Integrand size = 23, antiderivative size = 234 \[ \int \frac {x^3 \left (c+d x+e x^2\right )}{\left (a+b x^3\right )^3} \, dx=-\frac {x \left (c+d x+e x^2\right )}{6 b \left (a+b x^3\right )^2}-\frac {e}{6 b^2 \left (a+b x^3\right )}+\frac {x (c+2 d x)}{18 a b \left (a+b x^3\right )}-\frac {\left (\sqrt [3]{b} c+\sqrt [3]{a} d\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{5/3} b^{5/3}}+\frac {\left (\sqrt [3]{b} c-\sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{5/3} b^{5/3}}-\frac {\left (\sqrt [3]{b} c-\sqrt [3]{a} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{5/3} b^{5/3}} \] Output:
-1/6*x*(e*x^2+d*x+c)/b/(b*x^3+a)^2-1/6*e/b^2/(b*x^3+a)+1/18*x*(2*d*x+c)/a/ b/(b*x^3+a)-1/27*(b^(1/3)*c+a^(1/3)*d)*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)*3^ (1/2)/a^(1/3))*3^(1/2)/a^(5/3)/b^(5/3)+1/27*(b^(1/3)*c-a^(1/3)*d)*ln(a^(1/ 3)+b^(1/3)*x)/a^(5/3)/b^(5/3)-1/54*(b^(1/3)*c-a^(1/3)*d)*ln(a^(2/3)-a^(1/3 )*b^(1/3)*x+b^(2/3)*x^2)/a^(5/3)/b^(5/3)
Time = 0.14 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.91 \[ \int \frac {x^3 \left (c+d x+e x^2\right )}{\left (a+b x^3\right )^3} \, dx=\frac {\frac {9 a e-9 b x (c+d x)}{\left (a+b x^3\right )^2}+\frac {-18 a e+3 b x (c+2 d x)}{a \left (a+b x^3\right )}-\frac {2 \sqrt {3} \sqrt [3]{b} \left (\sqrt [3]{b} c+\sqrt [3]{a} d\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{5/3}}+\frac {2 \sqrt [3]{b} \left (\sqrt [3]{b} c-\sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{5/3}}+\frac {\sqrt [3]{b} \left (-\sqrt [3]{b} c+\sqrt [3]{a} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{5/3}}}{54 b^2} \] Input:
Integrate[(x^3*(c + d*x + e*x^2))/(a + b*x^3)^3,x]
Output:
((9*a*e - 9*b*x*(c + d*x))/(a + b*x^3)^2 + (-18*a*e + 3*b*x*(c + 2*d*x))/( a*(a + b*x^3)) - (2*Sqrt[3]*b^(1/3)*(b^(1/3)*c + a^(1/3)*d)*ArcTan[(1 - (2 *b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(5/3) + (2*b^(1/3)*(b^(1/3)*c - a^(1/3)*d )*Log[a^(1/3) + b^(1/3)*x])/a^(5/3) + (b^(1/3)*(-(b^(1/3)*c) + a^(1/3)*d)* Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(5/3))/(54*b^2)
Time = 0.92 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.97, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {2367, 25, 2393, 27, 2399, 16, 1142, 25, 27, 1082, 217, 1103}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {x^3 \left (c+d x+e x^2\right )}{\left (a+b x^3\right )^3} \, dx\) |
| \(\Big \downarrow \) 2367 |
| \(\displaystyle -\frac {\int -\frac {3 a b e x^2+2 a b d x+a b c}{\left (b x^3+a\right )^2}dx}{6 a b^2}-\frac {x \left (c+d x+e x^2\right )}{6 b \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {3 a b e x^2+2 a b d x+a b c}{\left (b x^3+a\right )^2}dx}{6 a b^2}-\frac {x \left (c+d x+e x^2\right )}{6 b \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 2393 |
| \(\displaystyle \frac {-\frac {\int -\frac {2 a b (c+d x)}{b x^3+a}dx}{3 a}-\frac {3 a e-b x (c+2 d x)}{3 \left (a+b x^3\right )}}{6 a b^2}-\frac {x \left (c+d x+e x^2\right )}{6 b \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2}{3} b \int \frac {c+d x}{b x^3+a}dx-\frac {3 a e-b x (c+2 d x)}{3 \left (a+b x^3\right )}}{6 a b^2}-\frac {x \left (c+d x+e x^2\right )}{6 b \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 2399 |
