\(\int (d+e x)^3 (b x+c x^2)^3 \, dx\) [245]

Optimal result

Integrand size = 19, antiderivative size = 162 \[ \int (d+e x)^3 \left (b x+c x^2\right )^3 \, dx=\frac {1}{4} b^3 d^3 x^4+\frac {3}{5} b^2 d^2 (c d+b e) x^5+\frac {1}{2} b d \left (c^2 d^2+3 b c d e+b^2 e^2\right ) x^6+\frac {1}{7} (c d+b e) \left (c^2 d^2+8 b c d e+b^2 e^2\right ) x^7+\frac {3}{8} c e \left (c^2 d^2+3 b c d e+b^2 e^2\right ) x^8+\frac {1}{3} c^2 e^2 (c d+b e) x^9+\frac {1}{10} c^3 e^3 x^{10} \]

[Out]

1/4*b^3*d^3*x^4+3/5*b^2*d^2*(b*e+c*d)*x^5+1/2*b*d*(b^2*e^2+3*b*c*d*e+c^2*d^2)*x^6+1/7*(b*e+c*d)*(b^2*e^2+8*b*c
*d*e+c^2*d^2)*x^7+3/8*c*e*(b^2*e^2+3*b*c*d*e+c^2*d^2)*x^8+1/3*c^2*e^2*(b*e+c*d)*x^9+1/10*c^3*e^3*x^10

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {712} \[ \int (d+e x)^3 \left (b x+c x^2\right )^3 \, dx=\frac {1}{4} b^3 d^3 x^4+\frac {3}{8} c e x^8 \left (b^2 e^2+3 b c d e+c^2 d^2\right )+\frac {1}{7} x^7 (b e+c d) \left (b^2 e^2+8 b c d e+c^2 d^2\right )+\frac {1}{2} b d x^6 \left (b^2 e^2+3 b c d e+c^2 d^2\right )+\frac {3}{5} b^2 d^2 x^5 (b e+c d)+\frac {1}{3} c^2 e^2 x^9 (b e+c d)+\frac {1}{10} c^3 e^3 x^{10} \]

[In]

Int[(d + e*x)^3*(b*x + c*x^2)^3,x]

[Out]

(b^3*d^3*x^4)/4 + (3*b^2*d^2*(c*d + b*e)*x^5)/5 + (b*d*(c^2*d^2 + 3*b*c*d*e + b^2*e^2)*x^6)/2 + ((c*d + b*e)*(
c^2*d^2 + 8*b*c*d*e + b^2*e^2)*x^7)/7 + (3*c*e*(c^2*d^2 + 3*b*c*d*e + b^2*e^2)*x^8)/8 + (c^2*e^2*(c*d + b*e)*x
^9)/3 + (c^3*e^3*x^10)/10

Rule 712

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] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps \begin{align*} \text {integral}& = \int \left (b^3 d^3 x^3+3 b^2 d^2 (c d+b e) x^4+3 b d \left (c^2 d^2+3 b c d e+b^2 e^2\right ) x^5+(c d+b e) \left (c^2 d^2+8 b c d e+b^2 e^2\right ) x^6+3 c e \left (c^2 d^2+3 b c d e+b^2 e^2\right ) x^7+3 c^2 e^2 (c d+b e) x^8+c^3 e^3 x^9\right ) \, dx \\ & = \frac {1}{4} b^3 d^3 x^4+\frac {3}{5} b^2 d^2 (c d+b e) x^5+\frac {1}{2} b d \left (c^2 d^2+3 b c d e+b^2 e^2\right ) x^6+\frac {1}{7} (c d+b e) \left (c^2 d^2+8 b c d e+b^2 e^2\right ) x^7+\frac {3}{8} c e \left (c^2 d^2+3 b c d e+b^2 e^2\right ) x^8+\frac {1}{3} c^2 e^2 (c d+b e) x^9+\frac {1}{10} c^3 e^3 x^{10} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.04 \[ \int (d+e x)^3 \left (b x+c x^2\right )^3 \, dx=\frac {1}{4} b^3 d^3 x^4+\frac {3}{5} b^2 d^2 (c d+b e) x^5+\frac {1}{2} b d \left (c^2 d^2+3 b c d e+b^2 e^2\right ) x^6+\frac {1}{7} \left (c^3 d^3+9 b c^2 d^2 e+9 b^2 c d e^2+b^3 e^3\right ) x^7+\frac {3}{8} c e \left (c^2 d^2+3 b c d e+b^2 e^2\right ) x^8+\frac {1}{3} c^2 e^2 (c d+b e) x^9+\frac {1}{10} c^3 e^3 x^{10} \]

