3.5.6 \(\int \frac {x (c+d x+e x^2+f x^3+g x^4+h x^5)}{a+b x^3} \, dx\) [406]

Optimal. Leaf size=275 \[ \frac {(b e-a h) x}{b^2}+\frac {f x^2}{2 b}+\frac {g x^3}{3 b}+\frac {h x^4}{4 b}-\frac {\left (b^{5/3} c-a^{2/3} b e-a b^{2/3} f+a^{5/3} h\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} \sqrt [3]{a} b^{7/3}}-\frac {\left (b^{2/3} (b c-a f)+a^{2/3} (b e-a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{7/3}}+\frac {\left (b^{2/3} (b c-a f)+a^{2/3} (b e-a h)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 \sqrt [3]{a} b^{7/3}}+\frac {(b d-a g) \log \left (a+b x^3\right )}{3 b^2} \]

[Out]

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

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Rubi [A]
time = 0.60, antiderivative size = 275, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {1850, 1901, 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 (-a^{2/3} b e+a^{5/3} h-a b^{2/3} f+b^{5/3} c\right )}{\sqrt {3} \sqrt [3]{a} b^{7/3}}+\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-a h)+b^{2/3} (b c-a f)\right )}{6 \sqrt [3]{a} b^{7/3}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^{2/3} (b e-a h)+b^{2/3} (b c-a f)\right )}{3 \sqrt [3]{a} b^{7/3}}+\frac {(b d-a g) \log \left (a+b x^3\right )}{3 b^2}+\frac {x (b e-a h)}{b^2}+\frac {f x^2}{2 b}+\frac {g x^3}{3 b}+\frac {h x^4}{4 b} \end {gather*}

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),x]

[Out]

