^2 + a^3*b^2*f^3 + 6*a^3*b^2*e^2*h + 12*a^4*b*e*h^2 + 8*a^5*h^3)/(a^7*b^7) - (8*b^5*c^3
+ 12*a*b^4*c^2*f - 12*a^4*b*e*h^2 - 8*a^5*h^3 + (f^3 - 6*e^2*h)*a^3*b^2 - (e^3 - 6*c*f^2)*a^2*b^3)/(a^7*b^7))^
(1/3)))^2 - 1/2*(a^4*b^4*e^2 + 4*a^5*b^3*e*h + 4*a^6*b^2*h^2)*((1/2)^(1/3)*(I*sqrt(3) + 1)*((8*b^5*c^3 + a^2*b
^3*e^3 + 12*a*b^4*c^2*f + 6*a^2*b^3*c*f^2 + a^3*b^2*f^3 + 6*a^3*b^2*e^2*h + 12*a^4*b*e*h^2 + 8*a^5*h^3)/(a^7*b
^7) - (8*b^5*c^3 + 12*a*b^4*c^2*f - 12*a^4*b*e*h^2 - 8*a^5*h^3 + (f^3 - 6*e^2*h)*a^3*b^2 - (e^3 - 6*c*f^2)*a^2
*b^3)/(a^7*b^7))^(1/3) - 2*(1/2)^(2/3)*(2*b^2*c*e + 2*a^2*f*h + (e*f + 4*c*h)*a*b)*(-I*sqrt(3) + 1)/(a^4*b^4*(
(8*b^5*c^3 + a^2*b^3*e^3 + 12*a*b^4*c^2*f + 6*a^2*b^3*c*f^2 + a^3*b^2*f^3 + 6*a^3*b^2*e^2*h + 12*a^4*b*e*h^2 +
8*a^5*h^3)/(a^7*b^7) - (8*b^5*c^3 + 12*a*b^4*c^2*f - 12*a^4*b*e*h^2 - 8*a^5*h^3 + (f^3 - 6*e^2*h)*a^3*b^2 - (
e^3 - 6*c*f^2)*a^2*b^3)/(a^7*b^7))^(1/3))) + 4*(4*a^2*b^3*c^2 + 4*a^3*b^2*c*f + a^4*b*f^2)*h + (8*b^5*c^3 + a^
2*b^3*e^3 + 12*a*b^4*c^2*f + 6*a^2*b^3*c*f^2 + a^3*b^2*f^3 + 6*a^3*b^2*e^2*h + 12*a^4*b*e*h^2 + 8*a^5*h^3)*x)
+ 12*(a^2*b*e + 2*a^3*h)*x - ((a^2*b^4*x^6 + 2*a^3*b^3*x^3 + a^4*b^2)*((1/2)^(1/3)*(I*sqrt(3) + 1)*((8*b^5*c^3
+ a^2*b^3*e^3 + 12*a*b^4*c^2*f + 6*a^2*b^3*c*f^2 + a^3*b^2*f^3 + 6*a^3*b^2*e^2*h + 12*a^4*b*e*h^2 + 8*a^5*h^3
)/(a^7*b^7) - (8*b^5*c^3 + 12*a*b^4*c^2*f - 12*a^4*b*e*h^2 - 8*a^5*h^3 + (f^3 - 6*e^2*h)*a^3*b^2 - (e^3 - 6*c*
f^2)*a^2*b^3)/(a^7*b^7))^(1/3) - 2*(1/2)^(2/3)*(2*b^2*c*e + 2*a^2*f*h + (e*f + 4*c*h)*a*b)*(-I*sqrt(3) + 1)/(a
^4*b^4*((8*b^5*c^3 + a^2*b^3*e^3 + 12*a*b^4*c^2*f + 6*a^2*b^3*c*f^2 + a^3*b^2*f^3 + 6*a^3*b^2*e^2*h + 12*a^4*b
*e*h^2 + 8*a^5*h^3)/(a^7*b^7) - (8*b^5*c^3 + 12*a*b^4*c^2*f - 12*a^4*b*e*h^2 - 8*a^5*h^3 + (f^3 - 6*e^2*h)*a^3
*b^2 - (e^3 - 6*c*f^2)*a^2*b^3)/(a^7*b^7))^(1/3))) + 3*sqrt(1/3)*(a^2*b^4*x^6 + 2*a^3*b^3*x^3 + a^4*b^2)*sqrt(
-(((1/2)^(1/3)*(I*sqrt(3) + 1)*((8*b^5*c^3 + a^2*b^3*e^3 + 12*a*b^4*c^2*f + 6*a^2*b^3*c*f^2 + a^3*b^2*f^3 + 6*
a^3*b^2*e^2*h + 12*a^4*b*e*h^2 + 8*a^5*h^3)/(a^7*b^7) - (8*b^5*c^3 + 12*a*b^4*c^2*f - 12*a^4*b*e*h^2 - 8*a^5*h
^3 + (f^3 - 6*e^2*h)*a^3*b^2 - (e^3 - 6*c*f^2)*a^2*b^3)/(a^7*b^7))^(1/3) - 2*(1/2)^(2/3)*(2*b^2*c*e + 2*a^2*f*
