Integrand size = 17, antiderivative size = 70 \[ \int \left (a+b x^n\right )^2 \left (c+d x^n\right ) \, dx=a^2 c x+\frac {a (2 b c+a d) x^{1+n}}{1+n}+\frac {b (b c+2 a d) x^{1+2 n}}{1+2 n}+\frac {b^2 d x^{1+3 n}}{1+3 n} \] Output:
a^2*c*x+a*(a*d+2*b*c)*x^(1+n)/(1+n)+b*(2*a*d+b*c)*x^(1+2*n)/(1+2*n)+b^2*d* x^(1+3*n)/(1+3*n)
Time = 0.16 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00 \[ \int \left (a+b x^n\right )^2 \left (c+d x^n\right ) \, dx=\frac {d x \left (a+b x^n\right )^3-(a d-b (c+3 c n)) x \left (a^2+\frac {2 a b x^n}{1+n}+\frac {b^2 x^{2 n}}{1+2 n}\right )}{b+3 b n} \] Input:
Integrate[(a + b*x^n)^2*(c + d*x^n),x]
Output:
(d*x*(a + b*x^n)^3 - (a*d - b*(c + 3*c*n))*x*(a^2 + (2*a*b*x^n)/(1 + n) + (b^2*x^(2*n))/(1 + 2*n)))/(b + 3*b*n)
Time = 0.37 (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.
\(\displaystyle \int \left (a+b x^n\right )^2 \left (c+d x^n\right ) \, dx\) |
\(\Big \downarrow \) 897 |
\(\displaystyle \int \left (a^2 c+b x^{2 n} (2 a d+b c)+a x^n (a d+2 b c)+b^2 d x^{3 n}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle a^2 c x+\frac {a x^{n+1} (a d+2 b c)}{n+1}+\frac {b x^{2 n+1} (2 a d+b c)}{2 n+1}+\frac {b^2 d x^{3 n+1}}{3 n+1}\) |
Input:
Int[(a + b*x^n)^2*(c + d*x^n),x]
Output:
a^2*c*x + (a*(2*b*c + a*d)*x^(1 + n))/(1 + n) + (b*(b*c + 2*a*d)*x^(1 + 2* n))/(1 + 2*n) + (b^2*d*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 = 0.72 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.97
method | result | size |
risch | \(a^{2} c x +\frac {a \left (a d +2 b c \right ) x \,x^{n}}{1+n}+\frac {b \left (2 a d +b c \right ) x \,x^{2 n}}{1+2 n}+\frac {d \,b^{2} x \,x^{3 n}}{1+3 n}\) | \(68\) |
norman | \(a^{2} c x +\frac {a \left (a d +2 b c \right ) x \,{\mathrm e}^{n \ln \left (x \right )}}{1+n}+\frac {b \left (2 a d +b c \right ) x \,{\mathrm e}^{2 n \ln \left (x \right )}}{1+2 n}+\frac {d \,b^{2} x \,{\mathrm e}^{3 n \ln \left (x \right )}}{1+3 n}\) | \(74\) |
parallelrisch | \(\frac {2 x \,x^{3 n} b^{2} d \,n^{2}+3 x \,x^{3 n} b^{2} d n +6 x \,x^{2 n} a b d \,n^{2}+3 x \,x^{2 n} b^{2} c \,n^{2}+d \,b^{2} x \,x^{3 n}+8 x \,x^{2 n} a b d n +4 x \,x^{2 n} b^{2} c n +6 x \,x^{n} a^{2} d \,n^{2}+12 x \,x^{n} a b c \,n^{2}+6 x \,a^{2} c \,n^{3}+2 x \,x^{2 n} a b d +x \,x^{2 n} b^{2} c +5 x \,x^{n} a^{2} d n +10 x \,x^{n} a b c n +11 x \,a^{2} c \,n^{2}+x \,x^{n} a^{2} d +2 x \,x^{n} a b c +6 x \,a^{2} c n +a^{2} c x}{\left (1+n \right ) \left (1+2 n \right ) \left (1+3 n \right )}\) | \(235\) |
