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

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

[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 = 0.92, 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 = {1836, 1887, 1871, 1860, 31, 634, 617, 204, 628, 260} \[ \frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^{2/3} (b e-a h)+b^{2/3} (b c-a f)\right )}{6 \sqrt [3]{a} b^{7/3}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^{2/3} (b e-a h)+b^{2/3} (b c-a f)\right )}{3 \sqrt [3]{a} b^{7/3}}-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (-a^{2/3} b e+a^{5/3} h-a b^{2/3} f+b^{5/3} c\right )}{\sqrt {3} \sqrt [3]{a} b^{7/3}}+\frac {(b d-a g) \log \left (a+b x^3\right )}{3 b^2}+\frac {x (b e-a h)}{b^2}+\frac {f x^2}{2 b}+\frac {g x^3}{3 b}+\frac {h x^4}{4 b} \]

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 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 1836

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 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 \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 ) \operatorname {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 = 0.48, size = 272, normalized size = 0.99 \[ \frac {\frac {2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^{2/3} b e+a^{5/3} (-h)-a b^{2/3} f+b^{5/3} c\right )}{\sqrt [3]{a}}+\frac {4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-a^{2/3} b e+a^{5/3} h+a b^{2/3} f-b^{5/3} c\right )}{\sqrt [3]{a}}-\frac {4 \sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right ) \left (-a^{2/3} b e+a^{5/3} h-a b^{2/3} f+b^{5/3} c\right )}{\sqrt [3]{a}}+4 \sqrt [3]{b} (b d-a g) \log \left (a+b x^3\right )+12 \sqrt [3]{b} x (b e-a h)+6 b^{4/3} f x^2+4 b^{4/3} g x^3+3 b^{4/3} h x^4}{12 b^{7/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Timed out

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giac [A]  time = 0.28, size = 295, normalized size = 1.07 \[ -\frac {\sqrt {3} {\left (a^{2} h - a b e - \left (-a b^{2}\right )^{\frac {1}{3}} b c + \left (-a b^{2}\right )^{\frac {1}{3}} a f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, \left (-a b^{2}\right )^{\frac {2}{3}} b} - \frac {{\left (a^{2} h - a b e + \left (-a b^{2}\right )^{\frac {1}{3}} b c - \left (-a b^{2}\right )^{\frac {1}{3}} a f\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, \left (-a b^{2}\right )^{\frac {2}{3}} b} + \frac {{\left (b d - a g\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{2}} + \frac {3 \, b^{3} h x^{4} + 4 \, b^{3} g x^{3} + 6 \, b^{3} f x^{2} - 12 \, a b^{2} h x + 12 \, b^{3} x e}{12 \, b^{4}} - \frac {{\left (b^{9} c \left (-\frac {a}{b}\right )^{\frac {1}{3}} - a b^{8} f \left (-\frac {a}{b}\right )^{\frac {1}{3}} + a^{2} b^{7} h - a b^{8} e\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a b^{9}} \]

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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maple [B]  time = 0.04, size = 455, normalized size = 1.65 \[ \frac {h \,x^{4}}{4 b}+\frac {g \,x^{3}}{3 b}+\frac {f \,x^{2}}{2 b}+\frac {\sqrt {3}\, a^{2} h \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{3}}+\frac {a^{2} h \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{3}}-\frac {a^{2} h \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{3}}-\frac {\sqrt {3}\, a e \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{2}}-\frac {a e \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{2}}+\frac {a e \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{2}}-\frac {\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 )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{2}}+\frac {a f \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{2}}-\frac {a f \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{2}}-\frac {a g \ln \left (b \,x^{3}+a \right )}{3 b^{2}}-\frac {a h x}{b^{2}}+\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 )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}} b}-\frac {c \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}} b}+\frac {c \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \left (\frac {a}{b}\right )^{\frac {1}{3}} b}+\frac {d \ln \left (b \,x^{3}+a \right )}{3 b}+\frac {e x}{b} \]

