3.17 \(\int (a+b x) (a c-b c x)^4 \, dx\)

Optimal. Leaf size=38 \[ \frac{c^4 (a-b x)^6}{6 b}-\frac{2 a c^4 (a-b x)^5}{5 b} \]

[Out]

(-2*a*c^4*(a - b*x)^5)/(5*b) + (c^4*(a - b*x)^6)/(6*b)

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Rubi [A]  time = 0.0114237, 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} \[ \frac{c^4 (a-b x)^6}{6 b}-\frac{2 a c^4 (a-b x)^5}{5 b} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)*(a*c - b*c*x)^4,x]

[Out]

(-2*a*c^4*(a - b*x)^5)/(5*b) + (c^4*(a - b*x)^6)/(6*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)^4 \, dx &=\int \left (2 a (a c-b c x)^4-\frac{(a c-b c x)^5}{c}\right ) \, dx\\ &=-\frac{2 a c^4 (a-b x)^5}{5 b}+\frac{c^4 (a-b x)^6}{6 b}\\ \end{align*}

Mathematica [A]  time = 0.0027008, size = 68, normalized size = 1.79 \[ c^4 \left (\frac{1}{2} a^2 b^3 x^4+\frac{2}{3} a^3 b^2 x^3-\frac{3}{2} a^4 b x^2+a^5 x-\frac{3}{5} a b^4 x^5+\frac{b^5 x^6}{6}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)*(a*c - b*c*x)^4,x]

[Out]

c^4*(a^5*x - (3*a^4*b*x^2)/2 + (2*a^3*b^2*x^3)/3 + (a^2*b^3*x^4)/2 - (3*a*b^4*x^5)/5 + (b^5*x^6)/6)

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Maple [B]  time = 0., size = 73, normalized size = 1.9 \begin{align*}{\frac{{b}^{5}{c}^{4}{x}^{6}}{6}}-{\frac{3\,a{b}^{4}{c}^{4}{x}^{5}}{5}}+{\frac{{a}^{2}{b}^{3}{c}^{4}{x}^{4}}{2}}+{\frac{2\,{a}^{3}{b}^{2}{c}^{4}{x}^{3}}{3}}-{\frac{3\,{a}^{4}b{c}^{4}{x}^{2}}{2}}+{a}^{5}{c}^{4}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(-b*c*x+a*c)^4,x)

[Out]

1/6*b^5*c^4*x^6-3/5*a*b^4*c^4*x^5+1/2*a^2*b^3*c^4*x^4+2/3*a^3*b^2*c^4*x^3-3/2*a^4*b*c^4*x^2+a^5*c^4*x

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Maxima [A]  time = 1.05802, size = 97, normalized size = 2.55 \begin{align*} \frac{1}{6} \, b^{5} c^{4} x^{6} - \frac{3}{5} \, a b^{4} c^{4} x^{5} + \frac{1}{2} \, a^{2} b^{3} c^{4} x^{4} + \frac{2}{3} \, a^{3} b^{2} c^{4} x^{3} - \frac{3}{2} \, a^{4} b c^{4} x^{2} + a^{5} c^{4} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^4,x, algorithm="maxima")

[Out]

1/6*b^5*c^4*x^6 - 3/5*a*b^4*c^4*x^5 + 1/2*a^2*b^3*c^4*x^4 + 2/3*a^3*b^2*c^4*x^3 - 3/2*a^4*b*c^4*x^2 + a^5*c^4*
x

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Fricas [A]  time = 1.73831, size = 153, normalized size = 4.03 \begin{align*} \frac{1}{6} x^{6} c^{4} b^{5} - \frac{3}{5} x^{5} c^{4} b^{4} a + \frac{1}{2} x^{4} c^{4} b^{3} a^{2} + \frac{2}{3} x^{3} c^{4} b^{2} a^{3} - \frac{3}{2} x^{2} c^{4} b a^{4} + x c^{4} a^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^4,x, algorithm="fricas")

[Out]

1/6*x^6*c^4*b^5 - 3/5*x^5*c^4*b^4*a + 1/2*x^4*c^4*b^3*a^2 + 2/3*x^3*c^4*b^2*a^3 - 3/2*x^2*c^4*b*a^4 + x*c^4*a^
5

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Sympy [B]  time = 0.079036, size = 82, normalized size = 2.16 \begin{align*} a^{5} c^{4} x - \frac{3 a^{4} b c^{4} x^{2}}{2} + \frac{2 a^{3} b^{2} c^{4} x^{3}}{3} + \frac{a^{2} b^{3} c^{4} x^{4}}{2} - \frac{3 a b^{4} c^{4} x^{5}}{5} + \frac{b^{5} c^{4} x^{6}}{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)**4,x)

[Out]

a**5*c**4*x - 3*a**4*b*c**4*x**2/2 + 2*a**3*b**2*c**4*x**3/3 + a**2*b**3*c**4*x**4/2 - 3*a*b**4*c**4*x**5/5 +
b**5*c**4*x**6/6

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Giac [A]  time = 1.21504, size = 97, normalized size = 2.55 \begin{align*} \frac{1}{6} \, b^{5} c^{4} x^{6} - \frac{3}{5} \, a b^{4} c^{4} x^{5} + \frac{1}{2} \, a^{2} b^{3} c^{4} x^{4} + \frac{2}{3} \, a^{3} b^{2} c^{4} x^{3} - \frac{3}{2} \, a^{4} b c^{4} x^{2} + a^{5} c^{4} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^4,x, algorithm="giac")

[Out]

1/6*b^5*c^4*x^6 - 3/5*a*b^4*c^4*x^5 + 1/2*a^2*b^3*c^4*x^4 + 2/3*a^3*b^2*c^4*x^3 - 3/2*a^4*b*c^4*x^2 + a^5*c^4*
x