Integrand size = 20, antiderivative size = 76 \[ \int x^2 \left (a+b x^n\right )^2 \left (A+B x^n\right ) \, dx=\frac {1}{3} a^2 A x^3+\frac {b^2 B x^{3 (1+n)}}{3 (1+n)}+\frac {a (2 A b+a B) x^{3+n}}{3+n}+\frac {b (A b+2 a B) x^{3+2 n}}{3+2 n} \] Output:
1/3*a^2*A*x^3+b^2*B*x^(3+3*n)/(3+3*n)+a*(2*A*b+B*a)*x^(3+n)/(3+n)+b*(A*b+2 *B*a)*x^(3+2*n)/(3+2*n)
Time = 0.12 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.92 \[ \int x^2 \left (a+b x^n\right )^2 \left (A+B x^n\right ) \, dx=\frac {1}{3} x^3 \left (a^2 A+\frac {3 a (2 A b+a B) x^n}{3+n}+\frac {3 b (A b+2 a B) x^{2 n}}{3+2 n}+\frac {b^2 B x^{3 n}}{1+n}\right ) \] Input:
Integrate[x^2*(a + b*x^n)^2*(A + B*x^n),x]
Output:
(x^3*(a^2*A + (3*a*(2*A*b + a*B)*x^n)/(3 + n) + (3*b*(A*b + 2*a*B)*x^(2*n) )/(3 + 2*n) + (b^2*B*x^(3*n))/(1 + n)))/3
Time = 0.38 (sec) , antiderivative size = 76, 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.
\(\displaystyle \int x^2 \left (a+b x^n\right )^2 \left (A+B x^n\right ) \, dx\) |
\(\Big \downarrow \) 950 |
\(\displaystyle \int \left (a^2 A x^2+b x^{2 (n+1)} (2 a B+A b)+a x^{n+2} (a B+2 A b)+b^2 B x^{3 n+2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} a^2 A x^3+\frac {a x^{n+3} (a B+2 A b)}{n+3}+\frac {b x^{2 n+3} (2 a B+A b)}{2 n+3}+\frac {b^2 B x^{3 (n+1)}}{3 (n+1)}\) |
Input:
Int[x^2*(a + b*x^n)^2*(A + B*x^n),x]
Output:
(a^2*A*x^3)/3 + (b^2*B*x^(3*(1 + n)))/(3*(1 + n)) + (a*(2*A*b + a*B)*x^(3 + n))/(3 + n) + (b*(A*b + 2*a*B)*x^(3 + 2*n))/(3 + 2*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 = 0.26 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00
method | result | size |
risch | \(\frac {a \left (2 A b +B a \right ) x^{3} x^{n}}{3+n}+\frac {b \left (A b +2 B a \right ) x^{3} x^{2 n}}{3+2 n}+\frac {a^{2} A \,x^{3}}{3}+\frac {b^{2} B \,x^{3} x^{3 n}}{3+3 n}\) | \(76\) |
norman | \(\frac {a \left (2 A b +B a \right ) x^{3} {\mathrm e}^{n \ln \left (x \right )}}{3+n}+\frac {b \left (A b +2 B a \right ) x^{3} {\mathrm e}^{2 n \ln \left (x \right )}}{3+2 n}+\frac {a^{2} A \,x^{3}}{3}+\frac {b^{2} B \,x^{3} {\mathrm e}^{3 n \ln \left (x \right )}}{3+3 n}\) | \(82\) |
parallelrisch | \(\frac {2 B \,x^{3} x^{3 n} b^{2} n^{2}+3 A \,x^{3} x^{2 n} b^{2} n^{2}+9 B \,x^{3} x^{3 n} b^{2} n +6 B \,x^{3} x^{2 n} a b \,n^{2}+12 A \,x^{3} x^{2 n} b^{2} n +12 A \,x^{3} x^{n} a b \,n^{2}+2 A \,x^{3} a^{2} n^{3}+9 b^{2} B \,x^{3} x^{3 n}+24 B \,x^{3} x^{2 n} a b n +6 B \,x^{3} x^{n} a^{2} n^{2}+9 A \,x^{3} x^{2 n} b^{2}+30 A \,x^{3} x^{n} a b n +11 A \,x^{3} a^{2} n^{2}+18 B \,x^{3} x^{2 n} a b +15 B \,x^{3} x^{n} a^{2} n +18 A \,x^{3} x^{n} a b +18 A \,x^{3} a^{2} n +9 B \,x^{3} x^{n} a^{2}+9 a^{2} A \,x^{3}}{3 \left (3+n \right ) \left (3+2 n \right ) \left (1+n \right )}\) | \(276\) |
