3.5.22 \(\int \frac {x^3 (c+d x+e x^2+f x^3+g x^4+h x^5)}{(a+b x^3)^3} \, dx\) [422]

Optimal. Leaf size=325 \[ -\frac {x \left (b c-a f+(b d-a g) x+(b e-a h) x^2\right )}{6 b^2 \left (a+b x^3\right )^2}+\frac {x \left (b c-7 a f+2 (b d-4 a g) x+3 (b e-3 a h) x^2\right )}{18 a b^2 \left (a+b x^3\right )}-\frac {\left (b^{4/3} c+\sqrt [3]{a} b d+2 a \sqrt [3]{b} f+5 a^{4/3} g\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{5/3} b^{8/3}}+\frac {\left (\sqrt [3]{b} (b c+2 a f)-\sqrt [3]{a} (b d+5 a g)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{5/3} b^{8/3}}-\frac {\left (\sqrt [3]{b} (b c+2 a f)-\sqrt [3]{a} (b d+5 a g)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{5/3} b^{8/3}}+\frac {h \log \left (a+b x^3\right )}{3 b^3} \]

[Out]

-1/6*x*(b*c-a*f+(-a*g+b*d)*x+(-a*h+b*e)*x^2)/b^2/(b*x^3+a)^2+1/18*x*(b*c-7*a*f+2*(-4*a*g+b*d)*x+3*(-3*a*h+b*e)
*x^2)/a/b^2/(b*x^3+a)+1/27*(b^(1/3)*(2*a*f+b*c)-a^(1/3)*(5*a*g+b*d))*ln(a^(1/3)+b^(1/3)*x)/a^(5/3)/b^(8/3)-1/5
4*(b^(1/3)*(2*a*f+b*c)-a^(1/3)*(5*a*g+b*d))*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(5/3)/b^(8/3)+1/3*h*ln
(b*x^3+a)/b^3-1/27*(b^(4/3)*c+a^(1/3)*b*d+2*a*b^(1/3)*f+5*a^(4/3)*g)*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)/a^(1/3)*
3^(1/2))/a^(5/3)/b^(8/3)*3^(1/2)

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Rubi [A]
time = 0.42, antiderivative size = 325, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {1842, 1872, 1885, 1874, 31, 648, 631, 210, 642, 266} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (5 a^{4/3} g+\sqrt [3]{a} b d+2 a \sqrt [3]{b} f+b^{4/3} c\right )}{9 \sqrt {3} a^{5/3} b^{8/3}}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (\sqrt [3]{b} (2 a f+b c)-\sqrt [3]{a} (5 a g+b d)\right )}{54 a^{5/3} b^{8/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (2 a f+b c)-\sqrt [3]{a} (5 a g+b d)\right )}{27 a^{5/3} b^{8/3}}+\frac {h \log \left (a+b x^3\right )}{3 b^3}+\frac {x \left (2 x (b d-4 a g)+3 x^2 (b e-3 a h)-7 a f+b c\right )}{18 a b^2 \left (a+b x^3\right )}-\frac {x \left (x (b d-a g)+x^2 (b e-a h)-a f+b c\right )}{6 b^2 \left (a+b x^3\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/6*(x*(b*c - a*f + (b*d - a*g)*x + (b*e - a*h)*x^2))/(b^2*(a + b*x^3)^2) + (x*(b*c - 7*a*f + 2*(b*d - 4*a*g)
*x + 3*(b*e - 3*a*h)*x^2))/(18*a*b^2*(a + b*x^3)) - ((b^(4/3)*c + a^(1/3)*b*d + 2*a*b^(1/3)*f + 5*a^(4/3)*g)*A
rcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(5/3)*b^(8/3)) + ((b^(1/3)*(b*c + 2*a*f) - a^(1
/3)*(b*d + 5*a*g))*Log[a^(1/3) + b^(1/3)*x])/(27*a^(5/3)*b^(8/3)) - ((b^(1/3)*(b*c + 2*a*f) - a^(1/3)*(b*d + 5
*a*g))*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(5/3)*b^(8/3)) + (h*Log[a + b*x^3])/(3*b^3)

