Integrand size = 23, antiderivative size = 69 \[ \int (c x)^m \left (A+B x^2+C x^4+D x^6\right ) \, dx=\frac {A (c x)^{1+m}}{c (1+m)}+\frac {B (c x)^{3+m}}{c^3 (3+m)}+\frac {C (c x)^{5+m}}{c^5 (5+m)}+\frac {D (c x)^{7+m}}{c^7 (7+m)} \] Output:
A*(c*x)^(1+m)/c/(1+m)+B*(c*x)^(3+m)/c^3/(3+m)+C*(c*x)^(5+m)/c^5/(5+m)+D*(c *x)^(7+m)/c^7/(7+m)
Time = 0.07 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.65 \[ \int (c x)^m \left (A+B x^2+C x^4+D x^6\right ) \, dx=x (c x)^m \left (\frac {A}{1+m}+\frac {B x^2}{3+m}+\frac {C x^4}{5+m}+\frac {D x^6}{7+m}\right ) \] Input:
Integrate[(c*x)^m*(A + B*x^2 + C*x^4 + D*x^6),x]
Output:
x*(c*x)^m*(A/(1 + m) + (B*x^2)/(3 + m) + (C*x^4)/(5 + m) + (D*x^6)/(7 + m) )
Time = 0.26 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, 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.
\(\displaystyle \int (c x)^m \left (A+B x^2+C x^4+D x^6\right ) \, dx\) |
\(\Big \downarrow \) 2010 |
\(\displaystyle \int \left (A (c x)^m+\frac {B (c x)^{m+2}}{c^2}+\frac {D (c x)^{m+6}}{c^6}+\frac {C (c x)^{m+4}}{c^4}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {A (c x)^{m+1}}{c (m+1)}+\frac {B (c x)^{m+3}}{c^3 (m+3)}+\frac {D (c x)^{m+7}}{c^7 (m+7)}+\frac {C (c x)^{m+5}}{c^5 (m+5)}\) |
Input:
Int[(c*x)^m*(A + B*x^2 + C*x^4 + D*x^6),x]
Output:
(A*(c*x)^(1 + m))/(c*(1 + m)) + (B*(c*x)^(3 + m))/(c^3*(3 + m)) + (C*(c*x) ^(5 + m))/(c^5*(5 + m)) + (D*(c*x)^(7 + m))/(c^7*(7 + m))
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]]
Time = 0.07 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.99
method | result | size |
norman | \(\frac {A x \,{\mathrm e}^{m \ln \left (c x \right )}}{1+m}+\frac {B \,x^{3} {\mathrm e}^{m \ln \left (c x \right )}}{3+m}+\frac {C \,x^{5} {\mathrm e}^{m \ln \left (c x \right )}}{5+m}+\frac {D x^{7} {\mathrm e}^{m \ln \left (c x \right )}}{7+m}\) | \(68\) |
gosper | \(\frac {x \left (D m^{3} x^{6}+9 D m^{2} x^{6}+C \,m^{3} x^{4}+23 D m \,x^{6}+11 C \,m^{2} x^{4}+15 D x^{6}+B \,m^{3} x^{2}+31 C m \,x^{4}+13 B \,m^{2} x^{2}+21 C \,x^{4}+A \,m^{3}+47 B m \,x^{2}+15 A \,m^{2}+35 x^{2} B +71 A m +105 A \right ) \left (c x \right )^{m}}{\left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\) | \(137\) |
