Integrand size = 17, antiderivative size = 70 \[ \int \left (a+b x^n\right )^2 \left (A+B x^n\right ) \, dx=a^2 A x+\frac {a (2 A b+a B) x^{1+n}}{1+n}+\frac {b (A b+2 a B) x^{1+2 n}}{1+2 n}+\frac {b^2 B x^{1+3 n}}{1+3 n} \] 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 = 0.11 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00 \[ \int \left (a+b x^n\right )^2 \left (A+B x^n\right ) \, dx=\frac {B x \left (a+b x^n\right )^3-(a B-A (b+3 b n)) x \left (a^2+\frac {2 a b x^n}{1+n}+\frac {b^2 x^{2 n}}{1+2 n}\right )}{b+3 b n} \] Input:
Integrate[(a + b*x^n)^2*(A + B*x^n),x]
Output:
(B*x*(a + b*x^n)^3 - (a*B - A*(b + 3*b*n))*x*(a^2 + (2*a*b*x^n)/(1 + n) + (b^2*x^(2*n))/(1 + 2*n)))/(b + 3*b*n)
Time = 0.35 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {897, 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.
\(\displaystyle \int \left (a+b x^n\right )^2 \left (A+B x^n\right ) \, dx\) |
\(\Big \downarrow \) 897 |
\(\displaystyle \int \left (a^2 A+b x^{2 n} (2 a B+A b)+a x^n (a B+2 A b)+b^2 B x^{3 n}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle a^2 A x+\frac {a x^{n+1} (a B+2 A b)}{n+1}+\frac {b x^{2 n+1} (2 a B+A b)}{2 n+1}+\frac {b^2 B x^{3 n+1}}{3 n+1}\) |
Input:
Int[(a + b*x^n)^2*(A + B*x^n),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[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol ] :> Int[ExpandIntegrand[(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b , c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && IGtQ[q, 0]
Time = 0.21 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.97
method | result | size |
risch | \(a^{2} A x +\frac {a \left (2 A b +B a \right ) x \,x^{n}}{1+n}+\frac {b \left (A b +2 B a \right ) x \,x^{2 n}}{1+2 n}+\frac {b^{2} B x \,x^{3 n}}{1+3 n}\) | \(68\) |
norman | \(a^{2} A x +\frac {a \left (2 A b +B a \right ) x \,{\mathrm e}^{n \ln \left (x \right )}}{1+n}+\frac {b \left (A b +2 B a \right ) x \,{\mathrm e}^{2 n \ln \left (x \right )}}{1+2 n}+\frac {b^{2} B x \,{\mathrm e}^{3 n \ln \left (x \right )}}{1+3 n}\) | \(74\) |
parallelrisch | \(\frac {2 B x \,x^{3 n} b^{2} n^{2}+3 A x \,x^{2 n} b^{2} n^{2}+3 B x \,x^{3 n} b^{2} n +6 B x \,x^{2 n} a b \,n^{2}+4 A x \,x^{2 n} b^{2} n +12 A x \,x^{n} a b \,n^{2}+6 A x \,a^{2} n^{3}+b^{2} B x \,x^{3 n}+8 B x \,x^{2 n} a b n +6 B x \,x^{n} a^{2} n^{2}+A x \,x^{2 n} b^{2}+10 A x \,x^{n} a b n +11 A x \,a^{2} n^{2}+2 B x \,x^{2 n} a b +5 B x \,x^{n} a^{2} n +2 A x \,x^{n} a b +6 A x \,a^{2} n +B x \,x^{n} a^{2}+a^{2} A x}{\left (1+n \right ) \left (1+2 n \right ) \left (1+3 n \right )}\) | \(235\) |
