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]

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

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Rubi [A]  time = 0.01, antiderivative size = 38, 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} \[ \frac {c^4 (a-b x)^6}{6 b}-\frac {2 a c^4 (a-b x)^5}{5 b} \]

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*}

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Mathematica [A]  time = 0.00, size = 68, normalized size = 1.79 \[ c^4 \left (a^5 x-\frac {3}{2} a^4 b x^2+\frac {2}{3} a^3 b^2 x^3+\frac {1}{2} a^2 b^3 x^4-\frac {3}{5} a b^4 x^5+\frac {b^5 x^6}{6}\right ) \]

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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fricas [A]  time = 0.48, size = 72, normalized size = 1.89 \[ \frac {1}{6} x^{6} c^{4} b^{5} - \frac {3}{5} x^{5} c^{4} b^{4} a + \frac {1}{2} x^{4} c^{4} b^{3} a^{2} + \frac {2}{3} x^{3} c^{4} b^{2} a^{3} - \frac {3}{2} x^{2} c^{4} b a^{4} + x c^{4} a^{5} \]

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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giac [A]  time = 1.23, size = 72, normalized size = 1.89 \[ \frac {1}{6} \, b^{5} c^{4} x^{6} - \frac {3}{5} \, a b^{4} c^{4} x^{5} + \frac {1}{2} \, a^{2} b^{3} c^{4} x^{4} + \frac {2}{3} \, a^{3} b^{2} c^{4} x^{3} - \frac {3}{2} \, a^{4} b c^{4} x^{2} + a^{5} c^{4} x \]

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

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maple [B]  time = 0.00, size = 73, normalized size = 1.92 \[ \frac {1}{6} b^{5} c^{4} x^{6}-\frac {3}{5} a \,b^{4} c^{4} x^{5}+\frac {1}{2} a^{2} b^{3} c^{4} x^{4}+\frac {2}{3} a^{3} b^{2} c^{4} x^{3}-\frac {3}{2} a^{4} b \,c^{4} x^{2}+a^{5} c^{4} x \]

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 = 1.16, size = 72, normalized size = 1.89 \[ \frac {1}{6} \, b^{5} c^{4} x^{6} - \frac {3}{5} \, a b^{4} c^{4} x^{5} + \frac {1}{2} \, a^{2} b^{3} c^{4} x^{4} + \frac {2}{3} \, a^{3} b^{2} c^{4} x^{3} - \frac {3}{2} \, a^{4} b c^{4} x^{2} + a^{5} c^{4} x \]

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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mupad [B]  time = 0.03, size = 72, normalized size = 1.89 \[ a^5\,c^4\,x-\frac {3\,a^4\,b\,c^4\,x^2}{2}+\frac {2\,a^3\,b^2\,c^4\,x^3}{3}+\frac {a^2\,b^3\,c^4\,x^4}{2}-\frac {3\,a\,b^4\,c^4\,x^5}{5}+\frac {b^5\,c^4\,x^6}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*c - b*c*x)^4*(a + b*x),x)

[Out]

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

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sympy [B]  time = 0.08, size = 82, normalized size = 2.16 \[ a^{5} c^{4} x - \frac {3 a^{4} b c^{4} x^{2}}{2} + \frac {2 a^{3} b^{2} c^{4} x^{3}}{3} + \frac {a^{2} b^{3} c^{4} x^{4}}{2} - \frac {3 a b^{4} c^{4} x^{5}}{5} + \frac {b^{5} c^{4} x^{6}}{6} \]

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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