Integrand size = 35, antiderivative size = 276 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{\left (a+b x^3\right )^2} \, dx=\frac {x \left (b c-a f+(b d-a g) x+(b e-a h) x^2\right )}{3 a b \left (a+b x^3\right )}-\frac {\left (2 b^{4/3} c+\sqrt [3]{a} b d+a \sqrt [3]{b} f+2 a^{4/3} g\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{5/3} b^{5/3}}+\frac {\left (\sqrt [3]{b} (2 b c+a f)-\sqrt [3]{a} (b d+2 a g)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{5/3}}-\frac {\left (\sqrt [3]{b} (2 b c+a f)-\sqrt [3]{a} (b d+2 a g)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{5/3} b^{5/3}}+\frac {h \log \left (a+b x^3\right )}{3 b^2} \]
1/3*x*(b*c-a*f+(-a*g+b*d)*x+(-a*h+b*e)*x^2)/a/b/(b*x^3+a)+1/9*(b^(1/3)*(a* f+2*b*c)-a^(1/3)*(2*a*g+b*d))*ln(a^(1/3)+b^(1/3)*x)/a^(5/3)/b^(5/3)-1/18*( b^(1/3)*(a*f+2*b*c)-a^(1/3)*(2*a*g+b*d))*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2 /3)*x^2)/a^(5/3)/b^(5/3)+1/3*h*ln(b*x^3+a)/b^2-1/9*(2*b^(4/3)*c+a^(1/3)*b* d+a*b^(1/3)*f+2*a^(4/3)*g)*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)/a^(1/3)*3^(1/2 ))/a^(5/3)/b^(5/3)*3^(1/2)
Time = 0.16 (sec) , antiderivative size = 268, normalized size of antiderivative = 0.97 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{\left (a+b x^3\right )^2} \, dx=\frac {\frac {6 \left (a^2 h+b^2 x (c+d x)-a b (e+x (f+g x))\right )}{a \left (a+b x^3\right )}-\frac {2 \sqrt {3} \sqrt [3]{b} \left (2 b^{4/3} c+\sqrt [3]{a} b d+a \sqrt [3]{b} f+2 a^{4/3} g\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{5/3}}+\frac {2 \sqrt [3]{b} \left (2 b^{4/3} c-\sqrt [3]{a} b d+a \sqrt [3]{b} f-2 a^{4/3} g\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{5/3}}+\frac {\sqrt [3]{b} \left (-2 b^{4/3} c+\sqrt [3]{a} b d-a \sqrt [3]{b} f+2 a^{4/3} g\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{5/3}}+6 h \log \left (a+b x^3\right )}{18 b^2} \]
((6*(a^2*h + b^2*x*(c + d*x) - a*b*(e + x*(f + g*x))))/(a*(a + b*x^3)) - ( 2*Sqrt[3]*b^(1/3)*(2*b^(4/3)*c + a^(1/3)*b*d + a*b^(1/3)*f + 2*a^(4/3)*g)* ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(5/3) + (2*b^(1/3)*(2*b^(4/ 3)*c - a^(1/3)*b*d + a*b^(1/3)*f - 2*a^(4/3)*g)*Log[a^(1/3) + b^(1/3)*x])/ a^(5/3) + (b^(1/3)*(-2*b^(4/3)*c + a^(1/3)*b*d - a*b^(1/3)*f + 2*a^(4/3)*g )*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(5/3) + 6*h*Log[a + b* x^3])/(18*b^2)
Time = 0.73 (sec) , antiderivative size = 274, normalized size of antiderivative = 0.99, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.371, Rules used = {2397, 25, 2410, 792, 2399, 16, 27, 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.
