Integrand size = 17, antiderivative size = 75 \[ \int (d+e x)^m \left (b x+c x^2\right ) \, dx=\frac {d (c d-b e) (d+e x)^{1+m}}{e^3 (1+m)}-\frac {(2 c d-b e) (d+e x)^{2+m}}{e^3 (2+m)}+\frac {c (d+e x)^{3+m}}{e^3 (3+m)} \]
d*(-b*e+c*d)*(e*x+d)^(1+m)/e^3/(1+m)-(-b*e+2*c*d)*(e*x+d)^(2+m)/e^3/(2+m)+ c*(e*x+d)^(3+m)/e^3/(3+m)
Time = 0.10 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00 \[ \int (d+e x)^m \left (b x+c x^2\right ) \, dx=\frac {d (c d-b e) (d+e x)^{1+m}}{e^3 (1+m)}-\frac {(2 c d-b e) (d+e x)^{2+m}}{e^3 (2+m)}+\frac {c (d+e x)^{3+m}}{e^3 (3+m)} \]
(d*(c*d - b*e)*(d + e*x)^(1 + m))/(e^3*(1 + m)) - ((2*c*d - b*e)*(d + e*x) ^(2 + m))/(e^3*(2 + m)) + (c*(d + e*x)^(3 + m))/(e^3*(3 + m))
Time = 0.22 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {1140, 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.
\(\displaystyle \int \left (b x+c x^2\right ) (d+e x)^m \, dx\) |
\(\Big \downarrow \) 1140 |
\(\displaystyle \int \left (\frac {d (c d-b e) (d+e x)^m}{e^2}+\frac {(b e-2 c d) (d+e x)^{m+1}}{e^2}+\frac {c (d+e x)^{m+2}}{e^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {d (c d-b e) (d+e x)^{m+1}}{e^3 (m+1)}-\frac {(2 c d-b e) (d+e x)^{m+2}}{e^3 (m+2)}+\frac {c (d+e x)^{m+3}}{e^3 (m+3)}\) |
(d*(c*d - b*e)*(d + e*x)^(1 + m))/(e^3*(1 + m)) - ((2*c*d - b*e)*(d + e*x) ^(2 + m))/(e^3*(2 + m)) + (c*(d + e*x)^(3 + m))/(e^3*(3 + m))
3.5.42.3.1 Defintions of rubi rules used
Int[((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x _Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 0]
Time = 2.14 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.55
method | result | size |
gosper | \(-\frac {\left (e x +d \right )^{1+m} \left (-c \,e^{2} m^{2} x^{2}-b \,e^{2} m^{2} x -3 c \,e^{2} m \,x^{2}-4 b \,e^{2} m x +2 c d e m x -2 c \,x^{2} e^{2}+b d e m -3 x b \,e^{2}+2 x c d e +3 b d e -2 c \,d^{2}\right )}{e^{3} \left (m^{3}+6 m^{2}+11 m +6\right )}\) | \(116\) |
norman | \(\frac {c \,x^{3} {\mathrm e}^{m \ln \left (e x +d \right )}}{3+m}+\frac {\left (b e m +c d m +3 b e \right ) x^{2} {\mathrm e}^{m \ln \left (e x +d \right )}}{e \left (m^{2}+5 m +6\right )}+\frac {m d \left (b e m +3 b e -2 c d \right ) x \,{\mathrm e}^{m \ln \left (e x +d \right )}}{e^{2} \left (m^{3}+6 m^{2}+11 m +6\right )}-\frac {d^{2} \left (b e m +3 b e -2 c d \right ) {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{3} \left (m^{3}+6 m^{2}+11 m +6\right )}\) | \(149\) |
