Integrand size = 15, antiderivative size = 64 \[ \int (c x)^m \left (a+b x^n\right )^2 \, dx=\frac {2 a b x^{1+n} (c x)^m}{1+m+n}+\frac {b^2 x^{1+2 n} (c x)^m}{1+m+2 n}+\frac {a^2 (c x)^{1+m}}{c (1+m)} \]
Time = 0.05 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.73 \[ \int (c x)^m \left (a+b x^n\right )^2 \, dx=x (c x)^m \left (\frac {a^2}{1+m}+\frac {2 a b x^n}{1+m+n}+\frac {b^2 x^{2 n}}{1+m+2 n}\right ) \]
Time = 0.20 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {802, 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.
\(\displaystyle \int (c x)^m \left (a+b x^n\right )^2 \, dx\) |
\(\Big \downarrow \) 802 |
\(\displaystyle \int \left (a^2 (c x)^m+2 a b x^n (c x)^m+b^2 x^{2 n} (c x)^m\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a^2 (c x)^{m+1}}{c (m+1)}+\frac {2 a b x^{n+1} (c x)^m}{m+n+1}+\frac {b^2 x^{2 n+1} (c x)^m}{m+2 n+1}\) |
(2*a*b*x^(1 + n)*(c*x)^m)/(1 + m + n) + (b^2*x^(1 + 2*n)*(c*x)^m)/(1 + m + 2*n) + (a^2*(c*x)^(1 + m))/(c*(1 + m))
3.28.55.3.1 Defintions of rubi rules used
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[Exp andIntegrand[(c*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 3.82 (sec) , antiderivative size = 201, normalized size of antiderivative = 3.14
method | result | size |
risch | \(\frac {x \left (b^{2} m^{2} x^{2 n}+b^{2} m n \,x^{2 n}+2 a b \,m^{2} x^{n}+4 a b m n \,x^{n}+2 m \,b^{2} x^{2 n}+b^{2} n \,x^{2 n}+a^{2} m^{2}+3 a^{2} m n +2 a^{2} n^{2}+4 m a b \,x^{n}+4 a b n \,x^{n}+b^{2} x^{2 n}+2 a^{2} m +3 a^{2} n +2 a b \,x^{n}+a^{2}\right ) x^{m} c^{m} {\mathrm e}^{\frac {i \operatorname {csgn}\left (i c x \right ) \pi m \left (\operatorname {csgn}\left (i c x \right )-\operatorname {csgn}\left (i x \right )\right ) \left (-\operatorname {csgn}\left (i c x \right )+\operatorname {csgn}\left (i c \right )\right )}{2}}}{\left (1+m \right ) \left (1+m +n \right ) \left (1+m +2 n \right )}\) | \(201\) |
parallelrisch | \(\frac {x \,x^{2 n} \left (c x \right )^{m} b^{2} m n +2 x \,x^{n} \left (c x \right )^{m} a b \,m^{2}+4 x \,x^{n} \left (c x \right )^{m} a b m +4 x \,x^{n} \left (c x \right )^{m} a b n +4 x \,x^{n} \left (c x \right )^{m} a b m n +x \,x^{2 n} \left (c x \right )^{m} b^{2}+x \left (c x \right )^{m} a^{2} m^{2}+2 x \left (c x \right )^{m} a^{2} n^{2}+2 x \left (c x \right )^{m} a^{2} m +3 x \left (c x \right )^{m} a^{2} n +x \left (c x \right )^{m} a^{2}+x \,x^{2 n} \left (c x \right )^{m} b^{2} m^{2}+2 x \,x^{2 n} \left (c x \right )^{m} b^{2} m +x \,x^{2 n} \left (c x \right )^{m} b^{2} n +3 x \left (c x \right )^{m} a^{2} m n +2 x \,x^{n} \left (c x \right )^{m} a b}{\left (1+m \right ) \left (1+m +n \right ) \left (1+m +2 n \right )}\) | \(251\) |
x*(b^2*m^2*(x^n)^2+b^2*m*n*(x^n)^2+2*a*b*m^2*x^n+4*a*b*m*n*x^n+2*m*b^2*(x^ n)^2+b^2*n*(x^n)^2+a^2*m^2+3*a^2*m*n+2*a^2*n^2+4*m*a*b*x^n+4*a*b*n*x^n+b^2 *(x^n)^2+2*a^2*m+3*a^2*n+2*a*b*x^n+a^2)/(1+m)/(1+m+n)/(1+m+2*n)*x^m*c^m*ex p(1/2*I*csgn(I*c*x)*Pi*m*(csgn(I*c*x)-csgn(I*x))*(-csgn(I*c*x)+csgn(I*c)))
Leaf count of result is larger than twice the leaf count of optimal. 172 vs. \(2 (64) = 128\).