Rule 31

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

Rule 210

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

Rule 266

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

Rule 631

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 642

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 648

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 1842

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 1872

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]] /
; GeQ[q, n]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1]

Rule 1874

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]

Rule 1885

Int[(P2_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{A = Coeff[P2, x, 0], B = Coeff[P2, x, 1], C = Coeff[P2, x,
2]}, Int[(A + B*x)/(a + b*x^3), x] + Dist[C, Int[x^2/(a + b*x^3), x], x] /; EqQ[a*B^3 - b*A^3, 0] ||  !Ration
alQ[a/b]] /; FreeQ[{a, b}, x] && PolyQ[P2, x, 2]

Rubi steps

\begin {align*} \int \frac {x^3 \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 (b c-a f+(b d-a g) x+(b e-a h) x^2\right )}{6 b^2 \left (a+b x^3\right )^2}-\frac {\int \frac {-a b (b c-a f)-2 a b (b d-a g) x-3 a b (b e-a h) x^2-6 a b^2 f x^3-6 a b^2 g x^4-6 a b^2 h x^5}{\left (a+b x^3\right )^2} \, dx}{6 a b^3}\\ &=-\frac {x \left (b c-a f+(b d-a g) x+(b e-a h) x^2\right )}{6 b^2 \left (a+b x^3\right )^2}+\frac {x \left (b c-7 a f+2 (b d-4 a g) x+3 (b e-3 a h) x^2\right )}{18 a b^2 \left (a+b x^3\right )}+\frac {\int \frac {2 a b^3 (b c+2 a f)+2 a b^3 (b d+5 a g) x+18 a^2 b^3 h x^2}{a+b x^3} \, dx}{18 a^2 b^5}\\ &=-\frac {x \left (b c-a f+(b d-a g) x+(b e-a h) x^2\right )}{6 b^2 \left (a+b x^3\right )^2}+\frac {x \left (b c-7 a f+2 (b d-4 a g) x+3 (b e-3 a h) x^2\right )}{18 a b^2 \left (a+b x^3\right )}+\frac {\int \frac {2 a b^3 (b c+2 a f)+2 a b^3 (b d+5 a g) x}{a+b x^3} \, dx}{18 a^2 b^5}+\frac {h \int \frac {x^2}{a+b x^3} \, dx}{b^2}\\ &=-\frac {x \left (b c-a f+(b d-a g) x+(b e-a h) x^2\right )}{6 b^2 \left (a+b x^3\right )^2}+\frac {x \left (b c-7 a f+2 (b d-4 a g) x+3 (b e-3 a h) x^2\right )}{18 a b^2 \left (a+b x^3\right )}+\frac {h \log \left (a+b x^3\right )}{3 b^3}+\frac {\int \frac {\sqrt [3]{a} \left (4 a b^{10/3} (b c+2 a f)+2 a^{4/3} b^3 (b d+5 a g)\right )+\sqrt [3]{b} \left (-2 a b^{10/3} (b c+2 a f)+2 a^{4/3} b^3 (b d+5 a g)\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{54 a^{8/3} b^{16/3}}+\frac {\left (\sqrt [3]{b} (b c+2 a f)-\sqrt [3]{a} (b d+5 a g)\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{5/3} b^{7/3}}\\ &=-\frac {x \left (b c-a f+(b d-a g) x+(b e-a h) x^2\right )}{6 b^2 \left (a+b x^3\right )^2}+\frac {x \left (b c-7 a f+2 (b d-4 a g) x+3 (b e-3 a h) x^2\right )}{18 a b^2 \left (a+b x^3\right )}+\frac {\left (\sqrt [3]{b} (b c+2 a f)-\sqrt [3]{a} (b d+5 a g)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{5/3} b^{8/3}}+\frac {h \log \left (a+b x^3\right )}{3 b^3}+\frac {\left (b^{4/3} c+\sqrt [3]{a} b d+2 a \sqrt [3]{b} f+5 a^{4/3} g\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{18 a^{4/3} b^{7/3}}-\frac {\left (\sqrt [3]{b} (b c+2 a f)-\sqrt [3]{a} (b d+5 a g)\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^{5/3} b^{8/3}}\\ &=-\frac {x \left (b c-a f+(b d-a g) x+(b e-a h) x^2\right )}{6 b^2 \left (a+b x^3\right )^2}+\frac {x \left (b c-7 a f+2 (b d-4 a g) x+3 (b e-3 a h) x^2\right )}{18 a b^2 \left (a+b x^3\right )}+\frac {\left (\sqrt [3]{b} (b c+2 a f)-\sqrt [3]{a} (b d+5 a g)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{5/3} b^{8/3}}-\frac {\left (\sqrt [3]{b} (b c+2 a f)-\sqrt [3]{a} (b d+5 a g)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{5/3} b^{8/3}}+\frac {h \log \left (a+b x^3\right )}{3 b^3}+\frac {\left (b^{4/3} c+\sqrt [3]{a} b d+2 a \sqrt [3]{b} f+5 a^{4/3} g\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{9 a^{5/3} b^{8/3}}\\ &=-\frac {x \left (b c-a f+(b d-a g) x+(b e-a h) x^2\right )}{6 b^2 \left (a+b x^3\right )^2}+\frac {x \left (b c-7 a f+2 (b d-4 a g) x+3 (b e-3 a h) x^2\right )}{18 a b^2 \left (a+b x^3\right )}-\frac {\left (b^{4/3} c+\sqrt [3]{a} b d+2 a \sqrt [3]{b} f+5 a^{4/3} g\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{5/3} b^{8/3}}+\frac {\left (\sqrt [3]{b} (b c+2 a f)-\sqrt [3]{a} (b d+5 a g)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{5/3} b^{8/3}}-\frac {\left (\sqrt [3]{b} (b c+2 a f)-\sqrt [3]{a} (b d+5 a g)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{5/3} b^{8/3}}+\frac {h \log \left (a+b x^3\right )}{3 b^3}\\ \end {align*}

