Integrand size = 22, antiderivative size = 265 \[ \int \frac {x^{3/2} \left (A+B x^3\right )}{\left (a+b x^3\right )^3} \, dx=\frac {(A b-a B) x^{5/2}}{6 a b \left (a+b x^3\right )^2}+\frac {(7 A b+5 a B) x^{5/2}}{36 a^2 b \left (a+b x^3\right )}-\frac {(7 A b+5 a B) \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{216 a^{13/6} b^{11/6}}+\frac {(7 A b+5 a B) \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{216 a^{13/6} b^{11/6}}+\frac {(7 A b+5 a B) \arctan \left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{108 a^{13/6} b^{11/6}}-\frac {(7 A b+5 a B) \text {arctanh}\left (\frac {\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}}{\sqrt [3]{a}+\sqrt [3]{b} x}\right )}{72 \sqrt {3} a^{13/6} b^{11/6}} \] Output:
1/6*(A*b-B*a)*x^(5/2)/a/b/(b*x^3+a)^2+1/36*(7*A*b+5*B*a)*x^(5/2)/a^2/b/(b* x^3+a)+1/216*(7*A*b+5*B*a)*arctan(-3^(1/2)+2*b^(1/6)*x^(1/2)/a^(1/6))/a^(1 3/6)/b^(11/6)+1/216*(7*A*b+5*B*a)*arctan(3^(1/2)+2*b^(1/6)*x^(1/2)/a^(1/6) )/a^(13/6)/b^(11/6)+1/108*(7*A*b+5*B*a)*arctan(b^(1/6)*x^(1/2)/a^(1/6))/a^ (13/6)/b^(11/6)-1/216*(7*A*b+5*B*a)*arctanh(3^(1/2)*a^(1/6)*b^(1/6)*x^(1/2 )/(a^(1/3)+b^(1/3)*x))*3^(1/2)/a^(13/6)/b^(11/6)
Time = 1.11 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.73 \[ \int \frac {x^{3/2} \left (A+B x^3\right )}{\left (a+b x^3\right )^3} \, dx=\frac {\frac {6 \sqrt [6]{a} b^{5/6} x^{5/2} \left (-a^2 B+7 A b^2 x^3+a b \left (13 A+5 B x^3\right )\right )}{\left (a+b x^3\right )^2}+2 (7 A b+5 a B) \arctan \left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )-(7 A b+5 a B) \arctan \left (\frac {\sqrt [3]{a}-\sqrt [3]{b} x}{\sqrt [6]{a} \sqrt [6]{b} \sqrt {x}}\right )-\sqrt {3} (7 A b+5 a B) \text {arctanh}\left (\frac {\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}}{\sqrt [3]{a}+\sqrt [3]{b} x}\right )}{216 a^{13/6} b^{11/6}} \] Input:
Integrate[(x^(3/2)*(A + B*x^3))/(a + b*x^3)^3,x]
Output:
((6*a^(1/6)*b^(5/6)*x^(5/2)*(-(a^2*B) + 7*A*b^2*x^3 + a*b*(13*A + 5*B*x^3) ))/(a + b*x^3)^2 + 2*(7*A*b + 5*a*B)*ArcTan[(b^(1/6)*Sqrt[x])/a^(1/6)] - ( 7*A*b + 5*a*B)*ArcTan[(a^(1/3) - b^(1/3)*x)/(a^(1/6)*b^(1/6)*Sqrt[x])] - S qrt[3]*(7*A*b + 5*a*B)*ArcTanh[(Sqrt[3]*a^(1/6)*b^(1/6)*Sqrt[x])/(a^(1/3) + b^(1/3)*x)])/(216*a^(13/6)*b^(11/6))
Time = 0.90 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.20, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {957, 819, 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.