orering | \(x \left (a +b \,x^{n}\right )^{2} \left (c +d \,x^{n}\right )-\frac {x^{2} \left (11 n^{2}+1\right ) \left (\frac {2 \left (a +b \,x^{n}\right ) \left (c +d \,x^{n}\right ) b \,x^{n} n}{x}+\frac {\left (a +b \,x^{n}\right )^{2} d \,x^{n} n}{x}\right )}{\left (2 n^{2}+3 n +1\right ) \left (1+3 n \right )}+\frac {2 x^{3} \left (-1+3 n \right ) \left (\frac {2 b^{2} x^{2 n} n^{2} \left (c +d \,x^{n}\right )}{x^{2}}+\frac {4 \left (a +b \,x^{n}\right ) d \,x^{2 n} n^{2} b}{x^{2}}+\frac {2 \left (a +b \,x^{n}\right ) \left (c +d \,x^{n}\right ) b \,x^{n} n^{2}}{x^{2}}-\frac {2 \left (a +b \,x^{n}\right ) \left (c +d \,x^{n}\right ) b \,x^{n} n}{x^{2}}+\frac {\left (a +b \,x^{n}\right )^{2} d \,x^{n} n^{2}}{x^{2}}-\frac {\left (a +b \,x^{n}\right )^{2} d \,x^{n} n}{x^{2}}\right )}{\left (2 n^{2}+3 n +1\right ) \left (1+3 n \right )}-\frac {x^{4} \left (\frac {6 b^{2} x^{2 n} n^{3} \left (c +d \,x^{n}\right )}{x^{3}}-\frac {6 b^{2} x^{2 n} n^{2} \left (c +d \,x^{n}\right )}{x^{3}}+\frac {6 x^{3 n} b^{2} n^{3} d}{x^{3}}+\frac {12 \left (a +b \,x^{n}\right ) d \,x^{2 n} n^{3} b}{x^{3}}-\frac {12 \left (a +b \,x^{n}\right ) d \,x^{2 n} n^{2} b}{x^{3}}+\frac {2 \left (a +b \,x^{n}\right ) \left (c +d \,x^{n}\right ) b \,x^{n} n^{3}}{x^{3}}-\frac {6 \left (a +b \,x^{n}\right ) \left (c +d \,x^{n}\right ) b \,x^{n} n^{2}}{x^{3}}+\frac {4 \left (a +b \,x^{n}\right ) \left (c +d \,x^{n}\right ) b \,x^{n} n}{x^{3}}+\frac {\left (a +b \,x^{n}\right )^{2} d \,x^{n} n^{3}}{x^{3}}-\frac {3 \left (a +b \,x^{n}\right )^{2} d \,x^{n} n^{2}}{x^{3}}+\frac {2 \left (a +b \,x^{n}\right )^{2} d \,x^{n} n}{x^{3}}\right )}{6 n^{3}+11 n^{2}+6 n +1}\) | \(524\) |
Input:
int((a+b*x^n)^2*(c+d*x^n),x,method=_RETURNVERBOSE)
Output:
a^2*c*x+a*(a*d+2*b*c)/(1+n)*x*x^n+b*(2*a*d+b*c)/(1+2*n)*x*(x^n)^2+d*b^2/(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 \[ \int \left (a+b x^n\right )^2 \left (c+d x^n\right ) \, dx=\frac {{\left (2 \, b^{2} d n^{2} + 3 \, b^{2} d n + b^{2} d\right )} x x^{3 \, n} + {\left (b^{2} c + 2 \, a b d + 3 \, {\left (b^{2} c + 2 \, a b d\right )} n^{2} + 4 \, {\left (b^{2} c + 2 \, a b d\right )} n\right )} x x^{2 \, n} + {\left (2 \, a b c + a^{2} d + 6 \, {\left (2 \, a b c + a^{2} d\right )} n^{2} + 5 \, {\left (2 \, a b c + a^{2} d\right )} n\right )} x x^{n} + {\left (6 \, a^{2} c n^{3} + 11 \, a^{2} c n^{2} + 6 \, a^{2} c n + a^{2} c\right )} x}{6 \, n^{3} + 11 \, n^{2} + 6 \, n + 1} \] Input:
