3.45 \(\int \frac {(a+b x) (a c-b c x)^6}{x^9} \, dx\)

Optimal. Leaf size=41 \[ -\frac {c^6 (a-b x)^7}{8 x^8}-\frac {9 b c^6 (a-b x)^7}{56 a x^7} \]

[Out]

-1/8*c^6*(-b*x+a)^7/x^8-9/56*b*c^6*(-b*x+a)^7/a/x^7

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Rubi [A]  time = 0.01, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {78, 37} \[ -\frac {9 b c^6 (a-b x)^7}{56 a x^7}-\frac {c^6 (a-b x)^7}{8 x^8} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(a*c - b*c*x)^6)/x^9,x]

[Out]

-(c^6*(a - b*x)^7)/(8*x^8) - (9*b*c^6*(a - b*x)^7)/(56*a*x^7)

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

Rubi steps

\begin {align*} \int \frac {(a+b x) (a c-b c x)^6}{x^9} \, dx &=-\frac {c^6 (a-b x)^7}{8 x^8}+\frac {1}{8} (9 b) \int \frac {(a c-b c x)^6}{x^8} \, dx\\ &=-\frac {c^6 (a-b x)^7}{8 x^8}-\frac {9 b c^6 (a-b x)^7}{56 a x^7}\\ \end {align*}

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Mathematica [B]  time = 0.02, size = 112, normalized size = 2.73 \[ -\frac {a^7 c^6}{8 x^8}+\frac {5 a^6 b c^6}{7 x^7}-\frac {3 a^5 b^2 c^6}{2 x^6}+\frac {a^4 b^3 c^6}{x^5}+\frac {5 a^3 b^4 c^6}{4 x^4}-\frac {3 a^2 b^5 c^6}{x^3}+\frac {5 a b^6 c^6}{2 x^2}-\frac {b^7 c^6}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(a*c - b*c*x)^6)/x^9,x]

[Out]

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

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fricas [B]  time = 0.56, size = 103, normalized size = 2.51 \[ -\frac {56 \, b^{7} c^{6} x^{7} - 140 \, a b^{6} c^{6} x^{6} + 168 \, a^{2} b^{5} c^{6} x^{5} - 70 \, a^{3} b^{4} c^{6} x^{4} - 56 \, a^{4} b^{3} c^{6} x^{3} + 84 \, a^{5} b^{2} c^{6} x^{2} - 40 \, a^{6} b c^{6} x + 7 \, a^{7} c^{6}}{56 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^6/x^9,x, algorithm="fricas")

[Out]

-1/56*(56*b^7*c^6*x^7 - 140*a*b^6*c^6*x^6 + 168*a^2*b^5*c^6*x^5 - 70*a^3*b^4*c^6*x^4 - 56*a^4*b^3*c^6*x^3 + 84
*a^5*b^2*c^6*x^2 - 40*a^6*b*c^6*x + 7*a^7*c^6)/x^8

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giac [B]  time = 0.97, size = 103, normalized size = 2.51 \[ -\frac {56 \, b^{7} c^{6} x^{7} - 140 \, a b^{6} c^{6} x^{6} + 168 \, a^{2} b^{5} c^{6} x^{5} - 70 \, a^{3} b^{4} c^{6} x^{4} - 56 \, a^{4} b^{3} c^{6} x^{3} + 84 \, a^{5} b^{2} c^{6} x^{2} - 40 \, a^{6} b c^{6} x + 7 \, a^{7} c^{6}}{56 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^6/x^9,x, algorithm="giac")

[Out]

-1/56*(56*b^7*c^6*x^7 - 140*a*b^6*c^6*x^6 + 168*a^2*b^5*c^6*x^5 - 70*a^3*b^4*c^6*x^4 - 56*a^4*b^3*c^6*x^3 + 84
*a^5*b^2*c^6*x^2 - 40*a^6*b*c^6*x + 7*a^7*c^6)/x^8

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maple [B]  time = 0.01, size = 83, normalized size = 2.02 \[ \left (-\frac {b^{7}}{x}+\frac {5 a \,b^{6}}{2 x^{2}}-\frac {3 a^{2} b^{5}}{x^{3}}+\frac {5 a^{3} b^{4}}{4 x^{4}}+\frac {a^{4} b^{3}}{x^{5}}-\frac {3 a^{5} b^{2}}{2 x^{6}}+\frac {5 a^{6} b}{7 x^{7}}-\frac {a^{7}}{8 x^{8}}\right ) c^{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(-b*c*x+a*c)^6/x^9,x)

[Out]

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

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maxima [B]  time = 0.99, size = 103, normalized size = 2.51 \[ -\frac {56 \, b^{7} c^{6} x^{7} - 140 \, a b^{6} c^{6} x^{6} + 168 \, a^{2} b^{5} c^{6} x^{5} - 70 \, a^{3} b^{4} c^{6} x^{4} - 56 \, a^{4} b^{3} c^{6} x^{3} + 84 \, a^{5} b^{2} c^{6} x^{2} - 40 \, a^{6} b c^{6} x + 7 \, a^{7} c^{6}}{56 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^6/x^9,x, algorithm="maxima")

[Out]

-1/56*(56*b^7*c^6*x^7 - 140*a*b^6*c^6*x^6 + 168*a^2*b^5*c^6*x^5 - 70*a^3*b^4*c^6*x^4 - 56*a^4*b^3*c^6*x^3 + 84
*a^5*b^2*c^6*x^2 - 40*a^6*b*c^6*x + 7*a^7*c^6)/x^8

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mupad [B]  time = 0.07, size = 102, normalized size = 2.49 \[ -\frac {\frac {a^7\,c^6}{8}-\frac {5\,a^6\,b\,c^6\,x}{7}+\frac {3\,a^5\,b^2\,c^6\,x^2}{2}-a^4\,b^3\,c^6\,x^3-\frac {5\,a^3\,b^4\,c^6\,x^4}{4}+3\,a^2\,b^5\,c^6\,x^5-\frac {5\,a\,b^6\,c^6\,x^6}{2}+b^7\,c^6\,x^7}{x^8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a*c - b*c*x)^6*(a + b*x))/x^9,x)

[Out]

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

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sympy [B]  time = 0.62, size = 110, normalized size = 2.68 \[ \frac {- 7 a^{7} c^{6} + 40 a^{6} b c^{6} x - 84 a^{5} b^{2} c^{6} x^{2} + 56 a^{4} b^{3} c^{6} x^{3} + 70 a^{3} b^{4} c^{6} x^{4} - 168 a^{2} b^{5} c^{6} x^{5} + 140 a b^{6} c^{6} x^{6} - 56 b^{7} c^{6} x^{7}}{56 x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)**6/x**9,x)

[Out]

(-7*a**7*c**6 + 40*a**6*b*c**6*x - 84*a**5*b**2*c**6*x**2 + 56*a**4*b**3*c**6*x**3 + 70*a**3*b**4*c**6*x**4 -
168*a**2*b**5*c**6*x**5 + 140*a*b**6*c**6*x**6 - 56*b**7*c**6*x**7)/(56*x**8)

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