\(\int (d+e x)^3 (b x+c x^2)^2 \, dx\) [232]

Optimal result

Integrand size = 19, antiderivative size = 127 \[ \int (d+e x)^3 \left (b x+c x^2\right )^2 \, dx=\frac {1}{3} b^2 d^3 x^3+\frac {1}{4} b d^2 (2 c d+3 b e) x^4+\frac {1}{5} d \left (c^2 d^2+6 b c d e+3 b^2 e^2\right ) x^5+\frac {1}{6} e \left (3 c^2 d^2+6 b c d e+b^2 e^2\right ) x^6+\frac {1}{7} c e^2 (3 c d+2 b e) x^7+\frac {1}{8} c^2 e^3 x^8 \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 127, 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} \[ \int (d+e x)^3 \left (b x+c x^2\right )^2 \, dx=\frac {1}{6} e x^6 \left (b^2 e^2+6 b c d e+3 c^2 d^2\right )+\frac {1}{5} d x^5 \left (3 b^2 e^2+6 b c d e+c^2 d^2\right )+\frac {1}{3} b^2 d^3 x^3+\frac {1}{4} b d^2 x^4 (3 b e+2 c d)+\frac {1}{7} c e^2 x^7 (2 b e+3 c d)+\frac {1}{8} c^2 e^3 x^8 \]

[In]

Int[(d + e*x)^3*(b*x + c*x^2)^2,x]

[Out]

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

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^2 d^3 x^2+b d^2 (2 c d+3 b e) x^3+d \left (c^2 d^2+6 b c d e+3 b^2 e^2\right ) x^4+e \left (3 c^2 d^2+6 b c d e+b^2 e^2\right ) x^5+c e^2 (3 c d+2 b e) x^6+c^2 e^3 x^7\right ) \, dx \\ & = \frac {1}{3} b^2 d^3 x^3+\frac {1}{4} b d^2 (2 c d+3 b e) x^4+\frac {1}{5} d \left (c^2 d^2+6 b c d e+3 b^2 e^2\right ) x^5+\frac {1}{6} e \left (3 c^2 d^2+6 b c d e+b^2 e^2\right ) x^6+\frac {1}{7} c e^2 (3 c d+2 b e) x^7+\frac {1}{8} c^2 e^3 x^8 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.00 \[ \int (d+e x)^3 \left (b x+c x^2\right )^2 \, dx=\frac {1}{3} b^2 d^3 x^3+\frac {1}{4} b d^2 (2 c d+3 b e) x^4+\frac {1}{5} d \left (c^2 d^2+6 b c d e+3 b^2 e^2\right ) x^5+\frac {1}{6} e \left (3 c^2 d^2+6 b c d e+b^2 e^2\right ) x^6+\frac {1}{7} c e^2 (3 c d+2 b e) x^7+\frac {1}{8} c^2 e^3 x^8 \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.86 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.98

