3.5.3 \(\int \frac {x^4 (c+d x+e x^2+f x^3+g x^4+h x^5)}{a+b x^3} \, dx\) [403]

Optimal. Leaf size=331 \[ -\frac {a (b e-a h) x}{b^3}+\frac {(b c-a f) x^2}{2 b^2}+\frac {(b d-a g) x^3}{3 b^2}+\frac {(b e-a h) x^4}{4 b^2}+\frac {f x^5}{5 b}+\frac {g x^6}{6 b}+\frac {h x^7}{7 b}+\frac {a^{2/3} \left (b^{5/3} c-a^{2/3} b e-a b^{2/3} f+a^{5/3} h\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} b^{10/3}}+\frac {a^{2/3} \left (b^{2/3} (b c-a f)+a^{2/3} (b e-a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{10/3}}-\frac {a^{2/3} \left (b^{2/3} (b c-a f)+a^{2/3} (b e-a h)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{10/3}}-\frac {a (b d-a g) \log \left (a+b x^3\right )}{3 b^3} \]

[Out]

-a*(-a*h+b*e)*x/b^3+1/2*(-a*f+b*c)*x^2/b^2+1/3*(-a*g+b*d)*x^3/b^2+1/4*(-a*h+b*e)*x^4/b^2+1/5*f*x^5/b+1/6*g*x^6
/b+1/7*h*x^7/b+1/3*a^(2/3)*(b^(2/3)*(-a*f+b*c)+a^(2/3)*(-a*h+b*e))*ln(a^(1/3)+b^(1/3)*x)/b^(10/3)-1/6*a^(2/3)*
(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)/b^(10/3)-1/3*a*(-a*g+b*d)*ln
(b*x^3+a)/b^3+1/3*a^(2/3)*(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))/b^(10/3)*3^(1/2)

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Rubi [A]
time = 0.68, antiderivative size = 331, 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 {a^{2/3} \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} b^{10/3}}-\frac {a^{2/3} \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 b^{10/3}}+\frac {a^{2/3} \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 b^{10/3}}-\frac {a (b d-a g) \log \left (a+b x^3\right )}{3 b^3}-\frac {a x (b e-a h)}{b^3}+\frac {x^2 (b c-a f)}{2 b^2}+\frac {x^3 (b d-a g)}{3 b^2}+\frac {x^4 (b e-a h)}{4 b^2}+\frac {f x^5}{5 b}+\frac {g x^6}{6 b}+\frac {h x^7}{7 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((a*(b*e - a*h)*x)/b^3) + ((b*c - a*f)*x^2)/(2*b^2) + ((b*d - a*g)*x^3)/(3*b^2) + ((b*e - a*h)*x^4)/(4*b^2) +
(f*x^5)/(5*b) + (g*x^6)/(6*b) + (h*x^7)/(7*b) + (a^(2/3)*(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]*b^(10/3)) + (a^(2/3)*(b^(2/3)*(b*c - a*f) + a^(2/3
)*(b*e - a*h))*Log[a^(1/3) + b^(1/3)*x])/(3*b^(10/3)) - (a^(2/3)*(b^(2/3)*(b*c - a*f) + a^(2/3)*(b*e - a*h))*L
og[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*b^(10/3)) - (a*(b*d - a*g)*Log[a + b*x^3])/(3*b^3)

