Integrand size = 33, antiderivative size = 52 \[ \int (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=-\frac {\left (c d^2-a e^2\right ) (d+e x)^{2+m}}{e^2 (2+m)}+\frac {c d (d+e x)^{3+m}}{e^2 (3+m)} \] Output:
-(-a*e^2+c*d^2)*(e*x+d)^(2+m)/e^2/(2+m)+c*d*(e*x+d)^(3+m)/e^2/(3+m)
Time = 0.07 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.87 \[ \int (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=\frac {(d+e x)^{2+m} \left (a e^2 (3+m)+c d (-d+e (2+m) x)\right )}{e^2 (2+m) (3+m)} \] Input:
Integrate[(d + e*x)^m*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2),x]
Output:
((d + e*x)^(2 + m)*(a*e^2*(3 + m) + c*d*(-d + e*(2 + m)*x)))/(e^2*(2 + m)* (3 + m))
Time = 0.21 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {1121, 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.
\(\displaystyle \int (d+e x)^m \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right ) \, dx\) |
\(\Big \downarrow \) 1121 |
\(\displaystyle \int \left (\frac {\left (a e^2-c d^2\right ) (d+e x)^{m+1}}{e}+\frac {c d (d+e x)^{m+2}}{e}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {c d (d+e x)^{m+3}}{e^2 (m+3)}-\frac {\left (c d^2-a e^2\right ) (d+e x)^{m+2}}{e^2 (m+2)}\) |
Input:
Int[(d + e*x)^m*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2),x]
Output:
-(((c*d^2 - a*e^2)*(d + e*x)^(2 + m))/(e^2*(2 + m))) + (c*d*(d + e*x)^(3 + m))/(e^2*(3 + m))
Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_ Symbol] :> Int[ExpandIntegrand[(d + e*x)^(m + p)*(a/d + (c/e)*x)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && (Int egerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && LtQ[c, 0]))
Time = 0.51 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.06
method | result | size |
gosper | \(\frac {\left (e x +d \right )^{2+m} \left (c d e m x +a \,e^{2} m +2 c d x e +3 a \,e^{2}-c \,d^{2}\right )}{e^{2} \left (m^{2}+5 m +6\right )}\) | \(55\) |
orering | \(\frac {\left (c d e m x +a \,e^{2} m +2 c d x e +3 a \,e^{2}-c \,d^{2}\right ) \left (e x +d \right ) \left (e x +d \right )^{m} \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}{e^{2} \left (m^{2}+5 m +6\right ) \left (c d x +a e \right )}\) | \(93\) |
risch | \(\frac {\left (c d \,e^{3} m \,x^{3}+a \,e^{4} m \,x^{2}+2 c \,d^{2} e^{2} m \,x^{2}+2 c d \,e^{3} x^{3}+2 a d \,e^{3} m x +3 a \,e^{4} x^{2}+c \,d^{3} e m x +3 c \,d^{2} e^{2} x^{2}+a \,d^{2} e^{2} m +6 a d \,e^{3} x +3 a \,d^{2} e^{2}-c \,d^{4}\right ) \left (e x +d \right )^{m}}{\left (2+m \right ) \left (3+m \right ) e^{2}}\) | \(135\) |
norman | \(\frac {\left (a \,e^{2} m +2 c \,d^{2} m +3 a \,e^{2}+3 c \,d^{2}\right ) x^{2} {\mathrm e}^{m \ln \left (e x +d \right )}}{m^{2}+5 m +6}+\frac {d^{2} \left (a \,e^{2} m +3 a \,e^{2}-c \,d^{2}\right ) {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{2} \left (m^{2}+5 m +6\right )}+\frac {d e c \,x^{3} {\mathrm e}^{m \ln \left (e x +d \right )}}{3+m}+\frac {d \left (2 a \,e^{2} m +c \,d^{2} m +6 a \,e^{2}\right ) x \,{\mathrm e}^{m \ln \left (e x +d \right )}}{e \left (m^{2}+5 m +6\right )}\) | \(162\) |
parallelrisch | \(\frac {x^{3} \left (e x +d \right )^{m} c d \,e^{3} m +2 x^{3} \left (e x +d \right )^{m} c d \,e^{3}+x^{2} \left (e x +d \right )^{m} a \,e^{4} m +2 x^{2} \left (e x +d \right )^{m} c \,d^{2} e^{2} m +3 x^{2} \left (e x +d \right )^{m} a \,e^{4}+3 x^{2} \left (e x +d \right )^{m} c \,d^{2} e^{2}+2 x \left (e x +d \right )^{m} a d \,e^{3} m +x \left (e x +d \right )^{m} c \,d^{3} e m +6 x \left (e x +d \right )^{m} a d \,e^{3}+\left (e x +d \right )^{m} a \,d^{2} e^{2} m +3 \left (e x +d \right )^{m} a \,d^{2} e^{2}-\left (e x +d \right )^{m} c \,d^{4}}{e^{2} \left (m^{2}+5 m +6\right )}\) | \(212\) |
Input:
int((e*x+d)^m*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e),x,method=_RETURNVERBOSE)
Output:
1/e^2*(e*x+d)^(2+m)/(m^2+5*m+6)*(c*d*e*m*x+a*e^2*m+2*c*d*e*x+3*a*e^2-c*d^2 )
Leaf count of result is larger than twice the leaf count of optimal. 136 vs. \(2 (52) = 104\).
