3.5.5 \(\int \frac {x^2 (c+d x+e x^2+f x^3+g x^4+h x^5)}{a+b x^3} \, dx\) [405]

Optimal. Leaf size=294 \[ \frac {(b d-a g) x}{b^2}+\frac {(b e-a h) x^2}{2 b^2}+\frac {f x^3}{3 b}+\frac {g x^4}{4 b}+\frac {h x^5}{5 b}+\frac {\sqrt [3]{a} \left (b^{4/3} d+\sqrt [3]{a} b e-a \sqrt [3]{b} g-a^{4/3} h\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} b^{8/3}}-\frac {\sqrt [3]{a} \left (\sqrt [3]{b} (b d-a g)-\sqrt [3]{a} (b e-a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{8/3}}+\frac {\sqrt [3]{a} \left (\sqrt [3]{b} (b d-a g)-\sqrt [3]{a} (b e-a h)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{8/3}}+\frac {(b c-a f) \log \left (a+b x^3\right )}{3 b^2} \]

[Out]

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

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Rubi [A]
time = 0.65, antiderivative size = 294, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {1850, 1901, 1885, 1874, 31, 648, 631, 210, 642, 266} \begin {gather*} \frac {\sqrt [3]{a} \text {ArcTan}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (a^{4/3} (-h)+\sqrt [3]{a} b e-a \sqrt [3]{b} g+b^{4/3} d\right )}{\sqrt {3} b^{8/3}}+\frac {\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (\sqrt [3]{b} (b d-a g)-\sqrt [3]{a} (b e-a h)\right )}{6 b^{8/3}}-\frac {\sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (b d-a g)-\sqrt [3]{a} (b e-a h)\right )}{3 b^{8/3}}+\frac {(b c-a f) \log \left (a+b x^3\right )}{3 b^2}+\frac {x (b d-a g)}{b^2}+\frac {x^2 (b e-a h)}{2 b^2}+\frac {f x^3}{3 b}+\frac {g x^4}{4 b}+\frac {h x^5}{5 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 1850

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, With[{Pqq =
Coeff[Pq, x, q]}, Dist[1/(b*(m + q + n*p + 1)), Int[(c*x)^m*ExpandToSum[b*(m + q + n*p + 1)*(Pq - Pqq*x^q) - a
*Pqq*(m + q - n + 1)*x^(q - n), x]*(a + b*x^n)^p, x], x] + Simp[Pqq*(c*x)^(m + q - n + 1)*((a + b*x^n)^(p + 1)
/(b*c^(q - n + 1)*(m + q + n*p + 1))), x]] /; NeQ[m + q + n*p + 1, 0] && q - n >= 0 && (IntegerQ[2*p] || Integ
erQ[p + (q + 1)/(2*n)])] /; FreeQ[{a, b, c, m, p}, x] && PolyQ[Pq, x] && IGtQ[n, 0]

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]