| \(\displaystyle \int \frac {x^{3/2} \left (A+B x^3\right )}{\left (a+b x^3\right )^3} \, dx\) |
| \(\Big \downarrow \) 957 |
| \(\displaystyle \frac {(5 a B+7 A b) \int \frac {x^{3/2}}{\left (b x^3+a\right )^2}dx}{12 a b}+\frac {x^{5/2} (A b-a B)}{6 a b \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 819 |
| \(\displaystyle \frac {(5 a B+7 A b) \left (\frac {\int \frac {x^{3/2}}{b x^3+a}dx}{6 a}+\frac {x^{5/2}}{3 a \left (a+b x^3\right )}\right )}{12 a b}+\frac {x^{5/2} (A b-a B)}{6 a b \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 851 |
| \(\displaystyle \frac {(5 a B+7 A b) \left (\frac {\int \frac {x^2}{b x^3+a}d\sqrt {x}}{3 a}+\frac {x^{5/2}}{3 a \left (a+b x^3\right )}\right )}{12 a b}+\frac {x^{5/2} (A b-a B)}{6 a b \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 824 |
| \(\displaystyle \frac {(5 a B+7 A b) \left (\frac {\frac {\int \frac {1}{\sqrt [3]{b} x+\sqrt [3]{a}}d\sqrt {x}}{3 b^{2/3}}+\frac {\int -\frac {\sqrt [6]{a}-\sqrt {3} \sqrt [6]{b} \sqrt {x}}{2 \left (\sqrt [3]{b} x-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}\right )}d\sqrt {x}}{3 \sqrt [6]{a} b^{2/3}}+\frac {\int -\frac {\sqrt {3} \sqrt [6]{b} \sqrt {x}+\sqrt [6]{a}}{2 \left (\sqrt [3]{b} x+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}\right )}d\sqrt {x}}{3 \sqrt [6]{a} b^{2/3}}}{3 a}+\frac {x^{5/2}}{3 a \left (a+b x^3\right )}\right )}{12 a b}+\frac {x^{5/2} (A b-a B)}{6 a b \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(5 a B+7 A b) \left (\frac {\frac {\int \frac {1}{\sqrt [3]{b} x+\sqrt [3]{a}}d\sqrt {x}}{3 b^{2/3}}-\frac {\int \frac {\sqrt [6]{a}-\sqrt {3} \sqrt [6]{b} \sqrt {x}}{\sqrt [3]{b} x-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{6 \sqrt [6]{a} b^{2/3}}-\frac {\int \frac {\sqrt {3} \sqrt [6]{b} \sqrt {x}+\sqrt [6]{a}}{\sqrt [3]{b} x+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{6 \sqrt [6]{a} b^{2/3}}}{3 a}+\frac {x^{5/2}}{3 a \left (a+b x^3\right )}\right )}{12 a b}+\frac {x^{5/2} (A b-a B)}{6 a b \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {(5 a B+7 A b) \left (\frac {-\frac {\int \frac {\sqrt [6]{a}-\sqrt {3} \sqrt [6]{b} \sqrt {x}}{\sqrt [3]{b} x-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{6 \sqrt [6]{a} b^{2/3}}-\frac {\int \frac {\sqrt {3} \sqrt [6]{b} \sqrt {x}+\sqrt [6]{a}}{\sqrt [3]{b} x+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{6 \sqrt [6]{a} b^{2/3}}+\frac {\arctan \left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{3 \sqrt [6]{a} b^{5/6}}}{3 a}+\frac {x^{5/2}}{3 a \left (a+b x^3\right )}\right )}{12 a b}+\frac {x^{5/2} (A b-a B)}{6 a b \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {(5 a B+7 A b) \left (\frac {-\frac {-\frac {1}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{b} x-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}-\frac {\sqrt {3} \int -\frac {\sqrt [6]{b} \left (\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} \sqrt {x}\right )}{\sqrt [3]{b} x-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{2 \sqrt [6]{b}}}{6 \sqrt [6]{a} b^{2/3}}-\frac {\frac {\sqrt {3} \int \frac {\sqrt [6]{b} \left (2 \sqrt [6]{b} \sqrt {x}+\sqrt {3} \sqrt [6]{a}\right )}{\sqrt [3]{b} x+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{2 \sqrt [6]{b}}-\frac {1}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{b} x+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{6 \sqrt [6]{a} b^{2/3}}+\frac {\arctan \left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{3 \sqrt [6]{a} b^{5/6}}}{3 a}+\frac {x^{5/2}}{3 a \left (a+b x^3\right )}\right )}{12 a b}+\frac {x^{5/2} (A b-a B)}{6 a b \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(5 a B+7 A b) \left (\frac {-\frac {\frac {\sqrt {3} \int \frac {\sqrt [6]{b} \left (\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} \sqrt {x}\right )}{\sqrt [3]{b} x-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{2 \sqrt [6]{b}}-\frac {1}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{b} x-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{6 \sqrt [6]{a} b^{2/3}}-\frac {\frac {\sqrt {3} \int \frac {\sqrt [6]{b} \left (2 \sqrt [6]{b} \sqrt {x}+\sqrt {3} \sqrt [6]{a}\right )}{\sqrt [3]{b} x+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{2 \sqrt [6]{b}}-\frac {1}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{b} x+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{6 \sqrt [6]{a} b^{2/3}}+\frac {\arctan \left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{3 \sqrt [6]{a} b^{5/6}}}{3 a}+\frac {x^{5/2}}{3 a \left (a+b x^3\right )}\right )}{12 a b}+\frac {x^{5/2} (A b-a B)}{6 a b \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(5 a B+7 A b) \left (\frac {-\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} \sqrt {x}}{\sqrt [3]{b} x-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}-\frac {1}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{b} x-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{6 \sqrt [6]{a} b^{2/3}}-\frac {\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [6]{b} \sqrt {x}+\sqrt {3} \sqrt [6]{a}}{\sqrt [3]{b} x+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}-\frac {1}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{b} x+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{6 \sqrt [6]{a} b^{2/3}}+\frac {\arctan \left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{3 \sqrt [6]{a} b^{5/6}}}{3 a}+\frac {x^{5/2}}{3 a \left (a+b x^3\right )}\right )}{12 a b}+\frac {x^{5/2} (A b-a B)}{6 a b \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {(5 a B+7 A b) \left (\frac {-\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} \sqrt {x}}{\sqrt [3]{b} x-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}-\frac {\int \frac {1}{-x-\frac {1}{3}}d\left (1-\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt {3} \sqrt [6]{a}}\right )}{\sqrt {3} \sqrt [6]{b}}}{6 \sqrt [6]{a} b^{2/3}}-\frac {\frac {\int \frac {1}{-x-\frac {1}{3}}d\left (\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt {3} \sqrt [6]{a}}+1\right )}{\sqrt {3} \sqrt [6]{b}}+\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [6]{b} \sqrt {x}+\sqrt {3} \sqrt [6]{a}}{\sqrt [3]{b} x+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{6 \sqrt [6]{a} b^{2/3}}+\frac {\arctan \left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{3 \sqrt [6]{a} b^{5/6}}}{3 a}+\frac {x^{5/2}}{3 a \left (a+b x^3\right )}\right )}{12 a b}+\frac {x^{5/2} (A b-a B)}{6 a b \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {(5 a B+7 A b) \left (\frac {-\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} \sqrt {x}}{\sqrt [3]{b} x-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}+\frac {\arctan \left (\sqrt {3} \left (1-\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt {3} \sqrt [6]{a}}\right )\right )}{\sqrt [6]{b}}}{6 \sqrt [6]{a} b^{2/3}}-\frac {\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [6]{b} \sqrt {x}+\sqrt {3} \sqrt [6]{a}}{\sqrt [3]{b} x+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}-\frac {\arctan \left (\sqrt {3} \left (\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt {3} \sqrt [6]{a}}+1\right )\right )}{\sqrt [6]{b}}}{6 \sqrt [6]{a} b^{2/3}}+\frac {\arctan \left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{3 \sqrt [6]{a} b^{5/6}}}{3 a}+\frac {x^{5/2}}{3 a \left (a+b x^3\right )}\right )}{12 a b}+\frac {x^{5/2} (A b-a B)}{6 a b \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {(5 a B+7 A b) \left (\frac {\frac {\arctan \left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{3 \sqrt [6]{a} b^{5/6}}-\frac {\frac {\arctan \left (\sqrt {3} \left (1-\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt {3} \sqrt [6]{a}}\right )\right )}{\sqrt [6]{b}}-\frac {\sqrt {3} \log \left (-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{2 \sqrt [6]{b}}}{6 \sqrt [6]{a} b^{2/3}}-\frac {\frac {\sqrt {3} \log \left (\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{2 \sqrt [6]{b}}-\frac {\arctan \left (\sqrt {3} \left (\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt {3} \sqrt [6]{a}}+1\right )\right )}{\sqrt [6]{b}}}{6 \sqrt [6]{a} b^{2/3}}}{3 a}+\frac {x^{5/2}}{3 a \left (a+b x^3\right )}\right )}{12 a b}+\frac {x^{5/2} (A b-a B)}{6 a b \left (a+b x^3\right )^2}\) |
Input:
Int[(x^(3/2)*(A + B*x^3))/(a + b*x^3)^3,x]
Output:
((A*b - a*B)*x^(5/2))/(6*a*b*(a + b*x^3)^2) + ((7*A*b + 5*a*B)*(x^(5/2)/(3 *a*(a + b*x^3)) + (ArcTan[(b^(1/6)*Sqrt[x])/a^(1/6)]/(3*a^(1/6)*b^(5/6)) - (ArcTan[Sqrt[3]*(1 - (2*b^(1/6)*Sqrt[x])/(Sqrt[3]*a^(1/6)))]/b^(1/6) - (S qrt[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])/(Sq rt[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)))/(3*a)))/(12*a*b)
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[((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_)^(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] :> 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 = 0.74 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.89
| method | result | size |
| derivativedivides | \(\frac {\frac {\left (7 A b +5 B a \right ) x^{\frac {11}{2}}}{36 a^{2}}+\frac {\left (13 A b -B a \right ) x^{\frac {5}{2}}}{36 a b}}{\left (b \,x^{3}+a \right )^{2}}+\frac {\left (7 A b +5 B a \right ) \left (\frac {\arctan \left (\frac {\sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{6}}}+\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {5}{6}} \ln \left (\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}-x -\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {\arctan \left (\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{6}}}-\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {5}{6}} \ln \left (x +\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {\arctan \left (\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{36 a^{2} b}\) | \(235\) |
| default | \(\frac {\frac {\left (7 A b +5 B a \right ) x^{\frac {11}{2}}}{36 a^{2}}+\frac {\left (13 A b -B a \right ) x^{\frac {5}{2}}}{36 a b}}{\left (b \,x^{3}+a \right )^{2}}+\frac {\left (7 A b +5 B a \right ) \left (\frac {\arctan \left (\frac {\sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{6}}}+\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {5}{6}} \ln \left (\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}-x -\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {\arctan \left (\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{6}}}-\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {5}{6}} \ln \left (x +\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {\arctan \left (\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{36 a^{2} b}\) | \(235\) |
Input:
int(x^(3/2)*(B*x^3+A)/(b*x^3+a)^3,x,method=_RETURNVERBOSE)
Output:
2*(1/72*(7*A*b+5*B*a)/a^2*x^(11/2)+1/72*(13*A*b-B*a)/a/b*x^(5/2))/(b*x^3+a )^2+1/36*(7*A*b+5*B*a)/a^2/b*(1/3/b/(a/b)^(1/6)*arctan(x^(1/2)/(a/b)^(1/6) )+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(2*x^(1/2)/(a/b)^(1/6)-3^(1/2))-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)))
Leaf count of result is larger than twice the leaf count of optimal. 1959 vs. \(2 (199) = 398\).
