Integrand size = 22, antiderivative size = 272 \[ \int \frac {A+B x^3}{x^{7/2} \left (a+b x^3\right )^3} \, dx=-\frac {2 A}{5 a^3 x^{5/2}}-\frac {(A b-a B) \sqrt {x}}{6 a^2 \left (a+b x^3\right )^2}-\frac {(23 A b-11 a B) \sqrt {x}}{36 a^3 \left (a+b x^3\right )}+\frac {11 (17 A b-5 a B) \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{216 a^{23/6} \sqrt [6]{b}}-\frac {11 (17 A b-5 a B) \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{216 a^{23/6} \sqrt [6]{b}}-\frac {11 (17 A b-5 a B) \arctan \left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{108 a^{23/6} \sqrt [6]{b}}-\frac {11 (17 A b-5 a B) \text {arctanh}\left (\frac {\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}}{\sqrt [3]{a}+\sqrt [3]{b} x}\right )}{72 \sqrt {3} a^{23/6} \sqrt [6]{b}} \] Output:
-2/5*A/a^3/x^(5/2)-1/6*(A*b-B*a)*x^(1/2)/a^2/(b*x^3+a)^2-1/36*(23*A*b-11*B *a)*x^(1/2)/a^3/(b*x^3+a)-11/216*(17*A*b-5*B*a)*arctan(-3^(1/2)+2*b^(1/6)* x^(1/2)/a^(1/6))/a^(23/6)/b^(1/6)-11/216*(17*A*b-5*B*a)*arctan(3^(1/2)+2*b ^(1/6)*x^(1/2)/a^(1/6))/a^(23/6)/b^(1/6)-11/108*(17*A*b-5*B*a)*arctan(b^(1 /6)*x^(1/2)/a^(1/6))/a^(23/6)/b^(1/6)-11/216*(17*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^(23/6)/b^(1/6)
Time = 1.14 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.77 \[ \int \frac {A+B x^3}{x^{7/2} \left (a+b x^3\right )^3} \, dx=\frac {-\frac {6 a^{5/6} \left (187 A b^2 x^6+a^2 \left (72 A-85 B x^3\right )+a b x^3 \left (289 A-55 B x^3\right )\right )}{x^{5/2} \left (a+b x^3\right )^2}+\frac {110 (-17 A b+5 a B) \arctan \left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{\sqrt [6]{b}}+\frac {55 (17 A b-5 a B) \arctan \left (\frac {\sqrt [3]{a}-\sqrt [3]{b} x}{\sqrt [6]{a} \sqrt [6]{b} \sqrt {x}}\right )}{\sqrt [6]{b}}+\frac {55 \sqrt {3} (-17 A b+5 a B) \text {arctanh}\left (\frac {\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}}{\sqrt [3]{a}+\sqrt [3]{b} x}\right )}{\sqrt [6]{b}}}{1080 a^{23/6}} \] Input:
Integrate[(A + B*x^3)/(x^(7/2)*(a + b*x^3)^3),x]
Output:
((-6*a^(5/6)*(187*A*b^2*x^6 + a^2*(72*A - 85*B*x^3) + a*b*x^3*(289*A - 55* B*x^3)))/(x^(5/2)*(a + b*x^3)^2) + (110*(-17*A*b + 5*a*B)*ArcTan[(b^(1/6)* Sqrt[x])/a^(1/6)])/b^(1/6) + (55*(17*A*b - 5*a*B)*ArcTan[(a^(1/3) - b^(1/3 )*x)/(a^(1/6)*b^(1/6)*Sqrt[x])])/b^(1/6) + (55*Sqrt[3]*(-17*A*b + 5*a*B)*A rcTanh[(Sqrt[3]*a^(1/6)*b^(1/6)*Sqrt[x])/(a^(1/3) + b^(1/3)*x)])/b^(1/6))/ (1080*a^(23/6))
Time = 0.94 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.20, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.591, Rules used = {957, 819, 847, 851, 753, 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.