Rule 1901

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^2 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{a+b x^3} \, dx &=\frac {h x^5}{5 b}+\frac {\int \frac {x^2 \left (5 b c+5 b d x+5 (b e-a h) x^2+5 b f x^3+5 b g x^4\right )}{a+b x^3} \, dx}{5 b}\\ &=\frac {g x^4}{4 b}+\frac {h x^5}{5 b}+\frac {\int \frac {x^2 \left (20 b^2 c+20 b (b d-a g) x+20 b (b e-a h) x^2+20 b^2 f x^3\right )}{a+b x^3} \, dx}{20 b^2}\\ &=\frac {f x^3}{3 b}+\frac {g x^4}{4 b}+\frac {h x^5}{5 b}+\frac {\int \frac {x^2 \left (60 b^2 (b c-a f)+60 b^2 (b d-a g) x+60 b^2 (b e-a h) x^2\right )}{a+b x^3} \, dx}{60 b^3}\\ &=\frac {f x^3}{3 b}+\frac {g x^4}{4 b}+\frac {h x^5}{5 b}+\frac {\int \left (60 b (b d-a g)+60 b (b e-a h) x-\frac {60 \left (a b (b d-a g)+a b (b e-a h) x-b^2 (b c-a f) x^2\right )}{a+b x^3}\right ) \, dx}{60 b^3}\\ &=\frac {(b d-a g) x}{b^2}+\frac {(b e-a h) x^2}{2 b^2}+\frac {f x^3}{3 b}+\frac {g x^4}{4 b}+\frac {h x^5}{5 b}-\frac {\int \frac {a b (b d-a g)+a b (b e-a h) x-b^2 (b c-a f) x^2}{a+b x^3} \, dx}{b^3}\\ &=\frac {(b d-a g) x}{b^2}+\frac {(b e-a h) x^2}{2 b^2}+\frac {f x^3}{3 b}+\frac {g x^4}{4 b}+\frac {h x^5}{5 b}-\frac {\int \frac {a b (b d-a g)+a b (b e-a h) x}{a+b x^3} \, dx}{b^3}+\frac {(b c-a f) \int \frac {x^2}{a+b x^3} \, dx}{b}\\ &=\frac {(b d-a g) x}{b^2}+\frac {(b e-a h) x^2}{2 b^2}+\frac {f x^3}{3 b}+\frac {g x^4}{4 b}+\frac {h x^5}{5 b}+\frac {(b c-a f) \log \left (a+b x^3\right )}{3 b^2}-\frac {\int \frac {\sqrt [3]{a} \left (2 a b^{4/3} (b d-a g)+a^{4/3} b (b e-a h)\right )+\sqrt [3]{b} \left (-a b^{4/3} (b d-a g)+a^{4/3} b (b e-a h)\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{3 a^{2/3} b^{10/3}}-\frac {\left (\sqrt [3]{a} \left (\sqrt [3]{b} (b d-a g)-\sqrt [3]{a} (b e-a h)\right )\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 b^{7/3}}\\ &=\frac {(b d-a g) x}{b^2}+\frac {(b e-a h) x^2}{2 b^2}+\frac {f x^3}{3 b}+\frac {g x^4}{4 b}+\frac {h x^5}{5 b}-\frac {\sqrt [3]{a} \left (\sqrt [3]{b} (b d-a g)-\sqrt [3]{a} (b e-a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{8/3}}+\frac {(b c-a f) \log \left (a+b x^3\right )}{3 b^2}-\frac {\left (a^{2/3} \left (b^{4/3} d+\sqrt [3]{a} b e-a \sqrt [3]{b} g-a^{4/3} h\right )\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{2 b^{7/3}}+\frac {\left (\sqrt [3]{a} \left (\sqrt [3]{b} (b d-a g)-\sqrt [3]{a} (b e-a h)\right )\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}{6 b^{8/3}}\\ &=\frac {(b d-a g) x}{b^2}+\frac {(b e-a h) x^2}{2 b^2}+\frac {f x^3}{3 b}+\frac {g x^4}{4 b}+\frac {h x^5}{5 b}-\frac {\sqrt [3]{a} \left (\sqrt [3]{b} (b d-a g)-\sqrt [3]{a} (b e-a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{8/3}}+\frac {\sqrt [3]{a} \left (\sqrt [3]{b} (b d-a g)-\sqrt [3]{a} (b e-a h)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{8/3}}+\frac {(b c-a f) \log \left (a+b x^3\right )}{3 b^2}-\frac {\left (\sqrt [3]{a} \left (b^{4/3} d+\sqrt [3]{a} b e-a \sqrt [3]{b} g-a^{4/3} h\right )\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{b^{8/3}}\\ &=\frac {(b d-a g) x}{b^2}+\frac {(b e-a h) x^2}{2 b^2}+\frac {f x^3}{3 b}+\frac {g x^4}{4 b}+\frac {h x^5}{5 b}+\frac {\sqrt [3]{a} \left (b^{4/3} d+\sqrt [3]{a} b e-a \sqrt [3]{b} g-a^{4/3} h\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} b^{8/3}}-\frac {\sqrt [3]{a} \left (\sqrt [3]{b} (b d-a g)-\sqrt [3]{a} (b e-a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{8/3}}+\frac {\sqrt [3]{a} \left (\sqrt [3]{b} (b d-a g)-\sqrt [3]{a} (b e-a h)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{8/3}}+\frac {(b c-a f) \log \left (a+b x^3\right )}{3 b^2}\\ \end {align*}

