Integrand size = 23, antiderivative size = 230 \[ \int \frac {x^5 \left (c+d x+e x^2\right )}{\left (a+b x^3\right )^3} \, dx=\frac {x \left (a d+a e x-b c x^2\right )}{6 b^2 \left (a+b x^3\right )^2}-\frac {3 c+7 d x+8 e x^2}{18 b^2 \left (a+b x^3\right )}-\frac {\left (2 \sqrt [3]{b} d+5 \sqrt [3]{a} e\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{2/3} b^{8/3}}+\frac {\left (2 \sqrt [3]{b} d-5 \sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{2/3} b^{8/3}}-\frac {\left (2 \sqrt [3]{b} d-5 \sqrt [3]{a} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{2/3} b^{8/3}} \] Output:
1/6*x*(-b*c*x^2+a*e*x+a*d)/b^2/(b*x^3+a)^2-1/18*(8*e*x^2+7*d*x+3*c)/b^2/(b *x^3+a)-1/27*(2*b^(1/3)*d+5*a^(1/3)*e)*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)*3^ (1/2)/a^(1/3))*3^(1/2)/a^(2/3)/b^(8/3)+1/27*(2*b^(1/3)*d-5*a^(1/3)*e)*ln(a ^(1/3)+b^(1/3)*x)/a^(2/3)/b^(8/3)-1/54*(2*b^(1/3)*d-5*a^(1/3)*e)*ln(a^(2/3 )-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(2/3)/b^(8/3)
Time = 0.23 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.90 \[ \int \frac {x^5 \left (c+d x+e x^2\right )}{\left (a+b x^3\right )^3} \, dx=\frac {\frac {9 a b^{2/3} (c+x (d+e x))}{\left (a+b x^3\right )^2}-\frac {3 b^{2/3} (6 c+x (7 d+8 e x))}{a+b x^3}-\frac {2 \sqrt {3} \left (2 \sqrt [3]{b} d+5 \sqrt [3]{a} e\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{2/3}}+\frac {2 \left (2 \sqrt [3]{b} d-5 \sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{2/3}}+\frac {\left (-2 \sqrt [3]{b} d+5 \sqrt [3]{a} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{2/3}}}{54 b^{8/3}} \] Input:
Integrate[(x^5*(c + d*x + e*x^2))/(a + b*x^3)^3,x]
Output:
((9*a*b^(2/3)*(c + x*(d + e*x)))/(a + b*x^3)^2 - (3*b^(2/3)*(6*c + x*(7*d + 8*e*x)))/(a + b*x^3) - (2*Sqrt[3]*(2*b^(1/3)*d + 5*a^(1/3)*e)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(2/3) + (2*(2*b^(1/3)*d - 5*a^(1/3)*e )*Log[a^(1/3) + b^(1/3)*x])/a^(2/3) + ((-2*b^(1/3)*d + 5*a^(1/3)*e)*Log[a^ (2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(2/3))/(54*b^(8/3))
Time = 1.04 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.07, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {2367, 2397, 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.
| \(\displaystyle \int \frac {x^5 \left (c+d x+e x^2\right )}{\left (a+b x^3\right )^3} \, dx\) |
| \(\Big \downarrow \) 2367 |
| \(\displaystyle \frac {x \left (a d+a e x-b c x^2\right )}{6 b^2 \left (a+b x^3\right )^2}-\frac {\int \frac {-6 a b^2 e x^4-6 a b^2 d x^3-3 a b^2 c x^2+2 a^2 b e x+a^2 b d}{\left (b x^3+a\right )^2}dx}{6 a b^3}\) |
| \(\Big \downarrow \) 2397 |
| \(\displaystyle \frac {x \left (a d+a e x-b c x^2\right )}{6 b^2 \left (a+b x^3\right )^2}-\frac {\frac {x \left (7 a b d+8 a b e x-3 b^2 c x^2\right )}{3 \left (a+b x^3\right )}-\frac {\int \frac {2 a^2 b^3 (2 d+5 e x)}{b x^3+a}dx}{3 a b^2}}{6 a b^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x \left (a d+a e x-b c x^2\right )}{6 b^2 \left (a+b x^3\right )^2}-\frac {\frac {x \left (7 a b d+8 a b e x-3 b^2 c x^2\right )}{3 \left (a+b x^3\right )}-\frac {2}{3} a b \int \frac {2 d+5 e x}{b x^3+a}dx}{6 a b^3}\) |
| \(\Big \downarrow \) 2399 |