| \(\displaystyle \int \frac {A+B x^3}{x^{7/2} \left (a+b x^3\right )^3} \, dx\) |
| \(\Big \downarrow \) 957 |
| \(\displaystyle \frac {(17 A b-5 a B) \int \frac {1}{x^{7/2} \left (b x^3+a\right )^2}dx}{12 a b}+\frac {A b-a B}{6 a b x^{5/2} \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 819 |
| \(\displaystyle \frac {(17 A b-5 a B) \left (\frac {11 \int \frac {1}{x^{7/2} \left (b x^3+a\right )}dx}{6 a}+\frac {1}{3 a x^{5/2} \left (a+b x^3\right )}\right )}{12 a b}+\frac {A b-a B}{6 a b x^{5/2} \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 847 |
| \(\displaystyle \frac {(17 A b-5 a B) \left (\frac {11 \left (-\frac {b \int \frac {1}{\sqrt {x} \left (b x^3+a\right )}dx}{a}-\frac {2}{5 a x^{5/2}}\right )}{6 a}+\frac {1}{3 a x^{5/2} \left (a+b x^3\right )}\right )}{12 a b}+\frac {A b-a B}{6 a b x^{5/2} \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 851 |
| \(\displaystyle \frac {(17 A b-5 a B) \left (\frac {11 \left (-\frac {2 b \int \frac {1}{b x^3+a}d\sqrt {x}}{a}-\frac {2}{5 a x^{5/2}}\right )}{6 a}+\frac {1}{3 a x^{5/2} \left (a+b x^3\right )}\right )}{12 a b}+\frac {A b-a B}{6 a b x^{5/2} \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 753 |
| \(\displaystyle \frac {(17 A b-5 a B) \left (\frac {11 \left (-\frac {2 b \left (\frac {\int \frac {1}{\sqrt [3]{b} x+\sqrt [3]{a}}d\sqrt {x}}{3 a^{2/3}}+\frac {\int \frac {2 \sqrt [6]{a}-\sqrt {3} \sqrt [6]{b} \sqrt {x}}{2 \left (\sqrt [3]{b} x-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}\right )}d\sqrt {x}}{3 a^{5/6}}+\frac {\int \frac {\sqrt {3} \sqrt [6]{b} \sqrt {x}+2 \sqrt [6]{a}}{2 \left (\sqrt [3]{b} x+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}\right )}d\sqrt {x}}{3 a^{5/6}}\right )}{a}-\frac {2}{5 a x^{5/2}}\right )}{6 a}+\frac {1}{3 a x^{5/2} \left (a+b x^3\right )}\right )}{12 a b}+\frac {A b-a B}{6 a b x^{5/2} \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(17 A b-5 a B) \left (\frac {11 \left (-\frac {2 b \left (\frac {\int \frac {1}{\sqrt [3]{b} x+\sqrt [3]{a}}d\sqrt {x}}{3 a^{2/3}}+\frac {\int \frac {2 \sqrt [6]{a}-\sqrt {3} \sqrt [6]{b} \sqrt {x}}{\sqrt [3]{b} x-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{6 a^{5/6}}+\frac {\int \frac {\sqrt {3} \sqrt [6]{b} \sqrt {x}+2 \sqrt [6]{a}}{\sqrt [3]{b} x+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{6 a^{5/6}}\right )}{a}-\frac {2}{5 a x^{5/2}}\right )}{6 a}+\frac {1}{3 a x^{5/2} \left (a+b x^3\right )}\right )}{12 a b}+\frac {A b-a B}{6 a b x^{5/2} \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {(17 A b-5 a B) \left (\frac {11 \left (-\frac {2 b \left (\frac {\int \frac {2 \sqrt [6]{a}-\sqrt {3} \sqrt [6]{b} \sqrt {x}}{\sqrt [3]{b} x-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{6 a^{5/6}}+\frac {\int \frac {\sqrt {3} \sqrt [6]{b} \sqrt {x}+2 \sqrt [6]{a}}{\sqrt [3]{b} x+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{6 a^{5/6}}+\frac {\arctan \left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{a}-\frac {2}{5 a x^{5/2}}\right )}{6 a}+\frac {1}{3 a x^{5/2} \left (a+b x^3\right )}\right )}{12 a b}+\frac {A b-a B}{6 a b x^{5/2} \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {(17 A b-5 a B) \left (\frac {11 \left (-\frac {2 b \left (\frac {\frac {1}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{b} x-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}-\frac {\sqrt {3} \int -\frac {\sqrt [6]{b} \left (\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} \sqrt {x}\right )}{\sqrt [3]{b} x-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{2 \sqrt [6]{b}}}{6 a^{5/6}}+\frac {\frac {1}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{b} x+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}+\frac {\sqrt {3} \int \frac {\sqrt [6]{b} \left (2 \sqrt [6]{b} \sqrt {x}+\sqrt {3} \sqrt [6]{a}\right )}{\sqrt [3]{b} x+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{2 \sqrt [6]{b}}}{6 a^{5/6}}+\frac {\arctan \left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{a}-\frac {2}{5 a x^{5/2}}\right )}{6 a}+\frac {1}{3 a x^{5/2} \left (a+b x^3\right )}\right )}{12 a b}+\frac {A b-a B}{6 a b x^{5/2} \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(17 A b-5 a B) \left (\frac {11 \left (-\frac {2 b \left (\frac {\frac {1}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{b} x-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}+\frac {\sqrt {3} \int \frac {\sqrt [6]{b} \left (\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} \sqrt {x}\right )}{\sqrt [3]{b} x-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{2 \sqrt [6]{b}}}{6 a^{5/6}}+\frac {\frac {1}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{b} x+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}+\frac {\sqrt {3} \int \frac {\sqrt [6]{b} \left (2 \sqrt [6]{b} \sqrt {x}+\sqrt {3} \sqrt [6]{a}\right )}{\sqrt [3]{b} x+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{2 \sqrt [6]{b}}}{6 a^{5/6}}+\frac {\arctan \left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{a}-\frac {2}{5 a x^{5/2}}\right )}{6 a}+\frac {1}{3 a x^{5/2} \left (a+b x^3\right )}\right )}{12 a b}+\frac {A b-a B}{6 a b x^{5/2} \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(17 A b-5 a B) \left (\frac {11 \left (-\frac {2 b \left (\frac {\frac {1}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{b} x-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}+\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} \sqrt {x}}{\sqrt [3]{b} x-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{6 a^{5/6}}+\frac {\frac {1}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{b} x+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}+\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [6]{b} \sqrt {x}+\sqrt {3} \sqrt [6]{a}}{\sqrt [3]{b} x+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{6 a^{5/6}}+\frac {\arctan \left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{a}-\frac {2}{5 a x^{5/2}}\right )}{6 a}+\frac {1}{3 a x^{5/2} \left (a+b x^3\right )}\right )}{12 a b}+\frac {A b-a B}{6 a b x^{5/2} \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {(17 A b-5 a B) \left (\frac {11 \left (-\frac {2 b \left (\frac {\frac {\int \frac {1}{-x-\frac {1}{3}}d\left (1-\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt {3} \sqrt [6]{a}}\right )}{\sqrt {3} \sqrt [6]{b}}+\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} \sqrt {x}}{\sqrt [3]{b} x-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{6 a^{5/6}}+\frac {\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [6]{b} \sqrt {x}+\sqrt {3} \sqrt [6]{a}}{\sqrt [3]{b} x+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}-\frac {\int \frac {1}{-x-\frac {1}{3}}d\left (\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt {3} \sqrt [6]{a}}+1\right )}{\sqrt {3} \sqrt [6]{b}}}{6 a^{5/6}}+\frac {\arctan \left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{a}-\frac {2}{5 a x^{5/2}}\right )}{6 a}+\frac {1}{3 a x^{5/2} \left (a+b x^3\right )}\right )}{12 a b}+\frac {A b-a B}{6 a b x^{5/2} \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {(17 A b-5 a B) \left (\frac {11 \left (-\frac {2 b \left (\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} \sqrt {x}}{\sqrt [3]{b} x-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}-\frac {\arctan \left (\sqrt {3} \left (1-\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt {3} \sqrt [6]{a}}\right )\right )}{\sqrt [6]{b}}}{6 a^{5/6}}+\frac {\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [6]{b} \sqrt {x}+\sqrt {3} \sqrt [6]{a}}{\sqrt [3]{b} x+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}+\frac {\arctan \left (\sqrt {3} \left (\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt {3} \sqrt [6]{a}}+1\right )\right )}{\sqrt [6]{b}}}{6 a^{5/6}}+\frac {\arctan \left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{a}-\frac {2}{5 a x^{5/2}}\right )}{6 a}+\frac {1}{3 a x^{5/2} \left (a+b x^3\right )}\right )}{12 a b}+\frac {A b-a B}{6 a b x^{5/2} \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {(17 A b-5 a B) \left (\frac {11 \left (-\frac {2 b \left (\frac {\arctan \left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt [6]{b}}+\frac {-\frac {\arctan \left (\sqrt {3} \left (1-\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt {3} \sqrt [6]{a}}\right )\right )}{\sqrt [6]{b}}-\frac {\sqrt {3} \log \left (-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{2 \sqrt [6]{b}}}{6 a^{5/6}}+\frac {\frac {\arctan \left (\sqrt {3} \left (\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt {3} \sqrt [6]{a}}+1\right )\right )}{\sqrt [6]{b}}+\frac {\sqrt {3} \log \left (\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{2 \sqrt [6]{b}}}{6 a^{5/6}}\right )}{a}-\frac {2}{5 a x^{5/2}}\right )}{6 a}+\frac {1}{3 a x^{5/2} \left (a+b x^3\right )}\right )}{12 a b}+\frac {A b-a B}{6 a b x^{5/2} \left (a+b x^3\right )^2}\) |
