\(\int \frac {x^4 (c+d x+e x^2+f x^3+g x^4+h x^5)}{(a+b x^3)^2} \, dx\) [412]

Optimal result

Integrand size = 38, antiderivative size = 337 \[ \int \frac {x^4 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{\left (a+b x^3\right )^2} \, dx=\frac {(b e-2 a h) x}{b^3}+\frac {f x^2}{2 b^2}+\frac {g x^3}{3 b^2}+\frac {h x^4}{4 b^2}+\frac {x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{3 b^3 \left (a+b x^3\right )}-\frac {\left (2 b^{5/3} c-4 a^{2/3} b e-5 a b^{2/3} f+7 a^{5/3} h\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} \sqrt [3]{a} b^{10/3}}-\frac {\left (b^{2/3} (2 b c-5 a f)+a^{2/3} (4 b e-7 a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 \sqrt [3]{a} b^{10/3}}+\frac {\left (b^{2/3} (2 b c-5 a f)+a^{2/3} (4 b e-7 a h)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 \sqrt [3]{a} b^{10/3}}+\frac {(b d-2 a g) \log \left (a+b x^3\right )}{3 b^3} \]

[Out]

(-2*a*h+b*e)*x/b^3+1/2*f*x^2/b^2+1/3*g*x^3/b^2+1/4*h*x^4/b^2+1/3*x*(a*(-a*h+b*e)-b*(-a*f+b*c)*x-b*(-a*g+b*d)*x
^2)/b^3/(b*x^3+a)-1/9*(b^(2/3)*(-5*a*f+2*b*c)+a^(2/3)*(-7*a*h+4*b*e))*ln(a^(1/3)+b^(1/3)*x)/a^(1/3)/b^(10/3)+1
/18*(b^(2/3)*(-5*a*f+2*b*c)+a^(2/3)*(-7*a*h+4*b*e))*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(1/3)/b^(10/3)
+1/3*(-2*a*g+b*d)*ln(b*x^3+a)/b^3-1/9*(2*b^(5/3)*c-4*a^(2/3)*b*e-5*a*b^(2/3)*f+7*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^(10/3)*3^(1/2)

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {1842, 1901, 1885, 1874, 31, 648, 631, 210, 642, 266} \[ \int \frac {x^4 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{\left (a+b x^3\right )^2} \, dx=-\frac {\arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (-4 a^{2/3} b e+7 a^{5/3} h-5 a b^{2/3} f+2 b^{5/3} c\right )}{3 \sqrt {3} \sqrt [3]{a} b^{10/3}}+\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^{2/3} (4 b e-7 a h)+b^{2/3} (2 b c-5 a f)\right )}{18 \sqrt [3]{a} b^{10/3}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^{2/3} (4 b e-7 a h)+b^{2/3} (2 b c-5 a f)\right )}{9 \sqrt [3]{a} b^{10/3}}+\frac {x \left (-b x (b c-a f)-b x^2 (b d-a g)+a (b e-a h)\right )}{3 b^3 \left (a+b x^3\right )}+\frac {(b d-2 a g) \log \left (a+b x^3\right )}{3 b^3}+\frac {x (b e-2 a h)}{b^3}+\frac {f x^2}{2 b^2}+\frac {g x^3}{3 b^2}+\frac {h x^4}{4 b^2} \]

[In]

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

[Out]

((b*e - 2*a*h)*x)/b^3 + (f*x^2)/(2*b^2) + (g*x^3)/(3*b^2) + (h*x^4)/(4*b^2) + (x*(a*(b*e - a*h) - b*(b*c - a*f
)*x - b*(b*d - a*g)*x^2))/(3*b^3*(a + b*x^3)) - ((2*b^(5/3)*c - 4*a^(2/3)*b*e - 5*a*b^(2/3)*f + 7*a^(5/3)*h)*A
rcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(1/3)*b^(10/3)) - ((b^(2/3)*(2*b*c - 5*a*f) + a
^(2/3)*(4*b*e - 7*a*h))*Log[a^(1/3) + b^(1/3)*x])/(9*a^(1/3)*b^(10/3)) + ((b^(2/3)*(2*b*c - 5*a*f) + a^(2/3)*(
4*b*e - 7*a*h))*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*a^(1/3)*b^(10/3)) + ((b*d - 2*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 1842