[In]

Integrate[(d + e*x)^3*(b*x + c*x^2)^3,x]

[Out]

(b^3*d^3*x^4)/4 + (3*b^2*d^2*(c*d + b*e)*x^5)/5 + (b*d*(c^2*d^2 + 3*b*c*d*e + b^2*e^2)*x^6)/2 + ((c^3*d^3 + 9*
b*c^2*d^2*e + 9*b^2*c*d*e^2 + b^3*e^3)*x^7)/7 + (3*c*e*(c^2*d^2 + 3*b*c*d*e + b^2*e^2)*x^8)/8 + (c^2*e^2*(c*d
+ b*e)*x^9)/3 + (c^3*e^3*x^10)/10

Maple [A] (verified)

Time = 1.98 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.09

method	result	size
norman	\(\frac {c^{3} e^{3} x^{10}}{10}+\left (\frac {1}{3} e^{3} b \,c^{2}+\frac {1}{3} d \,e^{2} c^{3}\right ) x^{9}+\left (\frac {3}{8} e^{3} b^{2} c +\frac {9}{8} d \,e^{2} b \,c^{2}+\frac {3}{8} d^{2} e \,c^{3}\right ) x^{8}+\left (\frac {1}{7} b^{3} e^{3}+\frac {9}{7} b^{2} d \,e^{2} c +\frac {9}{7} b \,c^{2} d^{2} e +\frac {1}{7} c^{3} d^{3}\right ) x^{7}+\left (\frac {1}{2} b^{3} d \,e^{2}+\frac {3}{2} d^{2} e \,b^{2} c +\frac {1}{2} d^{3} b \,c^{2}\right ) x^{6}+\left (\frac {3}{5} d^{2} e \,b^{3}+\frac {3}{5} b^{2} c \,d^{3}\right ) x^{5}+\frac {d^{3} x^{4} b^{3}}{4}\)	\(177\)
default	\(\frac {c^{3} e^{3} x^{10}}{10}+\frac {\left (3 e^{3} b \,c^{2}+3 d \,e^{2} c^{3}\right ) x^{9}}{9}+\frac {\left (3 e^{3} b^{2} c +9 d \,e^{2} b \,c^{2}+3 d^{2} e \,c^{3}\right ) x^{8}}{8}+\frac {\left (b^{3} e^{3}+9 b^{2} d \,e^{2} c +9 b \,c^{2} d^{2} e +c^{3} d^{3}\right ) x^{7}}{7}+\frac {\left (3 b^{3} d \,e^{2}+9 d^{2} e \,b^{2} c +3 d^{3} b \,c^{2}\right ) x^{6}}{6}+\frac {\left (3 d^{2} e \,b^{3}+3 b^{2} c \,d^{3}\right ) x^{5}}{5}+\frac {d^{3} x^{4} b^{3}}{4}\)	\(180\)
gosper	\(\frac {x^{4} \left (84 c^{3} e^{3} x^{6}+280 x^{5} e^{3} b \,c^{2}+280 x^{5} d \,e^{2} c^{3}+315 x^{4} e^{3} b^{2} c +945 x^{4} d \,e^{2} b \,c^{2}+315 x^{4} d^{2} e \,c^{3}+120 x^{3} b^{3} e^{3}+1080 x^{3} b^{2} d \,e^{2} c +1080 x^{3} b \,c^{2} d^{2} e +120 x^{3} c^{3} d^{3}+420 x^{2} b^{3} d \,e^{2}+1260 x^{2} d^{2} e \,b^{2} c +420 x^{2} d^{3} b \,c^{2}+504 x \,d^{2} e \,b^{3}+504 x \,b^{2} c \,d^{3}+210 b^{3} d^{3}\right )}{840}\)	\(192\)
risch	\(\frac {1}{10} c^{3} e^{3} x^{10}+\frac {1}{3} x^{9} e^{3} b \,c^{2}+\frac {1}{3} c^{3} d \,e^{2} x^{9}+\frac {3}{8} x^{8} e^{3} b^{2} c +\frac {9}{8} x^{8} d \,e^{2} b \,c^{2}+\frac {3}{8} x^{8} d^{2} e \,c^{3}+\frac {1}{7} x^{7} b^{3} e^{3}+\frac {9}{7} x^{7} b^{2} d \,e^{2} c +\frac {9}{7} x^{7} b \,c^{2} d^{2} e +\frac {1}{7} x^{7} c^{3} d^{3}+\frac {1}{2} x^{6} b^{3} d \,e^{2}+\frac {3}{2} x^{6} d^{2} e \,b^{2} c +\frac {1}{2} x^{6} d^{3} b \,c^{2}+\frac {3}{5} x^{5} d^{2} e \,b^{3}+\frac {3}{5} b^{2} c \,d^{3} x^{5}+\frac {1}{4} d^{3} x^{4} b^{3}\)	\(194\)
parallelrisch	\(\frac {1}{10} c^{3} e^{3} x^{10}+\frac {1}{3} x^{9} e^{3} b \,c^{2}+\frac {1}{3} c^{3} d \,e^{2} x^{9}+\frac {3}{8} x^{8} e^{3} b^{2} c +\frac {9}{8} x^{8} d \,e^{2} b \,c^{2}+\frac {3}{8} x^{8} d^{2} e \,c^{3}+\frac {1}{7} x^{7} b^{3} e^{3}+\frac {9}{7} x^{7} b^{2} d \,e^{2} c +\frac {9}{7} x^{7} b \,c^{2} d^{2} e +\frac {1}{7} x^{7} c^{3} d^{3}+\frac {1}{2} x^{6} b^{3} d \,e^{2}+\frac {3}{2} x^{6} d^{2} e \,b^{2} c +\frac {1}{2} x^{6} d^{3} b \,c^{2}+\frac {3}{5} x^{5} d^{2} e \,b^{3}+\frac {3}{5} b^{2} c \,d^{3} x^{5}+\frac {1}{4} d^{3} x^{4} b^{3}\)	\(194\)