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Mathematica [A]
time = 0.18, size = 290, normalized size = 0.99 \begin {gather*} \frac {60 b^{2/3} (b d-a g) x+30 b^{2/3} (b e-a h) x^2+20 b^{5/3} f x^3+15 b^{5/3} g x^4+12 b^{5/3} h x^5-20 \sqrt {3} \sqrt [3]{a} \left (-b^{4/3} d-\sqrt [3]{a} b e+a \sqrt [3]{b} g+a^{4/3} h\right ) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )+20 \sqrt [3]{a} \left (-b^{4/3} d+\sqrt [3]{a} b e+a \sqrt [3]{b} g-a^{4/3} h\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+10 \sqrt [3]{a} \left (b^{4/3} d-\sqrt [3]{a} b e-a \sqrt [3]{b} g+a^{4/3} h\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+20 b^{2/3} (b c-a f) \log \left (a+b x^3\right )}{60 b^{8/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(60*b^(2/3)*(b*d - a*g)*x + 30*b^(2/3)*(b*e - a*h)*x^2 + 20*b^(5/3)*f*x^3 + 15*b^(5/3)*g*x^4 + 12*b^(5/3)*h*x^
5 - 20*Sqrt[3]*a^(1/3)*(-(b^(4/3)*d) - a^(1/3)*b*e + a*b^(1/3)*g + a^(4/3)*h)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3
))/Sqrt[3]] + 20*a^(1/3)*(-(b^(4/3)*d) + a^(1/3)*b*e + a*b^(1/3)*g - a^(4/3)*h)*Log[a^(1/3) + b^(1/3)*x] + 10*
a^(1/3)*(b^(4/3)*d - a^(1/3)*b*e - a*b^(1/3)*g + a^(4/3)*h)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] + 2
0*b^(2/3)*(b*c - a*f)*Log[a + b*x^3])/(60*b^(8/3))

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Maple [A]
time = 0.37, size = 285, normalized size = 0.97

method	result	size
risch	\(\frac {h \,x^{5}}{5 b}+\frac {g \,x^{4}}{4 b}+\frac {f \,x^{3}}{3 b}-\frac {a h \,x^{2}}{2 b^{2}}+\frac {e \,x^{2}}{2 b}-\frac {a g x}{b^{2}}+\frac {d x}{b}+\frac {\munderset {\textit {\_R} =\RootOf \left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\left (b \left (-a f +b c \right ) \textit {\_R}^{2}+a \left (a h -b e \right ) \textit {\_R} +a^{2} g -a b d \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}}{3 b^{3}}\)	\(123\)
default	\(-\frac {-\frac {1}{5} b h \,x^{5}-\frac {1}{4} b g \,x^{4}-\frac {1}{3} f \,x^{3} b +\frac {1}{2} a h \,x^{2}-\frac {1}{2} b e \,x^{2}+a g x -x b d}{b^{2}}+\frac {\left (a^{2} g -a b d \right ) \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )+\left (a^{2} h -a b e \right ) \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )+\frac {\left (-a b f +b^{2} c \right ) \ln \left (b \,x^{3}+a \right )}{3 b}}{b^{2}}\)	\(285\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.49, size = 317, normalized size = 1.08 \begin {gather*} \frac {\sqrt {3} {\left (a^{2} h \left (\frac {a}{b}\right )^{\frac {2}{3}} - a b \left (\frac {a}{b}\right )^{\frac {2}{3}} e - a b d \left (\frac {a}{b}\right )^{\frac {1}{3}} + a^{2} g \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 )}{3 \, a b^{2}} + \frac {12 \, b h x^{5} + 15 \, b g x^{4} + 20 \, b f x^{3} - 30 \, {\left (a h - b e\right )} x^{2} + 60 \, {\left (b d - a g\right )} x}{60 \, b^{2}} + \frac {{\left (2 \, b^{2} c \left (\frac {a}{b}\right )^{\frac {2}{3}} - 2 \, a b f \left (\frac {a}{b}\right )^{\frac {2}{3}} + a^{2} h \left (\frac {a}{b}\right )^{\frac {1}{3}} - a b \left (\frac {a}{b}\right )^{\frac {1}{3}} e + a b d - a^{2} g\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (b^{2} c \left (\frac {a}{b}\right )^{\frac {2}{3}} - a b f \left (\frac {a}{b}\right )^{\frac {2}{3}} - a^{2} h \left (\frac {a}{b}\right )^{\frac {1}{3}} + a b \left (\frac {a}{b}\right )^{\frac {1}{3}} e - a b d + a^{2} g\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, 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^2*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^3+a),x, algorithm="maxima")

