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

Optimal. Leaf size=311 \[ -\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (\sqrt [3]{b} (b c-4 a f)-\sqrt [3]{a} (2 b d-5 a g)\right )}{18 a^{2/3} b^{8/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (b c-4 a f)-\sqrt [3]{a} (2 b d-5 a g)\right )}{9 a^{2/3} b^{8/3}}-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (-5 a^{4/3} g+2 \sqrt [3]{a} b d-4 a \sqrt [3]{b} f+b^{4/3} c\right )}{3 \sqrt {3} a^{2/3} b^{8/3}}+\frac {(b e-2 a h) \log \left (a+b x^3\right )}{3 b^3}-\frac {x \left (x (b d-a g)+x^2 (b e-a h)-a f+b c\right )}{3 b^2 \left (a+b x^3\right )}+\frac {f x}{b^2}+\frac {g x^2}{2 b^2}+\frac {h x^3}{3 b^2} \]

[Out]

f*x/b^2+1/2*g*x^2/b^2+1/3*h*x^3/b^2-1/3*x*(b*c-a*f+(-a*g+b*d)*x+(-a*h+b*e)*x^2)/b^2/(b*x^3+a)+1/9*(b^(1/3)*(-4
*a*f+b*c)-a^(1/3)*(-5*a*g+2*b*d))*ln(a^(1/3)+b^(1/3)*x)/a^(2/3)/b^(8/3)-1/18*(b^(1/3)*(-4*a*f+b*c)-a^(1/3)*(-5
*a*g+2*b*d))*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(2/3)/b^(8/3)+1/3*(-2*a*h+b*e)*ln(b*x^3+a)/b^3-1/9*(b
^(4/3)*c+2*a^(1/3)*b*d-4*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^(2/3)/b^
(8/3)*3^(1/2)

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Rubi [A]  time = 0.64, antiderivative size = 311, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {1828, 1887, 1871, 1860, 31, 634, 617, 204, 628, 260} \[ -\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (\sqrt [3]{b} (b c-4 a f)-\sqrt [3]{a} (2 b d-5 a g)\right )}{18 a^{2/3} b^{8/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (b c-4 a f)-\sqrt [3]{a} (2 b d-5 a g)\right )}{9 a^{2/3} b^{8/3}}-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (-5 a^{4/3} g+2 \sqrt [3]{a} b d-4 a \sqrt [3]{b} f+b^{4/3} c\right )}{3 \sqrt {3} a^{2/3} b^{8/3}}-\frac {x \left (x (b d-a g)+x^2 (b e-a h)-a f+b c\right )}{3 b^2 \left (a+b x^3\right )}+\frac {(b e-2 a h) \log \left (a+b x^3\right )}{3 b^3}+\frac {f x}{b^2}+\frac {g x^2}{2 b^2}+\frac {h x^3}{3 b^2} \]

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)^2,x]

[Out]

(f*x)/b^2 + (g*x^2)/(2*b^2) + (h*x^3)/(3*b^2) - (x*(b*c - a*f + (b*d - a*g)*x + (b*e - a*h)*x^2))/(3*b^2*(a +
b*x^3)) - ((b^(4/3)*c + 2*a^(1/3)*b*d - 4*a*b^(1/3)*f - 5*a^(4/3)*g)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a
^(1/3))])/(3*Sqrt[3]*a^(2/3)*b^(8/3)) + ((b^(1/3)*(b*c - 4*a*f) - a^(1/3)*(2*b*d - 5*a*g))*Log[a^(1/3) + b^(1/
3)*x])/(9*a^(2/3)*b^(8/3)) - ((b^(1/3)*(b*c - 4*a*f) - a^(1/3)*(2*b*d - 5*a*g))*Log[a^(2/3) - a^(1/3)*b^(1/3)*
x + b^(2/3)*x^2])/(18*a^(2/3)*b^(8/3)) + ((b*e - 2*a*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 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 260

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 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 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]

Rule 1871

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]

Rule 1887

Int[(Pq_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[Pq/(a + b*x^n), x], x] /; FreeQ[{a, b}, x
] && PolyQ[Pq, x] && IntegerQ[n]