[In]

int((e*x+d)^3*(c*x^2+b*x)^3,x,method=_RETURNVERBOSE)

[Out]

1/10*c^3*e^3*x^10+(1/3*e^3*b*c^2+1/3*d*e^2*c^3)*x^9+(3/8*e^3*b^2*c+9/8*d*e^2*b*c^2+3/8*d^2*e*c^3)*x^8+(1/7*b^3
*e^3+9/7*b^2*d*e^2*c+9/7*b*c^2*d^2*e+1/7*c^3*d^3)*x^7+(1/2*b^3*d*e^2+3/2*d^2*e*b^2*c+1/2*d^3*b*c^2)*x^6+(3/5*d
^2*e*b^3+3/5*b^2*c*d^3)*x^5+1/4*d^3*x^4*b^3

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.06 \[ \int (d+e x)^3 \left (b x+c x^2\right )^3 \, dx=\frac {1}{10} \, c^{3} e^{3} x^{10} + \frac {1}{4} \, b^{3} d^{3} x^{4} + \frac {1}{3} \, {\left (c^{3} d e^{2} + b c^{2} e^{3}\right )} x^{9} + \frac {3}{8} \, {\left (c^{3} d^{2} e + 3 \, b c^{2} d e^{2} + b^{2} c e^{3}\right )} x^{8} + \frac {1}{7} \, {\left (c^{3} d^{3} + 9 \, b c^{2} d^{2} e + 9 \, b^{2} c d e^{2} + b^{3} e^{3}\right )} x^{7} + \frac {1}{2} \, {\left (b c^{2} d^{3} + 3 \, b^{2} c d^{2} e + b^{3} d e^{2}\right )} x^{6} + \frac {3}{5} \, {\left (b^{2} c d^{3} + b^{3} d^{2} e\right )} x^{5} \]

