\(\int (d+e x)^4 (a d e+(c d^2+a e^2) x+c d e x^2) \, dx\) [1828]

Optimal result

Integrand size = 33, antiderivative size = 39 \[ \int (d+e x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=\frac {1}{6} \left (a-\frac {c d^2}{e^2}\right ) (d+e x)^6+\frac {c d (d+e x)^7}{7 e^2} \]

[Out]

1/6*(a-c*d^2/e^2)*(e*x+d)^6+1/7*c*d*(e*x+d)^7/e^2

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 39, 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} \[ \int (d+e x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=\frac {1}{6} (d+e x)^6 \left (a-\frac {c d^2}{e^2}\right )+\frac {c d (d+e x)^7}{7 e^2} \]

[In]

Int[(d + e*x)^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2),x]

[Out]

((a - (c*d^2)/e^2)*(d + e*x)^6)/6 + (c*d*(d + e*x)^7)/(7*e^2)

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)^5 \, dx \\ & = \int \left (\frac {\left (-c d^2+a e^2\right ) (d+e x)^5}{e}+\frac {c d (d+e x)^6}{e}\right ) \, dx \\ & = \frac {1}{6} \left (a-\frac {c d^2}{e^2}\right ) (d+e x)^6+\frac {c d (d+e x)^7}{7 e^2} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(117\) vs. \(2(39)=78\).

Time = 0.02 (sec) , antiderivative size = 117, normalized size of antiderivative = 3.00 \[ \int (d+e x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=\frac {1}{42} x \left (7 a e \left (6 d^5+15 d^4 e x+20 d^3 e^2 x^2+15 d^2 e^3 x^3+6 d e^4 x^4+e^5 x^5\right )+c d x \left (21 d^5+70 d^4 e x+105 d^3 e^2 x^2+84 d^2 e^3 x^3+35 d e^4 x^4+6 e^5 x^5\right )\right ) \]

[In]

Integrate[(d + e*x)^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2),x]

[Out]

(x*(7*a*e*(6*d^5 + 15*d^4*e*x + 20*d^3*e^2*x^2 + 15*d^2*e^3*x^3 + 6*d*e^4*x^4 + e^5*x^5) + c*d*x*(21*d^5 + 70*
d^4*e*x + 105*d^3*e^2*x^2 + 84*d^2*e^3*x^3 + 35*d*e^4*x^4 + 6*e^5*x^5)))/42

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(122\) vs. \(2(35)=70\).

Time = 2.11 (sec) , antiderivative size = 123, normalized size of antiderivative = 3.15

method	result	size
norman	\(\frac {c d \,e^{5} x^{7}}{7}+\left (\frac {1}{6} a \,e^{6}+\frac {5}{6} c \,d^{2} e^{4}\right ) x^{6}+\left (a \,e^{5} d +2 c \,d^{3} e^{3}\right ) x^{5}+\left (\frac {5}{2} a \,e^{4} d^{2}+\frac {5}{2} c \,d^{4} e^{2}\right ) x^{4}+\left (\frac {10}{3} a \,e^{3} d^{3}+\frac {5}{3} c \,d^{5} e \right ) x^{3}+\left (\frac {5}{2} a \,e^{2} d^{4}+\frac {1}{2} c \,d^{6}\right ) x^{2}+a e \,d^{5} x\)	\(123\)
gosper	\(\frac {x \left (6 c d \,e^{5} x^{6}+7 x^{5} a \,e^{6}+35 x^{5} c \,d^{2} e^{4}+42 a d \,e^{5} x^{4}+84 c \,d^{3} e^{3} x^{4}+105 x^{3} a \,e^{4} d^{2}+105 x^{3} c \,d^{4} e^{2}+140 x^{2} a \,e^{3} d^{3}+70 x^{2} c \,d^{5} e +105 x a \,e^{2} d^{4}+21 x c \,d^{6}+42 a e \,d^{5}\right )}{42}\)	\(128\)
risch	\(\frac {1}{7} c d \,e^{5} x^{7}+\frac {1}{6} x^{6} a \,e^{6}+\frac {5}{6} x^{6} c \,d^{2} e^{4}+a d \,e^{5} x^{5}+2 c \,d^{3} e^{3} x^{5}+\frac {5}{2} x^{4} a \,e^{4} d^{2}+\frac {5}{2} x^{4} c \,d^{4} e^{2}+\frac {10}{3} x^{3} a \,e^{3} d^{3}+\frac {5}{3} x^{3} c \,d^{5} e +\frac {5}{2} x^{2} a \,e^{2} d^{4}+\frac {1}{2} x^{2} c \,d^{6}+a e \,d^{5} x\)	\(128\)
parallelrisch	\(\frac {1}{7} c d \,e^{5} x^{7}+\frac {1}{6} x^{6} a \,e^{6}+\frac {5}{6} x^{6} c \,d^{2} e^{4}+a d \,e^{5} x^{5}+2 c \,d^{3} e^{3} x^{5}+\frac {5}{2} x^{4} a \,e^{4} d^{2}+\frac {5}{2} x^{4} c \,d^{4} e^{2}+\frac {10}{3} x^{3} a \,e^{3} d^{3}+\frac {5}{3} x^{3} c \,d^{5} e +\frac {5}{2} x^{2} a \,e^{2} d^{4}+\frac {1}{2} x^{2} c \,d^{6}+a e \,d^{5} x\)	\(128\)
default	\(\frac {c d \,e^{5} x^{7}}{7}+\frac {\left (4 c \,d^{2} e^{4}+e^{4} \left (e^{2} a +c \,d^{2}\right )\right ) x^{6}}{6}+\frac {\left (6 c \,d^{3} e^{3}+4 d \,e^{3} \left (e^{2} a +c \,d^{2}\right )+a \,e^{5} d \right ) x^{5}}{5}+\frac {\left (4 c \,d^{4} e^{2}+6 d^{2} e^{2} \left (e^{2} a +c \,d^{2}\right )+4 a \,e^{4} d^{2}\right ) x^{4}}{4}+\frac {\left (c \,d^{5} e +4 d^{3} e \left (e^{2} a +c \,d^{2}\right )+6 a \,e^{3} d^{3}\right ) x^{3}}{3}+\frac {\left (d^{4} \left (e^{2} a +c \,d^{2}\right )+4 a \,e^{2} d^{4}\right ) x^{2}}{2}+a e \,d^{5} x\)	\(198\)