| \(\displaystyle \frac {x \left (a d+a e x-b c x^2\right )}{6 b^2 \left (a+b x^3\right )^2}-\frac {\frac {x \left (7 a b d+8 a b e x-3 b^2 c x^2\right )}{3 \left (a+b x^3\right )}-\frac {2}{3} a b \left (\frac {\int \frac {\sqrt [3]{a} \left (4 \sqrt [3]{b} d+5 \sqrt [3]{a} e\right )-\sqrt [3]{b} \left (2 \sqrt [3]{b} d-5 \sqrt [3]{a} e\right ) x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3} \sqrt [3]{b}}+\frac {\left (2 d-\frac {5 \sqrt [3]{a} e}{\sqrt [3]{b}}\right ) \int \frac {1}{\sqrt [3]{b} x+\sqrt [3]{a}}dx}{3 a^{2/3}}\right )}{6 a b^3}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {x \left (a d+a e x-b c x^2\right )}{6 b^2 \left (a+b x^3\right )^2}-\frac {\frac {x \left (7 a b d+8 a b e x-3 b^2 c x^2\right )}{3 \left (a+b x^3\right )}-\frac {2}{3} a b \left (\frac {\int \frac {\sqrt [3]{a} \left (4 \sqrt [3]{b} d+5 \sqrt [3]{a} e\right )-\sqrt [3]{b} \left (2 \sqrt [3]{b} d-5 \sqrt [3]{a} e\right ) x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3} \sqrt [3]{b}}+\frac {\left (2 d-\frac {5 \sqrt [3]{a} e}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{6 a b^3}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {x \left (a d+a e x-b c x^2\right )}{6 b^2 \left (a+b x^3\right )^2}-\frac {\frac {x \left (7 a b d+8 a b e x-3 b^2 c x^2\right )}{3 \left (a+b x^3\right )}-\frac {2}{3} a b \left (\frac {\frac {3}{2} \sqrt [3]{a} \left (5 \sqrt [3]{a} e+2 \sqrt [3]{b} d\right ) \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {1}{2} \left (2 d-\frac {5 \sqrt [3]{a} e}{\sqrt [3]{b}}\right ) \int -\frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3} \sqrt [3]{b}}+\frac {\left (2 d-\frac {5 \sqrt [3]{a} e}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{6 a b^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {x \left (a d+a e x-b c x^2\right )}{6 b^2 \left (a+b x^3\right )^2}-\frac {\frac {x \left (7 a b d+8 a b e x-3 b^2 c x^2\right )}{3 \left (a+b x^3\right )}-\frac {2}{3} a b \left (\frac {\frac {3}{2} \sqrt [3]{a} \left (5 \sqrt [3]{a} e+2 \sqrt [3]{b} d\right ) \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {1}{2} \left (2 d-\frac {5 \sqrt [3]{a} e}{\sqrt [3]{b}}\right ) \int \frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3} \sqrt [3]{b}}+\frac {\left (2 d-\frac {5 \sqrt [3]{a} e}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{6 a b^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x \left (a d+a e x-b c x^2\right )}{6 b^2 \left (a+b x^3\right )^2}-\frac {\frac {x \left (7 a b d+8 a b e x-3 b^2 c x^2\right )}{3 \left (a+b x^3\right )}-\frac {2}{3} a b \left (\frac {\frac {3}{2} \sqrt [3]{a} \left (5 \sqrt [3]{a} e+2 \sqrt [3]{b} d\right ) \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {1}{2} \sqrt [3]{b} \left (2 d-\frac {5 \sqrt [3]{a} e}{\sqrt [3]{b}}\right ) \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3} \sqrt [3]{b}}+\frac {\left (2 d-\frac {5 \sqrt [3]{a} e}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{6 a b^3}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {x \left (a d+a e x-b c x^2\right )}{6 b^2 \left (a+b x^3\right )^2}-\frac {\frac {x \left (7 a b d+8 a b e x-3 b^2 c x^2\right )}{3 \left (a+b x^3\right )}-\frac {2}{3} a b \left (\frac {\frac {1}{2} \sqrt [3]{b} \left (2 d-\frac {5 \sqrt [3]{a} e}{\sqrt [3]{b}}\right ) \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {3 \left (5 \sqrt [3]{a} e+2 \sqrt [3]{b} d\right ) \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{\sqrt [3]{b}}}{3 