3.4 \(\int (a+b x) (a c-b c x)^3 \, dx\)

Optimal. Leaf size=38 \[ \frac{c^3 (a-b x)^5}{5 b}-\frac{a c^3 (a-b x)^4}{2 b} \]

[Out]

-(a*c^3*(a - b*x)^4)/(2*b) + (c^3*(a - b*x)^5)/(5*b)

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Rubi [A]  time = 0.013247, antiderivative size = 38, normalized size of antiderivative = 1., 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} \[ \frac{c^3 (a-b x)^5}{5 b}-\frac{a c^3 (a-b x)^4}{2 b} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)*(a*c - b*c*x)^3,x]

[Out]

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

Mathematica [A]  time = 0.0020905, size = 40, normalized size = 1.05 \[ c^3 \left (-a^3 b x^2+a^4 x+\frac{1}{2} a b^3 x^4-\frac{1}{5} b^4 x^5\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)*(a*c - b*c*x)^3,x]

[Out]

c^3*(a^4*x - a^3*b*x^2 + (a*b^3*x^4)/2 - (b^4*x^5)/5)

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Maple [A]  time = 0., size = 45, normalized size = 1.2 \begin{align*} -{\frac{{b}^{4}{c}^{3}{x}^{5}}{5}}+{\frac{a{b}^{3}{c}^{3}{x}^{4}}{2}}-{a}^{3}{c}^{3}b{x}^{2}+{a}^{4}{c}^{3}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(-b*c*x+a*c)^3,x)

[Out]

-1/5*b^4*c^3*x^5+1/2*a*b^3*c^3*x^4-a^3*c^3*b*x^2+a^4*c^3*x

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Maxima [A]  time = 0.987381, size = 59, normalized size = 1.55 \begin{align*} -\frac{1}{5} \, b^{4} c^{3} x^{5} + \frac{1}{2} \, a b^{3} c^{3} x^{4} - a^{3} b c^{3} x^{2} + a^{4} c^{3} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^3,x, algorithm="maxima")

[Out]

-1/5*b^4*c^3*x^5 + 1/2*a*b^3*c^3*x^4 - a^3*b*c^3*x^2 + a^4*c^3*x

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Fricas [A]  time = 1.72214, size = 89, normalized size = 2.34 \begin{align*} -\frac{1}{5} x^{5} c^{3} b^{4} + \frac{1}{2} x^{4} c^{3} b^{3} a - x^{2} c^{3} b a^{3} + x c^{3} a^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^3,x, algorithm="fricas")

[Out]

-1/5*x^5*c^3*b^4 + 1/2*x^4*c^3*b^3*a - x^2*c^3*b*a^3 + x*c^3*a^4

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Sympy [A]  time = 0.071989, size = 44, normalized size = 1.16 \begin{align*} a^{4} c^{3} x - a^{3} b c^{3} x^{2} + \frac{a b^{3} c^{3} x^{4}}{2} - \frac{b^{4} c^{3} x^{5}}{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)**3,x)

[Out]

a**4*c**3*x - a**3*b*c**3*x**2 + a*b**3*c**3*x**4/2 - b**4*c**3*x**5/5

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Giac [A]  time = 1.12726, size = 59, normalized size = 1.55 \begin{align*} -\frac{1}{5} \, b^{4} c^{3} x^{5} + \frac{1}{2} \, a b^{3} c^{3} x^{4} - a^{3} b c^{3} x^{2} + a^{4} c^{3} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^3,x, algorithm="giac")

[Out]

-1/5*b^4*c^3*x^5 + 1/2*a*b^3*c^3*x^4 - a^3*b*c^3*x^2 + a^4*c^3*x