\(\int (d+e x)^3 (a+b x+c x^2) \, dx\) [2108]

Optimal result

Integrand size = 18, antiderivative size = 69 \[ \int (d+e x)^3 \left (a+b x+c x^2\right ) \, dx=\frac {\left (c d^2-b d e+a e^2\right ) (d+e x)^4}{4 e^3}-\frac {(2 c d-b e) (d+e x)^5}{5 e^3}+\frac {c (d+e x)^6}{6 e^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {712} \[ \int (d+e x)^3 \left (a+b x+c x^2\right ) \, dx=\frac {(d+e x)^4 \left (a e^2-b d e+c d^2\right )}{4 e^3}-\frac {(d+e x)^5 (2 c d-b e)}{5 e^3}+\frac {c (d+e x)^6}{6 e^3} \]

[In]

Int[(d + e*x)^3*(a + b*x + c*x^2),x]

[Out]

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

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.51 \[ \int (d+e x)^3 \left (a+b x+c x^2\right ) \, dx=a d^3 x+\frac {1}{2} d^2 (b d+3 a e) x^2+\frac {1}{3} d \left (c d^2+3 b d e+3 a e^2\right ) x^3+\frac {1}{4} e \left (3 c d^2+3 b d e+a e^2\right ) x^4+\frac {1}{5} e^2 (3 c d+b e) x^5+\frac {1}{6} c e^3 x^6 \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 2.48 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.46