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)

[Out]

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

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maxima [A]  time = 3.03, size = 300, normalized size = 1.09 \[ \frac {\sqrt {3} {\left (b^{2} c \left (\frac {a}{b}\right )^{\frac {2}{3}} - a b f \left (\frac {a}{b}\right )^{\frac {2}{3}} - a b e \left (\frac {a}{b}\right )^{\frac {1}{3}} + a^{2} h \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a b^{2}} + \frac {3 \, b h x^{4} + 4 \, b g x^{3} + 6 \, b f x^{2} + 12 \, {\left (b e - a h\right )} x}{12 \, b^{2}} + \frac {{\left (2 \, b^{2} d \left (\frac {a}{b}\right )^{\frac {2}{3}} - 2 \, a b g \left (\frac {a}{b}\right )^{\frac {2}{3}} + b^{2} c \left (\frac {a}{b}\right )^{\frac {1}{3}} - a b f \left (\frac {a}{b}\right )^{\frac {1}{3}} + a b e - a^{2} h\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (b^{2} d \left (\frac {a}{b}\right )^{\frac {2}{3}} - a b g \left (\frac {a}{b}\right )^{\frac {2}{3}} - b^{2} c \left (\frac {a}{b}\right )^{\frac {1}{3}} + a b f \left (\frac {a}{b}\right )^{\frac {1}{3}} - a b e + a^{2} h\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*sqrt(3)*(b^2*c*(a/b)^(2/3) - a*b*f*(a/b)^(2/3) - a*b*e*(a/b)^(1/3) + a^2*h*(a/b)^(1/3))*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*(b*e - a*h)*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*b*e - a^2*h)*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*b*e + a^2*h)*log(x + (a/b)^(1/3))/(b^3*(a/b)^(2/3))

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mupad [B]  time = 4.99, size = 1161, normalized size = 4.22 \[ \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} \]

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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sympy [B]  time = 63.00, size = 811, normalized size = 2.95 \[ x \left (- \frac {a h}{b^{2}} + \frac {e}{b}\right ) + \operatorname {RootSum} {\left (27 t^{3} a b^{7} + t^{2} \left (27 a^{2} b^{5} g - 27 a b^{6} d\right ) + t \left (- 9 a^{3} b^{3} f h + 9 a^{3} b^{3} g^{2} + 9 a^{2} b^{4} c h - 18 a^{2} b^{4} d g + 9 a^{2} b^{4} e f - 9 a b^{5} c e + 9 a b^{5} d^{2}\right ) - a^{5} h^{3} + 3 a^{4} b e h^{2} - 3 a^{4} b f g h + a^{4} b g^{3} + 3 a^{3} b^{2} c g h + 3 a^{3} b^{2} d f h - 3 a^{3} b^{2} d g^{2} - 3 a^{3} b^{2} e^{2} h + 3 a^{3} b^{2} e f g - a^{3} b^{2} f^{3} - 3 a^{2} b^{3} c d h - 3 a^{2} b^{3} c e g + 3 a^{2} b^{3} c f^{2} + 3 a^{2} b^{3} d^{2} g - 3 a^{2} b^{3} d e f + a^{2} b^{3} e^{3} - 3 a b^{4} c^{2} f + 3 a b^{4} c d e - a b^{4} d^{3} + b^{5} c^{3}, \left (t \mapsto t \log {\left (x + \frac {- 9 t^{2} a^{2} b^{5} f + 9 t^{2} a b^{6} c + 3 t a^{4} b^{2} h^{2} - 6 t a^{3} b^{3} e h - 6 t a^{3} b^{3} f g + 6 t a^{2} b^{4} c g + 6 t a^{2} b^{4} d f + 3 t a^{2} b^{4} e^{2} - 6 t a b^{5} c d + a^{5} g h^{2} - a^{4} b d h^{2} - 2 a^{4} b e g h + 2 a^{4} b f^{2} h - a^{4} b f g^{2} - 4 a^{3} b^{2} c f h + a^{3} b^{2} c g^{2} + 2 a^{3} b^{2} d e h + 2 a^{3} b^{2} d f g + a^{3} b^{2} e^{2} g - 2 a^{3} b^{2} e f^{2} + 2 a^{2} b^{3} c^{2} h - 2 a^{2} b^{3} c d g + 4 a^{2} b^{3} c e f - a^{2} b^{3} d^{2} f - a^{2} b^{3} d e^{2} - 2 a b^{4} c^{2} e + a b^{4} c d^{2}}{a^{5} h^{3} - 3 a^{4} b e h^{2} + 3 a^{3} b^{2} e^{2} h - a^{3} b^{2} f^{3} + 3 a^{2} b^{3} c f^{2} - a^{2} b^{3} e^{3} - 3 a b^{4} c^{2} f + b^{5} c^{3}} \right )} \right )\right )} + \frac {f x^{2}}{2 b} + \frac {g x^{3}}{3 b} + \frac {h x^{4}}{4 b} \]