| \(\displaystyle \frac {\frac {2}{3} b \left (\frac {\int \frac {\sqrt [3]{a} \left (2 \sqrt [3]{b} c+\sqrt [3]{a} d\right )-\sqrt [3]{b} \left (\sqrt [3]{b} c-\sqrt [3]{a} d\right ) x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3} \sqrt [3]{b}}+\frac {\left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \int \frac {1}{\sqrt [3]{b} x+\sqrt [3]{a}}dx}{3 a^{2/3}}\right )-\frac {3 a e-b x (c+2 d x)}{3 \left (a+b x^3\right )}}{6 a b^2}-\frac {x \left (c+d x+e x^2\right )}{6 b \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {\frac {2}{3} b \left (\frac {\int \frac {\sqrt [3]{a} \left (2 \sqrt [3]{b} c+\sqrt [3]{a} d\right )-\sqrt [3]{b} \left (\sqrt [3]{b} c-\sqrt [3]{a} d\right ) x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3} \sqrt [3]{b}}+\frac {\left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )-\frac {3 a e-b x (c+2 d x)}{3 \left (a+b x^3\right )}}{6 a b^2}-\frac {x \left (c+d x+e x^2\right )}{6 b \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {\frac {2}{3} b \left (\frac {\frac {3}{2} \sqrt [3]{a} \left (\sqrt [3]{a} d+\sqrt [3]{b} c\right ) \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {1}{2} \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \int -\frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3} \sqrt [3]{b}}+\frac {\left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )-\frac {3 a e-b x (c+2 d x)}{3 \left (a+b x^3\right )}}{6 a b^2}-\frac {x \left (c+d x+e x^2\right )}{6 b \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2}{3} b \left (\frac {\frac {3}{2} \sqrt [3]{a} \left (\sqrt [3]{a} d+\sqrt [3]{b} c\right ) \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {1}{2} \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \int \frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3} \sqrt [3]{b}}+\frac {\left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )-\frac {3 a e-b x (c+2 d x)}{3 \left (a+b x^3\right )}}{6 a b^2}-\frac {x \left (c+d x+e x^2\right )}{6 b \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2}{3} b \left (\frac {\frac {3}{2} \sqrt [3]{a} \left (\sqrt [3]{a} d+\sqrt [3]{b} c\right ) \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {1}{2} \sqrt [3]{b} \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3} \sqrt [3]{b}}+\frac {\left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )-\frac {3 a e-b x (c+2 d x)}{3 \left (a+b x^3\right )}}{6 a b^2}-\frac {x \left (c+d x+e x^2\right )}{6 b \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {2}{3} b \left (\frac {\frac {1}{2} \sqrt [3]{b} \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {3 \left (\sqrt [3]{a} d+\sqrt [3]{b} c\right ) \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{\sqrt [3]{b}}}{3 a^{2/3} \sqrt [3]{b}}+\frac {\left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )-\frac {3 a e-b x (c+2 d x)}{3 \left (a+b x^3\right )}}{6 a b^2}-\frac {x \left (c+d x+e x^2\right )}{6 b \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {2}{3} b \left (\frac {\frac {1}{2} \sqrt [3]{b} \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right ) \left (\sqrt [3]{a} d+\sqrt [3]{b} c\right )}{\sqrt [3]{b}}}{3 a^{2/3} \sqrt [3]{b}}+\frac {\left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )-\frac {3 a e-b x (c+2 d x)}{3 \left (a+b x^3\right )}}{6 a b^2}-\frac {x \left (c+d x+e x^2\right )}{6 b \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {2}{3} b \left (\frac {-\frac {1}{2} \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right ) \left (\sqrt [3]{a} d+\sqrt [3]{b} c\right )}{\sqrt [3]{b}}}{3 a^{2/3} \sqrt [3]{b}}+\frac {\left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )-\frac {3 a e-b x (c+2 d x)}{3 \left (a+b x^3\right )}}{6 a b^2}-\frac {x \left (c+d x+e x^2\right )}{6 b \left (a+b x^3\right )^2}\) |