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = m + Expon[Pq, x]}, Module[{Q = Pol
ynomialQuotient[b^(Floor[(q - 1)/n] + 1)*x^m*Pq, a + b*x^n, x], R = PolynomialRemainder[b^(Floor[(q - 1)/n] +
1)*x^m*Pq, a + b*x^n, x]}, Dist[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)), Int[(a + b*x^n)^(p + 1)*ExpandToSum[
a*n*(p + 1)*Q + n*(p + 1)*R + D[x*R, x], x], x], x] + Simp[(-x)*R*((a + b*x^n)^(p + 1)/(a*n*(p + 1)*b^(Floor[(
q - 1)/n] + 1))), x]] /; GeQ[q, n]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1] && IGtQ[m,
0]

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

Mathematica [A] (verified)

Time = 0.43 (sec) , antiderivative size = 334, normalized size of antiderivative = 0.99 \[ \int \frac {x^4 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{\left (a+b x^3\right )^2} \, dx=\frac {36 b^{2/3} (b e-2 a h) x+18 b^{5/3} f x^2+12 b^{5/3} g x^3+9 b^{5/3} h x^4-\frac {12 b^{2/3} \left (b^2 c x^2+a^2 (g+h x)-a b (d+x (e+f x))\right )}{a+b x^3}-\frac {4 \sqrt {3} \left (2 b^2 c-4 a^{2/3} b^{4/3} e-5 a b f+7 a^{5/3} \sqrt [3]{b} h\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{a}}+\frac {4 \left (-2 b^2 c-4 a^{2/3} b^{4/3} e+5 a b f+7 a^{5/3} \sqrt [3]{b} h\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a}}+\frac {2 \left (2 b^2 c+4 a^{2/3} b^{4/3} e-5 a b f-7 a^{5/3} \sqrt [3]{b} h\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{\sqrt [3]{a}}+12 b^{2/3} (b d-2 a g) \log \left (a+b x^3\right )}{36 b^{11/3}} \]

[In]

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

[Out]

(36*b^(2/3)*(b*e - 2*a*h)*x + 18*b^(5/3)*f*x^2 + 12*b^(5/3)*g*x^3 + 9*b^(5/3)*h*x^4 - (12*b^(2/3)*(b^2*c*x^2 +
a^2*(g + h*x) - a*b*(d + x*(e + f*x))))/(a + b*x^3) - (4*Sqrt[3]*(2*b^2*c - 4*a^(2/3)*b^(4/3)*e - 5*a*b*f + 7
*a^(5/3)*b^(1/3)*h)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(1/3) + (4*(-2*b^2*c - 4*a^(2/3)*b^(4/3)*e
+ 5*a*b*f + 7*a^(5/3)*b^(1/3)*h)*Log[a^(1/3) + b^(1/3)*x])/a^(1/3) + (2*(2*b^2*c + 4*a^(2/3)*b^(4/3)*e - 5*a*b
*f - 7*a^(5/3)*b^(1/3)*h)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(1/3) + 12*b^(2/3)*(b*d - 2*a*g)*L
og[a + b*x^3])/(36*b^(11/3))

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 1.56 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.48

method	result	size
risch	\(\frac {h \,x^{4}}{4 b^{2}}+\frac {g \,x^{3}}{3 b^{2}}+\frac {f \,x^{2}}{2 b^{2}}-\frac {2 a h x}{b^{3}}+\frac {e x}{b^{2}}+\frac {\left (\frac {1}{3} a f b -\frac {1}{3} b^{2} c \right ) x^{2}+\left (-\frac {1}{3} a^{2} h +\frac {1}{3} a e b \right ) x -\frac {a \left (a g -b d \right )}{3}}{b^{3} \left (b \,x^{3}+a \right )}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\left (3 b \left (-2 a g +b d \right ) \textit {\_R}^{2}+b \left (-5 a f +2 b c \right ) \textit {\_R} +7 a^{2} h -4 a e b \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}}{9 b^{4}}\)	\(162\)
default	\(-\frac {-\frac {1}{4} b h \,x^{4}-\frac {1}{3} b g \,x^{3}-\frac {1}{2} b f \,x^{2}+2 a h x -b e x}{b^{3}}+\frac {\frac {\left (\frac {1}{3} a f b -\frac {1}{3} b^{2} c \right ) x^{2}+\left (-\frac {1}{3} a^{2} h +\frac {1}{3} a e b \right ) x -\frac {a \left (a g -b d \right )}{3}}{b \,x^{3}+a}+\frac {\left (7 a^{2} h -4 a e b \right ) \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{3}+\frac {\left (-5 a f b +2 b^{2} c \right ) \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3}+\frac {\left (-6 a b g +3 b^{2} d \right ) \ln \left (b \,x^{3}+a \right )}{9 b}}{b^{3}}\)	\(329\)

