3.1.29 \(\int x^2 (a+b x) (a c-b c x)^5 \, dx\)

Optimal. Leaf size=80 \[ -\frac {a^3 c^5 (a-b x)^6}{3 b^3}+\frac {5 a^2 c^5 (a-b x)^7}{7 b^3}+\frac {c^5 (a-b x)^9}{9 b^3}-\frac {a c^5 (a-b x)^8}{2 b^3} \]

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Rubi [A]  time = 0.04, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {75} \begin {gather*} \frac {5 a^2 c^5 (a-b x)^7}{7 b^3}-\frac {a^3 c^5 (a-b x)^6}{3 b^3}+\frac {c^5 (a-b x)^9}{9 b^3}-\frac {a c^5 (a-b x)^8}{2 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*x)*(a*c - b*c*x)^5,x]

[Out]

-(a^3*c^5*(a - b*x)^6)/(3*b^3) + (5*a^2*c^5*(a - b*x)^7)/(7*b^3) - (a*c^5*(a - b*x)^8)/(2*b^3) + (c^5*(a - b*x
)^9)/(9*b^3)

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^2 (a+b x) (a c-b c x)^5 \, dx &=\int \left (\frac {2 a^3 (a c-b c x)^5}{b^2}-\frac {5 a^2 (a c-b c x)^6}{b^2 c}+\frac {4 a (a c-b c x)^7}{b^2 c^2}-\frac {(a c-b c x)^8}{b^2 c^3}\right ) \, dx\\ &=-\frac {a^3 c^5 (a-b x)^6}{3 b^3}+\frac {5 a^2 c^5 (a-b x)^7}{7 b^3}-\frac {a c^5 (a-b x)^8}{2 b^3}+\frac {c^5 (a-b x)^9}{9 b^3}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 68, normalized size = 0.85 \begin {gather*} c^5 \left (\frac {a^6 x^3}{3}-a^5 b x^4+a^4 b^2 x^5-\frac {5}{7} a^2 b^4 x^7+\frac {1}{2} a b^5 x^8-\frac {1}{9} b^6 x^9\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a + b*x)*(a*c - b*c*x)^5,x]

[Out]

c^5*((a^6*x^3)/3 - a^5*b*x^4 + a^4*b^2*x^5 - (5*a^2*b^4*x^7)/7 + (a*b^5*x^8)/2 - (b^6*x^9)/9)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^2 (a+b x) (a c-b c x)^5 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x^2*(a + b*x)*(a*c - b*c*x)^5,x]

[Out]

IntegrateAlgebraic[x^2*(a + b*x)*(a*c - b*c*x)^5, x]

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fricas [A]  time = 1.03, size = 74, normalized size = 0.92 \begin {gather*} -\frac {1}{9} x^{9} c^{5} b^{6} + \frac {1}{2} x^{8} c^{5} b^{5} a - \frac {5}{7} x^{7} c^{5} b^{4} a^{2} + x^{5} c^{5} b^{2} a^{4} - x^{4} c^{5} b a^{5} + \frac {1}{3} x^{3} c^{5} a^{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*x+a)*(-b*c*x+a*c)^5,x, algorithm="fricas")

[Out]

-1/9*x^9*c^5*b^6 + 1/2*x^8*c^5*b^5*a - 5/7*x^7*c^5*b^4*a^2 + x^5*c^5*b^2*a^4 - x^4*c^5*b*a^5 + 1/3*x^3*c^5*a^6

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giac [A]  time = 1.03, size = 74, normalized size = 0.92 \begin {gather*} -\frac {1}{9} \, b^{6} c^{5} x^{9} + \frac {1}{2} \, a b^{5} c^{5} x^{8} - \frac {5}{7} \, a^{2} b^{4} c^{5} x^{7} + a^{4} b^{2} c^{5} x^{5} - a^{5} b c^{5} x^{4} + \frac {1}{3} \, a^{6} c^{5} x^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*x+a)*(-b*c*x+a*c)^5,x, algorithm="giac")

[Out]

-1/9*b^6*c^5*x^9 + 1/2*a*b^5*c^5*x^8 - 5/7*a^2*b^4*c^5*x^7 + a^4*b^2*c^5*x^5 - a^5*b*c^5*x^4 + 1/3*a^6*c^5*x^3

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maple [A]  time = 0.00, size = 75, normalized size = 0.94 \begin {gather*} -\frac {1}{9} b^{6} c^{5} x^{9}+\frac {1}{2} a \,b^{5} c^{5} x^{8}-\frac {5}{7} a^{2} b^{4} c^{5} x^{7}+a^{4} b^{2} c^{5} x^{5}-a^{5} b \,c^{5} x^{4}+\frac {1}{3} a^{6} c^{5} x^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(b*x+a)*(-b*c*x+a*c)^5,x)

[Out]

-1/9*b^6*c^5*x^9+1/2*a*b^5*c^5*x^8-5/7*a^2*c^5*b^4*x^7+a^4*c^5*b^2*x^5-a^5*c^5*b*x^4+1/3*a^6*c^5*x^3

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maxima [A]  time = 1.04, size = 74, normalized size = 0.92 \begin {gather*} -\frac {1}{9} \, b^{6} c^{5} x^{9} + \frac {1}{2} \, a b^{5} c^{5} x^{8} - \frac {5}{7} \, a^{2} b^{4} c^{5} x^{7} + a^{4} b^{2} c^{5} x^{5} - a^{5} b c^{5} x^{4} + \frac {1}{3} \, a^{6} c^{5} x^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*x+a)*(-b*c*x+a*c)^5,x, algorithm="maxima")

[Out]

-1/9*b^6*c^5*x^9 + 1/2*a*b^5*c^5*x^8 - 5/7*a^2*b^4*c^5*x^7 + a^4*b^2*c^5*x^5 - a^5*b*c^5*x^4 + 1/3*a^6*c^5*x^3

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mupad [B]  time = 0.03, size = 74, normalized size = 0.92 \begin {gather*} \frac {a^6\,c^5\,x^3}{3}-a^5\,b\,c^5\,x^4+a^4\,b^2\,c^5\,x^5-\frac {5\,a^2\,b^4\,c^5\,x^7}{7}+\frac {a\,b^5\,c^5\,x^8}{2}-\frac {b^6\,c^5\,x^9}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a*c - b*c*x)^5*(a + b*x),x)

[Out]

(a^6*c^5*x^3)/3 - (b^6*c^5*x^9)/9 - a^5*b*c^5*x^4 + (a*b^5*c^5*x^8)/2 + a^4*b^2*c^5*x^5 - (5*a^2*b^4*c^5*x^7)/
7

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sympy [A]  time = 0.09, size = 78, normalized size = 0.98 \begin {gather*} \frac {a^{6} c^{5} x^{3}}{3} - a^{5} b c^{5} x^{4} + a^{4} b^{2} c^{5} x^{5} - \frac {5 a^{2} b^{4} c^{5} x^{7}}{7} + \frac {a b^{5} c^{5} x^{8}}{2} - \frac {b^{6} c^{5} x^{9}}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(b*x+a)*(-b*c*x+a*c)**5,x)

[Out]

a**6*c**5*x**3/3 - a**5*b*c**5*x**4 + a**4*b**2*c**5*x**5 - 5*a**2*b**4*c**5*x**7/7 + a*b**5*c**5*x**8/2 - b**
6*c**5*x**9/9

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