orering | \(x \left (a +b \,x^{n}\right )^{2} \left (A +B \,x^{n}\right )-\frac {x^{2} \left (11 n^{2}+1\right ) \left (\frac {2 \left (a +b \,x^{n}\right ) \left (A +B \,x^{n}\right ) b \,x^{n} n}{x}+\frac {\left (a +b \,x^{n}\right )^{2} B \,x^{n} n}{x}\right )}{\left (2 n^{2}+3 n +1\right ) \left (1+3 n \right )}+\frac {2 x^{3} \left (-1+3 n \right ) \left (\frac {2 b^{2} x^{2 n} n^{2} \left (A +B \,x^{n}\right )}{x^{2}}+\frac {4 \left (a +b \,x^{n}\right ) B \,x^{2 n} n^{2} b}{x^{2}}+\frac {2 \left (a +b \,x^{n}\right ) \left (A +B \,x^{n}\right ) b \,x^{n} n^{2}}{x^{2}}-\frac {2 \left (a +b \,x^{n}\right ) \left (A +B \,x^{n}\right ) b \,x^{n} n}{x^{2}}+\frac {\left (a +b \,x^{n}\right )^{2} B \,x^{n} n^{2}}{x^{2}}-\frac {\left (a +b \,x^{n}\right )^{2} B \,x^{n} n}{x^{2}}\right )}{\left (2 n^{2}+3 n +1\right ) \left (1+3 n \right )}-\frac {x^{4} \left (\frac {6 b^{2} x^{2 n} n^{3} \left (A +B \,x^{n}\right )}{x^{3}}-\frac {6 b^{2} x^{2 n} n^{2} \left (A +B \,x^{n}\right )}{x^{3}}+\frac {6 b^{2} x^{3 n} n^{3} B}{x^{3}}+\frac {12 \left (a +b \,x^{n}\right ) B \,x^{2 n} n^{3} b}{x^{3}}-\frac {12 \left (a +b \,x^{n}\right ) B \,x^{2 n} n^{2} b}{x^{3}}+\frac {2 \left (a +b \,x^{n}\right ) \left (A +B \,x^{n}\right ) b \,x^{n} n^{3}}{x^{3}}-\frac {6 \left (a +b \,x^{n}\right ) \left (A +B \,x^{n}\right ) b \,x^{n} n^{2}}{x^{3}}+\frac {4 \left (a +b \,x^{n}\right ) \left (A +B \,x^{n}\right ) b \,x^{n} n}{x^{3}}+\frac {\left (a +b \,x^{n}\right )^{2} B \,x^{n} n^{3}}{x^{3}}-\frac {3 \left (a +b \,x^{n}\right )^{2} B \,x^{n} n^{2}}{x^{3}}+\frac {2 \left (a +b \,x^{n}\right )^{2} B \,x^{n} n}{x^{3}}\right )}{6 n^{3}+11 n^{2}+6 n +1}\) | \(524\) |
Input:
int((a+b*x^n)^2*(A+B*x^n),x,method=_RETURNVERBOSE)
Output:
a^2*A*x+a*(2*A*b+B*a)/(1+n)*x*x^n+b*(A*b+2*B*a)/(1+2*n)*x*(x^n)^2+b^2*B/(1 +3*n)*x*(x^n)^3
Leaf count of result is larger than twice the leaf count of optimal. 175 vs. \(2 (70) = 140\).
Time = 0.09 (sec) , antiderivative size = 175, normalized size of antiderivative = 2.50 \[ \int \left (a+b x^n\right )^2 \left (A+B x^n\right ) \, dx=\frac {{\left (2 \, B b^{2} n^{2} + 3 \, B b^{2} n + B b^{2}\right )} x x^{3 \, n} + {\left (2 \, B a b + A b^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} n^{2} + 4 \, {\left (2 \, B a b + A b^{2}\right )} n\right )} x x^{2 \, n} + {\left (B a^{2} + 2 \, A a b + 6 \, {\left (B a^{2} + 2 \, A a b\right )} n^{2} + 5 \, {\left (B a^{2} + 2 \, A a b\right )} n\right )} x x^{n} + {\left (6 \, A a^{2} n^{3} + 11 \, A a^{2} n^{2} + 6 \, A a^{2} n + A a^{2}\right )} x}{6 \, n^{3} + 11 \, n^{2} + 6 \, n + 1} \] Input:
integrate((a+b*x^n)^2*(A+B*x^n),x, algorithm="fricas")
Output:
((2*B*b^2*n^2 + 3*B*b^2*n + B*b^2)*x*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*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*x^n + (6*A*a^2*n^3 + 11*A *a^2*n^2 + 6*A*a^2*n + A*a^2)*x)/(6*n^3 + 11*n^2 + 6*n + 1)
Leaf count of result is larger than twice the leaf count of optimal. 726 vs. \(2 (63) = 126\).