method	result	size
norman	\(\frac {c^{2} e^{3} x^{8}}{8}+\left (\frac {2}{7} e^{3} b c +\frac {3}{7} d \,e^{2} c^{2}\right ) x^{7}+\left (\frac {1}{6} e^{3} b^{2}+b c d \,e^{2}+\frac {1}{2} d^{2} e \,c^{2}\right ) x^{6}+\left (\frac {3}{5} b^{2} d \,e^{2}+\frac {6}{5} b c e \,d^{2}+\frac {1}{5} c^{2} d^{3}\right ) x^{5}+\left (\frac {3}{4} b^{2} d^{2} e +\frac {1}{2} b c \,d^{3}\right ) x^{4}+\frac {b^{2} d^{3} x^{3}}{3}\)	\(125\)
default	\(\frac {c^{2} e^{3} x^{8}}{8}+\frac {\left (2 e^{3} b c +3 d \,e^{2} c^{2}\right ) x^{7}}{7}+\frac {\left (e^{3} b^{2}+6 b c d \,e^{2}+3 d^{2} e \,c^{2}\right ) x^{6}}{6}+\frac {\left (3 b^{2} d \,e^{2}+6 b c e \,d^{2}+c^{2} d^{3}\right ) x^{5}}{5}+\frac {\left (3 b^{2} d^{2} e +2 b c \,d^{3}\right ) x^{4}}{4}+\frac {b^{2} d^{3} x^{3}}{3}\)	\(128\)
gosper	\(\frac {x^{3} \left (105 c^{2} e^{3} x^{5}+240 x^{4} e^{3} b c +360 x^{4} d \,e^{2} c^{2}+140 x^{3} e^{3} b^{2}+840 x^{3} b c d \,e^{2}+420 x^{3} d^{2} e \,c^{2}+504 x^{2} b^{2} d \,e^{2}+1008 x^{2} b c e \,d^{2}+168 c^{2} d^{3} x^{2}+630 x \,b^{2} d^{2} e +420 b c \,d^{3} x +280 b^{2} d^{3}\right )}{840}\)	\(134\)
risch	\(\frac {1}{8} c^{2} e^{3} x^{8}+\frac {2}{7} x^{7} e^{3} b c +\frac {3}{7} x^{7} d \,e^{2} c^{2}+\frac {1}{6} x^{6} e^{3} b^{2}+x^{6} b c d \,e^{2}+\frac {1}{2} x^{6} d^{2} e \,c^{2}+\frac {3}{5} x^{5} b^{2} d \,e^{2}+\frac {6}{5} x^{5} b c e \,d^{2}+\frac {1}{5} x^{5} c^{2} d^{3}+\frac {3}{4} x^{4} b^{2} d^{2} e +\frac {1}{2} x^{4} b c \,d^{3}+\frac {1}{3} b^{2} d^{3} x^{3}\)	\(135\)
parallelrisch	\(\frac {1}{8} c^{2} e^{3} x^{8}+\frac {2}{7} x^{7} e^{3} b c +\frac {3}{7} x^{7} d \,e^{2} c^{2}+\frac {1}{6} x^{6} e^{3} b^{2}+x^{6} b c d \,e^{2}+\frac {1}{2} x^{6} d^{2} e \,c^{2}+\frac {3}{5} x^{5} b^{2} d \,e^{2}+\frac {6}{5} x^{5} b c e \,d^{2}+\frac {1}{5} x^{5} c^{2} d^{3}+\frac {3}{4} x^{4} b^{2} d^{2} e +\frac {1}{2} x^{4} b c \,d^{3}+\frac {1}{3} b^{2} d^{3} x^{3}\)	\(135\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.00 \[ \int (d+e x)^3 \left (b x+c x^2\right )^2 \, dx=\frac {1}{8} \, c^{2} e^{3} x^{8} + \frac {1}{3} \, b^{2} d^{3} x^{3} + \frac {1}{7} \, {\left (3 \, c^{2} d e^{2} + 2 \, b c e^{3}\right )} x^{7} + \frac {1}{6} \, {\left (3 \, c^{2} d^{2} e + 6 \, b c d e^{2} + b^{2} e^{3}\right )} x^{6} + \frac {1}{5} \, {\left (c^{2} d^{3} + 6 \, b c d^{2} e + 3 \, b^{2} d e^{2}\right )} x^{5} + \frac {1}{4} \, {\left (2 \, b c d^{3} + 3 \, b^{2} d^{2} e\right )} x^{4} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.09 \[ \int (d+e x)^3 \left (b x+c x^2\right )^2 \, dx=\frac {b^{2} d^{3} x^{3}}{3} + \frac {c^{2} e^{3} x^{8}}{8} + x^{7} \cdot \left (\frac {2 b c e^{3}}{7} + \frac {3 c^{2} d e^{2}}{7}\right ) + x^{6} \left (\frac {b^{2} e^{3}}{6} + b c d e^{2} + \frac {c^{2} d^{2} e}{2}\right ) + x^{5} \cdot \left (\frac {3 b^{2} d e^{2}}{5} + \frac {6 b c d^{2} e}{5} + \frac {c^{2} d^{3}}{5}\right ) + x^{4} \cdot \left (\frac {3 b^{2} d^{2} e}{4} + \frac {b c d^{3}}{2}\right ) \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.00 \[ \int (d+e x)^3 \left (b x+c x^2\right )^2 \, dx=\frac {1}{8} \, c^{2} e^{3} x^{8} + \frac {1}{3} \, b^{2} d^{3} x^{3} + \frac {1}{7} \, {\left (3 \, c^{2} d e^{2} + 2 \, b c e^{3}\right )} x^{7} + \frac {1}{6} \, {\left (3 \, c^{2} d^{2} e + 6 \, b c d e^{2} + b^{2} e^{3}\right )} x^{6} + \frac {1}{5} \, {\left (c^{2} d^{3} + 6 \, b c d^{2} e + 3 \, b^{2} d e^{2}\right )} x^{5} + \frac {1}{4} \, {\left (2 \, b c d^{3} + 3 \, b^{2} d^{2} e\right )} x^{4} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.06 \[ \int (d+e x)^3 \left (b x+c x^2\right )^2 \, dx=\frac {1}{8} \, c^{2} e^{3} x^{8} + \frac {3}{7} \, c^{2} d e^{2} x^{7} + \frac {2}{7} \, b c e^{3} x^{7} + \frac {1}{2} \, c^{2} d^{2} e x^{6} + b c d e^{2} x^{6} + \frac {1}{6} \, b^{2} e^{3} x^{6} + \frac {1}{5} \, c^{2} d^{3} x^{5} + \frac {6}{5} \, b c d^{2} e x^{5} + \frac {3}{5} \, b^{2} d e^{2} x^{5} + \frac {1}{2} \, b c d^{3} x^{4} + \frac {3}{4} \, b^{2} d^{2} e x^{4} + \frac {1}{3} \, b^{2} d^{3} x^{3} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 9.66 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.93 \[ \int (d+e x)^3 \left (b x+c x^2\right )^2 \, dx=x^5\,\left (\frac {3\,b^2\,d\,e^2}{5}+\frac {6\,b\,c\,d^2\,e}{5}+\frac {c^2\,d^3}{5}\right )+x^6\,\left (\frac {b^2\,e^3}{6}+b\,c\,d\,e^2+\frac {c^2\,d^2\,e}{2}\right )+\frac {b^2\,d^3\,x^3}{3}+\frac {c^2\,e^3\,x^8}{8}+\frac {b\,d^2\,x^4\,\left (3\,b\,e+2\,c\,d\right )}{4}+\frac {c\,e^2\,x^7\,\left (2\,b\,e+3\,c\,d\right )}{7} \]

[In]

int((b*x + c*x^2)^2*(d + e*x)^3,x)

[Out]

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