orering | \(\frac {x^{3} \left (6 n^{2}+49 n +65\right ) \left (a +b \,x^{n}\right )^{2} \left (A +B \,x^{n}\right )}{18 n^{2}+81 n +81}-\frac {x^{2} \left (11 n +25\right ) \left (2 x \left (a +b \,x^{n}\right )^{2} \left (A +B \,x^{n}\right )+2 x \left (a +b \,x^{n}\right ) \left (A +B \,x^{n}\right ) b \,x^{n} n +x \left (a +b \,x^{n}\right )^{2} B \,x^{n} n \right )}{9 \left (2 n^{2}+9 n +9\right )}+\frac {2 x^{3} \left (2 \left (a +b \,x^{n}\right )^{2} \left (A +B \,x^{n}\right )+6 \left (a +b \,x^{n}\right ) \left (A +B \,x^{n}\right ) b \,x^{n} n +3 \left (a +b \,x^{n}\right )^{2} B \,x^{n} n +2 b^{2} x^{2 n} n^{2} \left (A +B \,x^{n}\right )+4 \left (a +b \,x^{n}\right ) B \,x^{2 n} n^{2} b +2 \left (a +b \,x^{n}\right ) \left (A +B \,x^{n}\right ) b \,x^{n} n^{2}+\left (a +b \,x^{n}\right )^{2} B \,x^{n} n^{2}\right )}{3 \left (2 n^{2}+9 n +9\right )}-\frac {x^{4} \left (\frac {4 \left (a +b \,x^{n}\right ) \left (A +B \,x^{n}\right ) b \,x^{n} n}{x}+\frac {2 \left (a +b \,x^{n}\right )^{2} B \,x^{n} n}{x}+\frac {6 b^{2} x^{2 n} n^{2} \left (A +B \,x^{n}\right )}{x}+\frac {12 \left (a +b \,x^{n}\right ) B \,x^{2 n} n^{2} b}{x}+\frac {6 \left (a +b \,x^{n}\right ) \left (A +B \,x^{n}\right ) b \,x^{n} n^{2}}{x}+\frac {3 \left (a +b \,x^{n}\right )^{2} B \,x^{n} n^{2}}{x}+\frac {6 b^{2} x^{2 n} n^{3} \left (A +B \,x^{n}\right )}{x}+\frac {6 b^{2} x^{3 n} n^{3} B}{x}+\frac {12 \left (a +b \,x^{n}\right ) B \,x^{2 n} n^{3} b}{x}+\frac {2 \left (a +b \,x^{n}\right ) \left (A +B \,x^{n}\right ) b \,x^{n} n^{3}}{x}+\frac {\left (a +b \,x^{n}\right )^{2} B \,x^{n} n^{3}}{x}\right )}{9 \left (2 n^{3}+11 n^{2}+18 n +9\right )}\) | \(543\) |
Input:
int(x^2*(a+b*x^n)^2*(A+B*x^n),x,method=_RETURNVERBOSE)
Output:
a*(2*A*b+B*a)/(3+n)*x^3*x^n+b*(A*b+2*B*a)/(3+2*n)*x^3*(x^n)^2+1/3*a^2*A*x^ 3+1/3*b^2*B/(1+n)*x^3*(x^n)^3
Leaf count of result is larger than twice the leaf count of optimal. 189 vs. \(2 (72) = 144\).
Time = 0.11 (sec) , antiderivative size = 189, normalized size of antiderivative = 2.49 \[ \int x^2 \left (a+b x^n\right )^2 \left (A+B x^n\right ) \, dx=\frac {{\left (2 \, B b^{2} n^{2} + 9 \, B b^{2} n + 9 \, B b^{2}\right )} x^{3} x^{3 \, n} + 3 \, {\left (6 \, B a b + 3 \, A b^{2} + {\left (2 \, B a b + A b^{2}\right )} n^{2} + 4 \, {\left (2 \, B a b + A b^{2}\right )} n\right )} x^{3} x^{2 \, n} + 3 \, {\left (3 \, B a^{2} + 6 \, A a b + 2 \, {\left (B a^{2} + 2 \, A a b\right )} n^{2} + 5 \, {\left (B a^{2} + 2 \, A a b\right )} n\right )} x^{3} x^{n} + {\left (2 \, A a^{2} n^{3} + 11 \, A a^{2} n^{2} + 18 \, A a^{2} n + 9 \, A a^{2}\right )} x^{3}}{3 \, {\left (2 \, n^{3} + 11 \, n^{2} + 18 \, n + 9\right )}} \] Input:
integrate(x^2*(a+b*x^n)^2*(A+B*x^n),x, algorithm="fricas")
Output:
1/3*((2*B*b^2*n^2 + 9*B*b^2*n + 9*B*b^2)*x^3*x^(3*n) + 3*(6*B*a*b + 3*A*b^ 2 + (2*B*a*b + A*b^2)*n^2 + 4*(2*B*a*b + A*b^2)*n)*x^3*x^(2*n) + 3*(3*B*a^ 2 + 6*A*a*b + 2*(B*a^2 + 2*A*a*b)*n^2 + 5*(B*a^2 + 2*A*a*b)*n)*x^3*x^n + ( 2*A*a^2*n^3 + 11*A*a^2*n^2 + 18*A*a^2*n + 9*A*a^2)*x^3)/(2*n^3 + 11*n^2 + 18*n + 9)
Leaf count of result is larger than twice the leaf count of optimal. 770 vs. \(2 (66) = 132\).
