Integrand size = 22, antiderivative size = 1 \[ \int (d x)^m \left (a x^q+b x^r+c x^s\right ) \, dx=0 \] Output:
0
Result contains higher order function than in optimal. Order 3 vs. order 1 in optimal.
Time = 0.04 (sec) , antiderivative size = 41, normalized size of antiderivative = 41.00 \[ \int (d x)^m \left (a x^q+b x^r+c x^s\right ) \, dx=x (d x)^m \left (\frac {a x^q}{1+m+q}+\frac {b x^r}{1+m+r}+\frac {c x^s}{1+m+s}\right ) \] Input:
Integrate[(d*x)^m*(a*x^q + b*x^r + c*x^s),x]
Output:
x*(d*x)^m*((a*x^q)/(1 + m + q) + (b*x^r)/(1 + m + r) + (c*x^s)/(1 + m + s) )
Result contains higher order function than in optimal. Order 3 vs. order 1 in optimal.
Time = 0.22 (sec) , antiderivative size = 55, normalized size of antiderivative = 55.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, 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.
\(\displaystyle \int (d x)^m \left (a x^q+b x^r+c x^s\right ) \, dx\) |
\(\Big \downarrow \) 2010 |
\(\displaystyle \int \left (a x^q (d x)^m+b x^r (d x)^m+c x^s (d x)^m\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a x^{q+1} (d x)^m}{m+q+1}+\frac {b x^{r+1} (d x)^m}{m+r+1}+\frac {c x^{s+1} (d x)^m}{m+s+1}\) |
Input:
Int[(d*x)^m*(a*x^q + b*x^r + c*x^s),x]
Output:
(a*x^(1 + q)*(d*x)^m)/(1 + m + q) + (b*x^(1 + r)*(d*x)^m)/(1 + m + r) + (c *x^(1 + s)*(d*x)^m)/(1 + m + s)
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]]
Result contains higher order function than in optimal. Order 9 vs. order 1.
Time = 0.85 (sec) , antiderivative size = 227, normalized size of antiderivative = 227.00
method | result | size |
risch | \(\frac {x \left (a \,m^{2} x^{q}+a m r \,x^{q}+a m s \,x^{q}+a r s \,x^{q}+b \,m^{2} x^{r}+b m q \,x^{r}+b m s \,x^{r}+b q s \,x^{r}+c \,m^{2} x^{s}+c m q \,x^{s}+c m r \,x^{s}+c q r \,x^{s}+2 x^{q} a m +a \,x^{q} r +a \,x^{q} s +2 x^{r} b m +b q \,x^{r}+b \,x^{r} s +2 x^{s} c m +c q \,x^{s}+c r \,x^{s}+a \,x^{q}+b \,x^{r}+c \,x^{s}\right ) d^{m} x^{m} {\mathrm e}^{\frac {i \operatorname {csgn}\left (i d x \right ) \pi m \left (\operatorname {csgn}\left (i d x \right )-\operatorname {csgn}\left (i x \right )\right ) \left (-\operatorname {csgn}\left (i d x \right )+\operatorname {csgn}\left (i d \right )\right )}{2}}}{\left (1+m +q \right ) \left (1+m +r \right ) \left (1+s +m \right )}\) | \(227\) |
parallelrisch | \(\frac {x \,x^{q} \left (d x \right )^{m} a m r +x \,x^{q} \left (d x \right )^{m} a m s +x \,x^{q} \left (d x \right )^{m} a r s +x \,x^{r} \left (d x \right )^{m} b m q +x \,x^{r} \left (d x \right )^{m} b m s +x \,x^{r} \left (d x \right )^{m} b q s +x \,x^{s} \left (d x \right )^{m} c m q +x \,x^{s} \left (d x \right )^{m} c m r +x \,x^{s} \left (d x \right )^{m} c q r +x \,x^{q} \left (d x \right )^{m} a +x \,x^{r} \left (d x \right )^{m} b +x \,x^{s} \left (d x \right )^{m} c +x \,x^{q} \left (d x \right )^{m} a \,m^{2}+x \,x^{r} \left (d x \right )^{m} b \,m^{2}+x \,x^{s} \left (d x \right )^{m} c \,m^{2}+2 x \,x^{q} \left (d x \right )^{m} a m +x \,x^{q} \left (d x \right )^{m} a r +x \,x^{q} \left (d x \right )^{m} a s +2 x \,x^{r} \left (d x \right )^{m} b m +x \,x^{r} \left (d x \right )^{m} b q +x \,x^{r} \left (d x \right )^{m} b s +2 x \,x^{s} \left (d x \right )^{m} c m +x \,x^{s} \left (d x \right )^{m} c q +x \,x^{s} \left (d x \right )^{m} c r}{\left (1+m +q \right ) \left (1+m +r \right ) \left (1+s +m \right )}\) | \(324\) |
orering | \(\frac {x \left (3 m^{2}+2 q m +2 m r +2 m s +q r +q s +s r +3 m +q +r +s +1\right ) \left (d x \right )^{m} \left (a \,x^{q}+b \,x^{r}+c \,x^{s}\right )}{m^{3}+m^{2} q +m^{2} r +m^{2} s +m q r +m q s +m r s +q r s +3 m^{2}+2 q m +2 m r +2 m s +q r +q s +s r +3 m +q +r +s +1}-\frac {x^{2} \left (s +r +3 m +q \right ) \left (\frac {\left (d x \right )^{m} m \left (a \,x^{q}+b \,x^{r}+c \,x^{s}\right )}{x}+\left (d x \right )^{m} \left (\frac {a \,x^{q} q}{x}+\frac {b \,x^{r} r}{x}+\frac {c \,x^{s} s}{x}\right )\right )}{m^{3}+m^{2} q +m^{2} r +m^{2} s +m q r +m q s +m r s +q r s +3 m^{2}+2 q m +2 m r +2 m s +q r +q s +s r +3 m +q +r +s +1}+\frac {x^{3} \left (\frac {\left (d x \right )^{m} m^{2} \left (a \,x^{q}+b \,x^{r}+c \,x^{s}\right )}{x^{2}}-\frac {\left (d x \right )^{m} m \left (a \,x^{q}+b \,x^{r}+c \,x^{s}\right )}{x^{2}}+\frac {2 \left (d x \right )^{m} m \left (\frac {a \,x^{q} q}{x}+\frac {b \,x^{r} r}{x}+\frac {c \,x^{s} s}{x}\right )}{x}+\left (d x \right )^{m} \left (\frac {a \,x^{q} q^{2}}{x^{2}}-\frac {a \,x^{q} q}{x^{2}}+\frac {b \,x^{r} r^{2}}{x^{2}}-\frac {b \,x^{r} r}{x^{2}}+\frac {c \,x^{s} s^{2}}{x^{2}}-\frac {c \,x^{s} s}{x^{2}}\right )\right )}{m^{3}+m^{2} q +m^{2} r +m^{2} s +m q r +m q s +m r s +q r s +3 m^{2}+2 q m +2 m r +2 m s +q r +q s +s r +3 m +q +r +s +1}\) | \(511\) |
Input:
int((d*x)^m*(a*x^q+b*x^r+c*x^s),x,method=_RETURNVERBOSE)
Output:
x*(a*m^2*x^q+a*m*r*x^q+a*m*s*x^q+a*r*s*x^q+b*m^2*x^r+b*m*q*x^r+b*m*s*x^r+b *q*s*x^r+c*m^2*x^s+c*m*q*x^s+c*m*r*x^s+c*q*r*x^s+2*x^q*a*m+a*x^q*r+a*x^q*s +2*x^r*b*m+b*q*x^r+b*x^r*s+2*x^s*c*m+c*q*x^s+c*r*x^s+a*x^q+b*x^r+c*x^s)/(1 +m+q)/(1+m+r)/(1+s+m)*d^m*x^m*exp(1/2*I*csgn(I*d*x)*Pi*m*(csgn(I*d*x)-csgn (I*x))*(-csgn(I*d*x)+csgn(I*d)))
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.09 (sec) , antiderivative size = 192, normalized size of antiderivative = 192.00 \[ \int (d x)^m \left (a x^q+b x^r+c x^s\right ) \, dx=\frac {{\left (a m^{2} + 2 \, a m + {\left (a m + a\right )} r + {\left (a m + a r + a\right )} s + a\right )} x x^{q} e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )} + {\left (b m^{2} + 2 \, b m + {\left (b m + b\right )} q + {\left (b m + b q + b\right )} s + b\right )} x x^{r} e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )} + {\left (c m^{2} + 2 \, c m + {\left (c m + c\right )} q + {\left (c m + c q + c\right )} r + c\right )} x x^{s} e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )}}{m^{3} + 3 \, m^{2} + {\left (m^{2} + 2 \, m + 1\right )} q + {\left (m^{2} + {\left (m + 1\right )} q + 2 \, m + 1\right )} r + {\left (m^{2} + {\left (m + 1\right )} q + {\left (m + q + 1\right )} r + 2 \, m + 1\right )} s + 3 \, m + 1} \] Input:
integrate((d*x)^m*(a*x^q+b*x^r+c*x^s),x, algorithm="fricas")
Output:
((a*m^2 + 2*a*m + (a*m + a)*r + (a*m + a*r + a)*s + a)*x*x^q*e^(m*log(d) + m*log(x)) + (b*m^2 + 2*b*m + (b*m + b)*q + (b*m + b*q + b)*s + b)*x*x^r*e ^(m*log(d) + m*log(x)) + (c*m^2 + 2*c*m + (c*m + c)*q + (c*m + c*q + c)*r + c)*x*x^s*e^(m*log(d) + m*log(x)))/(m^3 + 3*m^2 + (m^2 + 2*m + 1)*q + (m^ 2 + (m + 1)*q + 2*m + 1)*r + (m^2 + (m + 1)*q + (m + q + 1)*r + 2*m + 1)*s + 3*m + 1)
Leaf count of result is larger than twice the leaf count of optimal. 2611 vs. \(2 (0) = 0\).
Time = 7.98 (sec) , antiderivative size = 2611, normalized size of antiderivative = 2611.00 \[ \int (d x)^m \left (a x^q+b x^r+c x^s\right ) \, dx=\text {Too large to display} \] Input:
integrate((d*x)**m*(a*x**q+b*x**r+c*x**s),x)
Output:
Piecewise((a*x*x**s*(d*x)**(-s - 1)*log(x) + b*x*x**s*(d*x)**(-s - 1)*log( x) + c*x*x**s*(d*x)**(-s - 1)*log(x), Eq(q, s) & Eq(r, s) & Eq(m, -s - 1)) , (a*x*x**q*(d*x)**(-q - 1)*log(x) + b*Piecewise((x*x**r*(d*x)**(-q - 1)/( -q + r), Ne(q - r, 0)), (x*x**r*(d*x)**(-q - 1)*log(x), True)) + c*Piecewi se((x*x**s*(d*x)**(-q - 1)/(-q + s), Ne(q - s, 0)), (x*x**s*(d*x)**(-q - 1 )*log(x), True)), Eq(m, -q - 1)), (a*Piecewise((x*x**q*(d*x)**(-r - 1)/(q - r), Ne(q - r, 0)), (x*x**q*(d*x)**(-r - 1)*log(x), True)) + b*x*x**r*(d* x)**(-r - 1)*log(x) + c*Piecewise((x*x**s*(d*x)**(-r - 1)/(-r + s), Ne(r - s, 0)), (x*x**s*(d*x)**(-r - 1)*log(x), True)), Eq(m, -r - 1)), (a*Piecew ise((x*x**q*(d*x)**(-s - 1)/(q - s), Ne(q - s, 0)), (x*x**q*(d*x)**(-s - 1 )*log(x), True)) + b*Piecewise((x*x**r*(d*x)**(-s - 1)/(r - s), Ne(r - s, 0)), (x*x**r*(d*x)**(-s - 1)*log(x), True)) + c*x*x**s*(d*x)**(-s - 1)*log (x), Eq(m, -s - 1)), (a*m**2*x*x**q*(d*x)**m/(m**3 + m**2*q + m**2*r + m** 2*s + 3*m**2 + m*q*r + m*q*s + 2*m*q + m*r*s + 2*m*r + 2*m*s + 3*m + q*r*s + q*r + q*s + q + r*s + r + s + 1) + a*m*r*x*x**q*(d*x)**m/(m**3 + m**2*q + m**2*r + m**2*s + 3*m**2 + m*q*r + m*q*s + 2*m*q + m*r*s + 2*m*r + 2*m* s + 3*m + q*r*s + q*r + q*s + q + r*s + r + s + 1) + a*m*s*x*x**q*(d*x)**m /(m**3 + m**2*q + m**2*r + m**2*s + 3*m**2 + m*q*r + m*q*s + 2*m*q + m*r*s + 2*m*r + 2*m*s + 3*m + q*r*s + q*r + q*s + q + r*s + r + s + 1) + 2*a*m* x*x**q*(d*x)**m/(m**3 + m**2*q + m**2*r + m**2*s + 3*m**2 + m*q*r + m*q...
