Integrand size = 22, antiderivative size = 318 \[ \int \frac {A+B x^3}{x^{3/2} \left (a+b x^3\right )^2} \, dx=-\frac {7 A b-a B}{3 a^2 b \sqrt {x}}+\frac {A b-a B}{3 a b \sqrt {x} \left (a+b x^3\right )}+\frac {(7 A b-a B) \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{18 a^{13/6} b^{5/6}}-\frac {(7 A b-a B) \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{18 a^{13/6} b^{5/6}}-\frac {(7 A b-a B) \arctan \left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{9 a^{13/6} b^{5/6}}-\frac {(7 A b-a B) \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{b} x\right )}{12 \sqrt {3} a^{13/6} b^{5/6}}+\frac {(7 A b-a B) \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{b} x\right )}{12 \sqrt {3} a^{13/6} b^{5/6}} \]
-1/9*(7*A*b-B*a)*arctan(b^(1/6)*x^(1/2)/a^(1/6))/a^(13/6)/b^(5/6)-1/18*(7* A*b-B*a)*arctan(-3^(1/2)+2*b^(1/6)*x^(1/2)/a^(1/6))/a^(13/6)/b^(5/6)-1/18* (7*A*b-B*a)*arctan(3^(1/2)+2*b^(1/6)*x^(1/2)/a^(1/6))/a^(13/6)/b^(5/6)-1/3 6*(7*A*b-B*a)*ln(a^(1/3)+b^(1/3)*x-a^(1/6)*b^(1/6)*3^(1/2)*x^(1/2))/a^(13/ 6)/b^(5/6)*3^(1/2)+1/36*(7*A*b-B*a)*ln(a^(1/3)+b^(1/3)*x+a^(1/6)*b^(1/6)*3 ^(1/2)*x^(1/2))/a^(13/6)/b^(5/6)*3^(1/2)+1/3*(-7*A*b+B*a)/a^2/b/x^(1/2)+1/ 3*(A*b-B*a)/a/b/(b*x^3+a)/x^(1/2)
Time = 0.84 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.58 \[ \int \frac {A+B x^3}{x^{3/2} \left (a+b x^3\right )^2} \, dx=\frac {\frac {6 \sqrt [6]{a} \left (-6 a A-7 A b x^3+a B x^3\right )}{\sqrt {x} \left (a+b x^3\right )}+\frac {2 (-7 A b+a B) \arctan \left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{b^{5/6}}+\frac {(7 A b-a B) \arctan \left (\frac {\sqrt [3]{a}-\sqrt [3]{b} x}{\sqrt [6]{a} \sqrt [6]{b} \sqrt {x}}\right )}{b^{5/6}}+\frac {\sqrt {3} (7 A b-a B) \text {arctanh}\left (\frac {\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}}{\sqrt [3]{a}+\sqrt [3]{b} x}\right )}{b^{5/6}}}{18 a^{13/6}} \]
((6*a^(1/6)*(-6*a*A - 7*A*b*x^3 + a*B*x^3))/(Sqrt[x]*(a + b*x^3)) + (2*(-7 *A*b + a*B)*ArcTan[(b^(1/6)*Sqrt[x])/a^(1/6)])/b^(5/6) + ((7*A*b - a*B)*Ar cTan[(a^(1/3) - b^(1/3)*x)/(a^(1/6)*b^(1/6)*Sqrt[x])])/b^(5/6) + (Sqrt[3]* (7*A*b - a*B)*ArcTanh[(Sqrt[3]*a^(1/6)*b^(1/6)*Sqrt[x])/(a^(1/3) + b^(1/3) *x)])/b^(5/6))/(18*a^(13/6))
Time = 0.52 (sec) , antiderivative size = 305, normalized size of antiderivative = 0.96, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {957, 847, 851, 824, 27, 218, 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.