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

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Mathematica [A]  time = 0.22, size = 294, normalized size = 0.95 \[ \frac {-\frac {\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (5 a^{4/3} g-2 \sqrt [3]{a} b d-4 a \sqrt [3]{b} f+b^{4/3} c\right )}{a^{2/3}}+\frac {2 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (5 a^{4/3} g-2 \sqrt [3]{a} b d-4 a \sqrt [3]{b} f+b^{4/3} c\right )}{a^{2/3}}+\frac {2 \sqrt {3} \sqrt [3]{b} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right ) \left (5 a^{4/3} g-2 \sqrt [3]{a} b d+4 a \sqrt [3]{b} f-b^{4/3} c\right )}{a^{2/3}}-\frac {6 \left (a^2 h-a b (e+x (f+g x))+b^2 x (c+d x)\right )}{a+b x^3}+6 (b e-2 a h) \log \left (a+b x^3\right )+18 b f x+9 b g x^2+6 b h x^3}{18 b^3} \]

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)^2,x]

[Out]

(18*b*f*x + 9*b*g*x^2 + 6*b*h*x^3 - (6*(a^2*h + b^2*x*(c + d*x) - a*b*(e + x*(f + g*x))))/(a + b*x^3) + (2*Sqr
t[3]*b^(1/3)*(-(b^(4/3)*c) - 2*a^(1/3)*b*d + 4*a*b^(1/3)*f + 5*a^(4/3)*g)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/S
qrt[3]])/a^(2/3) + (2*b^(1/3)*(b^(4/3)*c - 2*a^(1/3)*b*d - 4*a*b^(1/3)*f + 5*a^(4/3)*g)*Log[a^(1/3) + b^(1/3)*
x])/a^(2/3) - (b^(1/3)*(b^(4/3)*c - 2*a^(1/3)*b*d - 4*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^(2/3) + 6*(b*e - 2*a*h)*Log[a + b*x^3])/(18*b^3)

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

[Out]

Timed out

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giac [A]  time = 0.21, size = 330, normalized size = 1.06 \[ -\frac {\sqrt {3} {\left (b^{2} c - 4 \, a b f - 2 \, \left (-a b^{2}\right )^{\frac {1}{3}} b d + 5 \, \left (-a b^{2}\right )^{\frac {1}{3}} a g\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, \left (-a b^{2}\right )^{\frac {2}{3}} b^{2}} - \frac {{\left (b^{2} c - 4 \, a b f + 2 \, \left (-a b^{2}\right )^{\frac {1}{3}} b d - 5 \, \left (-a b^{2}\right )^{\frac {1}{3}} a g\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, \left (-a b^{2}\right )^{\frac {2}{3}} b^{2}} - \frac {{\left (2 \, a h - b e\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{3}} - \frac {a^{2} h + {\left (b^{2} d - a b g\right )} x^{2} - a b e + {\left (b^{2} c - a b f\right )} x}{3 \, {\left (b x^{3} + a\right )} b^{3}} - \frac {{\left (2 \, b^{4} d \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 5 \, a b^{3} g \left (-\frac {a}{b}\right )^{\frac {1}{3}} + b^{4} c - 4 \, a b^{3} f\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a b^{5}} + \frac {2 \, b^{4} h x^{3} + 3 \, b^{4} g x^{2} + 6 \, b^{4} f x}{6 \, b^{6}} \]

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)^2,x, algorithm="giac")

[Out]

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

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maple [B]  time = 0.06, size = 533, normalized size = 1.71 \[ \frac {a g \,x^{2}}{3 \left (b \,x^{3}+a \right ) b^{2}}-\frac {d \,x^{2}}{3 \left (b \,x^{3}+a \right ) b}+\frac {h \,x^{3}}{3 b^{2}}+\frac {a f x}{3 \left (b \,x^{3}+a \right ) b^{2}}-\frac {c x}{3 \left (b \,x^{3}+a \right ) b}+\frac {g \,x^{2}}{2 b^{2}}-\frac {a^{2} h}{3 \left (b \,x^{3}+a \right ) b^{3}}+\frac {a e}{3 \left (b \,x^{3}+a \right ) b^{2}}-\frac {4 \sqrt {3}\, a f \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{3}}-\frac {4 a f \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{3}}+\frac {2 a f \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{3}}-\frac {5 \sqrt {3}\, a g \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{3}}+\frac {5 a g \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{3}}-\frac {5 a g \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{3}}-\frac {2 a h \ln \left (b \,x^{3}+a \right )}{3 b^{3}}+\frac {\sqrt {3}\, c \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{2}}+\frac {c \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{2}}-\frac {c \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{2}}+\frac {2 \sqrt {3}\, d \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{2}}-\frac {2 d \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{2}}+\frac {d \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{2}}+\frac {e \ln \left (b \,x^{3}+a \right )}{3 b^{2}}+\frac {f x}{b^{2}} \]

