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

Optimal. Leaf size=59 \[ -\frac{2 a^2 c^4 (a-b x)^5}{5 b^2}-\frac{c^4 (a-b x)^7}{7 b^2}+\frac{a c^4 (a-b x)^6}{2 b^2} \]

[Out]

(-2*a^2*c^4*(a - b*x)^5)/(5*b^2) + (a*c^4*(a - b*x)^6)/(2*b^2) - (c^4*(a - b*x)^7)/(7*b^2)

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Rubi [A]  time = 0.0322142, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {75} \[ -\frac{2 a^2 c^4 (a-b x)^5}{5 b^2}-\frac{c^4 (a-b x)^7}{7 b^2}+\frac{a c^4 (a-b x)^6}{2 b^2} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x)*(a*c - b*c*x)^4,x]

[Out]

(-2*a^2*c^4*(a - b*x)^5)/(5*b^2) + (a*c^4*(a - b*x)^6)/(2*b^2) - (c^4*(a - b*x)^7)/(7*b^2)

Rule 75

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && EqQ[b*e + a*f, 0] &&  !(ILtQ[n
+ p + 2, 0] && GtQ[n + 2*p, 0])

Rubi steps

\begin{align*} \int x (a+b x) (a c-b c x)^4 \, dx &=\int \left (\frac{2 a^2 (a c-b c x)^4}{b}-\frac{3 a (a c-b c x)^5}{b c}+\frac{(a c-b c x)^6}{b c^2}\right ) \, dx\\ &=-\frac{2 a^2 c^4 (a-b x)^5}{5 b^2}+\frac{a c^4 (a-b x)^6}{2 b^2}-\frac{c^4 (a-b x)^7}{7 b^2}\\ \end{align*}

Mathematica [A]  time = 0.0030664, size = 85, normalized size = 1.44 \[ \frac{2}{5} a^2 b^3 c^4 x^5+\frac{1}{2} a^3 b^2 c^4 x^4-a^4 b c^4 x^3+\frac{1}{2} a^5 c^4 x^2-\frac{1}{2} a b^4 c^4 x^6+\frac{1}{7} b^5 c^4 x^7 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^5*c^4*x^2)/2 - a^4*b*c^4*x^3 + (a^3*b^2*c^4*x^4)/2 + (2*a^2*b^3*c^4*x^5)/5 - (a*b^4*c^4*x^6)/2 + (b^5*c^4*x
^7)/7

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Maple [A]  time = 0., size = 76, normalized size = 1.3 \begin{align*}{\frac{{b}^{5}{c}^{4}{x}^{7}}{7}}-{\frac{a{b}^{4}{c}^{4}{x}^{6}}{2}}+{\frac{2\,{a}^{2}{c}^{4}{b}^{3}{x}^{5}}{5}}+{\frac{{a}^{3}{c}^{4}{b}^{2}{x}^{4}}{2}}-{a}^{4}{c}^{4}b{x}^{3}+{\frac{{a}^{5}{c}^{4}{x}^{2}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*b^5*c^4*x^7-1/2*a*b^4*c^4*x^6+2/5*a^2*c^4*b^3*x^5+1/2*a^3*c^4*b^2*x^4-a^4*c^4*b*x^3+1/2*a^5*c^4*x^2

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Maxima [A]  time = 1.06573, size = 101, normalized size = 1.71 \begin{align*} \frac{1}{7} \, b^{5} c^{4} x^{7} - \frac{1}{2} \, a b^{4} c^{4} x^{6} + \frac{2}{5} \, a^{2} b^{3} c^{4} x^{5} + \frac{1}{2} \, a^{3} b^{2} c^{4} x^{4} - a^{4} b c^{4} x^{3} + \frac{1}{2} \, a^{5} c^{4} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*b^5*c^4*x^7 - 1/2*a*b^4*c^4*x^6 + 2/5*a^2*b^3*c^4*x^5 + 1/2*a^3*b^2*c^4*x^4 - a^4*b*c^4*x^3 + 1/2*a^5*c^4*
x^2

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Fricas [A]  time = 1.78383, size = 155, normalized size = 2.63 \begin{align*} \frac{1}{7} x^{7} c^{4} b^{5} - \frac{1}{2} x^{6} c^{4} b^{4} a + \frac{2}{5} x^{5} c^{4} b^{3} a^{2} + \frac{1}{2} x^{4} c^{4} b^{2} a^{3} - x^{3} c^{4} b a^{4} + \frac{1}{2} x^{2} c^{4} a^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*x^7*c^4*b^5 - 1/2*x^6*c^4*b^4*a + 2/5*x^5*c^4*b^3*a^2 + 1/2*x^4*c^4*b^2*a^3 - x^3*c^4*b*a^4 + 1/2*x^2*c^4*
a^5

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Sympy [A]  time = 0.078124, size = 80, normalized size = 1.36 \begin{align*} \frac{a^{5} c^{4} x^{2}}{2} - a^{4} b c^{4} x^{3} + \frac{a^{3} b^{2} c^{4} x^{4}}{2} + \frac{2 a^{2} b^{3} c^{4} x^{5}}{5} - \frac{a b^{4} c^{4} x^{6}}{2} + \frac{b^{5} c^{4} x^{7}}{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**5*c**4*x**2/2 - a**4*b*c**4*x**3 + a**3*b**2*c**4*x**4/2 + 2*a**2*b**3*c**4*x**5/5 - a*b**4*c**4*x**6/2 + b
**5*c**4*x**7/7

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Giac [A]  time = 1.20696, size = 101, normalized size = 1.71 \begin{align*} \frac{1}{7} \, b^{5} c^{4} x^{7} - \frac{1}{2} \, a b^{4} c^{4} x^{6} + \frac{2}{5} \, a^{2} b^{3} c^{4} x^{5} + \frac{1}{2} \, a^{3} b^{2} c^{4} x^{4} - a^{4} b c^{4} x^{3} + \frac{1}{2} \, a^{5} c^{4} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*b^5*c^4*x^7 - 1/2*a*b^4*c^4*x^6 + 2/5*a^2*b^3*c^4*x^5 + 1/2*a^3*b^2*c^4*x^4 - a^4*b*c^4*x^3 + 1/2*a^5*c^4*
x^2