Integrand size = 20, antiderivative size = 228 \[ \int \frac {A+B x^3}{x^5 \left (a+b x^3\right )^3} \, dx=-\frac {A}{4 a^3 x^4}+\frac {3 A b-a B}{a^4 x}+\frac {b (A b-a B) x^2}{6 a^3 \left (a+b x^3\right )^2}+\frac {b (8 A b-5 a B) x^2}{9 a^4 \left (a+b x^3\right )}-\frac {7 \sqrt [3]{b} (5 A b-2 a B) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{13/3}}-\frac {7 \sqrt [3]{b} (5 A b-2 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{13/3}}+\frac {7 \sqrt [3]{b} (5 A b-2 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{13/3}} \] Output:
-1/4*A/a^3/x^4+(3*A*b-B*a)/a^4/x+1/6*b*(A*b-B*a)*x^2/a^3/(b*x^3+a)^2+1/9*b *(8*A*b-5*B*a)*x^2/a^4/(b*x^3+a)-7/27*b^(1/3)*(5*A*b-2*B*a)*arctan(1/3*(a^ (1/3)-2*b^(1/3)*x)*3^(1/2)/a^(1/3))*3^(1/2)/a^(13/3)-7/27*b^(1/3)*(5*A*b-2 *B*a)*ln(a^(1/3)+b^(1/3)*x)/a^(13/3)+7/54*b^(1/3)*(5*A*b-2*B*a)*ln(a^(2/3) -a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(13/3)
Time = 0.20 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.94 \[ \int \frac {A+B x^3}{x^5 \left (a+b x^3\right )^3} \, dx=\frac {-\frac {27 a^{4/3} A}{x^4}-\frac {108 \sqrt [3]{a} (-3 A b+a B)}{x}-\frac {18 a^{4/3} b (-A b+a B) x^2}{\left (a+b x^3\right )^2}-\frac {12 \sqrt [3]{a} b (-8 A b+5 a B) x^2}{a+b x^3}-28 \sqrt {3} \sqrt [3]{b} (5 A b-2 a B) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )+28 \sqrt [3]{b} (-5 A b+2 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+14 \sqrt [3]{b} (5 A b-2 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{108 a^{13/3}} \] Input:
Integrate[(A + B*x^3)/(x^5*(a + b*x^3)^3),x]
Output:
((-27*a^(4/3)*A)/x^4 - (108*a^(1/3)*(-3*A*b + a*B))/x - (18*a^(4/3)*b*(-(A *b) + a*B)*x^2)/(a + b*x^3)^2 - (12*a^(1/3)*b*(-8*A*b + 5*a*B)*x^2)/(a + b *x^3) - 28*Sqrt[3]*b^(1/3)*(5*A*b - 2*a*B)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/ 3))/Sqrt[3]] + 28*b^(1/3)*(-5*A*b + 2*a*B)*Log[a^(1/3) + b^(1/3)*x] + 14*b ^(1/3)*(5*A*b - 2*a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(10 8*a^(13/3))
Time = 0.73 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.99, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {957, 819, 847, 847, 821, 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.