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)^2,x)

[Out]

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

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maxima [A]  time = 3.14, size = 329, normalized size = 1.06 \[ \frac {a b e - a^{2} h - {\left (b^{2} d - a b g\right )} x^{2} - {\left (b^{2} c - a b f\right )} x}{3 \, {\left (b^{4} x^{3} + a b^{3}\right )}} + \frac {2 \, h x^{3} + 3 \, g x^{2} + 6 \, f x}{6 \, b^{2}} + \frac {\sqrt {3} {\left (2 \, b^{2} d \left (\frac {a}{b}\right )^{\frac {2}{3}} - 5 \, a b g \left (\frac {a}{b}\right )^{\frac {2}{3}} + b^{2} c \left (\frac {a}{b}\right )^{\frac {1}{3}} - 4 \, a b f \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a b^{3}} + \frac {{\left (6 \, b e \left (\frac {a}{b}\right )^{\frac {2}{3}} - 12 \, a h \left (\frac {a}{b}\right )^{\frac {2}{3}} + 2 \, b d \left (\frac {a}{b}\right )^{\frac {1}{3}} - 5 \, a g \left (\frac {a}{b}\right )^{\frac {1}{3}} - b c + 4 \, a f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (3 \, b e \left (\frac {a}{b}\right )^{\frac {2}{3}} - 6 \, a h \left (\frac {a}{b}\right )^{\frac {2}{3}} - 2 \, b d \left (\frac {a}{b}\right )^{\frac {1}{3}} + 5 \, a g \left (\frac {a}{b}\right )^{\frac {1}{3}} + b c - 4 \, a f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]

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)^2,x, algorithm="maxima")

[Out]

1/3*(a*b*e - a^2*h - (b^2*d - a*b*g)*x^2 - (b^2*c - a*b*f)*x)/(b^4*x^3 + a*b^3) + 1/6*(2*h*x^3 + 3*g*x^2 + 6*f
*x)/b^2 + 1/9*sqrt(3)*(2*b^2*d*(a/b)^(2/3) - 5*a*b*g*(a/b)^(2/3) + b^2*c*(a/b)^(1/3) - 4*a*b*f*(a/b)^(1/3))*ar
ctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(a*b^3) + 1/18*(6*b*e*(a/b)^(2/3) - 12*a*h*(a/b)^(2/3) + 2*b
*d*(a/b)^(1/3) - 5*a*g*(a/b)^(1/3) - b*c + 4*a*f)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(b^3*(a/b)^(2/3)) + 1
/9*(3*b*e*(a/b)^(2/3) - 6*a*h*(a/b)^(2/3) - 2*b*d*(a/b)^(1/3) + 5*a*g*(a/b)^(1/3) + b*c - 4*a*f)*log(x + (a/b)
^(1/3))/(b^3*(a/b)^(2/3))