[In]

integrate((e*x+d)^3*(c*x^2+b*x)^3,x, algorithm="fricas")

[Out]

1/10*c^3*e^3*x^10 + 1/4*b^3*d^3*x^4 + 1/3*(c^3*d*e^2 + b*c^2*e^3)*x^9 + 3/8*(c^3*d^2*e + 3*b*c^2*d*e^2 + b^2*c
*e^3)*x^8 + 1/7*(c^3*d^3 + 9*b*c^2*d^2*e + 9*b^2*c*d*e^2 + b^3*e^3)*x^7 + 1/2*(b*c^2*d^3 + 3*b^2*c*d^2*e + b^3
*d*e^2)*x^6 + 3/5*(b^2*c*d^3 + b^3*d^2*e)*x^5

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.23 \[ \int (d+e x)^3 \left (b x+c x^2\right )^3 \, dx=\frac {b^{3} d^{3} x^{4}}{4} + \frac {c^{3} e^{3} x^{10}}{10} + x^{9} \left (\frac {b c^{2} e^{3}}{3} + \frac {c^{3} d e^{2}}{3}\right ) + x^{8} \cdot \left (\frac {3 b^{2} c e^{3}}{8} + \frac {9 b c^{2} d e^{2}}{8} + \frac {3 c^{3} d^{2} e}{8}\right ) + x^{7} \left (\frac {b^{3} e^{3}}{7} + \frac {9 b^{2} c d e^{2}}{7} + \frac {9 b c^{2} d^{2} e}{7} + \frac {c^{3} d^{3}}{7}\right ) + x^{6} \left (\frac {b^{3} d e^{2}}{2} + \frac {3 b^{2} c d^{2} e}{2} + \frac {b c^{2} d^{3}}{2}\right ) + x^{5} \cdot \left (\frac {3 b^{3} d^{2} e}{5} + \frac {3 b^{2} c d^{3}}{5}\right ) \]

[In]

integrate((e*x+d)**3*(c*x**2+b*x)**3,x)

[Out]

b**3*d**3*x**4/4 + c**3*e**3*x**10/10 + x**9*(b*c**2*e**3/3 + c**3*d*e**2/3) + x**8*(3*b**2*c*e**3/8 + 9*b*c**
2*d*e**2/8 + 3*c**3*d**2*e/8) + x**7*(b**3*e**3/7 + 9*b**2*c*d*e**2/7 + 9*b*c**2*d**2*e/7 + c**3*d**3/7) + x**
6*(b**3*d*e**2/2 + 3*b**2*c*d**2*e/2 + b*c**2*d**3/2) + x**5*(3*b**3*d**2*e/5 + 3*b**2*c*d**3/5)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.06 \[ \int (d+e x)^3 \left (b x+c x^2\right )^3 \, dx=\frac {1}{10} \, c^{3} e^{3} x^{10} + \frac {1}{4} \, b^{3} d^{3} x^{4} + \frac {1}{3} \, {\left (c^{3} d e^{2} + b c^{2} e^{3}\right )} x^{9} + \frac {3}{8} \, {\left (c^{3} d^{2} e + 3 \, b c^{2} d e^{2} + b^{2} c e^{3}\right )} x^{8} + \frac {1}{7} \, {\left (c^{3} d^{3} + 9 \, b c^{2} d^{2} e + 9 \, b^{2} c d e^{2} + b^{3} e^{3}\right )} x^{7} + \frac {1}{2} \, {\left (b c^{2} d^{3} + 3 \, b^{2} c d^{2} e + b^{3} d e^{2}\right )} x^{6} + \frac {3}{5} \, {\left (b^{2} c d^{3} + b^{3} d^{2} e\right )} x^{5} \]

[In]

integrate((e*x+d)^3*(c*x^2+b*x)^3,x, algorithm="maxima")

[Out]

1/10*c^3*e^3*x^10 + 1/4*b^3*d^3*x^4 + 1/3*(c^3*d*e^2 + b*c^2*e^3)*x^9 + 3/8*(c^3*d^2*e + 3*b*c^2*d*e^2 + b^2*c
*e^3)*x^8 + 1/7*(c^3*d^3 + 9*b*c^2*d^2*e + 9*b^2*c*d*e^2 + b^3*e^3)*x^7 + 1/2*(b*c^2*d^3 + 3*b^2*c*d^2*e + b^3
*d*e^2)*x^6 + 3/5*(b^2*c*d^3 + b^3*d^2*e)*x^5