[Out]

1/3*sqrt(3)*(a^2*h*(a/b)^(2/3) - a*b*(a/b)^(2/3)*e - a*b*d*(a/b)^(1/3) + a^2*g*(a/b)^(1/3))*arctan(1/3*sqrt(3)
*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(a*b^2) + 1/60*(12*b*h*x^5 + 15*b*g*x^4 + 20*b*f*x^3 - 30*(a*h - b*e)*x^2 +
60*(b*d - a*g)*x)/b^2 + 1/6*(2*b^2*c*(a/b)^(2/3) - 2*a*b*f*(a/b)^(2/3) + a^2*h*(a/b)^(1/3) - a*b*(a/b)^(1/3)*e
+ a*b*d - a^2*g)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(b^3*(a/b)^(2/3)) + 1/3*(b^2*c*(a/b)^(2/3) - a*b*f*(a
/b)^(2/3) - a^2*h*(a/b)^(1/3) + a*b*(a/b)^(1/3)*e - a*b*d + a^2*g)*log(x + (a/b)^(1/3))/(b^3*(a/b)^(2/3))

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Fricas [C] Result contains complex when optimal does not.
time = 1.74, size = 14746, normalized size = 50.16 \begin {gather*} \text {too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(12*b*h*x^5 + 15*b*g*x^4 + 20*b*f*x^3 - 10*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*((b*c - a*f)^2/b^4 - (b^3*c^2
+ a^3*g*h + (f^2 - e*g - d*h)*a^2*b + (d*e - 2*c*f)*a*b^2)/b^5)/(2*(b*c - a*f)^3/b^6 - 3*(b^3*c^2 + a^3*g*h +
(f^2 - e*g - d*h)*a^2*b + (d*e - 2*c*f)*a*b^2)*(b*c - a*f)/b^7 - (b^4*d^3 + a*b^3*e^3 - 3*a*b^3*d^2*g + 3*a^2*
b^2*d*g^2 - a^3*b*g^3 - 3*a^2*b^2*e^2*h + 3*a^3*b*e*h^2 - a^4*h^3)*a/b^8 + (b^5*c^3 - a^5*h^3 + (g^3 - 3*f*g*h
+ 3*e*h^2)*a^4*b - (f^3 - 3*e*f*g + 3*e^2*h - 3*c*g*h + 3*(g^2 - f*h)*d)*a^3*b^2 + (e^3 - 3*d*e*f + 3*d^2*g +
3*(f^2 - e*g - d*h)*c)*a^2*b^3 - (d^3 - 3*c*d*e + 3*c^2*f)*a*b^4)/b^8)^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(2
*(b*c - a*f)^3/b^6 - 3*(b^3*c^2 + a^3*g*h + (f^2 - e*g - d*h)*a^2*b + (d*e - 2*c*f)*a*b^2)*(b*c - a*f)/b^7 - (
b^4*d^3 + a*b^3*e^3 - 3*a*b^3*d^2*g + 3*a^2*b^2*d*g^2 - a^3*b*g^3 - 3*a^2*b^2*e^2*h + 3*a^3*b*e*h^2 - a^4*h^3)
*a/b^8 + (b^5*c^3 - a^5*h^3 + (g^3 - 3*f*g*h + 3*e*h^2)*a^4*b - (f^3 - 3*e*f*g + 3*e^2*h - 3*c*g*h + 3*(g^2 -
f*h)*d)*a^ ...

