Integrand size = 20, antiderivative size = 199 \[ \int \frac {x^6 \left (A+B x^3\right )}{\left (a+b x^3\right )^3} \, dx=\frac {B x}{b^3}+\frac {a (A b-a B) x}{6 b^3 \left (a+b x^3\right )^2}-\frac {(7 A b-13 a B) x}{18 b^3 \left (a+b x^3\right )}-\frac {2 (A b-7 a B) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{2/3} b^{10/3}}+\frac {2 (A b-7 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{2/3} b^{10/3}}-\frac {(A b-7 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{27 a^{2/3} b^{10/3}} \] Output:
B*x/b^3+1/6*a*(A*b-B*a)*x/b^3/(b*x^3+a)^2-1/18*(7*A*b-13*B*a)*x/b^3/(b*x^3 +a)-2/27*(A*b-7*B*a)*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)*3^(1/2)/a^(1/3))*3^( 1/2)/a^(2/3)/b^(10/3)+2/27*(A*b-7*B*a)*ln(a^(1/3)+b^(1/3)*x)/a^(2/3)/b^(10 /3)-1/27*(A*b-7*B*a)*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(2/3)/b^( 10/3)
Time = 0.17 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.94 \[ \int \frac {x^6 \left (A+B x^3\right )}{\left (a+b x^3\right )^3} \, dx=\frac {54 \sqrt [3]{b} B x+\frac {9 a \sqrt [3]{b} (A b-a B) x}{\left (a+b x^3\right )^2}-\frac {3 \sqrt [3]{b} (7 A b-13 a B) x}{a+b x^3}+\frac {4 \sqrt {3} (-A b+7 a B) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{2/3}}+\frac {4 (A b-7 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{2/3}}+\frac {2 (-A b+7 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{2/3}}}{54 b^{10/3}} \] Input:
Integrate[(x^6*(A + B*x^3))/(a + b*x^3)^3,x]
Output:
(54*b^(1/3)*B*x + (9*a*b^(1/3)*(A*b - a*B)*x)/(a + b*x^3)^2 - (3*b^(1/3)*( 7*A*b - 13*a*B)*x)/(a + b*x^3) + (4*Sqrt[3]*(-(A*b) + 7*a*B)*ArcTan[(1 - ( 2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(2/3) + (4*(A*b - 7*a*B)*Log[a^(1/3) + b ^(1/3)*x])/a^(2/3) + (2*(-(A*b) + 7*a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(2/3))/(54*b^(10/3))
Time = 0.64 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {957, 817, 843, 750, 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.
| \(\displaystyle \int \frac {x^6 \left (A+B x^3\right )}{\left (a+b x^3\right )^3} \, dx\) |
| \(\Big \downarrow \) 957 |
| \(\displaystyle \frac {x^7 (A b-a B)}{6 a b \left (a+b x^3\right )^2}-\frac {(A b-7 a B) \int \frac {x^6}{\left (b x^3+a\right )^2}dx}{6 a b}\) |
| \(\Big \downarrow \) 817 |
| \(\displaystyle \frac {x^7 (A b-a B)}{6 a b \left (a+b x^3\right )^2}-\frac {(A b-7 a B) \left (\frac {4 \int \frac {x^3}{b x^3+a}dx}{3 b}-\frac {x^4}{3 b \left (a+b x^3\right )}\right )}{6 a b}\) |
| \(\Big \downarrow \) 843 |
| \(\displaystyle \frac {x^7 (A b-a B)}{6 a b \left (a+b x^3\right )^2}-\frac {(A b-7 a B) \left (\frac {4 \left (\frac {x}{b}-\frac {a \int \frac {1}{b x^3+a}dx}{b}\right )}{3 b}-\frac {x^4}{3 b \left (a+b x^3\right )}\right )}{6 a b}\) |
| \(\Big \downarrow \) 750 |