Input:
Int[(x^3*(c + d*x + e*x^2))/(a + b*x^3)^3,x]
Output:
-1/6*(x*(c + d*x + e*x^2))/(b*(a + b*x^3)^2) + (-1/3*(3*a*e - b*x*(c + 2*d *x))/(a + b*x^3) + (2*b*(((c - (a^(1/3)*d)/b^(1/3))*Log[a^(1/3) + b^(1/3)* x])/(3*a^(2/3)*b^(1/3)) + (-((Sqrt[3]*(b^(1/3)*c + a^(1/3)*d)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(1/3)) - ((c - (a^(1/3)*d)/b^(1/3))*Log [a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/2)/(3*a^(2/3)*b^(1/3))))/3)/( 6*a*b^2)
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[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])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[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]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = m + Expon[Pq, x]}, Module[{Q = PolynomialQuotient[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]}, Simp[(-x)*R*((a + b*x^n)^(p + 1)/(a*n*(p + 1)*b^(Floo r[(q - 1)/n] + 1))), x] + Simp[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)) I nt[(a + b*x^n)^(p + 1)*ExpandToSum[a*n*(p + 1)*Q + n*(p + 1)*R + D[x*R, x], x], x], x]] /; GeQ[q, n]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0 ] && LtQ[p, -1] && IGtQ[m, 0]
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], i}, Simp[(a*Coeff[Pq, x, q] - b*x*ExpandToSum[Pq - Coeff[Pq, x, q]*x^q , x])*((a + b*x^n)^(p + 1)/(a*b*n*(p + 1))), x] + Simp[1/(a*n*(p + 1)) In t[Sum[(n*(p + 1) + i + 1)*Coeff[Pq, x, i]*x^i, {i, 0, q - 1}]*(a + b*x^n)^( p + 1), x], x] /; q == n - 1] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n , 0] && LtQ[p, -1]
Int[((A_) + (B_.)*(x_))/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{r = Numer ator[Rt[a/b, 3]], s = Denominator[Rt[a/b, 3]]}, Simp[(-r)*((B*r - A*s)/(3*a *s)) Int[1/(r + s*x), x], x] + Simp[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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.14 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.41
| method | result | size |
| risch | \(\frac {\frac {d \,x^{5}}{9 a}+\frac {c \,x^{4}}{18 a}-\frac {e \,x^{3}}{3 b}-\frac {d \,x^{2}}{18 b}-\frac {c x}{9 b}-\frac {a e}{6 b^{2}}}{\left (b \,x^{3}+a \right )^{2}}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3} b +a \right )}{\sum }\frac {\left (d \textit {\_R} +c \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}}{27 b^{2} a}\) | \(97\) |
| default | \(\frac {\frac {d \,x^{5}}{9 a}+\frac {c \,x^{4}}{18 a}-\frac {e \,x^{3}}{3 b}-\frac {d \,x^{2}}{18 b}-\frac {c x}{9 b}-\frac {a e}{6 b^{2}}}{\left (b \,x^{3}+a \right )^{2}}+\frac {c \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )+d \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 b a}\) | \(256\) |
Input:
int(x^3*(e*x^2+d*x+c)/(b*x^3+a)^3,x,method=_RETURNVERBOSE)
Output:
(1/9*d/a*x^5+1/18*c/a*x^4-1/3*e*x^3/b-1/18*d*x^2/b-1/9*c*x/b-1/6*a*e/b^2)/ (b*x^3+a)^2+1/27/b^2/a*sum((_R*d+c)/_R^2*ln(x-_R),_R=RootOf(_Z^3*b+a))
Result contains complex when optimal does not.