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Mathematica [A]
time = 0.17, size = 315, normalized size = 0.97 \begin {gather*} \frac {-\frac {9 \left (a^2 h+b^2 x (c+d x)-a b (e+x (f+g x))\right )}{\left (a+b x^3\right )^2}+\frac {36 a^2 h+3 b^2 x (c+2 d x)-3 a b (6 e+x (7 f+8 g x))}{a \left (a+b x^3\right )}-\frac {2 \sqrt {3} \sqrt [3]{b} \left (b^{4/3} c+\sqrt [3]{a} b d+2 a \sqrt [3]{b} f+5 a^{4/3} g\right ) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{5/3}}+\frac {2 \sqrt [3]{b} \left (b^{4/3} c-\sqrt [3]{a} b d+2 a \sqrt [3]{b} f-5 a^{4/3} g\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{5/3}}+\frac {\sqrt [3]{b} \left (-b^{4/3} c+\sqrt [3]{a} b d-2 a \sqrt [3]{b} f+5 a^{4/3} g\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{5/3}}+18 h \log \left (a+b x^3\right )}{54 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-9*(a^2*h + b^2*x*(c + d*x) - a*b*(e + x*(f + g*x))))/(a + b*x^3)^2 + (36*a^2*h + 3*b^2*x*(c + 2*d*x) - 3*a*
b*(6*e + x*(7*f + 8*g*x)))/(a*(a + b*x^3)) - (2*Sqrt[3]*b^(1/3)*(b^(4/3)*c + a^(1/3)*b*d + 2*a*b^(1/3)*f + 5*a
^(4/3)*g)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(5/3) + (2*b^(1/3)*(b^(4/3)*c - a^(1/3)*b*d + 2*a*b^(
1/3)*f - 5*a^(4/3)*g)*Log[a^(1/3) + b^(1/3)*x])/a^(5/3) + (b^(1/3)*(-(b^(4/3)*c) + a^(1/3)*b*d - 2*a*b^(1/3)*f
+ 5*a^(4/3)*g)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(5/3) + 18*h*Log[a + b*x^3])/(54*b^3)