[In]

int((e*x+d)^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2),x,method=_RETURNVERBOSE)

[Out]

1/7*c*d*e^5*x^7+(1/6*a*e^6+5/6*c*d^2*e^4)*x^6+(a*d*e^5+2*c*d^3*e^3)*x^5+(5/2*a*e^4*d^2+5/2*c*d^4*e^2)*x^4+(10/
3*a*e^3*d^3+5/3*c*d^5*e)*x^3+(5/2*a*e^2*d^4+1/2*c*d^6)*x^2+a*e*d^5*x

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 121 vs. \(2 (35) = 70\).

Time = 0.25 (sec) , antiderivative size = 121, normalized size of antiderivative = 3.10 \[ \int (d+e x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=\frac {1}{7} \, c d e^{5} x^{7} + a d^{5} e x + \frac {1}{6} \, {\left (5 \, c d^{2} e^{4} + a e^{6}\right )} x^{6} + {\left (2 \, c d^{3} e^{3} + a d e^{5}\right )} x^{5} + \frac {5}{2} \, {\left (c d^{4} e^{2} + a d^{2} e^{4}\right )} x^{4} + \frac {5}{3} \, {\left (c d^{5} e + 2 \, a d^{3} e^{3}\right )} x^{3} + \frac {1}{2} \, {\left (c d^{6} + 5 \, a d^{4} e^{2}\right )} x^{2} \]

[In]

integrate((e*x+d)^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2),x, algorithm="fricas")

[Out]

1/7*c*d*e^5*x^7 + a*d^5*e*x + 1/6*(5*c*d^2*e^4 + a*e^6)*x^6 + (2*c*d^3*e^3 + a*d*e^5)*x^5 + 5/2*(c*d^4*e^2 + a
*d^2*e^4)*x^4 + 5/3*(c*d^5*e + 2*a*d^3*e^3)*x^3 + 1/2*(c*d^6 + 5*a*d^4*e^2)*x^2

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 136 vs. \(2 (34) = 68\).

Time = 0.03 (sec) , antiderivative size = 136, normalized size of antiderivative = 3.49 \[ \int (d+e x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=a d^{5} e x + \frac {c d e^{5} x^{7}}{7} + x^{6} \left (\frac {a e^{6}}{6} + \frac {5 c d^{2} e^{4}}{6}\right ) + x^{5} \left (a d e^{5} + 2 c d^{3} e^{3}\right ) + x^{4} \cdot \left (\frac {5 a d^{2} e^{4}}{2} + \frac {5 c d^{4} e^{2}}{2}\right ) + x^{3} \cdot \left (\frac {10 a d^{3} e^{3}}{3} + \frac {5 c d^{5} e}{3}\right ) + x^{2} \cdot \left (\frac {5 a d^{4} e^{2}}{2} + \frac {c d^{6}}{2}\right ) \]

[In]

integrate((e*x+d)**4*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2),x)

[Out]

a*d**5*e*x + c*d*e**5*x**7/7 + x**6*(a*e**6/6 + 5*c*d**2*e**4/6) + x**5*(a*d*e**5 + 2*c*d**3*e**3) + x**4*(5*a
*d**2*e**4/2 + 5*c*d**4*e**2/2) + x**3*(10*a*d**3*e**3/3 + 5*c*d**5*e/3) + x**2*(5*a*d**4*e**2/2 + c*d**6/2)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 121 vs. \(2 (35) = 70\).