| \(\displaystyle \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{\left (a+b x^3\right )^2} \, dx\) |
| \(\Big \downarrow \) 2397 |
| \(\displaystyle \frac {x \left (x (b d-a g)+x^2 (b e-a h)-a f+b c\right )}{3 a b \left (a+b x^3\right )}-\frac {\int -\frac {3 a b h x^2+b (b d+2 a g) x+b (2 b c+a f)}{b x^3+a}dx}{3 a b^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {3 a b h x^2+b (b d+2 a g) x+b (2 b c+a f)}{b x^3+a}dx}{3 a b^2}+\frac {x \left (x (b d-a g)+x^2 (b e-a h)-a f+b c\right )}{3 a b \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 2410 |
| \(\displaystyle \frac {\int \frac {b (2 b c+a f)+b (b d+2 a g) x}{b x^3+a}dx+3 a b h \int \frac {x^2}{b x^3+a}dx}{3 a b^2}+\frac {x \left (x (b d-a g)+x^2 (b e-a h)-a f+b c\right )}{3 a b \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 792 |
| \(\displaystyle \frac {\int \frac {b (2 b c+a f)+b (b d+2 a g) x}{b x^3+a}dx+a h \log \left (a+b x^3\right )}{3 a b^2}+\frac {x \left (x (b d-a g)+x^2 (b e-a h)-a f+b c\right )}{3 a b \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 2399 |
| \(\displaystyle \frac {\frac {\int \frac {b \left (\sqrt [3]{a} \left (2 \sqrt [3]{b} (2 b c+a f)+\sqrt [3]{a} (b d+2 a g)\right )-\sqrt [3]{b} \left (\sqrt [3]{b} (2 b c+a f)-\sqrt [3]{a} (b d+2 a g)\right ) x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3} \sqrt [3]{b}}+\frac {b^{2/3} \left (\sqrt [3]{b} (a f+2 b c)-\sqrt [3]{a} (2 a g+b d)\right ) \int \frac {1}{\sqrt [3]{b} x+\sqrt [3]{a}}dx}{3 a^{2/3}}+a h \log \left (a+b x^3\right )}{3 a b^2}+\frac {x \left (x (b d-a g)+x^2 (b e-a h)-a f+b c\right )}{3 a b \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {\frac {\int \frac {b \left (\sqrt [3]{a} \left (2 \sqrt [3]{b} (2 b c+a f)+\sqrt [3]{a} (b d+2 a g)\right )-\sqrt [3]{b} \left (\sqrt [3]{b} (2 b c+a f)-\sqrt [3]{a} (b d+2 a g)\right ) x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3} \sqrt [3]{b}}+\frac {\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (a f+2 b c)-\sqrt [3]{a} (2 a g+b d)\right )}{3 a^{2/3}}+a h \log \left (a+b x^3\right )}{3 a b^2}+\frac {x \left (x (b d-a g)+x^2 (b e-a h)-a f+b c\right )}{3 a b \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {b^{2/3} \int \frac {\sqrt [3]{a} \left (2 \sqrt [3]{b} (2 b c+a f)+\sqrt [3]{a} (b d+2 a g)\right )-\sqrt [3]{b} \left (\sqrt [3]{b} (2 b c+a f)-\sqrt [3]{a} (b d+2 a g)\right ) x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3}}+\frac {\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (a f+2 b c)-\sqrt [3]{a} (2 a g+b d)\right )}{3 a^{2/3}}+a h \log \left (a+b x^3\right )}{3 a b^2}+\frac {x \left (x (b d-a g)+x^2 (b e-a h)-a f+b c\right )}{3 a b \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {\frac {b^{2/3} \left (\frac {3}{2} \sqrt [3]{a} \left (2 a^{4/3} g+\sqrt [3]{a} b d+a \sqrt [3]{b} f+2 b^{4/3} c\right ) \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {1}{2} \left (-\frac {\sqrt [3]{a} (2 a g+b d)}{\sqrt [3]{b}}+a f+2 b c\right ) \int -\frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx\right )}{3 a^{2/3}}+\frac {\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (a f+2 b c)-\sqrt [3]{a} (2 a g+b d)\right )}{3 a^{2/3}}+a h \log \left (a+b x^3\right )}{3 a b^2}+\frac {x \left (x (b d-a g)+x^2 (b e-a h)-a f+b c\right )}{3 a b \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {b^{2/3} \left (\frac {3}{2} \sqrt [3]{a} \left (2 a^{4/3} g+\sqrt [3]{a} b d+a \sqrt [3]{b} f+2 b^{4/3} c\right ) \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {1}{2} \left (-\frac {\sqrt [3]{a} (2 a g+b d)}{\sqrt [3]{b}}+a f+2 b c\right ) \int \frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx\right )}{3 a^{2/3}}+\frac {\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (a f+2 b c)-\sqrt [3]{a} (2 a g+b d)\right )}{3 a^{2/3}}+a h \log \left (a+b x^3\right )}{3 a b^2}+\frac {x \left (x (b d-a g)+x^2 (b e-a h)-a f+b c\right )}{3 a b \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {b^{2/3} \left (\frac {3}{2} \sqrt [3]{a} \left (2 a^{4/3} g+\sqrt [3]{a} b d+a \sqrt [3]{b} f+2 b^{4/3} c\right ) \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {1}{2} \sqrt [3]{b} \left (-\frac {\sqrt [3]{a} (2 a g+b d)}{\sqrt [3]{b}}+a f+2 b c\right ) \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx\right )}{3 a^{2/3}}+\frac {\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (a f+2 b c)-\sqrt [3]{a} (2 a g+b d)\right )}{3 a^{2/3}}+a h \log \left (a+b x^3\right )}{3 a b^2}+\frac {x \left (x (b d-a g)+x^2 (b e-a h)-a f+b c\right )}{3 a b \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {b^{2/3} \left (\frac {1}{2} \sqrt [3]{b} \left (-\frac {\sqrt [3]{a} (2 a g+b d)}{\sqrt [3]{b}}+a f+2 b c\right ) \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {3 \left (2 a^{4/3} g+\sqrt [3]{a} b d+a \sqrt [3]{b} f+2 b^{4/3} c\right ) \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{\sqrt [3]{b}}\right )}{3 a^{2/3}}+\frac {\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (a f+2 b c)-\sqrt [3]{a} (2 a g+b d)\right )}{3 a^{2/3}}+a h \log \left (a+b x^3\right )}{3 a b^2}+\frac {x \left (x (b d-a g)+x^2 (b e-a h)-a f+b c\right )}{3 a b \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {b^{2/3} \left (\frac {1}{2} \sqrt [3]{b} \left (-\frac {\sqrt [3]{a} (2 a g+b d)}{\sqrt [3]{b}}+a f+2 b c\right ) \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right ) \left (2 a^{4/3} g+\sqrt [3]{a} b d+a \sqrt [3]{b} f+2 b^{4/3} c\right )}{\sqrt [3]{b}}\right )}{3 a^{2/3}}+\frac {\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (a f+2 b c)-\sqrt [3]{a} (2 a g+b d)\right )}{3 a^{2/3}}+a h \log \left (a+b x^3\right )}{3 a b^2}+\frac {x \left (x (b d-a g)+x^2 (b e-a h)-a f+b c\right )}{3 a b \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {b^{2/3} \left (-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right ) \left (2 a^{4/3} g+\sqrt [3]{a} b d+a \sqrt [3]{b} f+2 b^{4/3} c\right )}{\sqrt [3]{b}}-\frac {1}{2} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-\frac {\sqrt [3]{a} (2 a g+b d)}{\sqrt [3]{b}}+a f+2 b c\right )\right )}{3 a^{2/3}}+\frac {\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (a f+2 b c)-\sqrt [3]{a} (2 a g+b d)\right )}{3 a^{2/3}}+a h \log \left (a+b x^3\right )}{3 a b^2}+\frac {x \left (x (b d-a g)+x^2 (b e-a h)-a f+b c\right )}{3 a b \left (a+b x^3\right )}\) |