Time = 0.69 (sec) , antiderivative size = 770, normalized size of antiderivative = 10.13 \[ \int x^2 \left (a+b x^n\right )^2 \left (A+B x^n\right ) \, dx=\begin {cases} \frac {A a^{2} x^{3}}{3} + 2 A a b \log {\left (x \right )} - \frac {A b^{2}}{3 x^{3}} + B a^{2} \log {\left (x \right )} - \frac {2 B a b}{3 x^{3}} - \frac {B b^{2}}{6 x^{6}} & \text {for}\: n = -3 \\\frac {A a^{2} x^{3}}{3} + \frac {4 A a b x^{\frac {3}{2}}}{3} + A b^{2} \log {\left (x \right )} + \frac {2 B a^{2} x^{\frac {3}{2}}}{3} + 2 B a b \log {\left (x \right )} - \frac {2 B b^{2}}{3 x^{\frac {3}{2}}} & \text {for}\: n = - \frac {3}{2} \\\frac {A a^{2} x^{3}}{3} + A a b x^{2} + A b^{2} x + \frac {B a^{2} x^{2}}{2} + 2 B a b x + B b^{2} \log {\left (x \right )} & \text {for}\: n = -1 \\\frac {2 A a^{2} n^{3} x^{3}}{6 n^{3} + 33 n^{2} + 54 n + 27} + \frac {11 A a^{2} n^{2} x^{3}}{6 n^{3} + 33 n^{2} + 54 n + 27} + \frac {18 A a^{2} n x^{3}}{6 n^{3} + 33 n^{2} + 54 n + 27} + \frac {9 A a^{2} x^{3}}{6 n^{3} + 33 n^{2} + 54 n + 27} + \frac {12 A a b n^{2} x^{3} x^{n}}{6 n^{3} + 33 n^{2} + 54 n + 27} + \frac {30 A a b n x^{3} x^{n}}{6 n^{3} + 33 n^{2} + 54 n + 27} + \frac {18 A a b x^{3} x^{n}}{6 n^{3} + 33 n^{2} + 54 n + 27} + \frac {3 A b^{2} n^{2} x^{3} x^{2 n}}{6 n^{3} + 33 n^{2} + 54 n + 27} + \frac {12 A b^{2} n x^{3} x^{2 n}}{6 n^{3} + 33 n^{2} + 54 n + 27} + \frac {9 A b^{2} x^{3} x^{2 n}}{6 n^{3} + 33 n^{2} + 54 n + 27} + \frac {6 B a^{2} n^{2} x^{3} x^{n}}{6 n^{3} + 33 n^{2} + 54 n + 27} + \frac {15 B a^{2} n x^{3} x^{n}}{6 n^{3} + 33 n^{2} + 54 n + 27} + \frac {9 B a^{2} x^{3} x^{n}}{6 n^{3} + 33 n^{2} + 54 n + 27} + \frac {6 B a b n^{2} x^{3} x^{2 n}}{6 n^{3} + 33 n^{2} + 54 n + 27} + \frac {24 B a b n x^{3} x^{2 n}}{6 n^{3} + 33 n^{2} + 54 n + 27} + \frac {18 B a b x^{3} x^{2 n}}{6 n^{3} + 33 n^{2} + 54 n + 27} + \frac {2 B b^{2} n^{2} x^{3} x^{3 n}}{6 n^{3} + 33 n^{2} + 54 n + 27} + \frac {9 B b^{2} n x^{3} x^{3 n}}{6 n^{3} + 33 n^{2} + 54 n + 27} + \frac {9 B b^{2} x^{3} x^{3 n}}{6 n^{3} + 33 n^{2} + 54 n + 27} & \text {otherwise} \end {cases} \] Input:
integrate(x**2*(a+b*x**n)**2*(A+B*x**n),x)
Output:
Piecewise((A*a**2*x**3/3 + 2*A*a*b*log(x) - A*b**2/(3*x**3) + B*a**2*log(x ) - 2*B*a*b/(3*x**3) - B*b**2/(6*x**6), Eq(n, -3)), (A*a**2*x**3/3 + 4*A*a *b*x**(3/2)/3 + A*b**2*log(x) + 2*B*a**2*x**(3/2)/3 + 2*B*a*b*log(x) - 2*B *b**2/(3*x**(3/2)), Eq(n, -3/2)), (A*a**2*x**3/3 + A*a*b*x**2 + A*b**2*x + B*a**2*x**2/2 + 2*B*a*b*x + B*b**2*log(x), Eq(n, -1)), (2*A*a**2*n**3*x** 3/(6*n**3 + 33*n**2 + 54*n + 27) + 11*A*a**2*n**2*x**3/(6*n**3 + 33*n**2 + 54*n + 27) + 18*A*a**2*n*x**3/(6*n**3 + 33*n**2 + 54*n + 27) + 9*A*a**2*x **3/(6*n**3 + 33*n**2 + 54*n + 27) + 12*A*a*b*n**2*x**3*x**n/(6*n**3 + 33* n**2 + 54*n + 27) + 30*A*a*b*n*x**3*x**n/(6*n**3 + 33*n**2 + 54*n + 27) + 18*A*a*b*x**3*x**n/(6*n**3 + 33*n**2 + 54*n + 27) + 3*A*b**2*n**2*x**3*x** (2*n)/(6*n**3 + 33*n**2 + 54*n + 27) + 12*A*b**2*n*x**3*x**(2*n)/(6*n**3 + 33*n**2 + 54*n + 27) + 9*A*b**2*x**3*x**(2*n)/(6*n**3 + 33*n**2 + 54*n + 27) + 6*B*a**2*n**2*x**3*x**n/(6*n**3 + 33*n**2 + 54*n + 27) + 15*B*a**2*n *x**3*x**n/(6*n**3 + 33*n**2 + 54*n + 27) + 9*B*a**2*x**3*x**n/(6*n**3 + 3 3*n**2 + 54*n + 27) + 6*B*a*b*n**2*x**3*x**(2*n)/(6*n**3 + 33*n**2 + 54*n + 27) + 24*B*a*b*n*x**3*x**(2*n)/(6*n**3 + 33*n**2 + 54*n + 27) + 18*B*a*b *x**3*x**(2*n)/(6*n**3 + 33*n**2 + 54*n + 27) + 2*B*b**2*n**2*x**3*x**(3*n )/(6*n**3 + 33*n**2 + 54*n + 27) + 9*B*b**2*n*x**3*x**(3*n)/(6*n**3 + 33*n **2 + 54*n + 27) + 9*B*b**2*x**3*x**(3*n)/(6*n**3 + 33*n**2 + 54*n + 27), True))
Time = 0.03 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.26 \[ \int x^2 \left (a+b x^n\right )^2 \left (A+B x^n\right ) \, dx=\frac {1}{3} \, A a^{2} x^{3} + \frac {B b^{2} x^{3 \, n + 3}}{3 \, {\left (n + 1\right )}} + \frac {2 \, B a b x^{2 \, n + 3}}{2 \, n + 3} + \frac {A b^{2} x^{2 \, n + 3}}{2 \, n + 3} + \frac {B a^{2} x^{n + 3}}{n + 3} + \frac {2 \, A a b x^{n + 3}}{n + 3} \] Input:
integrate(x^2*(a+b*x^n)^2*(A+B*x^n),x, algorithm="maxima")
Output:
1/3*A*a^2*x^3 + 1/3*B*b^2*x^(3*n + 3)/(n + 1) + 2*B*a*b*x^(2*n + 3)/(2*n + 3) + A*b^2*x^(2*n + 3)/(2*n + 3) + B*a^2*x^(n + 3)/(n + 3) + 2*A*a*b*x^(n + 3)/(n + 3)
Leaf count of result is larger than twice the leaf count of optimal. 275 vs. \(2 (72) = 144\).
