Integrand size = 35, antiderivative size = 313 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{\left (a+b x^3\right )^3} \, dx=\frac {x \left (b c-a f+(b d-a g) x+(b e-a h) x^2\right )}{6 a b \left (a+b x^3\right )^2}-\frac {3 a (b e+a h)-b x (5 b c+a f+2 (2 b d+a g) x)}{18 a^2 b^2 \left (a+b x^3\right )}-\frac {\left (5 b^{4/3} c+2 \sqrt [3]{a} b d+a \sqrt [3]{b} f+a^{4/3} g\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{8/3} b^{5/3}}+\frac {\left (\sqrt [3]{b} (5 b c+a f)-\sqrt [3]{a} (2 b d+a g)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{8/3} b^{5/3}}-\frac {\left (\sqrt [3]{b} (5 b c+a f)-\sqrt [3]{a} (2 b d+a g)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{8/3} b^{5/3}} \]
1/6*x*(b*c-a*f+(-a*g+b*d)*x+(-a*h+b*e)*x^2)/a/b/(b*x^3+a)^2+1/18*(-3*a*(a* h+b*e)+b*x*(5*b*c+a*f+2*(a*g+2*b*d)*x))/a^2/b^2/(b*x^3+a)+1/27*(b^(1/3)*(a *f+5*b*c)-a^(1/3)*(a*g+2*b*d))*ln(a^(1/3)+b^(1/3)*x)/a^(8/3)/b^(5/3)-1/54* (b^(1/3)*(a*f+5*b*c)-a^(1/3)*(a*g+2*b*d))*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^( 2/3)*x^2)/a^(8/3)/b^(5/3)-1/27*(5*b^(4/3)*c+2*a^(1/3)*b*d+a*b^(1/3)*f+a^(4 /3)*g)*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)/a^(1/3)*3^(1/2))/a^(8/3)/b^(5/3)*3 ^(1/2)
Time = 0.25 (sec) , antiderivative size = 295, normalized size of antiderivative = 0.94 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{\left (a+b x^3\right )^3} \, dx=\frac {\frac {3 a^{2/3} \left (-6 a^2 h+b^2 x (5 c+4 d x)+a b x (f+2 g x)\right )}{a+b x^3}+\frac {9 a^{5/3} \left (a^2 h+b^2 x (c+d x)-a b (e+x (f+g x))\right )}{\left (a+b x^3\right )^2}-2 \sqrt {3} \sqrt [3]{b} \left (5 b^{4/3} c+2 \sqrt [3]{a} b d+a \sqrt [3]{b} f+a^{4/3} g\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )+2 \sqrt [3]{b} \left (5 b^{4/3} c-2 \sqrt [3]{a} b d+a \sqrt [3]{b} f-a^{4/3} g\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+\sqrt [3]{b} \left (-5 b^{4/3} c+2 \sqrt [3]{a} b d-a \sqrt [3]{b} f+a^{4/3} g\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{8/3} b^2} \]
((3*a^(2/3)*(-6*a^2*h + b^2*x*(5*c + 4*d*x) + a*b*x*(f + 2*g*x)))/(a + b*x ^3) + (9*a^(5/3)*(a^2*h + b^2*x*(c + d*x) - a*b*(e + x*(f + g*x))))/(a + b *x^3)^2 - 2*Sqrt[3]*b^(1/3)*(5*b^(4/3)*c + 2*a^(1/3)*b*d + a*b^(1/3)*f + a ^(4/3)*g)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 2*b^(1/3)*(5*b^(4/ 3)*c - 2*a^(1/3)*b*d + a*b^(1/3)*f - a^(4/3)*g)*Log[a^(1/3) + b^(1/3)*x] + b^(1/3)*(-5*b^(4/3)*c + 2*a^(1/3)*b*d - a*b^(1/3)*f + a^(4/3)*g)*Log[a^(2 /3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(8/3)*b^2)
Time = 0.74 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.02, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.343, Rules used = {2397, 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.