| \(\displaystyle \int \frac {A+B x^3}{x^{3/2} \left (a+b x^3\right )^2} \, dx\) |
| \(\Big \downarrow \) 957 |
| \(\displaystyle \frac {(7 A b-a B) \int \frac {1}{x^{3/2} \left (b x^3+a\right )}dx}{6 a b}+\frac {A b-a B}{3 a b \sqrt {x} \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 847 |
| \(\displaystyle \frac {(7 A b-a B) \left (-\frac {b \int \frac {x^{3/2}}{b x^3+a}dx}{a}-\frac {2}{a \sqrt {x}}\right )}{6 a b}+\frac {A b-a B}{3 a b \sqrt {x} \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 851 |
| \(\displaystyle \frac {(7 A b-a B) \left (-\frac {2 b \int \frac {x^2}{b x^3+a}d\sqrt {x}}{a}-\frac {2}{a \sqrt {x}}\right )}{6 a b}+\frac {A b-a B}{3 a b \sqrt {x} \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 824 |
| \(\displaystyle \frac {(7 A b-a B) \left (-\frac {2 b \left (\frac {\int \frac {1}{\sqrt [3]{b} x+\sqrt [3]{a}}d\sqrt {x}}{3 b^{2/3}}+\frac {\int -\frac {\sqrt [6]{a}-\sqrt {3} \sqrt [6]{b} \sqrt {x}}{2 \left (\sqrt [3]{b} x-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}\right )}d\sqrt {x}}{3 \sqrt [6]{a} b^{2/3}}+\frac {\int -\frac {\sqrt {3} \sqrt [6]{b} \sqrt {x}+\sqrt [6]{a}}{2 \left (\sqrt [3]{b} x+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}\right )}d\sqrt {x}}{3 \sqrt [6]{a} b^{2/3}}\right )}{a}-\frac {2}{a \sqrt {x}}\right )}{6 a b}+\frac {A b-a B}{3 a b \sqrt {x} \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(7 A b-a B) \left (-\frac {2 b \left (\frac {\int \frac {1}{\sqrt [3]{b} x+\sqrt [3]{a}}d\sqrt {x}}{3 b^{2/3}}-\frac {\int \frac {\sqrt [6]{a}-\sqrt {3} \sqrt [6]{b} \sqrt {x}}{\sqrt [3]{b} x-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{6 \sqrt [6]{a} b^{2/3}}-\frac {\int \frac {\sqrt {3} \sqrt [6]{b} \sqrt {x}+\sqrt [6]{a}}{\sqrt [3]{b} x+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{6 \sqrt [6]{a} b^{2/3}}\right )}{a}-\frac {2}{a \sqrt {x}}\right )}{6 a b}+\frac {A b-a B}{3 a b \sqrt {x} \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {(7 A b-a B) \left (-\frac {2 b \left (-\frac {\int \frac {\sqrt [6]{a}-\sqrt {3} \sqrt [6]{b} \sqrt {x}}{\sqrt [3]{b} x-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{6 \sqrt [6]{a} b^{2/3}}-\frac {\int \frac {\sqrt {3} \sqrt [6]{b} \sqrt {x}+\sqrt [6]{a}}{\sqrt [3]{b} x+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{6 \sqrt [6]{a} b^{2/3}}+\frac {\arctan \left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{3 \sqrt [6]{a} b^{5/6}}\right )}{a}-\frac {2}{a \sqrt {x}}\right )}{6 a b}+\frac {A b-a B}{3 a b \sqrt {x} \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {(7 A b-a B) \left (-\frac {2 b \left (-\frac {-\frac {1}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{b} x-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}-\frac {\sqrt {3} \int -\frac {\sqrt [6]{b} \left (\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} \sqrt {x}\right )}{\sqrt [3]{b} x-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{2 \sqrt [6]{b}}}{6 \sqrt [6]{a} b^{2/3}}-\frac {\frac {\sqrt {3} \int \frac {\sqrt [6]{b} \left (2 \sqrt [6]{b} \sqrt {x}+\sqrt {3} \sqrt [6]{a}\right )}{\sqrt [3]{b} x+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{2 \sqrt [6]{b}}-\frac {1}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{b} x+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{6 \sqrt [6]{a} b^{2/3}}+\frac {\arctan \left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{3 \sqrt [6]{a} b^{5/6}}\right )}{a}-\frac {2}{a \sqrt {x}}\right )}{6 a b}+\frac {A b-a B}{3 a b \sqrt {x} \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(7 A b-a B) \left (-\frac {2 b \left (-\frac {\frac {\sqrt {3} \int \frac {\sqrt [6]{b} \left (\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} \sqrt {x}\right )}{\sqrt [3]{b} x-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{2 \sqrt [6]{b}}-\frac {1}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{b} x-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{6 \sqrt [6]{a} b^{2/3}}-\frac {\frac {\sqrt {3} \int \frac {\sqrt [6]{b} \left (2 \sqrt [6]{b} \sqrt {x}+\sqrt {3} \sqrt [6]{a}\right )}{\sqrt [3]{b} x+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{2 \sqrt [6]{b}}-\frac {1}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{b} x+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{6 \sqrt [6]{a} b^{2/3}}+\frac {\arctan \left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{3 \sqrt [6]{a} b^{5/6}}\right )}{a}-\frac {2}{a \sqrt {x}}\right )}{6 a b}+\frac {A b-a B}{3 a b \sqrt {x} \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(7 A b-a B) \left (-\frac {2 b \left (-\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} \sqrt {x}}{\sqrt [3]{b} x-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}-\frac {1}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{b} x-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{6 \sqrt [6]{a} b^{2/3}}-\frac {\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [6]{b} \sqrt {x}+\sqrt {3} \sqrt [6]{a}}{\sqrt [3]{b} x+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}-\frac {1}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{b} x+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{6 \sqrt [6]{a} b^{2/3}}+\frac {\arctan \left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{3 \sqrt [6]{a} b^{5/6}}\right )}{a}-\frac {2}{a \sqrt {x}}\right )}{6 a b}+\frac {A b-a B}{3 a b \sqrt {x} \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {(7 A b-a B) \left (-\frac {2 b \left (-\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} \sqrt {x}}{\sqrt [3]{b} x-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}-\frac {\int \frac {1}{-x-\frac {1}{3}}d\left (1-\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt {3} \sqrt [6]{a}}\right )}{\sqrt {3} \sqrt [6]{b}}}{6 \sqrt [6]{a} b^{2/3}}-\frac {\frac {\int \frac {1}{-x-\frac {1}{3}}d\left (\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt {3} \sqrt [6]{a}}+1\right )}{\sqrt {3} \sqrt [6]{b}}+\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [6]{b} \sqrt {x}+\sqrt {3} \sqrt [6]{a}}{\sqrt [3]{b} x+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{6 \sqrt [6]{a} b^{2/3}}+\frac {\arctan \left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{3 \sqrt [6]{a} b^{5/6}}\right )}{a}-\frac {2}{a \sqrt {x}}\right )}{6 a b}+\frac {A b-a B}{3 a b \sqrt {x} \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {(7 A b-a B) \left (-\frac {2 b \left (-\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} \sqrt {x}}{\sqrt [3]{b} x-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}+\frac {\arctan \left (\sqrt {3} \left (1-\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt {3} \sqrt [6]{a}}\right )\right )}{\sqrt [6]{b}}}{6 \sqrt [6]{a} b^{2/3}}-\frac {\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [6]{b} \sqrt {x}+\sqrt {3} \sqrt [6]{a}}{\sqrt [3]{b} x+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}-\frac {\arctan \left (\sqrt {3} \left (\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt {3} \sqrt [6]{a}}+1\right )\right )}{\sqrt [6]{b}}}{6 \sqrt [6]{a} b^{2/3}}+\frac {\arctan \left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{3 \sqrt [6]{a} b^{5/6}}\right )}{a}-\frac {2}{a \sqrt {x}}\right )}{6 a b}+\frac {A b-a B}{3 a b \sqrt {x} \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {(7 A b-a B) \left (-\frac {2 b \left (\frac {\arctan \left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{3 \sqrt [6]{a} b^{5/6}}-\frac {\frac {\arctan \left (\sqrt {3} \left (1-\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt {3} \sqrt [6]{a}}\right )\right )}{\sqrt [6]{b}}-\frac {\sqrt {3} \log \left (-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{2 \sqrt [6]{b}}}{6 \sqrt [6]{a} b^{2/3}}-\frac {\frac {\sqrt {3} \log \left (\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{2 \sqrt [6]{b}}-\frac {\arctan \left (\sqrt {3} \left (\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt {3} \sqrt [6]{a}}+1\right )\right )}{\sqrt [6]{b}}}{6 \sqrt [6]{a} b^{2/3}}\right )}{a}-\frac {2}{a \sqrt {x}}\right )}{6 a b}+\frac {A b-a B}{3 a b \sqrt {x} \left (a+b x^3\right )}\) |