a^{2/3} \sqrt [3]{b}}+\frac {\left (2 d-\frac {5 \sqrt [3]{a} e}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{6 a b^3}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {x \left (a d+a e x-b c x^2\right )}{6 b^2 \left (a+b x^3\right )^2}-\frac {\frac {x \left (7 a b d+8 a b e x-3 b^2 c x^2\right )}{3 \left (a+b x^3\right )}-\frac {2}{3} a b \left (\frac {\frac {1}{2} \sqrt [3]{b} \left (2 d-\frac {5 \sqrt [3]{a} e}{\sqrt [3]{b}}\right ) \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right ) \left (5 \sqrt [3]{a} e+2 \sqrt [3]{b} d\right )}{\sqrt [3]{b}}}{3 a^{2/3} \sqrt [3]{b}}+\frac {\left (2 d-\frac {5 \sqrt [3]{a} e}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{6 a b^3}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {x \left (a d+a e x-b c x^2\right )}{6 b^2 \left (a+b x^3\right )^2}-\frac {\frac {x \left (7 a b d+8 a b e x-3 b^2 c x^2\right )}{3 \left (a+b x^3\right )}-\frac {2}{3} a b \left (\frac {-\frac {1}{2} \left (2 d-\frac {5 \sqrt [3]{a} e}{\sqrt [3]{b}}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right ) \left (5 \sqrt [3]{a} e+2 \sqrt [3]{b} d\right )}{\sqrt [3]{b}}}{3 a^{2/3} \sqrt [3]{b}}+\frac {\left (2 d-\frac {5 \sqrt [3]{a} e}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{6 a b^3}\) |
Input:
Int[(x^5*(c + d*x + e*x^2))/(a + b*x^3)^3,x]
Output:
(x*(a*d + a*e*x - b*c*x^2))/(6*b^2*(a + b*x^3)^2) - ((x*(7*a*b*d + 8*a*b*e *x - 3*b^2*c*x^2))/(3*(a + b*x^3)) - (2*a*b*(((2*d - (5*a^(1/3)*e)/b^(1/3) )*Log[a^(1/3) + b^(1/3)*x])/(3*a^(2/3)*b^(1/3)) + (-((Sqrt[3]*(2*b^(1/3)*d + 5*a^(1/3)*e)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(1/3)) - (( 2*d - (5*a^(1/3)*e)/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^3)
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] :> With[{q = Expon[Pq, x]}, Module[{Q = PolynomialQuotient[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, x], R = PolynomialRemainder[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, x]}, S imp[(-x)*R*((a + b*x^n)^(p + 1)/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1))), x] + Simp[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]] /; GeQ[q, n]] /; 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 = 0.13 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.43
| method | result | size |
| risch | \(\frac {-\frac {4 e \,x^{5}}{9 b}-\frac {7 d \,x^{4}}{18 b}-\frac {c \,x^{3}}{3 b}-\frac {5 a e \,x^{2}}{18 b^{2}}-\frac {2 d a x}{9 b^{2}}-\frac {a c}{6 b^{2}}}{\left (b \,x^{3}+a \right )^{2}}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3} b +a \right )}{\sum }\frac {\left (5 e \textit {\_R} +2 d \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}}{27 b^{3}}\) | \(99\) |
| default | \(\frac {-\frac {4 e \,x^{5}}{9 b}-\frac {7 d \,x^{4}}{18 b}-\frac {c \,x^{3}}{3 b}-\frac {5 a e \,x^{2}}{18 b^{2}}-\frac {2 d a x}{9 b^{2}}-\frac {a c}{6 b^{2}}}{\left (b \,x^{3}+a \right )^{2}}+\frac {2 d \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )+5 e \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 b^{2}}\) | \(257\) |
Input:
int(x^5*(e*x^2+d*x+c)/(b*x^3+a)^3,x,method=_RETURNVERBOSE)
Output:
(-4/9*e/b*x^5-7/18*d/b*x^4-1/3/b*c*x^3-5/18*a*e/b^2*x^2-2/9/b^2*d*a*x-1/6* a*c/b^2)/(b*x^3+a)^2+1/27/b^3*sum((5*_R*e+2*d)/_R^2*ln(x-_R),_R=RootOf(_Z^ 3*b+a))
Result contains complex when optimal does not.