h + (e*f + 4*c*h)*a*b)*(-I*sqrt(3) + 1)/(a^4*b^4*((8*b^5*c^3 + a^2*b^3*e^3 + 12*a*b^4*c^2*f + 6*a^2*b^3*c*f^2
+ a^3*b^2*f^3 + 6*a^3*b^2*e^2*h + 12*a^4*b*e*h^2 + 8*a^5*h^3)/(a^7*b^7) - (8*b^5*c^3 + 12*a*b^4*c^2*f - 12*a^4
*b*e*h^2 - 8*a^5*h^3 + (f^3 - 6*e^2*h)*a^3*b^2 - (e^3 - 6*c*f^2)*a^2*b^3)/(a^7*b^7))^(1/3)))^2*a^4*b^4 + 32*b^
2*c*e + 16*a*b*e*f + 32*(2*a*b*c + a^2*f)*h)/(a^4*b^4)))*log(-8*a*b^4*c^2*e - 8*a^2*b^3*c*e*f - 2*a^3*b^2*e*f^
2 - 1/4*(2*a^5*b^6*c + a^6*b^5*f)*((1/2)^(1/3)*(I*sqrt(3) + 1)*((8*b^5*c^3 + a^2*b^3*e^3 + 12*a*b^4*c^2*f + 6*
a^2*b^3*c*f^2 + a^3*b^2*f^3 + 6*a^3*b^2*e^2*h + 12*a^4*b*e*h^2 + 8*a^5*h^3)/(a^7*b^7) - (8*b^5*c^3 + 12*a*b^4*
c^2*f - 12*a^4*b*e*h^2 - 8*a^5*h^3 + (f^3 - 6*e^2*h)*a^3*b^2 - (e^3 - 6*c*f^2)*a^2*b^3)/(a^7*b^7))^(1/3) - 2*(
1/2)^(2/3)*(2*b^2*c*e + 2*a^2*f*h + (e*f + 4*c*h)*a*b)*(-I*sqrt(3) + 1)/(a^4*b^4*((8*b^5*c^3 + a^2*b^3*e^3 + 1
2*a*b^4*c^2*f + 6*a^2*b^3*c*f^2 + a^3*b^2*f^3 + 6*a^3*b^2*e^2*h + 12*a^4*b*e*h^2 + 8*a^5*h^3)/(a^7*b^7) - (8*b
^5*c^3 + 12*a*b^4*c^2*f - 12*a^4*b*e*h^2 - 8*a^5*h^3 + (f^3 - 6*e^2*h)*a^3*b^2 - (e^3 - 6*c*f^2)*a^2*b^3)/(a^7
*b^7))^(1/3)))^2 + 1/2*(a^4*b^4*e^2 + 4*a^5*b^3*e*h + 4*a^6*b^2*h^2)*((1/2)^(1/3)*(I*sqrt(3) + 1)*((8*b^5*c^3
+ a^2*b^3*e^3 + 12*a*b^4*c^2*f + 6*a^2*b^3*c*f^2 + a^3*b^2*f^3 + 6*a^3*b^2*e^2*h + 12*a^4*b*e*h^2 + 8*a^5*h^3)
/(a^7*b^7) - (8*b^5*c^3 + 12*a*b^4*c^2*f - 12*a^4*b*e*h^2 - 8*a^5*h^3 + (f^3 - 6*e^2*h)*a^3*b^2 - (e^3 - 6*c*f
^2)*a^2*b^3)/(a^7*b^7))^(1/3) - 2*(1/2)^(2/3)*(2*b^2*c*e + 2*a^2*f*h + (e*f + 4*c*h)*a*b)*(-I*sqrt(3) + 1)/(a^
4*b^4*((8*b^5*c^3 + a^2*b^3*e^3 + 12*a*b^4*c^2*f + 6*a^2*b^3*c*f^2 + a^3*b^2*f^3 + 6*a^3*b^2*e^2*h + 12*a^4*b*
e*h^2 + 8*a^5*h^3)/(a^7*b^7) - (8*b^5*c^3 + 12*a*b^4*c^2*f - 12*a^4*b*e*h^2 - 8*a^5*h^3 + (f^3 - 6*e^2*h)*a^3*
b^2 - (e^3 - 6*c*f^2)*a^2*b^3)/(a^7*b^7))^(1/3))) - 4*(4*a^2*b^3*c^2 + 4*a^3*b^2*c*f + a^4*b*f^2)*h + 2*(8*b^5
*c^3 + a^2*b^3*e^3 + 12*a*b^4*c^2*f + 6*a^2*b^3*c*f^2 + a^3*b^2*f^3 + 6*a^3*b^2*e^2*h + 12*a^4*b*e*h^2 + 8*a^5
*h^3)*x + 3/4*sqrt(1/3)*(2*a^4*b^4*e^2 + 8*a^5*b^3*e*h + 8*a^6*b^2*h^2 + (2*a^5*b^6*c + a^6*b^5*f)*((1/2)^(1/3
)*(I*sqrt(3) + 1)*((8*b^5*c^3 + a^2*b^3*e^3 + 12*a*b^4*c^2*f + 6*a^2*b^3*c*f^2 + a^3*b^2*f^3 + 6*a^3*b^2*e^2*h