| \(\displaystyle \frac {x^7 (A b-a B)}{6 a b \left (a+b x^3\right )^2}-\frac {(A b-7 a B) \left (\frac {4 \left (\frac {x}{b}-\frac {a \left (\frac {\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3}}+\frac {\int \frac {1}{\sqrt [3]{b} x+\sqrt [3]{a}}dx}{3 a^{2/3}}\right )}{b}\right )}{3 b}-\frac {x^4}{3 b \left (a+b x^3\right )}\right )}{6 a b}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {x^7 (A b-a B)}{6 a b \left (a+b x^3\right )^2}-\frac {(A b-7 a B) \left (\frac {4 \left (\frac {x}{b}-\frac {a \left (\frac {\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b}\right )}{3 b}-\frac {x^4}{3 b \left (a+b x^3\right )}\right )}{6 a b}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {x^7 (A b-a B)}{6 a b \left (a+b x^3\right )^2}-\frac {(A b-7 a B) \left (\frac {4 \left (\frac {x}{b}-\frac {a \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {\int -\frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{2 \sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b}\right )}{3 b}-\frac {x^4}{3 b \left (a+b x^3\right )}\right )}{6 a b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {x^7 (A b-a B)}{6 a b \left (a+b x^3\right )^2}-\frac {(A b-7 a B) \left (\frac {4 \left (\frac {x}{b}-\frac {a \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {\int \frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{2 \sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b}\right )}{3 b}-\frac {x^4}{3 b \left (a+b x^3\right )}\right )}{6 a b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^7 (A b-a B)}{6 a b \left (a+b x^3\right )^2}-\frac {(A b-7 a B) \left (\frac {4 \left (\frac {x}{b}-\frac {a \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b}\right )}{3 b}-\frac {x^4}{3 b \left (a+b x^3\right )}\right )}{6 a b}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {x^7 (A b-a B)}{6 a b \left (a+b x^3\right )^2}-\frac {(A b-7 a B) \left (\frac {4 \left (\frac {x}{b}-\frac {a \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b}\right )}{3 b}-\frac {x^4}{3 b \left (a+b x^3\right )}\right )}{6 a b}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {x^7 (A b-a B)}{6 a b \left (a+b x^3\right )^2}-\frac {(A b-7 a B) \left (\frac {4 \left (\frac {x}{b}-\frac {a \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b}\right )}{3 b}-\frac {x^4}{3 b \left (a+b x^3\right )}\right )}{6 a b}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {x^7 (A b-a B)}{6 a b \left (a+b x^3\right )^2}-\frac {(A b-7 a B) \left (\frac {4 \left (\frac {x}{b}-\frac {a \left (\frac {-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{b}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b}\right )}{3 b}-\frac {x^4}{3 b \left (a+b x^3\right )}\right )}{6 a b}\) |
Input:
Int[(x^6*(A + B*x^3))/(a + b*x^3)^3,x]
Output:
((A*b - a*B)*x^7)/(6*a*b*(a + b*x^3)^2) - ((A*b - 7*a*B)*(-1/3*x^4/(b*(a + b*x^3)) + (4*(x/b - (a*(Log[a^(1/3) + b^(1/3)*x]/(3*a^(2/3)*b^(1/3)) + (- ((Sqrt[3]*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(1/3)) - Log[a^(2 /3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2]/(2*b^(1/3)))/(3*a^(2/3))))/b))/(3*b )))/(6*a*b)