(A*b - a*B)/(3*a*b*Sqrt[x]*(a + b*x^3)) + ((7*A*b - a*B)*(-2/(a*Sqrt[x]) - (2*b*(ArcTan[(b^(1/6)*Sqrt[x])/a^(1/6)]/(3*a^(1/6)*b^(5/6)) - (ArcTan[Sqr t[3]*(1 - (2*b^(1/6)*Sqrt[x])/(Sqrt[3]*a^(1/6)))]/b^(1/6) - (Sqrt[3]*Log[a ^(1/3) - Sqrt[3]*a^(1/6)*b^(1/6)*Sqrt[x] + b^(1/3)*x])/(2*b^(1/6)))/(6*a^( 1/6)*b^(2/3)) - (-(ArcTan[Sqrt[3]*(1 + (2*b^(1/6)*Sqrt[x])/(Sqrt[3]*a^(1/6 )))]/b^(1/6)) + (Sqrt[3]*Log[a^(1/3) + Sqrt[3]*a^(1/6)*b^(1/6)*Sqrt[x] + b ^(1/3)*x])/(2*b^(1/6)))/(6*a^(1/6)*b^(2/3))))/a))/(6*a*b)
3.2.68.3.1 Defintions of rubi rules used
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_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Module[{r = Numerator [Rt[a/b, n]], s = Denominator[Rt[a/b, n]], k, u}, Simp[u = Int[(r*Cos[(2*k - 1)*m*(Pi/n)] - s*Cos[(2*k - 1)*(m + 1)*(Pi/n)]*x)/(r^2 - 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2*x^2), x] + Int[(r*Cos[(2*k - 1)*m*(Pi/n)] + s*Cos[(2*k - 1)*(m + 1)*(Pi/n)]*x)/(r^2 + 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2*x^2), x] ; 2*(-1)^(m/2)*(r^(m + 2)/(a*n*s^m)) Int[1/(r^2 + s^2*x^2), x] + 2*(r^(m + 1)/(a*n*s^m)) Sum[u, {k, 1, (n - 2)/4}], x]] /; FreeQ[{a, b}, x] && IGt Q[(n - 2)/4, 0] && IGtQ[m, 0] && LtQ[m, n - 1] && PosQ[a/b]
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[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^ n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && FractionQ[m] && 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 = 4.50 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.68
| method | result | size |
| derivativedivides | \(-\frac {2 \left (\frac {\left (\frac {A b}{6}-\frac {B a}{6}\right ) x^{\frac {5}{2}}}{b \,x^{3}+a}+\left (\frac {7 A b}{6}-\frac {B a}{6}\right ) \left (\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {5}{6}} \ln \left (\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}-x -\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {\arctan \left (-\sqrt {3}+\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{6}}}-\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {5}{6}} \ln \left (x +\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {\arctan \left (\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{6}}}+\frac {\arctan \left (\frac {\sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )\right )}{a^{2}}-\frac {2 A}{a^{2} \sqrt {x}}\) | \(216\) |
| default | \(-\frac {2 \left (\frac {\left (\frac {A b}{6}-\frac {B a}{6}\right ) x^{\frac {5}{2}}}{b \,x^{3}+a}+\left (\frac {7 A b}{6}-\frac {B a}{6}\right ) \left (\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {5}{6}} \ln \left (\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}-x -\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {\arctan \left (-\sqrt {3}+\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{6}}}-\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {5}{6}} \ln \left (x +\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {\arctan \left (\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{6}}}+\frac {\arctan \left (\frac {\sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )\right )}{a^{2}}-\frac {2 A}{a^{2} \sqrt {x}}\) | \(216\) |
| risch | \(-\frac {2 A}{a^{2} \sqrt {x}}-\frac {\frac {2 \left (\frac {A b}{6}-\frac {B a}{6}\right ) x^{\frac {5}{2}}}{b \,x^{3}+a}+2 \left (\frac {7 A b}{6}-\frac {B a}{6}\right ) \left (\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {5}{6}} \ln \left (\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}-x -\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {\arctan \left (-\sqrt {3}+\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{6}}}-\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {5}{6}} \ln \left (x +\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {\arctan \left (\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{6}}}+\frac {\arctan \left (\frac {\sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{a^{2}}\) | \(218\) |
-2/a^2*((1/6*A*b-1/6*B*a)*x^(5/2)/(b*x^3+a)+(7/6*A*b-1/6*B*a)*(1/12/a*3^(1 /2)*(a/b)^(5/6)*ln(3^(1/2)*(a/b)^(1/6)*x^(1/2)-x-(a/b)^(1/3))+1/6/b/(a/b)^ (1/6)*arctan(-3^(1/2)+2*x^(1/2)/(a/b)^(1/6))-1/12/a*3^(1/2)*(a/b)^(5/6)*ln (x+3^(1/2)*(a/b)^(1/6)*x^(1/2)+(a/b)^(1/3))+1/6/b/(a/b)^(1/6)*arctan(2*x^( 1/2)/(a/b)^(1/6)+3^(1/2))+1/3/b/(a/b)^(1/6)*arctan(x^(1/2)/(a/b)^(1/6))))- 2*A/a^2/x^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 1788 vs. \(2 (226) = 452\).