+ 12*a^4*b*e*h^2 + 8*a^5*h^3)/(a^7*b^7) - (8*b^5*c^3 + 12*a*b^4*c^2*f - 12*a^4*b*e*h^2 - 8*a^5*h^3 + (f^3 - 6
*e^2*h)*a^3*b^2 - (e^3 - 6*c*f^2)*a^2*b^3)/(a^7*b^7))^(1/3) - 2*(1/2)^(2/3)*(2*b^2*c*e + 2*a^2*f*h + (e*f + 4*
c*h)*a*b)*(-I*sqrt(3) + 1)/(a^4*b^4*((8*b^5*c^3 + a^2*b^3*e^3 + 12*a*b^4*c^2*f + 6*a^2*b^3*c*f^2 + a^3*b^2*f^3
+ 6*a^3*b^2*e^2*h + 12*a^4*b*e*h^2 + 8*a^5*h^3)/(a^7*b^7) - (8*b^5*c^3 + 12*a*b^4*c^2*f - 12*a^4*b*e*h^2 - 8*
a^5*h^3 + (f^3 - 6*e^2*h)*a^3*b^2 - (e^3 - 6*c*f^2)*a^2*b^3)/(a^7*b^7))^(1/3))))*sqrt(-(((1/2)^(1/3)*(I*sqrt(3
) + 1)*((8*b^5*c^3 + a^2*b^3*e^3 + 12*a*b^4*c^2*f + 6*a^2*b^3*c*f^2 + a^3*b^2*f^3 + 6*a^3*b^2*e^2*h + 12*a^4*b
*e*h^2 + 8*a^5*h^3)/(a^7*b^7) - (8*b^5*c^3 + 12*a*b^4*c^2*f - 12*a^4*b*e*h^2 - 8*a^5*h^3 + (f^3 - 6*e^2*h)*a^3
*b^2 - (e^3 - 6*c*f^2)*a^2*b^3)/(a^7*b^7))^(1/3) - 2*(1/2)^(2/3)*(2*b^2*c*e + 2*a^2*f*h + (e*f + 4*c*h)*a*b)*(
-I*sqrt(3) + 1)/(a^4*b^4*((8*b^5*c^3 + a^2*b^3*e^3 + 12*a*b^4*c^2*f + 6*a^2*b^3*c*f^2 + a^3*b^2*f^3 + 6*a^3*b^
2*e^2*h + 12*a^4*b*e*h^2 + 8*a^5*h^3)/(a^7*b^7) - (8*b^5*c^3 + 12*a*b^4*c^2*f - 12*a^4*b*e*h^2 - 8*a^5*h^3 + (
f^3 - 6*e^2*h)*a^3*b^2 - (e^3 - 6*c*f^2)*a^2*b^3)/(a^7*b^7))^(1/3)))^2*a^4*b^4 + 32*b^2*c*e + 16*a*b*e*f + 32*
(2*a*b*c + a^2*f)*h)/(a^4*b^4))) - ((a^2*b^4*x^6 + 2*a^3*b^3*x^3 + a^4*b^2)*((1/2)^(1/3)*(I*sqrt(3) + 1)*((8*b
^5*c^3 + a^2*b^3*e^3 + 12*a*b^4*c^2*f + 6*a^2*b^3*c*f^2 + a^3*b^2*f^3 + 6*a^3*b^2*e^2*h + 12*a^4*b*e*h^2 + 8*a
^5*h^3)/(a^7*b^7) - (8*b^5*c^3 + 12*a*b^4*c^2*f - 12*a^4*b*e*h^2 - 8*a^5*h^3 + (f^3 - 6*e^2*h)*a^3*b^2 - (e^3
- 6*c*f^2)*a^2*b^3)/(a^7*b^7))^(1/3) - 2*(1/2)^(2/3)*(2*b^2*c*e + 2*a^2*f*h + (e*f + 4*c*h)*a*b)*(-I*sqrt(3) +
1)/(a^4*b^4*((8*b^5*c^3 + a^2*b^3*e^3 + 12*a*b^4*c^2*f + 6*a^2*b^3*c*f^2 + a^3*b^2*f^3 + 6*a^3*b^2*e^2*h + 12
*a^4*b*e*h^2 + 8*a^5*h^3)/(a^7*b^7) - (8*b^5*c^3 + 12*a*b^4*c^2*f - 12*a^4*b*e*h^2 - 8*a^5*h^3 + (f^3 - 6*e^2*
h)*a^3*b^2 - (e^3 - 6*c*f^2)*a^2*b^3)/(a^7*b^7))^(1/3))) - 3*sqrt(1/3)*(a^2*b^4*x^6 + 2*a^3*b^3*x^3 + a^4*b^2)
*sqrt(-(((1/2)^(1/3)*(I*sqrt(3) + 1)*((8*b^5*c^3 + a^2*b^3*e^3 + 12*a*b^4*c^2*f + 6*a^2*b^3*c*f^2 + a^3*b^2*f^
3 + 6*a^3*b^2*e^2*h + 12*a^4*b*e*h^2 + 8*a^5*h^3)/(a^7*b^7) - (8*b^5*c^3 + 12*a*b^4*c^2*f - 12*a^4*b*e*h^2 - 8
*a^5*h^3 + (f^3 - 6*e^2*h)*a^3*b^2 - (e^3 - 6*c*f^2)*a^2*b^3)/(a^7*b^7))^(1/3) - 2*(1/2)^(2/3)*(2*b^2*c*e + 2*