| \(\displaystyle \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{\left (a+b x^3\right )^3} \, dx\) |
| \(\Big \downarrow \) 2397 |
| \(\displaystyle \frac {x \left (x (b d-a g)+x^2 (b e-a h)-a f+b c\right )}{6 a b \left (a+b x^3\right )^2}-\frac {\int -\frac {3 b (b e+a h) x^2+2 b (2 b d+a g) x+b (5 b c+a f)}{\left (b x^3+a\right )^2}dx}{6 a b^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {3 b (b e+a h) x^2+2 b (2 b d+a g) x+b (5 b c+a f)}{\left (b x^3+a\right )^2}dx}{6 a b^2}+\frac {x \left (x (b d-a g)+x^2 (b e-a h)-a f+b c\right )}{6 a b \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 2393 |
| \(\displaystyle \frac {-\frac {\int -\frac {2 b (5 b c+a f+(2 b d+a g) x)}{b x^3+a}dx}{3 a}-\frac {3 a (a h+b e)-b x (2 x (a g+2 b d)+a f+5 b c)}{3 a \left (a+b x^3\right )}}{6 a b^2}+\frac {x \left (x (b d-a g)+x^2 (b e-a h)-a f+b c\right )}{6 a b \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2 b \int \frac {5 b c+a f+(2 b d+a g) x}{b x^3+a}dx}{3 a}-\frac {3 a (a h+b e)-b x (2 x (a g+2 b d)+a f+5 b c)}{3 a \left (a+b x^3\right )}}{6 a b^2}+\frac {x \left (x (b d-a g)+x^2 (b e-a h)-a f+b c\right )}{6 a b \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 2399 |
| \(\displaystyle \frac {\frac {2 b \left (\frac {\int \frac {\sqrt [3]{a} \left (2 \sqrt [3]{b} (5 b c+a f)+\sqrt [3]{a} (2 b d+a g)\right )-\sqrt [3]{b} \left (\sqrt [3]{b} (5 b c+a f)-\sqrt [3]{a} (2 b d+a g)\right ) x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3} \sqrt [3]{b}}+\frac {\left (-\frac {\sqrt [3]{a} (a g+2 b d)}{\sqrt [3]{b}}+a f+5 b c\right ) \int \frac {1}{\sqrt [3]{b} x+\sqrt [3]{a}}dx}{3 a^{2/3}}\right )}{3 a}-\frac {3 a (a h+b e)-b x (2 x (a g+2 b d)+a f+5 b c)}{3 a \left (a+b x^3\right )}}{6 a b^2}+\frac {x \left (x (b d-a g)+x^2 (b e-a h)-a f+b c\right )}{6 a b \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {\frac {2 b \left (\frac {\int \frac {\sqrt [3]{a} \left (2 \sqrt [3]{b} (5 b c+a f)+\sqrt [3]{a} (2 b d+a g)\right )-\sqrt [3]{b} \left (\sqrt [3]{b} (5 b c+a f)-\sqrt [3]{a} (2 b d+a g)\right ) x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3} \sqrt [3]{b}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-\frac {\sqrt [3]{a} (a g+2 b d)}{\sqrt [3]{b}}+a f+5 b c\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{3 a}-\frac {3 a (a h+b e)-b x (2 x (a g+2 b d)+a f+5 b c)}{3 a \left (a+b x^3\right )}}{6 a b^2}+\frac {x \left (x (b d-a g)+x^2 (b e-a h)-a f+b c\right )}{6 a b \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {\frac {2 b \left (\frac {\frac {3}{2} \sqrt [3]{a} \left (a^{4/3} g+2 \sqrt [3]{a} b d+a \sqrt [3]{b} f+5 b^{4/3} c\right ) \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {1}{2} \left (-\frac {\sqrt [3]{a} (a g+2 b d)}{\sqrt [3]{b}}+a f+5 b c\right ) \int -\frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3} \sqrt [3]{b}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-\frac {\sqrt [3]{a} (a g+2 b d)}{\sqrt [3]{b}}+a f+5 b c\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{3 a}-\frac {3 a (a