Rule 31

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

Rule 210

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

Rule 266

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

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 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^4 \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^7}{7 b}+\frac {\int \frac {x^4 \left (7 b c+7 b d x+7 (b e-a h) x^2+7 b f x^3+7 b g x^4\right )}{a+b x^3} \, dx}{7 b}\\ &=\frac {g x^6}{6 b}+\frac {h x^7}{7 b}+\frac {\int \frac {x^4 \left (42 b^2 c+42 b (b d-a g) x+42 b (b e-a h) x^2+42 b^2 f x^3\right )}{a+b x^3} \, dx}{42 b^2}\\ &=\frac {f x^5}{5 b}+\frac {g x^6}{6 b}+\frac {h x^7}{7 b}+\frac {\int \frac {x^4 \left (210 b^2 (b c-a f)+210 b^2 (b d-a g) x+210 b^2 (b e-a h) x^2\right )}{a+b x^3} \, dx}{210 b^3}\\ &=\frac {f x^5}{5 b}+\frac {g x^6}{6 b}+\frac {h x^7}{7 b}+\frac {\int \left (-210 a (b e-a h)+210 b (b c-a f) x+210 b (b d-a g) x^2+210 b (b e-a h) x^3+\frac {210 \left (a^2 (b e-a h)-a b (b c-a f) x-a b (b d-a g) x^2\right )}{a+b x^3}\right ) \, dx}{210 b^3}\\ &=-\frac {a (b e-a h) x}{b^3}+\frac {(b c-a f) x^2}{2 b^2}+\frac {(b d-a g) x^3}{3 b^2}+\frac {(b e-a h) x^4}{4 b^2}+\frac {f x^5}{5 b}+\frac {g x^6}{6 b}+\frac {h x^7}{7 b}+\frac {\int \frac {a^2 (b e-a h)-a b (b c-a f) x-a b (b d-a g) x^2}{a+b x^3} \, dx}{b^3}\\ &=-\frac {a (b e-a h) x}{b^3}+\frac {(b c-a f) x^2}{2 b^2}+\frac {(b d-a g) x^3}{3 b^2}+\frac {(b e-a h) x^4}{4 b^2}+\frac {f x^5}{5 b}+\frac {g x^6}{6 b}+\frac {h x^7}{7 b}+\frac {\int \frac {a^2 (b e-a h)-a b (b c-a f) x}{a+b x^3} \, dx}{b^3}-\frac {(a (b d-a g)) \int \frac {x^2}{a+b x^3} \, dx}{b^2}\\ &=-\frac {a (b e-a h) x}{b^3}+\frac {(b c-a f) x^2}{2 b^2}+\frac {(b d-a g) x^3}{3 b^2}+\frac {(b e-a h) x^4}{4 b^2}+\frac {f x^5}{5 b}+\frac {g x^6}{6 b}+\frac {h x^7}{7 b}-\frac {a (b d-a g) \log \left (a+b x^3\right )}{3 b^3}+\frac {\int \frac {\sqrt [3]{a} \left (-a^{4/3} b (b c-a f)+2 a^2 \sqrt [3]{b} (b e-a h)\right )+\sqrt [3]{b} \left (-a^{4/3} b (b c-a f)-a^2 \sqrt [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 (a^{2/3} \left (b^{2/3} (b c-a f)+a^{2/3} (b e-a h)\right )\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 b^3}\\ &=-\frac {a (b e-a h) x}{b^3}+\frac {(b c-a f) x^2}{2 b^2}+\frac {(b d-a g) x^3}{3 b^2}+\frac {(b e-a h) x^4}{4 b^2}+\frac {f x^5}{5 b}+\frac {g x^6}{6 b}+\frac {h x^7}{7 b}+\frac {a^{2/3} \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 b^{10/3}}-\frac {a (b d-a g) \log \left (a+b x^3\right )}{3 b^3}-\frac {\left (a \left (b^{5/3} c-a^{2/3} b e-a b^{2/3} f+a^{5/3} h\right )\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{2 b^3}-\frac {\left (a^{2/3} \left (b^{2/3} (b c-a f)+a^{2/3} (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^{10/3}}\\ &=-\frac {a (b e-a h) x}{b^3}+\frac {(b c-a f) x^2}{2 b^2}+\frac {(b d-a g) x^3}{3 b^2}+\frac {(b e-a h) x^4}{4 b^2}+\frac {f x^5}{5 b}+\frac {g x^6}{6 b}+\frac {h x^7}{7 b}+\frac {a^{2/3} \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 b^{10/3}}-\frac {a^{2/3} \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 b^{10/3}}-\frac {a (b d-a g) \log \left (a+b x^3\right )}{3 b^3}-\frac {\left (a^{2/3} \left (b^{5/3} c-a^{2/3} b e-a b^{2/3} f+a^{5/3} 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^{10/3}}\\ &=-\frac {a (b e-a h) x}{b^3}+\frac {(b c-a f) x^2}{2 b^2}+\frac {(b d-a g) x^3}{3 b^2}+\frac {(b e-a h) x^4}{4 b^2}+\frac {f x^5}{5 b}+\frac {g x^6}{6 b}+\frac {h x^7}{7 b}+\frac {a^{2/3} \left (b^{5/3} c-a^{2/3} b e-a b^{2/3} f+a^{5/3} h\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} b^{10/3}}+\frac {a^{2/3} \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 b^{10/3}}-\frac {a^{2/3} \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 b^{10/3}}-\frac {a (b d-a g) \log \left (a+b x^3\right )}{3 b^3}\\ \end {align*}