orering | \(\frac {x \left (D m^{3} x^{6}+9 D m^{2} x^{6}+C \,m^{3} x^{4}+23 D m \,x^{6}+11 C \,m^{2} x^{4}+15 D x^{6}+B \,m^{3} x^{2}+31 C m \,x^{4}+13 B \,m^{2} x^{2}+21 C \,x^{4}+A \,m^{3}+47 B m \,x^{2}+15 A \,m^{2}+35 x^{2} B +71 A m +105 A \right ) \left (c x \right )^{m}}{\left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\) | \(137\) |
parallelrisch | \(\frac {D x^{7} \left (c x \right )^{m} m^{3}+9 D x^{7} \left (c x \right )^{m} m^{2}+C \,x^{5} \left (c x \right )^{m} m^{3}+23 D x^{7} \left (c x \right )^{m} m +11 C \,x^{5} \left (c x \right )^{m} m^{2}+15 D x^{7} \left (c x \right )^{m}+B \,x^{3} \left (c x \right )^{m} m^{3}+31 C \,x^{5} \left (c x \right )^{m} m +13 B \,x^{3} \left (c x \right )^{m} m^{2}+21 C \,x^{5} \left (c x \right )^{m}+A x \left (c x \right )^{m} m^{3}+47 B \,x^{3} \left (c x \right )^{m} m +15 A x \left (c x \right )^{m} m^{2}+35 B \,x^{3} \left (c x \right )^{m}+71 A x \left (c x \right )^{m} m +105 A x \left (c x \right )^{m}}{\left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\) | \(215\) |
Input:
int((c*x)^m*(D*x^6+C*x^4+B*x^2+A),x,method=_RETURNVERBOSE)
Output:
A/(1+m)*x*exp(m*ln(c*x))+B/(3+m)*x^3*exp(m*ln(c*x))+C/(5+m)*x^5*exp(m*ln(c *x))+D/(7+m)*x^7*exp(m*ln(c*x))
Time = 0.08 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.70 \[ \int (c x)^m \left (A+B x^2+C x^4+D x^6\right ) \, dx=\frac {{\left ({\left (D m^{3} + 9 \, D m^{2} + 23 \, D m + 15 \, D\right )} x^{7} + {\left (C m^{3} + 11 \, C m^{2} + 31 \, C m + 21 \, C\right )} x^{5} + {\left (B m^{3} + 13 \, B m^{2} + 47 \, B m + 35 \, B\right )} x^{3} + {\left (A m^{3} + 15 \, A m^{2} + 71 \, A m + 105 \, A\right )} x\right )} \left (c x\right )^{m}}{m^{4} + 16 \, m^{3} + 86 \, m^{2} + 176 \, m + 105} \] Input:
integrate((c*x)^m*(D*x^6+C*x^4+B*x^2+A),x, algorithm="fricas")
Output:
((D*m^3 + 9*D*m^2 + 23*D*m + 15*D)*x^7 + (C*m^3 + 11*C*m^2 + 31*C*m + 21*C )*x^5 + (B*m^3 + 13*B*m^2 + 47*B*m + 35*B)*x^3 + (A*m^3 + 15*A*m^2 + 71*A* m + 105*A)*x)*(c*x)^m/(m^4 + 16*m^3 + 86*m^2 + 176*m + 105)
Leaf count of result is larger than twice the leaf count of optimal. 624 vs. \(2 (58) = 116\).