h+b e)-b x (2 x (a g+2 b d)+a f+5 b c)}{3 a \left (a+b x^3\right )}}{6 a b^2}+\frac {x \left (x (b d-a g)+x^2 (b e-a h)-a f+b c\right )}{6 a b \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 b \left (\frac {\frac {3}{2} \sqrt [3]{a} \left (a^{4/3} g+2 \sqrt [3]{a} b d+a \sqrt [3]{b} f+5 b^{4/3} c\right ) \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {1}{2} \left (-\frac {\sqrt [3]{a} (a g+2 b d)}{\sqrt [3]{b}}+a f+5 b c\right ) \int \frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3} \sqrt [3]{b}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-\frac {\sqrt [3]{a} (a g+2 b d)}{\sqrt [3]{b}}+a f+5 b c\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{3 a}-\frac {3 a (a h+b e)-b x (2 x (a g+2 b d)+a f+5 b c)}{3 a \left (a+b x^3\right )}}{6 a b^2}+\frac {x \left (x (b d-a g)+x^2 (b e-a h)-a f+b c\right )}{6 a b \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2 b \left (\frac {\frac {3}{2} \sqrt [3]{a} \left (a^{4/3} g+2 \sqrt [3]{a} b d+a \sqrt [3]{b} f+5 b^{4/3} c\right ) \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {1}{2} \sqrt [3]{b} \left (-\frac {\sqrt [3]{a} (a g+2 b d)}{\sqrt [3]{b}}+a f+5 b c\right ) \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3} \sqrt [3]{b}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-\frac {\sqrt [3]{a} (a g+2 b d)}{\sqrt [3]{b}}+a f+5 b c\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{3 a}-\frac {3 a (a h+b e)-b x (2 x (a g+2 b d)+a f+5 b c)}{3 a \left (a+b x^3\right )}}{6 a b^2}+\frac {x \left (x (b d-a g)+x^2 (b e-a h)-a f+b c\right )}{6 a b \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {2 b \left (\frac {\frac {1}{2} \sqrt [3]{b} \left (-\frac {\sqrt [3]{a} (a g+2 b d)}{\sqrt [3]{b}}+a f+5 b c\right ) \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {3 \left (a^{4/3} g+2 \sqrt [3]{a} b d+a \sqrt [3]{b} f+5 b^{4/3} c\right ) \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{\sqrt [3]{b}}}{3 a^{2/3} \sqrt [3]{b}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-\frac {\sqrt [3]{a} (a g+2 b d)}{\sqrt [3]{b}}+a f+5 b c\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{3 a}-\frac {3 a (a h+b e)-b x (2 x (a g+2 b d)+a f+5 b c)}{3 a \left (a+b x^3\right )}}{6 a b^2}+\frac {x \left (x (b d-a g)+x^2 (b e-a h)-a f+b c\right )}{6 a b \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {2 b \left (\frac {\frac {1}{2} \sqrt [3]{b} \left (-\frac {\sqrt [3]{a} (a g+2 b d)}{\sqrt [3]{b}}+a f+5 b c\right ) \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right ) \left (a^{4/3} g+2 \sqrt [3]{a} b d+a \sqrt [3]{b} f+5 b^{4/3} c\right )}{\sqrt [3]{b}}}{3 a^{2/3} \sqrt [3]{b}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-\frac {\sqrt [3]{a} (a g+2 b d)}{\sqrt [3]{b}}+a f+5 b c\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{3 a}-\frac {3 a (a h+b e)-b x (2 x (a g+2 b d)+a f+5 b c)}{3 a \left (a+b x^3\right )}}{6 a b^2}+\frac {x \left (x (b d-a g)+x^2 (b e-a h)-a f+b c\right )}{6 a b \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {2 b \left (\frac {-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right ) \left (a^{4/3} g+2 \sqrt [3]{a} b d+a \sqrt [3]{b} f+5 b^{4/3} c\right )}{\sqrt [3]{b}}-\frac {1}{2} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-\frac {\sqrt [3]{a} (a g+2 b d)}{\sqrt [3]{b}}+a f+5 b c\right )}{3 a^{2/3} \sqrt [3]{b}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-\frac {\sqrt [3]{a} (a g+2 b d)}{\sqrt [3]{b}}+a f+5 b c\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{3 a}-\frac {3 a (a h+b e)-b x (2 x (a g+2 b d)+a f+5 b c)}{3 a \left (a+b x^3\right )}}{6 a b^2}+\frac {x \left (x (b d-a g)+x^2 (b e-a h)-a f+b c\right )}{6 a b \left (a+b x^3\right )^2}\) |
(x*(b*c - a*f + (b*d - a*g)*x + (b*e - a*h)*x^2))/(6*a*b*(a + b*x^3)^2) + (-1/3*(3*a*(b*e + a*h) - b*x*(5*b*c + a*f + 2*(2*b*d + a*g)*x))/(a*(a + b* x^3)) + (2*b*(((5*b*c + a*f - (a^(1/3)*(2*b*d + a*g))/b^(1/3))*Log[a^(1/3) + b^(1/3)*x])/(3*a^(2/3)*b^(1/3)) + (-((Sqrt[3]*(5*b^(4/3)*c + 2*a^(1/3)* b*d + a*b^(1/3)*f + a^(4/3)*g)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] )/b^(1/3)) - ((5*b*c + a*f - (a^(1/3)*(2*b*d + 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)*b^(1/3))))/(3*a))/(6*a*b ^2)
3.5.25.3.1 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[((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] :> 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[(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 = 1.71 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.48
| method | result | size |
| risch | \(\frac {\frac {\left (a g +2 b d \right ) x^{5}}{9 a^{2}}+\frac {\left (a f +5 b c \right ) x^{4}}{18 a^{2}}-\frac {h \,x^{3}}{3 b}-\frac {\left (a g -7 b d \right ) x^{2}}{18 a b}-\frac {\left (a f -4 b c \right ) x}{9 a b}-\frac {a h +b e}{6 b^{2}}}{\left (b \,x^{3}+a \right )^{2}}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\left (\left (a g +2 b d \right ) \textit {\_R} +a f +5 b c \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}}{27 a^{2} b^{2}}\) | \(149\) |
| default | \(\frac {\frac {\left (a g +2 b d \right ) x^{5}}{9 a^{2}}+\frac {\left (a f +5 b c \right ) x^{4}}{18 a^{2}}-\frac {h \,x^{3}}{3 b}-\frac {\left (a g -7 b d \right ) x^{2}}{18 a b}-\frac {\left (a f -4 b c \right ) x}{9 a b}-\frac {a h +b e}{6 b^{2}}}{\left (b \,x^{3}+a \right )^{2}}+\frac {\left (a f +5 b c \right ) \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )+\left (a g +2 b d \right ) \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 a^{2} b}\) | \(309\) |
(1/9*(a*g+2*b*d)/a^2*x^5+1/18*(a*f+5*b*c)/a^2*x^4-1/3*h*x^3/b-1/18*(a*g-7* b*d)/a/b*x^2-1/9*(a*f-4*b*c)/a/b*x-1/6*(a*h+b*e)/b^2)/(b*x^3+a)^2+1/27/a^2 /b^2*sum(((a*g+2*b*d)*_R+a*f+5*b*c)/_R^2*ln(x-_R),_R=RootOf(_Z^3*b+a))
Result contains complex when optimal does not.
Time = 1.52 (sec) , antiderivative size = 6984, normalized size of antiderivative = 22.31 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{\left (a+b x^3\right )^3} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{\left (a+b x^3\right )^3} \, dx=\text {Timed out} \]