Time = 0.81 (sec) , antiderivative size = 2246, normalized size of antiderivative = 9.77 \[ \int \frac {x^5 \left (c+d x+e x^2\right )}{\left (a+b x^3\right )^3} \, dx=\text {Too large to display} \] Input:
integrate(x^5*(e*x^2+d*x+c)/(b*x^3+a)^3,x, algorithm="fricas")
Output:
-1/108*(48*b*e*x^5 + 42*b*d*x^4 + 36*b*c*x^3 + 30*a*e*x^2 + 24*a*d*x + 2*( b^4*x^6 + 2*a*b^3*x^3 + a^2*b^2)*((1/2)^(1/3)*(I*sqrt(3) + 1)*((8*b*d^3 + 125*a*e^3)/(a^2*b^8) + (8*b*d^3 - 125*a*e^3)/(a^2*b^8))^(1/3) - 20*(1/2)^( 2/3)*d*e*(-I*sqrt(3) + 1)/(a*b^5*((8*b*d^3 + 125*a*e^3)/(a^2*b^8) + (8*b*d ^3 - 125*a*e^3)/(a^2*b^8))^(1/3)))*log(5/4*((1/2)^(1/3)*(I*sqrt(3) + 1)*(( 8*b*d^3 + 125*a*e^3)/(a^2*b^8) + (8*b*d^3 - 125*a*e^3)/(a^2*b^8))^(1/3) - 20*(1/2)^(2/3)*d*e*(-I*sqrt(3) + 1)/(a*b^5*((8*b*d^3 + 125*a*e^3)/(a^2*b^8 ) + (8*b*d^3 - 125*a*e^3)/(a^2*b^8))^(1/3)))^2*a^2*b^5*e - 2*((1/2)^(1/3)* (I*sqrt(3) + 1)*((8*b*d^3 + 125*a*e^3)/(a^2*b^8) + (8*b*d^3 - 125*a*e^3)/( a^2*b^8))^(1/3) - 20*(1/2)^(2/3)*d*e*(-I*sqrt(3) + 1)/(a*b^5*((8*b*d^3 + 1 25*a*e^3)/(a^2*b^8) + (8*b*d^3 - 125*a*e^3)/(a^2*b^8))^(1/3)))*a*b^3*d^2 + 100*a*d*e^2 + (8*b*d^3 + 125*a*e^3)*x) + 18*a*c - ((b^4*x^6 + 2*a*b^3*x^3 + a^2*b^2)*((1/2)^(1/3)*(I*sqrt(3) + 1)*((8*b*d^3 + 125*a*e^3)/(a^2*b^8) + (8*b*d^3 - 125*a*e^3)/(a^2*b^8))^(1/3) - 20*(1/2)^(2/3)*d*e*(-I*sqrt(3) + 1)/(a*b^5*((8*b*d^3 + 125*a*e^3)/(a^2*b^8) + (8*b*d^3 - 125*a*e^3)/(a^2* b^8))^(1/3))) + 3*sqrt(1/3)*(b^4*x^6 + 2*a*b^3*x^3 + a^2*b^2)*sqrt(-(((1/2 )^(1/3)*(I*sqrt(3) + 1)*((8*b*d^3 + 125*a*e^3)/(a^2*b^8) + (8*b*d^3 - 125* a*e^3)/(a^2*b^8))^(1/3) - 20*(1/2)^(2/3)*d*e*(-I*sqrt(3) + 1)/(a*b^5*((8*b *d^3 + 125*a*e^3)/(a^2*b^8) + (8*b*d^3 - 125*a*e^3)/(a^2*b^8))^(1/3)))^2*a *b^5 + 160*d*e)/(a*b^5)))*log(-5/4*((1/2)^(1/3)*(I*sqrt(3) + 1)*((8*b*d...