[In]

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

[Out]

1/4*h*x^4/b^2+1/3*g*x^3/b^2+1/2*f*x^2/b^2-2/b^3*a*h*x+1/b^2*e*x+((1/3*a*f*b-1/3*b^2*c)*x^2+(-1/3*a^2*h+1/3*a*e
*b)*x-1/3*a*(a*g-b*d))/b^3/(b*x^3+a)+1/9/b^4*sum((3*b*(-2*a*g+b*d)*_R^2+b*(-5*a*f+2*b*c)*_R+7*a^2*h-4*a*e*b)/_
R^2*ln(x-_R),_R=RootOf(_Z^3*b+a))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.96 (sec) , antiderivative size = 16147, normalized size of antiderivative = 47.91 \[ \int \frac {x^4 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{\left (a+b x^3\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F(-1)]

Timed out. \[ \int \frac {x^4 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{\left (a+b x^3\right )^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.08 \[ \int \frac {x^4 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{\left (a+b x^3\right )^2} \, dx=\frac {a b d - a^{2} g - {\left (b^{2} c - a b f\right )} x^{2} + {\left (a b e - a^{2} h\right )} x}{3 \, {\left (b^{4} x^{3} + a b^{3}\right )}} + \frac {\sqrt {3} {\left (2 \, b^{2} c \left (\frac {a}{b}\right )^{\frac {2}{3}} - 5 \, a b f \left (\frac {a}{b}\right )^{\frac {2}{3}} - 4 \, a b e \left (\frac {a}{b}\right )^{\frac {1}{3}} + 7 \, a^{2} h \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a b^{3}} + \frac {3 \, b h x^{4} + 4 \, b g x^{3} + 6 \, b f x^{2} + 12 \, {\left (b e - 2 \, a h\right )} x}{12 \, b^{3}} + \frac {{\left (6 \, b^{2} d \left (\frac {a}{b}\right )^{\frac {2}{3}} - 12 \, a b g \left (\frac {a}{b}\right )^{\frac {2}{3}} + 2 \, b^{2} c \left (\frac {a}{b}\right )^{\frac {1}{3}} - 5 \, a b f \left (\frac {a}{b}\right )^{\frac {1}{3}} + 4 \, a b e - 7 \, a^{2} h\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, b^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (3 \, b^{2} d \left (\frac {a}{b}\right )^{\frac {2}{3}} - 6 \, a b g \left (\frac {a}{b}\right )^{\frac {2}{3}} - 2 \, b^{2} c \left (\frac {a}{b}\right )^{\frac {1}{3}} + 5 \, a b f \left (\frac {a}{b}\right )^{\frac {1}{3}} - 4 \, a b e + 7 \, a^{2} h\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, b^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]

[In]

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

[Out]

1/3*(a*b*d - a^2*g - (b^2*c - a*b*f)*x^2 + (a*b*e - a^2*h)*x)/(b^4*x^3 + a*b^3) + 1/9*sqrt(3)*(2*b^2*c*(a/b)^(
2/3) - 5*a*b*f*(a/b)^(2/3) - 4*a*b*e*(a/b)^(1/3) + 7*a^2*h*(a/b)^(1/3))*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))
/(a/b)^(1/3))/(a*b^3) + 1/12*(3*b*h*x^4 + 4*b*g*x^3 + 6*b*f*x^2 + 12*(b*e - 2*a*h)*x)/b^3 + 1/18*(6*b^2*d*(a/b
)^(2/3) - 12*a*b*g*(a/b)^(2/3) + 2*b^2*c*(a/b)^(1/3) - 5*a*b*f*(a/b)^(1/3) + 4*a*b*e - 7*a^2*h)*log(x^2 - x*(a
/b)^(1/3) + (a/b)^(2/3))/(b^4*(a/b)^(2/3)) + 1/9*(3*b^2*d*(a/b)^(2/3) - 6*a*b*g*(a/b)^(2/3) - 2*b^2*c*(a/b)^(1
/3) + 5*a*b*f*(a/b)^(1/3) - 4*a*b*e + 7*a^2*h)*log(x + (a/b)^(1/3))/(b^4*(a/b)^(2/3))