Time = 0.37 (sec) , antiderivative size = 1788, normalized size of antiderivative = 5.62 \[ \int \frac {A+B x^3}{x^{3/2} \left (a+b x^3\right )^2} \, dx=\text {Too large to display} \]
-1/36*(2*(a^2*b*x^4 + a^3*x)*(-(B^6*a^6 - 42*A*B^5*a^5*b + 735*A^2*B^4*a^4 *b^2 - 6860*A^3*B^3*a^3*b^3 + 36015*A^4*B^2*a^2*b^4 - 100842*A^5*B*a*b^5 + 117649*A^6*b^6)/(a^13*b^5))^(1/6)*log(a^11*b^4*(-(B^6*a^6 - 42*A*B^5*a^5* b + 735*A^2*B^4*a^4*b^2 - 6860*A^3*B^3*a^3*b^3 + 36015*A^4*B^2*a^2*b^4 - 1 00842*A^5*B*a*b^5 + 117649*A^6*b^6)/(a^13*b^5))^(5/6) - (B^5*a^5 - 35*A*B^ 4*a^4*b + 490*A^2*B^3*a^3*b^2 - 3430*A^3*B^2*a^2*b^3 + 12005*A^4*B*a*b^4 - 16807*A^5*b^5)*sqrt(x)) - 2*(a^2*b*x^4 + a^3*x)*(-(B^6*a^6 - 42*A*B^5*a^5 *b + 735*A^2*B^4*a^4*b^2 - 6860*A^3*B^3*a^3*b^3 + 36015*A^4*B^2*a^2*b^4 - 100842*A^5*B*a*b^5 + 117649*A^6*b^6)/(a^13*b^5))^(1/6)*log(-a^11*b^4*(-(B^ 6*a^6 - 42*A*B^5*a^5*b + 735*A^2*B^4*a^4*b^2 - 6860*A^3*B^3*a^3*b^3 + 3601 5*A^4*B^2*a^2*b^4 - 100842*A^5*B*a*b^5 + 117649*A^6*b^6)/(a^13*b^5))^(5/6) - (B^5*a^5 - 35*A*B^4*a^4*b + 490*A^2*B^3*a^3*b^2 - 3430*A^3*B^2*a^2*b^3 + 12005*A^4*B*a*b^4 - 16807*A^5*b^5)*sqrt(x)) + (a^2*b*x^4 + a^3*x - sqrt( -3)*(a^2*b*x^4 + a^3*x))*(-(B^6*a^6 - 42*A*B^5*a^5*b + 735*A^2*B^4*a^4*b^2 - 6860*A^3*B^3*a^3*b^3 + 36015*A^4*B^2*a^2*b^4 - 100842*A^5*B*a*b^5 + 117 649*A^6*b^6)/(a^13*b^5))^(1/6)*log(1/2*(sqrt(-3)*a^11*b^4 + a^11*b^4)*(-(B ^6*a^6 - 42*A*B^5*a^5*b + 735*A^2*B^4*a^4*b^2 - 6860*A^3*B^3*a^3*b^3 + 360 15*A^4*B^2*a^2*b^4 - 100842*A^5*B*a*b^5 + 117649*A^6*b^6)/(a^13*b^5))^(5/6 ) - (B^5*a^5 - 35*A*B^4*a^4*b + 490*A^2*B^3*a^3*b^2 - 3430*A^3*B^2*a^2*b^3 + 12005*A^4*B*a*b^4 - 16807*A^5*b^5)*sqrt(x)) - (a^2*b*x^4 + a^3*x - s...