Time = 0.85 (sec) , antiderivative size = 2191, normalized size of antiderivative = 9.36 \[ \int \frac {x^3 \left (c+d x+e x^2\right )}{\left (a+b x^3\right )^3} \, dx=\text {Too large to display} \] Input:
integrate(x^3*(e*x^2+d*x+c)/(b*x^3+a)^3,x, algorithm="fricas")
Output:
1/108*(12*b^2*d*x^5 + 6*b^2*c*x^4 - 36*a*b*e*x^3 - 6*a*b*d*x^2 - 12*a*b*c* x - 18*a^2*e - 2*(a*b^4*x^6 + 2*a^2*b^3*x^3 + a^3*b^2)*((1/2)^(1/3)*(I*sqr t(3) + 1)*((b*c^3 + a*d^3)/(a^5*b^5) + (b*c^3 - a*d^3)/(a^5*b^5))^(1/3) - 2*(1/2)^(2/3)*c*d*(-I*sqrt(3) + 1)/(a^3*b^3*((b*c^3 + a*d^3)/(a^5*b^5) + ( b*c^3 - a*d^3)/(a^5*b^5))^(1/3)))*log(1/4*((1/2)^(1/3)*(I*sqrt(3) + 1)*((b *c^3 + a*d^3)/(a^5*b^5) + (b*c^3 - a*d^3)/(a^5*b^5))^(1/3) - 2*(1/2)^(2/3) *c*d*(-I*sqrt(3) + 1)/(a^3*b^3*((b*c^3 + a*d^3)/(a^5*b^5) + (b*c^3 - a*d^3 )/(a^5*b^5))^(1/3)))^2*a^4*b^3*d - 1/2*((1/2)^(1/3)*(I*sqrt(3) + 1)*((b*c^ 3 + a*d^3)/(a^5*b^5) + (b*c^3 - a*d^3)/(a^5*b^5))^(1/3) - 2*(1/2)^(2/3)*c* d*(-I*sqrt(3) + 1)/(a^3*b^3*((b*c^3 + a*d^3)/(a^5*b^5) + (b*c^3 - a*d^3)/( a^5*b^5))^(1/3)))*a^2*b^2*c^2 + 2*a*c*d^2 + (b*c^3 + a*d^3)*x) + ((a*b^4*x ^6 + 2*a^2*b^3*x^3 + a^3*b^2)*((1/2)^(1/3)*(I*sqrt(3) + 1)*((b*c^3 + a*d^3 )/(a^5*b^5) + (b*c^3 - a*d^3)/(a^5*b^5))^(1/3) - 2*(1/2)^(2/3)*c*d*(-I*sqr t(3) + 1)/(a^3*b^3*((b*c^3 + a*d^3)/(a^5*b^5) + (b*c^3 - a*d^3)/(a^5*b^5)) ^(1/3))) + 3*sqrt(1/3)*(a*b^4*x^6 + 2*a^2*b^3*x^3 + a^3*b^2)*sqrt(-(((1/2) ^(1/3)*(I*sqrt(3) + 1)*((b*c^3 + a*d^3)/(a^5*b^5) + (b*c^3 - a*d^3)/(a^5*b ^5))^(1/3) - 2*(1/2)^(2/3)*c*d*(-I*sqrt(3) + 1)/(a^3*b^3*((b*c^3 + a*d^3)/ (a^5*b^5) + (b*c^3 - a*d^3)/(a^5*b^5))^(1/3)))^2*a^3*b^3 + 16*c*d)/(a^3*b^ 3)))*log(-1/4*((1/2)^(1/3)*(I*sqrt(3) + 1)*((b*c^3 + a*d^3)/(a^5*b^5) + (b *c^3 - a*d^3)/(a^5*b^5))^(1/3) - 2*(1/2)^(2/3)*c*d*(-I*sqrt(3) + 1)/(a^...