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

[Out]

Timed out

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Giac [A]
time = 0.52, size = 333, normalized size = 1.13 \begin {gather*} \frac {{\left (b c - a f\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{2}} - \frac {\sqrt {3} {\left (\left (-a b^{2}\right )^{\frac {1}{3}} b^{2} d - \left (-a b^{2}\right )^{\frac {1}{3}} a b g + \left (-a b^{2}\right )^{\frac {2}{3}} a h - \left (-a b^{2}\right )^{\frac {2}{3}} b e\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 )}{3 \, b^{4}} - \frac {{\left (\left (-a b^{2}\right )^{\frac {1}{3}} b^{2} d - \left (-a b^{2}\right )^{\frac {1}{3}} a b g - \left (-a b^{2}\right )^{\frac {2}{3}} a h + \left (-a b^{2}\right )^{\frac {2}{3}} b e\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, b^{4}} + \frac {12 \, b^{4} h x^{5} + 15 \, b^{4} g x^{4} + 20 \, b^{4} f x^{3} - 30 \, a b^{3} h x^{2} + 30 \, b^{4} x^{2} e + 60 \, b^{4} d x - 60 \, a b^{3} g x}{60 \, b^{5}} - \frac {{\left (a^{2} b^{9} h \left (-\frac {a}{b}\right )^{\frac {1}{3}} - a b^{10} \left (-\frac {a}{b}\right )^{\frac {1}{3}} e - a b^{10} d + a^{2} b^{9} g\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a b^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(b*c - a*f)*log(abs(b*x^3 + a))/b^2 - 1/3*sqrt(3)*((-a*b^2)^(1/3)*b^2*d - (-a*b^2)^(1/3)*a*b*g + (-a*b^2)^
(2/3)*a*h - (-a*b^2)^(2/3)*b*e)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/b^4 - 1/6*((-a*b^2)^(1/3
)*b^2*d - (-a*b^2)^(1/3)*a*b*g - (-a*b^2)^(2/3)*a*h + (-a*b^2)^(2/3)*b*e)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2
/3))/b^4 + 1/60*(12*b^4*h*x^5 + 15*b^4*g*x^4 + 20*b^4*f*x^3 - 30*a*b^3*h*x^2 + 30*b^4*x^2*e + 60*b^4*d*x - 60*
a*b^3*g*x)/b^5 - 1/3*(a^2*b^9*h*(-a/b)^(1/3) - a*b^10*(-a/b)^(1/3)*e - a*b^10*d + a^2*b^9*g)*(-a/b)^(1/3)*log(
abs(x - (-a/b)^(1/3)))/(a*b^11)