(x*(b*c - a*f + (b*d - a*g)*x + (b*e - a*h)*x^2))/(3*a*b*(a + b*x^3)) + (( b^(1/3)*(b^(1/3)*(2*b*c + a*f) - a^(1/3)*(b*d + 2*a*g))*Log[a^(1/3) + b^(1 /3)*x])/(3*a^(2/3)) + (b^(2/3)*(-((Sqrt[3]*(2*b^(4/3)*c + a^(1/3)*b*d + a* b^(1/3)*f + 2*a^(4/3)*g)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(1 /3)) - ((2*b*c + a*f - (a^(1/3)*(b*d + 2*a*g))/b^(1/3))*Log[a^(2/3) - a^(1 /3)*b^(1/3)*x + b^(2/3)*x^2])/2))/(3*a^(2/3)) + a*h*Log[a + b*x^3])/(3*a*b ^2)
3.5.16.3.1 Defintions of rubi rules used
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[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveConten t[a + b*x^n, x]]/(b*n), x] /; FreeQ[{a, b, m, n}, x] && EqQ[m, n - 1]
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_)*((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]
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] + Si mp[C Int[x^2/(a + b*x^3), x], x] /; EqQ[a*B^3 - b*A^3, 0] || !RationalQ[ a/b]] /; FreeQ[{a, b}, x] && PolyQ[P2, x, 2]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.52 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.43
| method | result | size |
| risch | \(\frac {-\frac {\left (a g -b d \right ) x^{2}}{3 a b}-\frac {\left (a f -b c \right ) x}{3 a b}+\frac {a h -b e}{3 b^{2}}}{b \,x^{3}+a}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\left (3 h \,\textit {\_R}^{2}+\frac {\left (2 a g +b d \right ) \textit {\_R}}{a}+\frac {a f +2 b c}{a}\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}}{9 b^{2}}\) | \(120\) |
| default | \(\frac {-\frac {\left (a g -b d \right ) x^{2}}{3 a b}-\frac {\left (a f -b c \right ) x}{3 a b}+\frac {a h -b e}{3 b^{2}}}{b \,x^{3}+a}+\frac {\left (a f +2 b c \right ) \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )+\left (2 a g +b d \right ) \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )+\frac {a h \ln \left (b \,x^{3}+a \right )}{b}}{3 b a}\) | \(283\) |
(-1/3*(a*g-b*d)/a/b*x^2-1/3*(a*f-b*c)/a/b*x+1/3*(a*h-b*e)/b^2)/(b*x^3+a)+1 /9/b^2*sum((3*h*_R^2+1/a*(2*a*g+b*d)*_R+(a*f+2*b*c)/a)/_R^2*ln(x-_R),_R=Ro otOf(_Z^3*b+a))
Result contains complex when optimal does not.
Time = 1.52 (sec) , antiderivative size = 12636, normalized size of antiderivative = 45.78 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{\left (a+b x^3\right )^2} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{\left (a+b x^3\right )^2} \, dx=\text {Timed out} \]
Time = 0.29 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.06 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{\left (a+b x^3\right )^2} \, dx=-\frac {a b e - a^{2} h - {\left (b^{2} d - a b g\right )} x^{2} - {\left (b^{2} c - a b f\right )} x}{3 \, {\left (a b^{3} x^{3} + a^{2} b^{2}\right )}} + \frac {\sqrt {3} {\left (b^{2} d \left (\frac {a}{b}\right )^{\frac {2}{3}} + 2 \, a b g \left (\frac {a}{b}\right )^{\frac {2}{3}} + 2 \, b^{2} c \left (\frac {a}{b}\right )^{\frac {1}{3}} + a b f \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a^{2} b^{2}} + \frac {{\left (6 \, a h \left (\frac {a}{b}\right )^{\frac {2}{3}} + b d \left (\frac {a}{b}\right )^{\frac {1}{3}} + 2 \, a g \left (\frac {a}{b}\right )^{\frac {1}{3}} - 2 \, b c - a f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, a b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (3 \, a h \left (\frac {a}{b}\right )^{\frac {2}{3}} - b d \left (\frac {a}{b}\right )^{\frac {1}{3}} - 2 \, a g \left (\frac {a}{b}\right )^{\frac {1}{3}} + 2 \, b c + a f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, a b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]
-1/3*(a*b*e - a^2*h - (b^2*d - a*b*g)*x^2 - (b^2*c - a*b*f)*x)/(a*b^3*x^3 + a^2*b^2) + 1/9*sqrt(3)*(b^2*d*(a/b)^(2/3) + 2*a*b*g*(a/b)^(2/3) + 2*b^2* c*(a/b)^(1/3) + a*b*f*(a/b)^(1/3))*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/ (a/b)^(1/3))/(a^2*b^2) + 1/18*(6*a*h*(a/b)^(2/3) + b*d*(a/b)^(1/3) + 2*a*g *(a/b)^(1/3) - 2*b*c - a*f)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a*b^2* (a/b)^(2/3)) + 1/9*(3*a*h*(a/b)^(2/3) - b*d*(a/b)^(1/3) - 2*a*g*(a/b)^(1/3 ) + 2*b*c + a*f)*log(x + (a/b)^(1/3))/(a*b^2*(a/b)^(2/3))
Time = 0.27 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.09 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{\left (a+b x^3\right )^2} \, dx=\frac {h \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{2}} - \frac {\sqrt {3} {\left (2 \, b^{2} c + a b f - \left (-a b^{2}\right )^{\frac {1}{3}} b d - 2 \, \left (-a b^{2}\right )^{\frac {1}{3}} a g\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, \left (-a b^{2}\right )^{\frac {2}{3}} a b} - \frac {{\left (2 \, b^{2} c + a b f + \left (-a b^{2}\right )^{\frac {1}{3}} b d + 2 \, \left (-a b^{2}\right )^{\frac {1}{3}} a g\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, \left (-a b^{2}\right )^{\frac {2}{3}} a b} + \frac {{\left (b d - a g\right )} x^{2} + {\left (b c - a f\right )} x - \frac {a b e - a^{2} h}{b}}{3 \, {\left (b x^{3} + a\right )} a b} - \frac {{\left (a b^{3} d \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 2 \, a^{2} b^{2} g \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 2 \, a b^{3} c + a^{2} b^{2} f\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a^{3} b^{3}} \]
1/3*h*log(abs(b*x^3 + a))/b^2 - 1/9*sqrt(3)*(2*b^2*c + a*b*f - (-a*b^2)^(1 /3)*b*d - 2*(-a*b^2)^(1/3)*a*g)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(- a/b)^(1/3))/((-a*b^2)^(2/3)*a*b) - 1/18*(2*b^2*c + a*b*f + (-a*b^2)^(1/3)* b*d + 2*(-a*b^2)^(1/3)*a*g)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/((-a* b^2)^(2/3)*a*b) + 1/3*((b*d - a*g)*x^2 + (b*c - a*f)*x - (a*b*e - a^2*h)/b )/((b*x^3 + a)*a*b) - 1/9*(a*b^3*d*(-a/b)^(1/3) + 2*a^2*b^2*g*(-a/b)^(1/3) + 2*a*b^3*c + a^2*b^2*f)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a^3*b^3 )