a^2*f*h + (e*f + 4*c*h)*a*b)*(-I*sqrt(3) + 1)/(a^4*b^4*((8*b^5*c^3 + a^2*b^3*e^3 + 12*a*b^4*c^2*f + 6*a^2*b^3*
c*f^2 + a^3*b^2*f^3 + 6*a^3*b^2*e^2*h + 12*a^4*b*e*h^2 + 8*a^5*h^3)/(a^7*b^7) - (8*b^5*c^3 + 12*a*b^4*c^2*f -
12*a^4*b*e*h^2 - 8*a^5*h^3 + (f^3 - 6*e^2*h)*a^3*b^2 - (e^3 - 6*c*f^2)*a^2*b^3)/(a^7*b^7))^(1/3)))^2*a^4*b^4 +
32*b^2*c*e + 16*a*b*e*f + 32*(2*a*b*c + a^2*f)*h)/(a^4*b^4)))*log(-8*a*b^4*c^2*e - 8*a^2*b^3*c*e*f - 2*a^3*b^
2*e*f^2 - 1/4*(2*a^5*b^6*c + a^6*b^5*f)*((1/2)^(1/3)*(I*sqrt(3) + 1)*((8*b^5*c^3 + a^2*b^3*e^3 + 12*a*b^4*c^2*
f + 6*a^2*b^3*c*f^2 + a^3*b^2*f^3 + 6*a^3*b^2*e^2*h + 12*a^4*b*e*h^2 + 8*a^5*h^3)/(a^7*b^7) - (8*b^5*c^3 + 12*
a*b^4*c^2*f - 12*a^4*b*e*h^2 - 8*a^5*h^3 + (f^3 - 6*e^2*h)*a^3*b^2 - (e^3 - 6*c*f^2)*a^2*b^3)/(a^7*b^7))^(1/3)
- 2*(1/2)^(2/3)*(2*b^2*c*e + 2*a^2*f*h + (e*f + 4*c*h)*a*b)*(-I*sqrt(3) + 1)/(a^4*b^4*((8*b^5*c^3 + a^2*b^3*e
^3 + 12*a*b^4*c^2*f + 6*a^2*b^3*c*f^2 + a^3*b^2*f^3 + 6*a^3*b^2*e^2*h + 12*a^4*b*e*h^2 + 8*a^5*h^3)/(a^7*b^7)
- (8*b^5*c^3 + 12*a*b^4*c^2*f - 12*a^4*b*e*h^2 - 8*a^5*h^3 + (f^3 - 6*e^2*h)*a^3*b^2 - (e^3 - 6*c*f^2)*a^2*b^3
)/(a^7*b^7))^(1/3)))^2 + 1/2*(a^4*b^4*e^2 + 4*a^5*b^3*e*h + 4*a^6*b^2*h^2)*((1/2)^(1/3)*(I*sqrt(3) + 1)*((8*b^
5*c^3 + a^2*b^3*e^3 + 12*a*b^4*c^2*f + 6*a^2*b^3*c*f^2 + a^3*b^2*f^3 + 6*a^3*b^2*e^2*h + 12*a^4*b*e*h^2 + 8*a^
5*h^3)/(a^7*b^7) - (8*b^5*c^3 + 12*a*b^4*c^2*f - 12*a^4*b*e*h^2 - 8*a^5*h^3 + (f^3 - 6*e^2*h)*a^3*b^2 - (e^3 -
6*c*f^2)*a^2*b^3)/(a^7*b^7))^(1/3) - 2*(1/2)^(2/3)*(2*b^2*c*e + 2*a^2*f*h + (e*f + 4*c*h)*a*b)*(-I*sqrt(3) +
1)/(a^4*b^4*((8*b^5*c^3 + a^2*b^3*e^3 + 12*a*b^4*c^2*f + 6*a^2*b^3*c*f^2 + a^3*b^2*f^3 + 6*a^3*b^2*e^2*h + 12*
a^4*b*e*h^2 + 8*a^5*h^3)/(a^7*b^7) - (8*b^5*c^3 + 12*a*b^4*c^2*f - 12*a^4*b*e*h^2 - 8*a^5*h^3 + (f^3 - 6*e^2*h
)*a^3*b^2 - (e^3 - 6*c*f^2)*a^2*b^3)/(a^7*b^7))^(1/3))) - 4*(4*a^2*b^3*c^2 + 4*a^3*b^2*c*f + a^4*b*f^2)*h + 2*
(8*b^5*c^3 + a^2*b^3*e^3 + 12*a*b^4*c^2*f + 6*a^2*b^3*c*f^2 + a^3*b^2*f^3 + 6*a^3*b^2*e^2*h + 12*a^4*b*e*h^2 +
8*a^5*h^3)*x - 3/4*sqrt(1/3)*(2*a^4*b^4*e^2 + 8*a^5*b^3*e*h + 8*a^6*b^2*h^2 + (2*a^5*b^6*c + a^6*b^5*f)*((1/2
)^(1/3)*(I*sqrt(3) + 1)*((8*b^5*c^3 + a^2*b^3*e^3 + 12*a*b^4*c^2*f + 6*a^2*b^3*c*f^2 + a^3*b^2*f^3 + 6*a^3*b^2
