Integrand size = 20, antiderivative size = 78 \[ \int \frac {\left (a+b x^n\right )^2 \left (A+B x^n\right )}{x^2} \, dx=-\frac {a^2 A}{x}-\frac {a (2 A b+a B) x^{-1+n}}{1-n}-\frac {b (A b+2 a B) x^{-1+2 n}}{1-2 n}-\frac {b^2 B x^{-1+3 n}}{1-3 n} \] Output:
-a^2*A/x-a*(2*A*b+B*a)*x^(-1+n)/(1-n)-b*(A*b+2*B*a)*x^(-1+2*n)/(1-2*n)-b^2 *B*x^(-1+3*n)/(1-3*n)
Time = 0.13 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+b x^n\right )^2 \left (A+B x^n\right )}{x^2} \, dx=\frac {-a^2 A+\frac {a (2 A b+a B) x^n}{-1+n}+\frac {b (A b+2 a B) x^{2 n}}{-1+2 n}+\frac {b^2 B x^{3 n}}{-1+3 n}}{x} \] Input:
Integrate[((a + b*x^n)^2*(A + B*x^n))/x^2,x]
Output:
(-(a^2*A) + (a*(2*A*b + a*B)*x^n)/(-1 + n) + (b*(A*b + 2*a*B)*x^(2*n))/(-1 + 2*n) + (b^2*B*x^(3*n))/(-1 + 3*n))/x
Time = 0.38 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {950, 2009}
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.
\(\displaystyle \int \frac {\left (a+b x^n\right )^2 \left (A+B x^n\right )}{x^2} \, dx\) |
\(\Big \downarrow \) 950 |
\(\displaystyle \int \left (\frac {a^2 A}{x^2}+a x^{n-2} (a B+2 A b)+b x^{2 (n-1)} (2 a B+A b)+b^2 B x^{3 n-2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {a^2 A}{x}-\frac {a x^{n-1} (a B+2 A b)}{1-n}-\frac {b x^{2 n-1} (2 a B+A b)}{1-2 n}-\frac {b^2 B x^{3 n-1}}{1-3 n}\) |
Input:
Int[((a + b*x^n)^2*(A + B*x^n))/x^2,x]
Output:
-((a^2*A)/x) - (a*(2*A*b + a*B)*x^(-1 + n))/(1 - n) - (b*(A*b + 2*a*B)*x^( -1 + 2*n))/(1 - 2*n) - (b^2*B*x^(-1 + 3*n))/(1 - 3*n)
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n _))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(a + b*x^n)^p*(c + d*x^ n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] && IGt Q[p, 0] && IGtQ[q, 0]
Time = 0.23 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.96
method | result | size |
norman | \(\frac {\frac {a \left (2 A b +B a \right ) {\mathrm e}^{n \ln \left (x \right )}}{-1+n}+\frac {b \left (A b +2 B a \right ) {\mathrm e}^{2 n \ln \left (x \right )}}{-1+2 n}+\frac {b^{2} B \,{\mathrm e}^{3 n \ln \left (x \right )}}{-1+3 n}-a^{2} A}{x}\) | \(75\) |
risch | \(-\frac {a^{2} A}{x}+\frac {b^{2} B \,x^{3 n}}{\left (-1+3 n \right ) x}+\frac {b \left (A b +2 B a \right ) x^{2 n}}{\left (-1+2 n \right ) x}+\frac {a \left (2 A b +B a \right ) x^{n}}{\left (-1+n \right ) x}\) | \(77\) |