Time = 0.10 (sec) , antiderivative size = 136, normalized size of antiderivative = 2.62 \[ \int (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=\frac {{\left (a d^{2} e^{2} m - c d^{4} + 3 \, a d^{2} e^{2} + {\left (c d e^{3} m + 2 \, c d e^{3}\right )} x^{3} + {\left (3 \, c d^{2} e^{2} + 3 \, a e^{4} + {\left (2 \, c d^{2} e^{2} + a e^{4}\right )} m\right )} x^{2} + {\left (6 \, a d e^{3} + {\left (c d^{3} e + 2 \, a d e^{3}\right )} m\right )} x\right )} {\left (e x + d\right )}^{m}}{e^{2} m^{2} + 5 \, e^{2} m + 6 \, e^{2}} \] Input:
integrate((e*x+d)^m*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2),x, algorithm="fricas ")
Output:
(a*d^2*e^2*m - c*d^4 + 3*a*d^2*e^2 + (c*d*e^3*m + 2*c*d*e^3)*x^3 + (3*c*d^ 2*e^2 + 3*a*e^4 + (2*c*d^2*e^2 + a*e^4)*m)*x^2 + (6*a*d*e^3 + (c*d^3*e + 2 *a*d*e^3)*m)*x)*(e*x + d)^m/(e^2*m^2 + 5*e^2*m + 6*e^2)
Leaf count of result is larger than twice the leaf count of optimal. 556 vs. \(2 (42) = 84\).
Time = 0.42 (sec) , antiderivative size = 556, normalized size of antiderivative = 10.69 \[ \int (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=\begin {cases} \frac {c d^{2} d^{m} x^{2}}{2} & \text {for}\: e = 0 \\- \frac {a e^{2}}{d e^{2} + e^{3} x} + \frac {c d^{2} \log {\left (\frac {d}{e} + x \right )}}{d e^{2} + e^{3} x} + \frac {c d^{2}}{d e^{2} + e^{3} x} + \frac {c d e x \log {\left (\frac {d}{e} + x \right )}}{d e^{2} + e^{3} x} & \text {for}\: m = -3 \\a \log {\left (\frac {d}{e} + x \right )} - \frac {c d^{2} \log {\left (\frac {d}{e} + x \right )}}{e^{2}} + \frac {c d x}{e} & \text {for}\: m = -2 \\\frac {a d^{2} e^{2} m \left (d + e x\right )^{m}}{e^{2} m^{2} + 5 e^{2} m + 6 e^{2}} + \frac {3 a d^{2} e^{2} \left (d + e x\right )^{m}}{e^{2} m^{2} + 5 e^{2} m + 6 e^{2}} + \frac {2 a d e^{3} m x \left (d + e x\right )^{m}}{e^{2} m^{2} + 5 e^{2} m + 6 e^{2}} + \frac {6 a d e^{3} x \left (d + e x\right )^{m}}{e^{2} m^{2} + 5 e^{2} m + 6 e^{2}} + \frac {a e^{4} m x^{2} \left (d + e x\right )^{m}}{e^{2} m^{2} + 5 e^{2} m + 6 e^{2}} + \frac {3 a e^{4} x^{2} \left (d + e x\right )^{m}}{e^{2} m^{2} + 5 e^{2} m + 6 e^{2}} - \frac {c d^{4} \left (d + e x\right )^{m}}{e^{2} m^{2} + 5 e^{2} m + 6 e^{2}} + \frac {c d^{3} e m x \left (d + e x\right )^{m}}{e^{2} m^{2} + 5 e^{2} m + 6 e^{2}} + \frac {2 c d^{2} e^{2} m x^{2} \left (d + e x\right )^{m}}{e^{2} m^{2} + 5 e^{2} m + 6 e^{2}} + \frac {3 c d^{2} e^{2} x^{2} \left (d + e x\right )^{m}}{e^{2} m^{2} + 5 e^{2} m + 6 e^{2}} + \frac {c d e^{3} m x^{3} \left (d + e x\right )^{m}}{e^{2} m^{2} + 5 e^{2} m + 6 e^{2}} + \frac {2 c d e^{3} x^{3} \left (d + e x\right )^{m}}{e^{2} m^{2} + 5 e^{2} m + 6 e^{2}} & \text {otherwise} \end {cases} \] Input:
integrate((e*x+d)**m*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2),x)
Output:
Piecewise((c*d**2*d**m*x**2/2, Eq(e, 0)), (-a*e**2/(d*e**2 + e**3*x) + c*d **2*log(d/e + x)/(d*e**2 + e**3*x) + c*d**2/(d*e**2 + e**3*x) + c*d*e*x*lo g(d/e + x)/(d*e**2 + e**3*x), Eq(m, -3)), (a*log(d/e + x) - c*d**2*log(d/e + x)/e**2 + c*d*x/e, Eq(m, -2)), (a*d**2*e**2*m*(d + e*x)**m/(e**2*m**2 + 5*e**2*m + 6*e**2) + 3*a*d**2*e**2*(d + e*x)**m/(e**2*m**2 + 5*e**2*m + 6 *e**2) + 2*a*d*e**3*m*x*(d + e*x)**m/(e**2*m**2 + 5*e**2*m + 6*e**2) + 6*a *d*e**3*x*(d + e*x)**m/(e**2*m**2 + 5*e**2*m + 6*e**2) + a*e**4*m*x**2*(d + e*x)**m/(e**2*m**2 + 5*e**2*m + 6*e**2) + 3*a*e**4*x**2*(d + e*x)**m/(e* *2*m**2 + 5*e**2*m + 6*e**2) - c*d**4*(d + e*x)**m/(e**2*m**2 + 5*e**2*m + 6*e**2) + c*d**3*e*m*x*(d + e*x)**m/(e**2*m**2 + 5*e**2*m + 6*e**2) + 2*c *d**2*e**2*m*x**2*(d + e*x)**m/(e**2*m**2 + 5*e**2*m + 6*e**2) + 3*c*d**2* e**2*x**2*(d + e*x)**m/(e**2*m**2 + 5*e**2*m + 6*e**2) + c*d*e**3*m*x**3*( d + e*x)**m/(e**2*m**2 + 5*e**2*m + 6*e**2) + 2*c*d*e**3*x**3*(d + e*x)**m /(e**2*m**2 + 5*e**2*m + 6*e**2), True))
Leaf count of result is larger than twice the leaf count of optimal. 174 vs. \(2 (52) = 104\).
Time = 0.04 (sec) , antiderivative size = 174, normalized size of antiderivative = 3.35 \[ \int (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=\frac {{\left (e^{2} {\left (m + 1\right )} x^{2} + d e m x - d^{2}\right )} {\left (e x + d\right )}^{m} a}{m^{2} + 3 \, m + 2} + \frac {{\left (e^{2} {\left (m + 1\right )} x^{2} + d e m x - d^{2}\right )} {\left (e x + d\right )}^{m} c d^{2}}{{\left (m^{2} + 3 \, m + 2\right )} e^{2}} + \frac {{\left (e x + d\right )}^{m + 1} a d}{m + 1} + \frac {{\left ({\left (m^{2} + 3 \, m + 2\right )} e^{3} x^{3} + {\left (m^{2} + m\right )} d e^{2} x^{2} - 2 \, d^{2} e m x + 2 \, d^{3}\right )} {\left (e x + d\right )}^{m} c d}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{2}} \] Input:
integrate((e*x+d)^m*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2),x, algorithm="maxima ")
Output:
(e^2*(m + 1)*x^2 + d*e*m*x - d^2)*(e*x + d)^m*a/(m^2 + 3*m + 2) + (e^2*(m + 1)*x^2 + d*e*m*x - d^2)*(e*x + d)^m*c*d^2/((m^2 + 3*m + 2)*e^2) + (e*x + d)^(m + 1)*a*d/(m + 1) + ((m^2 + 3*m + 2)*e^3*x^3 + (m^2 + m)*d*e^2*x^2 - 2*d^2*e*m*x + 2*d^3)*(e*x + d)^m*c*d/((m^3 + 6*m^2 + 11*m + 6)*e^2)
Leaf count of result is larger than twice the leaf count of optimal. 219 vs. \(2 (52) = 104\).