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Mathematica [A]
time = 0.29, size = 334, normalized size = 1.01 \begin {gather*} \frac {a (-b e+a h) x}{b^3}+\frac {(b c-a f) x^2}{2 b^2}+\frac {(b d-a g) x^3}{3 b^2}+\frac {(b e-a h) x^4}{4 b^2}+\frac {f x^5}{5 b}+\frac {g x^6}{6 b}+\frac {h x^7}{7 b}+\frac {a^{2/3} \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} b^{10/3}}+\frac {a^{2/3} \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 )}{3 b^{10/3}}+\frac {a^{2/3} \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 )}{6 b^{10/3}}+\frac {a (-b d+a g) \log \left (a+b x^3\right )}{3 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a*(-(b*e) + a*h)*x)/b^3 + ((b*c - a*f)*x^2)/(2*b^2) + ((b*d - a*g)*x^3)/(3*b^2) + ((b*e - a*h)*x^4)/(4*b^2) +
(f*x^5)/(5*b) + (g*x^6)/(6*b) + (h*x^7)/(7*b) + (a^(2/3)*(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]])/(Sqrt[3]*b^(10/3)) + (a^(2/3)*(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])/(3*b^(10/3)) + (a^(2/3)*(-(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])/(6*b^(10/3)) + (a*(-(b*d) + a*g)*Log[a + b*x^3])/(
3*b^3)

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Maple [A]
time = 0.38, size = 332, normalized size = 1.00

method	result	size
risch	\(\frac {h \,x^{7}}{7 b}+\frac {g \,x^{6}}{6 b}+\frac {f \,x^{5}}{5 b}-\frac {a h \,x^{4}}{4 b^{2}}+\frac {e \,x^{4}}{4 b}-\frac {a g \,x^{3}}{3 b^{2}}+\frac {d \,x^{3}}{3 b}-\frac {a f \,x^{2}}{2 b^{2}}+\frac {c \,x^{2}}{2 b}+\frac {a^{2} h x}{b^{3}}-\frac {a e x}{b^{2}}+\frac {a \left (\munderset {\textit {\_R} =\RootOf \left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\left (b \left (a g -b d \right ) \textit {\_R}^{2}+b \left (a f -b c \right ) \textit {\_R} -a^{2} h +a b e \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}\right )}{3 b^{4}}\)	\(165\)
default	\(\frac {\frac {1}{7} b^{2} h \,x^{7}+\frac {1}{6} b^{2} g \,x^{6}+\frac {1}{5} f \,x^{5} b^{2}-\frac {1}{4} a b h \,x^{4}+\frac {1}{4} b^{2} e \,x^{4}-\frac {1}{3} a b g \,x^{3}+\frac {1}{3} b^{2} d \,x^{3}-\frac {1}{2} a b f \,x^{2}+\frac {1}{2} b^{2} c \,x^{2}+a^{2} h x -a b e x}{b^{3}}-\frac {\left (\left (a^{2} h -a b e \right ) \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )+\left (-a b f +b^{2} c \right ) \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )+\frac {\left (-a b g +b^{2} d \right ) \ln \left (b \,x^{3}+a \right )}{3 b}\right ) a}{b^{3}}\)	\(332\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Maxima [A]
time = 0.56, size = 383, normalized size = 1.16 \begin {gather*} -\frac {\sqrt {3} {\left (a b^{2} c \left (\frac {a}{b}\right )^{\frac {2}{3}} - a^{2} b f \left (\frac {a}{b}\right )^{\frac {2}{3}} + a^{3} h \left (\frac {a}{b}\right )^{\frac {1}{3}} - a^{2} 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^{3}} + \frac {60 \, b^{2} h x^{7} + 70 \, b^{2} g x^{6} + 84 \, b^{2} f x^{5} - 105 \, {\left (a b h - b^{2} e\right )} x^{4} + 140 \, {\left (b^{2} d - a b g\right )} x^{3} + 210 \, {\left (b^{2} c - a b f\right )} x^{2} + 420 \, {\left (a^{2} h - a b e\right )} x}{420 \, b^{3}} - \frac {{\left (2 \, a b^{2} d \left (\frac {a}{b}\right )^{\frac {2}{3}} - 2 \, a^{2} b g \left (\frac {a}{b}\right )^{\frac {2}{3}} + a b^{2} c \left (\frac {a}{b}\right )^{\frac {1}{3}} - a^{2} b f \left (\frac {a}{b}\right )^{\frac {1}{3}} - a^{3} h + a^{2} 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} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (a b^{2} d \left (\frac {a}{b}\right )^{\frac {2}{3}} - a^{2} b g \left (\frac {a}{b}\right )^{\frac {2}{3}} - a b^{2} c \left (\frac {a}{b}\right )^{\frac {1}{3}} + a^{2} b f \left (\frac {a}{b}\right )^{\frac {1}{3}} + a^{3} h - a^{2} b e\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, b^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(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*b^2*c*(a/b)^(2/3) - a^2*b*f*(a/b)^(2/3) + a^3*h*(a/b)^(1/3) - a^2*b*(a/b)^(1/3)*e)*arctan(1/3*
sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(a*b^3) + 1/420*(60*b^2*h*x^7 + 70*b^2*g*x^6 + 84*b^2*f*x^5 - 105*(a*
b*h - b^2*e)*x^4 + 140*(b^2*d - a*b*g)*x^3 + 210*(b^2*c - a*b*f)*x^2 + 420*(a^2*h - a*b*e)*x)/b^3 - 1/6*(2*a*b
^2*d*(a/b)^(2/3) - 2*a^2*b*g*(a/b)^(2/3) + a*b^2*c*(a/b)^(1/3) - a^2*b*f*(a/b)^(1/3) - a^3*h + a^2*b*e)*log(x^
2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(b^4*(a/b)^(2/3)) - 1/3*(a*b^2*d*(a/b)^(2/3) - a^2*b*g*(a/b)^(2/3) - a*b^2*c*
(a/b)^(1/3) + a^2*b*f*(a/b)^(1/3) + a^3*h - a^2*b*e)*log(x + (a/b)^(1/3))/(b^4*(a/b)^(2/3))