parallelrisch | \(\frac {2 B \,x^{3 n} b^{2} n^{2}+3 A \,x^{2 n} b^{2} n^{2}-3 B \,x^{3 n} b^{2} n +6 B \,x^{2 n} a b \,n^{2}-4 A \,x^{2 n} b^{2} n +12 A \,x^{n} a b \,n^{2}-6 A \,a^{2} n^{3}+b^{2} B \,x^{3 n}-8 B \,x^{2 n} a b n +6 B \,x^{n} a^{2} n^{2}+A \,b^{2} x^{2 n}-10 A \,x^{n} a b n +11 A \,a^{2} n^{2}+2 B a b \,x^{2 n}-5 B \,x^{n} a^{2} n +2 a b A \,x^{n}-6 A \,a^{2} n +a^{2} B \,x^{n}+a^{2} A}{x \left (-1+3 n \right ) \left (-1+2 n \right ) \left (-1+n \right )}\) | \(219\) |
orering | \(-\frac {3 \left (2 n^{2}-9 n +5\right ) \left (a +b \,x^{n}\right )^{2} \left (A +B \,x^{n}\right )}{x \left (-1+3 n \right ) \left (-1+2 n \right )}+\frac {\left (11 n -25\right ) x^{2} \left (\frac {2 \left (a +b \,x^{n}\right ) \left (A +B \,x^{n}\right ) b \,x^{n} n}{x^{3}}+\frac {\left (a +b \,x^{n}\right )^{2} B \,x^{n} n}{x^{3}}-\frac {2 \left (a +b \,x^{n}\right )^{2} \left (A +B \,x^{n}\right )}{x^{3}}\right )}{6 n^{2}-5 n +1}-\frac {2 x^{3} \left (-5+3 n \right ) \left (\frac {2 b^{2} x^{2 n} n^{2} \left (A +B \,x^{n}\right )}{x^{4}}+\frac {4 \left (a +b \,x^{n}\right ) B \,x^{2 n} n^{2} b}{x^{4}}+\frac {2 \left (a +b \,x^{n}\right ) \left (A +B \,x^{n}\right ) b \,x^{n} n^{2}}{x^{4}}-\frac {10 \left (a +b \,x^{n}\right ) \left (A +B \,x^{n}\right ) b \,x^{n} n}{x^{4}}+\frac {\left (a +b \,x^{n}\right )^{2} B \,x^{n} n^{2}}{x^{4}}-\frac {5 \left (a +b \,x^{n}\right )^{2} B \,x^{n} n}{x^{4}}+\frac {6 \left (a +b \,x^{n}\right )^{2} \left (A +B \,x^{n}\right )}{x^{4}}\right )}{6 n^{3}-11 n^{2}+6 n -1}+\frac {x^{4} \left (\frac {6 b^{2} x^{2 n} n^{3} \left (A +B \,x^{n}\right )}{x^{5}}-\frac {18 b^{2} x^{2 n} n^{2} \left (A +B \,x^{n}\right )}{x^{5}}+\frac {6 b^{2} x^{3 n} n^{3} B}{x^{5}}+\frac {12 \left (a +b \,x^{n}\right ) B \,x^{2 n} n^{3} b}{x^{5}}-\frac {36 \left (a +b \,x^{n}\right ) B \,x^{2 n} n^{2} b}{x^{5}}+\frac {2 \left (a +b \,x^{n}\right ) \left (A +B \,x^{n}\right ) b \,x^{n} n^{3}}{x^{5}}-\frac {18 \left (a +b \,x^{n}\right ) \left (A +B \,x^{n}\right ) b \,x^{n} n^{2}}{x^{5}}+\frac {52 \left (a +b \,x^{n}\right ) \left (A +B \,x^{n}\right ) b \,x^{n} n}{x^{5}}+\frac {\left (a +b \,x^{n}\right )^{2} B \,x^{n} n^{3}}{x^{5}}-\frac {9 \left (a +b \,x^{n}\right )^{2} B \,x^{n} n^{2}}{x^{5}}+\frac {26 \left (a +b \,x^{n}\right )^{2} B \,x^{n} n}{x^{5}}-\frac {24 \left (a +b \,x^{n}\right )^{2} \left (A +B \,x^{n}\right )}{x^{5}}\right )}{6 n^{3}-11 n^{2}+6 n -1}\) | \(601\) |
Input:
int((a+b*x^n)^2*(A+B*x^n)/x^2,x,method=_RETURNVERBOSE)
Output:
(a*(2*A*b+B*a)/(-1+n)*exp(n*ln(x))+b*(A*b+2*B*a)/(-1+2*n)*exp(n*ln(x))^2+b ^2*B/(-1+3*n)*exp(n*ln(x))^3-a^2*A)/x
Leaf count of result is larger than twice the leaf count of optimal. 177 vs. \(2 (73) = 146\).