risch | \(-\frac {\left (-c \,e^{3} m^{2} x^{3}-b \,e^{3} m^{2} x^{2}-c d \,e^{2} m^{2} x^{2}-3 c \,e^{3} m \,x^{3}-b d \,e^{2} m^{2} x -4 b \,e^{3} m \,x^{2}-c d \,e^{2} m \,x^{2}-2 c \,x^{3} e^{3}-3 b d \,e^{2} m x -3 b \,e^{3} x^{2}+2 c \,d^{2} e m x +b \,d^{2} e m +3 b \,d^{2} e -2 d^{3} c \right ) \left (e x +d \right )^{m}}{\left (2+m \right ) \left (3+m \right ) \left (1+m \right ) e^{3}}\) | \(164\) |
parallelrisch | \(\frac {x^{3} \left (e x +d \right )^{m} c \,e^{3} m^{2}+3 x^{3} \left (e x +d \right )^{m} c \,e^{3} m +x^{2} \left (e x +d \right )^{m} b \,e^{3} m^{2}+x^{2} \left (e x +d \right )^{m} c d \,e^{2} m^{2}+2 x^{3} \left (e x +d \right )^{m} c \,e^{3}+4 x^{2} \left (e x +d \right )^{m} b \,e^{3} m +x^{2} \left (e x +d \right )^{m} c d \,e^{2} m +x \left (e x +d \right )^{m} b d \,e^{2} m^{2}+3 x^{2} \left (e x +d \right )^{m} b \,e^{3}+3 x \left (e x +d \right )^{m} b d \,e^{2} m -2 x \left (e x +d \right )^{m} c \,d^{2} e m -\left (e x +d \right )^{m} b \,d^{2} e m -3 \left (e x +d \right )^{m} b \,d^{2} e +2 \left (e x +d \right )^{m} c \,d^{3}}{e^{3} \left (m^{3}+6 m^{2}+11 m +6\right )}\) | \(250\) |
-1/e^3*(e*x+d)^(1+m)/(m^3+6*m^2+11*m+6)*(-c*e^2*m^2*x^2-b*e^2*m^2*x-3*c*e^ 2*m*x^2-4*b*e^2*m*x+2*c*d*e*m*x-2*c*e^2*x^2+b*d*e*m-3*b*e^2*x+2*c*d*e*x+3* b*d*e-2*c*d^2)
Leaf count of result is larger than twice the leaf count of optimal. 160 vs. \(2 (75) = 150\).
Time = 0.26 (sec) , antiderivative size = 160, normalized size of antiderivative = 2.13 \[ \int (d+e x)^m \left (b x+c x^2\right ) \, dx=-\frac {{\left (b d^{2} e m - 2 \, c d^{3} + 3 \, b d^{2} e - {\left (c e^{3} m^{2} + 3 \, c e^{3} m + 2 \, c e^{3}\right )} x^{3} - {\left (3 \, b e^{3} + {\left (c d e^{2} + b e^{3}\right )} m^{2} + {\left (c d e^{2} + 4 \, b e^{3}\right )} m\right )} x^{2} - {\left (b d e^{2} m^{2} - {\left (2 \, c d^{2} e - 3 \, b d e^{2}\right )} m\right )} x\right )} {\left (e x + d\right )}^{m}}{e^{3} m^{3} + 6 \, e^{3} m^{2} + 11 \, e^{3} m + 6 \, e^{3}} \]
-(b*d^2*e*m - 2*c*d^3 + 3*b*d^2*e - (c*e^3*m^2 + 3*c*e^3*m + 2*c*e^3)*x^3 - (3*b*e^3 + (c*d*e^2 + b*e^3)*m^2 + (c*d*e^2 + 4*b*e^3)*m)*x^2 - (b*d*e^2 *m^2 - (2*c*d^2*e - 3*b*d*e^2)*m)*x)*(e*x + d)^m/(e^3*m^3 + 6*e^3*m^2 + 11 *e^3*m + 6*e^3)
Leaf count of result is larger than twice the leaf count of optimal. 1095 vs. \(2 (63) = 126\).
