Integrand size = 26, antiderivative size = 55 \[ \int (d x)^m \left (a x^n+b x^{1+n}+c x^{2+n}\right ) \, dx=\frac {a x^{1+n} (d x)^m}{1+m+n}+\frac {b x^{2+n} (d x)^m}{2+m+n}+\frac {c x^{3+n} (d x)^m}{3+m+n} \] Output:
a*x^(1+n)*(d*x)^m/(1+m+n)+b*x^(2+n)*(d*x)^m/(2+m+n)+c*x^(3+n)*(d*x)^m/(3+m +n)
Time = 0.12 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.73 \[ \int (d x)^m \left (a x^n+b x^{1+n}+c x^{2+n}\right ) \, dx=x^{1+n} (d x)^m \left (\frac {a}{1+m+n}+x \left (\frac {b}{2+m+n}+\frac {c x}{3+m+n}\right )\right ) \] Input:
Integrate[(d*x)^m*(a*x^n + b*x^(1 + n) + c*x^(2 + n)),x]
Output:
x^(1 + n)*(d*x)^m*(a/(1 + m + n) + x*(b/(2 + m + n) + (c*x)/(3 + m + n)))
Time = 0.22 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2010, 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.
\(\displaystyle \int (d x)^m \left (a x^n+b x^{n+1}+c x^{n+2}\right ) \, dx\) |
\(\Big \downarrow \) 2010 |
\(\displaystyle \int \left (a x^n (d x)^m+b x^{n+1} (d x)^m+c x^{n+2} (d x)^m\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a x^{n+1} (d x)^m}{m+n+1}+\frac {b x^{n+2} (d x)^m}{m+n+2}+\frac {c x^{n+3} (d x)^m}{m+n+3}\) |
Input:
Int[(d*x)^m*(a*x^n + b*x^(1 + n) + c*x^(2 + n)),x]
Output:
(a*x^(1 + n)*(d*x)^m)/(1 + m + n) + (b*x^(2 + n)*(d*x)^m)/(2 + m + n) + (c *x^(3 + n)*(d*x)^m)/(3 + m + n)
Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x] , x] /; FreeQ[{c, m}, x] && SumQ[u] && !LinearQ[u, x] && !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
Leaf count of result is larger than twice the leaf count of optimal. \(160\) vs. \(2(55)=110\).
Time = 0.46 (sec) , antiderivative size = 161, normalized size of antiderivative = 2.93
method | result | size |
orering | \(\frac {\left (c \,m^{2} x^{2}+2 c m n \,x^{2}+c \,n^{2} x^{2}+b \,m^{2} x +2 b m n x +b \,n^{2} x +3 x^{2} c m +3 c \,x^{2} n +a \,m^{2}+2 a m n +a \,n^{2}+4 b x m +4 b x n +2 c \,x^{2}+5 a m +5 a n +3 b x +6 a \right ) x \left (d x \right )^{m} \left (x^{n} a +b \,x^{1+n}+c \,x^{2+n}\right )}{\left (1+m +n \right ) \left (2+m +n \right ) \left (3+m +n \right ) \left (c \,x^{2}+b x +a \right )}\) | \(161\) |
risch | \(\frac {x \left (c \,m^{2} x^{2}+2 c m n \,x^{2}+c \,n^{2} x^{2}+b \,m^{2} x +2 b m n x +b \,n^{2} x +3 x^{2} c m +3 c \,x^{2} n +a \,m^{2}+2 a m n +a \,n^{2}+4 b x m +4 b x n +2 c \,x^{2}+5 a m +5 a n +3 b x +6 a \right ) x^{n} x^{m} d^{m} {\mathrm e}^{\frac {i \operatorname {csgn}\left (i d x \right ) \pi m \left (\operatorname {csgn}\left (i d x \right )-\operatorname {csgn}\left (i x \right )\right ) \left (-\operatorname {csgn}\left (i d x \right )+\operatorname {csgn}\left (i d \right )\right )}{2}}}{\left (1+m +n \right ) \left (2+m +n \right ) \left (3+m +n \right )}\) | \(173\) |