Time = 0.45 (sec) , antiderivative size = 624, normalized size of antiderivative = 9.04 \[ \int (c x)^m \left (A+B x^2+C x^4+D x^6\right ) \, dx=\begin {cases} \frac {- \frac {A}{6 x^{6}} - \frac {B}{4 x^{4}} - \frac {C}{2 x^{2}} + D \log {\left (x \right )}}{c^{7}} & \text {for}\: m = -7 \\\frac {- \frac {A}{4 x^{4}} - \frac {B}{2 x^{2}} + C \log {\left (x \right )} + \frac {D x^{2}}{2}}{c^{5}} & \text {for}\: m = -5 \\\frac {- \frac {A}{2 x^{2}} + B \log {\left (x \right )} + \frac {C x^{2}}{2} + \frac {D x^{4}}{4}}{c^{3}} & \text {for}\: m = -3 \\\frac {A \log {\left (x \right )} + \frac {B x^{2}}{2} + \frac {C x^{4}}{4} + \frac {D x^{6}}{6}}{c} & \text {for}\: m = -1 \\\frac {A m^{3} x \left (c x\right )^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {15 A m^{2} x \left (c x\right )^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {71 A m x \left (c x\right )^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {105 A x \left (c x\right )^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {B m^{3} x^{3} \left (c x\right )^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {13 B m^{2} x^{3} \left (c x\right )^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {47 B m x^{3} \left (c x\right )^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {35 B x^{3} \left (c x\right )^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {C m^{3} x^{5} \left (c x\right )^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {11 C m^{2} x^{5} \left (c x\right )^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {31 C m x^{5} \left (c x\right )^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {21 C x^{5} \left (c x\right )^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {D m^{3} x^{7} \left (c x\right )^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {9 D m^{2} x^{7} \left (c x\right )^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {23 D m x^{7} \left (c x\right )^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {15 D x^{7} \left (c x\right )^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} & \text {otherwise} \end {cases} \] Input:
integrate((c*x)**m*(D*x**6+C*x**4+B*x**2+A),x)
Output:
Piecewise(((-A/(6*x**6) - B/(4*x**4) - C/(2*x**2) + D*log(x))/c**7, Eq(m, -7)), ((-A/(4*x**4) - B/(2*x**2) + C*log(x) + D*x**2/2)/c**5, Eq(m, -5)), ((-A/(2*x**2) + B*log(x) + C*x**2/2 + D*x**4/4)/c**3, Eq(m, -3)), ((A*log( x) + B*x**2/2 + C*x**4/4 + D*x**6/6)/c, Eq(m, -1)), (A*m**3*x*(c*x)**m/(m* *4 + 16*m**3 + 86*m**2 + 176*m + 105) + 15*A*m**2*x*(c*x)**m/(m**4 + 16*m* *3 + 86*m**2 + 176*m + 105) + 71*A*m*x*(c*x)**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 105*A*x*(c*x)**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + B*m**3*x**3*(c*x)**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 13*B*m* *2*x**3*(c*x)**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 47*B*m*x**3*(c *x)**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 35*B*x**3*(c*x)**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + C*m**3*x**5*(c*x)**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 11*C*m**2*x**5*(c*x)**m/(m**4 + 16*m**3 + 86*m **2 + 176*m + 105) + 31*C*m*x**5*(c*x)**m/(m**4 + 16*m**3 + 86*m**2 + 176* m + 105) + 21*C*x**5*(c*x)**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + D *m**3*x**7*(c*x)**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 9*D*m**2*x* *7*(c*x)**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 23*D*m*x**7*(c*x)** m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 15*D*x**7*(c*x)**m/(m**4 + 16 *m**3 + 86*m**2 + 176*m + 105), True))
Time = 0.04 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.96 \[ \int (c x)^m \left (A+B x^2+C x^4+D x^6\right ) \, dx=\frac {D c^{m} x^{7} x^{m}}{m + 7} + \frac {C c^{m} x^{5} x^{m}}{m + 5} + \frac {B c^{m} x^{3} x^{m}}{m + 3} + \frac {\left (c x\right )^{m + 1} A}{c {\left (m + 1\right )}} \] Input:
integrate((c*x)^m*(D*x^6+C*x^4+B*x^2+A),x, algorithm="maxima")
Output:
D*c^m*x^7*x^m/(m + 7) + C*c^m*x^5*x^m/(m + 5) + B*c^m*x^3*x^m/(m + 3) + (c *x)^(m + 1)*A/(c*(m + 1))
Leaf count of result is larger than twice the leaf count of optimal. 214 vs. \(2 (69) = 138\).
