3.39 \(\int \frac{(a+b x) (a c-b c x)^5}{x^8} \, dx\)

Optimal. Leaf size=41 \[ -\frac{4 b c^5 (a-b x)^6}{21 a x^6}-\frac{c^5 (a-b x)^6}{7 x^7} \]

[Out]

-(c^5*(a - b*x)^6)/(7*x^7) - (4*b*c^5*(a - b*x)^6)/(21*a*x^6)

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Rubi [A]  time = 0.0077355, 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} \[ -\frac{4 b c^5 (a-b x)^6}{21 a x^6}-\frac{c^5 (a-b x)^6}{7 x^7} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(a*c - b*c*x)^5)/x^8,x]

[Out]

-(c^5*(a - b*x)^6)/(7*x^7) - (4*b*c^5*(a - b*x)^6)/(21*a*x^6)

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)^5}{x^8} \, dx &=-\frac{c^5 (a-b x)^6}{7 x^7}+\frac{1}{7} (8 b) \int \frac{(a c-b c x)^5}{x^7} \, dx\\ &=-\frac{c^5 (a-b x)^6}{7 x^7}-\frac{4 b c^5 (a-b x)^6}{21 a x^6}\\ \end{align*}

Mathematica [A]  time = 0.006867, size = 66, normalized size = 1.61 \[ c^5 \left (-\frac{a^4 b^2}{x^5}+\frac{5 a^2 b^4}{3 x^3}+\frac{2 a^5 b}{3 x^6}-\frac{a^6}{7 x^7}-\frac{2 a b^5}{x^2}+\frac{b^6}{x}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(a*c - b*c*x)^5)/x^8,x]

[Out]

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

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Maple [A]  time = 0.004, size = 61, normalized size = 1.5 \begin{align*}{c}^{5} \left ({\frac{5\,{a}^{2}{b}^{4}}{3\,{x}^{3}}}-{\frac{{a}^{4}{b}^{2}}{{x}^{5}}}-2\,{\frac{a{b}^{5}}{{x}^{2}}}+{\frac{2\,{a}^{5}b}{3\,{x}^{6}}}-{\frac{{a}^{6}}{7\,{x}^{7}}}+{\frac{{b}^{6}}{x}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(-b*c*x+a*c)^5/x^8,x)

[Out]

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

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Maxima [A]  time = 1.03788, size = 101, normalized size = 2.46 \begin{align*} \frac{21 \, b^{6} c^{5} x^{6} - 42 \, a b^{5} c^{5} x^{5} + 35 \, a^{2} b^{4} c^{5} x^{4} - 21 \, a^{4} b^{2} c^{5} x^{2} + 14 \, a^{5} b c^{5} x - 3 \, a^{6} c^{5}}{21 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^5/x^8,x, algorithm="maxima")

[Out]

1/21*(21*b^6*c^5*x^6 - 42*a*b^5*c^5*x^5 + 35*a^2*b^4*c^5*x^4 - 21*a^4*b^2*c^5*x^2 + 14*a^5*b*c^5*x - 3*a^6*c^5
)/x^7

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Fricas [A]  time = 1.56565, size = 158, normalized size = 3.85 \begin{align*} \frac{21 \, b^{6} c^{5} x^{6} - 42 \, a b^{5} c^{5} x^{5} + 35 \, a^{2} b^{4} c^{5} x^{4} - 21 \, a^{4} b^{2} c^{5} x^{2} + 14 \, a^{5} b c^{5} x - 3 \, a^{6} c^{5}}{21 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^5/x^8,x, algorithm="fricas")

[Out]

1/21*(21*b^6*c^5*x^6 - 42*a*b^5*c^5*x^5 + 35*a^2*b^4*c^5*x^4 - 21*a^4*b^2*c^5*x^2 + 14*a^5*b*c^5*x - 3*a^6*c^5
)/x^7

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Sympy [B]  time = 0.666901, size = 80, normalized size = 1.95 \begin{align*} \frac{- 3 a^{6} c^{5} + 14 a^{5} b c^{5} x - 21 a^{4} b^{2} c^{5} x^{2} + 35 a^{2} b^{4} c^{5} x^{4} - 42 a b^{5} c^{5} x^{5} + 21 b^{6} c^{5} x^{6}}{21 x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)**5/x**8,x)

[Out]

(-3*a**6*c**5 + 14*a**5*b*c**5*x - 21*a**4*b**2*c**5*x**2 + 35*a**2*b**4*c**5*x**4 - 42*a*b**5*c**5*x**5 + 21*
b**6*c**5*x**6)/(21*x**7)

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Giac [A]  time = 1.3303, size = 101, normalized size = 2.46 \begin{align*} \frac{21 \, b^{6} c^{5} x^{6} - 42 \, a b^{5} c^{5} x^{5} + 35 \, a^{2} b^{4} c^{5} x^{4} - 21 \, a^{4} b^{2} c^{5} x^{2} + 14 \, a^{5} b c^{5} x - 3 \, a^{6} c^{5}}{21 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^5/x^8,x, algorithm="giac")

[Out]

1/21*(21*b^6*c^5*x^6 - 42*a*b^5*c^5*x^5 + 35*a^2*b^4*c^5*x^4 - 21*a^4*b^2*c^5*x^2 + 14*a^5*b*c^5*x - 3*a^6*c^5
)/x^7