Input:
Int[(A + B*x^3)/(x^(7/2)*(a + b*x^3)^3),x]
Output:
(A*b - a*B)/(6*a*b*x^(5/2)*(a + b*x^3)^2) + ((17*A*b - 5*a*B)*(1/(3*a*x^(5 /2)*(a + b*x^3)) + (11*(-2/(5*a*x^(5/2)) - (2*b*(ArcTan[(b^(1/6)*Sqrt[x])/ a^(1/6)]/(3*a^(5/6)*b^(1/6)) + (-(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^(5/6)) + (ArcTan[Sqrt[3]*(1 + (2*b^(1/6)*Sqrt[x])/(Sqrt[3]*a^(1/6)))]/b^(1/6) + (Sqrt[3]*Log[a^(1/3) + S qrt[3]*a^(1/6)*b^(1/6)*Sqrt[x] + b^(1/3)*x])/(2*b^(1/6)))/(6*a^(5/6))))/a) )/(6*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[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> Module[{r = Numerator[Rt[a/ b, n]], s = Denominator[Rt[a/b, n]], k, u, v}, Simp[u = Int[(r - s*Cos[(2*k - 1)*(Pi/n)]*x)/(r^2 - 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2*x^2), x] + Int[ (r + s*Cos[(2*k - 1)*(Pi/n)]*x)/(r^2 + 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2* x^2), x]; 2*(r^2/(a*n)) Int[1/(r^2 + s^2*x^2), x] + 2*(r/(a*n)) Sum[u, {k, 1, (n - 2)/4}], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] && 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[((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 = 0.86 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.87
| method | result | size |
| derivativedivides | \(-\frac {2 A}{5 a^{3} x^{\frac {5}{2}}}-\frac {2 \left (\frac {\left (\frac {23}{72} b^{2} A -\frac {11}{72} a b B \right ) x^{\frac {7}{2}}+\frac {a \left (29 A b -17 B a \right ) \sqrt {x}}{72}}{\left (b \,x^{3}+a \right )^{2}}+\frac {11 \left (17 A b -5 B a \right ) \left (\frac {\left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {\sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{3 a}+\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \ln \left (x +\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{6 a}-\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \ln \left (\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}-x -\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{6 a}\right )}{72}\right )}{a^{3}}\) | \(237\) |
| default | \(-\frac {2 A}{5 a^{3} x^{\frac {5}{2}}}-\frac {2 \left (\frac {\left (\frac {23}{72} b^{2} A -\frac {11}{72} a b B \right ) x^{\frac {7}{2}}+\frac {a \left (29 A b -17 B a \right ) \sqrt {x}}{72}}{\left (b \,x^{3}+a \right )^{2}}+\frac {11 \left (17 A b -5 B a \right ) \left (\frac {\left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {\sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{3 a}+\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \ln \left (x +\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{6 a}-\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \ln \left (\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}-x -\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{6 a}\right )}{72}\right )}{a^{3}}\) | \(237\) |
| risch | \(-\frac {2 A}{5 a^{3} x^{\frac {5}{2}}}-\frac {\frac {2 \left (\frac {23}{72} b^{2} A -\frac {11}{72} a b B \right ) x^{\frac {7}{2}}+\frac {a \left (29 A b -17 B a \right ) \sqrt {x}}{36}}{\left (b \,x^{3}+a \right )^{2}}+\frac {11 \left (17 A b -5 B a \right ) \left (\frac {\left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {\sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{3 a}+\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \ln \left (x +\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{6 a}-\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \ln \left (\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}-x -\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{6 a}\right )}{36}}{a^{3}}\) | \(238\) |
Input:
int((B*x^3+A)/x^(7/2)/(b*x^3+a)^3,x,method=_RETURNVERBOSE)
Output:
-2/5*A/a^3/x^(5/2)-2/a^3*(((23/72*b^2*A-11/72*a*b*B)*x^(7/2)+1/72*a*(29*A* b-17*B*a)*x^(1/2))/(b*x^3+a)^2+11/72*(17*A*b-5*B*a)*(1/3/a*(a/b)^(1/6)*arc tan(x^(1/2)/(a/b)^(1/6))+1/12/a*3^(1/2)*(a/b)^(1/6)*ln(x+3^(1/2)*(a/b)^(1/ 6)*x^(1/2)+(a/b)^(1/3))+1/6/a*(a/b)^(1/6)*arctan(2*x^(1/2)/(a/b)^(1/6)+3^( 1/2))-1/12/a*3^(1/2)*(a/b)^(1/6)*ln(3^(1/2)*(a/b)^(1/6)*x^(1/2)-x-(a/b)^(1 /3))+1/6/a*(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. 1608 vs. \(2 (202) = 404\).