| \(\displaystyle \int \frac {A+B x^3}{x^5 \left (a+b x^3\right )^3} \, dx\) |
| \(\Big \downarrow \) 957 |
| \(\displaystyle \frac {(5 A b-2 a B) \int \frac {1}{x^5 \left (b x^3+a\right )^2}dx}{3 a b}+\frac {A b-a B}{6 a b x^4 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 819 |
| \(\displaystyle \frac {(5 A b-2 a B) \left (\frac {7 \int \frac {1}{x^5 \left (b x^3+a\right )}dx}{3 a}+\frac {1}{3 a x^4 \left (a+b x^3\right )}\right )}{3 a b}+\frac {A b-a B}{6 a b x^4 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 847 |
| \(\displaystyle \frac {(5 A b-2 a B) \left (\frac {7 \left (-\frac {b \int \frac {1}{x^2 \left (b x^3+a\right )}dx}{a}-\frac {1}{4 a x^4}\right )}{3 a}+\frac {1}{3 a x^4 \left (a+b x^3\right )}\right )}{3 a b}+\frac {A b-a B}{6 a b x^4 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 847 |
| \(\displaystyle \frac {(5 A b-2 a B) \left (\frac {7 \left (-\frac {b \left (-\frac {b \int \frac {x}{b x^3+a}dx}{a}-\frac {1}{a x}\right )}{a}-\frac {1}{4 a x^4}\right )}{3 a}+\frac {1}{3 a x^4 \left (a+b x^3\right )}\right )}{3 a b}+\frac {A b-a B}{6 a b x^4 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 821 |
| \(\displaystyle \frac {(5 A b-2 a B) \left (\frac {7 \left (-\frac {b \left (-\frac {b \left (\frac {\int \frac {\sqrt [3]{b} x+\sqrt [3]{a}}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\int \frac {1}{\sqrt [3]{b} x+\sqrt [3]{a}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}\right )}{a}-\frac {1}{a x}\right )}{a}-\frac {1}{4 a x^4}\right )}{3 a}+\frac {1}{3 a x^4 \left (a+b x^3\right )}\right )}{3 a b}+\frac {A b-a B}{6 a b x^4 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {(5 A b-2 a B) \left (\frac {7 \left (-\frac {b \left (-\frac {b \left (\frac {\int \frac {\sqrt [3]{b} x+\sqrt [3]{a}}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{a}-\frac {1}{a x}\right )}{a}-\frac {1}{4 a x^4}\right )}{3 a}+\frac {1}{3 a x^4 \left (a+b x^3\right )}\right )}{3 a b}+\frac {A b-a B}{6 a b x^4 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {(5 A b-2 a B) \left (\frac {7 \left (-\frac {b \left (-\frac {b \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {\int -\frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{2 \sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{a}-\frac {1}{a x}\right )}{a}-\frac {1}{4 a x^4}\right )}{3 a}+\frac {1}{3 a x^4 \left (a+b x^3\right )}\right )}{3 a b}+\frac {A b-a B}{6 a b x^4 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(5 A b-2 a B) \left (\frac {7 \left (-\frac {b \left (-\frac {b \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {\int \frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{2 \sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{a}-\frac {1}{a x}\right )}{a}-\frac {1}{4 a x^4}\right )}{3 a}+\frac {1}{3 a x^4 \left (a+b x^3\right )}\right )}{3 a b}+\frac {A b-a B}{6 a b x^4 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(5 A b-2 a B) \left (\frac {7 \left (-\frac {b \left (-\frac {b \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{a}-\frac {1}{a x}\right )}{a}-\frac {1}{4 a x^4}\right )}{3 a}+\frac {1}{3 a x^4 \left (a+b x^3\right )}\right )}{3 a b}+\frac {A b-a B}{6 a b x^4 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {(5 