Time = 0.13 (sec) , antiderivative size = 1959, normalized size of antiderivative = 7.39 \[ \int \frac {x^{3/2} \left (A+B x^3\right )}{\left (a+b x^3\right )^3} \, dx=\text {Too large to display} \] Input:
integrate(x^(3/2)*(B*x^3+A)/(b*x^3+a)^3,x, algorithm="fricas")
Output:
1/432*(2*(a^2*b^3*x^6 + 2*a^3*b^2*x^3 + a^4*b)*(-(15625*B^6*a^6 + 131250*A *B^5*a^5*b + 459375*A^2*B^4*a^4*b^2 + 857500*A^3*B^3*a^3*b^3 + 900375*A^4* B^2*a^2*b^4 + 504210*A^5*B*a*b^5 + 117649*A^6*b^6)/(a^13*b^11))^(1/6)*log( a^11*b^9*(-(15625*B^6*a^6 + 131250*A*B^5*a^5*b + 459375*A^2*B^4*a^4*b^2 + 857500*A^3*B^3*a^3*b^3 + 900375*A^4*B^2*a^2*b^4 + 504210*A^5*B*a*b^5 + 117 649*A^6*b^6)/(a^13*b^11))^(5/6) + (3125*B^5*a^5 + 21875*A*B^4*a^4*b + 6125 0*A^2*B^3*a^3*b^2 + 85750*A^3*B^2*a^2*b^3 + 60025*A^4*B*a*b^4 + 16807*A^5* b^5)*sqrt(x)) - 2*(a^2*b^3*x^6 + 2*a^3*b^2*x^3 + a^4*b)*(-(15625*B^6*a^6 + 131250*A*B^5*a^5*b + 459375*A^2*B^4*a^4*b^2 + 857500*A^3*B^3*a^3*b^3 + 90 0375*A^4*B^2*a^2*b^4 + 504210*A^5*B*a*b^5 + 117649*A^6*b^6)/(a^13*b^11))^( 1/6)*log(-a^11*b^9*(-(15625*B^6*a^6 + 131250*A*B^5*a^5*b + 459375*A^2*B^4* a^4*b^2 + 857500*A^3*B^3*a^3*b^3 + 900375*A^4*B^2*a^2*b^4 + 504210*A^5*B*a *b^5 + 117649*A^6*b^6)/(a^13*b^11))^(5/6) + (3125*B^5*a^5 + 21875*A*B^4*a^ 4*b + 61250*A^2*B^3*a^3*b^2 + 85750*A^3*B^2*a^2*b^3 + 60025*A^4*B*a*b^4 + 16807*A^5*b^5)*sqrt(x)) + (a^2*b^3*x^6 + 2*a^3*b^2*x^3 + a^4*b - sqrt(-3)* (a^2*b^3*x^6 + 2*a^3*b^2*x^3 + a^4*b))*(-(15625*B^6*a^6 + 131250*A*B^5*a^5 *b + 459375*A^2*B^4*a^4*b^2 + 857500*A^3*B^3*a^3*b^3 + 900375*A^4*B^2*a^2* b^4 + 504210*A^5*B*a*b^5 + 117649*A^6*b^6)/(a^13*b^11))^(1/6)*log(1/2*(sqr t(-3)*a^11*b^9 + a^11*b^9)*(-(15625*B^6*a^6 + 131250*A*B^5*a^5*b + 459375* A^2*B^4*a^4*b^2 + 857500*A^3*B^3*a^3*b^3 + 900375*A^4*B^2*a^2*b^4 + 504...
