\(\int (d+e x)^m (a d e+(c d^2+a e^2) x+c d e x^2) \, dx\) [2086]

Optimal result

Integrand size = 33, antiderivative size = 52 \[ \int (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=-\frac {\left (c d^2-a e^2\right ) (d+e x)^{2+m}}{e^2 (2+m)}+\frac {c d (d+e x)^{3+m}}{e^2 (3+m)} \]

[Out]

-(-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)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {640, 45} \[ \int (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=\frac {c d (d+e x)^{m+3}}{e^2 (m+3)}-\frac {\left (c d^2-a e^2\right ) (d+e x)^{m+2}}{e^2 (m+2)} \]

[In]

Int[(d + e*x)^m*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2),x]

[Out]

-(((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))

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 640

Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a
/d + (c/e)*x)^p, x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&
IntegerQ[p]

Rubi steps \begin{align*} \text {integral}& = \int (a e+c d x) (d+e x)^{1+m} \, dx \\ & = \int \left (\frac {\left (-c d^2+a e^2\right ) (d+e x)^{1+m}}{e}+\frac {c d (d+e x)^{2+m}}{e}\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)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.87 \[ \int (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=\frac {(d+e x)^{2+m} \left (a e^2 (3+m)+c d (-d+e (2+m) x)\right )}{e^2 (2+m) (3+m)} \]

[In]

Integrate[(d + e*x)^m*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2),x]

[Out]

((d + e*x)^(2 + m)*(a*e^2*(3 + m) + c*d*(-d + e*(2 + m)*x)))/(e^2*(2 + m)*(3 + m))

Maple [A] (verified)

Time = 4.10 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.06

method	result	size
gosper	\(\frac {\left (e x +d \right )^{2+m} \left (c d e m x +a \,e^{2} m +2 x c d e +3 e^{2} a -c \,d^{2}\right )}{e^{2} \left (m^{2}+5 m +6\right )}\)	\(55\)
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	\(\frac {\left (a \,e^{2} m +2 c \,d^{2} m +3 e^{2} a +3 c \,d^{2}\right ) x^{2} {\mathrm e}^{m \ln \left (e x +d \right )}}{m^{2}+5 m +6}+\frac {d^{2} \left (a \,e^{2} m +3 e^{2} a -c \,d^{2}\right ) {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{2} \left (m^{2}+5 m +6\right )}+\frac {c d e \,x^{3} {\mathrm e}^{m \ln \left (e x +d \right )}}{3+m}+\frac {d \left (2 a \,e^{2} m +c \,d^{2} m +6 e^{2} a \right ) x \,{\mathrm e}^{m \ln \left (e x +d \right )}}{e \left (m^{2}+5 m +6\right )}\)	\(162\)
parallelrisch	\(\frac {x^{3} \left (e x +d \right )^{m} c d \,e^{3} m +2 x^{3} \left (e x +d \right )^{m} c d \,e^{3}+x^{2} \left (e x +d \right )^{m} a \,e^{4} m +2 x^{2} \left (e x +d \right )^{m} c \,d^{2} e^{2} m +3 x^{2} \left (e x +d \right )^{m} a \,e^{4}+3 x^{2} \left (e x +d \right )^{m} c \,d^{2} e^{2}+2 x \left (e x +d \right )^{m} a d \,e^{3} m +x \left (e x +d \right )^{m} c \,d^{3} e m +6 x \left (e x +d \right )^{m} a d \,e^{3}+\left (e x +d \right )^{m} a \,d^{2} e^{2} m +3 \left (e x +d \right )^{m} a \,d^{2} e^{2}-\left (e x +d \right )^{m} c \,d^{4}}{e^{2} \left (m^{2}+5 m +6\right )}\)	\(212\)

[In]

int((e*x+d)^m*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2),x,method=_RETURNVERBOSE)

[Out]

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)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 136 vs. \(2 (52) = 104\).

Time = 0.40 (sec) , antiderivative size = 136, normalized size of antiderivative = 2.62 \[ \int (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=\frac {{\left (a d^{2} e^{2} m - c d^{4} + 3 \, a d^{2} e^{2} + {\left (c d e^{3} m + 2 \, c d e^{3}\right )} x^{3} + {\left (3 \, c d^{2} e^{2} + 3 \, a e^{4} + {\left (2 \, c d^{2} e^{2} + a e^{4}\right )} m\right )} x^{2} + {\left (6 \, a d e^{3} + {\left (c d^{3} e + 2 \, a d e^{3}\right )} m\right )} x\right )} {\left (e x + d\right )}^{m}}{e^{2} m^{2} + 5 \, e^{2} m + 6 \, e^{2}} \]

[In]

integrate((e*x+d)^m*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2),x, algorithm="fricas")

[Out]

(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)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 556 vs. \(2 (42) = 84\).