Time = 5.64 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.71 \[ \int \frac {x^5 \left (c+d x+e x^2\right )}{\left (a+b x^3\right )^3} \, dx=\operatorname {RootSum} {\left (19683 t^{3} a^{2} b^{8} + 810 t a b^{3} d e + 125 a e^{3} - 8 b d^{3}, \left ( t \mapsto t \log {\left (x + \frac {3645 t^{2} a^{2} b^{5} e + 108 t a b^{3} d^{2} + 100 a d e^{2}}{125 a e^{3} + 8 b d^{3}} \right )} \right )\right )} + \frac {- 3 a c - 4 a d x - 5 a e x^{2} - 6 b c x^{3} - 7 b d x^{4} - 8 b e x^{5}}{18 a^{2} b^{2} + 36 a b^{3} x^{3} + 18 b^{4} x^{6}} \] Input:
integrate(x**5*(e*x**2+d*x+c)/(b*x**3+a)**3,x)
Output:
RootSum(19683*_t**3*a**2*b**8 + 810*_t*a*b**3*d*e + 125*a*e**3 - 8*b*d**3, Lambda(_t, _t*log(x + (3645*_t**2*a**2*b**5*e + 108*_t*a*b**3*d**2 + 100* a*d*e**2)/(125*a*e**3 + 8*b*d**3)))) + (-3*a*c - 4*a*d*x - 5*a*e*x**2 - 6* b*c*x**3 - 7*b*d*x**4 - 8*b*e*x**5)/(18*a**2*b**2 + 36*a*b**3*x**3 + 18*b* *4*x**6)
Time = 0.11 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.90 \[ \int \frac {x^5 \left (c+d x+e x^2\right )}{\left (a+b x^3\right )^3} \, dx=-\frac {8 \, b e x^{5} + 7 \, b d x^{4} + 6 \, b c x^{3} + 5 \, a e x^{2} + 4 \, a d x + 3 \, a c}{18 \, {\left (b^{4} x^{6} + 2 \, a b^{3} x^{3} + a^{2} b^{2}\right )}} + \frac {\sqrt {3} {\left (5 \, e \left (\frac {a}{b}\right )^{\frac {1}{3}} + 2 \, d\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (5 \, e \left (\frac {a}{b}\right )^{\frac {1}{3}} - 2 \, d\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (5 \, e \left (\frac {a}{b}\right )^{\frac {1}{3}} - 2 \, d\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \] Input:
integrate(x^5*(e*x^2+d*x+c)/(b*x^3+a)^3,x, algorithm="maxima")
Output:
-1/18*(8*b*e*x^5 + 7*b*d*x^4 + 6*b*c*x^3 + 5*a*e*x^2 + 4*a*d*x + 3*a*c)/(b ^4*x^6 + 2*a*b^3*x^3 + a^2*b^2) + 1/27*sqrt(3)*(5*e*(a/b)^(1/3) + 2*d)*arc tan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(b^3*(a/b)^(2/3)) + 1/54* (5*e*(a/b)^(1/3) - 2*d)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(b^3*(a/b)^ (2/3)) - 1/27*(5*e*(a/b)^(1/3) - 2*d)*log(x + (a/b)^(1/3))/(b^3*(a/b)^(2/3 ))
Time = 0.14 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.90 \[ \int \frac {x^5 \left (c+d x+e x^2\right )}{\left (a+b x^3\right )^3} \, dx=-\frac {\sqrt {3} {\left (2 \, b d - 5 \, \left (-a b^{2}\right )^{\frac {1}{3}} e\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, \left (-a b^{2}\right )^{\frac {2}{3}} b^{2}} - \frac {{\left (2 \, b d + 5 \, \left (-a b^{2}\right )^{\frac {1}{3}} e\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, \left (-a b^{2}\right )^{\frac {2}{3}} b^{2}} - \frac {{\left (5 \, e \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 2 \, d\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a b^{2}} - \frac {8 \, b e x^{5} + 7 \, b d x^{4} + 6 \, b c x^{3} + 5 \, a e x^{2} + 4 \, a d x + 3 \, a c}{18 \, {\left (b x^{3} + a\right )}^{2} b^{2}} \] Input:
integrate(x^5*(e*x^2+d*x+c)/(b*x^3+a)^3,x, algorithm="giac")
Output:
-1/27*sqrt(3)*(2*b*d - 5*(-a*b^2)^(1/3)*e)*arctan(1/3*sqrt(3)*(2*x + (-a/b )^(1/3))/(-a/b)^(1/3))/((-a*b^2)^(2/3)*b^2) - 1/54*(2*b*d + 5*(-a*b^2)^(1/ 3)*e)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/((-a*b^2)^(2/3)*b^2) - 1/27 *(5*e*(-a/b)^(1/3) + 2*d)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a*b^2) - 1/18*(8*b*e*x^5 + 7*b*d*x^4 + 6*b*c*x^3 + 5*a*e*x^2 + 4*a*d*x + 3*a*c)/( (b*x^3 + a)^2*b^2)