Time = 0.11 (sec) , antiderivative size = 1608, normalized size of antiderivative = 5.91 \[ \int \frac {A+B x^3}{x^{7/2} \left (a+b x^3\right )^3} \, dx=\text {Too large to display} \] Input:
integrate((B*x^3+A)/x^(7/2)/(b*x^3+a)^3,x, algorithm="fricas")
Output:
-1/2160*(110*(a^3*b^2*x^9 + 2*a^4*b*x^6 + a^5*x^3)*(-(15625*B^6*a^6 - 3187 50*A*B^5*a^5*b + 2709375*A^2*B^4*a^4*b^2 - 12282500*A^3*B^3*a^3*b^3 + 3132 0375*A^4*B^2*a^2*b^4 - 42595710*A^5*B*a*b^5 + 24137569*A^6*b^6)/(a^23*b))^ (1/6)*log(11*a^4*(-(15625*B^6*a^6 - 318750*A*B^5*a^5*b + 2709375*A^2*B^4*a ^4*b^2 - 12282500*A^3*B^3*a^3*b^3 + 31320375*A^4*B^2*a^2*b^4 - 42595710*A^ 5*B*a*b^5 + 24137569*A^6*b^6)/(a^23*b))^(1/6) - 11*(5*B*a - 17*A*b)*sqrt(x )) - 110*(a^3*b^2*x^9 + 2*a^4*b*x^6 + a^5*x^3)*(-(15625*B^6*a^6 - 318750*A *B^5*a^5*b + 2709375*A^2*B^4*a^4*b^2 - 12282500*A^3*B^3*a^3*b^3 + 31320375 *A^4*B^2*a^2*b^4 - 42595710*A^5*B*a*b^5 + 24137569*A^6*b^6)/(a^23*b))^(1/6 )*log(-11*a^4*(-(15625*B^6*a^6 - 318750*A*B^5*a^5*b + 2709375*A^2*B^4*a^4* b^2 - 12282500*A^3*B^3*a^3*b^3 + 31320375*A^4*B^2*a^2*b^4 - 42595710*A^5*B *a*b^5 + 24137569*A^6*b^6)/(a^23*b))^(1/6) - 11*(5*B*a - 17*A*b)*sqrt(x)) + 55*(a^3*b^2*x^9 + 2*a^4*b*x^6 + a^5*x^3 + sqrt(-3)*(a^3*b^2*x^9 + 2*a^4* b*x^6 + a^5*x^3))*(-(15625*B^6*a^6 - 318750*A*B^5*a^5*b + 2709375*A^2*B^4* a^4*b^2 - 12282500*A^3*B^3*a^3*b^3 + 31320375*A^4*B^2*a^2*b^4 - 42595710*A ^5*B*a*b^5 + 24137569*A^6*b^6)/(a^23*b))^(1/6)*log(-11*(5*B*a - 17*A*b)*sq rt(x) + 11/2*(sqrt(-3)*a^4 + a^4)*(-(15625*B^6*a^6 - 318750*A*B^5*a^5*b + 2709375*A^2*B^4*a^4*b^2 - 12282500*A^3*B^3*a^3*b^3 + 31320375*A^4*B^2*a^2* b^4 - 42595710*A^5*B*a*b^5 + 24137569*A^6*b^6)/(a^23*b))^(1/6)) - 55*(a^3* b^2*x^9 + 2*a^4*b*x^6 + a^5*x^3 + sqrt(-3)*(a^3*b^2*x^9 + 2*a^4*b*x^6 +...