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]

x*(-a*h/b**2 + e/b) + RootSum(27*_t**3*a*b**7 + _t**2*(27*a**2*b**5*g - 27*a*b**6*d) + _t*(-9*a**3*b**3*f*h +
9*a**3*b**3*g**2 + 9*a**2*b**4*c*h - 18*a**2*b**4*d*g + 9*a**2*b**4*e*f - 9*a*b**5*c*e + 9*a*b**5*d**2) - a**5
*h**3 + 3*a**4*b*e*h**2 - 3*a**4*b*f*g*h + a**4*b*g**3 + 3*a**3*b**2*c*g*h + 3*a**3*b**2*d*f*h - 3*a**3*b**2*d
*g**2 - 3*a**3*b**2*e**2*h + 3*a**3*b**2*e*f*g - a**3*b**2*f**3 - 3*a**2*b**3*c*d*h - 3*a**2*b**3*c*e*g + 3*a*
*2*b**3*c*f**2 + 3*a**2*b**3*d**2*g - 3*a**2*b**3*d*e*f + a**2*b**3*e**3 - 3*a*b**4*c**2*f + 3*a*b**4*c*d*e -
a*b**4*d**3 + b**5*c**3, Lambda(_t, _t*log(x + (-9*_t**2*a**2*b**5*f + 9*_t**2*a*b**6*c + 3*_t*a**4*b**2*h**2
- 6*_t*a**3*b**3*e*h - 6*_t*a**3*b**3*f*g + 6*_t*a**2*b**4*c*g + 6*_t*a**2*b**4*d*f + 3*_t*a**2*b**4*e**2 - 6*
_t*a*b**5*c*d + a**5*g*h**2 - a**4*b*d*h**2 - 2*a**4*b*e*g*h + 2*a**4*b*f**2*h - a**4*b*f*g**2 - 4*a**3*b**2*c
*f*h + a**3*b**2*c*g**2 + 2*a**3*b**2*d*e*h + 2*a**3*b**2*d*f*g + a**3*b**2*e**2*g - 2*a**3*b**2*e*f**2 + 2*a*
*2*b**3*c**2*h - 2*a**2*b**3*c*d*g + 4*a**2*b**3*c*e*f - a**2*b**3*d**2*f - a**2*b**3*d*e**2 - 2*a*b**4*c**2*e
+ a*b**4*c*d**2)/(a**5*h**3 - 3*a**4*b*e*h**2 + 3*a**3*b**2*e**2*h - a**3*b**2*f**3 + 3*a**2*b**3*c*f**2 - a*
*2*b**3*e**3 - 3*a*b**4*c**2*f + b**5*c**3)))) + f*x**2/(2*b) + g*x**3/(3*b) + h*x**4/(4*b)

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