Time = 0.28 (sec) , antiderivative size = 172, normalized size of antiderivative = 2.69 \[ \int (c x)^m \left (a+b x^n\right )^2 \, dx=\frac {{\left (b^{2} m^{2} + 2 \, b^{2} m + b^{2} + {\left (b^{2} m + b^{2}\right )} n\right )} x x^{2 \, n} e^{\left (m \log \left (c\right ) + m \log \left (x\right )\right )} + 2 \, {\left (a b m^{2} + 2 \, a b m + a b + 2 \, {\left (a b m + a b\right )} n\right )} x x^{n} e^{\left (m \log \left (c\right ) + m \log \left (x\right )\right )} + {\left (a^{2} m^{2} + 2 \, a^{2} n^{2} + 2 \, a^{2} m + a^{2} + 3 \, {\left (a^{2} m + a^{2}\right )} n\right )} x e^{\left (m \log \left (c\right ) + m \log \left (x\right )\right )}}{m^{3} + 2 \, {\left (m + 1\right )} n^{2} + 3 \, m^{2} + 3 \, {\left (m^{2} + 2 \, m + 1\right )} n + 3 \, m + 1} \]
((b^2*m^2 + 2*b^2*m + b^2 + (b^2*m + b^2)*n)*x*x^(2*n)*e^(m*log(c) + m*log (x)) + 2*(a*b*m^2 + 2*a*b*m + a*b + 2*(a*b*m + a*b)*n)*x*x^n*e^(m*log(c) + m*log(x)) + (a^2*m^2 + 2*a^2*n^2 + 2*a^2*m + a^2 + 3*(a^2*m + a^2)*n)*x*e ^(m*log(c) + m*log(x)))/(m^3 + 2*(m + 1)*n^2 + 3*m^2 + 3*(m^2 + 2*m + 1)*n + 3*m + 1)
Leaf count of result is larger than twice the leaf count of optimal. 1148 vs. \(2 (56) = 112\).
Time = 1.45 (sec) , antiderivative size = 1148, normalized size of antiderivative = 17.94 \[ \int (c x)^m \left (a+b x^n\right )^2 \, dx=\text {Too large to display} \]
Piecewise(((a + b)**2*log(x)/c, Eq(m, -1) & Eq(n, 0)), ((a**2*log(x) + 2*a *b*x**n/n + b**2*x**(2*n)/(2*n))/c, Eq(m, -1)), (a**2*Piecewise((0**(-2*n - 1)*x, Eq(c, 0)), (Piecewise((-1/(2*n*(c*x)**(2*n)), Ne(n, 0)), (log(c*x) , True))/c, True)) + 2*a*b*Piecewise((-x*x**n*(c*x)**(-2*n - 1)/n, Ne(n, 0 )), (x*x**n*(c*x)**(-2*n - 1)*log(x), True)) + b**2*x*x**(2*n)*(c*x)**(-2* n - 1)*log(x), Eq(m, -2*n - 1)), (a**2*Piecewise((0**(-n - 1)*x, Eq(c, 0)) , (Piecewise((-1/(n*(c*x)**n), Ne(n, 0)), (log(c*x), True))/c, True)) + 2* a*b*x*x**n*(c*x)**(-n - 1)*log(x) + b**2*Piecewise((x*x**(2*n)*(c*x)**(-n - 1)/n, Ne(n, 0)), (x*x**(2*n)*(c*x)**(-n - 1)*log(x), True)), Eq(m, -n - 1)), (a**2*m**2*x*(c*x)**m/(m**3 + 3*m**2*n + 3*m**2 + 2*m*n**2 + 6*m*n + 3*m + 2*n**2 + 3*n + 1) + 3*a**2*m*n*x*(c*x)**m/(m**3 + 3*m**2*n + 3*m**2 + 2*m*n**2 + 6*m*n + 3*m + 2*n**2 + 3*n + 1) + 2*a**2*m*x*(c*x)**m/(m**3 + 3*m**2*n + 3*m**2 + 2*m*n**2 + 6*m*n + 3*m + 2*n**2 + 3*n + 1) + 2*a**2*n **2*x*(c*x)**m/(m**3 + 3*m**2*n + 3*m**2 + 2*m*n**2 + 6*m*n + 3*m + 2*n**2 + 3*n + 1) + 3*a**2*n*x*(c*x)**m/(m**3 + 3*m**2*n + 3*m**2 + 2*m*n**2 + 6 *m*n + 3*m + 2*n**2 + 3*n + 1) + a**2*x*(c*x)**m/(m**3 + 3*m**2*n + 3*m**2 + 2*m*n**2 + 6*m*n + 3*m + 2*n**2 + 3*n + 1) + 2*a*b*m**2*x*x**n*(c*x)**m /(m**3 + 3*m**2*n + 3*m**2 + 2*m*n**2 + 6*m*n + 3*m + 2*n**2 + 3*n + 1) + 4*a*b*m*n*x*x**n*(c*x)**m/(m**3 + 3*m**2*n + 3*m**2 + 2*m*n**2 + 6*m*n + 3 *m + 2*n**2 + 3*n + 1) + 4*a*b*m*x*x**n*(c*x)**m/(m**3 + 3*m**2*n + 3*m...