Time = 0.54 (sec) , antiderivative size = 1095, normalized size of antiderivative = 14.60 \[ \int (d+e x)^m \left (b x+c x^2\right ) \, dx=\begin {cases} d^{m} \left (\frac {b x^{2}}{2} + \frac {c x^{3}}{3}\right ) & \text {for}\: e = 0 \\- \frac {b d e}{2 d^{2} e^{3} + 4 d e^{4} x + 2 e^{5} x^{2}} - \frac {2 b e^{2} x}{2 d^{2} e^{3} + 4 d e^{4} x + 2 e^{5} x^{2}} + \frac {2 c d^{2} \log {\left (\frac {d}{e} + x \right )}}{2 d^{2} e^{3} + 4 d e^{4} x + 2 e^{5} x^{2}} + \frac {3 c d^{2}}{2 d^{2} e^{3} + 4 d e^{4} x + 2 e^{5} x^{2}} + \frac {4 c d e x \log {\left (\frac {d}{e} + x \right )}}{2 d^{2} e^{3} + 4 d e^{4} x + 2 e^{5} x^{2}} + \frac {4 c d e x}{2 d^{2} e^{3} + 4 d e^{4} x + 2 e^{5} x^{2}} + \frac {2 c e^{2} x^{2} \log {\left (\frac {d}{e} + x \right )}}{2 d^{2} e^{3} + 4 d e^{4} x + 2 e^{5} x^{2}} & \text {for}\: m = -3 \\\frac {b d e \log {\left (\frac {d}{e} + x \right )}}{d e^{3} + e^{4} x} + \frac {b d e}{d e^{3} + e^{4} x} + \frac {b e^{2} x \log {\left (\frac {d}{e} + x \right )}}{d e^{3} + e^{4} x} - \frac {2 c d^{2} \log {\left (\frac {d}{e} + x \right )}}{d e^{3} + e^{4} x} - \frac {2 c d^{2}}{d e^{3} + e^{4} x} - \frac {2 c d e x \log {\left (\frac {d}{e} + x \right )}}{d e^{3} + e^{4} x} + \frac {c e^{2} x^{2}}{d e^{3} + e^{4} x} & \text {for}\: m = -2 \\- \frac {b d \log {\left (\frac {d}{e} + x \right )}}{e^{2}} + \frac {b x}{e} + \frac {c d^{2} \log {\left (\frac {d}{e} + x \right )}}{e^{3}} - \frac {c d x}{e^{2}} + \frac {c x^{2}}{2 e} & \text {for}\: m = -1 \\- \frac {b d^{2} e m \left (d + e x\right )^{m}}{e^{3} m^{3} + 6 e^{3} m^{2} + 11 e^{3} m + 6 e^{3}} - \frac {3 b d^{2} e \left (d + e x\right )^{m}}{e^{3} m^{3} + 6 e^{3} m^{2} + 11 e^{3} m + 6 e^{3}} + \frac {b d e^{2} m^{2} x \left (d + e x\right )^{m}}{e^{3} m^{3} + 6 e^{3} m^{2} + 11 e^{3} m + 6 e^{3}} + \frac {3 b d e^{2} m x \left (d + e x\right )^{m}}{e^{3} m^{3} + 6 e^{3} m^{2} + 11 e^{3} m + 6 e^{3}} + \frac {b e^{3} m^{2} x^{2} \left (d + e x\right )^{m}}{e^{3} m^{3} + 6 e^{3} m^{2} + 11 e^{3} m + 6 e^{3}} + \frac {4 b e^{3} m x^{2} \left (d + e x\right )^{m}}{e^{3} m^{3} + 6 e^{3} m^{2} + 11 e^{3} m + 6 e^{3}} + \frac {3 b e^{3} x^{2} \left (d + e x\right )^{m}}{e^{3} m^{3} + 6 e^{3} m^{2} + 11 e^{3} m + 6 e^{3}} + \frac {2 c d^{3} \left (d + e x\right )^{m}}{e^{3} m^{3} + 