*e^2*h + 12*a^4*b*e*h^2 + 8*a^5*h^3)/(a^7*b^7) - (8*b^5*c^3 + 12*a*b^4*c^2*f - 12*a^4*b*e*h^2 - 8*a^5*h^3 + (f
^3 - 6*e^2*h)*a^3*b^2 - (e^3 - 6*c*f^2)*a^2*b^3)/(a^7*b^7))^(1/3) - 2*(1/2)^(2/3)*(2*b^2*c*e + 2*a^2*f*h + (e*
f + 4*c*h)*a*b)*(-I*sqrt(3) + 1)/(a^4*b^4*((8*b^5*c^3 + a^2*b^3*e^3 + 12*a*b^4*c^2*f + 6*a^2*b^3*c*f^2 + a^3*b
^2*f^3 + 6*a^3*b^2*e^2*h + 12*a^4*b*e*h^2 + 8*a^5*h^3)/(a^7*b^7) - (8*b^5*c^3 + 12*a*b^4*c^2*f - 12*a^4*b*e*h^
2 - 8*a^5*h^3 + (f^3 - 6*e^2*h)*a^3*b^2 - (e^3 - 6*c*f^2)*a^2*b^3)/(a^7*b^7))^(1/3))))*sqrt(-(((1/2)^(1/3)*(I*
sqrt(3) + 1)*((8*b^5*c^3 + a^2*b^3*e^3 + 12*a*b^4*c^2*f + 6*a^2*b^3*c*f^2 + a^3*b^2*f^3 + 6*a^3*b^2*e^2*h + 12
*a^4*b*e*h^2 + 8*a^5*h^3)/(a^7*b^7) - (8*b^5*c^3 + 12*a*b^4*c^2*f - 12*a^4*b*e*h^2 - 8*a^5*h^3 + (f^3 - 6*e^2*
h)*a^3*b^2 - (e^3 - 6*c*f^2)*a^2*b^3)/(a^7*b^7))^(1/3) - 2*(1/2)^(2/3)*(2*b^2*c*e + 2*a^2*f*h + (e*f + 4*c*h)*
a*b)*(-I*sqrt(3) + 1)/(a^4*b^4*((8*b^5*c^3 + a^2*b^3*e^3 + 12*a*b^4*c^2*f + 6*a^2*b^3*c*f^2 + a^3*b^2*f^3 + 6*
a^3*b^2*e^2*h + 12*a^4*b*e*h^2 + 8*a^5*h^3)/(a^7*b^7) - (8*b^5*c^3 + 12*a*b^4*c^2*f - 12*a^4*b*e*h^2 - 8*a^5*h
^3 + (f^3 - 6*e^2*h)*a^3*b^2 - (e^3 - 6*c*f^2)*a^2*b^3)/(a^7*b^7))^(1/3)))^2*a^4*b^4 + 32*b^2*c*e + 16*a*b*e*f
+ 32*(2*a*b*c + a^2*f)*h)/(a^4*b^4))))/(a^2*b^4*x^6 + 2*a^3*b^3*x^3 + a^4*b^2)

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giac [A]  time = 0.21, size = 340, normalized size = 1.05 \[ -\frac {\sqrt {3} {\left (2 \, a^{2} h + a b e - 2 \, \left (-a b^{2}\right )^{\frac {1}{3}} b c - \left (-a b^{2}\right )^{\frac {1}{3}} a f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{2} b} - \frac {{\left (2 \, a^{2} h + a b e + 2 \, \left (-a b^{2}\right )^{\frac {1}{3}} b c + \left (-a b^{2}\right )^{\frac {1}{3}} a f\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{2} b} - \frac {{\left (2 \, b^{2} c \left (-\frac {a}{b}\right )^{\frac {1}{3}} + a b f \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 2 \, a^{2} h + a b e\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a^{3} b^{2}} + \frac {4 \, b^{3} c x^{5} + 2 \, a b^{2} f x^{5} - 7 \, a^{2} b h x^{4} + a b^{2} x^{4} e - 6 \, a^{2} b g x^{3} + 7 \, a b^{2} c x^{2} - a^{2} b f x^{2} - 4 \, a^{3} h x - 2 \, a^{2} b x e - 3 \, a^{2} b d - 3 \, a^{3} g}{18 \, {\left (b x^{3} + a\right )}^{2} a^{2} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^3+a)^3,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.06, size = 498, normalized size = 1.54 \[ \frac {\sqrt {3}\, e \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{27 \left (\frac {a}{b}\right )^{\frac {2}{3}} a \,b^{2}}+\frac {e \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {2}{3}} a \,b^{2}}-\frac {e \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \left (\frac {a}{b}\right )^{\frac {2}{3}} a \,b^{2}}+\frac {\sqrt {3}\, f \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} a \,b^{2}}-\frac {f \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} a \,b^{2}}+\frac {f \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \left (\frac {a}{b}\right )^{\frac {1}{3}} a \,b^{2}}+\frac {2 \sqrt {3}\, c \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{2} b}-\frac {2 c \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{2} b}+\frac {c \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{2} b}+\frac {2 \sqrt {3}\, h \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{27 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{3}}+\frac {2 h \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{3}}-\frac {h \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{3}}+\frac {\frac {\left (a f +2 b c \right ) x^{5}}{9 a^{2}}-\frac {g \,x^{3}}{3 b}-\frac {\left (7 a h -b e \right ) x^{4}}{18 a b}-\frac {\left (a f -7 b c \right ) x^{2}}{18 a b}-\frac {\left (2 a h +b e \right ) x}{9 b^{2}}-\frac {a g +b d}{6 b^{2}}}{\left (b \,x^{3}+a \right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^3+a)^3,x)

[Out]

(1/9*(a*f+2*b*c)/a^2*x^5-1/18*(7*a*h-b*e)/a/b*x^4-1/3/b*g*x^3-1/18*(a*f-7*b*c)/a/b*x^2-1/9*(2*a*h+b*e)/b^2*x-1
/6*(a*g+b*d)/b^2)/(b*x^3+a)^2+2/27/b^3/(a/b)^(2/3)*ln(x+(a/b)^(1/3))*h+1/27/(a/b)^(2/3)/a/b^2*e*ln(x+(a/b)^(1/
3))-1/27/b^3/(a/b)^(2/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))*h-1/54/(a/b)^(2/3)/a/b^2*e*ln(x^2-(a/b)^(1/3)*x+(a/
b)^(2/3))+2/27/b^3/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*h+1/27/(a/b)^(2/3)*3^(1/2)/a/b^