Timed out. \[ \int \frac {A+B x^3}{x^{7/2} \left (a+b x^3\right )^3} \, dx=\text {Timed out} \] Input:
integrate((B*x**3+A)/x**(7/2)/(b*x**3+a)**3,x)
Output:
Timed out
Time = 0.12 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.27 \[ \int \frac {A+B x^3}{x^{7/2} \left (a+b x^3\right )^3} \, dx=\frac {11 \, {\left (5 \, B a b - 17 \, A b^{2}\right )} x^{6} + 17 \, {\left (5 \, B a^{2} - 17 \, A a b\right )} x^{3} - 72 \, A a^{2}}{180 \, {\left (a^{3} b^{2} x^{\frac {17}{2}} + 2 \, a^{4} b x^{\frac {11}{2}} + a^{5} x^{\frac {5}{2}}\right )}} + \frac {11 \, {\left (\frac {\sqrt {3} {\left (5 \, B a - 17 \, A b\right )} \log \left (\sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} \sqrt {x} + b^{\frac {1}{3}} x + a^{\frac {1}{3}}\right )}{a^{\frac {5}{6}} b^{\frac {1}{6}}} - \frac {\sqrt {3} {\left (5 \, B a - 17 \, A b\right )} \log \left (-\sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} \sqrt {x} + b^{\frac {1}{3}} x + a^{\frac {1}{3}}\right )}{a^{\frac {5}{6}} b^{\frac {1}{6}}} + \frac {4 \, {\left (5 \, B a b^{\frac {1}{3}} - 17 \, A b^{\frac {4}{3}}\right )} \arctan \left (\frac {b^{\frac {1}{3}} \sqrt {x}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{a^{\frac {2}{3}} b^{\frac {1}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}} + \frac {2 \, {\left (5 \, B a^{\frac {4}{3}} b^{\frac {1}{3}} - 17 \, A a^{\frac {1}{3}} b^{\frac {4}{3}}\right )} \arctan \left (\frac {\sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} + 2 \, b^{\frac {1}{3}} \sqrt {x}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{a b^{\frac {1}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}} + \frac {2 \, {\left (5 \, B a^{\frac {4}{3}} b^{\frac {1}{3}} - 17 \, A a^{\frac {1}{3}} b^{\frac {4}{3}}\right )} \arctan \left (-\frac {\sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} - 2 \, b^{\frac {1}{3}} \sqrt {x}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{a b^{\frac {1}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}}{432 \, a^{3}} \] Input:
integrate((B*x^3+A)/x^(7/2)/(b*x^3+a)^3,x, algorithm="maxima")
Output:
1/180*(11*(5*B*a*b - 17*A*b^2)*x^6 + 17*(5*B*a^2 - 17*A*a*b)*x^3 - 72*A*a^ 2)/(a^3*b^2*x^(17/2) + 2*a^4*b*x^(11/2) + a^5*x^(5/2)) + 11/432*(sqrt(3)*( 5*B*a - 17*A*b)*log(sqrt(3)*a^(1/6)*b^(1/6)*sqrt(x) + b^(1/3)*x + a^(1/3)) /(a^(5/6)*b^(1/6)) - sqrt(3)*(5*B*a - 17*A*b)*log(-sqrt(3)*a^(1/6)*b^(1/6) *sqrt(x) + b^(1/3)*x + a^(1/3))/(a^(5/6)*b^(1/6)) + 4*(5*B*a*b^(1/3) - 17* A*b^(4/3))*arctan(b^(1/3)*sqrt(x)/sqrt(a^(1/3)*b^(1/3)))/(a^(2/3)*b^(1/3)* sqrt(a^(1/3)*b^(1/3))) + 2*(5*B*a^(4/3)*b^(1/3) - 17*A*a^(1/3)*b^(4/3))*ar ctan((sqrt(3)*a^(1/6)*b^(1/6) + 2*b^(1/3)*sqrt(x))/sqrt(a^(1/3)*b^(1/3)))/ (a*b^(1/3)*sqrt(a^(1/3)*b^(1/3))) + 2*(5*B*a^(4/3)*b^(1/3) - 17*A*a^(1/3)* b^(4/3))*arctan(-(sqrt(3)*a^(1/6)*b^(1/6) - 2*b^(1/3)*sqrt(x))/sqrt(a^(1/3 )*b^(1/3)))/(a*b^(1/3)*sqrt(a^(1/3)*b^(1/3))))/a^3