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Mupad [B]
time = 5.02, size = 1170, normalized size = 3.98 \begin {gather*} x^2\,\left (\frac {e}{2\,b}-\frac {a\,h}{2\,b^2}\right )+\left (\sum _{k=1}^3\ln \left (\mathrm {root}\left (27\,b^8\,z^3+27\,a\,b^6\,f\,z^2-27\,b^7\,c\,z^2-18\,a\,b^5\,c\,f\,z+9\,a\,b^5\,d\,e\,z+9\,a^3\,b^3\,g\,h\,z-9\,a^2\,b^4\,e\,g\,z-9\,a^2\,b^4\,d\,h\,z+9\,a^2\,b^4\,f^2\,z+9\,b^6\,c^2\,z+3\,a^4\,b\,f\,g\,h-3\,a\,b^4\,c\,d\,e-3\,a^3\,b^2\,e\,f\,g-3\,a^3\,b^2\,d\,f\,h-3\,a^3\,b^2\,c\,g\,h+3\,a^2\,b^3\,d\,e\,f+3\,a^2\,b^3\,c\,e\,g+3\,a^2\,b^3\,c\,d\,h-3\,a^4\,b\,e\,h^2+3\,a\,b^4\,c^2\,f+3\,a^3\,b^2\,e^2\,h+3\,a^3\,b^2\,d\,g^2-3\,a^2\,b^3\,d^2\,g-3\,a^2\,b^3\,c\,f^2+a^3\,b^2\,f^3+a\,b^4\,d^3+a^5\,h^3-a^2\,b^3\,e^3-a^4\,b\,g^3-b^5\,c^3,z,k\right )\,\left (\frac {6\,a^2\,b^3\,f-6\,a\,b^4\,c}{b^3}+\frac {x\,\left (3\,a^2\,b^3\,g-3\,a\,b^4\,d\right )}{b^3}+\mathrm {root}\left (27\,b^8\,z^3+27\,a\,b^6\,f\,z^2-27\,b^7\,c\,z^2-18\,a\,b^5\,c\,f\,z+9\,a\,b^5\,d\,e\,z+9\,a^3\,b^3\,g\,h\,z-9\,a^2\,b^4\,e\,g\,z-9\,a^2\,b^4\,d\,h\,z+9\,a^2\,b^4\,f^2\,z+9\,b^6\,c^2\,z+3\,a^4\,b\,f\,g\,h-3\,a\,b^4\,c\,d\,e-3\,a^3\,b^2\,e\,f\,g-3\,a^3\,b^2\,d\,f\,h-3\,a^3\,b^2\,c\,g\,h+3\,a^2\,b^3\,d\,e\,f+3\,a^2\,b^3\,c\,e\,g+3\,a^2\,b^3\,c\,d\,h-3\,a^4\,b\,e\,h^2+3\,a\,b^4\,c^2\,f+3\,a^3\,b^2\,e^2\,h+3\,a^3\,b^2\,d\,g^2-3\,a^2\,b^3\,d^2\,g-3\,a^2\,b^3\,c\,f^2+a^3\,b^2\,f^3+a\,b^4\,d^3+a^5\,h^3-a^2\,b^3\,e^3-a^4\,b\,g^3-b^5\,c^3,z,k\right )\,a\,b^2\,9\right )+\frac {a\,b^3\,c^2+a^3\,b\,f^2+a^4\,g\,h-a^3\,b\,d\,h-a^3\,b\,e\,g-2\,a^2\,b^2\,c\,f+a^2\,b^2\,d\,e}{b^3}+\frac {x\,\left (a^4\,h^2+a^2\,b^2\,e^2+a\,b^3\,c\,d-2\,a^3\,b\,e\,h+a^3\,b\,f\,g-a^2\,b^2\,c\,g-a^2\,b^2\,d\,f\right )}{b^3}\right )\,\mathrm {root}\left (27\,b^8\,z^3+27\,a\,b^6\,f\,z^2-27\,b^7\,c\,z^2-18\,a\,b^5\,c\,f\,z+9\,a\,b^5\,d\,e\,z+9\,a^3\,b^3\,g\,h\,z-9\,a^2\,b^4\,e\,g\,z-9\,a^2\,b^4\,d\,h\,z+9\,a^2\,b^4\,f^2\,z+9\,b^6\,c^2\,z+3\,a^4\,b\,f\,g\,h-3\,a\,b^4\,c\,d\,e-3\,a^3\,b^2\,e\,f\,g-3\,a^3\,b^2\,d\,f\,h-3\,a^3\,b^2\,c\,g\,h+3\,a^2\,b^3\,d\,e\,f+3\,a^2\,b^3\,c\,e\,g+3\,a^2\,b^3\,c\,d\,h-3\,a^4\,b\,e\,h^2+3\,a\,b^4\,c^2\,f+3\,a^3\,b^2\,e^2\,h+3\,a^3\,b^2\,d\,g^2-3\,a^2\,b^3\,d^2\,g-3\,a^2\,b^3\,c\,f^2+a^3\,b^2\,f^3+a\,b^4\,d^3+a^5\,h^3-a^2\,b^3\,e^3-a^4\,b\,g^3-b^5\,c^3,z,k\right )\right )+x\,\left (\frac {d}{b}-\frac {a\,g}{b^2}\right )+\frac {f\,x^3}{3\,b}+\frac {g\,x^4}{4\,b}+\frac {h\,x^5}{5\,b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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