2*e*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-1/27/(a/b)^(1/3)/a/b^2*f*ln(x+(a/b)^(1/3))-2/27/(a/b)^(1/3)/a^2/b*
c*ln(x+(a/b)^(1/3))+1/54/b^2/a/(a/b)^(1/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))*f+1/27/(a/b)^(1/3)/a^2/b*c*ln(x^2
-(a/b)^(1/3)*x+(a/b)^(2/3))+1/27/b^2/a*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*f+2/27*3^(1
/2)/(a/b)^(1/3)/a^2/b*c*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))

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maxima [A]  time = 3.07, size = 344, normalized size = 1.07 \[ -\frac {6 \, a^{2} b g x^{3} - 2 \, {\left (2 \, b^{3} c + a b^{2} f\right )} x^{5} - {\left (a b^{2} e - 7 \, a^{2} b h\right )} x^{4} + 3 \, a^{2} b d + 3 \, a^{3} g - {\left (7 \, a b^{2} c - a^{2} b f\right )} x^{2} + 2 \, {\left (a^{2} b e + 2 \, a^{3} h\right )} x}{18 \, {\left (a^{2} b^{4} x^{6} + 2 \, a^{3} b^{3} x^{3} + a^{4} b^{2}\right )}} + \frac {\sqrt {3} {\left (2 \, b^{2} c \left (\frac {a}{b}\right )^{\frac {1}{3}} + a b f \left (\frac {a}{b}\right )^{\frac {1}{3}} + a b e + 2 \, a^{2} h\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{2} b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (2 \, b^{2} c \left (\frac {a}{b}\right )^{\frac {1}{3}} + a b f \left (\frac {a}{b}\right )^{\frac {1}{3}} - a b e - 2 \, a^{2} h\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a^{2} b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (2 \, b^{2} c \left (\frac {a}{b}\right )^{\frac {1}{3}} + a b f \left (\frac {a}{b}\right )^{\frac {1}{3}} - a b e - 2 \, a^{2} h\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, a^{2} b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^3+a)^3,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 5.36, size = 640, normalized size = 1.98 \[ \left (\sum _{k=1}^3\ln \left (\mathrm {root}\left (19683\,a^7\,b^7\,z^3+162\,a^5\,b^3\,f\,h\,z+324\,a^4\,b^4\,c\,h\,z+81\,a^4\,b^4\,e\,f\,z+162\,a^3\,b^5\,c\,e\,z-12\,a^4\,b\,e\,h^2+12\,a\,b^4\,c^2\,f-6\,a^3\,b^2\,e^2\,h+6\,a^2\,b^3\,c\,f^2+a^3\,b^2\,f^3-8\,a^5\,h^3+8\,b^5\,c^3-a^2\,b^3\,e^3,z,k\right )\,\left (\mathrm {root}\left (19683\,a^7\,b^7\,z^3+162\,a^5\,b^3\,f\,h\,z+324\,a^4\,b^4\,c\,h\,z+81\,a^4\,b^4\,e\,f\,z+162\,a^3\,b^5\,c\,e\,z-12\,a^4\,b\,e\,h^2+12\,a\,b^4\,c^2\,f-6\,a^3\,b^2\,e^2\,h+6\,a^2\,b^3\,c\,f^2+a^3\,b^2\,f^3-8\,a^5\,h^3+8\,b^5\,c^3-a^2\,b^3\,e^3,z,k\right )\,a\,b^2\,9+\frac {x\,\left (54\,h\,a^4\,b+27\,e\,a^3\,b^2\right )}{81\,a^4\,b}\right )+\frac {2\,b^2\,c\,e+2\,a^2\,f\,h+4\,a\,b\,c\,h+a\,b\,e\,f}{81\,a^3\,b^2}+\frac {x\,\left (a^2\,f^2+4\,a\,b\,c\,f+4\,b^2\,c^2\right )}{81\,a^4\,b}\right )\,\mathrm {root}\left (19683\,a^7\,b^7\,z^3+162\,a^5\,b^3\,f\,h\,z+324\,a^4\,b^4\,c\,h\,z+81\,a^4\,b^4\,e\,f\,z+162\,a^3\,b^5\,c\,e\,z-12\,a^4\,b\,e\,h^2+12\,a\,b^4\,c^2\,f-6\,a^3\,b^2\,e^2\,h+6\,a^2\,b^3\,c\,f^2+a^3\,b^2\,f^3-8\,a^5\,h^3+8\,b^5\,c^3-a^2\,b^3\,e^3,z,k\right )\right )-\frac {\frac {b\,d+a\,g}{6\,b^2}+\frac {x\,\left (b\,e+2\,a\,h\right )}{9\,b^2}+\frac {g\,x^3}{3\,b}-\frac {x^5\,\left (2\,b\,c+a\,f\right )}{9\,a^2}-\frac {x^2\,\left (7\,b\,c-a\,f\right )}{18\,a\,b}-\frac {x^4\,\left (b\,e-7\,a\,h\right )}{18\,a\,b}}{a^2+2\,a\,b\,x^3+b^2\,x^6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5))/(a + b*x^3)^3,x)

[Out]

symsum(log(root(19683*a^7*b^7*z^3 + 162*a^5*b^3*f*h*z + 324*a^4*b^4*c*h*z + 81*a^4*b^4*e*f*z + 162*a^3*b^5*c*e
*z - 12*a^4*b*e*h^2 + 12*a*b^4*c^2*f - 6*a^3*b^2*e^2*h + 6*a^2*b^3*c*f^2 + a^3*b^2*f^3 - 8*a^5*h^3 + 8*b^5*c^3
- a^2*b^3*e^3, z, k)*(9*root(19683*a^7*b^7*z^3 + 162*a^5*b^3*f*h*z + 324*a^4*b^4*c*h*z + 81*a^4*b^4*e*f*z + 1
62*a^3*b^5*c*e*z - 12*a^4*b*e*h^2 + 12*a*b^4*c^2*f - 6*a^3*b^2*e^2*h + 6*a^2*b^3*c*f^2 + a^3*b^2*f^3 - 8*a^5*h
^3 + 8*b^5*c^3 - a^2*b^3*e^3, z, k)*a*b^2 + (x*(27*a^3*b^2*e + 54*a^4*b*h))/(81*a^4*b)) + (2*b^2*c*e + 2*a^2*f
*h + 4*a*b*c*h + a*b*e*f)/(81*a^3*b^2) + (x*(4*b^2*c^2 + a^2*f^2 + 4*a*b*c*f))/(81*a^4*b))*root(19683*a^7*b^7*
z^3 + 162*a^5*b^3*f*h*z + 324*a^4*b^4*c*h*z + 81*a^4*b^4*e*f*z + 162*a^3*b^5*c*e*z - 12*a^4*b*e*h^2 + 12*a*b^4
*c^2*f - 6*a^3*b^2*e^2*h + 6*a^2*b^3*c*f^2 + a^3*b^2*f^3 - 8*a^5*h^3 + 8*b^5*c^3 - a^2*b^3*e^3, z, k), k, 1, 3
) - ((b*d + a*g)/(6*b^2) + (x*(b*e + 2*a*h))/(9*b^2) + (g*x^3)/(3*b) - (x^5*(2*b*c + a*f))/(9*a^2) - (x^2*(7*b
*c - a*f))/(18*a*b) - (x^4*(b*e - 7*a*h))/(18*a*b))/(a^2 + b^2*x^6 + 2*a*b*x^3)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(h*x**5+g*x**4+f*x**3+e*x**2+d*x+c)/(b*x**3+a)**3,x)

[Out]

Timed out

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