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mupad [B]  time = 0.15, size = 1229, normalized size = 3.95 \[ \left (\sum _{k=1}^3\ln \left (\frac {36\,a^3\,h^2+9\,a\,b^2\,e^2+2\,b^3\,c\,d-5\,a\,b^2\,c\,g-8\,a\,b^2\,d\,f-36\,a^2\,b\,e\,h+20\,a^2\,b\,f\,g}{9\,b^4}+\mathrm {root}\left (729\,a^2\,b^9\,z^3+1458\,a^3\,b^6\,h\,z^2-729\,a^2\,b^7\,e\,z^2+54\,a\,b^6\,c\,d\,z-972\,a^3\,b^4\,e\,h\,z+540\,a^3\,b^4\,f\,g\,z-216\,a^2\,b^5\,d\,f\,z-135\,a^2\,b^5\,c\,g\,z+972\,a^4\,b^3\,h^2\,z+243\,a^2\,b^5\,e^2\,z+360\,a^4\,b\,f\,g\,h-18\,a\,b^4\,c\,d\,e-180\,a^3\,b^2\,e\,f\,g-144\,a^3\,b^2\,d\,f\,h-90\,a^3\,b^2\,c\,g\,h+72\,a^2\,b^3\,d\,e\,f+45\,a^2\,b^3\,c\,e\,g+36\,a^2\,b^3\,c\,d\,h-324\,a^4\,b\,e\,h^2+12\,a\,b^4\,c^2\,f+162\,a^3\,b^2\,e^2\,h+150\,a^3\,b^2\,d\,g^2-60\,a^2\,b^3\,d^2\,g-48\,a^2\,b^3\,c\,f^2+64\,a^3\,b^2\,f^3-27\,a^2\,b^3\,e^3-125\,a^4\,b\,g^3+8\,a\,b^4\,d^3+216\,a^5\,h^3-b^5\,c^3,z,k\right )\,\left (\frac {108\,a^2\,b^3\,h-54\,a\,b^4\,e}{9\,b^4}+\frac {x\,\left (9\,b^4\,c-36\,a\,b^3\,f\right )}{9\,b^3}+\mathrm {root}\left (729\,a^2\,b^9\,z^3+1458\,a^3\,b^6\,h\,z^2-729\,a^2\,b^7\,e\,z^2+54\,a\,b^6\,c\,d\,z-972\,a^3\,b^4\,e\,h\,z+540\,a^3\,b^4\,f\,g\,z-216\,a^2\,b^5\,d\,f\,z-135\,a^2\,b^5\,c\,g\,z+972\,a^4\,b^3\,h^2\,z+243\,a^2\,b^5\,e^2\,z+360\,a^4\,b\,f\,g\,h-18\,a\,b^4\,c\,d\,e-180\,a^3\,b^2\,e\,f\,g-144\,a^3\,b^2\,d\,f\,h-90\,a^3\,b^2\,c\,g\,h+72\,a^2\,b^3\,d\,e\,f+45\,a^2\,b^3\,c\,e\,g+36\,a^2\,b^3\,c\,d\,h-324\,a^4\,b\,e\,h^2+12\,a\,b^4\,c^2\,f+162\,a^3\,b^2\,e^2\,h+150\,a^3\,b^2\,d\,g^2-60\,a^2\,b^3\,d^2\,g-48\,a^2\,b^3\,c\,f^2+64\,a^3\,b^2\,f^3-27\,a^2\,b^3\,e^3-125\,a^4\,b\,g^3+8\,a\,b^4\,d^3+216\,a^5\,h^3-b^5\,c^3,z,k\right )\,a\,b^2\,9\right )+\frac {x\,\left (4\,b^2\,d^2+25\,a^2\,g^2-3\,b^2\,c\,e-24\,a^2\,f\,h+6\,a\,b\,c\,h-20\,a\,b\,d\,g+12\,a\,b\,e\,f\right )}{9\,b^3}\right )\,\mathrm {root}\left (729\,a^2\,b^9\,z^3+1458\,a^3\,b^6\,h\,z^2-729\,a^2\,b^7\,e\,z^2+54\,a\,b^6\,c\,d\,z-972\,a^3\,b^4\,e\,h\,z+540\,a^3\,b^4\,f\,g\,z-216\,a^2\,b^5\,d\,f\,z-135\,a^2\,b^5\,c\,g\,z+972\,a^4\,b^3\,h^2\,z+243\,a^2\,b^5\,e^2\,z+360\,a^4\,b\,f\,g\,h-18\,a\,b^4\,c\,d\,e-180\,a^3\,b^2\,e\,f\,g-144\,a^3\,b^2\,d\,f\,h-90\,a^3\,b^2\,c\,g\,h+72\,a^2\,b^3\,d\,e\,f+45\,a^2\,b^3\,c\,e\,g+36\,a^2\,b^3\,c\,d\,h-324\,a^4\,b\,e\,h^2+12\,a\,b^4\,c^2\,f+162\,a^3\,b^2\,e^2\,h+150\,a^3\,b^2\,d\,g^2-60\,a^2\,b^3\,d^2\,g-48\,a^2\,b^3\,c\,f^2+64\,a^3\,b^2\,f^3-27\,a^2\,b^3\,e^3-125\,a^4\,b\,g^3+8\,a\,b^4\,d^3+216\,a^5\,h^3-b^5\,c^3,z,k\right )\right )-\frac {x\,\left (\frac {b\,c}{3}-\frac {a\,f}{3}\right )+\frac {a^2\,h-a\,b\,e}{3\,b}+x^2\,\left (\frac {b\,d}{3}-\frac {a\,g}{3}\right )}{b^3\,x^3+a\,b^2}+\frac {g\,x^2}{2\,b^2}+\frac {h\,x^3}{3\,b^2}+\frac {f\,x}{b^2} \]

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)^2,x)