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.04 (sec) , antiderivative size = 67, normalized size of antiderivative = 67.00 \[ \int (d x)^m \left (a x^q+b x^r+c x^s\right ) \, dx=\frac {a d^{m} x e^{\left (m \log \left (x\right ) + q \log \left (x\right )\right )}}{m + q + 1} + \frac {b d^{m} x e^{\left (m \log \left (x\right ) + r \log \left (x\right )\right )}}{m + r + 1} + \frac {c d^{m} x e^{\left (m \log \left (x\right ) + s \log \left (x\right )\right )}}{m + s + 1} \] Input:
integrate((d*x)^m*(a*x^q+b*x^r+c*x^s),x, algorithm="maxima")
Output:
a*d^m*x*e^(m*log(x) + q*log(x))/(m + q + 1) + b*d^m*x*e^(m*log(x) + r*log( x))/(m + r + 1) + c*d^m*x*e^(m*log(x) + s*log(x))/(m + s + 1)
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.13 (sec) , antiderivative size = 729, normalized size of antiderivative = 729.00 \[ \int (d x)^m \left (a x^q+b x^r+c x^s\right ) \, dx=\text {Too large to display} \] Input:
integrate((d*x)^m*(a*x^q+b*x^r+c*x^s),x, algorithm="giac")
Output:
(a*m^2*x*x^q*e^(m*log(d) + m*log(x)) + a*m*r*x*x^q*e^(m*log(d) + m*log(x)) + a*m*s*x*x^q*e^(m*log(d) + m*log(x)) + a*r*s*x*x^q*e^(m*log(d) + m*log(x )) + b*m^2*x*x^r*e^(m*log(d) + m*log(x)) + b*m*q*x*x^r*e^(m*log(d) + m*log (x)) + b*m*s*x*x^r*e^(m*log(d) + m*log(x)) + b*q*s*x*x^r*e^(m*log(d) + m*l og(x)) + c*m^2*x*x^s*e^(m*log(d) + m*log(x)) + c*m*q*x*x^s*e^(m*log(d) + m *log(x)) + c*m*r*x*x^s*e^(m*log(d) + m*log(x)) + c*q*r*x*x^s*e^(m*log(d) + m*log(x)) + b*m^2*x*e^(m*log(d) + m*log(x)) + c*m^2*x*e^(m*log(d) + m*log (x)) + b*m*q*x*e^(m*log(d) + m*log(x)) + c*m*q*x*e^(m*log(d) + m*log(x)) + c*m*r*x*e^(m*log(d) + m*log(x)) + c*q*r*x*e^(m*log(d) + m*log(x)) + b*m*s *x*e^(m*log(d) + m*log(x)) + b*q*s*x*e^(m*log(d) + m*log(x)) + 2*a*m*x*x^q *e^(m*log(d) + m*log(x)) + a*r*x*x^q*e^(m*log(d) + m*log(x)) + a*s*x*x^q*e ^(m*log(d) + m*log(x)) + 2*b*m*x*x^r*e^(m*log(d) + m*log(x)) + b*q*x*x^r*e ^(m*log(d) + m*log(x)) + b*s*x*x^r*e^(m*log(d) + m*log(x)) + 2*c*m*x*x^s*e ^(m*log(d) + m*log(x)) + c*q*x*x^s*e^(m*log(d) + m*log(x)) + c*r*x*x^s*e^( m*log(d) + m*log(x)) + 2*b*m*x*e^(m*log(d) + m*log(x)) + 2*c*m*x*e^(m*log( d) + m*log(x)) + b*q*x*e^(m*log(d) + m*log(x)) + c*q*x*e^(m*log(d) + m*log (x)) + c*r*x*e^(m*log(d) + m*log(x)) + b*s*x*e^(m*log(d) + m*log(x)) + a*x *x^q*e^(m*log(d) + m*log(x)) + b*x*x^r*e^(m*log(d) + m*log(x)) + c*x*x^s*e ^(m*log(d) + m*log(x)) + b*x*e^(m*log(d) + m*log(x)) + c*x*e^(m*log(d) + m *log(x)))/(m^3 + m^2*q + m^2*r + m*q*r + m^2*s + m*q*s + m*r*s + q*r*s ...
Time = 12.13 (sec) , antiderivative size = 46, normalized size of antiderivative = 46.00 \[ \int (d x)^m \left (a x^q+b x^r+c x^s\right ) \, dx={\left (d\,x\right )}^m\,\left (\frac {a\,x^{q+1}}{m+q+1}+\frac {b\,x^{r+1}}{m+r+1}+\frac {c\,x^{s+1}}{m+s+1}\right ) \] Input:
int((d*x)^m*(a*x^q + b*x^r + c*x^s),x)
Output:
(d*x)^m*((a*x^(q + 1))/(m + q + 1) + (b*x^(r + 1))/(m + r + 1) + (c*x^(s + 1))/(m + s + 1))
Time = 0.21 (sec) , antiderivative size = 238, normalized size of antiderivative = 238.00 \[ \int (d x)^m \left (a x^q+b x^r+c x^s\right ) \, dx=\frac {x^{m} d^{m} x \left (x^{q} a \,m^{2}+x^{q} a m r +x^{q} a m s +2 x^{q} a m +x^{q} a r s +x^{q} a r +x^{q} a s +x^{q} a +x^{r} b \,m^{2}+x^{r} b m q +x^{r} b m s +2 x^{r} b m +x^{r} b q s +x^{r} b q +x^{r} b s +x^{r} b +x^{s} c \,m^{2}+x^{s} c m q +x^{s} c m r +2 x^{s} c m +x^{s} c q r +x^{s} c q +x^{s} c r +x^{s} c \right )}{m^{3}+m^{2} q +m^{2} r +m^{2} s +m q r +m q s +m r s +q r s +3 m^{2}+2 m q +2 m r +2 m s +q r +q s +r s +3 m +q +r +s +1} \] Input:
int((d*x)^m*(a*x^q+b*x^r+c*x^s),x)
Output:
(x**m*d**m*x*(x**q*a*m**2 + x**q*a*m*r + x**q*a*m*s + 2*x**q*a*m + x**q*a* r*s + x**q*a*r + x**q*a*s + x**q*a + x**r*b*m**2 + x**r*b*m*q + x**r*b*m*s + 2*x**r*b*m + x**r*b*q*s + x**r*b*q + x**r*b*s + x**r*b + x**s*c*m**2 + x**s*c*m*q + x**s*c*m*r + 2*x**s*c*m + x**s*c*q*r + x**s*c*q + x**s*c*r + x**s*c))/(m**3 + m**2*q + m**2*r + m**2*s + 3*m**2 + m*q*r + m*q*s + 2*m*q + m*r*s + 2*m*r + 2*m*s + 3*m + q*r*s + q*r + q*s + q + r*s + r + s + 1)