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Fricas [C] Result contains complex when optimal does not.
time = 1.99, size = 15635, normalized size = 47.24 \begin {gather*} \text {too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1260*(180*b^2*h*x^7 + 210*b^2*g*x^6 + 252*b^2*f*x^5 + 315*(b^2*e - a*b*h)*x^4 - 70*((-I*sqrt(3) + 1)*((a*b*d
- a^2*g)^2/b^6 - ((g^2 - f*h)*a^4 + (e*f - 2*d*g + c*h)*a^3*b + (d^2 - c*e)*a^2*b^2)/b^6)/(-1/27*(a*b*d - a^2
*g)^3/b^9 + 1/54*(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^2/b^10 + 1/18*((g^2 - f*h)*a^4 + (e*f - 2*d*g + c*h)*a^3*b + (d^2 - c*e)*a^2*b^2)*(a*
b*d - a^2*g)/b^9 + 1/54*(a^2*b^5*c^3 - a^7*h^3 + (g^3 - 3*f*g*h + 3*e*h^2)*a^6*b - (f^3 - 3*e*f*g + 3*e^2*h -
3*c*g*h + 3*(g^2 - f*h)*d)*a^5*b^2 + (e^3 - 3*d*e*f + 3*d^2*g + 3*(f^2 - e*g - d*h)*c)*a^4*b^3 - (d^3 - 3*c*d*
e + 3*c^2*f)*a^3*b^4)/b^10)^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27*(a*b*d - a^2*g)^3/b^9 + 1/54*(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^2/b^10 + 1/1
8*((g^2 - f*h)*a^4 + (e*f - 2*d*g + c*h)*a^3*b + (d^2 - c*e)*a^2*b^2)*(a*b*d - a^2*g)/b^9 + 1/54*(a^2*b^5*c^3
- a^7*h^3  ...

