3.40 \(\int \frac{(a+b x) (a c-b c x)^5}{x^9} \, dx\)

Optimal. Leaf size=65 \[ -\frac{5 b^2 c^5 (a-b x)^6}{168 a^2 x^6}-\frac{5 b c^5 (a-b x)^6}{28 a x^7}-\frac{c^5 (a-b x)^6}{8 x^8} \]

[Out]

-(c^5*(a - b*x)^6)/(8*x^8) - (5*b*c^5*(a - b*x)^6)/(28*a*x^7) - (5*b^2*c^5*(a - b*x)^6)/(168*a^2*x^6)

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Rubi [A]  time = 0.0166091, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {78, 45, 37} \[ -\frac{5 b^2 c^5 (a-b x)^6}{168 a^2 x^6}-\frac{5 b c^5 (a-b x)^6}{28 a x^7}-\frac{c^5 (a-b x)^6}{8 x^8} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(a*c - b*c*x)^5)/x^9,x]

[Out]

-(c^5*(a - b*x)^6)/(8*x^8) - (5*b*c^5*(a - b*x)^6)/(28*a*x^7) - (5*b^2*c^5*(a - b*x)^6)/(168*a^2*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 45

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] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

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^9} \, dx &=-\frac{c^5 (a-b x)^6}{8 x^8}+\frac{1}{4} (5 b) \int \frac{(a c-b c x)^5}{x^8} \, dx\\ &=-\frac{c^5 (a-b x)^6}{8 x^8}-\frac{5 b c^5 (a-b x)^6}{28 a x^7}+\frac{\left (5 b^2\right ) \int \frac{(a c-b c x)^5}{x^7} \, dx}{28 a}\\ &=-\frac{c^5 (a-b x)^6}{8 x^8}-\frac{5 b c^5 (a-b x)^6}{28 a x^7}-\frac{5 b^2 c^5 (a-b x)^6}{168 a^2 x^6}\\ \end{align*}

Mathematica [A]  time = 0.0064724, size = 73, normalized size = 1.12 \[ c^5 \left (-\frac{5 a^4 b^2}{6 x^6}+\frac{5 a^2 b^4}{4 x^4}+\frac{4 a^5 b}{7 x^7}-\frac{a^6}{8 x^8}-\frac{4 a b^5}{3 x^3}+\frac{b^6}{2 x^2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(a*c - b*c*x)^5)/x^9,x]

[Out]

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

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Maple [A]  time = 0.004, size = 62, normalized size = 1. \begin{align*}{c}^{5} \left ( -{\frac{4\,a{b}^{5}}{3\,{x}^{3}}}+{\frac{5\,{a}^{2}{b}^{4}}{4\,{x}^{4}}}-{\frac{{a}^{6}}{8\,{x}^{8}}}+{\frac{{b}^{6}}{2\,{x}^{2}}}-{\frac{5\,{a}^{4}{b}^{2}}{6\,{x}^{6}}}+{\frac{4\,{a}^{5}b}{7\,{x}^{7}}} \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^9,x)

[Out]

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

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Maxima [A]  time = 1.02189, size = 101, normalized size = 1.55 \begin{align*} \frac{84 \, b^{6} c^{5} x^{6} - 224 \, a b^{5} c^{5} x^{5} + 210 \, a^{2} b^{4} c^{5} x^{4} - 140 \, a^{4} b^{2} c^{5} x^{2} + 96 \, a^{5} b c^{5} x - 21 \, a^{6} c^{5}}{168 \, x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^5/x^9,x, algorithm="maxima")

[Out]

1/168*(84*b^6*c^5*x^6 - 224*a*b^5*c^5*x^5 + 210*a^2*b^4*c^5*x^4 - 140*a^4*b^2*c^5*x^2 + 96*a^5*b*c^5*x - 21*a^
6*c^5)/x^8

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Fricas [A]  time = 1.69759, size = 165, normalized size = 2.54 \begin{align*} \frac{84 \, b^{6} c^{5} x^{6} - 224 \, a b^{5} c^{5} x^{5} + 210 \, a^{2} b^{4} c^{5} x^{4} - 140 \, a^{4} b^{2} c^{5} x^{2} + 96 \, a^{5} b c^{5} x - 21 \, a^{6} c^{5}}{168 \, x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^5/x^9,x, algorithm="fricas")

[Out]

1/168*(84*b^6*c^5*x^6 - 224*a*b^5*c^5*x^5 + 210*a^2*b^4*c^5*x^4 - 140*a^4*b^2*c^5*x^2 + 96*a^5*b*c^5*x - 21*a^
6*c^5)/x^8

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Sympy [A]  time = 0.732513, size = 80, normalized size = 1.23 \begin{align*} \frac{- 21 a^{6} c^{5} + 96 a^{5} b c^{5} x - 140 a^{4} b^{2} c^{5} x^{2} + 210 a^{2} b^{4} c^{5} x^{4} - 224 a b^{5} c^{5} x^{5} + 84 b^{6} c^{5} x^{6}}{168 x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)**5/x**9,x)

[Out]

(-21*a**6*c**5 + 96*a**5*b*c**5*x - 140*a**4*b**2*c**5*x**2 + 210*a**2*b**4*c**5*x**4 - 224*a*b**5*c**5*x**5 +
84*b**6*c**5*x**6)/(168*x**8)

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Giac [A]  time = 1.21318, size = 101, normalized size = 1.55 \begin{align*} \frac{84 \, b^{6} c^{5} x^{6} - 224 \, a b^{5} c^{5} x^{5} + 210 \, a^{2} b^{4} c^{5} x^{4} - 140 \, a^{4} b^{2} c^{5} x^{2} + 96 \, a^{5} b c^{5} x - 21 \, a^{6} c^{5}}{168 \, x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^5/x^9,x, algorithm="giac")

[Out]

1/168*(84*b^6*c^5*x^6 - 224*a*b^5*c^5*x^5 + 210*a^2*b^4*c^5*x^4 - 140*a^4*b^2*c^5*x^2 + 96*a^5*b*c^5*x - 21*a^
6*c^5)/x^8