3.17.75 \(\int (a+b x) (a^2+2 a b x+b^2 x^2) \, dx\)

Optimal. Leaf size=14 \[ \frac {(a+b x)^4}{4 b} \]

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Rubi [A]  time = 0.00, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {27, 32} \begin {gather*} \frac {(a+b x)^4}{4 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a + b*x)^4/(4*b)

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 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 14, normalized size = 1.00 \begin {gather*} \frac {(a+b x)^4}{4 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a + b*x)^4/(4*b)

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

[Out]

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

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fricas [B]  time = 0.37, size = 31, normalized size = 2.21 \begin {gather*} \frac {1}{4} x^{4} b^{3} + x^{3} b^{2} a + \frac {3}{2} x^{2} b a^{2} + x a^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.15, size = 31, normalized size = 2.21 \begin {gather*} \frac {1}{2} \, {\left (b x^{2} + 2 \, a x\right )} a^{2} + \frac {1}{4} \, {\left (b x^{2} + 2 \, a x\right )}^{2} b \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.05, size = 32, normalized size = 2.29 \begin {gather*} \frac {1}{4} b^{3} x^{4}+a \,b^{2} x^{3}+\frac {3}{2} a^{2} b \,x^{2}+a^{3} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.49, size = 23, normalized size = 1.64 \begin {gather*} \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{2}}{4 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(b^2*x^2 + 2*a*b*x + a^2)^2/b

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mupad [B]  time = 0.04, size = 31, normalized size = 2.21 \begin {gather*} a^3\,x+\frac {3\,a^2\,b\,x^2}{2}+a\,b^2\,x^3+\frac {b^3\,x^4}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [B]  time = 0.07, size = 32, normalized size = 2.29 \begin {gather*} a^{3} x + \frac {3 a^{2} b x^{2}}{2} + a b^{2} x^{3} + \frac {b^{3} x^{4}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**3*x + 3*a**2*b*x**2/2 + a*b**2*x**3 + b**3*x**4/4

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