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

Optimal. Leaf size=59 \[ -\frac {a^2 c^5 (a-b x)^6}{3 b^2}-\frac {c^5 (a-b x)^8}{8 b^2}+\frac {3 a c^5 (a-b x)^7}{7 b^2} \]

[Out]

-1/3*a^2*c^5*(-b*x+a)^6/b^2+3/7*a*c^5*(-b*x+a)^7/b^2-1/8*c^5*(-b*x+a)^8/b^2

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Rubi [A]  time = 0.03, antiderivative size = 59, normalized size of antiderivative = 1.00, 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} \[ -\frac {a^2 c^5 (a-b x)^6}{3 b^2}-\frac {c^5 (a-b x)^8}{8 b^2}+\frac {3 a c^5 (a-b x)^7}{7 b^2} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x)*(a*c - b*c*x)^5,x]

[Out]

-(a^2*c^5*(a - b*x)^6)/(3*b^2) + (3*a*c^5*(a - b*x)^7)/(7*b^2) - (c^5*(a - b*x)^8)/(8*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)^5 \, dx &=\int \left (\frac {2 a^2 (a c-b c x)^5}{b}-\frac {3 a (a c-b c x)^6}{b c}+\frac {(a c-b c x)^7}{b c^2}\right ) \, dx\\ &=-\frac {a^2 c^5 (a-b x)^6}{3 b^2}+\frac {3 a c^5 (a-b x)^7}{7 b^2}-\frac {c^5 (a-b x)^8}{8 b^2}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 73, normalized size = 1.24 \[ c^5 \left (\frac {a^6 x^2}{2}-\frac {4}{3} a^5 b x^3+\frac {5}{4} a^4 b^2 x^4-\frac {5}{6} a^2 b^4 x^6+\frac {4}{7} a b^5 x^7-\frac {1}{8} b^6 x^8\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

c^5*((a^6*x^2)/2 - (4*a^5*b*x^3)/3 + (5*a^4*b^2*x^4)/4 - (5*a^2*b^4*x^6)/6 + (4*a*b^5*x^7)/7 - (b^6*x^8)/8)

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fricas [A]  time = 0.63, size = 75, normalized size = 1.27 \[ -\frac {1}{8} x^{8} c^{5} b^{6} + \frac {4}{7} x^{7} c^{5} b^{5} a - \frac {5}{6} x^{6} c^{5} b^{4} a^{2} + \frac {5}{4} x^{4} c^{5} b^{2} a^{4} - \frac {4}{3} x^{3} c^{5} b a^{5} + \frac {1}{2} x^{2} c^{5} a^{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*x^8*c^5*b^6 + 4/7*x^7*c^5*b^5*a - 5/6*x^6*c^5*b^4*a^2 + 5/4*x^4*c^5*b^2*a^4 - 4/3*x^3*c^5*b*a^5 + 1/2*x^2
*c^5*a^6

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giac [A]  time = 1.12, size = 75, normalized size = 1.27 \[ -\frac {1}{8} \, b^{6} c^{5} x^{8} + \frac {4}{7} \, a b^{5} c^{5} x^{7} - \frac {5}{6} \, a^{2} b^{4} c^{5} x^{6} + \frac {5}{4} \, a^{4} b^{2} c^{5} x^{4} - \frac {4}{3} \, a^{5} b c^{5} x^{3} + \frac {1}{2} \, a^{6} c^{5} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*b^6*c^5*x^8 + 4/7*a*b^5*c^5*x^7 - 5/6*a^2*b^4*c^5*x^6 + 5/4*a^4*b^2*c^5*x^4 - 4/3*a^5*b*c^5*x^3 + 1/2*a^6
*c^5*x^2

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maple [A]  time = 0.00, size = 76, normalized size = 1.29 \[ -\frac {1}{8} b^{6} c^{5} x^{8}+\frac {4}{7} a \,b^{5} c^{5} x^{7}-\frac {5}{6} a^{2} b^{4} c^{5} x^{6}+\frac {5}{4} a^{4} b^{2} c^{5} x^{4}-\frac {4}{3} a^{5} b \,c^{5} x^{3}+\frac {1}{2} a^{6} c^{5} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*b^6*c^5*x^8+4/7*a*b^5*c^5*x^7-5/6*a^2*c^5*b^4*x^6+5/4*a^4*c^5*b^2*x^4-4/3*a^5*c^5*b*x^3+1/2*a^6*c^5*x^2

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maxima [A]  time = 1.04, size = 75, normalized size = 1.27 \[ -\frac {1}{8} \, b^{6} c^{5} x^{8} + \frac {4}{7} \, a b^{5} c^{5} x^{7} - \frac {5}{6} \, a^{2} b^{4} c^{5} x^{6} + \frac {5}{4} \, a^{4} b^{2} c^{5} x^{4} - \frac {4}{3} \, a^{5} b c^{5} x^{3} + \frac {1}{2} \, a^{6} c^{5} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*b^6*c^5*x^8 + 4/7*a*b^5*c^5*x^7 - 5/6*a^2*b^4*c^5*x^6 + 5/4*a^4*b^2*c^5*x^4 - 4/3*a^5*b*c^5*x^3 + 1/2*a^6
*c^5*x^2

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mupad [B]  time = 0.03, size = 75, normalized size = 1.27 \[ \frac {a^6\,c^5\,x^2}{2}-\frac {4\,a^5\,b\,c^5\,x^3}{3}+\frac {5\,a^4\,b^2\,c^5\,x^4}{4}-\frac {5\,a^2\,b^4\,c^5\,x^6}{6}+\frac {4\,a\,b^5\,c^5\,x^7}{7}-\frac {b^6\,c^5\,x^8}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a*c - b*c*x)^5*(a + b*x),x)

[Out]

(a^6*c^5*x^2)/2 - (b^6*c^5*x^8)/8 - (4*a^5*b*c^5*x^3)/3 + (4*a*b^5*c^5*x^7)/7 + (5*a^4*b^2*c^5*x^4)/4 - (5*a^2
*b^4*c^5*x^6)/6

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sympy [A]  time = 0.08, size = 87, normalized size = 1.47 \[ \frac {a^{6} c^{5} x^{2}}{2} - \frac {4 a^{5} b c^{5} x^{3}}{3} + \frac {5 a^{4} b^{2} c^{5} x^{4}}{4} - \frac {5 a^{2} b^{4} c^{5} x^{6}}{6} + \frac {4 a b^{5} c^{5} x^{7}}{7} - \frac {b^{6} c^{5} x^{8}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**6*c**5*x**2/2 - 4*a**5*b*c**5*x**3/3 + 5*a**4*b**2*c**5*x**4/4 - 5*a**2*b**4*c**5*x**6/6 + 4*a*b**5*c**5*x*
*7/7 - b**6*c**5*x**8/8

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