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.04 \[ \int \frac {x^4 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{\left (a+b x^3\right )^2} \, dx=\frac {\sqrt {3} {\left (4 \, a b e - 7 \, a^{2} h + 2 \, \left (-a b^{2}\right )^{\frac {1}{3}} b c - 5 \, \left (-a b^{2}\right )^{\frac {1}{3}} a f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, \left (-a b^{2}\right )^{\frac {2}{3}} b^{2}} + \frac {{\left (4 \, a b e - 7 \, a^{2} h - 2 \, \left (-a b^{2}\right )^{\frac {1}{3}} b c + 5 \, \left (-a b^{2}\right )^{\frac {1}{3}} a f\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, \left (-a b^{2}\right )^{\frac {2}{3}} b^{2}} + \frac {{\left (b d - 2 \, a g\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{3}} + \frac {a b d - a^{2} g - {\left (b^{2} c - a b f\right )} x^{2} + {\left (a b e - a^{2} h\right )} x}{3 \, {\left (b x^{3} + a\right )} b^{3}} - \frac {{\left (2 \, b^{6} c \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 5 \, a b^{5} f \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 4 \, a b^{5} e + 7 \, a^{2} b^{4} h\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a b^{7}} + \frac {3 \, b^{6} h x^{4} + 4 \, b^{6} g x^{3} + 6 \, b^{6} f x^{2} + 12 \, b^{6} e x - 24 \, a b^{5} h x}{12 \, b^{8}} \]

[In]

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

[Out]

1/9*sqrt(3)*(4*a*b*e - 7*a^2*h + 2*(-a*b^2)^(1/3)*b*c - 5*(-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^2) + 1/18*(4*a*b*e - 7*a^2*h - 2*(-a*b^2)^(1/3)*b*c + 5*(-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^2) + 1/3*(b*d - 2*a*g)*log(abs(b*x^3 + a))/b^
3 + 1/3*(a*b*d - a^2*g - (b^2*c - a*b*f)*x^2 + (a*b*e - a^2*h)*x)/((b*x^3 + a)*b^3) - 1/9*(2*b^6*c*(-a/b)^(1/3
) - 5*a*b^5*f*(-a/b)^(1/3) - 4*a*b^5*e + 7*a^2*b^4*h)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a*b^7) + 1/12*(
3*b^6*h*x^4 + 4*b^6*g*x^3 + 6*b^6*f*x^2 + 12*b^6*e*x - 24*a*b^5*h*x)/b^8

Mupad [B] (verification not implemented)