((b*e - a*h)*x)/b^2 + (f*x^2)/(2*b) + (g*x^3)/(3*b) + (h*x^4)/(4*b) - ((b^(5/3)*c - a^(2/3)*b*e - a*b^(2/3)*f
+ a^(5/3)*h)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(1/3)*b^(7/3)) - ((b^(2/3)*(b*c - a
*f) + a^(2/3)*(b*e - a*h))*Log[a^(1/3) + b^(1/3)*x])/(3*a^(1/3)*b^(7/3)) + ((b^(2/3)*(b*c - a*f) + a^(2/3)*(b*
e - a*h))*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*a^(1/3)*b^(7/3)) + ((b*d - a*g)*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 \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^4}{4 b}+\frac {\int \frac {x \left (4 b c+4 b d x+4 (b e-a h) x^2+4 b f x^3+4 b g x^4\right )}{a+b x^3} \, dx}{4 b}\\ &=\frac {g x^3}{3 b}+\frac {h x^4}{4 b}+\frac {\int \frac {x \left (12 b^2 c+12 b (b d-a g) x+12 b (b e-a h) x^2+12 b^2 f x^3\right )}{a+b x^3} \, dx}{12 b^2}\\ &=\frac {f x^2}{2 b}+\frac {g x^3}{3 b}+\frac {h x^4}{4 b}+\frac {\int \frac {x \left (24 b^2 (b c-a f)+24 b^2 (b d-a g) x+24 b^2 (b e-a h) x^2\right )}{a+b x^3} \, dx}{24 b^3}\\ &=\frac {f x^2}{2 b}+\frac {g x^3}{3 b}+\frac {h x^4}{4 b}+\frac {\int \left (24 b (b e-a h)-\frac {24 \left (a b (b e-a h)-b^2 (b c-a f) x-b^2 (b d-a g) x^2\right )}{a+b x^3}\right ) \, dx}{24 b^3}\\ &=\frac {(b e-a h) x}{b^2}+\frac {f x^2}{2 b}+\frac {g x^3}{3 b}+\frac {h x^4}{4 b}-\frac {\int \frac {a b (b e-a h)-b^2 (b c-a f) x-b^2 (b d-a g) x^2}{a+b x^3} \, dx}{b^3}\\ &=\frac {(b e-a h) x}{b^2}+\frac {f x^2}{2 b}+\frac {g x^3}{3 b}+\frac {h x^4}{4 b}-\frac {\int \frac {a b (b e-a h)-b^2 (b c-a f) x}{a+b x^3} \, dx}{b^3}+\frac {(b d-a g) \int \frac {x^2}{a+b x^3} \, dx}{b}\\ &=\frac {(b e-a h) x}{b^2}+\frac {f x^2}{2 b}+\frac {g x^3}{3 b}+\frac {h x^4}{4 b}+\frac {(b d-a g) \log \left (a+b x^3\right )}{3 b^2}-\frac {\int \frac {\sqrt [3]{a} \left (-\sqrt [3]{a} b^2 (b c-a f)+2 a b^{4/3} (b e-a h)\right )+\sqrt [3]{b} \left (-\sqrt [3]{a} b^2 (b c-a f)-a b^{4/3} (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 (b^{2/3} (b c-a f)+a^{2/3} (b e-a h)\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 \sqrt [3]{a} b^2}\\ &=\frac {(b e-a h) x}{b^2}+\frac {f x^2}{2 b}+\frac {g x^3}{3 b}+\frac {h x^4}{4 b}-\frac {\left (b^{2/3} (b c-a f)+a^{2/3} (b e-a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{7/3}}+\frac {(b d-a g) \log \left (a+b x^3\right )}{3 b^2}+\frac {\left (b^{5/3} c-a^{2/3} b e-a b^{2/3} f+a^{5/3} h\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{2 b^2}+\frac {\left (b^{2/3} (b c-a f)+a^{2/3} (b e-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}{6 \sqrt [3]{a} b^{7/3}}\\ &=\frac {(b e-a h) x}{b^2}+\frac {f x^2}{2 b}+\frac {g x^3}{3 b}+\frac {h x^4}{4 b}-\frac {\left (b^{2/3} (b c-a f)+a^{2/3} (b e-a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{7/3}}+\frac {\left (b^{2/3} (b c-a f)+a^{2/3} (b e-a h)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 \sqrt [3]{a} b^{7/3}}+\frac {(b d-a g) \log \left (a+b x^3\right )}{3 b^2}+\frac {\left (b^{5/3} c-a^{2/3} b e-a b^{2/3} f+a^{5/3} h\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{\sqrt [3]{a} b^{7/3}}\\ &=\frac {(b e-a h) x}{b^2}+\frac {f x^2}{2 b}+\frac {g x^3}{3 b}+\frac {h x^4}{4 b}-\frac {\left (b^{5/3} c-a^{2/3} b e-a b^{2/3} f+a^{5/3} h\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} \sqrt [3]{a} b^{7/3}}-\frac {\left (b^{2/3} (b c-a f)+a^{2/3} (b e-a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{7/3}}+\frac {\left (b^{2/3} (b c-a f)+a^{2/3} (b e-a h)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 \sqrt [3]{a} b^{7/3}}+\frac {(b d-a g) \log \left (a+b x^3\right )}{3 b^2}\\ \end {align*}

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Mathematica [A]
time = 0.26, size = 272, normalized size = 0.99 \begin {gather*} \frac {12 \sqrt [3]{b} (b e-a h) x+6 b^{4/3} f x^2+4 b^{4/3} g x^3+3 b^{4/3} h x^4-\frac {4 \sqrt {3} \left (b^{5/3} c-a^{2/3} b e-a b^{2/3} f+a^{5/3} h\right ) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{a}}+\frac {4 \left (-b^{5/3} c-a^{2/3} b e+a b^{2/3} f+a^{5/3} h\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a}}+\frac {2 \left (b^{5/3} c+a^{2/3} b e-a b^{2/3} f-a^{5/3} h\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{\sqrt [3]{a}}+4 \sqrt [3]{b} (b d-a g) \log \left (a+b x^3\right )}{12 b^{7/3}} \end {gather*}

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),x]

[Out]

(12*b^(1/3)*(b*e - a*h)*x + 6*b^(4/3)*f*x^2 + 4*b^(4/3)*g*x^3 + 3*b^(4/3)*h*x^4 - (4*Sqrt[3]*(b^(5/3)*c - a^(2
/3)*b*e - a*b^(2/3)*f + a^(5/3)*h)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(1/3) + (4*(-(b^(5/3)*c) - a
^(2/3)*b*e + a*b^(2/3)*f + a^(5/3)*h)*Log[a^(1/3) + b^(1/3)*x])/a^(1/3) + (2*(b^(5/3)*c + a^(2/3)*b*e - a*b^(2
/3)*f - a^(5/3)*h)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(1/3) + 4*b^(1/3)*(b*d - a*g)*Log[a + b*x
^3])/(12*b^(7/3))