parallelrisch | \(\frac {2 x \,x^{n} \left (d x \right )^{m} a m n +2 x \,x^{1+n} \left (d x \right )^{m} b m n +2 x \,x^{2+n} \left (d x \right )^{m} c m n +6 x \,x^{n} \left (d x \right )^{m} a +3 x \,x^{1+n} \left (d x \right )^{m} b +2 x \,x^{2+n} \left (d x \right )^{m} c +x \,x^{n} \left (d x \right )^{m} a \,m^{2}+x \,x^{n} \left (d x \right )^{m} a \,n^{2}+x \,x^{1+n} \left (d x \right )^{m} b \,m^{2}+x \,x^{1+n} \left (d x \right )^{m} b \,n^{2}+x \,x^{2+n} \left (d x \right )^{m} c \,m^{2}+x \,x^{2+n} \left (d x \right )^{m} c \,n^{2}+5 x \,x^{n} \left (d x \right )^{m} a m +5 x \,x^{n} \left (d x \right )^{m} a n +4 x \,x^{1+n} \left (d x \right )^{m} b m +4 x \,x^{1+n} \left (d x \right )^{m} b n +3 x \,x^{2+n} \left (d x \right )^{m} c m +3 x \,x^{2+n} \left (d x \right )^{m} c n}{\left (1+m +n \right ) \left (2+m +n \right ) \left (3+m +n \right )}\) | \(285\) |
Input:
int((d*x)^m*(x^n*a+b*x^(1+n)+c*x^(2+n)),x,method=_RETURNVERBOSE)
Output:
(c*m^2*x^2+2*c*m*n*x^2+c*n^2*x^2+b*m^2*x+2*b*m*n*x+b*n^2*x+3*c*m*x^2+3*c*n *x^2+a*m^2+2*a*m*n+a*n^2+4*b*m*x+4*b*n*x+2*c*x^2+5*a*m+5*a*n+3*b*x+6*a)/(1 +m+n)/(2+m+n)/(3+m+n)*x/(c*x^2+b*x+a)*(d*x)^m*(x^n*a+b*x^(1+n)+c*x^(2+n))
Leaf count of result is larger than twice the leaf count of optimal. 147 vs. \(2 (55) = 110\).
Time = 0.09 (sec) , antiderivative size = 147, normalized size of antiderivative = 2.67 \[ \int (d x)^m \left (a x^n+b x^{1+n}+c x^{2+n}\right ) \, dx=\frac {{\left (a m^{2} + a n^{2} + {\left (c m^{2} + c n^{2} + 3 \, c m + {\left (2 \, c m + 3 \, c\right )} n + 2 \, c\right )} x^{2} + 5 \, a m + {\left (2 \, a m + 5 \, a\right )} n + {\left (b m^{2} + b n^{2} + 4 \, b m + 2 \, {\left (b m + 2 \, b\right )} n + 3 \, b\right )} x + 6 \, a\right )} x^{n + 2} e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )}}{{\left (m^{3} + 3 \, {\left (m + 2\right )} n^{2} + n^{3} + 6 \, m^{2} + {\left (3 \, m^{2} + 12 \, m + 11\right )} n + 11 \, m + 6\right )} x} \] Input:
integrate((d*x)^m*(a*x^n+b*x^(1+n)+c*x^(2+n)),x, algorithm="fricas")
Output:
(a*m^2 + a*n^2 + (c*m^2 + c*n^2 + 3*c*m + (2*c*m + 3*c)*n + 2*c)*x^2 + 5*a *m + (2*a*m + 5*a)*n + (b*m^2 + b*n^2 + 4*b*m + 2*(b*m + 2*b)*n + 3*b)*x + 6*a)*x^(n + 2)*e^(m*log(d) + m*log(x))/((m^3 + 3*(m + 2)*n^2 + n^3 + 6*m^ 2 + (3*m^2 + 12*m + 11)*n + 11*m + 6)*x)
Leaf count of result is larger than twice the leaf count of optimal. 1258 vs. \(2 (49) = 98\).