Time = 0.08 (sec) , antiderivative size = 177, normalized size of antiderivative = 2.27 \[ \int \frac {\left (a+b x^n\right )^2 \left (A+B x^n\right )}{x^2} \, dx=-\frac {6 \, A a^{2} n^{3} - 11 \, A a^{2} n^{2} + 6 \, A a^{2} n - A a^{2} - {\left (2 \, B b^{2} n^{2} - 3 \, B b^{2} n + B b^{2}\right )} x^{3 \, n} - {\left (2 \, B a b + A b^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} n^{2} - 4 \, {\left (2 \, B a b + A b^{2}\right )} n\right )} x^{2 \, n} - {\left (B a^{2} + 2 \, A a b + 6 \, {\left (B a^{2} + 2 \, A a b\right )} n^{2} - 5 \, {\left (B a^{2} + 2 \, A a b\right )} n\right )} x^{n}}{{\left (6 \, n^{3} - 11 \, n^{2} + 6 \, n - 1\right )} x} \] Input:
integrate((a+b*x^n)^2*(A+B*x^n)/x^2,x, algorithm="fricas")
Output:
-(6*A*a^2*n^3 - 11*A*a^2*n^2 + 6*A*a^2*n - A*a^2 - (2*B*b^2*n^2 - 3*B*b^2* n + B*b^2)*x^(3*n) - (2*B*a*b + A*b^2 + 3*(2*B*a*b + A*b^2)*n^2 - 4*(2*B*a *b + A*b^2)*n)*x^(2*n) - (B*a^2 + 2*A*a*b + 6*(B*a^2 + 2*A*a*b)*n^2 - 5*(B *a^2 + 2*A*a*b)*n)*x^n)/((6*n^3 - 11*n^2 + 6*n - 1)*x)
Leaf count of result is larger than twice the leaf count of optimal. 785 vs. \(2 (65) = 130\).
Time = 0.48 (sec) , antiderivative size = 785, normalized size of antiderivative = 10.06 \[ \int \frac {\left (a+b x^n\right )^2 \left (A+B x^n\right )}{x^2} \, dx=\begin {cases} - \frac {A a^{2}}{x} - \frac {3 A a b}{x^{\frac {2}{3}}} - \frac {3 A b^{2}}{\sqrt [3]{x}} - \frac {3 B a^{2}}{2 x^{\frac {2}{3}}} - \frac {6 B a b}{\sqrt [3]{x}} + B b^{2} \log {\left (x \right )} & \text {for}\: n = \frac {1}{3} \\- \frac {A a^{2}}{x} - \frac {4 A a b}{\sqrt {x}} + A b^{2} \log {\left (x \right )} - \frac {2 B a^{2}}{\sqrt {x}} + 2 B a b \log {\left (x \right )} + 2 B b^{2} \sqrt {x} & \text {for}\: n = \frac {1}{2} \\- \frac {A a^{2}}{x} + 2 A a b \log {\left (x \right )} + A b^{2} x + B a^{2} \log {\left (x \right )} + 2 B a b x + \frac {B b^{2} x^{2}}{2} & \text {for}\: n = 1 \\- \frac {6 A a^{2} n^{3}}{6 n^{3} x - 11 n^{2} x + 6 n x - x} + \frac {11 A a^{2} n^{2}}{6 n^{3} x - 11 n^{2} x + 6 n x - x} - \frac {6 A a^{2} n}{6 n^{3} x - 11 n^{2} x + 6 n x - x} + \frac {A a^{2}}{6 n^{3} x - 11 n^{2} x + 6 n x - x} + \frac {12 A a b n^{2} x^{n}}{6 n^{3} x - 11 n^{2} x + 6 n x - x} - \frac {10 A a b n x^{n}}{6 n^{3} x - 11 n^{2} x + 6 n x - x} + \frac {2 A a b x^{n}}{6 n^{3} x - 11 n^{2} x + 6 n x - x} + \frac {3 A b^{2} n^{2} x^{2 n}}{6 n^{3} x - 11 n^{2} x + 6 n x - x} - \frac {4 A b^{2} n x^{2 n}}{6 n^{3} x - 11 n^{2} x + 6 n x - x} + \frac {A b^{2} x^{2 n}}{6 n^{3} x - 11 n^{2} x + 6 n x - x} + \frac {6 B a^{2} n^{2} x^{n}}{6 n^{3} x - 11 n^{2} x + 6 n x - x} - \frac {5 B a^{2} n x^{n}}{6 n^{3} x - 11 n^{2} x + 6 n x - x} + \frac {B a^{2} x^{n}}{6 n^{3} x - 11 n^{2} x + 6 n x - x} + \frac {6 B a b n^{2} x^{2 n}}{6 n^{3} x - 11 n^{2} x + 6 n x - x} - \frac {8 B a b n x^{2 n}}{6 n^{3} x - 11 n^{2} x + 6 n x - x} + \frac {2 B a b x^{2 n}}{6 n^{3} x - 11 n^{2} x + 6 n x - x} + \frac {2 B b^{2} n^{2} x^{3 n}}{6 n^{3} x - 11 n^{2} x + 6 n x - x} - \frac {3 B b^{2} n x^{3 n}}{6 n^{3} x - 11 n^{2} x + 6 n x - x} + \frac {B b^{2} x^{3 n}}{6 n^{3} x - 11 n^{2} x + 6 n x - x} & \text {otherwise} \end {cases} \] Input:
integrate((a+b*x**n)**2*(A+B*x**n)/x**2,x)
Output:
Piecewise((-A*a**2/x - 3*A*a*b/x**(2/3) - 3*A*b**2/x**(1/3) - 3*B*a**2/(2* x**(2/3)) - 6*B*a*b/x**(1/3) + B*b**2*log(x), Eq(n, 1/3)), (-A*a**2/x - 4* A*a*b/sqrt(x) + A*b**2*log(x) - 2*B*a**2/sqrt(x) + 2*B*a*b*log(x) + 2*B*b* *2*sqrt(x), Eq(n, 1/2)), (-A*a**2/x + 2*A*a*b*log(x) + A*b**2*x + B*a**2*l og(x) + 2*B*a*b*x + B*b**2*x**2/2, Eq(n, 1)), (-6*A*a**2*n**3/(6*n**3*x - 11*n**2*x + 6*n*x - x) + 11*A*a**2*n**2/(6*n**3*x - 11*n**2*x + 6*n*x - x) - 6*A*a**2*n/(6*n**3*x - 11*n**2*x + 6*n*x - x) + A*a**2/(6*n**3*x - 11*n **2*x + 6*n*x - x) + 12*A*a*b*n**2*x**n/(6*n**3*x - 11*n**2*x + 6*n*x - x) - 10*A*a*b*n*x**n/(6*n**3*x - 11*n**2*x + 6*n*x - x) + 2*A*a*b*x**n/(6*n* *3*x - 11*n**2*x + 6*n*x - x) + 3*A*b**2*n**2*x**(2*n)/(6*n**3*x - 11*n**2 *x + 6*n*x - x) - 4*A*b**2*n*x**(2*n)/(6*n**3*x - 11*n**2*x + 6*n*x - x) + A*b**2*x**(2*n)/(6*n**3*x - 11*n**2*x + 6*n*x - x) + 6*B*a**2*n**2*x**n/( 6*n**3*x - 11*n**2*x + 6*n*x - x) - 5*B*a**2*n*x**n/(6*n**3*x - 11*n**2*x + 6*n*x - x) + B*a**2*x**n/(6*n**3*x - 11*n**2*x + 6*n*x - x) + 6*B*a*b*n* *2*x**(2*n)/(6*n**3*x - 11*n**2*x + 6*n*x - x) - 8*B*a*b*n*x**(2*n)/(6*n** 3*x - 11*n**2*x + 6*n*x - x) + 2*B*a*b*x**(2*n)/(6*n**3*x - 11*n**2*x + 6* n*x - x) + 2*B*b**2*n**2*x**(3*n)/(6*n**3*x - 11*n**2*x + 6*n*x - x) - 3*B *b**2*n*x**(3*n)/(6*n**3*x - 11*n**2*x + 6*n*x - x) + B*b**2*x**(3*n)/(6*n **3*x - 11*n**2*x + 6*n*x - x), True))