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Maple [A]
time = 0.37, size = 271, normalized size = 0.99

method	result	size
risch	\(\frac {h \,x^{4}}{4 b}+\frac {g \,x^{3}}{3 b}+\frac {f \,x^{2}}{2 b}-\frac {a h x}{b^{2}}+\frac {e x}{b}+\frac {\munderset {\textit {\_R} =\RootOf \left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\left (b \left (-a g +b d \right ) \textit {\_R}^{2}+b \left (-a f +b c \right ) \textit {\_R} +a^{2} h -a b e \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}}{3 b^{3}}\)	\(104\)
default	\(-\frac {-\frac {1}{4} b h \,x^{4}-\frac {1}{3} b g \,x^{3}-\frac {1}{2} b f \,x^{2}+a h x -b e x}{b^{2}}+\frac {\left (a^{2} h -a b e \right ) \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )+\left (-a b f +b^{2} c \right ) \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )+\frac {\left (-a b g +b^{2} d \right ) \ln \left (b \,x^{3}+a \right )}{3 b}}{b^{2}}\)	\(271\)

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),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.51, size = 304, normalized size = 1.11 \begin {gather*} \frac {\sqrt {3} {\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\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 {3 \, b h x^{4} + 4 \, b g x^{3} + 6 \, b f x^{2} - 12 \, {\left (a h - b e\right )} x}{12 \, b^{2}} + \frac {{\left (2 \, b^{2} d \left (\frac {a}{b}\right )^{\frac {2}{3}} - 2 \, a b g \left (\frac {a}{b}\right )^{\frac {2}{3}} + b^{2} c \left (\frac {a}{b}\right )^{\frac {1}{3}} - a b f \left (\frac {a}{b}\right )^{\frac {1}{3}} - a^{2} h + a 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^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (b^{2} d \left (\frac {a}{b}\right )^{\frac {2}{3}} - a b g \left (\frac {a}{b}\right )^{\frac {2}{3}} - b^{2} c \left (\frac {a}{b}\right )^{\frac {1}{3}} + a b f \left (\frac {a}{b}\right )^{\frac {1}{3}} + a^{2} h - a b e\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*(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)*(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)*arctan(1/3*sqrt(3)
*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(a*b^2) + 1/12*(3*b*h*x^4 + 4*b*g*x^3 + 6*b*f*x^2 - 12*(a*h - b*e)*x)/b^2 +
1/6*(2*b^2*d*(a/b)^(2/3) - 2*a*b*g*(a/b)^(2/3) + b^2*c*(a/b)^(1/3) - a*b*f*(a/b)^(1/3) - a^2*h + a*b*e)*log(x^
2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(b^3*(a/b)^(2/3)) + 1/3*(b^2*d*(a/b)^(2/3) - a*b*g*(a/b)^(2/3) - b^2*c*(a/b)^
(1/3) + a*b*f*(a/b)^(1/3) + a^2*h - a*b*e)*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 = 1.72, size = 14875, normalized size = 54.09 \begin {gather*} \text {too large to display} \end {gather*}

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),x, algorithm="fricas")

[Out]

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

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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*(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 = 0.50, size = 295, normalized size = 1.07 \begin {gather*} -\frac {\sqrt {3} {\left (a^{2} h - a b e - \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 )}{3 \, \left (-a b^{2}\right )^{\frac {2}{3}} b} - \frac {{\left (a^{2} h - a b e + \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 )}{6 \, \left (-a b^{2}\right )^{\frac {2}{3}} b} + \frac {{\left (b d - a g\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{2}} + \frac {3 \, b^{3} h x^{4} + 4 \, b^{3} g x^{3} + 6 \, b^{3} f x^{2} - 12 \, a b^{2} h x + 12 \, b^{3} x e}{12 \, b^{4}} - \frac {{\left (b^{9} c \left (-\frac {a}{b}\right )^{\frac {1}{3}} - a b^{8} f \left (-\frac {a}{b}\right )^{\frac {1}{3}} + a^{2} b^{7} h - a b^{8} e\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^{9}} \end {gather*}

