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

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

[Out]

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

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Rubi [A]  time = 0.0075868, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, 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 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]))))

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]

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*}

Mathematica [B]  time = 0.0078445, size = 112, normalized size = 2.73 \[ -\frac{3 a^5 b^2 c^6}{2 x^6}+\frac{a^4 b^3 c^6}{x^5}+\frac{5 a^3 b^4 c^6}{4 x^4}-\frac{3 a^2 b^5 c^6}{x^3}+\frac{5 a^6 b c^6}{7 x^7}-\frac{a^7 c^6}{8 x^8}+\frac{5 a b^6 c^6}{2 x^2}-\frac{b^7 c^6}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a^7*c^6)/(8*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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Maple [B]  time = 0.004, size = 83, normalized size = 2. \begin{align*}{c}^{6} \left ( -3\,{\frac{{a}^{2}{b}^{5}}{{x}^{3}}}+{\frac{{a}^{4}{b}^{3}}{{x}^{5}}}+{\frac{5\,{a}^{3}{b}^{4}}{4\,{x}^{4}}}-{\frac{{a}^{7}}{8\,{x}^{8}}}+{\frac{5\,a{b}^{6}}{2\,{x}^{2}}}-{\frac{3\,{a}^{5}{b}^{2}}{2\,{x}^{6}}}+{\frac{5\,{a}^{6}b}{7\,{x}^{7}}}-{\frac{{b}^{7}}{x}} \right ) \end{align*}

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

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Maxima [B]  time = 1.1001, size = 139, normalized size = 3.39 \begin{align*} -\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}} \end{align*}

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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Fricas [B]  time = 1.51988, size = 219, normalized size = 5.34 \begin{align*} -\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}} \end{align*}

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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Sympy [B]  time = 0.843013, size = 112, normalized size = 2.73 \begin{align*} - \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}} \end{align*}

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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Giac [B]  time = 1.21786, size = 139, normalized size = 3.39 \begin{align*} -\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}} \end{align*}

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