Timed out. \[ \int \frac {x^{3/2} \left (A+B x^3\right )}{\left (a+b x^3\right )^3} \, dx=\text {Timed out} \] Input:
integrate(x**(3/2)*(B*x**3+A)/(b*x**3+a)**3,x)
Output:
Timed out
Time = 0.13 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.02 \[ \int \frac {x^{3/2} \left (A+B x^3\right )}{\left (a+b x^3\right )^3} \, dx=\frac {{\left (5 \, B a b + 7 \, A b^{2}\right )} x^{\frac {11}{2}} - {\left (B a^{2} - 13 \, A a b\right )} x^{\frac {5}{2}}}{36 \, {\left (a^{2} b^{3} x^{6} + 2 \, a^{3} b^{2} x^{3} + a^{4} b\right )}} - \frac {{\left (5 \, B a + 7 \, A b\right )} {\left (\frac {\sqrt {3} \log \left (\sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} \sqrt {x} + b^{\frac {1}{3}} x + a^{\frac {1}{3}}\right )}{a^{\frac {1}{6}} b^{\frac {5}{6}}} - \frac {\sqrt {3} \log \left (-\sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} \sqrt {x} + b^{\frac {1}{3}} x + a^{\frac {1}{3}}\right )}{a^{\frac {1}{6}} b^{\frac {5}{6}}} - \frac {2 \, \arctan \left (\frac {\sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} + 2 \, b^{\frac {1}{3}} \sqrt {x}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{b^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}} - \frac {2 \, \arctan \left (-\frac {\sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} - 2 \, b^{\frac {1}{3}} \sqrt {x}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{b^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}} - \frac {4 \, \arctan \left (\frac {b^{\frac {1}{3}} \sqrt {x}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{b^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}}{432 \, a^{2} b} \] Input:
integrate(x^(3/2)*(B*x^3+A)/(b*x^3+a)^3,x, algorithm="maxima")
Output:
1/36*((5*B*a*b + 7*A*b^2)*x^(11/2) - (B*a^2 - 13*A*a*b)*x^(5/2))/(a^2*b^3* x^6 + 2*a^3*b^2*x^3 + a^4*b) - 1/432*(5*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*b)
Time = 0.54 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.18 \[ \int \frac {x^{3/2} \left (A+B x^3\right )}{\left (a+b x^3\right )^3} \, dx=\frac {{\left (5 \, B a + 7 \, A b\right )} \arctan \left (\frac {\sqrt {3} \left (\frac {a}{b}\right )^{\frac {1}{6}} + 2 \, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{216 \, \left (a b^{5}\right )^{\frac {1}{6}} a^{2} b} + \frac {{\left (5 \, B a + 7 \, A b\right )} \arctan \left (-\frac {\sqrt {3} \left (\frac {a}{b}\right )^{\frac {1}{6}} - 2 \, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{216 \, \left (a b^{5}\right )^{\frac {1}{6}} a^{2} b} + \frac {{\left (5 \, B a \left (\frac {a}{b}\right )^{\frac {5}{6}} + 7 \, A b \left (\frac {a}{b}\right )^{\frac {5}{6}}\right )} \arctan \left (\frac {\sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{108 \, a^{3} b} + \frac {5 \, B a b x^{\frac {11}{2}} + 7 \, A b^{2} x^{\frac {11}{2}} - B a^{2} x^{\frac {5}{2}} + 13 \, A a b x^{\frac {5}{2}}}{36 \, {\left (b x^{3} + a\right )}^{2} a^{2} b} - \frac {\sqrt {3} {\left (5 \, \left (a b^{5}\right )^{\frac {5}{6}} B a + 7 \, \left (a b^{5}\right )^{\frac {5}{6}} A b\right )} \log \left (\sqrt {3} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{6}} + x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{432 \, a^{3} b^{6}} + \frac {\sqrt {3} {\left (5 \, \left (a b^{5}\right )^{\frac {5}{6}} B a + 7 \, \left (a b^{5}\right )^{\frac {5}{6}} A b\right )} \log \left (-\sqrt {3} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{6}} + x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{432 \, a^{3} b^{6}} \] Input:
integrate(x^(3/2)*(B*x^3+A)/(b*x^3+a)^3,x, algorithm="giac")
Output:
1/216*(5*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*b) + 1/216*(5*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*b) + 1/108*(5*B*a*(a/b)^(5 /6) + 7*A*b*(a/b)^(5/6))*arctan(sqrt(x)/(a/b)^(1/6))/(a^3*b) + 1/36*(5*B*a *b*x^(11/2) + 7*A*b^2*x^(11/2) - B*a^2*x^(5/2) + 13*A*a*b*x^(5/2))/((b*x^3 + a)^2*a^2*b) - 1/432*sqrt(3)*(5*(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^6) + 1/432*sqrt (3)*(5*(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^6)
Time = 1.18 (sec) , antiderivative size = 1672, normalized size of antiderivative = 6.31 \[ \int \frac {x^{3/2} \left (A+B x^3\right )}{\left (a+b x^3\right )^3} \, dx=\text {Too large to display} \] Input:
int((x^(3/2)*(A + B*x^3))/(a + b*x^3)^3,x)
Output:
((x^(11/2)*(7*A*b + 5*B*a))/(36*a^2) + (x^(5/2)*(13*A*b - B*a))/(36*a*b))/ (a^2 + b^2*x^6 + 2*a*b*x^3) + (atan(((((343*A^3*b^3 + 125*B^3*a^3 + 525*A* B^2*a^2*b + 735*A^2*B*a*b^2)/(1296*a^3) - (x^(1/2)*(7*A*b + 5*B*a)*(49*A^2 *b^4 + 25*B^2*a^2*b^2 + 70*A*B*a*b^3))/(1296*(-a)^(19/6)*b^(11/6)))*(7*A*b + 5*B*a)^2*1i)/(46656*(-a)^(13/3)*b^(11/3)) - (((343*A^3*b^3 + 125*B^3*a^ 3 + 525*A*B^2*a^2*b + 735*A^2*B*a*b^2)/(1296*a^3) + (x^(1/2)*(7*A*b + 5*B* a)*(49*A^2*b^4 + 25*B^2*a^2*b^2 + 70*A*B*a*b^3))/(1296*(-a)^(19/6)*b^(11/6 )))*(7*A*b + 5*B*a)^2*1i)/(46656*(-a)^(13/3)*b^(11/3)))/((((343*A^3*b^3 + 125*B^3*a^3 + 525*A*B^2*a^2*b + 735*A^2*B*a*b^2)/(1296*a^3) - (x^(1/2)*(7* A*b + 5*B*a)*(49*A^2*b^4 + 25*B^2*a^2*b^2 + 70*A*B*a*b^3))/(1296*(-a)^(19/ 6)*b^(11/6)))*(7*A*b + 5*B*a)^2)/(46656*(-a)^(13/3)*b^(11/3)) + (((343*A^3 *b^3 + 125*B^3*a^3 + 525*A*B^2*a^2*b + 735*A^2*B*a*b^2)/(1296*a^3) + (x^(1 /2)*(7*A*b + 5*B*a)*(49*A^2*b^4 + 25*B^2*a^2*b^2 + 70*A*B*a*b^3))/(1296*(- a)^(19/6)*b^(11/6)))*(7*A*b + 5*B*a)^2)/(46656*(-a)^(13/3)*b^(11/3))))*(7* A*b + 5*B*a)*1i)/(108*(-a)^(13/6)*b^(11/6)) + (atan(((((3^(1/2)*1i)/2 - 1/ 2)^2*(7*A*b + 5*B*a)^2*((343*A^3*b^3 + 125*B^3*a^3 + 525*A*B^2*a^2*b + 735 *A^2*B*a*b^2)/(1296*a^3) - (x^(1/2)*((3^(1/2)*1i)/2 - 1/2)*(7*A*b + 5*B*a) *(49*A^2*b^4 + 25*B^2*a^2*b^2 + 70*A*B*a*b^3))/(1296*(-a)^(19/6)*b^(11/6)) )*1i)/(46656*(-a)^(13/3)*b^(11/3)) - (((3^(1/2)*1i)/2 - 1/2)^2*(7*A*b + 5* B*a)^2*((343*A^3*b^3 + 125*B^3*a^3 + 525*A*B^2*a^2*b + 735*A^2*B*a*b^2)...