Exception generated. \[ \int \frac {\left (a+b x^n\right )^2 \left (A+B x^n\right )}{x^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((a+b*x^n)^2*(A+B*x^n)/x^2,x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(n-2>0)', see `assume?` for more details)Is
\[ \int \frac {\left (a+b x^n\right )^2 \left (A+B x^n\right )}{x^2} \, dx=\int { \frac {{\left (B x^{n} + A\right )} {\left (b x^{n} + a\right )}^{2}}{x^{2}} \,d x } \] Input:
integrate((a+b*x^n)^2*(A+B*x^n)/x^2,x, algorithm="giac")
Output:
integrate((B*x^n + A)*(b*x^n + a)^2/x^2, x)
Time = 3.70 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.03 \[ \int \frac {\left (a+b x^n\right )^2 \left (A+B x^n\right )}{x^2} \, dx=\frac {x^{2\,n}\,\left (A\,b^2+2\,B\,a\,b\right )}{x\,\left (2\,n-1\right )}-\frac {A\,a^2}{x}+\frac {x^n\,\left (B\,a^2+2\,A\,b\,a\right )}{x\,\left (n-1\right )}+\frac {B\,b^2\,x^{3\,n}}{x\,\left (3\,n-1\right )} \] Input:
int(((A + B*x^n)*(a + b*x^n)^2)/x^2,x)
Output:
(x^(2*n)*(A*b^2 + 2*B*a*b))/(x*(2*n - 1)) - (A*a^2)/x + (x^n*(B*a^2 + 2*A* a*b))/(x*(n - 1)) + (B*b^2*x^(3*n))/(x*(3*n - 1))
Time = 0.21 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.90 \[ \int \frac {\left (a+b x^n\right )^2 \left (A+B x^n\right )}{x^2} \, dx=\frac {2 x^{3 n} b^{3} n^{2}-3 x^{3 n} b^{3} n +x^{3 n} b^{3}+9 x^{2 n} a \,b^{2} n^{2}-12 x^{2 n} a \,b^{2} n +3 x^{2 n} a \,b^{2}+18 x^{n} a^{2} b \,n^{2}-15 x^{n} a^{2} b n +3 x^{n} a^{2} b -6 a^{3} n^{3}+11 a^{3} n^{2}-6 a^{3} n +a^{3}}{x \left (6 n^{3}-11 n^{2}+6 n -1\right )} \] Input:
int((a+b*x^n)^2*(A+B*x^n)/x^2,x)
Output:
(2*x**(3*n)*b**3*n**2 - 3*x**(3*n)*b**3*n + x**(3*n)*b**3 + 9*x**(2*n)*a*b **2*n**2 - 12*x**(2*n)*a*b**2*n + 3*x**(2*n)*a*b**2 + 18*x**n*a**2*b*n**2 - 15*x**n*a**2*b*n + 3*x**n*a**2*b - 6*a**3*n**3 + 11*a**3*n**2 - 6*a**3*n + a**3)/(x*(6*n**3 - 11*n**2 + 6*n - 1))