Time = 0.19 (sec) , antiderivative size = 121, normalized size of antiderivative = 3.10 \[ \int (d+e x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=\frac {1}{7} \, c d e^{5} x^{7} + a d^{5} e x + \frac {1}{6} \, {\left (5 \, c d^{2} e^{4} + a e^{6}\right )} x^{6} + {\left (2 \, c d^{3} e^{3} + a d e^{5}\right )} x^{5} + \frac {5}{2} \, {\left (c d^{4} e^{2} + a d^{2} e^{4}\right )} x^{4} + \frac {5}{3} \, {\left (c d^{5} e + 2 \, a d^{3} e^{3}\right )} x^{3} + \frac {1}{2} \, {\left (c d^{6} + 5 \, a d^{4} e^{2}\right )} x^{2} \]

[In]

integrate((e*x+d)^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2),x, algorithm="maxima")

[Out]

1/7*c*d*e^5*x^7 + a*d^5*e*x + 1/6*(5*c*d^2*e^4 + a*e^6)*x^6 + (2*c*d^3*e^3 + a*d*e^5)*x^5 + 5/2*(c*d^4*e^2 + a
*d^2*e^4)*x^4 + 5/3*(c*d^5*e + 2*a*d^3*e^3)*x^3 + 1/2*(c*d^6 + 5*a*d^4*e^2)*x^2

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 127 vs. \(2 (35) = 70\).

Time = 0.27 (sec) , antiderivative size = 127, normalized size of antiderivative = 3.26 \[ \int (d+e x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=\frac {1}{7} \, c d e^{5} x^{7} + \frac {5}{6} \, c d^{2} e^{4} x^{6} + \frac {1}{6} \, a e^{6} x^{6} + 2 \, c d^{3} e^{3} x^{5} + a d e^{5} x^{5} + \frac {5}{2} \, c d^{4} e^{2} x^{4} + \frac {5}{2} \, a d^{2} e^{4} x^{4} + \frac {5}{3} \, c d^{5} e x^{3} + \frac {10}{3} \, a d^{3} e^{3} x^{3} + \frac {1}{2} \, c d^{6} x^{2} + \frac {5}{2} \, a d^{4} e^{2} x^{2} + a d^{5} e x \]

[In]

integrate((e*x+d)^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2),x, algorithm="giac")

[Out]

1/7*c*d*e^5*x^7 + 5/6*c*d^2*e^4*x^6 + 1/6*a*e^6*x^6 + 2*c*d^3*e^3*x^5 + a*d*e^5*x^5 + 5/2*c*d^4*e^2*x^4 + 5/2*
a*d^2*e^4*x^4 + 5/3*c*d^5*e*x^3 + 10/3*a*d^3*e^3*x^3 + 1/2*c*d^6*x^2 + 5/2*a*d^4*e^2*x^2 + a*d^5*e*x

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 122, normalized size of antiderivative = 3.13 \[ \int (d+e x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=x^4\,\left (\frac {5\,c\,d^4\,e^2}{2}+\frac {5\,a\,d^2\,e^4}{2}\right )+x^2\,\left (\frac {c\,d^6}{2}+\frac {5\,a\,d^4\,e^2}{2}\right )+x^6\,\left (\frac {5\,c\,d^2\,e^4}{6}+\frac {a\,e^6}{6}\right )+x^5\,\left (2\,c\,d^3\,e^3+a\,d\,e^5\right )+x^3\,\left (\frac {5\,c\,d^5\,e}{3}+\frac {10\,a\,d^3\,e^3}{3}\right )+a\,d^5\,e\,x+\frac {c\,d\,e^5\,x^7}{7} \]

[In]

int((d + e*x)^4*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2),x)

[Out]

x^4*((5*a*d^2*e^4)/2 + (5*c*d^4*e^2)/2) + x^2*((c*d^6)/2 + (5*a*d^4*e^2)/2) + x^6*((a*e^6)/6 + (5*c*d^2*e^4)/6
) + x^5*(2*c*d^3*e^3 + a*d*e^5) + x^3*((10*a*d^3*e^3)/3 + (5*c*d^5*e)/3) + a*d^5*e*x + (c*d*e^5*x^7)/7