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),x, algorithm="giac")

[Out]

-1/3*sqrt(3)*(a^2*h - a*b*e - (-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)*b) - 1/6*(a^2*h - a*b*e + (-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)*b) + 1/3*(b*d - a*g)*log(abs(b*x^3 + a))/b^2 + 1/12*(3*b^3*h*x^4
+ 4*b^3*g*x^3 + 6*b^3*f*x^2 - 12*a*b^2*h*x + 12*b^3*x*e)/b^4 - 1/3*(b^9*c*(-a/b)^(1/3) - a*b^8*f*(-a/b)^(1/3)
+ a^2*b^7*h - a*b^8*e)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a*b^9)

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Mupad [B]
time = 4.99, size = 1161, normalized size = 4.22 \begin {gather*} \left (\sum _{k=1}^3\ln \left (\mathrm {root}\left (27\,a\,b^7\,z^3-27\,a\,b^6\,d\,z^2+27\,a^2\,b^5\,g\,z^2-9\,a\,b^5\,c\,e\,z-9\,a^3\,b^3\,f\,h\,z-18\,a^2\,b^4\,d\,g\,z+9\,a^2\,b^4\,e\,f\,z+9\,a^2\,b^4\,c\,h\,z+9\,a\,b^5\,d^2\,z+9\,a^3\,b^3\,g^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^2\,b^3\,e^3+a^4\,b\,g^3+b^5\,c^3-a^3\,b^2\,f^3-a\,b^4\,d^3-a^5\,h^3,z,k\right )\,\left (\frac {6\,a^2\,b^2\,g-6\,a\,b^3\,d}{b^2}+\frac {x\,\left (3\,a^2\,b^2\,h-3\,a\,b^3\,e\right )}{b^2}+\mathrm {root}\left (27\,a\,b^7\,z^3-27\,a\,b^6\,d\,z^2+27\,a^2\,b^5\,g\,z^2-9\,a\,b^5\,c\,e\,z-9\,a^3\,b^3\,f\,h\,z-18\,a^2\,b^4\,d\,g\,z+9\,a^2\,b^4\,e\,f\,z+9\,a^2\,b^4\,c\,h\,z+9\,a\,b^5\,d^2\,z+9\,a^3\,b^3\,g^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^2\,b^3\,e^3+a^4\,b\,g^3+b^5\,c^3-a^3\,b^2\,f^3-a\,b^4\,d^3-a^5\,h^3,z,k\right )\,a\,b^2\,9\right )+\frac {a^3\,g^2+a\,b^2\,d^2-a^3\,f\,h-a\,b^2\,c\,e+a^2\,b\,c\,h-2\,a^2\,b\,d\,g+a^2\,b\,e\,f}{b^2}+\frac {x\,\left (b^3\,c^2+a^2\,b\,f^2+a^3\,g\,h-2\,a\,b^2\,c\,f+a\,b^2\,d\,e-a^2\,b\,d\,h-a^2\,b\,e\,g\right )}{b^2}\right )\,\mathrm {root}\left (27\,a\,b^7\,z^3-27\,a\,b^6\,d\,z^2+27\,a^2\,b^5\,g\,z^2-9\,a\,b^5\,c\,e\,z-9\,a^3\,b^3\,f\,h\,z-18\,a^2\,b^4\,d\,g\,z+9\,a^2\,b^4\,e\,f\,z+9\,a^2\,b^4\,c\,h\,z+9\,a\,b^5\,d^2\,z+9\,a^3\,b^3\,g^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^2\,b^3\,e^3+a^4\,b\,g^3+b^5\,c^3-a^3\,b^2\,f^3-a\,b^4\,d^3-a^5\,h^3,z,k\right )\right )+x\,\left (\frac {e}{b}-\frac {a\,h}{b^2}\right )+\frac {f\,x^2}{2\,b}+\frac {g\,x^3}{3\,b}+\frac {h\,x^4}{4\,b} \end {gather*}

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),x)

[Out]

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

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