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Maple [A]
time = 0.40, size = 337, normalized size = 1.04

method	result	size
risch	\(\frac {-\frac {\left (4 a g -b d \right ) x^{5}}{9 a b}-\frac {\left (7 a f -b c \right ) x^{4}}{18 a b}+\frac {\left (2 a h -b e \right ) x^{3}}{3 b^{2}}-\frac {\left (5 a g +b d \right ) x^{2}}{18 b^{2}}-\frac {\left (2 a f +b c \right ) x}{9 b^{2}}+\frac {a \left (3 a h -b e \right )}{6 b^{3}}}{\left (b \,x^{3}+a \right )^{2}}+\frac {\munderset {\textit {\_R} =\RootOf \left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\left (9 h \,\textit {\_R}^{2}+\frac {\left (5 a g +b d \right ) \textit {\_R}}{a}+\frac {2 a f +b c}{a}\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}}{27 b^{3}}\)	\(173\)
default	\(\frac {-\frac {\left (4 a g -b d \right ) x^{5}}{9 a b}-\frac {\left (7 a f -b c \right ) x^{4}}{18 a b}+\frac {\left (2 a h -b e \right ) x^{3}}{3 b^{2}}-\frac {\left (5 a g +b d \right ) x^{2}}{18 b^{2}}-\frac {\left (2 a f +b c \right ) x}{9 b^{2}}+\frac {a \left (3 a h -b e \right )}{6 b^{3}}}{\left (b \,x^{3}+a \right )^{2}}+\frac {\left (2 a f +b c \right ) \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )+\left (5 a g +b d \right ) \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )+\frac {3 a h \ln \left (b \,x^{3}+a \right )}{b}}{9 a \,b^{2}}\)	\(337\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.53, size = 369, normalized size = 1.14 \begin {gather*} \frac {2 \, {\left (b^{3} d - 4 \, a b^{2} g\right )} x^{5} + {\left (b^{3} c - 7 \, a b^{2} f\right )} x^{4} + 9 \, a^{3} h + 6 \, {\left (2 \, a^{2} b h - a b^{2} e\right )} x^{3} - 3 \, a^{2} b e - {\left (a b^{2} d + 5 \, a^{2} b g\right )} x^{2} - 2 \, {\left (a b^{2} c + 2 \, a^{2} b f\right )} x}{18 \, {\left (a b^{5} x^{6} + 2 \, a^{2} b^{4} x^{3} + a^{3} b^{3}\right )}} + \frac {\sqrt {3} {\left (b^{2} d \left (\frac {a}{b}\right )^{\frac {2}{3}} + 5 \, a b g \left (\frac {a}{b}\right )^{\frac {2}{3}} + b^{2} c \left (\frac {a}{b}\right )^{\frac {1}{3}} + 2 \, a b f \left (\frac {a}{b}\right )^{\frac {1}{3}}\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}} + \frac {{\left (18 \, a h \left (\frac {a}{b}\right )^{\frac {2}{3}} + b d \left (\frac {a}{b}\right )^{\frac {1}{3}} + 5 \, a g \left (\frac {a}{b}\right )^{\frac {1}{3}} - b c - 2 \, 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 \, a b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (9 \, a h \left (\frac {a}{b}\right )^{\frac {2}{3}} - b d \left (\frac {a}{b}\right )^{\frac {1}{3}} - 5 \, a g \left (\frac {a}{b}\right )^{\frac {1}{3}} + b c + 2 \, a f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, a b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Fricas [C] Result contains complex when optimal does not.
time = 2.48, size = 12939, normalized size = 39.81 \begin {gather*} \text {too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(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*(12*(b^3*d - 4*a*b^2*g)*x^5 + 6*(b^3*c - 7*a*b^2*f)*x^4 - 18*a^2*b*e + 54*a^3*h - 36*(a*b^2*e - 2*a^2*b*
h)*x^3 - 6*(a*b^2*d + 5*a^2*b*g)*x^2 - 2*(a*b^5*x^6 + 2*a^2*b^4*x^3 + a^3*b^3)*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)
*(81*h^2/b^6 - (b^3*c*d + 10*a^2*b*f*g + 81*a^3*h^2 + (2*d*f + 5*c*g)*a*b^2)/(a^3*b^6))/(1458*h^3/b^9 - 27*(b^
3*c*d + 10*a^2*b*f*g + 81*a^3*h^2 + (2*d*f + 5*c*g)*a*b^2)*h/(a^3*b^9) + (b^4*c^3 + a*b^3*d^3 + 6*a*b^3*c^2*f
+ 12*a^2*b^2*c*f^2 + 8*a^3*b*f^3 + 15*a^2*b^2*d^2*g + 75*a^3*b*d*g^2 + 125*a^4*g^3)/(a^5*b^8) + (b^5*c^3 + 729
*a^5*h^3 - 5*(25*g^3 - 54*f*g*h)*a^4*b + (8*f^3 + 135*c*g*h - 3*(25*g^2 - 18*f*h)*d)*a^3*b^2 - 3*(5*d^2*g - (4
*f^2 + 9*d*h)*c)*a^2*b^3 - (d^3 - 6*c^2*f)*a*b^4)/(a^5*b^9))^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(1458*h^3/b^9
- 27*(b^3*c*d + 10*a^2*b*f*g + 81*a^3*h^2 + (2*d*f + 5*c*g)*a*b^2)*h/(a^3*b^9) + (b^4*c^3 + a*b^3*d^3 + 6*a*b
^3*c^2*f + 12*a^2*b^2*c*f^2 + 8*a^3*b*f^3 + 15*a^2*b^2*d^2*g + 75*a^3*b*d*g^2 + 125*a^4*g^3)/(a^5*b^8) + (b^5*
c^3 + 729* ...