Time = 9.31 (sec) , antiderivative size = 1241, normalized size of antiderivative = 3.68 \[ \int \frac {x^4 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{\left (a+b x^3\right )^2} \, dx=\left (\sum _{k=1}^3\ln \left (\mathrm {root}\left (729\,a\,b^{10}\,z^3-729\,a\,b^8\,d\,z^2+1458\,a^2\,b^7\,g\,z^2-216\,a\,b^6\,c\,e\,z-945\,a^3\,b^4\,f\,h\,z-972\,a^2\,b^5\,d\,g\,z+540\,a^2\,b^5\,e\,f\,z+378\,a^2\,b^5\,c\,h\,z+243\,a\,b^6\,d^2\,z+972\,a^3\,b^4\,g^2\,z-630\,a^4\,b\,f\,g\,h+72\,a\,b^4\,c\,d\,e+360\,a^3\,b^2\,e\,f\,g+315\,a^3\,b^2\,d\,f\,h+252\,a^3\,b^2\,c\,g\,h-180\,a^2\,b^3\,d\,e\,f-144\,a^2\,b^3\,c\,e\,g-126\,a^2\,b^3\,c\,d\,h+588\,a^4\,b\,e\,h^2-60\,a\,b^4\,c^2\,f-336\,a^3\,b^2\,e^2\,h-324\,a^3\,b^2\,d\,g^2+162\,a^2\,b^3\,d^2\,g+150\,a^2\,b^3\,c\,f^2-125\,a^3\,b^2\,f^3+64\,a^2\,b^3\,e^3+216\,a^4\,b\,g^3-27\,a\,b^4\,d^3-343\,a^5\,h^3+8\,b^5\,c^3,z,k\right )\,\left (\frac {108\,a^2\,b^3\,g-54\,a\,b^4\,d}{9\,b^4}+\frac {x\,\left (63\,a^2\,b^3\,h-36\,a\,b^4\,e\right )}{9\,b^4}+\mathrm {root}\left (729\,a\,b^{10}\,z^3-729\,a\,b^8\,d\,z^2+1458\,a^2\,b^7\,g\,z^2-216\,a\,b^6\,c\,e\,z-945\,a^3\,b^4\,f\,h\,z-972\,a^2\,b^5\,d\,g\,z+540\,a^2\,b^5\,e\,f\,z+378\,a^2\,b^5\,c\,h\,z+243\,a\,b^6\,d^2\,z+972\,a^3\,b^4\,g^2\,z-630\,a^4\,b\,f\,g\,h+72\,a\,b^4\,c\,d\,e+360\,a^3\,b^2\,e\,f\,g+315\,a^3\,b^2\,d\,f\,h+252\,a^3\,b^2\,c\,g\,h-180\,a^2\,b^3\,d\,e\,f-144\,a^2\,b^3\,c\,e\,g-126\,a^2\,b^3\,c\,d\,h+588\,a^4\,b\,e\,h^2-60\,a\,b^4\,c^2\,f-336\,a^3\,b^2\,e^2\,h-324\,a^3\,b^2\,d\,g^2+162\,a^2\,b^3\,d^2\,g+150\,a^2\,b^3\,c\,f^2-125\,a^3\,b^2\,f^3+64\,a^2\,b^3\,e^3+216\,a^4\,b\,g^3-27\,a\,b^4\,d^3-343\,a^5\,h^3+8\,b^5\,c^3,z,k\right )\,a\,b^2\,9\right )+\frac {36\,a^3\,g^2+9\,a\,b^2\,d^2-35\,a^3\,f\,h-8\,a\,b^2\,c\,e+14\,a^2\,b\,c\,h-36\,a^2\,b\,d\,g+20\,a^2\,b\,e\,f}{9\,b^4}+\frac {x\,\left (4\,b^3\,c^2+25\,a^2\,b\,f^2+42\,a^3\,g\,h-20\,a\,b^2\,c\,f+12\,a\,b^2\,d\,e-21\,a^2\,b\,d\,h-24\,a^2\,b\,e\,g\right )}{9\,b^4}\right )\,\mathrm {root}\left (729\,a\,b^{10}\,z^3-729\,a\,b^8\,d\,z^2+1458\,a^2\,b^7\,g\,z^2-216\,a\,b^6\,c\,e\,z-945\,a^3\,b^4\,f\,h\,z-972\,a^2\,b^5\,d\,g\,z+540\,a^2\,b^5\,e\,f\,z+378\,a^2\,b^5\,c\,h\,z+243\,a\,b^6\,d^2\,z+972\,a^3\,b^4\,g^2\,z-630\,a^4\,b\,f\,g\,h+72\,a\,b^4\,c\,d\,e+360\,a^3\,b^2\,e\,f\,g+315\,a^3\,b^2\,d\,f\,h+252\,a^3\,b^2\,c\,g\,h-180\,a^2\,b^3\,d\,e\,f-144\,a^2\,b^3\,c\,e\,g-126\,a^2\,b^3\,c\,d\,h+588\,a^4\,b\,e\,h^2-60\,a\,b^4\,c^2\,f-336\,a^3\,b^2\,e^2\,h-324\,a^3\,b^2\,d\,g^2+162\,a^2\,b^3\,d^2\,g+150\,a^2\,b^3\,c\,f^2-125\,a^3\,b^2\,f^3+64\,a^2\,b^3\,e^3+216\,a^4\,b\,g^3-27\,a\,b^4\,d^3-343\,a^5\,h^3+8\,b^5\,c^3,z,k\right )\right )+x\,\left (\frac {e}{b^2}-\frac {2\,a\,h}{b^3}\right )-\frac {x\,\left (\frac {a^2\,h}{3}-\frac {a\,b\,e}{3}\right )+\frac {a^2\,g}{3}+x^2\,\left (\frac {b^2\,c}{3}-\frac {a\,b\,f}{3}\right )-\frac {a\,b\,d}{3}}{b^4\,x^3+a\,b^3}+\frac {f\,x^2}{2\,b^2}+\frac {g\,x^3}{3\,b^2}+\frac {h\,x^4}{4\,b^2} \]

[In]