[Out]

symsum(log((36*a^3*h^2 + 9*a*b^2*e^2 + 2*b^3*c*d - 5*a*b^2*c*g - 8*a*b^2*d*f - 36*a^2*b*e*h + 20*a^2*b*f*g)/(9
*b^4) + root(729*a^2*b^9*z^3 + 1458*a^3*b^6*h*z^2 - 729*a^2*b^7*e*z^2 + 54*a*b^6*c*d*z - 972*a^3*b^4*e*h*z + 5
40*a^3*b^4*f*g*z - 216*a^2*b^5*d*f*z - 135*a^2*b^5*c*g*z + 972*a^4*b^3*h^2*z + 243*a^2*b^5*e^2*z + 360*a^4*b*f
*g*h - 18*a*b^4*c*d*e - 180*a^3*b^2*e*f*g - 144*a^3*b^2*d*f*h - 90*a^3*b^2*c*g*h + 72*a^2*b^3*d*e*f + 45*a^2*b
^3*c*e*g + 36*a^2*b^3*c*d*h - 324*a^4*b*e*h^2 + 12*a*b^4*c^2*f + 162*a^3*b^2*e^2*h + 150*a^3*b^2*d*g^2 - 60*a^
2*b^3*d^2*g - 48*a^2*b^3*c*f^2 + 64*a^3*b^2*f^3 - 27*a^2*b^3*e^3 - 125*a^4*b*g^3 + 8*a*b^4*d^3 + 216*a^5*h^3 -
b^5*c^3, z, k)*((108*a^2*b^3*h - 54*a*b^4*e)/(9*b^4) + (x*(9*b^4*c - 36*a*b^3*f))/(9*b^3) + 9*root(729*a^2*b^
9*z^3 + 1458*a^3*b^6*h*z^2 - 729*a^2*b^7*e*z^2 + 54*a*b^6*c*d*z - 972*a^3*b^4*e*h*z + 540*a^3*b^4*f*g*z - 216*
a^2*b^5*d*f*z - 135*a^2*b^5*c*g*z + 972*a^4*b^3*h^2*z + 243*a^2*b^5*e^2*z + 360*a^4*b*f*g*h - 18*a*b^4*c*d*e -
180*a^3*b^2*e*f*g - 144*a^3*b^2*d*f*h - 90*a^3*b^2*c*g*h + 72*a^2*b^3*d*e*f + 45*a^2*b^3*c*e*g + 36*a^2*b^3*c
*d*h - 324*a^4*b*e*h^2 + 12*a*b^4*c^2*f + 162*a^3*b^2*e^2*h + 150*a^3*b^2*d*g^2 - 60*a^2*b^3*d^2*g - 48*a^2*b^
3*c*f^2 + 64*a^3*b^2*f^3 - 27*a^2*b^3*e^3 - 125*a^4*b*g^3 + 8*a*b^4*d^3 + 216*a^5*h^3 - b^5*c^3, z, k)*a*b^2)
+ (x*(4*b^2*d^2 + 25*a^2*g^2 - 3*b^2*c*e - 24*a^2*f*h + 6*a*b*c*h - 20*a*b*d*g + 12*a*b*e*f))/(9*b^3))*root(72
9*a^2*b^9*z^3 + 1458*a^3*b^6*h*z^2 - 729*a^2*b^7*e*z^2 + 54*a*b^6*c*d*z - 972*a^3*b^4*e*h*z + 540*a^3*b^4*f*g*
z - 216*a^2*b^5*d*f*z - 135*a^2*b^5*c*g*z + 972*a^4*b^3*h^2*z + 243*a^2*b^5*e^2*z + 360*a^4*b*f*g*h - 18*a*b^4
*c*d*e - 180*a^3*b^2*e*f*g - 144*a^3*b^2*d*f*h - 90*a^3*b^2*c*g*h + 72*a^2*b^3*d*e*f + 45*a^2*b^3*c*e*g + 36*a
^2*b^3*c*d*h - 324*a^4*b*e*h^2 + 12*a*b^4*c^2*f + 162*a^3*b^2*e^2*h + 150*a^3*b^2*d*g^2 - 60*a^2*b^3*d^2*g - 4
8*a^2*b^3*c*f^2 + 64*a^3*b^2*f^3 - 27*a^2*b^3*e^3 - 125*a^4*b*g^3 + 8*a*b^4*d^3 + 216*a^5*h^3 - b^5*c^3, z, k)
, k, 1, 3) - (x*((b*c)/3 - (a*f)/3) + (a^2*h - a*b*e)/(3*b) + x^2*((b*d)/3 - (a*g)/3))/(a*b^2 + b^3*x^3) + (g*
x^2)/(2*b^2) + (h*x^3)/(3*b^2) + (f*x)/b^2

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

[Out]

Timed out

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