Time = 0.13 (sec) , antiderivative size = 275, normalized size of antiderivative = 3.62 \[ \int x^2 \left (a+b x^n\right )^2 \left (A+B x^n\right ) \, dx=\frac {2 \, B b^{2} n^{2} x^{3} x^{3 \, n} + 6 \, B a b n^{2} x^{3} x^{2 \, n} + 3 \, A b^{2} n^{2} x^{3} x^{2 \, n} + 6 \, B a^{2} n^{2} x^{3} x^{n} + 12 \, A a b n^{2} x^{3} x^{n} + 2 \, A a^{2} n^{3} x^{3} + 9 \, B b^{2} n x^{3} x^{3 \, n} + 24 \, B a b n x^{3} x^{2 \, n} + 12 \, A b^{2} n x^{3} x^{2 \, n} + 15 \, B a^{2} n x^{3} x^{n} + 30 \, A a b n x^{3} x^{n} + 11 \, A a^{2} n^{2} x^{3} + 9 \, B b^{2} x^{3} x^{3 \, n} + 18 \, B a b x^{3} x^{2 \, n} + 9 \, A b^{2} x^{3} x^{2 \, n} + 9 \, B a^{2} x^{3} x^{n} + 18 \, A a b x^{3} x^{n} + 18 \, A a^{2} n x^{3} + 9 \, A a^{2} x^{3}}{3 \, {\left (2 \, n^{3} + 11 \, n^{2} + 18 \, n + 9\right )}} \] Input:
integrate(x^2*(a+b*x^n)^2*(A+B*x^n),x, algorithm="giac")
Output:
1/3*(2*B*b^2*n^2*x^3*x^(3*n) + 6*B*a*b*n^2*x^3*x^(2*n) + 3*A*b^2*n^2*x^3*x ^(2*n) + 6*B*a^2*n^2*x^3*x^n + 12*A*a*b*n^2*x^3*x^n + 2*A*a^2*n^3*x^3 + 9* B*b^2*n*x^3*x^(3*n) + 24*B*a*b*n*x^3*x^(2*n) + 12*A*b^2*n*x^3*x^(2*n) + 15 *B*a^2*n*x^3*x^n + 30*A*a*b*n*x^3*x^n + 11*A*a^2*n^2*x^3 + 9*B*b^2*x^3*x^( 3*n) + 18*B*a*b*x^3*x^(2*n) + 9*A*b^2*x^3*x^(2*n) + 9*B*a^2*x^3*x^n + 18*A *a*b*x^3*x^n + 18*A*a^2*n*x^3 + 9*A*a^2*x^3)/(2*n^3 + 11*n^2 + 18*n + 9)
Time = 3.73 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.05 \[ \int x^2 \left (a+b x^n\right )^2 \left (A+B x^n\right ) \, dx=\frac {A\,a^2\,x^3}{3}+\frac {x^{2\,n}\,x^3\,\left (A\,b^2+2\,B\,a\,b\right )}{2\,n+3}+\frac {x^n\,x^3\,\left (B\,a^2+2\,A\,b\,a\right )}{n+3}+\frac {B\,b^2\,x^{3\,n}\,x^3}{3\,n+3} \] Input:
int(x^2*(A + B*x^n)*(a + b*x^n)^2,x)
Output:
(A*a^2*x^3)/3 + (x^(2*n)*x^3*(A*b^2 + 2*B*a*b))/(2*n + 3) + (x^n*x^3*(B*a^ 2 + 2*A*a*b))/(n + 3) + (B*b^2*x^(3*n)*x^3)/(3*n + 3)
Time = 0.20 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.99 \[ \int x^2 \left (a+b x^n\right )^2 \left (A+B x^n\right ) \, dx=\frac {x^{3} \left (2 x^{3 n} b^{3} n^{2}+9 x^{3 n} b^{3} n +9 x^{3 n} b^{3}+9 x^{2 n} a \,b^{2} n^{2}+36 x^{2 n} a \,b^{2} n +27 x^{2 n} a \,b^{2}+18 x^{n} a^{2} b \,n^{2}+45 x^{n} a^{2} b n +27 x^{n} a^{2} b +2 a^{3} n^{3}+11 a^{3} n^{2}+18 a^{3} n +9 a^{3}\right )}{6 n^{3}+33 n^{2}+54 n +27} \] Input:
int(x^2*(a+b*x^n)^2*(A+B*x^n),x)
Output:
(x**3*(2*x**(3*n)*b**3*n**2 + 9*x**(3*n)*b**3*n + 9*x**(3*n)*b**3 + 9*x**( 2*n)*a*b**2*n**2 + 36*x**(2*n)*a*b**2*n + 27*x**(2*n)*a*b**2 + 18*x**n*a** 2*b*n**2 + 45*x**n*a**2*b*n + 27*x**n*a**2*b + 2*a**3*n**3 + 11*a**3*n**2 + 18*a**3*n + 9*a**3))/(3*(2*n**3 + 11*n**2 + 18*n + 9))