method	result	size
norman	\(\frac {e^{3} c \,x^{6}}{6}+\left (\frac {1}{5} e^{3} b +\frac {3}{5} d \,e^{2} c \right ) x^{5}+\left (\frac {1}{4} a \,e^{3}+\frac {3}{4} b d \,e^{2}+\frac {3}{4} d^{2} e c \right ) x^{4}+\left (a d \,e^{2}+b \,d^{2} e +\frac {1}{3} d^{3} c \right ) x^{3}+\left (\frac {3}{2} a e \,d^{2}+\frac {1}{2} b \,d^{3}\right ) x^{2}+a \,d^{3} x\)	\(101\)
default	\(\frac {e^{3} c \,x^{6}}{6}+\frac {\left (e^{3} b +3 d \,e^{2} c \right ) x^{5}}{5}+\frac {\left (a \,e^{3}+3 b d \,e^{2}+3 d^{2} e c \right ) x^{4}}{4}+\frac {\left (3 a d \,e^{2}+3 b \,d^{2} e +d^{3} c \right ) x^{3}}{3}+\frac {\left (3 a e \,d^{2}+b \,d^{3}\right ) x^{2}}{2}+a \,d^{3} x\)	\(103\)
gosper	\(\frac {1}{6} e^{3} c \,x^{6}+\frac {1}{5} x^{5} e^{3} b +\frac {3}{5} d \,e^{2} c \,x^{5}+\frac {1}{4} x^{4} a \,e^{3}+\frac {3}{4} x^{4} b d \,e^{2}+\frac {3}{4} x^{4} d^{2} e c +x^{3} a d \,e^{2}+x^{3} b \,d^{2} e +\frac {1}{3} c \,d^{3} x^{3}+\frac {3}{2} a e \,d^{2} x^{2}+\frac {1}{2} b \,d^{3} x^{2}+a \,d^{3} x\)	\(111\)
risch	\(\frac {1}{6} e^{3} c \,x^{6}+\frac {1}{5} x^{5} e^{3} b +\frac {3}{5} d \,e^{2} c \,x^{5}+\frac {1}{4} x^{4} a \,e^{3}+\frac {3}{4} x^{4} b d \,e^{2}+\frac {3}{4} x^{4} d^{2} e c +x^{3} a d \,e^{2}+x^{3} b \,d^{2} e +\frac {1}{3} c \,d^{3} x^{3}+\frac {3}{2} a e \,d^{2} x^{2}+\frac {1}{2} b \,d^{3} x^{2}+a \,d^{3} x\)	\(111\)
parallelrisch	\(\frac {1}{6} e^{3} c \,x^{6}+\frac {1}{5} x^{5} e^{3} b +\frac {3}{5} d \,e^{2} c \,x^{5}+\frac {1}{4} x^{4} a \,e^{3}+\frac {3}{4} x^{4} b d \,e^{2}+\frac {3}{4} x^{4} d^{2} e c +x^{3} a d \,e^{2}+x^{3} b \,d^{2} e +\frac {1}{3} c \,d^{3} x^{3}+\frac {3}{2} a e \,d^{2} x^{2}+\frac {1}{2} b \,d^{3} x^{2}+a \,d^{3} x\)	\(111\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.48 \[ \int (d+e x)^3 \left (a+b x+c x^2\right ) \, dx=\frac {1}{6} \, c e^{3} x^{6} + \frac {1}{5} \, {\left (3 \, c d e^{2} + b e^{3}\right )} x^{5} + a d^{3} x + \frac {1}{4} \, {\left (3 \, c d^{2} e + 3 \, b d e^{2} + a e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (c d^{3} + 3 \, b d^{2} e + 3 \, a d e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (b d^{3} + 3 \, a d^{2} e\right )} x^{2} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.59 \[ \int (d+e x)^3 \left (a+b x+c x^2\right ) \, dx=a d^{3} x + \frac {c e^{3} x^{6}}{6} + x^{5} \left (\frac {b e^{3}}{5} + \frac {3 c d e^{2}}{5}\right ) + x^{4} \left (\frac {a e^{3}}{4} + \frac {3 b d e^{2}}{4} + \frac {3 c d^{2} e}{4}\right ) + x^{3} \left (a d e^{2} + b d^{2} e + \frac {c d^{3}}{3}\right ) + x^{2} \cdot \left (\frac {3 a d^{2} e}{2} + \frac {b d^{3}}{2}\right ) \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.48 \[ \int (d+e x)^3 \left (a+b x+c x^2\right ) \, dx=\frac {1}{6} \, c e^{3} x^{6} + \frac {1}{5} \, {\left (3 \, c d e^{2} + b e^{3}\right )} x^{5} + a d^{3} x + \frac {1}{4} \, {\left (3 \, c d^{2} e + 3 \, b d e^{2} + a e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (c d^{3} + 3 \, b d^{2} e + 3 \, a d e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (b d^{3} + 3 \, a d^{2} e\right )} x^{2} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.59 \[ \int (d+e x)^3 \left (a+b x+c x^2\right ) \, dx=\frac {1}{6} \, c e^{3} x^{6} + \frac {3}{5} \, c d e^{2} x^{5} + \frac {1}{5} \, b e^{3} x^{5} + \frac {3}{4} \, c d^{2} e x^{4} + \frac {3}{4} \, b d e^{2} x^{4} + \frac {1}{4} \, a e^{3} x^{4} + \frac {1}{3} \, c d^{3} x^{3} + b d^{2} e x^{3} + a d e^{2} x^{3} + \frac {1}{2} \, b d^{3} x^{2} + \frac {3}{2} \, a d^{2} e x^{2} + a d^{3} x \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.45 \[ \int (d+e x)^3 \left (a+b x+c x^2\right ) \, dx=x^2\,\left (\frac {b\,d^3}{2}+\frac {3\,a\,e\,d^2}{2}\right )+x^5\,\left (\frac {b\,e^3}{5}+\frac {3\,c\,d\,e^2}{5}\right )+x^3\,\left (\frac {c\,d^3}{3}+b\,d^2\,e+a\,d\,e^2\right )+x^4\,\left (\frac {3\,c\,d^2\,e}{4}+\frac {3\,b\,d\,e^2}{4}+\frac {a\,e^3}{4}\right )+\frac {c\,e^3\,x^6}{6}+a\,d^3\,x \]

[In]

int((d + e*x)^3*(a + b*x + c*x^2),x)

[Out]

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