Time = 0.28 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.27 \[ \int \frac {x^{3/2} \left (A+B x^3\right )}{\left (a+b x^3\right )^3} \, dx=\frac {-2 b^{\frac {1}{6}} a^{\frac {7}{6}} \mathit {atan} \left (\frac {b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}-2 \sqrt {x}\, b^{\frac {1}{3}}}{b^{\frac {1}{6}} a^{\frac {1}{6}}}\right )-2 b^{\frac {7}{6}} a^{\frac {1}{6}} \mathit {atan} \left (\frac {b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}-2 \sqrt {x}\, b^{\frac {1}{3}}}{b^{\frac {1}{6}} a^{\frac {1}{6}}}\right ) x^{3}+2 b^{\frac {1}{6}} a^{\frac {7}{6}} \mathit {atan} \left (\frac {b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+2 \sqrt {x}\, b^{\frac {1}{3}}}{b^{\frac {1}{6}} a^{\frac {1}{6}}}\right )+2 b^{\frac {7}{6}} a^{\frac {1}{6}} \mathit {atan} \left (\frac {b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+2 \sqrt {x}\, b^{\frac {1}{3}}}{b^{\frac {1}{6}} a^{\frac {1}{6}}}\right ) x^{3}+4 b^{\frac {1}{6}} a^{\frac {7}{6}} \mathit {atan} \left (\frac {\sqrt {x}\, b^{\frac {1}{6}}}{a^{\frac {1}{6}}}\right )+4 b^{\frac {7}{6}} a^{\frac {1}{6}} \mathit {atan} \left (\frac {\sqrt {x}\, b^{\frac {1}{6}}}{a^{\frac {1}{6}}}\right ) x^{3}+b^{\frac {1}{6}} a^{\frac {7}{6}} \sqrt {3}\, \mathrm {log}\left (-\sqrt {x}\, b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right )+b^{\frac {7}{6}} a^{\frac {1}{6}} \sqrt {3}\, \mathrm {log}\left (-\sqrt {x}\, b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) x^{3}-b^{\frac {1}{6}} a^{\frac {7}{6}} \sqrt {3}\, \mathrm {log}\left (\sqrt {x}\, b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right )-b^{\frac {7}{6}} a^{\frac {1}{6}} \sqrt {3}\, \mathrm {log}\left (\sqrt {x}\, b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) x^{3}+12 \sqrt {x}\, a^{\frac {1}{3}} b \,x^{2}}{36 a^{\frac {4}{3}} b \left (b \,x^{3}+a \right )} \] Input:
int(x^(3/2)*(B*x^3+A)/(b*x^3+a)^3,x)
Output:
( - 2*b**(1/6)*a**(1/6)*atan((b**(1/6)*a**(1/6)*sqrt(3) - 2*sqrt(x)*b**(1/ 3))/(b**(1/6)*a**(1/6)))*a - 2*b**(1/6)*a**(1/6)*atan((b**(1/6)*a**(1/6)*s qrt(3) - 2*sqrt(x)*b**(1/3))/(b**(1/6)*a**(1/6)))*b*x**3 + 2*b**(1/6)*a**( 1/6)*atan((b**(1/6)*a**(1/6)*sqrt(3) + 2*sqrt(x)*b**(1/3))/(b**(1/6)*a**(1 /6)))*a + 2*b**(1/6)*a**(1/6)*atan((b**(1/6)*a**(1/6)*sqrt(3) + 2*sqrt(x)* b**(1/3))/(b**(1/6)*a**(1/6)))*b*x**3 + 4*b**(1/6)*a**(1/6)*atan((sqrt(x)* b**(1/3))/(b**(1/6)*a**(1/6)))*a + 4*b**(1/6)*a**(1/6)*atan((sqrt(x)*b**(1 /3))/(b**(1/6)*a**(1/6)))*b*x**3 + b**(1/6)*a**(1/6)*sqrt(3)*log( - sqrt(x )*b**(1/6)*a**(1/6)*sqrt(3) + a**(1/3) + b**(1/3)*x)*a + b**(1/6)*a**(1/6) *sqrt(3)*log( - sqrt(x)*b**(1/6)*a**(1/6)*sqrt(3) + a**(1/3) + b**(1/3)*x) *b*x**3 - b**(1/6)*a**(1/6)*sqrt(3)*log(sqrt(x)*b**(1/6)*a**(1/6)*sqrt(3) + a**(1/3) + b**(1/3)*x)*a - b**(1/6)*a**(1/6)*sqrt(3)*log(sqrt(x)*b**(1/6 )*a**(1/6)*sqrt(3) + a**(1/3) + b**(1/3)*x)*b*x**3 + 12*sqrt(x)*a**(1/3)*b *x**2)/(36*a**(1/3)*a*b*(a + b*x**3))