3.11.30 \(\int (a+b x) (a c-b c x) \, dx\) [1030]

Optimal. Leaf size=18 \[ a^2 c x-\frac {1}{3} b^2 c x^3 \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {41} \begin {gather*} a^2 c x-\frac {1}{3} b^2 c x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)*(a*c - b*c*x),x]

[Out]

a^2*c*x - (b^2*c*x^3)/3

Rule 41

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[(a*c + b*d*x^2)^m, x] /; FreeQ[{a, b
, c, d, m}, x] && EqQ[b*c + a*d, 0] && (IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))

Rubi steps

\begin {align*} \int (a+b x) (a c-b c x) \, dx &=\int \left (a^2 c-b^2 c x^2\right ) \, dx\\ &=a^2 c x-\frac {1}{3} b^2 c x^3\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 18, normalized size = 1.00 \begin {gather*} c \left (a^2 x-\frac {b^2 x^3}{3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)*(a*c - b*c*x),x]

[Out]

c*(a^2*x - (b^2*x^3)/3)

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Maple [A]
time = 0.04, size = 17, normalized size = 0.94

method	result	size
default	\(a^{2} c x -\frac {1}{3} b^{2} c \,x^{3}\)	\(17\)
norman	\(a^{2} c x -\frac {1}{3} b^{2} c \,x^{3}\)	\(17\)
risch	\(a^{2} c x -\frac {1}{3} b^{2} c \,x^{3}\)	\(17\)
gosper	\(\frac {c x \left (-x^{2} b^{2}+3 a^{2}\right )}{3}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(-b*c*x+a*c),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.29, size = 16, normalized size = 0.89 \begin {gather*} -\frac {1}{3} \, b^{2} c x^{3} + a^{2} c x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*b^2*c*x^3 + a^2*c*x

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Fricas [A]
time = 0.66, size = 16, normalized size = 0.89 \begin {gather*} -\frac {1}{3} \, b^{2} c x^{3} + a^{2} c x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*b^2*c*x^3 + a^2*c*x

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Sympy [A]
time = 0.01, size = 15, normalized size = 0.83 \begin {gather*} a^{2} c x - \frac {b^{2} c x^{3}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*c*x - b**2*c*x**3/3

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Giac [A]
time = 1.76, size = 16, normalized size = 0.89 \begin {gather*} -\frac {1}{3} \, b^{2} c x^{3} + a^{2} c x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*b^2*c*x^3 + a^2*c*x

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Mupad [B]
time = 0.02, size = 18, normalized size = 1.00 \begin {gather*} \frac {c\,x\,\left (3\,a^2-b^2\,x^2\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*c - b*c*x)*(a + b*x),x)

[Out]

(c*x*(3*a^2 - b^2*x^2))/3

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