Leaf count of result is larger than twice the leaf count of optimal. 2200 vs. \(2 (299) = 598\).
Time = 151.56 (sec) , antiderivative size = 2200, normalized size of antiderivative = 6.92 \[ \int \frac {A+B x^3}{x^{3/2} \left (a+b x^3\right )^2} \, dx=\text {Too large to display} \]
Piecewise((zoo*(-2*A/(13*x**(13/2)) - 2*B/(7*x**(7/2))), Eq(a, 0) & Eq(b, 0)), ((-2*A/sqrt(x) + 2*B*x**(5/2)/5)/a**2, Eq(b, 0)), ((-2*A/(13*x**(13/2 )) - 2*B/(7*x**(7/2)))/b**2, Eq(a, 0)), (-14*A*a*b*sqrt(x)*log(sqrt(x) - ( -a/b)**(1/6))/(36*a**3*b*sqrt(x)*(-a/b)**(1/6) + 36*a**2*b**2*x**(7/2)*(-a /b)**(1/6)) + 14*A*a*b*sqrt(x)*log(sqrt(x) + (-a/b)**(1/6))/(36*a**3*b*sqr t(x)*(-a/b)**(1/6) + 36*a**2*b**2*x**(7/2)*(-a/b)**(1/6)) - 7*A*a*b*sqrt(x )*log(-4*sqrt(x)*(-a/b)**(1/6) + 4*x + 4*(-a/b)**(1/3))/(36*a**3*b*sqrt(x) *(-a/b)**(1/6) + 36*a**2*b**2*x**(7/2)*(-a/b)**(1/6)) + 7*A*a*b*sqrt(x)*lo g(4*sqrt(x)*(-a/b)**(1/6) + 4*x + 4*(-a/b)**(1/3))/(36*a**3*b*sqrt(x)*(-a/ b)**(1/6) + 36*a**2*b**2*x**(7/2)*(-a/b)**(1/6)) - 14*sqrt(3)*A*a*b*sqrt(x )*atan(2*sqrt(3)*sqrt(x)/(3*(-a/b)**(1/6)) - sqrt(3)/3)/(36*a**3*b*sqrt(x) *(-a/b)**(1/6) + 36*a**2*b**2*x**(7/2)*(-a/b)**(1/6)) - 14*sqrt(3)*A*a*b*s qrt(x)*atan(2*sqrt(3)*sqrt(x)/(3*(-a/b)**(1/6)) + sqrt(3)/3)/(36*a**3*b*sq rt(x)*(-a/b)**(1/6) + 36*a**2*b**2*x**(7/2)*(-a/b)**(1/6)) - 72*A*a*b*(-a/ b)**(1/6)/(36*a**3*b*sqrt(x)*(-a/b)**(1/6) + 36*a**2*b**2*x**(7/2)*(-a/b)* *(1/6)) - 14*A*b**2*x**(7/2)*log(sqrt(x) - (-a/b)**(1/6))/(36*a**3*b*sqrt( x)*(-a/b)**(1/6) + 36*a**2*b**2*x**(7/2)*(-a/b)**(1/6)) + 14*A*b**2*x**(7/ 2)*log(sqrt(x) + (-a/b)**(1/6))/(36*a**3*b*sqrt(x)*(-a/b)**(1/6) + 36*a**2 *b**2*x**(7/2)*(-a/b)**(1/6)) - 7*A*b**2*x**(7/2)*log(-4*sqrt(x)*(-a/b)**( 1/6) + 4*x + 4*(-a/b)**(1/3))/(36*a**3*b*sqrt(x)*(-a/b)**(1/6) + 36*a**...