Time = 3.72 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.72 \[ \int \frac {x^3 \left (c+d x+e x^2\right )}{\left (a+b x^3\right )^3} \, dx=\operatorname {RootSum} {\left (19683 t^{3} a^{5} b^{5} + 81 t a^{2} b^{2} c d + a d^{3} - b c^{3}, \left ( t \mapsto t \log {\left (x + \frac {729 t^{2} a^{4} b^{3} d + 27 t a^{2} b^{2} c^{2} + 2 a c d^{2}}{a d^{3} + b c^{3}} \right )} \right )\right )} + \frac {- 3 a^{2} e - 2 a b c x - a b d x^{2} - 6 a b e x^{3} + b^{2} c x^{4} + 2 b^{2} d x^{5}}{18 a^{3} b^{2} + 36 a^{2} b^{3} x^{3} + 18 a b^{4} x^{6}} \] Input:
integrate(x**3*(e*x**2+d*x+c)/(b*x**3+a)**3,x)
Output:
RootSum(19683*_t**3*a**5*b**5 + 81*_t*a**2*b**2*c*d + a*d**3 - b*c**3, Lam bda(_t, _t*log(x + (729*_t**2*a**4*b**3*d + 27*_t*a**2*b**2*c**2 + 2*a*c*d **2)/(a*d**3 + b*c**3)))) + (-3*a**2*e - 2*a*b*c*x - a*b*d*x**2 - 6*a*b*e* x**3 + b**2*c*x**4 + 2*b**2*d*x**5)/(18*a**3*b**2 + 36*a**2*b**3*x**3 + 18 *a*b**4*x**6)
Time = 0.12 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.94 \[ \int \frac {x^3 \left (c+d x+e x^2\right )}{\left (a+b x^3\right )^3} \, dx=\frac {2 \, b^{2} d x^{5} + b^{2} c x^{4} - 6 \, a b e x^{3} - a b d x^{2} - 2 \, a b c x - 3 \, a^{2} e}{18 \, {\left (a b^{4} x^{6} + 2 \, a^{2} b^{3} x^{3} + a^{3} b^{2}\right )}} + \frac {\sqrt {3} {\left (d \left (\frac {a}{b}\right )^{\frac {1}{3}} + c\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (d \left (\frac {a}{b}\right )^{\frac {1}{3}} - c\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (d \left (\frac {a}{b}\right )^{\frac {1}{3}} - c\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, a b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \] Input:
integrate(x^3*(e*x^2+d*x+c)/(b*x^3+a)^3,x, algorithm="maxima")
Output:
1/18*(2*b^2*d*x^5 + b^2*c*x^4 - 6*a*b*e*x^3 - a*b*d*x^2 - 2*a*b*c*x - 3*a^ 2*e)/(a*b^4*x^6 + 2*a^2*b^3*x^3 + a^3*b^2) + 1/27*sqrt(3)*(d*(a/b)^(1/3) + c)*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(a*b^2*(a/b)^(2/3) ) + 1/54*(d*(a/b)^(1/3) - c)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a*b^2 *(a/b)^(2/3)) - 1/27*(d*(a/b)^(1/3) - c)*log(x + (a/b)^(1/3))/(a*b^2*(a/b) ^(2/3))
Time = 0.14 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.94 \[ \int \frac {x^3 \left (c+d x+e x^2\right )}{\left (a+b x^3\right )^3} \, dx=-\frac {\sqrt {3} {\left (b c - \left (-a b^{2}\right )^{\frac {1}{3}} d\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, \left (-a b^{2}\right )^{\frac {2}{3}} a b} - \frac {{\left (b c + \left (-a b^{2}\right )^{\frac {1}{3}} d\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, \left (-a b^{2}\right )^{\frac {2}{3}} a b} - \frac {{\left (d \left (-\frac {a}{b}\right )^{\frac {1}{3}} + c\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a^{2} b} + \frac {2 \, b^{2} d x^{5} + b^{2} c x^{4} - 6 \, a b e x^{3} - a b d x^{2} - 2 \, a b c x - 3 \, a^{2} e}{18 \, {\left (b x^{3} + a\right )}^{2} a b^{2}} \] Input:
integrate(x^3*(e*x^2+d*x+c)/(b*x^3+a)^3,x, algorithm="giac")
Output:
-1/27*sqrt(3)*(b*c - (-a*b^2)^(1/3)*d)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1 /3))/(-a/b)^(1/3))/((-a*b^2)^(2/3)*a*b) - 1/54*(b*c + (-a*b^2)^(1/3)*d)*lo g(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/((-a*b^2)^(2/3)*a*b) - 1/27*(d*(-a/ b)^(1/3) + c)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a^2*b) + 1/18*(2*b^ 2*d*x^5 + b^2*c*x^4 - 6*a*b*e*x^3 - a*b*d*x^2 - 2*a*b*c*x - 3*a^2*e)/((b*x ^3 + a)^2*a*b^2)