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

[Out]

symsum(log(root(729*a*b^10*z^3 - 729*a*b^8*d*z^2 + 1458*a^2*b^7*g*z^2 - 216*a*b^6*c*e*z - 945*a^3*b^4*f*h*z -
972*a^2*b^5*d*g*z + 540*a^2*b^5*e*f*z + 378*a^2*b^5*c*h*z + 243*a*b^6*d^2*z + 972*a^3*b^4*g^2*z - 630*a^4*b*f*
g*h + 72*a*b^4*c*d*e + 360*a^3*b^2*e*f*g + 315*a^3*b^2*d*f*h + 252*a^3*b^2*c*g*h - 180*a^2*b^3*d*e*f - 144*a^2
*b^3*c*e*g - 126*a^2*b^3*c*d*h + 588*a^4*b*e*h^2 - 60*a*b^4*c^2*f - 336*a^3*b^2*e^2*h - 324*a^3*b^2*d*g^2 + 16
2*a^2*b^3*d^2*g + 150*a^2*b^3*c*f^2 - 125*a^3*b^2*f^3 + 64*a^2*b^3*e^3 + 216*a^4*b*g^3 - 27*a*b^4*d^3 - 343*a^
5*h^3 + 8*b^5*c^3, z, k)*((108*a^2*b^3*g - 54*a*b^4*d)/(9*b^4) + (x*(63*a^2*b^3*h - 36*a*b^4*e))/(9*b^4) + 9*r
oot(729*a*b^10*z^3 - 729*a*b^8*d*z^2 + 1458*a^2*b^7*g*z^2 - 216*a*b^6*c*e*z - 945*a^3*b^4*f*h*z - 972*a^2*b^5*
d*g*z + 540*a^2*b^5*e*f*z + 378*a^2*b^5*c*h*z + 243*a*b^6*d^2*z + 972*a^3*b^4*g^2*z - 630*a^4*b*f*g*h + 72*a*b
^4*c*d*e + 360*a^3*b^2*e*f*g + 315*a^3*b^2*d*f*h + 252*a^3*b^2*c*g*h - 180*a^2*b^3*d*e*f - 144*a^2*b^3*c*e*g -
126*a^2*b^3*c*d*h + 588*a^4*b*e*h^2 - 60*a*b^4*c^2*f - 336*a^3*b^2*e^2*h - 324*a^3*b^2*d*g^2 + 162*a^2*b^3*d^
2*g + 150*a^2*b^3*c*f^2 - 125*a^3*b^2*f^3 + 64*a^2*b^3*e^3 + 216*a^4*b*g^3 - 27*a*b^4*d^3 - 343*a^5*h^3 + 8*b^
5*c^3, z, k)*a*b^2) + (36*a^3*g^2 + 9*a*b^2*d^2 - 35*a^3*f*h - 8*a*b^2*c*e + 14*a^2*b*c*h - 36*a^2*b*d*g + 20*
a^2*b*e*f)/(9*b^4) + (x*(4*b^3*c^2 + 25*a^2*b*f^2 + 42*a^3*g*h - 20*a*b^2*c*f + 12*a*b^2*d*e - 21*a^2*b*d*h -
24*a^2*b*e*g))/(9*b^4))*root(729*a*b^10*z^3 - 729*a*b^8*d*z^2 + 1458*a^2*b^7*g*z^2 - 216*a*b^6*c*e*z - 945*a^3
*b^4*f*h*z - 972*a^2*b^5*d*g*z + 540*a^2*b^5*e*f*z + 378*a^2*b^5*c*h*z + 243*a*b^6*d^2*z + 972*a^3*b^4*g^2*z -
630*a^4*b*f*g*h + 72*a*b^4*c*d*e + 360*a^3*b^2*e*f*g + 315*a^3*b^2*d*f*h + 252*a^3*b^2*c*g*h - 180*a^2*b^3*d*
e*f - 144*a^2*b^3*c*e*g - 126*a^2*b^3*c*d*h + 588*a^4*b*e*h^2 - 60*a*b^4*c^2*f - 336*a^3*b^2*e^2*h - 324*a^3*b
^2*d*g^2 + 162*a^2*b^3*d^2*g + 150*a^2*b^3*c*f^2 - 125*a^3*b^2*f^3 + 64*a^2*b^3*e^3 + 216*a^4*b*g^3 - 27*a*b^4
*d^3 - 343*a^5*h^3 + 8*b^5*c^3, z, k), k, 1, 3) + x*(e/b^2 - (2*a*h)/b^3) - (x*((a^2*h)/3 - (a*b*e)/3) + (a^2*
g)/3 + x^2*((b^2*c)/3 - (a*b*f)/3) - (a*b*d)/3)/(a*b^3 + b^4*x^3) + (f*x^2)/(2*b^2) + (g*x^3)/(3*b^2) + (h*x^4
)/(4*b^2)