integrate((a+b*x^n)^2*(c+d*x^n),x, algorithm="fricas")
Output:
((2*b^2*d*n^2 + 3*b^2*d*n + b^2*d)*x*x^(3*n) + (b^2*c + 2*a*b*d + 3*(b^2*c + 2*a*b*d)*n^2 + 4*(b^2*c + 2*a*b*d)*n)*x*x^(2*n) + (2*a*b*c + a^2*d + 6* (2*a*b*c + a^2*d)*n^2 + 5*(2*a*b*c + a^2*d)*n)*x*x^n + (6*a^2*c*n^3 + 11*a ^2*c*n^2 + 6*a^2*c*n + a^2*c)*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 = 0.39 (sec) , antiderivative size = 726, normalized size of antiderivative = 10.37 \[ \int \left (a+b x^n\right )^2 \left (c+d x^n\right ) \, dx=\begin {cases} a^{2} c x + a^{2} d \log {\left (x \right )} + 2 a b c \log {\left (x \right )} - \frac {2 a b d}{x} - \frac {b^{2} c}{x} - \frac {b^{2} d}{2 x^{2}} & \text {for}\: n = -1 \\a^{2} c x + 2 a^{2} d \sqrt {x} + 4 a b c \sqrt {x} + 2 a b d \log {\left (x \right )} + b^{2} c \log {\left (x \right )} - \frac {2 b^{2} d}{\sqrt {x}} & \text {for}\: n = - \frac {1}{2} \\a^{2} c x + \frac {3 a^{2} d x^{\frac {2}{3}}}{2} + 3 a b c x^{\frac {2}{3}} + 6 a b d \sqrt [3]{x} + 3 b^{2} c \sqrt [3]{x} + b^{2} d \log {\left (x \right )} & \text {for}\: n = - \frac {1}{3} \\\frac {6 a^{2} c n^{3} x}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {11 a^{2} c n^{2} x}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {6 a^{2} c n x}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {a^{2} c x}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {6 a^{2} d n^{2} x x^{n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {5 a^{2} d n x x^{n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {a^{2} d x x^{n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {12 a b c n^{2} x x^{n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {10 a b c n x x^{n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {2 a b c x x^{n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {6 a b d n^{2} x x^{2 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {8 a b d n x x^{2 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {2 a b d x x^{2 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {3 b^{2} c n^{2} x x^{2 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {4 b^{2} c n x x^{2 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {b^{2} c x x^{2 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {2 b^{2} d n^{2} x x^{3 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {3 b^{2} d n x x^{3 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {b^{2} d x x^{3 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} & \text {otherwise} \end {cases} \] Input:
integrate((a+b*x**n)**2*(c+d*x**n),x)
Output:
Piecewise((a**2*c*x + a**2*d*log(x) + 2*a*b*c*log(x) - 2*a*b*d/x - b**2*c/ x - b**2*d/(2*x**2), Eq(n, -1)), (a**2*c*x + 2*a**2*d*sqrt(x) + 4*a*b*c*sq rt(x) + 2*a*b*d*log(x) + b**2*c*log(x) - 2*b**2*d/sqrt(x), Eq(n, -1/2)), ( a**2*c*x + 3*a**2*d*x**(2/3)/2 + 3*a*b*c*x**(2/3) + 6*a*b*d*x**(1/3) + 3*b **2*c*x**(1/3) + b**2*d*log(x), Eq(n, -1/3)), (6*a**2*c*n**3*x/(6*n**3 + 1 1*n**2 + 6*n + 1) + 11*a**2*c*n**2*x/(6*n**3 + 11*n**2 + 6*n + 1) + 6*a**2 *c*n*x/(6*n**3 + 11*n**2 + 6*n + 1) + a**2*c*x/(6*n**3 + 11*n**2 + 6*n + 1 ) + 6*a**2*d*n**2*x*x**n/(6*n**3 + 11*n**2 + 6*n + 1) + 5*a**2*d*n*x*x**n/ (6*n**3 + 11*n**2 + 6*n + 1) + a**2*d*x*x**n/(6*n**3 + 11*n**2 + 6*n + 1) + 12*a*b*c*n**2*x*x**n/(6*n**3 + 11*n**2 + 6*n + 1) + 10*a*b*c*n*x*x**n/(6 *n**3 + 11*n**2 + 6*n + 1) + 2*a*b*c*x*x**n/(6*n**3 + 11*n**2 + 6*n + 1) + 6*a*b*d*n**2*x*x**(2*n)/(6*n**3 + 11*n**2 + 6*n + 1) + 8*a*b*d*n*x*x**(2* n)/(6*n**3 + 11*n**2 + 6*n + 1) + 2*a*b*d*x*x**(2*n)/(6*n**3 + 11*n**2 + 6 *n + 1) + 3*b**2*c*n**2*x*x**(2*n)/(6*n**3 + 11*n**2 + 6*n + 1) + 4*b**2*c *n*x*x**(2*n)/(6*n**3 + 11*n**2 + 6*n + 1) + b**2*c*x*x**(2*n)/(6*n**3 + 1 1*n**2 + 6*n + 1) + 2*b**2*d*n**2*x*x**(3*n)/(6*n**3 + 11*n**2 + 6*n + 1) + 3*b**2*d*n*x*x**(3*n)/(6*n**3 + 11*n**2 + 6*n + 1) + b**2*d*x*x**(3*n)/( 6*n**3 + 11*n**2 + 6*n + 1), True))
Time = 0.03 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.34 \[ \int \left (a+b x^n\right )^2 \left (c+d x^n\right ) \, dx=a^{2} c x + \frac {b^{2} d x^{3 \, n + 1}}{3 \, n + 1} + \frac {b^{2} c x^{2 \, n + 1}}{2 \, n + 1} + \frac {2 \, a b d x^{2 \, n + 1}}{2 \, n + 1} + \frac {2 \, a b c x^{n + 1}}{n + 1} + \frac {a^{2} d x^{n + 1}}{n + 1} \] Input:
integrate((a+b*x^n)^2*(c+d*x^n),x, algorithm="maxima")
Output:
a^2*c*x + b^2*d*x^(3*n + 1)/(3*n + 1) + b^2*c*x^(2*n + 1)/(2*n + 1) + 2*a* b*d*x^(2*n + 1)/(2*n + 1) + 2*a*b*c*x^(n + 1)/(n + 1) + a^2*d*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 \[ \int \left (a+b x^n\right )^2 \left (c+d x^n\right ) \, dx=\frac {6 \, a^{2} c n^{3} x + 2 \, b^{2} d n^{2} x x^{3 \, n} + 3 \, b^{2} c n^{2} x x^{2 \, n} + 6 \, a b d n^{2} x x^{2 \, n} + 12 \, a b c n^{2} x x^{n} + 6 \, a^{2} d n^{2} x x^{n} + 11 \, a^{2} c n^{2} x + 3 \, b^{2} d n x x^{3 \, n} + 4 \, b^{2} c n x x^{2 \, n} + 8 \, a b d n x x^{2 \, n} + 10 \, a b c n x x^{n} + 5 \, a^{2} d n x x^{n} + 6 \, a^{2} c n x + b^{2} d x x^{3 \, n} + b^{2} c x x^{2 \, n} + 2 \, a b d x x^{2 \, n} + 2 \, a b c x x^{n} + a^{2} d x x^{n} + a^{2} c x}{6 \, n^{3} + 11 \, n^{2} + 6 \, n + 1} \] Input:
integrate((a+b*x^n)^2*(c+d*x^n),x, algorithm="giac")
Output:
(6*a^2*c*n^3*x + 2*b^2*d*n^2*x*x^(3*n) + 3*b^2*c*n^2*x*x^(2*n) + 6*a*b*d*n ^2*x*x^(2*n) + 12*a*b*c*n^2*x*x^n + 6*a^2*d*n^2*x*x^n + 11*a^2*c*n^2*x + 3 *b^2*d*n*x*x^(3*n) + 4*b^2*c*n*x*x^(2*n) + 8*a*b*d*n*x*x^(2*n) + 10*a*b*c* n*x*x^n + 5*a^2*d*n*x*x^n + 6*a^2*c*n*x + b^2*d*x*x^(3*n) + b^2*c*x*x^(2*n ) + 2*a*b*d*x*x^(2*n) + 2*a*b*c*x*x^n + a^2*d*x*x^n + a^2*c*x)/(6*n^3 + 11 *n^2 + 6*n + 1)
Time = 0.78 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.01 \[ \int \left (a+b x^n\right )^2 \left (c+d x^n\right ) \, dx=a^2\,c\,x+\frac {x\,x^{2\,n}\,\left (c\,b^2+2\,a\,d\,b\right )}{2\,n+1}+\frac {x\,x^n\,\left (d\,a^2+2\,b\,c\,a\right )}{n+1}+\frac {b^2\,d\,x\,x^{3\,n}}{3\,n+1} \] Input:
int((a + b*x^n)^2*(c + d*x^n),x)
Output:
a^2*c*x + (x*x^(2*n)*(b^2*c + 2*a*b*d))/(2*n + 1) + (x*x^n*(a^2*d + 2*a*b* c))/(n + 1) + (b^2*d*x*x^(3*n))/(3*n + 1)
Time = 0.26 (sec) , antiderivative size = 214, normalized size of antiderivative = 3.06 \[ \int \left (a+b x^n\right )^2 \left (c+d x^n\right ) \, dx=\frac {x \left (2 x^{3 n} b^{2} d \,n^{2}+3 x^{3 n} b^{2} d n +x^{3 n} b^{2} d +6 x^{2 n} a b d \,n^{2}+8 x^{2 n} a b d n +2 x^{2 n} a b d +3 x^{2 n} b^{2} c \,n^{2}+4 x^{2 n} b^{2} c n +x^{2 n} b^{2} c +6 x^{n} a^{2} d \,n^{2}+5 x^{n} a^{2} d n +x^{n} a^{2} d +12 x^{n} a b c \,n^{2}+10 x^{n} a b c n +2 x^{n} a b c +6 a^{2} c \,n^{3}+11 a^{2} c \,n^{2}+6 a^{2} c n +a^{2} c \right )}{6 n^{3}+11 n^{2}+6 n +1} \] Input:
int((a+b*x^n)^2*(c+d*x^n),x)
Output:
(x*(2*x**(3*n)*b**2*d*n**2 + 3*x**(3*n)*b**2*d*n + x**(3*n)*b**2*d + 6*x** (2*n)*a*b*d*n**2 + 8*x**(2*n)*a*b*d*n + 2*x**(2*n)*a*b*d + 3*x**(2*n)*b**2 *c*n**2 + 4*x**(2*n)*b**2*c*n + x**(2*n)*b**2*c + 6*x**n*a**2*d*n**2 + 5*x **n*a**2*d*n + x**n*a**2*d + 12*x**n*a*b*c*n**2 + 10*x**n*a*b*c*n + 2*x**n *a*b*c + 6*a**2*c*n**3 + 11*a**2*c*n**2 + 6*a**2*c*n + a**2*c))/(6*n**3 + 11*n**2 + 6*n + 1)