3.1452 \(\int (d+e x)^3 (a^2+2 a b x+b^2 x^2) \, dx\)

Optimal. Leaf size=65 \[ -\frac {2 b (d+e x)^5 (b d-a e)}{5 e^3}+\frac {(d+e x)^4 (b d-a e)^2}{4 e^3}+\frac {b^2 (d+e x)^6}{6 e^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.07, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {27, 43} \[ -\frac {2 b (d+e x)^5 (b d-a e)}{5 e^3}+\frac {(d+e x)^4 (b d-a e)^2}{4 e^3}+\frac {b^2 (d+e x)^6}{6 e^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 43

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])

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 122, normalized size = 1.88 \[ \frac {1}{4} e x^4 \left (a^2 e^2+6 a b d e+3 b^2 d^2\right )+\frac {1}{3} d x^3 \left (3 a^2 e^2+6 a b d e+b^2 d^2\right )+a^2 d^3 x+\frac {1}{2} a d^2 x^2 (3 a e+2 b d)+\frac {1}{5} b e^2 x^5 (2 a e+3 b d)+\frac {1}{6} b^2 e^3 x^6 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [B]  time = 1.17, size = 130, normalized size = 2.00 \[ \frac {1}{6} x^{6} e^{3} b^{2} + \frac {3}{5} x^{5} e^{2} d b^{2} + \frac {2}{5} x^{5} e^{3} b a + \frac {3}{4} x^{4} e d^{2} b^{2} + \frac {3}{2} x^{4} e^{2} d b a + \frac {1}{4} x^{4} e^{3} a^{2} + \frac {1}{3} x^{3} d^{3} b^{2} + 2 x^{3} e d^{2} b a + x^{3} e^{2} d a^{2} + x^{2} d^{3} b a + \frac {3}{2} x^{2} e d^{2} a^{2} + x d^{3} a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [B]  time = 0.16, size = 127, normalized size = 1.95 \[ \frac {1}{6} \, b^{2} x^{6} e^{3} + \frac {3}{5} \, b^{2} d x^{5} e^{2} + \frac {3}{4} \, b^{2} d^{2} x^{4} e + \frac {1}{3} \, b^{2} d^{3} x^{3} + \frac {2}{5} \, a b x^{5} e^{3} + \frac {3}{2} \, a b d x^{4} e^{2} + 2 \, a b d^{2} x^{3} e + a b d^{3} x^{2} + \frac {1}{4} \, a^{2} x^{4} e^{3} + a^{2} d x^{3} e^{2} + \frac {3}{2} \, a^{2} d^{2} x^{2} e + a^{2} d^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [B]  time = 0.04, size = 125, normalized size = 1.92 \[ \frac {b^{2} e^{3} x^{6}}{6}+a^{2} d^{3} x +\frac {\left (2 a b \,e^{3}+3 b^{2} d \,e^{2}\right ) x^{5}}{5}+\frac {\left (a^{2} e^{3}+6 d \,e^{2} a b +3 d^{2} e \,b^{2}\right ) x^{4}}{4}+\frac {\left (3 d \,e^{2} a^{2}+6 d^{2} e a b +d^{3} b^{2}\right ) x^{3}}{3}+\frac {\left (3 d^{2} e \,a^{2}+2 d^{3} a b \right ) x^{2}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^3*(b^2*x^2+2*a*b*x+a^2),x)

[Out]

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

________________________________________________________________________________________

maxima [B]  time = 1.37, size = 124, normalized size = 1.91 \[ \frac {1}{6} \, b^{2} e^{3} x^{6} + a^{2} d^{3} x + \frac {1}{5} \, {\left (3 \, b^{2} d e^{2} + 2 \, a b e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (3 \, b^{2} d^{2} e + 6 \, a b d e^{2} + a^{2} e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (b^{2} d^{3} + 6 \, a b d^{2} e + 3 \, a^{2} d e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (2 \, a b d^{3} + 3 \, a^{2} d^{2} e\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 0.51, size = 115, normalized size = 1.77 \[ x^3\,\left (a^2\,d\,e^2+2\,a\,b\,d^2\,e+\frac {b^2\,d^3}{3}\right )+x^4\,\left (\frac {a^2\,e^3}{4}+\frac {3\,a\,b\,d\,e^2}{2}+\frac {3\,b^2\,d^2\,e}{4}\right )+a^2\,d^3\,x+\frac {b^2\,e^3\,x^6}{6}+\frac {a\,d^2\,x^2\,\left (3\,a\,e+2\,b\,d\right )}{2}+\frac {b\,e^2\,x^5\,\left (2\,a\,e+3\,b\,d\right )}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^3*(a^2 + b^2*x^2 + 2*a*b*x),x)

[Out]

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

________________________________________________________________________________________

sympy [B]  time = 0.09, size = 133, normalized size = 2.05 \[ a^{2} d^{3} x + \frac {b^{2} e^{3} x^{6}}{6} + x^{5} \left (\frac {2 a b e^{3}}{5} + \frac {3 b^{2} d e^{2}}{5}\right ) + x^{4} \left (\frac {a^{2} e^{3}}{4} + \frac {3 a b d e^{2}}{2} + \frac {3 b^{2} d^{2} e}{4}\right ) + x^{3} \left (a^{2} d e^{2} + 2 a b d^{2} e + \frac {b^{2} d^{3}}{3}\right ) + x^{2} \left (\frac {3 a^{2} d^{2} e}{2} + a b d^{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________