Time = 0.28 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.04 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{\left (a+b x^3\right )^3} \, dx=-\frac {6 \, a^{2} b h x^{3} - 2 \, {\left (2 \, b^{3} d + a b^{2} g\right )} x^{5} - {\left (5 \, b^{3} c + a b^{2} f\right )} x^{4} + 3 \, a^{2} b e + 3 \, a^{3} h - {\left (7 \, a b^{2} d - a^{2} b g\right )} x^{2} - 2 \, {\left (4 \, a b^{2} c - a^{2} b f\right )} x}{18 \, {\left (a^{2} b^{4} x^{6} + 2 \, a^{3} b^{3} x^{3} + a^{4} b^{2}\right )}} + \frac {\sqrt {3} {\left (2 \, b d \left (\frac {a}{b}\right )^{\frac {1}{3}} + a g \left (\frac {a}{b}\right )^{\frac {1}{3}} + 5 \, b c + a f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{2} b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (2 \, b d \left (\frac {a}{b}\right )^{\frac {1}{3}} + a g \left (\frac {a}{b}\right )^{\frac {1}{3}} - 5 \, b c - a f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a^{2} b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (2 \, b d \left (\frac {a}{b}\right )^{\frac {1}{3}} + a g \left (\frac {a}{b}\right )^{\frac {1}{3}} - 5 \, b c - a f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, a^{2} b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]
-1/18*(6*a^2*b*h*x^3 - 2*(2*b^3*d + a*b^2*g)*x^5 - (5*b^3*c + a*b^2*f)*x^4 + 3*a^2*b*e + 3*a^3*h - (7*a*b^2*d - a^2*b*g)*x^2 - 2*(4*a*b^2*c - a^2*b* f)*x)/(a^2*b^4*x^6 + 2*a^3*b^3*x^3 + a^4*b^2) + 1/27*sqrt(3)*(2*b*d*(a/b)^ (1/3) + a*g*(a/b)^(1/3) + 5*b*c + a*f)*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/ 3))/(a/b)^(1/3))/(a^2*b^2*(a/b)^(2/3)) + 1/54*(2*b*d*(a/b)^(1/3) + a*g*(a/ b)^(1/3) - 5*b*c - a*f)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a^2*b^2*(a /b)^(2/3)) - 1/27*(2*b*d*(a/b)^(1/3) + a*g*(a/b)^(1/3) - 5*b*c - a*f)*log( x + (a/b)^(1/3))/(a^2*b^2*(a/b)^(2/3))
Time = 0.29 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.05 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{\left (a+b x^3\right )^3} \, dx=-\frac {\sqrt {3} {\left (5 \, b^{2} c + a b f - 2 \, \left (-a b^{2}\right )^{\frac {1}{3}} b d - \left (-a b^{2}\right )^{\frac {1}{3}} a g\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{2} b} - \frac {{\left (5 \, b^{2} c + a b f + 2 \, \left (-a b^{2}\right )^{\frac {1}{3}} b d + \left (-a b^{2}\right )^{\frac {1}{3}} a g\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{2} b} - \frac {{\left (2 \, b d \left (-\frac {a}{b}\right )^{\frac {1}{3}} + a g \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 5 \, b c + a f\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a^{3} b} + \frac {4 \, b^{3} d x^{5} + 2 \, a b^{2} g x^{5} + 5 \, b^{3} c x^{4} + a b^{2} f x^{4} - 6 \, a^{2} b h x^{3} + 7 \, a b^{2} d x^{2} - a^{2} b g x^{2} + 8 \, a b^{2} c x - 2 \, a^{2} b f x - 3 \, a^{2} b e - 3 \, a^{3} h}{18 \, {\left (b x^{3} + a\right )}^{2} a^{2} b^{2}} \]