Time = 0.25 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.96 \[ \int \frac {x^5 \left (c+d x+e x^2\right )}{\left (a+b x^3\right )^3} \, dx=\left (\sum _{k=1}^3\ln \left (\frac {10\,d\,e+25\,e^2\,x+{\mathrm {root}\left (19683\,a^2\,b^8\,z^3+810\,a\,b^3\,d\,e\,z+125\,a\,e^3-8\,b\,d^3,z,k\right )}^2\,a\,b^5\,729+\mathrm {root}\left (19683\,a^2\,b^8\,z^3+810\,a\,b^3\,d\,e\,z+125\,a\,e^3-8\,b\,d^3,z,k\right )\,b^3\,d\,x\,54}{b^3\,81}\right )\,\mathrm {root}\left (19683\,a^2\,b^8\,z^3+810\,a\,b^3\,d\,e\,z+125\,a\,e^3-8\,b\,d^3,z,k\right )\right )-\frac {\frac {c\,x^3}{3\,b}+\frac {7\,d\,x^4}{18\,b}+\frac {4\,e\,x^5}{9\,b}+\frac {a\,c}{6\,b^2}+\frac {2\,a\,d\,x}{9\,b^2}+\frac {5\,a\,e\,x^2}{18\,b^2}}{a^2+2\,a\,b\,x^3+b^2\,x^6} \] Input:
int((x^5*(c + d*x + e*x^2))/(a + b*x^3)^3,x)
Output:
symsum(log((10*d*e + 25*e^2*x + 729*root(19683*a^2*b^8*z^3 + 810*a*b^3*d*e *z + 125*a*e^3 - 8*b*d^3, z, k)^2*a*b^5 + 54*root(19683*a^2*b^8*z^3 + 810* a*b^3*d*e*z + 125*a*e^3 - 8*b*d^3, z, k)*b^3*d*x)/(81*b^3))*root(19683*a^2 *b^8*z^3 + 810*a*b^3*d*e*z + 125*a*e^3 - 8*b*d^3, z, k), k, 1, 3) - ((c*x^ 3)/(3*b) + (7*d*x^4)/(18*b) + (4*e*x^5)/(9*b) + (a*c)/(6*b^2) + (2*a*d*x)/ (9*b^2) + (5*a*e*x^2)/(18*b^2))/(a^2 + b^2*x^6 + 2*a*b*x^3)
Time = 0.15 (sec) , antiderivative size = 576, normalized size of antiderivative = 2.50 \[ \int \frac {x^5 \left (c+d x+e x^2\right )}{\left (a+b x^3\right )^3} \, dx =\text {Too large to display} \] Input:
int(x^5*(e*x^2+d*x+c)/(b*x^3+a)^3,x)
Output:
( - 4*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*d - 8*b**(1/3)*a**(2/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)* x)/(a**(1/3)*sqrt(3)))*a*b*d*x**3 - 4*b**(1/3)*a**(2/3)*sqrt(3)*atan((a**( 1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*b**2*d*x**6 - 10*sqrt(3)*atan((a* *(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a**3*e - 20*sqrt(3)*atan((a**(1 /3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a**2*b*e*x**3 - 10*sqrt(3)*atan((a **(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a*b**2*e*x**6 - 2*b**(1/3)*a** (2/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a**2*d - 4*b**(1 /3)*a**(2/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a*b*d*x** 3 - 2*b**(1/3)*a**(2/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2 )*b**2*d*x**6 + 4*b**(1/3)*a**(2/3)*log(a**(1/3) + b**(1/3)*x)*a**2*d + 8* b**(1/3)*a**(2/3)*log(a**(1/3) + b**(1/3)*x)*a*b*d*x**3 + 4*b**(1/3)*a**(2 /3)*log(a**(1/3) + b**(1/3)*x)*b**2*d*x**6 - 12*b**(2/3)*a**(1/3)*a**2*d*x - 15*b**(2/3)*a**(1/3)*a**2*e*x**2 - 21*b**(2/3)*a**(1/3)*a*b*d*x**4 - 24 *b**(2/3)*a**(1/3)*a*b*e*x**5 + 9*b**(2/3)*a**(1/3)*b**2*c*x**6 + 5*log(a* *(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a**3*e + 10*log(a**(2/3) - b **(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a**2*b*e*x**3 + 5*log(a**(2/3) - b**(1 /3)*a**(1/3)*x + b**(2/3)*x**2)*a*b**2*e*x**6 - 10*log(a**(1/3) + b**(1/3) *x)*a**3*e - 20*log(a**(1/3) + b**(1/3)*x)*a**2*b*e*x**3 - 10*log(a**(1/3) + b**(1/3)*x)*a*b**2*e*x**6)/(54*b**(2/3)*a**(1/3)*a*b**2*(a**2 + 2*a*...