Time = 0.47 (sec) , antiderivative size = 556, normalized size of antiderivative = 10.69 \[ \int (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=\begin {cases} \frac {c d^{2} d^{m} x^{2}}{2} & \text {for}\: e = 0 \\- \frac {a e^{2}}{d e^{2} + e^{3} x} + \frac {c d^{2} \log {\left (\frac {d}{e} + x \right )}}{d e^{2} + e^{3} x} + \frac {c d^{2}}{d e^{2} + e^{3} x} + \frac {c d e x \log {\left (\frac {d}{e} + x \right )}}{d e^{2} + e^{3} x} & \text {for}\: m = -3 \\a \log {\left (\frac {d}{e} + x \right )} - \frac {c d^{2} \log {\left (\frac {d}{e} + x \right )}}{e^{2}} + \frac {c d x}{e} & \text {for}\: m = -2 \\\frac {a d^{2} e^{2} m \left (d + e x\right )^{m}}{e^{2} m^{2} + 5 e^{2} m + 6 e^{2}} + \frac {3 a d^{2} e^{2} \left (d + e x\right )^{m}}{e^{2} m^{2} + 5 e^{2} m + 6 e^{2}} + \frac {2 a d e^{3} m x \left (d + e x\right )^{m}}{e^{2} m^{2} + 5 e^{2} m + 6 e^{2}} + \frac {6 a d e^{3} x \left (d + e x\right )^{m}}{e^{2} m^{2} + 5 e^{2} m + 6 e^{2}} + \frac {a e^{4} m x^{2} \left (d + e x\right )^{m}}{e^{2} m^{2} + 5 e^{2} m + 6 e^{2}} + \frac {3 a e^{4} x^{2} \left (d + e x\right )^{m}}{e^{2} m^{2} + 5 e^{2} m + 6 e^{2}} - \frac {c d^{4} \left (d + e x\right )^{m}}{e^{2} m^{2} + 5 e^{2} m + 6 e^{2}} + \frac {c d^{3} e m x \left (d + e x\right )^{m}}{e^{2} m^{2} + 5 e^{2} m + 6 e^{2}} + \frac {2 c d^{2} e^{2} m x^{2} \left (d + e x\right )^{m}}{e^{2} m^{2} + 5 e^{2} m + 6 e^{2}} + \frac {3 c d^{2} e^{2} x^{2} \left (d + e x\right )^{m}}{e^{2} m^{2} + 5 e^{2} m + 6 e^{2}} + \frac {c d e^{3} m x^{3} \left (d + e x\right )^{m}}{e^{2} m^{2} + 5 e^{2} m + 6 e^{2}} + \frac {2 c d e^{3} x^{3} \left (d + e x\right )^{m}}{e^{2} m^{2} + 5 e^{2} m + 6 e^{2}} & \text {otherwise} \end {cases} \]

[In]

integrate((e*x+d)**m*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2),x)

[Out]

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*log(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))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 174 vs. \(2 (52) = 104\).

Time = 0.20 (sec) , antiderivative size = 174, normalized size of antiderivative = 3.35 \[ \int (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=\frac {{\left (e^{2} {\left (m + 1\right )} x^{2} + d e m x - d^{2}\right )} {\left (e x + d\right )}^{m} a}{m^{2} + 3 \, m + 2} + \frac {{\left (e^{2} {\left (m + 1\right )} x^{2} + d e m x - d^{2}\right )} {\left (e x + d\right )}^{m} c d^{2}}{{\left (m^{2} + 3 \, m + 2\right )} e^{2}} + \frac {{\left (e x + d\right )}^{m + 1} a d}{m + 1} + \frac {{\left ({\left (m^{2} + 3 \, m + 2\right )} e^{3} x^{3} + {\left (m^{2} + m\right )} d e^{2} x^{2} - 2 \, d^{2} e m x + 2 \, d^{3}\right )} {\left (e x + d\right )}^{m} c d}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{2}} \]

[In]

integrate((e*x+d)^m*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2),x, algorithm="maxima")

[Out]

(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)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 219 vs. \(2 (52) = 104\).

Time = 0.26 (sec) , antiderivative size = 219, normalized size of antiderivative = 4.21 \[ \int (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=\frac {{\left (e x + d\right )}^{m} c d e^{3} m x^{3} + 2 \, {\left (e x + d\right )}^{m} c d^{2} e^{2} m x^{2} + {\left (e x + d\right )}^{m} a e^{4} m x^{2} + 2 \, {\left (e x + d\right )}^{m} c d e^{3} x^{3} + {\left (e x + d\right )}^{m} c d^{3} e m x + 2 \, {\left (e x + d\right )}^{m} a d e^{3} m x + 3 \, {\left (e x + d\right )}^{m} c d^{2} e^{2} x^{2} + 3 \, {\left (e x + d\right )}^{m} a e^{4} x^{2} + {\left (e x + d\right )}^{m} a d^{2} e^{2} m + 6 \, {\left (e x + d\right )}^{m} a d e^{3} x - {\left (e x + d\right )}^{m} c d^{4} + 3 \, {\left (e x + d\right )}^{m} a d^{2} e^{2}}{e^{2} m^{2} + 5 \, e^{2} m + 6 \, e^{2}} \]

[In]

integrate((e*x+d)^m*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2),x, algorithm="giac")

[Out]

((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)

Mupad [B] (verification not implemented)

Time = 9.86 (sec) , antiderivative size = 141, normalized size of antiderivative = 2.71 \[ \int (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx={\left (d+e\,x\right )}^m\,\left (\frac {x^2\,\left (3\,a\,e^2+3\,c\,d^2+a\,e^2\,m+2\,c\,d^2\,m\right )}{m^2+5\,m+6}+\frac {d^2\,\left (3\,a\,e^2-c\,d^2+a\,e^2\,m\right )}{e^2\,\left (m^2+5\,m+6\right )}+\frac {d\,x\,\left (6\,a\,e^2+2\,a\,e^2\,m+c\,d^2\,m\right )}{e\,\left (m^2+5\,m+6\right )}+\frac {c\,d\,e\,x^3\,\left (m+2\right )}{m^2+5\,m+6}\right ) \]

[In]

int((d + e*x)^m*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2),x)

[Out]

(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))