-1/27*sqrt(3)*(5*b^2*c + a*b*f - 2*(-a*b^2)^(1/3)*b*d - (-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^ 2*b) - 1/54*(5*b^2*c + a*b*f + 2*(-a*b^2)^(1/3)*b*d + (-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^2*b) - 1/27*(2* b*d*(-a/b)^(1/3) + a*g*(-a/b)^(1/3) + 5*b*c + a*f)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a^3*b) + 1/18*(4*b^3*d*x^5 + 2*a*b^2*g*x^5 + 5*b^3*c*x^4 + a*b^2*f*x^4 - 6*a^2*b*h*x^3 + 7*a*b^2*d*x^2 - a^2*b*g*x^2 + 8*a*b^2*c*x - 2*a^2*b*f*x - 3*a^2*b*e - 3*a^3*h)/((b*x^3 + a)^2*a^2*b^2)
Time = 9.29 (sec) , antiderivative size = 630, normalized size of antiderivative = 2.01 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{\left (a+b x^3\right )^3} \, dx=\frac {\frac {x^4\,\left (5\,b\,c+a\,f\right )}{18\,a^2}-\frac {h\,x^3}{3\,b}-\frac {b\,e+a\,h}{6\,b^2}+\frac {x^5\,\left (2\,b\,d+a\,g\right )}{9\,a^2}+\frac {x\,\left (4\,b\,c-a\,f\right )}{9\,a\,b}+\frac {x^2\,\left (7\,b\,d-a\,g\right )}{18\,a\,b}}{a^2+2\,a\,b\,x^3+b^2\,x^6}+\left (\sum _{k=1}^3\ln \left (\mathrm {root}\left (19683\,a^8\,b^5\,z^3+81\,a^5\,b^2\,f\,g\,z+405\,a^4\,b^3\,c\,g\,z+162\,a^4\,b^3\,d\,f\,z+810\,a^3\,b^4\,c\,d\,z+6\,a^3\,b\,d\,g^2-75\,a\,b^3\,c^2\,f+12\,a^2\,b^2\,d^2\,g-15\,a^2\,b^2\,c\,f^2+8\,a\,b^3\,d^3+a^4\,g^3-125\,b^4\,c^3-a^3\,b\,f^3,z,k\right )\,\left (\mathrm {root}\left (19683\,a^8\,b^5\,z^3+81\,a^5\,b^2\,f\,g\,z+405\,a^4\,b^3\,c\,g\,z+162\,a^4\,b^3\,d\,f\,z+810\,a^3\,b^4\,c\,d\,z+6\,a^3\,b\,d\,g^2-75\,a\,b^3\,c^2\,f+12\,a^2\,b^2\,d^2\,g-15\,a^2\,b^2\,c\,f^2+8\,a\,b^3\,d^3+a^4\,g^3-125\,b^4\,c^3-a^3\,b\,f^3,z,k\right )\,a\,b^2\,9+\frac {x\,\left (27\,f\,a^3\,b^2+135\,c\,a^2\,b^3\right )}{81\,a^4\,b}\right )+\frac {10\,b^2\,c\,d+a^2\,f\,g+5\,a\,b\,c\,g+2\,a\,b\,d\,f}{81\,a^4\,b}+\frac {x\,\left (a^2\,g^2+4\,a\,b\,d\,g+4\,b^2\,d^2\right )}{81\,a^4\,b}\right )\,\mathrm {root}\left (19683\,a^8\,b^5\,z^3+81\,a^5\,b^2\,f\,g\,z+405\,a^4\,b^3\,c\,g\,z+162\,a^4\,b^3\,d\,f\,z+810\,a^3\,b^4\,c\,d\,z+6\,a^3\,b\,d\,g^2-75\,a\,b^3\,c^2\,f+12\,a^2\,b^2\,d^2\,g-15\,a^2\,b^2\,c\,f^2+8\,a\,b^3\,d^3+a^4\,g^3-125\,b^4\,c^3-a^3\,b\,f^3,z,k\right )\right ) \]
((x^4*(5*b*c + a*f))/(18*a^2) - (h*x^3)/(3*b) - (b*e + a*h)/(6*b^2) + (x^5 *(2*b*d + a*g))/(9*a^2) + (x*(4*b*c - a*f))/(9*a*b) + (x^2*(7*b*d - a*g))/ (18*a*b))/(a^2 + b^2*x^6 + 2*a*b*x^3) + symsum(log(root(19683*a^8*b^5*z^3 + 81*a^5*b^2*f*g*z + 405*a^4*b^3*c*g*z + 162*a^4*b^3*d*f*z + 810*a^3*b^4*c *d*z + 6*a^3*b*d*g^2 - 75*a*b^3*c^2*f + 12*a^2*b^2*d^2*g - 15*a^2*b^2*c*f^ 2 + 8*a*b^3*d^3 + a^4*g^3 - 125*b^4*c^3 - a^3*b*f^3, z, k)*(9*root(19683*a ^8*b^5*z^3 + 81*a^5*b^2*f*g*z + 405*a^4*b^3*c*g*z + 162*a^4*b^3*d*f*z + 81 0*a^3*b^4*c*d*z + 6*a^3*b*d*g^2 - 75*a*b^3*c^2*f + 12*a^2*b^2*d^2*g - 15*a ^2*b^2*c*f^2 + 8*a*b^3*d^3 + a^4*g^3 - 125*b^4*c^3 - a^3*b*f^3, z, k)*a*b^ 2 + (x*(135*a^2*b^3*c + 27*a^3*b^2*f))/(81*a^4*b)) + (10*b^2*c*d + a^2*f*g + 5*a*b*c*g + 2*a*b*d*f)/(81*a^4*b) + (x*(4*b^2*d^2 + a^2*g^2 + 4*a*b*d*g ))/(81*a^4*b))*root(19683*a^8*b^5*z^3 + 81*a^5*b^2*f*g*z + 405*a^4*b^3*c*g *z + 162*a^4*b^3*d*f*z + 810*a^3*b^4*c*d*z + 6*a^3*b*d*g^2 - 75*a*b^3*c^2* f + 12*a^2*b^2*d^2*g - 15*a^2*b^2*c*f^2 + 8*a*b^3*d^3 + a^4*g^3 - 125*b^4* c^3 - a^3*b*f^3, z, k), k, 1, 3)