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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**4*(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 = 0.53, size = 380, normalized size = 1.15 \begin {gather*} -\frac {{\left (a b d - a^{2} g\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{3}} - \frac {\sqrt {3} {\left (\left (-a b^{2}\right )^{\frac {1}{3}} a^{2} h - \left (-a b^{2}\right )^{\frac {1}{3}} a b e - \left (-a b^{2}\right )^{\frac {2}{3}} b c + \left (-a b^{2}\right )^{\frac {2}{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 \, b^{4}} - \frac {{\left (\left (-a b^{2}\right )^{\frac {1}{3}} a^{2} h - \left (-a b^{2}\right )^{\frac {1}{3}} a b e + \left (-a b^{2}\right )^{\frac {2}{3}} b c - \left (-a b^{2}\right )^{\frac {2}{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 \, b^{4}} + \frac {60 \, b^{6} h x^{7} + 70 \, b^{6} g x^{6} + 84 \, b^{6} f x^{5} - 105 \, a b^{5} h x^{4} + 105 \, b^{6} x^{4} e + 140 \, b^{6} d x^{3} - 140 \, a b^{5} g x^{3} + 210 \, b^{6} c x^{2} - 210 \, a b^{5} f x^{2} + 420 \, a^{2} b^{4} h x - 420 \, a b^{5} x e}{420 \, b^{7}} + \frac {{\left (a b^{14} c \left (-\frac {a}{b}\right )^{\frac {1}{3}} - a^{2} b^{13} f \left (-\frac {a}{b}\right )^{\frac {1}{3}} + a^{3} b^{12} h - a^{2} b^{13} 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^{15}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(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*(a*b*d - a^2*g)*log(abs(b*x^3 + a))/b^3 - 1/3*sqrt(3)*((-a*b^2)^(1/3)*a^2*h - (-a*b^2)^(1/3)*a*b*e - (-a*
b^2)^(2/3)*b*c + (-a*b^2)^(2/3)*a*f)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/b^4 - 1/6*((-a*b^2)
^(1/3)*a^2*h - (-a*b^2)^(1/3)*a*b*e + (-a*b^2)^(2/3)*b*c - (-a*b^2)^(2/3)*a*f)*log(x^2 + x*(-a/b)^(1/3) + (-a/
b)^(2/3))/b^4 + 1/420*(60*b^6*h*x^7 + 70*b^6*g*x^6 + 84*b^6*f*x^5 - 105*a*b^5*h*x^4 + 105*b^6*x^4*e + 140*b^6*
d*x^3 - 140*a*b^5*g*x^3 + 210*b^6*c*x^2 - 210*a*b^5*f*x^2 + 420*a^2*b^4*h*x - 420*a*b^5*x*e)/b^7 + 1/3*(a*b^14
*c*(-a/b)^(1/3) - a^2*b^13*f*(-a/b)^(1/3) + a^3*b^12*h - a^2*b^13*e)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(
a*b^15)

