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

Optimal. Leaf size=38 \[ \frac {(a+b x)^3 (b d-a e)}{3 b^2}+\frac {e (a+b x)^4}{4 b^2} \]

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Rubi [A]  time = 0.03, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {27, 43} \begin {gather*} \frac {(a+b x)^3 (b d-a e)}{3 b^2}+\frac {e (a+b x)^4}{4 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]  time = 0.01, size = 46, normalized size = 1.21 \begin {gather*} \frac {1}{12} x \left (6 a^2 (2 d+e x)+4 a b x (3 d+2 e x)+b^2 x^2 (4 d+3 e x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(6*a^2*(2*d + e*x) + 4*a*b*x*(3*d + 2*e*x) + b^2*x^2*(4*d + 3*e*x)))/12

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (d+e x) \left (a^2+2 a b x+b^2 x^2\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.35, size = 49, normalized size = 1.29 \begin {gather*} \frac {1}{4} x^{4} e b^{2} + \frac {1}{3} x^{3} d b^{2} + \frac {2}{3} x^{3} e b a + x^{2} d b a + \frac {1}{2} x^{2} e a^{2} + x d a^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.15, size = 52, normalized size = 1.37 \begin {gather*} \frac {1}{4} \, b^{2} x^{4} e + \frac {1}{3} \, b^{2} d x^{3} + \frac {2}{3} \, a b x^{3} e + a b d x^{2} + \frac {1}{2} \, a^{2} x^{2} e + a^{2} d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.04, size = 49, normalized size = 1.29 \begin {gather*} \frac {b^{2} e \,x^{4}}{4}+a^{2} d x +\frac {\left (2 b e a +b^{2} d \right ) x^{3}}{3}+\frac {\left (e \,a^{2}+2 b d a \right ) x^{2}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.34, size = 48, normalized size = 1.26 \begin {gather*} \frac {1}{4} \, b^{2} e x^{4} + a^{2} d x + \frac {1}{3} \, {\left (b^{2} d + 2 \, a b e\right )} x^{3} + \frac {1}{2} \, {\left (2 \, a b d + a^{2} e\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.05, size = 47, normalized size = 1.24 \begin {gather*} x^2\,\left (\frac {e\,a^2}{2}+b\,d\,a\right )+x^3\,\left (\frac {d\,b^2}{3}+\frac {2\,a\,e\,b}{3}\right )+\frac {b^2\,e\,x^4}{4}+a^2\,d\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.07, size = 49, normalized size = 1.29 \begin {gather*} a^{2} d x + \frac {b^{2} e x^{4}}{4} + x^{3} \left (\frac {2 a b e}{3} + \frac {b^{2} d}{3}\right ) + x^{2} \left (\frac {a^{2} e}{2} + a b d\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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