A b-2 a B) \left (\frac {7 \left (-\frac {b \left (-\frac {b \left (\frac {\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{\sqrt [3]{b}}-\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{a}-\frac {1}{a x}\right )}{a}-\frac {1}{4 a x^4}\right )}{3 a}+\frac {1}{3 a x^4 \left (a+b x^3\right )}\right )}{3 a b}+\frac {A b-a B}{6 a b x^4 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {(5 A b-2 a B) \left (\frac {7 \left (-\frac {b \left (-\frac {b \left (\frac {-\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{a}-\frac {1}{a x}\right )}{a}-\frac {1}{4 a x^4}\right )}{3 a}+\frac {1}{3 a x^4 \left (a+b x^3\right )}\right )}{3 a b}+\frac {A b-a B}{6 a b x^4 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {(5 A b-2 a B) \left (\frac {7 \left (-\frac {b \left (-\frac {b \left (\frac {\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{b}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{a}-\frac {1}{a x}\right )}{a}-\frac {1}{4 a x^4}\right )}{3 a}+\frac {1}{3 a x^4 \left (a+b x^3\right )}\right )}{3 a b}+\frac {A b-a B}{6 a b x^4 \left (a+b x^3\right )^2}\) |
Input:
Int[(A + B*x^3)/(x^5*(a + b*x^3)^3),x]
Output:
(A*b - a*B)/(6*a*b*x^4*(a + b*x^3)^2) + ((5*A*b - 2*a*B)*(1/(3*a*x^4*(a + b*x^3)) + (7*(-1/4*1/(a*x^4) - (b*(-(1/(a*x)) - (b*(-1/3*Log[a^(1/3) + b^( 1/3)*x]/(a^(1/3)*b^(2/3)) + (-((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^(1/3)*b^(1/3))))/a))/a))/(3*a)))/(3*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[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-( c*x)^(m + 1))*((a + b*x^n)^(p + 1)/(a*c*n*(p + 1))), x] + Simp[(m + n*(p + 1) + 1)/(a*n*(p + 1)) Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a , b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p , x]
Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> Simp[-(3*Rt[a, 3]*Rt[b, 3])^(- 1) Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]*Rt[b, 3]) Int[(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*x )^(m + 1)*((a + b*x^n)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))) Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a , b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && 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]
Time = 0.85 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.77
| method | result | size |
| default | \(-\frac {A}{4 a^{3} x^{4}}-\frac {-3 A b +B a}{a^{4} x}+\frac {b \left (\frac {\left (\frac {8}{9} b^{2} A -\frac {5}{9} a b B \right ) x^{5}+\frac {a \left (19 A b -13 B a \right ) x^{2}}{18}}{\left (b \,x^{3}+a \right )^{2}}+\left (\frac {35 A b}{9}-\frac {14 B a}{9}\right ) \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )\right )}{a^{4}}\) | \(175\) |
| risch | \(\frac {\frac {7 b^{2} \left (5 A b -2 B a \right ) x^{9}}{9 a^{4}}+\frac {49 b \left (5 A b -2 B a \right ) x^{6}}{36 a^{3}}+\frac {\left (5 A b -2 B a \right ) x^{3}}{2 a^{2}}-\frac {A}{4 a}}{x^{4} \left (b \,x^{3}+a \right )^{2}}+\frac {7 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{13} \textit {\_Z}^{3}+125 A^{3} b^{4}-150 A^{2} B a \,b^{3}+60 A \,B^{2} a^{2} b^{2}-8 B^{3} a^{3} b \right )}{\sum }\textit {\_R} \ln \left (\left (-4 \textit {\_R}^{3} a^{13}-375 A^{3} b^{4}+450 A^{2} B a \,b^{3}-180 A \,B^{2} a^{2} b^{2}+24 B^{3} a^{3} b \right ) x +\left (5 a^{9} b A -2 a^{10} B \right ) \textit {\_R}^{2}\right )\right )}{27}\) | \(202\) |