Time = 20.02 (sec) , antiderivative size = 1258, normalized size of antiderivative = 22.87 \[ \int (d x)^m \left (a x^n+b x^{1+n}+c x^{2+n}\right ) \, dx =\text {Too large to display} \] Input:
integrate((d*x)**m*(a*x**n+b*x**(1+n)+c*x**(2+n)),x)
Output:
Piecewise((-a*x*x**n*(d*x)**(-n - 3)/2 - b*x*x**(n + 1)*(d*x)**(-n - 3) + c*x*x**(n + 2)*(d*x)**(-n - 3)*log(x), Eq(m, -n - 3)), (-a*x*x**n*(d*x)**( -n - 2) + b*x*x**(n + 1)*(d*x)**(-n - 2)*log(x) + c*x*x**(n + 2)*(d*x)**(- n - 2), Eq(m, -n - 2)), (a*x*x**n*(d*x)**(-n - 1)*log(x) + b*x*x**(n + 1)* (d*x)**(-n - 1) + c*x*x**(n + 2)*(d*x)**(-n - 1)/2, Eq(m, -n - 1)), (a*m** 2*x*x**n*(d*x)**m/(m**3 + 3*m**2*n + 6*m**2 + 3*m*n**2 + 12*m*n + 11*m + n **3 + 6*n**2 + 11*n + 6) + 2*a*m*n*x*x**n*(d*x)**m/(m**3 + 3*m**2*n + 6*m* *2 + 3*m*n**2 + 12*m*n + 11*m + n**3 + 6*n**2 + 11*n + 6) + 5*a*m*x*x**n*( d*x)**m/(m**3 + 3*m**2*n + 6*m**2 + 3*m*n**2 + 12*m*n + 11*m + n**3 + 6*n* *2 + 11*n + 6) + a*n**2*x*x**n*(d*x)**m/(m**3 + 3*m**2*n + 6*m**2 + 3*m*n* *2 + 12*m*n + 11*m + n**3 + 6*n**2 + 11*n + 6) + 5*a*n*x*x**n*(d*x)**m/(m* *3 + 3*m**2*n + 6*m**2 + 3*m*n**2 + 12*m*n + 11*m + n**3 + 6*n**2 + 11*n + 6) + 6*a*x*x**n*(d*x)**m/(m**3 + 3*m**2*n + 6*m**2 + 3*m*n**2 + 12*m*n + 11*m + n**3 + 6*n**2 + 11*n + 6) + b*m**2*x*x**(n + 1)*(d*x)**m/(m**3 + 3* m**2*n + 6*m**2 + 3*m*n**2 + 12*m*n + 11*m + n**3 + 6*n**2 + 11*n + 6) + 2 *b*m*n*x*x**(n + 1)*(d*x)**m/(m**3 + 3*m**2*n + 6*m**2 + 3*m*n**2 + 12*m*n + 11*m + n**3 + 6*n**2 + 11*n + 6) + 4*b*m*x*x**(n + 1)*(d*x)**m/(m**3 + 3*m**2*n + 6*m**2 + 3*m*n**2 + 12*m*n + 11*m + n**3 + 6*n**2 + 11*n + 6) + b*n**2*x*x**(n + 1)*(d*x)**m/(m**3 + 3*m**2*n + 6*m**2 + 3*m*n**2 + 12*m* n + 11*m + n**3 + 6*n**2 + 11*n + 6) + 4*b*n*x*x**(n + 1)*(d*x)**m/(m**...