Time = 6.13 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.97 \[ \int \frac {x^3 \left (c+d x+e x^2\right )}{\left (a+b x^3\right )^3} \, dx=\left (\sum _{k=1}^3\ln \left (\frac {c\,d+d^2\,x+{\mathrm {root}\left (19683\,a^5\,b^5\,z^3+81\,a^2\,b^2\,c\,d\,z+a\,d^3-b\,c^3,z,k\right )}^2\,a^3\,b^3\,729+\mathrm {root}\left (19683\,a^5\,b^5\,z^3+81\,a^2\,b^2\,c\,d\,z+a\,d^3-b\,c^3,z,k\right )\,a\,b^2\,c\,x\,27}{a^2\,b\,81}\right )\,\mathrm {root}\left (19683\,a^5\,b^5\,z^3+81\,a^2\,b^2\,c\,d\,z+a\,d^3-b\,c^3,z,k\right )\right )-\frac {\frac {d\,x^2}{18\,b}-\frac {d\,x^5}{9\,a}-\frac {c\,x^4}{18\,a}+\frac {e\,x^3}{3\,b}+\frac {a\,e}{6\,b^2}+\frac {c\,x}{9\,b}}{a^2+2\,a\,b\,x^3+b^2\,x^6} \] Input:
int((x^3*(c + d*x + e*x^2))/(a + b*x^3)^3,x)
Output:
symsum(log((c*d + d^2*x + 729*root(19683*a^5*b^5*z^3 + 81*a^2*b^2*c*d*z + a*d^3 - b*c^3, z, k)^2*a^3*b^3 + 27*root(19683*a^5*b^5*z^3 + 81*a^2*b^2*c* d*z + a*d^3 - b*c^3, z, k)*a*b^2*c*x)/(81*a^2*b))*root(19683*a^5*b^5*z^3 + 81*a^2*b^2*c*d*z + a*d^3 - b*c^3, z, k), k, 1, 3) - ((d*x^2)/(18*b) - (d* x^5)/(9*a) - (c*x^4)/(18*a) + (e*x^3)/(3*b) + (a*e)/(6*b^2) + (c*x)/(9*b)) /(a^2 + b^2*x^6 + 2*a*b*x^3)
Time = 0.16 (sec) , antiderivative size = 574, normalized size of antiderivative = 2.45 \[ \int \frac {x^3 \left (c+d x+e x^2\right )}{\left (a+b x^3\right )^3} \, dx =\text {Too large to display} \] Input:
int(x^3*(e*x^2+d*x+c)/(b*x^3+a)^3,x)
Output:
( - 2*b**(1/3)*a**(2/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*s qrt(3)))*a**2*c - 4*b**(1/3)*a**(2/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)* x)/(a**(1/3)*sqrt(3)))*a*b*c*x**3 - 2*b**(1/3)*a**(2/3)*sqrt(3)*atan((a**( 1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*b**2*c*x**6 - 2*sqrt(3)*atan((a** (1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a**3*d - 4*sqrt(3)*atan((a**(1/3 ) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a**2*b*d*x**3 - 2*sqrt(3)*atan((a**( 1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a*b**2*d*x**6 - b**(1/3)*a**(2/3) *log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a**2*c - 2*b**(1/3)*a **(2/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a*b*c*x**3 - b **(1/3)*a**(2/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*b**2* c*x**6 + 2*b**(1/3)*a**(2/3)*log(a**(1/3) + b**(1/3)*x)*a**2*c + 4*b**(1/3 )*a**(2/3)*log(a**(1/3) + b**(1/3)*x)*a*b*c*x**3 + 2*b**(1/3)*a**(2/3)*log (a**(1/3) + b**(1/3)*x)*b**2*c*x**6 - 6*b**(2/3)*a**(1/3)*a**2*c*x - 3*b** (2/3)*a**(1/3)*a**2*d*x**2 + 3*b**(2/3)*a**(1/3)*a*b*c*x**4 + 6*b**(2/3)*a **(1/3)*a*b*d*x**5 + 9*b**(2/3)*a**(1/3)*a*b*e*x**6 + log(a**(2/3) - b**(1 /3)*a**(1/3)*x + b**(2/3)*x**2)*a**3*d + 2*log(a**(2/3) - b**(1/3)*a**(1/3 )*x + b**(2/3)*x**2)*a**2*b*d*x**3 + log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a*b**2*d*x**6 - 2*log(a**(1/3) + b**(1/3)*x)*a**3*d - 4*log (a**(1/3) + b**(1/3)*x)*a**2*b*d*x**3 - 2*log(a**(1/3) + b**(1/3)*x)*a*b** 2*d*x**6)/(54*b**(2/3)*a**(1/3)*a**2*b*(a**2 + 2*a*b*x**3 + b**2*x**6))