Time = 0.55 (sec) , antiderivative size = 726, normalized size of antiderivative = 10.37 \[ \int \left (a+b x^n\right )^2 \left (A+B x^n\right ) \, dx=\begin {cases} A a^{2} x + 2 A a b \log {\left (x \right )} - \frac {A b^{2}}{x} + B a^{2} \log {\left (x \right )} - \frac {2 B a b}{x} - \frac {B b^{2}}{2 x^{2}} & \text {for}\: n = -1 \\A a^{2} x + 4 A a b \sqrt {x} + A b^{2} \log {\left (x \right )} + 2 B a^{2} \sqrt {x} + 2 B a b \log {\left (x \right )} - \frac {2 B b^{2}}{\sqrt {x}} & \text {for}\: n = - \frac {1}{2} \\A a^{2} x + 3 A a b x^{\frac {2}{3}} + 3 A b^{2} \sqrt [3]{x} + \frac {3 B a^{2} x^{\frac {2}{3}}}{2} + 6 B a b \sqrt [3]{x} + B b^{2} \log {\left (x \right )} & \text {for}\: n = - \frac {1}{3} \\\frac {6 A a^{2} n^{3} x}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {11 A a^{2} n^{2} x}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {6 A a^{2} n x}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {A a^{2} x}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {12 A a b n^{2} x x^{n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {10 A a b n x x^{n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {2 A a b x x^{n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {3 A b^{2} n^{2} x x^{2 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {4 A b^{2} n x x^{2 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {A b^{2} x x^{2 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {6 B a^{2} n^{2} x x^{n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {5 B a^{2} n x x^{n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {B a^{2} x x^{n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {6 B a b n^{2} x x^{2 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {8 B a b n x x^{2 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {2 B a b x x^{2 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {2 B b^{2} n^{2} x x^{3 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {3 B b^{2} n x x^{3 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {B b^{2} x x^{3 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} & \text {otherwise} \end {cases} \] Input:
integrate((a+b*x**n)**2*(A+B*x**n),x)
Output:
Piecewise((A*a**2*x + 2*A*a*b*log(x) - A*b**2/x + B*a**2*log(x) - 2*B*a*b/ x - B*b**2/(2*x**2), Eq(n, -1)), (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 + 3*A*a*b*x**(2/3) + 3*A*b**2*x**(1/3) + 3*B*a**2*x**(2/3)/2 + 6* B*a*b*x**(1/3) + B*b**2*log(x), Eq(n, -1/3)), (6*A*a**2*n**3*x/(6*n**3 + 1 1*n**2 + 6*n + 1) + 11*A*a**2*n**2*x/(6*n**3 + 11*n**2 + 6*n + 1) + 6*A*a* *2*n*x/(6*n**3 + 11*n**2 + 6*n + 1) + A*a**2*x/(6*n**3 + 11*n**2 + 6*n + 1 ) + 12*A*a*b*n**2*x*x**n/(6*n**3 + 11*n**2 + 6*n + 1) + 10*A*a*b*n*x*x**n/ (6*n**3 + 11*n**2 + 6*n + 1) + 2*A*a*b*x*x**n/(6*n**3 + 11*n**2 + 6*n + 1) + 3*A*b**2*n**2*x*x**(2*n)/(6*n**3 + 11*n**2 + 6*n + 1) + 4*A*b**2*n*x*x* *(2*n)/(6*n**3 + 11*n**2 + 6*n + 1) + A*b**2*x*x**(2*n)/(6*n**3 + 11*n**2 + 6*n + 1) + 6*B*a**2*n**2*x*x**n/(6*n**3 + 11*n**2 + 6*n + 1) + 5*B*a**2* n*x*x**n/(6*n**3 + 11*n**2 + 6*n + 1) + B*a**2*x*x**n/(6*n**3 + 11*n**2 + 6*n + 1) + 6*B*a*b*n**2*x*x**(2*n)/(6*n**3 + 11*n**2 + 6*n + 1) + 8*B*a*b* n*x*x**(2*n)/(6*n**3 + 11*n**2 + 6*n + 1) + 2*B*a*b*x*x**(2*n)/(6*n**3 + 1 1*n**2 + 6*n + 1) + 2*B*b**2*n**2*x*x**(3*n)/(6*n**3 + 11*n**2 + 6*n + 1) + 3*B*b**2*n*x*x**(3*n)/(6*n**3 + 11*n**2 + 6*n + 1) + B*b**2*x*x**(3*n)/( 6*n**3 + 11*n**2 + 6*n + 1), True))
Time = 0.06 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.34 \[ \int \left (a+b x^n\right )^2 \left (A+B x^n\right ) \, dx=A a^{2} x + \frac {B b^{2} x^{3 \, n + 1}}{3 \, n + 1} + \frac {2 \, B a b x^{2 \, n + 1}}{2 \, n + 1} + \frac {A b^{2} x^{2 \, n + 1}}{2 \, n + 1} + \frac {B a^{2} x^{n + 1}}{n + 1} + \frac {2 \, A a b x^{n + 1}}{n + 1} \] Input:
integrate((a+b*x^n)^2*(A+B*x^n),x, algorithm="maxima")
Output:
A*a^2*x + B*b^2*x^(3*n + 1)/(3*n + 1) + 2*B*a*b*x^(2*n + 1)/(2*n + 1) + A* b^2*x^(2*n + 1)/(2*n + 1) + B*a^2*x^(n + 1)/(n + 1) + 2*A*a*b*x^(n + 1)/(n + 1)
Leaf count of result is larger than twice the leaf count of optimal. 232 vs. \(2 (70) = 140\).