6 e^{3} m^{2} + 11 e^{3} m + 6 e^{3}} - \frac {2 c d^{2} e m x \left (d + e x\right )^{m}}{e^{3} m^{3} + 6 e^{3} m^{2} + 11 e^{3} m + 6 e^{3}} + \frac {c d e^{2} m^{2} x^{2} \left (d + e x\right )^{m}}{e^{3} m^{3} + 6 e^{3} m^{2} + 11 e^{3} m + 6 e^{3}} + \frac {c d e^{2} m x^{2} \left (d + e x\right )^{m}}{e^{3} m^{3} + 6 e^{3} m^{2} + 11 e^{3} m + 6 e^{3}} + \frac {c e^{3} m^{2} x^{3} \left (d + e x\right )^{m}}{e^{3} m^{3} + 6 e^{3} m^{2} + 11 e^{3} m + 6 e^{3}} + \frac {3 c e^{3} m x^{3} \left (d + e x\right )^{m}}{e^{3} m^{3} + 6 e^{3} m^{2} + 11 e^{3} m + 6 e^{3}} + \frac {2 c e^{3} x^{3} \left (d + e x\right )^{m}}{e^{3} m^{3} + 6 e^{3} m^{2} + 11 e^{3} m + 6 e^{3}} & \text {otherwise} \end {cases} \]
Piecewise((d**m*(b*x**2/2 + c*x**3/3), Eq(e, 0)), (-b*d*e/(2*d**2*e**3 + 4 *d*e**4*x + 2*e**5*x**2) - 2*b*e**2*x/(2*d**2*e**3 + 4*d*e**4*x + 2*e**5*x **2) + 2*c*d**2*log(d/e + x)/(2*d**2*e**3 + 4*d*e**4*x + 2*e**5*x**2) + 3* c*d**2/(2*d**2*e**3 + 4*d*e**4*x + 2*e**5*x**2) + 4*c*d*e*x*log(d/e + x)/( 2*d**2*e**3 + 4*d*e**4*x + 2*e**5*x**2) + 4*c*d*e*x/(2*d**2*e**3 + 4*d*e** 4*x + 2*e**5*x**2) + 2*c*e**2*x**2*log(d/e + x)/(2*d**2*e**3 + 4*d*e**4*x + 2*e**5*x**2), Eq(m, -3)), (b*d*e*log(d/e + x)/(d*e**3 + e**4*x) + b*d*e/ (d*e**3 + e**4*x) + b*e**2*x*log(d/e + x)/(d*e**3 + e**4*x) - 2*c*d**2*log (d/e + x)/(d*e**3 + e**4*x) - 2*c*d**2/(d*e**3 + e**4*x) - 2*c*d*e*x*log(d /e + x)/(d*e**3 + e**4*x) + c*e**2*x**2/(d*e**3 + e**4*x), Eq(m, -2)), (-b *d*log(d/e + x)/e**2 + b*x/e + c*d**2*log(d/e + x)/e**3 - c*d*x/e**2 + c*x **2/(2*e), Eq(m, -1)), (-b*d**2*e*m*(d + e*x)**m/(e**3*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) - 3*b*d**2*e*(d + e*x)**m/(e**3*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) + b*d*e**2*m**2*x*(d + e*x)**m/(e**3*m**3 + 6*e**3*m* *2 + 11*e**3*m + 6*e**3) + 3*b*d*e**2*m*x*(d + e*x)**m/(e**3*m**3 + 6*e**3 *m**2 + 11*e**3*m + 6*e**3) + b*e**3*m**2*x**2*(d + e*x)**m/(e**3*m**3 + 6 *e**3*m**2 + 11*e**3*m + 6*e**3) + 4*b*e**3*m*x**2*(d + e*x)**m/(e**3*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) + 3*b*e**3*x**2*(d + e*x)**m/(e**3*m* *3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) + 2*c*d**3*(d + e*x)**m/(e**3*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) - 2*c*d**2*e*m*x*(d + e*x)**m/(e**3...