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_)^3)^(-1), x_Symbol] :> Simp[1/(3*Rt[a, 3]^2) Int[1/ (Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]^2) Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x] /; FreeQ[{a, b}, x]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^( n - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*n*(p + 1))), x] - Simp[c^n *((m - n + 1)/(b*n*(p + 1))) Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x], x ] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] && ! ILtQ[(m + n*(p + 1) + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*(m + n*p + 1))), x] - Simp[ a*c^n*((m - n + 1)/(b*(m + n*p + 1))) Int[(c*x)^(m - n)*(a + b*x^n)^p, x] , x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n* p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n _)), x_Symbol] :> Simp[(-(b*c - a*d))*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a *b*e*n*(p + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*b*n* (p + 1)) Int[(e*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && (( !IntegerQ[p + 1/2] && N eQ[p, -5/4]) || !RationalQ[m] || (IGtQ[n, 0] && ILtQ[p + 1/2, 0] && LeQ[-1 , m, (-n)*(p + 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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.87 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.43
| method | result | size |
| risch | \(\frac {B x}{b^{3}}+\frac {\left (-\frac {7}{18} b^{2} A +\frac {13}{18} a b B \right ) x^{4}-\frac {a \left (2 A b -5 B a \right ) x}{9}}{b^{3} \left (b \,x^{3}+a \right )^{2}}+\frac {2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\left (A b -7 B a \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}\right )}{27 b^{4}}\) | \(85\) |
| default | \(\frac {B x}{b^{3}}+\frac {\frac {\left (-\frac {7}{18} b^{2} A +\frac {13}{18} a b B \right ) x^{4}-\frac {a \left (2 A b -5 B a \right ) x}{9}}{\left (b \,x^{3}+a \right )^{2}}+\frac {2 \left (A b -7 B a \right ) \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{9}}{b^{3}}\) | \(153\) |
Input:
int(x^6*(B*x^3+A)/(b*x^3+a)^3,x,method=_RETURNVERBOSE)
Output:
B*x/b^3+((-7/18*b^2*A+13/18*a*b*B)*x^4-1/9*a*(2*A*b-5*B*a)*x)/b^3/(b*x^3+a )^2+2/27/b^4*sum((A*b-7*B*a)/_R^2*ln(x-_R),_R=RootOf(_Z^3*b+a))
Leaf count of result is larger than twice the leaf count of optimal. 374 vs. \(2 (161) = 322\).
Time = 0.14 (sec) , antiderivative size = 789, normalized size of antiderivative = 3.96 \[ \int \frac {x^6 \left (A+B x^3\right )}{\left (a+b x^3\right )^3} \, dx =\text {Too large to display} \] Input:
integrate(x^6*(B*x^3+A)/(b*x^3+a)^3,x, algorithm="fricas")
Output:
[1/54*(54*B*a^2*b^3*x^7 + 21*(7*B*a^3*b^2 - A*a^2*b^3)*x^4 - 6*sqrt(1/3)*( (7*B*a^2*b^3 - A*a*b^4)*x^6 + 7*B*a^4*b - A*a^3*b^2 + 2*(7*B*a^3*b^2 - A*a ^2*b^3)*x^3)*sqrt(-(a^2*b)^(1/3)/b)*log((2*a*b*x^3 - 3*(a^2*b)^(1/3)*a*x - a^2 + 3*sqrt(1/3)*(2*a*b*x^2 + (a^2*b)^(2/3)*x - (a^2*b)^(1/3)*a)*sqrt(-( a^2*b)^(1/3)/b))/(b*x^3 + a)) + 2*((7*B*a*b^2 - A*b^3)*x^6 + 7*B*a^3 - A*a ^2*b + 2*(7*B*a^2*b - A*a*b^2)*x^3)*(a^2*b)^(2/3)*log(a*b*x^2 - (a^2*b)^(2 /3)*x + (a^2*b)^(1/3)*a) - 4*((7*B*a*b^2 - A*b^3)*x^6 + 7*B*a^3 - A*a^2*b + 2*(7*B*a^2*b - A*a*b^2)*x^3)*(a^2*b)^(2/3)*log(a*b*x + (a^2*b)^(2/3)) + 12*(7*B*a^4*b - A*a^3*b^2)*x)/(a^2*b^6*x^6 + 2*a^3*b^5*x^3 + a^4*b^4), 1/5 4*(54*B*a^2*b^3*x^7 + 21*(7*B*a^3*b^2 - A*a^2*b^3)*x^4 - 12*sqrt(1/3)*((7* B*a^2*b^3 - A*a*b^4)*x^6 + 7*B*a^4*b - A*a^3*b^2 + 2*(7*B*a^3*b^2 - A*a^2* b^3)*x^3)*sqrt((a^2*b)^(1/3)/b)*arctan(sqrt(1/3)*(2*(a^2*b)^(2/3)*x - (a^2 *b)^(1/3)*a)*sqrt((a^2*b)^(1/3)/b)/a^2) + 2*((7*B*a*b^2 - A*b^3)*x^6 + 7*B *a^3 - A*a^2*b + 2*(7*B*a^2*b - A*a*b^2)*x^3)*(a^2*b)^(2/3)*log(a*b*x^2 - (a^2*b)^(2/3)*x + (a^2*b)^(1/3)*a) - 4*((7*B*a*b^2 - A*b^3)*x^6 + 7*B*a^3 - A*a^2*b + 2*(7*B*a^2*b - A*a*b^2)*x^3)*(a^2*b)^(2/3)*log(a*b*x + (a^2*b) ^(2/3)) + 12*(7*B*a^4*b - A*a^3*b^2)*x)/(a^2*b^6*x^6 + 2*a^3*b^5*x^3 + a^4 *b^4)]
Time = 0.98 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.71 \[ \int \frac {x^6 \left (A+B x^3\right )}{\left (a+b x^3\right )^3} \, dx=\frac {B x}{b^{3}} + \frac {x^{4} \left (- 7 A b^{2} + 13 B a b\right ) + x \left (- 4 A a b + 10 B a^{2}\right )}{18 a^{2} b^{3} + 36 a b^{4} x^{3} + 18 b^{5} x^{6}} + \operatorname {RootSum} {\left (19683 t^{3} a^{2} b^{10} - 8 A^{3} b^{3} + 168 A^{2} B a b^{2} - 1176 A B^{2} a^{2} b + 2744 B^{3} a^{3}, \left ( t \mapsto t \log {\left (- \frac {27 t a b^{3}}{- 2 A b + 14 B a} + x \right )} \right )\right )} \] Input:
integrate(x**6*(B*x**3+A)/(b*x**3+a)**3,x)
Output:
B*x/b**3 + (x**4*(-7*A*b**2 + 13*B*a*b) + x*(-4*A*a*b + 10*B*a**2))/(18*a* *2*b**3 + 36*a*b**4*x**3 + 18*b**5*x**6) + RootSum(19683*_t**3*a**2*b**10 - 8*A**3*b**3 + 168*A**2*B*a*b**2 - 1176*A*B**2*a**2*b + 2744*B**3*a**3, L ambda(_t, _t*log(-27*_t*a*b**3/(-2*A*b + 14*B*a) + x)))
Time = 0.11 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.96 \[ \int \frac {x^6 \left (A+B x^3\right )}{\left (a+b x^3\right )^3} \, dx=\frac {{\left (13 \, B a b - 7 \, A b^{2}\right )} x^{4} + 2 \, {\left (5 \, B a^{2} - 2 \, A a b\right )} x}{18 \, {\left (b^{5} x^{6} + 2 \, a b^{4} x^{3} + a^{2} b^{3}\right )}} + \frac {B x}{b^{3}} - \frac {2 \, \sqrt {3} {\left (7 \, B a - A b\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, b^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (7 \, B a - A b\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{27 \, b^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {2 \, {\left (7 \, B a - A b\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, b^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \] Input:
integrate(x^6*(B*x^3+A)/(b*x^3+a)^3,x, algorithm="maxima")
Output:
1/18*((13*B*a*b - 7*A*b^2)*x^4 + 2*(5*B*a^2 - 2*A*a*b)*x)/(b^5*x^6 + 2*a*b ^4*x^3 + a^2*b^3) + B*x/b^3 - 2/27*sqrt(3)*(7*B*a - A*b)*arctan(1/3*sqrt(3 )*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(b^4*(a/b)^(2/3)) + 1/27*(7*B*a - A*b)* log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(b^4*(a/b)^(2/3)) - 2/27*(7*B*a - A *b)*log(x + (a/b)^(1/3))/(b^4*(a/b)^(2/3))
Time = 0.13 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.94 \[ \int \frac {x^6 \left (A+B x^3\right )}{\left (a+b x^3\right )^3} \, dx=\frac {2 \, \sqrt {3} {\left (7 \, B a - A b\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, \left (-a b^{2}\right )^{\frac {2}{3}} b^{2}} + \frac {{\left (7 \, B a - A b\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{27 \, \left (-a b^{2}\right )^{\frac {2}{3}} b^{2}} + \frac {B x}{b^{3}} + \frac {2 \, {\left (7 \, B a - A b\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a b^{3}} + \frac {13 \, B a b x^{4} - 7 \, A b^{2} x^{4} + 10 \, B a^{2} x - 4 \, A a b x}{18 \, {\left (b x^{3} + a\right )}^{2} b^{3}} \] Input:
integrate(x^6*(B*x^3+A)/(b*x^3+a)^3,x, algorithm="giac")
Output:
2/27*sqrt(3)*(7*B*a - A*b)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^ (1/3))/((-a*b^2)^(2/3)*b^2) + 1/27*(7*B*a - A*b)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/((-a*b^2)^(2/3)*b^2) + B*x/b^3 + 2/27*(7*B*a - A*b)*(-a/b) ^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a*b^3) + 1/18*(13*B*a*b*x^4 - 7*A*b^2*x ^4 + 10*B*a^2*x - 4*A*a*b*x)/((b*x^3 + a)^2*b^3)
Time = 0.94 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.92 \[ \int \frac {x^6 \left (A+B x^3\right )}{\left (a+b x^3\right )^3} \, dx=\frac {B\,x}{b^3}-\frac {x^4\,\left (\frac {7\,A\,b^2}{18}-\frac {13\,B\,a\,b}{18}\right )-x\,\left (\frac {5\,B\,a^2}{9}-\frac {2\,A\,a\,b}{9}\right )}{a^2\,b^3+2\,a\,b^4\,x^3+b^5\,x^6}+\frac {2\,\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (A\,b-7\,B\,a\right )}{27\,a^{2/3}\,b^{10/3}}-\frac {2\,\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (A\,b-7\,B\,a\right )}{27\,a^{2/3}\,b^{10/3}}+\frac {2\,\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (A\,b-7\,B\,a\right )}{27\,a^{2/3}\,b^{10/3}} \] Input:
int((x^6*(A + B*x^3))/(a + b*x^3)^3,x)
Output:
(B*x)/b^3 - (x^4*((7*A*b^2)/18 - (13*B*a*b)/18) - x*((5*B*a^2)/9 - (2*A*a* b)/9))/(a^2*b^3 + b^5*x^6 + 2*a*b^4*x^3) + (2*log(b^(1/3)*x + a^(1/3))*(A* b - 7*B*a))/(27*a^(2/3)*b^(10/3)) - (2*log(3^(1/2)*a^(1/3)*1i - 2*b^(1/3)* x + a^(1/3))*((3^(1/2)*1i)/2 + 1/2)*(A*b - 7*B*a))/(27*a^(2/3)*b^(10/3)) + (2*log(3^(1/2)*a^(1/3)*1i + 2*b^(1/3)*x - a^(1/3))*((3^(1/2)*1i)/2 - 1/2) *(A*b - 7*B*a))/(27*a^(2/3)*b^(10/3))
Time = 0.24 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.88 \[ \int \frac {x^6 \left (A+B x^3\right )}{\left (a+b x^3\right )^3} \, dx=\frac {4 a^{\frac {4}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x}{a^{\frac {1}{3}} \sqrt {3}}\right )+4 a^{\frac {1}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x}{a^{\frac {1}{3}} \sqrt {3}}\right ) b \,x^{3}+2 a^{\frac {4}{3}} \mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right )+2 a^{\frac {1}{3}} \mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) b \,x^{3}-4 a^{\frac {4}{3}} \mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right )-4 a^{\frac {1}{3}} \mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) b \,x^{3}+12 b^{\frac {1}{3}} a x +9 b^{\frac {4}{3}} x^{4}}{9 b^{\frac {7}{3}} \left (b \,x^{3}+a \right )} \] Input:
int(x^6*(B*x^3+A)/(b*x^3+a)^3,x)
Output:
(4*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a + 4*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*b*x **3 + 2*a**(1/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a + 2 *a**(1/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*b*x**3 - 4*a **(1/3)*log(a**(1/3) + b**(1/3)*x)*a - 4*a**(1/3)*log(a**(1/3) + b**(1/3)* x)*b*x**3 + 12*b**(1/3)*a*x + 9*b**(1/3)*b*x**4)/(9*b**(1/3)*b**2*(a + b*x **3))