Time = 0.12 (sec) , antiderivative size = 219, normalized size of antiderivative = 4.21 \[ \int (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=\frac {{\left (e x + d\right )}^{m} c d e^{3} m x^{3} + 2 \, {\left (e x + d\right )}^{m} c d^{2} e^{2} m x^{2} + {\left (e x + d\right )}^{m} a e^{4} m x^{2} + 2 \, {\left (e x + d\right )}^{m} c d e^{3} x^{3} + {\left (e x + d\right )}^{m} c d^{3} e m x + 2 \, {\left (e x + d\right )}^{m} a d e^{3} m x + 3 \, {\left (e x + d\right )}^{m} c d^{2} e^{2} x^{2} + 3 \, {\left (e x + d\right )}^{m} a e^{4} x^{2} + {\left (e x + d\right )}^{m} a d^{2} e^{2} m + 6 \, {\left (e x + d\right )}^{m} a d e^{3} x - {\left (e x + d\right )}^{m} c d^{4} + 3 \, {\left (e x + d\right )}^{m} a d^{2} e^{2}}{e^{2} m^{2} + 5 \, e^{2} m + 6 \, e^{2}} \] Input:
integrate((e*x+d)^m*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2),x, algorithm="giac")
Output:
((e*x + d)^m*c*d*e^3*m*x^3 + 2*(e*x + d)^m*c*d^2*e^2*m*x^2 + (e*x + d)^m*a *e^4*m*x^2 + 2*(e*x + d)^m*c*d*e^3*x^3 + (e*x + d)^m*c*d^3*e*m*x + 2*(e*x + d)^m*a*d*e^3*m*x + 3*(e*x + d)^m*c*d^2*e^2*x^2 + 3*(e*x + d)^m*a*e^4*x^2 + (e*x + d)^m*a*d^2*e^2*m + 6*(e*x + d)^m*a*d*e^3*x - (e*x + d)^m*c*d^4 + 3*(e*x + d)^m*a*d^2*e^2)/(e^2*m^2 + 5*e^2*m + 6*e^2)
Time = 5.38 (sec) , antiderivative size = 141, normalized size of antiderivative = 2.71 \[ \int (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx={\left (d+e\,x\right )}^m\,\left (\frac {x^2\,\left (3\,a\,e^2+3\,c\,d^2+a\,e^2\,m+2\,c\,d^2\,m\right )}{m^2+5\,m+6}+\frac {d^2\,\left (3\,a\,e^2-c\,d^2+a\,e^2\,m\right )}{e^2\,\left (m^2+5\,m+6\right )}+\frac {d\,x\,\left (6\,a\,e^2+2\,a\,e^2\,m+c\,d^2\,m\right )}{e\,\left (m^2+5\,m+6\right )}+\frac {c\,d\,e\,x^3\,\left (m+2\right )}{m^2+5\,m+6}\right ) \] Input:
int((d + e*x)^m*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2),x)
Output:
(d + e*x)^m*((x^2*(3*a*e^2 + 3*c*d^2 + a*e^2*m + 2*c*d^2*m))/(5*m + m^2 + 6) + (d^2*(3*a*e^2 - c*d^2 + a*e^2*m))/(e^2*(5*m + m^2 + 6)) + (d*x*(6*a*e ^2 + 2*a*e^2*m + c*d^2*m))/(e*(5*m + m^2 + 6)) + (c*d*e*x^3*(m + 2))/(5*m + m^2 + 6))
Time = 0.19 (sec) , antiderivative size = 134, normalized size of antiderivative = 2.58 \[ \int (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=\frac {\left (e x +d \right )^{m} \left (c d \,e^{3} m \,x^{3}+a \,e^{4} m \,x^{2}+2 c \,d^{2} e^{2} m \,x^{2}+2 c d \,e^{3} x^{3}+2 a d \,e^{3} m x +3 a \,e^{4} x^{2}+c \,d^{3} e m x +3 c \,d^{2} e^{2} x^{2}+a \,d^{2} e^{2} m +6 a d \,e^{3} x +3 a \,d^{2} e^{2}-c \,d^{4}\right )}{e^{2} \left (m^{2}+5 m +6\right )} \] Input:
int((e*x+d)^m*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2),x)
Output:
((d + e*x)**m*(a*d**2*e**2*m + 3*a*d**2*e**2 + 2*a*d*e**3*m*x + 6*a*d*e**3 *x + a*e**4*m*x**2 + 3*a*e**4*x**2 - c*d**4 + c*d**3*e*m*x + 2*c*d**2*e**2 *m*x**2 + 3*c*d**2*e**2*x**2 + c*d*e**3*m*x**3 + 2*c*d*e**3*x**3))/(e**2*( m**2 + 5*m + 6))