Time = 0.14 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.23 \[ \int \frac {A+B x^3}{x^{7/2} \left (a+b x^3\right )^3} \, dx=\frac {11 \, \sqrt {3} {\left (5 \, \left (a b^{5}\right )^{\frac {1}{6}} B a - 17 \, \left (a b^{5}\right )^{\frac {1}{6}} A b\right )} \log \left (\sqrt {3} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{6}} + x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{432 \, a^{4} b} - \frac {11 \, \sqrt {3} {\left (5 \, \left (a b^{5}\right )^{\frac {1}{6}} B a - 17 \, \left (a b^{5}\right )^{\frac {1}{6}} A b\right )} \log \left (-\sqrt {3} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{6}} + x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{432 \, a^{4} b} + \frac {11 \, {\left (5 \, \left (a b^{5}\right )^{\frac {1}{6}} B a - 17 \, \left (a b^{5}\right )^{\frac {1}{6}} A b\right )} \arctan \left (\frac {\sqrt {3} \left (\frac {a}{b}\right )^{\frac {1}{6}} + 2 \, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{216 \, a^{4} b} + \frac {11 \, {\left (5 \, \left (a b^{5}\right )^{\frac {1}{6}} B a - 17 \, \left (a b^{5}\right )^{\frac {1}{6}} A b\right )} \arctan \left (-\frac {\sqrt {3} \left (\frac {a}{b}\right )^{\frac {1}{6}} - 2 \, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{216 \, a^{4} b} + \frac {11 \, {\left (5 \, \left (a b^{5}\right )^{\frac {1}{6}} B a - 17 \, \left (a b^{5}\right )^{\frac {1}{6}} A b\right )} \arctan \left (\frac {\sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{108 \, a^{4} b} + \frac {11 \, B a b x^{\frac {7}{2}} - 23 \, A b^{2} x^{\frac {7}{2}} + 17 \, B a^{2} \sqrt {x} - 29 \, A a b \sqrt {x}}{36 \, {\left (b x^{3} + a\right )}^{2} a^{3}} - \frac {2 \, A}{5 \, a^{3} x^{\frac {5}{2}}} \] Input:
integrate((B*x^3+A)/x^(7/2)/(b*x^3+a)^3,x, algorithm="giac")
Output:
11/432*sqrt(3)*(5*(a*b^5)^(1/6)*B*a - 17*(a*b^5)^(1/6)*A*b)*log(sqrt(3)*sq rt(x)*(a/b)^(1/6) + x + (a/b)^(1/3))/(a^4*b) - 11/432*sqrt(3)*(5*(a*b^5)^( 1/6)*B*a - 17*(a*b^5)^(1/6)*A*b)*log(-sqrt(3)*sqrt(x)*(a/b)^(1/6) + x + (a /b)^(1/3))/(a^4*b) + 11/216*(5*(a*b^5)^(1/6)*B*a - 17*(a*b^5)^(1/6)*A*b)*a rctan((sqrt(3)*(a/b)^(1/6) + 2*sqrt(x))/(a/b)^(1/6))/(a^4*b) + 11/216*(5*( a*b^5)^(1/6)*B*a - 17*(a*b^5)^(1/6)*A*b)*arctan(-(sqrt(3)*(a/b)^(1/6) - 2* sqrt(x))/(a/b)^(1/6))/(a^4*b) + 11/108*(5*(a*b^5)^(1/6)*B*a - 17*(a*b^5)^( 1/6)*A*b)*arctan(sqrt(x)/(a/b)^(1/6))/(a^4*b) + 1/36*(11*B*a*b*x^(7/2) - 2 3*A*b^2*x^(7/2) + 17*B*a^2*sqrt(x) - 29*A*a*b*sqrt(x))/((b*x^3 + a)^2*a^3) - 2/5*A/(a^3*x^(5/2))
Time = 1.29 (sec) , antiderivative size = 2109, normalized size of antiderivative = 7.75 \[ \int \frac {A+B x^3}{x^{7/2} \left (a+b x^3\right )^3} \, dx=\text {Too large to display} \] Input:
int((A + B*x^3)/(x^(7/2)*(a + b*x^3)^3),x)
Output:
- ((2*A)/(5*a) + (17*x^3*(17*A*b - 5*B*a))/(180*a^2) + (11*b*x^6*(17*A*b - 5*B*a))/(180*a^3))/(a^2*x^(5/2) + b^2*x^(17/2) + 2*a*b*x^(11/2)) - (atan( (((x^(1/2)*(443639472636450816*A^4*a^15*b^9 + 3319819810560000*B^4*a^19*b^ 5 + 230262702060441600*A^2*B^2*a^17*b^7 - 45149549423616000*A*B^3*a^18*b^6 - 521928791337000960*A^3*B*a^16*b^8) - (11*(17*A*b - 5*B*a)*(512439176949 055488*A^3*a^19*b^8 - 13037837801472000*B^3*a^22*b^5 + 132985945575014400* A*B^2*a^21*b^6 - 452152214955048960*A^2*B*a^20*b^7))/(216*(-a)^(23/6)*b^(1 /6)))*(17*A*b - 5*B*a)*11i)/(216*(-a)^(23/6)*b^(1/6)) + ((x^(1/2)*(4436394 72636450816*A^4*a^15*b^9 + 3319819810560000*B^4*a^19*b^5 + 230262702060441 600*A^2*B^2*a^17*b^7 - 45149549423616000*A*B^3*a^18*b^6 - 5219287913370009 60*A^3*B*a^16*b^8) + (11*(17*A*b - 5*B*a)*(512439176949055488*A^3*a^19*b^8 - 13037837801472000*B^3*a^22*b^5 + 132985945575014400*A*B^2*a^21*b^6 - 45 2152214955048960*A^2*B*a^20*b^7))/(216*(-a)^(23/6)*b^(1/6)))*(17*A*b - 5*B *a)*11i)/(216*(-a)^(23/6)*b^(1/6)))/((11*(x^(1/2)*(443639472636450816*A^4* a^15*b^9 + 3319819810560000*B^4*a^19*b^5 + 230262702060441600*A^2*B^2*a^17 *b^7 - 45149549423616000*A*B^3*a^18*b^6 - 521928791337000960*A^3*B*a^16*b^ 8) - (11*(17*A*b - 5*B*a)*(512439176949055488*A^3*a^19*b^8 - 1303783780147 2000*B^3*a^22*b^5 + 132985945575014400*A*B^2*a^21*b^6 - 452152214955048960 *A^2*B*a^20*b^7))/(216*(-a)^(23/6)*b^(1/6)))*(17*A*b - 5*B*a))/(216*(-a)^( 23/6)*b^(1/6)) - (11*(x^(1/2)*(443639472636450816*A^4*a^15*b^9 + 331981...
Time = 0.28 (sec) , antiderivative size = 386, normalized size of antiderivative = 1.42 \[ \int \frac {A+B x^3}{x^{7/2} \left (a+b x^3\right )^3} \, dx=\frac {110 \sqrt {x}\, b^{\frac {7}{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 ) x^{2}+110 \sqrt {x}\, b^{\frac {13}{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^{5}-110 \sqrt {x}\, b^{\frac {7}{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 ) x^{2}-110 \sqrt {x}\, b^{\frac {13}{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^{5}-220 \sqrt {x}\, b^{\frac {7}{6}} a^{\frac {7}{6}} \mathit {atan} \left (\frac {\sqrt {x}\, b^{\frac {1}{6}}}{a^{\frac {1}{6}}}\right ) x^{2}-220 \sqrt {x}\, b^{\frac {13}{6}} a^{\frac {1}{6}} \mathit {atan} \left (\frac {\sqrt {x}\, b^{\frac {1}{6}}}{a^{\frac {1}{6}}}\right ) x^{5}+55 \sqrt {x}\, b^{\frac {7}{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 ) x^{2}+55 \sqrt {x}\, b^{\frac {13}{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^{5}-55 \sqrt {x}\, b^{\frac {7}{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 ) x^{2}-55 \sqrt {x}\, b^{\frac {13}{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^{5}-72 b^{\frac {1}{3}} a^{2}-132 b^{\frac {4}{3}} a \,x^{3}}{180 \sqrt {x}\, b^{\frac {1}{3}} a^{3} x^{2} \left (b \,x^{3}+a \right )} \] Input:
int((B*x^3+A)/x^(7/2)/(b*x^3+a)^3,x)
Output:
(110*sqrt(x)*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*b*x**2 + 110*sqrt(x)*b**(1/6)*a**(1/6)*a tan((b**(1/6)*a**(1/6)*sqrt(3) - 2*sqrt(x)*b**(1/3))/(b**(1/6)*a**(1/6)))* b**2*x**5 - 110*sqrt(x)*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*b*x**2 - 110*sqrt(x)*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**2*x**5 - 220*sqrt(x)*b**(1/6)*a**(1/6)*atan((sqrt(x)*b**(1/3 ))/(b**(1/6)*a**(1/6)))*a*b*x**2 - 220*sqrt(x)*b**(1/6)*a**(1/6)*atan((sqr t(x)*b**(1/3))/(b**(1/6)*a**(1/6)))*b**2*x**5 + 55*sqrt(x)*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*x**2 + 55*sqrt(x)*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**2*x**5 - 55*sqrt(x)*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*x**2 - 55*sqrt(x)*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**2*x**5 - 72*b**(1/3)*a**2 - 132*b**(1/3)*a*b*x**3)/(180*sqrt(x)*b**(1/3)*a**3*x**2*(a + b*x**3))