Time = 0.30 (sec) , antiderivative size = 240, normalized size of antiderivative = 0.75 \[ \int \frac {A+B x^3}{x^{3/2} \left (a+b x^3\right )^2} \, dx=\frac {{\left (B a - 7 \, A b\right )} x^{3} - 6 \, A a}{3 \, {\left (a^{2} b x^{\frac {7}{2}} + a^{3} \sqrt {x}\right )}} - \frac {{\left (B a - 7 \, A b\right )} {\left (\frac {\sqrt {3} \log \left (\sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} \sqrt {x} + b^{\frac {1}{3}} x + a^{\frac {1}{3}}\right )}{a^{\frac {1}{6}} b^{\frac {5}{6}}} - \frac {\sqrt {3} \log \left (-\sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} \sqrt {x} + b^{\frac {1}{3}} x + a^{\frac {1}{3}}\right )}{a^{\frac {1}{6}} b^{\frac {5}{6}}} - \frac {2 \, \arctan \left (\frac {\sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} + 2 \, b^{\frac {1}{3}} \sqrt {x}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{b^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}} - \frac {2 \, \arctan \left (-\frac {\sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} - 2 \, b^{\frac {1}{3}} \sqrt {x}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{b^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}} - \frac {4 \, \arctan \left (\frac {b^{\frac {1}{3}} \sqrt {x}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{b^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}}{36 \, a^{2}} \]
1/3*((B*a - 7*A*b)*x^3 - 6*A*a)/(a^2*b*x^(7/2) + a^3*sqrt(x)) - 1/36*(B*a - 7*A*b)*(sqrt(3)*log(sqrt(3)*a^(1/6)*b^(1/6)*sqrt(x) + b^(1/3)*x + a^(1/3 ))/(a^(1/6)*b^(5/6)) - sqrt(3)*log(-sqrt(3)*a^(1/6)*b^(1/6)*sqrt(x) + b^(1 /3)*x + a^(1/3))/(a^(1/6)*b^(5/6)) - 2*arctan((sqrt(3)*a^(1/6)*b^(1/6) + 2 *b^(1/3)*sqrt(x))/sqrt(a^(1/3)*b^(1/3)))/(b^(2/3)*sqrt(a^(1/3)*b^(1/3))) - 2*arctan(-(sqrt(3)*a^(1/6)*b^(1/6) - 2*b^(1/3)*sqrt(x))/sqrt(a^(1/3)*b^(1 /3)))/(b^(2/3)*sqrt(a^(1/3)*b^(1/3))) - 4*arctan(b^(1/3)*sqrt(x)/sqrt(a^(1 /3)*b^(1/3)))/(b^(2/3)*sqrt(a^(1/3)*b^(1/3))))/a^2
Time = 0.68 (sec) , antiderivative size = 284, normalized size of antiderivative = 0.89 \[ \int \frac {A+B x^3}{x^{3/2} \left (a+b x^3\right )^2} \, dx=\frac {{\left (B a - 7 \, A b\right )} \arctan \left (\frac {\sqrt {3} \left (\frac {a}{b}\right )^{\frac {1}{6}} + 2 \, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{18 \, \left (a b^{5}\right )^{\frac {1}{6}} a^{2}} + \frac {{\left (B a - 7 \, A b\right )} \arctan \left (-\frac {\sqrt {3} \left (\frac {a}{b}\right )^{\frac {1}{6}} - 2 \, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{18 \, \left (a b^{5}\right )^{\frac {1}{6}} a^{2}} + \frac {{\left (B a \left (\frac {a}{b}\right )^{\frac {5}{6}} - 7 \, A b \left (\frac {a}{b}\right )^{\frac {5}{6}}\right )} \arctan \left (\frac {\sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{9 \, a^{3}} + \frac {B a x^{3} - 7 \, A b x^{3} - 6 \, A a}{3 \, {\left (b x^{\frac {7}{2}} + a \sqrt {x}\right )} a^{2}} - \frac {\sqrt {3} {\left (\left (a b^{5}\right )^{\frac {5}{6}} B a - 7 \, \left (a b^{5}\right )^{\frac {5}{6}} A b\right )} \log \left (\sqrt {3} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{6}} + x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{36 \, a^{3} b^{5}} + \frac {\sqrt {3} {\left (\left (a b^{5}\right )^{\frac {5}{6}} B a - 7 \, \left (a b^{5}\right )^{\frac {5}{6}} A b\right )} \log \left (-\sqrt {3} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{6}} + x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{36 \, a^{3} b^{5}} \]
1/18*(B*a - 7*A*b)*arctan((sqrt(3)*(a/b)^(1/6) + 2*sqrt(x))/(a/b)^(1/6))/( (a*b^5)^(1/6)*a^2) + 1/18*(B*a - 7*A*b)*arctan(-(sqrt(3)*(a/b)^(1/6) - 2*s qrt(x))/(a/b)^(1/6))/((a*b^5)^(1/6)*a^2) + 1/9*(B*a*(a/b)^(5/6) - 7*A*b*(a /b)^(5/6))*arctan(sqrt(x)/(a/b)^(1/6))/a^3 + 1/3*(B*a*x^3 - 7*A*b*x^3 - 6* A*a)/((b*x^(7/2) + a*sqrt(x))*a^2) - 1/36*sqrt(3)*((a*b^5)^(5/6)*B*a - 7*( a*b^5)^(5/6)*A*b)*log(sqrt(3)*sqrt(x)*(a/b)^(1/6) + x + (a/b)^(1/3))/(a^3* b^5) + 1/36*sqrt(3)*((a*b^5)^(5/6)*B*a - 7*(a*b^5)^(5/6)*A*b)*log(-sqrt(3) *sqrt(x)*(a/b)^(1/6) + x + (a/b)^(1/3))/(a^3*b^5)
Time = 7.17 (sec) , antiderivative size = 1757, normalized size of antiderivative = 5.53 \[ \int \frac {A+B x^3}{x^{3/2} \left (a+b x^3\right )^2} \, dx=\text {Too large to display} \]
(atan((((7*A*b - B*a)^2*(81*B^3*a^18*b^3 - 27783*A^3*a^15*b^6 - 1701*A*B^2 *a^17*b^4 + 11907*A^2*B*a^16*b^5 + (x^(1/2)*(7*A*b - B*a)*(23147208*A^2*a^ 17*b^6 + 472392*B^2*a^19*b^4 - 6613488*A*B*a^18*b^5))/(5832*(-a)^(13/6)*b^ (5/6)))*1i)/((-a)^(13/3)*b^(5/3)) + ((7*A*b - B*a)^2*(27783*A^3*a^15*b^6 - 81*B^3*a^18*b^3 + 1701*A*B^2*a^17*b^4 - 11907*A^2*B*a^16*b^5 + (x^(1/2)*( 7*A*b - B*a)*(23147208*A^2*a^17*b^6 + 472392*B^2*a^19*b^4 - 6613488*A*B*a^ 18*b^5))/(5832*(-a)^(13/6)*b^(5/6)))*1i)/((-a)^(13/3)*b^(5/3)))/(((7*A*b - B*a)^2*(81*B^3*a^18*b^3 - 27783*A^3*a^15*b^6 - 1701*A*B^2*a^17*b^4 + 1190 7*A^2*B*a^16*b^5 + (x^(1/2)*(7*A*b - B*a)*(23147208*A^2*a^17*b^6 + 472392* B^2*a^19*b^4 - 6613488*A*B*a^18*b^5))/(5832*(-a)^(13/6)*b^(5/6))))/((-a)^( 13/3)*b^(5/3)) - ((7*A*b - B*a)^2*(27783*A^3*a^15*b^6 - 81*B^3*a^18*b^3 + 1701*A*B^2*a^17*b^4 - 11907*A^2*B*a^16*b^5 + (x^(1/2)*(7*A*b - B*a)*(23147 208*A^2*a^17*b^6 + 472392*B^2*a^19*b^4 - 6613488*A*B*a^18*b^5))/(5832*(-a) ^(13/6)*b^(5/6))))/((-a)^(13/3)*b^(5/3))))*(7*A*b - B*a)*1i)/(9*(-a)^(13/6 )*b^(5/6)) - ((2*A)/a + (x^3*(7*A*b - B*a))/(3*a^2))/(a*x^(1/2) + b*x^(7/2 )) + (atan(((((3^(1/2)*1i)/2 - 1/2)^2*(7*A*b - B*a)^2*(81*B^3*a^18*b^3 - 2 7783*A^3*a^15*b^6 - 1701*A*B^2*a^17*b^4 + 11907*A^2*B*a^16*b^5 + (x^(1/2)* ((3^(1/2)*1i)/2 - 1/2)*(7*A*b - B*a)*(23147208*A^2*a^17*b^6 + 472392*B^2*a ^19*b^4 - 6613488*A*B*a^18*b^5))/(5832*(-a)^(13/6)*b^(5/6)))*1i)/((-a)^(13 /3)*b^(5/3)) + (((3^(1/2)*1i)/2 - 1/2)^2*(7*A*b - B*a)^2*(27783*A^3*a^1...