3.10.71 \(\int (a+b x)^2 (a c-b c x) \, dx\)

Optimal. Leaf size=32 \[ \frac {2 a c (a+b x)^3}{3 b}-\frac {c (a+b x)^4}{4 b} \]

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Rubi [A]  time = 0.01, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {43} \begin {gather*} \frac {2 a c (a+b x)^3}{3 b}-\frac {c (a+b x)^4}{4 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^2*(a*c - b*c*x),x]

[Out]

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

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

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Mathematica [A]  time = 0.00, size = 40, normalized size = 1.25 \begin {gather*} c \left (a^3 x+\frac {1}{2} a^2 b x^2-\frac {1}{3} a b^2 x^3-\frac {1}{4} b^3 x^4\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.13, size = 36, normalized size = 1.12 \begin {gather*} -\frac {1}{4} x^{4} c b^{3} - \frac {1}{3} x^{3} c b^{2} a + \frac {1}{2} x^{2} c b a^{2} + x c a^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 1.02, size = 36, normalized size = 1.12 \begin {gather*} -\frac {1}{4} \, b^{3} c x^{4} - \frac {1}{3} \, a b^{2} c x^{3} + \frac {1}{2} \, a^{2} b c x^{2} + a^{3} c x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 37, normalized size = 1.16 \begin {gather*} -\frac {1}{4} b^{3} c \,x^{4}-\frac {1}{3} a \,b^{2} c \,x^{3}+\frac {1}{2} a^{2} b c \,x^{2}+a^{3} c x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.32, size = 36, normalized size = 1.12 \begin {gather*} -\frac {1}{4} \, b^{3} c x^{4} - \frac {1}{3} \, a b^{2} c x^{3} + \frac {1}{2} \, a^{2} b c x^{2} + a^{3} c x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.05, size = 36, normalized size = 1.12 \begin {gather*} c\,a^3\,x+\frac {c\,a^2\,b\,x^2}{2}-\frac {c\,a\,b^2\,x^3}{3}-\frac {c\,b^3\,x^4}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*c - b*c*x)*(a + b*x)^2,x)

[Out]

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

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sympy [A]  time = 0.07, size = 39, normalized size = 1.22 \begin {gather*} a^{3} c x + \frac {a^{2} b c x^{2}}{2} - \frac {a b^{2} c x^{3}}{3} - \frac {b^{3} c x^{4}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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