Time = 0.20 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.51 \[ \int (d+e x)^m \left (b x+c x^2\right ) \, dx=\frac {{\left (e^{2} {\left (m + 1\right )} x^{2} + d e m x - d^{2}\right )} {\left (e x + d\right )}^{m} b}{{\left (m^{2} + 3 \, m + 2\right )} e^{2}} + \frac {{\left ({\left (m^{2} + 3 \, m + 2\right )} e^{3} x^{3} + {\left (m^{2} + m\right )} d e^{2} x^{2} - 2 \, d^{2} e m x + 2 \, d^{3}\right )} {\left (e x + d\right )}^{m} c}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{3}} \]
(e^2*(m + 1)*x^2 + d*e*m*x - d^2)*(e*x + d)^m*b/((m^2 + 3*m + 2)*e^2) + (( 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/((m^3 + 6*m^2 + 11*m + 6)*e^3)
Leaf count of result is larger than twice the leaf count of optimal. 260 vs. \(2 (75) = 150\).
Time = 0.28 (sec) , antiderivative size = 260, normalized size of antiderivative = 3.47 \[ \int (d+e x)^m \left (b x+c x^2\right ) \, dx=\frac {{\left (e x + d\right )}^{m} c e^{3} m^{2} x^{3} + {\left (e x + d\right )}^{m} c d e^{2} m^{2} x^{2} + {\left (e x + d\right )}^{m} b e^{3} m^{2} x^{2} + 3 \, {\left (e x + d\right )}^{m} c e^{3} m x^{3} + {\left (e x + d\right )}^{m} b d e^{2} m^{2} x + {\left (e x + d\right )}^{m} c d e^{2} m x^{2} + 4 \, {\left (e x + d\right )}^{m} b e^{3} m x^{2} + 2 \, {\left (e x + d\right )}^{m} c e^{3} x^{3} - 2 \, {\left (e x + d\right )}^{m} c d^{2} e m x + 3 \, {\left (e x + d\right )}^{m} b d e^{2} m x + 3 \, {\left (e x + d\right )}^{m} b e^{3} x^{2} - {\left (e x + d\right )}^{m} b d^{2} e m + 2 \, {\left (e x + d\right )}^{m} c d^{3} - 3 \, {\left (e x + d\right )}^{m} b d^{2} e}{e^{3} m^{3} + 6 \, e^{3} m^{2} + 11 \, e^{3} m + 6 \, e^{3}} \]
((e*x + d)^m*c*e^3*m^2*x^3 + (e*x + d)^m*c*d*e^2*m^2*x^2 + (e*x + d)^m*b*e ^3*m^2*x^2 + 3*(e*x + d)^m*c*e^3*m*x^3 + (e*x + d)^m*b*d*e^2*m^2*x + (e*x + d)^m*c*d*e^2*m*x^2 + 4*(e*x + d)^m*b*e^3*m*x^2 + 2*(e*x + d)^m*c*e^3*x^3 - 2*(e*x + d)^m*c*d^2*e*m*x + 3*(e*x + d)^m*b*d*e^2*m*x + 3*(e*x + d)^m*b *e^3*x^2 - (e*x + d)^m*b*d^2*e*m + 2*(e*x + d)^m*c*d^3 - 3*(e*x + d)^m*b*d ^2*e)/(e^3*m^3 + 6*e^3*m^2 + 11*e^3*m + 6*e^3)
Time = 9.63 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.95 \[ \int (d+e x)^m \left (b x+c x^2\right ) \, dx={\left (d+e\,x\right )}^m\,\left (\frac {c\,x^3\,\left (m^2+3\,m+2\right )}{m^3+6\,m^2+11\,m+6}-\frac {d^2\,\left (3\,b\,e-2\,c\,d+b\,e\,m\right )}{e^3\,\left (m^3+6\,m^2+11\,m+6\right )}+\frac {x^2\,\left (m+1\right )\,\left (3\,b\,e+b\,e\,m+c\,d\,m\right )}{e\,\left (m^3+6\,m^2+11\,m+6\right )}+\frac {d\,m\,x\,\left (3\,b\,e-2\,c\,d+b\,e\,m\right )}{e^2\,\left (m^3+6\,m^2+11\,m+6\right )}\right ) \]