Input:
int((B*x^3+A)/x^5/(b*x^3+a)^3,x,method=_RETURNVERBOSE)
Output:
-1/4*A/a^3/x^4-(-3*A*b+B*a)/a^4/x+1/a^4*b*(((8/9*b^2*A-5/9*a*b*B)*x^5+1/18 *a*(19*A*b-13*B*a)*x^2)/(b*x^3+a)^2+(35/9*A*b-14/9*B*a)*(-1/3/b/(a/b)^(1/3 )*ln(x+(a/b)^(1/3))+1/6/b/(a/b)^(1/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))+1/ 3*3^(1/2)/b/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))))
Time = 0.15 (sec) , antiderivative size = 366, normalized size of antiderivative = 1.61 \[ \int \frac {A+B x^3}{x^5 \left (a+b x^3\right )^3} \, dx=-\frac {84 \, {\left (2 \, B a b^{2} - 5 \, A b^{3}\right )} x^{9} + 147 \, {\left (2 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{6} + 27 \, A a^{3} + 54 \, {\left (2 \, B a^{3} - 5 \, A a^{2} b\right )} x^{3} + 28 \, \sqrt {3} {\left ({\left (2 \, B a b^{2} - 5 \, A b^{3}\right )} x^{10} + 2 \, {\left (2 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{7} + {\left (2 \, B a^{3} - 5 \, A a^{2} b\right )} x^{4}\right )} \left (-\frac {b}{a}\right )^{\frac {1}{3}} \arctan \left (\frac {2}{3} \, \sqrt {3} x \left (-\frac {b}{a}\right )^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) - 14 \, {\left ({\left (2 \, B a b^{2} - 5 \, A b^{3}\right )} x^{10} + 2 \, {\left (2 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{7} + {\left (2 \, B a^{3} - 5 \, A a^{2} b\right )} x^{4}\right )} \left (-\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b x^{2} - a x \left (-\frac {b}{a}\right )^{\frac {2}{3}} - a \left (-\frac {b}{a}\right )^{\frac {1}{3}}\right ) + 28 \, {\left ({\left (2 \, B a b^{2} - 5 \, A b^{3}\right )} x^{10} + 2 \, {\left (2 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{7} + {\left (2 \, B a^{3} - 5 \, A a^{2} b\right )} x^{4}\right )} \left (-\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b x + a \left (-\frac {b}{a}\right )^{\frac {2}{3}}\right )}{108 \, {\left (a^{4} b^{2} x^{10} + 2 \, a^{5} b x^{7} + a^{6} x^{4}\right )}} \] Input:
integrate((B*x^3+A)/x^5/(b*x^3+a)^3,x, algorithm="fricas")
Output:
-1/108*(84*(2*B*a*b^2 - 5*A*b^3)*x^9 + 147*(2*B*a^2*b - 5*A*a*b^2)*x^6 + 2 7*A*a^3 + 54*(2*B*a^3 - 5*A*a^2*b)*x^3 + 28*sqrt(3)*((2*B*a*b^2 - 5*A*b^3) *x^10 + 2*(2*B*a^2*b - 5*A*a*b^2)*x^7 + (2*B*a^3 - 5*A*a^2*b)*x^4)*(-b/a)^ (1/3)*arctan(2/3*sqrt(3)*x*(-b/a)^(1/3) + 1/3*sqrt(3)) - 14*((2*B*a*b^2 - 5*A*b^3)*x^10 + 2*(2*B*a^2*b - 5*A*a*b^2)*x^7 + (2*B*a^3 - 5*A*a^2*b)*x^4) *(-b/a)^(1/3)*log(b*x^2 - a*x*(-b/a)^(2/3) - a*(-b/a)^(1/3)) + 28*((2*B*a* b^2 - 5*A*b^3)*x^10 + 2*(2*B*a^2*b - 5*A*a*b^2)*x^7 + (2*B*a^3 - 5*A*a^2*b )*x^4)*(-b/a)^(1/3)*log(b*x + a*(-b/a)^(2/3)))/(a^4*b^2*x^10 + 2*a^5*b*x^7 + a^6*x^4)
Time = 0.60 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.83 \[ \int \frac {A+B x^3}{x^5 \left (a+b x^3\right )^3} \, dx=\operatorname {RootSum} {\left (19683 t^{3} a^{13} + 42875 A^{3} b^{4} - 51450 A^{2} B a b^{3} + 20580 A B^{2} a^{2} b^{2} - 2744 B^{3} a^{3} b, \left ( t \mapsto t \log {\left (\frac {729 t^{2} a^{9}}{1225 A^{2} b^{3} - 980 A B a b^{2} + 196 B^{2} a^{2} b} + x \right )} \right )\right )} + \frac {- 9 A a^{3} + x^{9} \cdot \left (140 A b^{3} - 56 B a b^{2}\right ) + x^{6} \cdot \left (245 A a b^{2} - 98 B a^{2} b\right ) + x^{3} \cdot \left (90 A a^{2} b - 36 B a^{3}\right )}{36 a^{6} x^{4} + 72 a^{5} b x^{7} + 36 a^{4} b^{2} x^{10}} \] Input:
integrate((B*x**3+A)/x**5/(b*x**3+a)**3,x)
Output:
RootSum(19683*_t**3*a**13 + 42875*A**3*b**4 - 51450*A**2*B*a*b**3 + 20580* A*B**2*a**2*b**2 - 2744*B**3*a**3*b, Lambda(_t, _t*log(729*_t**2*a**9/(122 5*A**2*b**3 - 980*A*B*a*b**2 + 196*B**2*a**2*b) + x))) + (-9*A*a**3 + x**9 *(140*A*b**3 - 56*B*a*b**2) + x**6*(245*A*a*b**2 - 98*B*a**2*b) + x**3*(90 *A*a**2*b - 36*B*a**3))/(36*a**6*x**4 + 72*a**5*b*x**7 + 36*a**4*b**2*x**1 0)
Time = 0.11 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.97 \[ \int \frac {A+B x^3}{x^5 \left (a+b x^3\right )^3} \, dx=-\frac {28 \, {\left (2 \, B a b^{2} - 5 \, A b^{3}\right )} x^{9} + 49 \, {\left (2 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{6} + 9 \, A a^{3} + 18 \, {\left (2 \, B a^{3} - 5 \, A a^{2} b\right )} x^{3}}{36 \, {\left (a^{4} b^{2} x^{10} + 2 \, a^{5} b x^{7} + a^{6} x^{4}\right )}} - \frac {7 \, \sqrt {3} {\left (2 \, B a - 5 \, A b\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{4} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {7 \, {\left (2 \, B a - 5 \, A b\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a^{4} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {7 \, {\left (2 \, B a - 5 \, A b\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, a^{4} \left (\frac {a}{b}\right )^{\frac {1}{3}}} \] Input:
integrate((B*x^3+A)/x^5/(b*x^3+a)^3,x, algorithm="maxima")
Output:
-1/36*(28*(2*B*a*b^2 - 5*A*b^3)*x^9 + 49*(2*B*a^2*b - 5*A*a*b^2)*x^6 + 9*A *a^3 + 18*(2*B*a^3 - 5*A*a^2*b)*x^3)/(a^4*b^2*x^10 + 2*a^5*b*x^7 + a^6*x^4 ) - 7/27*sqrt(3)*(2*B*a - 5*A*b)*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a /b)^(1/3))/(a^4*(a/b)^(1/3)) - 7/54*(2*B*a - 5*A*b)*log(x^2 - x*(a/b)^(1/3 ) + (a/b)^(2/3))/(a^4*(a/b)^(1/3)) + 7/27*(2*B*a - 5*A*b)*log(x + (a/b)^(1 /3))/(a^4*(a/b)^(1/3))
Time = 0.14 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.11 \[ \int \frac {A+B x^3}{x^5 \left (a+b x^3\right )^3} \, dx=\frac {7 \, {\left (2 \, B a b \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 5 \, A b^{2} \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a^{5}} + \frac {7 \, \sqrt {3} {\left (2 \, \left (-a b^{2}\right )^{\frac {2}{3}} B a - 5 \, \left (-a b^{2}\right )^{\frac {2}{3}} A b\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{5} b} - \frac {7 \, {\left (2 \, \left (-a b^{2}\right )^{\frac {2}{3}} B a - 5 \, \left (-a b^{2}\right )^{\frac {2}{3}} A b\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a^{5} b} - \frac {10 \, B a b^{2} x^{5} - 16 \, A b^{3} x^{5} + 13 \, B a^{2} b x^{2} - 19 \, A a b^{2} x^{2}}{18 \, {\left (b x^{3} + a\right )}^{2} a^{4}} - \frac {4 \, B a x^{3} - 12 \, A b x^{3} + A a}{4 \, a^{4} x^{4}} \] Input:
integrate((B*x^3+A)/x^5/(b*x^3+a)^3,x, algorithm="giac")
Output:
7/27*(2*B*a*b*(-a/b)^(1/3) - 5*A*b^2*(-a/b)^(1/3))*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/a^5 + 7/27*sqrt(3)*(2*(-a*b^2)^(2/3)*B*a - 5*(-a*b^2)^(2/ 3)*A*b)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a^5*b) - 7/ 54*(2*(-a*b^2)^(2/3)*B*a - 5*(-a*b^2)^(2/3)*A*b)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^5*b) - 1/18*(10*B*a*b^2*x^5 - 16*A*b^3*x^5 + 13*B*a^2*b *x^2 - 19*A*a*b^2*x^2)/((b*x^3 + a)^2*a^4) - 1/4*(4*B*a*x^3 - 12*A*b*x^3 + A*a)/(a^4*x^4)
Time = 1.02 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.05 \[ \int \frac {A+B x^3}{x^5 \left (a+b x^3\right )^3} \, dx=\frac {\frac {x^3\,\left (5\,A\,b-2\,B\,a\right )}{2\,a^2}-\frac {A}{4\,a}+\frac {7\,b^2\,x^9\,\left (5\,A\,b-2\,B\,a\right )}{9\,a^4}+\frac {49\,b\,x^6\,\left (5\,A\,b-2\,B\,a\right )}{36\,a^3}}{a^2\,x^4+2\,a\,b\,x^7+b^2\,x^{10}}+\frac {7\,{\left (-b\right )}^{1/3}\,\ln \left (a^{1/3}\,{\left (-b\right )}^{8/3}+b^3\,x\right )\,\left (5\,A\,b-2\,B\,a\right )}{27\,a^{13/3}}+\frac {7\,{\left (-b\right )}^{1/3}\,\ln \left (a^{1/3}\,{\left (-b\right )}^{8/3}-2\,b^3\,x+\sqrt {3}\,a^{1/3}\,{\left (-b\right )}^{8/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (5\,A\,b-2\,B\,a\right )}{27\,a^{13/3}}-\frac {7\,{\left (-b\right )}^{1/3}\,\ln \left (2\,b^3\,x-a^{1/3}\,{\left (-b\right )}^{8/3}+\sqrt {3}\,a^{1/3}\,{\left (-b\right )}^{8/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (5\,A\,b-2\,B\,a\right )}{27\,a^{13/3}} \] Input:
int((A + B*x^3)/(x^5*(a + b*x^3)^3),x)
Output:
((x^3*(5*A*b - 2*B*a))/(2*a^2) - A/(4*a) + (7*b^2*x^9*(5*A*b - 2*B*a))/(9* a^4) + (49*b*x^6*(5*A*b - 2*B*a))/(36*a^3))/(a^2*x^4 + b^2*x^10 + 2*a*b*x^ 7) + (7*(-b)^(1/3)*log(a^(1/3)*(-b)^(8/3) + b^3*x)*(5*A*b - 2*B*a))/(27*a^ (13/3)) + (7*(-b)^(1/3)*log(a^(1/3)*(-b)^(8/3) - 2*b^3*x + 3^(1/2)*a^(1/3) *(-b)^(8/3)*1i)*((3^(1/2)*1i)/2 - 1/2)*(5*A*b - 2*B*a))/(27*a^(13/3)) - (7 *(-b)^(1/3)*log(2*b^3*x - a^(1/3)*(-b)^(8/3) + 3^(1/2)*a^(1/3)*(-b)^(8/3)* 1i)*((3^(1/2)*1i)/2 + 1/2)*(5*A*b - 2*B*a))/(27*a^(13/3))
Time = 0.25 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.90 \[ \int \frac {A+B x^3}{x^5 \left (a+b x^3\right )^3} \, dx=\frac {-28 \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x}{a^{\frac {1}{3}} \sqrt {3}}\right ) a \,b^{2} x^{4}-28 \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x}{a^{\frac {1}{3}} \sqrt {3}}\right ) b^{3} x^{7}-9 b^{\frac {2}{3}} a^{\frac {7}{3}}+63 b^{\frac {5}{3}} a^{\frac {4}{3}} x^{3}+84 b^{\frac {8}{3}} a^{\frac {1}{3}} x^{6}+14 \,\mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) a \,b^{2} x^{4}+14 \,\mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) b^{3} x^{7}-28 \,\mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) a \,b^{2} x^{4}-28 \,\mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) b^{3} x^{7}}{36 b^{\frac {2}{3}} a^{\frac {10}{3}} x^{4} \left (b \,x^{3}+a \right )} \] Input:
int((B*x^3+A)/x^5/(b*x^3+a)^3,x)
Output:
( - 28*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a*b**2*x **4 - 28*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*b**3*x **7 - 9*b**(2/3)*a**(1/3)*a**2 + 63*b**(2/3)*a**(1/3)*a*b*x**3 + 84*b**(2/ 3)*a**(1/3)*b**2*x**6 + 14*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x **2)*a*b**2*x**4 + 14*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)* b**3*x**7 - 28*log(a**(1/3) + b**(1/3)*x)*a*b**2*x**4 - 28*log(a**(1/3) + b**(1/3)*x)*b**3*x**7)/(36*b**(2/3)*a**(1/3)*a**3*x**4*(a + b*x**3))