Time = 0.13 (sec) , antiderivative size = 232, normalized size of antiderivative = 3.31 \[ \int \left (a+b x^n\right )^2 \left (A+B x^n\right ) \, dx=\frac {6 \, A a^{2} n^{3} x + 2 \, B b^{2} n^{2} x x^{3 \, n} + 6 \, B a b n^{2} x x^{2 \, n} + 3 \, A b^{2} n^{2} x x^{2 \, n} + 6 \, B a^{2} n^{2} x x^{n} + 12 \, A a b n^{2} x x^{n} + 11 \, A a^{2} n^{2} x + 3 \, B b^{2} n x x^{3 \, n} + 8 \, B a b n x x^{2 \, n} + 4 \, A b^{2} n x x^{2 \, n} + 5 \, B a^{2} n x x^{n} + 10 \, A a b n x x^{n} + 6 \, A a^{2} n x + B b^{2} x x^{3 \, n} + 2 \, B a b x x^{2 \, n} + A b^{2} x x^{2 \, n} + B a^{2} x x^{n} + 2 \, A a b x x^{n} + A a^{2} x}{6 \, n^{3} + 11 \, n^{2} + 6 \, n + 1} \] Input:
integrate((a+b*x^n)^2*(A+B*x^n),x, algorithm="giac")
Output:
(6*A*a^2*n^3*x + 2*B*b^2*n^2*x*x^(3*n) + 6*B*a*b*n^2*x*x^(2*n) + 3*A*b^2*n ^2*x*x^(2*n) + 6*B*a^2*n^2*x*x^n + 12*A*a*b*n^2*x*x^n + 11*A*a^2*n^2*x + 3 *B*b^2*n*x*x^(3*n) + 8*B*a*b*n*x*x^(2*n) + 4*A*b^2*n*x*x^(2*n) + 5*B*a^2*n *x*x^n + 10*A*a*b*n*x*x^n + 6*A*a^2*n*x + B*b^2*x*x^(3*n) + 2*B*a*b*x*x^(2 *n) + A*b^2*x*x^(2*n) + B*a^2*x*x^n + 2*A*a*b*x*x^n + A*a^2*x)/(6*n^3 + 11 *n^2 + 6*n + 1)
Time = 3.75 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.01 \[ \int \left (a+b x^n\right )^2 \left (A+B x^n\right ) \, dx=A\,a^2\,x+\frac {x\,x^{2\,n}\,\left (A\,b^2+2\,B\,a\,b\right )}{2\,n+1}+\frac {x\,x^n\,\left (B\,a^2+2\,A\,b\,a\right )}{n+1}+\frac {B\,b^2\,x\,x^{3\,n}}{3\,n+1} \] Input:
int((A + B*x^n)*(a + b*x^n)^2,x)
Output:
A*a^2*x + (x*x^(2*n)*(A*b^2 + 2*B*a*b))/(2*n + 1) + (x*x^n*(B*a^2 + 2*A*a* b))/(n + 1) + (B*b^2*x*x^(3*n))/(3*n + 1)
Time = 0.24 (sec) , antiderivative size = 146, normalized size of antiderivative = 2.09 \[ \int \left (a+b x^n\right )^2 \left (A+B x^n\right ) \, dx=\frac {x \left (2 x^{3 n} b^{3} n^{2}+3 x^{3 n} b^{3} n +x^{3 n} b^{3}+9 x^{2 n} a \,b^{2} n^{2}+12 x^{2 n} a \,b^{2} n +3 x^{2 n} a \,b^{2}+18 x^{n} a^{2} b \,n^{2}+15 x^{n} a^{2} b n +3 x^{n} a^{2} b +6 a^{3} n^{3}+11 a^{3} n^{2}+6 a^{3} n +a^{3}\right )}{6 n^{3}+11 n^{2}+6 n +1} \] Input:
int((a+b*x^n)^2*(A+B*x^n),x)
Output:
(x*(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))/(6*n**3 + 11*n**2 + 6*n + 1)