Time = 9.77 (sec) , antiderivative size = 835, normalized size of antiderivative = 3.03 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{\left (a+b x^3\right )^2} \, dx=\left (\sum _{k=1}^3\ln \left (\frac {\mathrm {root}\left (729\,a^5\,b^6\,z^3-729\,a^5\,b^4\,h\,z^2+54\,a^4\,b^3\,f\,g\,z+108\,a^3\,b^4\,c\,g\,z+27\,a^3\,b^4\,d\,f\,z+54\,a^2\,b^5\,c\,d\,z+243\,a^5\,b^2\,h^2\,z-18\,a^4\,b\,f\,g\,h-36\,a^3\,b^2\,c\,g\,h-9\,a^3\,b^2\,d\,f\,h-18\,a^2\,b^3\,c\,d\,h-12\,a\,b^4\,c^2\,f+12\,a^3\,b^2\,d\,g^2+6\,a^2\,b^3\,d^2\,g-6\,a^2\,b^3\,c\,f^2+8\,a^4\,b\,g^3+a\,b^4\,d^3-27\,a^5\,h^3-8\,b^5\,c^3-a^3\,b^2\,f^3,z,k\right )\,\left (-6\,a^2\,h+\mathrm {root}\left (729\,a^5\,b^6\,z^3-729\,a^5\,b^4\,h\,z^2+54\,a^4\,b^3\,f\,g\,z+108\,a^3\,b^4\,c\,g\,z+27\,a^3\,b^4\,d\,f\,z+54\,a^2\,b^5\,c\,d\,z+243\,a^5\,b^2\,h^2\,z-18\,a^4\,b\,f\,g\,h-36\,a^3\,b^2\,c\,g\,h-9\,a^3\,b^2\,d\,f\,h-18\,a^2\,b^3\,c\,d\,h-12\,a\,b^4\,c^2\,f+12\,a^3\,b^2\,d\,g^2+6\,a^2\,b^3\,d^2\,g-6\,a^2\,b^3\,c\,f^2+8\,a^4\,b\,g^3+a\,b^4\,d^3-27\,a^5\,h^3-8\,b^5\,c^3-a^3\,b^2\,f^3,z,k\right )\,a^2\,b^2\,9+2\,b^2\,c\,x+a\,b\,f\,x\right )}{a}+\frac {9\,a^3\,h^2+2\,b^3\,c\,d+4\,a\,b^2\,c\,g+a\,b^2\,d\,f+2\,a^2\,b\,f\,g}{9\,a^2\,b^2}+\frac {x\,\left (4\,a^2\,g^2-3\,f\,h\,a^2+4\,a\,b\,d\,g-6\,c\,h\,a\,b+b^2\,d^2\right )}{9\,a^2\,b}\right )\,\mathrm {root}\left (729\,a^5\,b^6\,z^3-729\,a^5\,b^4\,h\,z^2+54\,a^4\,b^3\,f\,g\,z+108\,a^3\,b^4\,c\,g\,z+27\,a^3\,b^4\,d\,f\,z+54\,a^2\,b^5\,c\,d\,z+243\,a^5\,b^2\,h^2\,z-18\,a^4\,b\,f\,g\,h-36\,a^3\,b^2\,c\,g\,h-9\,a^3\,b^2\,d\,f\,h-18\,a^2\,b^3\,c\,d\,h-12\,a\,b^4\,c^2\,f+12\,a^3\,b^2\,d\,g^2+6\,a^2\,b^3\,d^2\,g-6\,a^2\,b^3\,c\,f^2+8\,a^4\,b\,g^3+a\,b^4\,d^3-27\,a^5\,h^3-8\,b^5\,c^3-a^3\,b^2\,f^3,z,k\right )\right )+\frac {\frac {x\,\left (b\,c-a\,f\right )}{3\,a\,b}-\frac {b\,e-a\,h}{3\,b^2}+\frac {x^2\,\left (b\,d-a\,g\right )}{3\,a\,b}}{b\,x^3+a} \]
symsum(log((root(729*a^5*b^6*z^3 - 729*a^5*b^4*h*z^2 + 54*a^4*b^3*f*g*z + 108*a^3*b^4*c*g*z + 27*a^3*b^4*d*f*z + 54*a^2*b^5*c*d*z + 243*a^5*b^2*h^2* z - 18*a^4*b*f*g*h - 36*a^3*b^2*c*g*h - 9*a^3*b^2*d*f*h - 18*a^2*b^3*c*d*h - 12*a*b^4*c^2*f + 12*a^3*b^2*d*g^2 + 6*a^2*b^3*d^2*g - 6*a^2*b^3*c*f^2 + 8*a^4*b*g^3 + a*b^4*d^3 - 27*a^5*h^3 - 8*b^5*c^3 - a^3*b^2*f^3, z, k)*(9* root(729*a^5*b^6*z^3 - 729*a^5*b^4*h*z^2 + 54*a^4*b^3*f*g*z + 108*a^3*b^4* c*g*z + 27*a^3*b^4*d*f*z + 54*a^2*b^5*c*d*z + 243*a^5*b^2*h^2*z - 18*a^4*b *f*g*h - 36*a^3*b^2*c*g*h - 9*a^3*b^2*d*f*h - 18*a^2*b^3*c*d*h - 12*a*b^4* c^2*f + 12*a^3*b^2*d*g^2 + 6*a^2*b^3*d^2*g - 6*a^2*b^3*c*f^2 + 8*a^4*b*g^3 + a*b^4*d^3 - 27*a^5*h^3 - 8*b^5*c^3 - a^3*b^2*f^3, z, k)*a^2*b^2 - 6*a^2 *h + 2*b^2*c*x + a*b*f*x))/a + (9*a^3*h^2 + 2*b^3*c*d + 4*a*b^2*c*g + a*b^ 2*d*f + 2*a^2*b*f*g)/(9*a^2*b^2) + (x*(b^2*d^2 + 4*a^2*g^2 - 3*a^2*f*h - 6 *a*b*c*h + 4*a*b*d*g))/(9*a^2*b))*root(729*a^5*b^6*z^3 - 729*a^5*b^4*h*z^2 + 54*a^4*b^3*f*g*z + 108*a^3*b^4*c*g*z + 27*a^3*b^4*d*f*z + 54*a^2*b^5*c* d*z + 243*a^5*b^2*h^2*z - 18*a^4*b*f*g*h - 36*a^3*b^2*c*g*h - 9*a^3*b^2*d* f*h - 18*a^2*b^3*c*d*h - 12*a*b^4*c^2*f + 12*a^3*b^2*d*g^2 + 6*a^2*b^3*d^2 *g - 6*a^2*b^3*c*f^2 + 8*a^4*b*g^3 + a*b^4*d^3 - 27*a^5*h^3 - 8*b^5*c^3 - a^3*b^2*f^3, z, k), k, 1, 3) + ((x*(b*c - a*f))/(3*a*b) - (b*e - a*h)/(3*b ^2) + (x^2*(b*d - a*g))/(3*a*b))/(a + b*x^3)