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.19 \[ \int (d+e x)^3 \left (b x+c x^2\right )^3 \, dx=\frac {1}{10} \, c^{3} e^{3} x^{10} + \frac {1}{3} \, c^{3} d e^{2} x^{9} + \frac {1}{3} \, b c^{2} e^{3} x^{9} + \frac {3}{8} \, c^{3} d^{2} e x^{8} + \frac {9}{8} \, b c^{2} d e^{2} x^{8} + \frac {3}{8} \, b^{2} c e^{3} x^{8} + \frac {1}{7} \, c^{3} d^{3} x^{7} + \frac {9}{7} \, b c^{2} d^{2} e x^{7} + \frac {9}{7} \, b^{2} c d e^{2} x^{7} + \frac {1}{7} \, b^{3} e^{3} x^{7} + \frac {1}{2} \, b c^{2} d^{3} x^{6} + \frac {3}{2} \, b^{2} c d^{2} e x^{6} + \frac {1}{2} \, b^{3} d e^{2} x^{6} + \frac {3}{5} \, b^{2} c d^{3} x^{5} + \frac {3}{5} \, b^{3} d^{2} e x^{5} + \frac {1}{4} \, b^{3} d^{3} x^{4} \]

[In]

integrate((e*x+d)^3*(c*x^2+b*x)^3,x, algorithm="giac")

[Out]

1/10*c^3*e^3*x^10 + 1/3*c^3*d*e^2*x^9 + 1/3*b*c^2*e^3*x^9 + 3/8*c^3*d^2*e*x^8 + 9/8*b*c^2*d*e^2*x^8 + 3/8*b^2*
c*e^3*x^8 + 1/7*c^3*d^3*x^7 + 9/7*b*c^2*d^2*e*x^7 + 9/7*b^2*c*d*e^2*x^7 + 1/7*b^3*e^3*x^7 + 1/2*b*c^2*d^3*x^6
+ 3/2*b^2*c*d^2*e*x^6 + 1/2*b^3*d*e^2*x^6 + 3/5*b^2*c*d^3*x^5 + 3/5*b^3*d^2*e*x^5 + 1/4*b^3*d^3*x^4

Mupad [B] (verification not implemented)

Time = 9.51 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.96 \[ \int (d+e x)^3 \left (b x+c x^2\right )^3 \, dx=x^7\,\left (\frac {b^3\,e^3}{7}+\frac {9\,b^2\,c\,d\,e^2}{7}+\frac {9\,b\,c^2\,d^2\,e}{7}+\frac {c^3\,d^3}{7}\right )+\frac {b^3\,d^3\,x^4}{4}+\frac {c^3\,e^3\,x^{10}}{10}+\frac {b\,d\,x^6\,\left (b^2\,e^2+3\,b\,c\,d\,e+c^2\,d^2\right )}{2}+\frac {3\,c\,e\,x^8\,\left (b^2\,e^2+3\,b\,c\,d\,e+c^2\,d^2\right )}{8}+\frac {3\,b^2\,d^2\,x^5\,\left (b\,e+c\,d\right )}{5}+\frac {c^2\,e^2\,x^9\,\left (b\,e+c\,d\right )}{3} \]

[In]

int((b*x + c*x^2)^3*(d + e*x)^3,x)

[Out]

x^7*((b^3*e^3)/7 + (c^3*d^3)/7 + (9*b*c^2*d^2*e)/7 + (9*b^2*c*d*e^2)/7) + (b^3*d^3*x^4)/4 + (c^3*e^3*x^10)/10
+ (b*d*x^6*(b^2*e^2 + c^2*d^2 + 3*b*c*d*e))/2 + (3*c*e*x^8*(b^2*e^2 + c^2*d^2 + 3*b*c*d*e))/8 + (3*b^2*d^2*x^5
*(b*e + c*d))/5 + (c^2*e^2*x^9*(b*e + c*d))/3