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Mupad [B]
time = 5.09, size = 1271, normalized size = 3.84 \begin {gather*} x^2\,\left (\frac {c}{2\,b}-\frac {a\,f}{2\,b^2}\right )+x^3\,\left (\frac {d}{3\,b}-\frac {a\,g}{3\,b^2}\right )+x^4\,\left (\frac {e}{4\,b}-\frac {a\,h}{4\,b^2}\right )+\left (\sum _{k=1}^3\ln \left (\mathrm {root}\left (27\,b^{10}\,z^3+27\,a\,b^8\,d\,z^2-27\,a^2\,b^7\,g\,z^2-9\,a^4\,b^4\,f\,h\,z-18\,a^3\,b^5\,d\,g\,z+9\,a^3\,b^5\,e\,f\,z+9\,a^3\,b^5\,c\,h\,z-9\,a^2\,b^6\,c\,e\,z+9\,a^4\,b^4\,g^2\,z+9\,a^2\,b^6\,d^2\,z+3\,a^6\,b\,f\,g\,h-3\,a^5\,b^2\,e\,f\,g-3\,a^5\,b^2\,d\,f\,h-3\,a^5\,b^2\,c\,g\,h+3\,a^4\,b^3\,d\,e\,f+3\,a^4\,b^3\,c\,e\,g+3\,a^4\,b^3\,c\,d\,h-3\,a^3\,b^4\,c\,d\,e-3\,a^6\,b\,e\,h^2+3\,a^5\,b^2\,e^2\,h+3\,a^5\,b^2\,d\,g^2-3\,a^4\,b^3\,d^2\,g-3\,a^4\,b^3\,c\,f^2+3\,a^3\,b^4\,c^2\,f+a^5\,b^2\,f^3+a^3\,b^4\,d^3+a^7\,h^3-a^4\,b^3\,e^3-a^2\,b^5\,c^3-a^6\,b\,g^3,z,k\right )\,\left (\frac {6\,a^2\,b^4\,d-6\,a^3\,b^3\,g}{b^4}+\frac {x\,\left (3\,a^2\,b^4\,e-3\,a^3\,b^3\,h\right )}{b^4}+\mathrm {root}\left (27\,b^{10}\,z^3+27\,a\,b^8\,d\,z^2-27\,a^2\,b^7\,g\,z^2-9\,a^4\,b^4\,f\,h\,z-18\,a^3\,b^5\,d\,g\,z+9\,a^3\,b^5\,e\,f\,z+9\,a^3\,b^5\,c\,h\,z-9\,a^2\,b^6\,c\,e\,z+9\,a^4\,b^4\,g^2\,z+9\,a^2\,b^6\,d^2\,z+3\,a^6\,b\,f\,g\,h-3\,a^5\,b^2\,e\,f\,g-3\,a^5\,b^2\,d\,f\,h-3\,a^5\,b^2\,c\,g\,h+3\,a^4\,b^3\,d\,e\,f+3\,a^4\,b^3\,c\,e\,g+3\,a^4\,b^3\,c\,d\,h-3\,a^3\,b^4\,c\,d\,e-3\,a^6\,b\,e\,h^2+3\,a^5\,b^2\,e^2\,h+3\,a^5\,b^2\,d\,g^2-3\,a^4\,b^3\,d^2\,g-3\,a^4\,b^3\,c\,f^2+3\,a^3\,b^4\,c^2\,f+a^5\,b^2\,f^3+a^3\,b^4\,d^3+a^7\,h^3-a^4\,b^3\,e^3-a^2\,b^5\,c^3-a^6\,b\,g^3,z,k\right )\,a\,b^2\,9\right )+\frac {a^5\,g^2+a^3\,b^2\,d^2-a^5\,f\,h+a^4\,b\,c\,h-2\,a^4\,b\,d\,g+a^4\,b\,e\,f-a^3\,b^2\,c\,e}{b^4}+\frac {x\,\left (a^4\,b\,f^2+a^2\,b^3\,c^2+a^5\,g\,h-a^4\,b\,d\,h-a^4\,b\,e\,g-2\,a^3\,b^2\,c\,f+a^3\,b^2\,d\,e\right )}{b^4}\right )\,\mathrm {root}\left (27\,b^{10}\,z^3+27\,a\,b^8\,d\,z^2-27\,a^2\,b^7\,g\,z^2-9\,a^4\,b^4\,f\,h\,z-18\,a^3\,b^5\,d\,g\,z+9\,a^3\,b^5\,e\,f\,z+9\,a^3\,b^5\,c\,h\,z-9\,a^2\,b^6\,c\,e\,z+9\,a^4\,b^4\,g^2\,z+9\,a^2\,b^6\,d^2\,z+3\,a^6\,b\,f\,g\,h-3\,a^5\,b^2\,e\,f\,g-3\,a^5\,b^2\,d\,f\,h-3\,a^5\,b^2\,c\,g\,h+3\,a^4\,b^3\,d\,e\,f+3\,a^4\,b^3\,c\,e\,g+3\,a^4\,b^3\,c\,d\,h-3\,a^3\,b^4\,c\,d\,e-3\,a^6\,b\,e\,h^2+3\,a^5\,b^2\,e^2\,h+3\,a^5\,b^2\,d\,g^2-3\,a^4\,b^3\,d^2\,g-3\,a^4\,b^3\,c\,f^2+3\,a^3\,b^4\,c^2\,f+a^5\,b^2\,f^3+a^3\,b^4\,d^3+a^7\,h^3-a^4\,b^3\,e^3-a^2\,b^5\,c^3-a^6\,b\,g^3,z,k\right )\right )+\frac {f\,x^5}{5\,b}+\frac {g\,x^6}{6\,b}+\frac {h\,x^7}{7\,b}-\frac {a\,x\,\left (\frac {e}{b}-\frac {a\,h}{b^2}\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^2*(c/(2*b) - (a*f)/(2*b^2)) + x^3*(d/(3*b) - (a*g)/(3*b^2)) + x^4*(e/(4*b) - (a*h)/(4*b^2)) + symsum(log(roo
t(27*b^10*z^3 + 27*a*b^8*d*z^2 - 27*a^2*b^7*g*z^2 - 9*a^4*b^4*f*h*z - 18*a^3*b^5*d*g*z + 9*a^3*b^5*e*f*z + 9*a
^3*b^5*c*h*z - 9*a^2*b^6*c*e*z + 9*a^4*b^4*g^2*z + 9*a^2*b^6*d^2*z + 3*a^6*b*f*g*h - 3*a^5*b^2*e*f*g - 3*a^5*b
^2*d*f*h - 3*a^5*b^2*c*g*h + 3*a^4*b^3*d*e*f + 3*a^4*b^3*c*e*g + 3*a^4*b^3*c*d*h - 3*a^3*b^4*c*d*e - 3*a^6*b*e
*h^2 + 3*a^5*b^2*e^2*h + 3*a^5*b^2*d*g^2 - 3*a^4*b^3*d^2*g - 3*a^4*b^3*c*f^2 + 3*a^3*b^4*c^2*f + a^5*b^2*f^3 +
a^3*b^4*d^3 + a^7*h^3 - a^4*b^3*e^3 - a^2*b^5*c^3 - a^6*b*g^3, z, k)*((6*a^2*b^4*d - 6*a^3*b^3*g)/b^4 + (x*(3
*a^2*b^4*e - 3*a^3*b^3*h))/b^4 + 9*root(27*b^10*z^3 + 27*a*b^8*d*z^2 - 27*a^2*b^7*g*z^2 - 9*a^4*b^4*f*h*z - 18
*a^3*b^5*d*g*z + 9*a^3*b^5*e*f*z + 9*a^3*b^5*c*h*z - 9*a^2*b^6*c*e*z + 9*a^4*b^4*g^2*z + 9*a^2*b^6*d^2*z + 3*a
^6*b*f*g*h - 3*a^5*b^2*e*f*g - 3*a^5*b^2*d*f*h - 3*a^5*b^2*c*g*h + 3*a^4*b^3*d*e*f + 3*a^4*b^3*c*e*g + 3*a^4*b
^3*c*d*h - 3*a^3*b^4*c*d*e - 3*a^6*b*e*h^2 + 3*a^5*b^2*e^2*h + 3*a^5*b^2*d*g^2 - 3*a^4*b^3*d^2*g - 3*a^4*b^3*c
*f^2 + 3*a^3*b^4*c^2*f + a^5*b^2*f^3 + a^3*b^4*d^3 + a^7*h^3 - a^4*b^3*e^3 - a^2*b^5*c^3 - a^6*b*g^3, z, k)*a*
b^2) + (a^5*g^2 + a^3*b^2*d^2 - a^5*f*h + a^4*b*c*h - 2*a^4*b*d*g + a^4*b*e*f - a^3*b^2*c*e)/b^4 + (x*(a^4*b*f
^2 + a^2*b^3*c^2 + a^5*g*h - a^4*b*d*h - a^4*b*e*g - 2*a^3*b^2*c*f + a^3*b^2*d*e))/b^4)*root(27*b^10*z^3 + 27*
a*b^8*d*z^2 - 27*a^2*b^7*g*z^2 - 9*a^4*b^4*f*h*z - 18*a^3*b^5*d*g*z + 9*a^3*b^5*e*f*z + 9*a^3*b^5*c*h*z - 9*a^
2*b^6*c*e*z + 9*a^4*b^4*g^2*z + 9*a^2*b^6*d^2*z + 3*a^6*b*f*g*h - 3*a^5*b^2*e*f*g - 3*a^5*b^2*d*f*h - 3*a^5*b^
2*c*g*h + 3*a^4*b^3*d*e*f + 3*a^4*b^3*c*e*g + 3*a^4*b^3*c*d*h - 3*a^3*b^4*c*d*e - 3*a^6*b*e*h^2 + 3*a^5*b^2*e^
2*h + 3*a^5*b^2*d*g^2 - 3*a^4*b^3*d^2*g - 3*a^4*b^3*c*f^2 + 3*a^3*b^4*c^2*f + a^5*b^2*f^3 + a^3*b^4*d^3 + a^7*
h^3 - a^4*b^3*e^3 - a^2*b^5*c^3 - a^6*b*g^3, z, k), k, 1, 3) + (f*x^5)/(5*b) + (g*x^6)/(6*b) + (h*x^7)/(7*b) -
(a*x*(e/b - (a*h)/b^2))/b

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