3.31 \(\int (a+b x) (a c-b c x)^5 \, dx\)

Optimal. Leaf size=38 \[ \frac {c^5 (a-b x)^7}{7 b}-\frac {a c^5 (a-b x)^6}{3 b} \]

[Out]

-1/3*a*c^5*(-b*x+a)^6/b+1/7*c^5*(-b*x+a)^7/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} \[ \frac {c^5 (a-b x)^7}{7 b}-\frac {a c^5 (a-b x)^6}{3 b} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)*(a*c - b*c*x)^5,x]

[Out]

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

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Mathematica [A]  time = 0.01, size = 64, normalized size = 1.68 \[ c^5 \left (a^6 x-2 a^5 b x^2+\frac {5}{3} a^4 b^2 x^3-a^2 b^4 x^5+\frac {2}{3} a b^5 x^6-\frac {1}{7} b^6 x^7\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)*(a*c - b*c*x)^5,x]

[Out]

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

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fricas [A]  time = 0.40, size = 72, normalized size = 1.89 \[ -\frac {1}{7} x^{7} c^{5} b^{6} + \frac {2}{3} x^{6} c^{5} b^{5} a - x^{5} c^{5} b^{4} a^{2} + \frac {5}{3} x^{3} c^{5} b^{2} a^{4} - 2 x^{2} c^{5} b a^{5} + x c^{5} a^{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^5,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.98, size = 72, normalized size = 1.89 \[ -\frac {1}{7} \, b^{6} c^{5} x^{7} + \frac {2}{3} \, a b^{5} c^{5} x^{6} - a^{2} b^{4} c^{5} x^{5} + \frac {5}{3} \, a^{4} b^{2} c^{5} x^{3} - 2 \, a^{5} b c^{5} x^{2} + a^{6} c^{5} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^5,x, algorithm="giac")

[Out]

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

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maple [B]  time = 0.00, size = 73, normalized size = 1.92 \[ -\frac {1}{7} b^{6} c^{5} x^{7}+\frac {2}{3} a \,b^{5} c^{5} x^{6}-a^{2} b^{4} c^{5} x^{5}+\frac {5}{3} a^{4} b^{2} c^{5} x^{3}-2 a^{5} b \,c^{5} x^{2}+a^{6} c^{5} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(-b*c*x+a*c)^5,x)

[Out]

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

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maxima [A]  time = 1.03, size = 72, normalized size = 1.89 \[ -\frac {1}{7} \, b^{6} c^{5} x^{7} + \frac {2}{3} \, a b^{5} c^{5} x^{6} - a^{2} b^{4} c^{5} x^{5} + \frac {5}{3} \, a^{4} b^{2} c^{5} x^{3} - 2 \, a^{5} b c^{5} x^{2} + a^{6} c^{5} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^5,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.03, size = 72, normalized size = 1.89 \[ a^6\,c^5\,x-2\,a^5\,b\,c^5\,x^2+\frac {5\,a^4\,b^2\,c^5\,x^3}{3}-a^2\,b^4\,c^5\,x^5+\frac {2\,a\,b^5\,c^5\,x^6}{3}-\frac {b^6\,c^5\,x^7}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*c - b*c*x)^5*(a + b*x),x)

[Out]

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

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sympy [B]  time = 0.09, size = 78, normalized size = 2.05 \[ a^{6} c^{5} x - 2 a^{5} b c^{5} x^{2} + \frac {5 a^{4} b^{2} c^{5} x^{3}}{3} - a^{2} b^{4} c^{5} x^{5} + \frac {2 a b^{5} c^{5} x^{6}}{3} - \frac {b^{6} c^{5} x^{7}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)**5,x)

[Out]

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

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