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(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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Giac [A]
time = 0.51, size = 363, normalized size = 1.12 \begin {gather*} \frac {h \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{3}} - \frac {\sqrt {3} {\left (b^{2} c + 2 \, a b f - \left (-a b^{2}\right )^{\frac {1}{3}} b d - 5 \, \left (-a b^{2}\right )^{\frac {1}{3}} a g\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 b^{2}} - \frac {{\left (b^{2} c + 2 \, a b f + \left (-a b^{2}\right )^{\frac {1}{3}} b d + 5 \, \left (-a b^{2}\right )^{\frac {1}{3}} a g\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 b^{2}} + \frac {2 \, {\left (b^{2} d - 4 \, a b g\right )} x^{5} + {\left (b^{2} c - 7 \, a b f\right )} x^{4} + 6 \, {\left (2 \, a^{2} h - a b e\right )} x^{3} - {\left (a b d + 5 \, a^{2} g\right )} x^{2} - 2 \, {\left (a b c + 2 \, a^{2} f\right )} x + \frac {3 \, {\left (3 \, a^{3} h - a^{2} b e\right )}}{b}}{18 \, {\left (b x^{3} + a\right )}^{2} a b^{2}} - \frac {{\left (a b^{4} d \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 5 \, a^{2} b^{3} g \left (-\frac {a}{b}\right )^{\frac {1}{3}} + a b^{4} c + 2 \, a^{2} b^{3} f\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^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 5.66, size = 908, normalized size = 2.79 \begin {gather*} \frac {\frac {3\,a^2\,h-a\,b\,e}{6\,b^3}-\frac {x\,\left (b\,c+2\,a\,f\right )}{9\,b^2}-\frac {x^2\,\left (b\,d+5\,a\,g\right )}{18\,b^2}-\frac {x^3\,\left (b\,e-2\,a\,h\right )}{3\,b^2}+\frac {x^4\,\left (b\,c-7\,a\,f\right )}{18\,a\,b}+\frac {x^5\,\left (b\,d-4\,a\,g\right )}{9\,a\,b}}{a^2+2\,a\,b\,x^3+b^2\,x^6}+\left (\sum _{k=1}^3\ln \left (\mathrm {root}\left (19683\,a^5\,b^9\,z^3-19683\,a^5\,b^6\,h\,z^2+810\,a^4\,b^4\,f\,g\,z+405\,a^3\,b^5\,c\,g\,z+162\,a^3\,b^5\,d\,f\,z+81\,a^2\,b^6\,c\,d\,z+6561\,a^5\,b^3\,h^2\,z-270\,a^4\,b\,f\,g\,h-135\,a^3\,b^2\,c\,g\,h-54\,a^3\,b^2\,d\,f\,h-27\,a^2\,b^3\,c\,d\,h-6\,a\,b^4\,c^2\,f+75\,a^3\,b^2\,d\,g^2+15\,a^2\,b^3\,d^2\,g-12\,a^2\,b^3\,c\,f^2-8\,a^3\,b^2\,f^3+125\,a^4\,b\,g^3+a\,b^4\,d^3-729\,a^5\,h^3-b^5\,c^3,z,k\right )\,\left (\mathrm {root}\left (19683\,a^5\,b^9\,z^3-19683\,a^5\,b^6\,h\,z^2+810\,a^4\,b^4\,f\,g\,z+405\,a^3\,b^5\,c\,g\,z+162\,a^3\,b^5\,d\,f\,z+81\,a^2\,b^6\,c\,d\,z+6561\,a^5\,b^3\,h^2\,z-270\,a^4\,b\,f\,g\,h-135\,a^3\,b^2\,c\,g\,h-54\,a^3\,b^2\,d\,f\,h-27\,a^2\,b^3\,c\,d\,h-6\,a\,b^4\,c^2\,f+75\,a^3\,b^2\,d\,g^2+15\,a^2\,b^3\,d^2\,g-12\,a^2\,b^3\,c\,f^2-8\,a^3\,b^2\,f^3+125\,a^4\,b\,g^3+a\,b^4\,d^3-729\,a^5\,h^3-b^5\,c^3,z,k\right )\,a\,b^2\,9-\frac {6\,a\,h}{b}+\frac {x\,\left (54\,f\,a^2\,b^3+27\,c\,a\,b^4\right )}{81\,a^2\,b^3}\right )+\frac {81\,a^3\,h^2+b^3\,c\,d+5\,a\,b^2\,c\,g+2\,a\,b^2\,d\,f+10\,a^2\,b\,f\,g}{81\,a^2\,b^4}+\frac {x\,\left (25\,a^2\,g^2-18\,f\,h\,a^2+10\,a\,b\,d\,g-9\,c\,h\,a\,b+b^2\,d^2\right )}{81\,a^2\,b^3}\right )\,\mathrm {root}\left (19683\,a^5\,b^9\,z^3-19683\,a^5\,b^6\,h\,z^2+810\,a^4\,b^4\,f\,g\,z+405\,a^3\,b^5\,c\,g\,z+162\,a^3\,b^5\,d\,f\,z+81\,a^2\,b^6\,c\,d\,z+6561\,a^5\,b^3\,h^2\,z-270\,a^4\,b\,f\,g\,h-135\,a^3\,b^2\,c\,g\,h-54\,a^3\,b^2\,d\,f\,h-27\,a^2\,b^3\,c\,d\,h-6\,a\,b^4\,c^2\,f+75\,a^3\,b^2\,d\,g^2+15\,a^2\,b^3\,d^2\,g-12\,a^2\,b^3\,c\,f^2-8\,a^3\,b^2\,f^3+125\,a^4\,b\,g^3+a\,b^4\,d^3-729\,a^5\,h^3-b^5\,c^3,z,k\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((3*a^2*h - a*b*e)/(6*b^3) - (x*(b*c + 2*a*f))/(9*b^2) - (x^2*(b*d + 5*a*g))/(18*b^2) - (x^3*(b*e - 2*a*h))/(3
*b^2) + (x^4*(b*c - 7*a*f))/(18*a*b) + (x^5*(b*d - 4*a*g))/(9*a*b))/(a^2 + b^2*x^6 + 2*a*b*x^3) + symsum(log(r
oot(19683*a^5*b^9*z^3 - 19683*a^5*b^6*h*z^2 + 810*a^4*b^4*f*g*z + 405*a^3*b^5*c*g*z + 162*a^3*b^5*d*f*z + 81*a
^2*b^6*c*d*z + 6561*a^5*b^3*h^2*z - 270*a^4*b*f*g*h - 135*a^3*b^2*c*g*h - 54*a^3*b^2*d*f*h - 27*a^2*b^3*c*d*h
- 6*a*b^4*c^2*f + 75*a^3*b^2*d*g^2 + 15*a^2*b^3*d^2*g - 12*a^2*b^3*c*f^2 - 8*a^3*b^2*f^3 + 125*a^4*b*g^3 + a*b
^4*d^3 - 729*a^5*h^3 - b^5*c^3, z, k)*(9*root(19683*a^5*b^9*z^3 - 19683*a^5*b^6*h*z^2 + 810*a^4*b^4*f*g*z + 40
5*a^3*b^5*c*g*z + 162*a^3*b^5*d*f*z + 81*a^2*b^6*c*d*z + 6561*a^5*b^3*h^2*z - 270*a^4*b*f*g*h - 135*a^3*b^2*c*
g*h - 54*a^3*b^2*d*f*h - 27*a^2*b^3*c*d*h - 6*a*b^4*c^2*f + 75*a^3*b^2*d*g^2 + 15*a^2*b^3*d^2*g - 12*a^2*b^3*c
*f^2 - 8*a^3*b^2*f^3 + 125*a^4*b*g^3 + a*b^4*d^3 - 729*a^5*h^3 - b^5*c^3, z, k)*a*b^2 - (6*a*h)/b + (x*(54*a^2
*b^3*f + 27*a*b^4*c))/(81*a^2*b^3)) + (81*a^3*h^2 + b^3*c*d + 5*a*b^2*c*g + 2*a*b^2*d*f + 10*a^2*b*f*g)/(81*a^
2*b^4) + (x*(b^2*d^2 + 25*a^2*g^2 - 18*a^2*f*h - 9*a*b*c*h + 10*a*b*d*g))/(81*a^2*b^3))*root(19683*a^5*b^9*z^3
- 19683*a^5*b^6*h*z^2 + 810*a^4*b^4*f*g*z + 405*a^3*b^5*c*g*z + 162*a^3*b^5*d*f*z + 81*a^2*b^6*c*d*z + 6561*a
^5*b^3*h^2*z - 270*a^4*b*f*g*h - 135*a^3*b^2*c*g*h - 54*a^3*b^2*d*f*h - 27*a^2*b^3*c*d*h - 6*a*b^4*c^2*f + 75*
a^3*b^2*d*g^2 + 15*a^2*b^3*d^2*g - 12*a^2*b^3*c*f^2 - 8*a^3*b^2*f^3 + 125*a^4*b*g^3 + a*b^4*d^3 - 729*a^5*h^3
- b^5*c^3, z, k), k, 1, 3)

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