Time = 0.05 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.29 \[ \int (d x)^m \left (a x^n+b x^{1+n}+c x^{2+n}\right ) \, dx=\frac {c d^{m} x^{3} e^{\left (m \log \left (x\right ) + n \log \left (x\right )\right )}}{m + n + 3} + \frac {b d^{m} x^{2} e^{\left (m \log \left (x\right ) + n \log \left (x\right )\right )}}{m + n + 2} + \frac {a d^{m} x e^{\left (m \log \left (x\right ) + n \log \left (x\right )\right )}}{m + n + 1} \] Input:
integrate((d*x)^m*(a*x^n+b*x^(1+n)+c*x^(2+n)),x, algorithm="maxima")
Output:
c*d^m*x^3*e^(m*log(x) + n*log(x))/(m + n + 3) + b*d^m*x^2*e^(m*log(x) + n* log(x))/(m + n + 2) + a*d^m*x*e^(m*log(x) + n*log(x))/(m + n + 1)
Leaf count of result is larger than twice the leaf count of optimal. 398 vs. \(2 (55) = 110\).
Time = 0.13 (sec) , antiderivative size = 398, normalized size of antiderivative = 7.24 \[ \int (d x)^m \left (a x^n+b x^{1+n}+c x^{2+n}\right ) \, dx=\frac {c m^{2} x^{3} x^{n} e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )} + 2 \, c m n x^{3} x^{n} e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )} + c n^{2} x^{3} x^{n} e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )} + b m^{2} x^{2} x^{n} e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )} + 2 \, b m n x^{2} x^{n} e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )} + b n^{2} x^{2} x^{n} e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )} + 3 \, c m x^{3} x^{n} e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )} + 3 \, c n x^{3} x^{n} e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )} + a m^{2} x x^{n} e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )} + 2 \, a m n x x^{n} e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )} + a n^{2} x x^{n} e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )} + 4 \, b m x^{2} x^{n} e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )} + 4 \, b n x^{2} x^{n} e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )} + 2 \, c x^{3} x^{n} e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )} + 5 \, a m x x^{n} e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )} + 5 \, a n x x^{n} e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )} + 3 \, b x^{2} x^{n} e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )} + 6 \, a x x^{n} e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )}}{m^{3} + 3 \, m^{2} n + 3 \, m n^{2} + n^{3} + 6 \, m^{2} + 12 \, m n + 6 \, n^{2} + 11 \, m + 11 \, n + 6} \] Input:
integrate((d*x)^m*(a*x^n+b*x^(1+n)+c*x^(2+n)),x, algorithm="giac")
Output:
(c*m^2*x^3*x^n*e^(m*log(d) + m*log(x)) + 2*c*m*n*x^3*x^n*e^(m*log(d) + m*l og(x)) + c*n^2*x^3*x^n*e^(m*log(d) + m*log(x)) + b*m^2*x^2*x^n*e^(m*log(d) + m*log(x)) + 2*b*m*n*x^2*x^n*e^(m*log(d) + m*log(x)) + b*n^2*x^2*x^n*e^( m*log(d) + m*log(x)) + 3*c*m*x^3*x^n*e^(m*log(d) + m*log(x)) + 3*c*n*x^3*x ^n*e^(m*log(d) + m*log(x)) + a*m^2*x*x^n*e^(m*log(d) + m*log(x)) + 2*a*m*n *x*x^n*e^(m*log(d) + m*log(x)) + a*n^2*x*x^n*e^(m*log(d) + m*log(x)) + 4*b *m*x^2*x^n*e^(m*log(d) + m*log(x)) + 4*b*n*x^2*x^n*e^(m*log(d) + m*log(x)) + 2*c*x^3*x^n*e^(m*log(d) + m*log(x)) + 5*a*m*x*x^n*e^(m*log(d) + m*log(x )) + 5*a*n*x*x^n*e^(m*log(d) + m*log(x)) + 3*b*x^2*x^n*e^(m*log(d) + m*log (x)) + 6*a*x*x^n*e^(m*log(d) + m*log(x)))/(m^3 + 3*m^2*n + 3*m*n^2 + n^3 + 6*m^2 + 12*m*n + 6*n^2 + 11*m + 11*n + 6)
Time = 12.48 (sec) , antiderivative size = 218, normalized size of antiderivative = 3.96 \[ \int (d x)^m \left (a x^n+b x^{1+n}+c x^{2+n}\right ) \, dx=\frac {a\,x\,x^n\,{\left (d\,x\right )}^m\,\left (m^2+2\,m\,n+5\,m+n^2+5\,n+6\right )}{m^3+3\,m^2\,n+6\,m^2+3\,m\,n^2+12\,m\,n+11\,m+n^3+6\,n^2+11\,n+6}+\frac {b\,x\,x^{n+1}\,{\left (d\,x\right )}^m\,\left (m^2+2\,m\,n+4\,m+n^2+4\,n+3\right )}{m^3+3\,m^2\,n+6\,m^2+3\,m\,n^2+12\,m\,n+11\,m+n^3+6\,n^2+11\,n+6}+\frac {c\,x\,x^{n+2}\,{\left (d\,x\right )}^m\,\left (m^2+2\,m\,n+3\,m+n^2+3\,n+2\right )}{m^3+3\,m^2\,n+6\,m^2+3\,m\,n^2+12\,m\,n+11\,m+n^3+6\,n^2+11\,n+6} \] Input:
int((d*x)^m*(a*x^n + b*x^(n + 1) + c*x^(n + 2)),x)
Output:
(a*x*x^n*(d*x)^m*(5*m + 5*n + 2*m*n + m^2 + n^2 + 6))/(11*m + 11*n + 12*m* n + 3*m*n^2 + 3*m^2*n + 6*m^2 + m^3 + 6*n^2 + n^3 + 6) + (b*x*x^(n + 1)*(d *x)^m*(4*m + 4*n + 2*m*n + m^2 + n^2 + 3))/(11*m + 11*n + 12*m*n + 3*m*n^2 + 3*m^2*n + 6*m^2 + m^3 + 6*n^2 + n^3 + 6) + (c*x*x^(n + 2)*(d*x)^m*(3*m + 3*n + 2*m*n + m^2 + n^2 + 2))/(11*m + 11*n + 12*m*n + 3*m*n^2 + 3*m^2*n + 6*m^2 + m^3 + 6*n^2 + n^3 + 6)
Time = 0.20 (sec) , antiderivative size = 155, normalized size of antiderivative = 2.82 \[ \int (d x)^m \left (a x^n+b x^{1+n}+c x^{2+n}\right ) \, dx=\frac {x^{m +n} d^{m} x \left (c \,m^{2} x^{2}+2 c m n \,x^{2}+c \,n^{2} x^{2}+b \,m^{2} x +2 b m n x +b \,n^{2} x +3 c m \,x^{2}+3 c n \,x^{2}+a \,m^{2}+2 a m n +a \,n^{2}+4 b m x +4 b n x +2 c \,x^{2}+5 a m +5 a n +3 b x +6 a \right )}{m^{3}+3 m^{2} n +3 m \,n^{2}+n^{3}+6 m^{2}+12 m n +6 n^{2}+11 m +11 n +6} \] Input:
int((d*x)^m*(a*x^n+b*x^(1+n)+c*x^(2+n)),x)
Output:
(x**(m + n)*d**m*x*(a*m**2 + 2*a*m*n + 5*a*m + a*n**2 + 5*a*n + 6*a + b*m* *2*x + 2*b*m*n*x + 4*b*m*x + b*n**2*x + 4*b*n*x + 3*b*x + c*m**2*x**2 + 2* c*m*n*x**2 + 3*c*m*x**2 + c*n**2*x**2 + 3*c*n*x**2 + 2*c*x**2))/(m**3 + 3* m**2*n + 6*m**2 + 3*m*n**2 + 12*m*n + 11*m + n**3 + 6*n**2 + 11*n + 6)