Time = 0.13 (sec) , antiderivative size = 214, normalized size of antiderivative = 3.10 \[ \int (c x)^m \left (A+B x^2+C x^4+D x^6\right ) \, dx=\frac {\left (c x\right )^{m} D m^{3} x^{7} + 9 \, \left (c x\right )^{m} D m^{2} x^{7} + \left (c x\right )^{m} C m^{3} x^{5} + 23 \, \left (c x\right )^{m} D m x^{7} + 11 \, \left (c x\right )^{m} C m^{2} x^{5} + 15 \, \left (c x\right )^{m} D x^{7} + \left (c x\right )^{m} B m^{3} x^{3} + 31 \, \left (c x\right )^{m} C m x^{5} + 13 \, \left (c x\right )^{m} B m^{2} x^{3} + 21 \, \left (c x\right )^{m} C x^{5} + \left (c x\right )^{m} A m^{3} x + 47 \, \left (c x\right )^{m} B m x^{3} + 15 \, \left (c x\right )^{m} A m^{2} x + 35 \, \left (c x\right )^{m} B x^{3} + 71 \, \left (c x\right )^{m} A m x + 105 \, \left (c x\right )^{m} A x}{m^{4} + 16 \, m^{3} + 86 \, m^{2} + 176 \, m + 105} \] Input:
integrate((c*x)^m*(D*x^6+C*x^4+B*x^2+A),x, algorithm="giac")
Output:
((c*x)^m*D*m^3*x^7 + 9*(c*x)^m*D*m^2*x^7 + (c*x)^m*C*m^3*x^5 + 23*(c*x)^m* D*m*x^7 + 11*(c*x)^m*C*m^2*x^5 + 15*(c*x)^m*D*x^7 + (c*x)^m*B*m^3*x^3 + 31 *(c*x)^m*C*m*x^5 + 13*(c*x)^m*B*m^2*x^3 + 21*(c*x)^m*C*x^5 + (c*x)^m*A*m^3 *x + 47*(c*x)^m*B*m*x^3 + 15*(c*x)^m*A*m^2*x + 35*(c*x)^m*B*x^3 + 71*(c*x) ^m*A*m*x + 105*(c*x)^m*A*x)/(m^4 + 16*m^3 + 86*m^2 + 176*m + 105)
Timed out. \[ \int (c x)^m \left (A+B x^2+C x^4+D x^6\right ) \, dx=\int {\left (c\,x\right )}^m\,\left (A+B\,x^2+C\,x^4+x^6\,D\right ) \,d x \] Input:
int((c*x)^m*(A + B*x^2 + C*x^4 + x^6*D),x)
Output:
int((c*x)^m*(A + B*x^2 + C*x^4 + x^6*D), x)
Time = 0.15 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.99 \[ \int (c x)^m \left (A+B x^2+C x^4+D x^6\right ) \, dx=\frac {x^{m} c^{m} x \left (d \,m^{3} x^{6}+9 d \,m^{2} x^{6}+c \,m^{3} x^{4}+23 d m \,x^{6}+11 c \,m^{2} x^{4}+15 d \,x^{6}+b \,m^{3} x^{2}+31 c m \,x^{4}+13 b \,m^{2} x^{2}+21 c \,x^{4}+a \,m^{3}+47 b m \,x^{2}+15 a \,m^{2}+35 b \,x^{2}+71 a m +105 a \right )}{m^{4}+16 m^{3}+86 m^{2}+176 m +105} \] Input:
int((c*x)^m*(D*x^6+C*x^4+B*x^2+A),x)
Output:
(x**m*c**m*x*(a*m**3 + 15*a*m**2 + 71*a*m + 105*a + b*m**3*x**2 + 13*b*m** 2*x**2 + 47*b*m*x**2 + 35*b*x**2 + c*m**3*x**4 + 11*c*m**2*x**4 + 31*c*m*x **4 + 21*c*x**4 + d*m**3*x**6 + 9*d*m**2*x**6 + 23*d*m*x**6 + 15*d*x**6))/ (m**4 + 16*m**3 + 86*m**2 + 176*m + 105)