Time = 0.22 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.11 \[ \int (c x)^m \left (a+b x^n\right )^2 \, dx=\frac {b^{2} c^{m} x e^{\left (m \log \left (x\right ) + 2 \, n \log \left (x\right )\right )}}{m + 2 \, n + 1} + \frac {2 \, a b c^{m} x e^{\left (m \log \left (x\right ) + n \log \left (x\right )\right )}}{m + n + 1} + \frac {\left (c x\right )^{m + 1} a^{2}}{c {\left (m + 1\right )}} \]
b^2*c^m*x*e^(m*log(x) + 2*n*log(x))/(m + 2*n + 1) + 2*a*b*c^m*x*e^(m*log(x ) + n*log(x))/(m + n + 1) + (c*x)^(m + 1)*a^2/(c*(m + 1))
Leaf count of result is larger than twice the leaf count of optimal. 613 vs. \(2 (64) = 128\).
Time = 0.29 (sec) , antiderivative size = 613, normalized size of antiderivative = 9.58 \[ \int (c x)^m \left (a+b x^n\right )^2 \, dx=\frac {b^{2} m^{2} x x^{2 \, n} e^{\left (m \log \left (c\right ) + m \log \left (x\right )\right )} + b^{2} m n x x^{2 \, n} e^{\left (m \log \left (c\right ) + m \log \left (x\right )\right )} + 2 \, a b m^{2} x x^{n} e^{\left (m \log \left (c\right ) + m \log \left (x\right )\right )} + b^{2} m^{2} x x^{n} e^{\left (m \log \left (c\right ) + m \log \left (x\right )\right )} + 4 \, a b m n x x^{n} e^{\left (m \log \left (c\right ) + m \log \left (x\right )\right )} + b^{2} m n x x^{n} e^{\left (m \log \left (c\right ) + m \log \left (x\right )\right )} + a^{2} m^{2} x e^{\left (m \log \left (c\right ) + m \log \left (x\right )\right )} + 2 \, a b m^{2} x e^{\left (m \log \left (c\right ) + m \log \left (x\right )\right )} + b^{2} m^{2} x e^{\left (m \log \left (c\right ) + m \log \left (x\right )\right )} + 3 \, a^{2} m n x e^{\left (m \log \left (c\right ) + m \log \left (x\right )\right )} + 4 \, a b m n x e^{\left (m \log \left (c\right ) + m \log \left (x\right )\right )} + b^{2} m n x e^{\left (m \log \left (c\right ) + m \log \left (x\right )\right )} + 2 \, a^{2} n^{2} x e^{\left (m \log \left (c\right ) + m \log \left (x\right )\right )} + 2 \, b^{2} m x x^{2 \, n} e^{\left (m \log \left (c\right ) + m \log \left (x\right )\right )} + b^{2} n x x^{2 \, n} e^{\left (m \log \left (c\right ) + m \log \left (x\right )\right )} + 4 \, a b m x x^{n} e^{\left (m \log \left (c\right ) + m \log \left (x\right )\right )} + 2 \, b^{2} m x x^{n} e^{\left (m \log \left (c\right ) + m \log \left (x\right )\right )} + 4 \, a b n x x^{n} e^{\left (m \log \left (c\right ) + m \log \left (x\right )\right )} + b^{2} n x x^{n} e^{\left (m \log \left (c\right ) + m \log \left (x\right )\right )} + 2 \, a^{2} m x e^{\left (m \log \left (c\right ) + m \log \left (x\right )\right )} + 4 \, a b m x e^{\left (m \log \left (c\right ) + m \log \left (x\right )\right )} + 2 \, b^{2} m x e^{\left (m \log \left (c\right ) + m \log \left (x\right )\right )} + 3 \, a^{2} n x e^{\left (m \log \left (c\right ) + m \log \left (x\right )\right )} + 4 \, a b n x e^{\left (m \log \left (c\right ) + m \log \left (x\right )\right )} + b^{2} n x e^{\left (m \log \left (c\right ) + m \log \left (x\right )\right )} + b^{2} x x^{2 \, n} e^{\left (m \log \left (c\right ) + m \log \left (x\right )\right )} + 2 \, a b x x^{n} e^{\left (m \log \left (c\right ) + m \log \left (x\right )\right )} + b^{2} x x^{n} e^{\left (m \log \left (c\right ) + m \log \left (x\right )\right )} + a^{2} x e^{\left (m \log \left (c\right ) + m \log \left (x\right )\right )} + 2 \, a b x e^{\left (m \log \left (c\right ) + m \log \left (x\right )\right )} + b^{2} x e^{\left (m \log \left (c\right ) + m \log \left (x\right )\right )}}{m^{3} + 3 \, m^{2} n + 2 \, m n^{2} + 3 \, m^{2} + 6 \, m n + 2 \, n^{2} + 3 \, m + 3 \, n + 1} \]
(b^2*m^2*x*x^(2*n)*e^(m*log(c) + m*log(x)) + b^2*m*n*x*x^(2*n)*e^(m*log(c) + m*log(x)) + 2*a*b*m^2*x*x^n*e^(m*log(c) + m*log(x)) + b^2*m^2*x*x^n*e^( m*log(c) + m*log(x)) + 4*a*b*m*n*x*x^n*e^(m*log(c) + m*log(x)) + b^2*m*n*x *x^n*e^(m*log(c) + m*log(x)) + a^2*m^2*x*e^(m*log(c) + m*log(x)) + 2*a*b*m ^2*x*e^(m*log(c) + m*log(x)) + b^2*m^2*x*e^(m*log(c) + m*log(x)) + 3*a^2*m *n*x*e^(m*log(c) + m*log(x)) + 4*a*b*m*n*x*e^(m*log(c) + m*log(x)) + b^2*m *n*x*e^(m*log(c) + m*log(x)) + 2*a^2*n^2*x*e^(m*log(c) + m*log(x)) + 2*b^2 *m*x*x^(2*n)*e^(m*log(c) + m*log(x)) + b^2*n*x*x^(2*n)*e^(m*log(c) + m*log (x)) + 4*a*b*m*x*x^n*e^(m*log(c) + m*log(x)) + 2*b^2*m*x*x^n*e^(m*log(c) + m*log(x)) + 4*a*b*n*x*x^n*e^(m*log(c) + m*log(x)) + b^2*n*x*x^n*e^(m*log( c) + m*log(x)) + 2*a^2*m*x*e^(m*log(c) + m*log(x)) + 4*a*b*m*x*e^(m*log(c) + m*log(x)) + 2*b^2*m*x*e^(m*log(c) + m*log(x)) + 3*a^2*n*x*e^(m*log(c) + m*log(x)) + 4*a*b*n*x*e^(m*log(c) + m*log(x)) + b^2*n*x*e^(m*log(c) + m*l og(x)) + b^2*x*x^(2*n)*e^(m*log(c) + m*log(x)) + 2*a*b*x*x^n*e^(m*log(c) + m*log(x)) + b^2*x*x^n*e^(m*log(c) + m*log(x)) + a^2*x*e^(m*log(c) + m*log (x)) + 2*a*b*x*e^(m*log(c) + m*log(x)) + b^2*x*e^(m*log(c) + m*log(x)))/(m ^3 + 3*m^2*n + 2*m*n^2 + 3*m^2 + 6*m*n + 2*n^2 + 3*m + 3*n + 1)
Time = 5.71 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.39 \[ \int (c x)^m \left (a+b x^n\right )^2 \, dx={\left (c\,x\right )}^m\,\left (\frac {a^2\,x}{m+1}+\frac {b^2\,x\,x^{2\,n}\,\left (m+n+1\right )}{m^2+3\,m\,n+2\,m+2\,n^2+3\,n+1}+\frac {2\,a\,b\,x\,x^n\,\left (m+2\,n+1\right )}{m^2+3\,m\,n+2\,m+2\,n^2+3\,n+1}\right ) \]