3.46 \(\int \frac{(a+b x) (a c-b c x)^6}{x^{10}} \, dx\)

Optimal. Leaf size=65 \[ -\frac{11 b^2 c^6 (a-b x)^7}{504 a^2 x^7}-\frac{11 b c^6 (a-b x)^7}{72 a x^8}-\frac{c^6 (a-b x)^7}{9 x^9} \]

[Out]

-(c^6*(a - b*x)^7)/(9*x^9) - (11*b*c^6*(a - b*x)^7)/(72*a*x^8) - (11*b^2*c^6*(a - b*x)^7)/(504*a^2*x^7)

________________________________________________________________________________________

Rubi [A]  time = 0.0160827, 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} \[ -\frac{11 b^2 c^6 (a-b x)^7}{504 a^2 x^7}-\frac{11 b c^6 (a-b x)^7}{72 a x^8}-\frac{c^6 (a-b x)^7}{9 x^9} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(a*c - b*c*x)^6)/x^10,x]

[Out]

-(c^6*(a - b*x)^7)/(9*x^9) - (11*b*c^6*(a - b*x)^7)/(72*a*x^8) - (11*b^2*c^6*(a - b*x)^7)/(504*a^2*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 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)^6}{x^{10}} \, dx &=-\frac{c^6 (a-b x)^7}{9 x^9}+\frac{1}{9} (11 b) \int \frac{(a c-b c x)^6}{x^9} \, dx\\ &=-\frac{c^6 (a-b x)^7}{9 x^9}-\frac{11 b c^6 (a-b x)^7}{72 a x^8}+\frac{\left (11 b^2\right ) \int \frac{(a c-b c x)^6}{x^8} \, dx}{72 a}\\ &=-\frac{c^6 (a-b x)^7}{9 x^9}-\frac{11 b c^6 (a-b x)^7}{72 a x^8}-\frac{11 b^2 c^6 (a-b x)^7}{504 a^2 x^7}\\ \end{align*}

Mathematica [A]  time = 0.0077399, size = 116, normalized size = 1.78 \[ -\frac{9 a^5 b^2 c^6}{7 x^7}+\frac{5 a^4 b^3 c^6}{6 x^6}+\frac{a^3 b^4 c^6}{x^5}-\frac{9 a^2 b^5 c^6}{4 x^4}+\frac{5 a^6 b c^6}{8 x^8}-\frac{a^7 c^6}{9 x^9}+\frac{5 a b^6 c^6}{3 x^3}-\frac{b^7 c^6}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(a*c - b*c*x)^6)/x^10,x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.006, size = 83, normalized size = 1.3 \begin{align*}{c}^{6} \left ({\frac{5\,a{b}^{6}}{3\,{x}^{3}}}+{\frac{{a}^{3}{b}^{4}}{{x}^{5}}}-{\frac{9\,{a}^{2}{b}^{5}}{4\,{x}^{4}}}+{\frac{5\,{a}^{6}b}{8\,{x}^{8}}}-{\frac{{b}^{7}}{2\,{x}^{2}}}+{\frac{5\,{a}^{4}{b}^{3}}{6\,{x}^{6}}}-{\frac{9\,{a}^{5}{b}^{2}}{7\,{x}^{7}}}-{\frac{{a}^{7}}{9\,{x}^{9}}} \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^10,x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.0465, size = 139, normalized size = 2.14 \begin{align*} -\frac{252 \, b^{7} c^{6} x^{7} - 840 \, a b^{6} c^{6} x^{6} + 1134 \, a^{2} b^{5} c^{6} x^{5} - 504 \, a^{3} b^{4} c^{6} x^{4} - 420 \, a^{4} b^{3} c^{6} x^{3} + 648 \, a^{5} b^{2} c^{6} x^{2} - 315 \, a^{6} b c^{6} x + 56 \, a^{7} c^{6}}{504 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^6/x^10,x, algorithm="maxima")

[Out]

-1/504*(252*b^7*c^6*x^7 - 840*a*b^6*c^6*x^6 + 1134*a^2*b^5*c^6*x^5 - 504*a^3*b^4*c^6*x^4 - 420*a^4*b^3*c^6*x^3
+ 648*a^5*b^2*c^6*x^2 - 315*a^6*b*c^6*x + 56*a^7*c^6)/x^9

________________________________________________________________________________________

Fricas [A]  time = 1.85926, size = 230, normalized size = 3.54 \begin{align*} -\frac{252 \, b^{7} c^{6} x^{7} - 840 \, a b^{6} c^{6} x^{6} + 1134 \, a^{2} b^{5} c^{6} x^{5} - 504 \, a^{3} b^{4} c^{6} x^{4} - 420 \, a^{4} b^{3} c^{6} x^{3} + 648 \, a^{5} b^{2} c^{6} x^{2} - 315 \, a^{6} b c^{6} x + 56 \, a^{7} c^{6}}{504 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^6/x^10,x, algorithm="fricas")

[Out]

-1/504*(252*b^7*c^6*x^7 - 840*a*b^6*c^6*x^6 + 1134*a^2*b^5*c^6*x^5 - 504*a^3*b^4*c^6*x^4 - 420*a^4*b^3*c^6*x^3
+ 648*a^5*b^2*c^6*x^2 - 315*a^6*b*c^6*x + 56*a^7*c^6)/x^9

________________________________________________________________________________________

Sympy [A]  time = 0.902041, size = 112, normalized size = 1.72 \begin{align*} - \frac{56 a^{7} c^{6} - 315 a^{6} b c^{6} x + 648 a^{5} b^{2} c^{6} x^{2} - 420 a^{4} b^{3} c^{6} x^{3} - 504 a^{3} b^{4} c^{6} x^{4} + 1134 a^{2} b^{5} c^{6} x^{5} - 840 a b^{6} c^{6} x^{6} + 252 b^{7} c^{6} x^{7}}{504 x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)**6/x**10,x)

[Out]

-(56*a**7*c**6 - 315*a**6*b*c**6*x + 648*a**5*b**2*c**6*x**2 - 420*a**4*b**3*c**6*x**3 - 504*a**3*b**4*c**6*x*
*4 + 1134*a**2*b**5*c**6*x**5 - 840*a*b**6*c**6*x**6 + 252*b**7*c**6*x**7)/(504*x**9)

________________________________________________________________________________________

Giac [A]  time = 1.16943, size = 139, normalized size = 2.14 \begin{align*} -\frac{252 \, b^{7} c^{6} x^{7} - 840 \, a b^{6} c^{6} x^{6} + 1134 \, a^{2} b^{5} c^{6} x^{5} - 504 \, a^{3} b^{4} c^{6} x^{4} - 420 \, a^{4} b^{3} c^{6} x^{3} + 648 \, a^{5} b^{2} c^{6} x^{2} - 315 \, a^{6} b c^{6} x + 56 \, a^{7} c^{6}}{504 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^6/x^10,x, algorithm="giac")

[Out]

-1/504*(252*b^7*c^6*x^7 - 840*a*b^6*c^6*x^6 + 1134*a^2*b^5*c^6*x^5 - 504*a^3*b^4*c^6*x^4 - 420*a^4*b^3*c^6*x^3
+ 648*a^5*b^2*c^6*x^2 - 315*a^6*b*c^6*x + 56*a^7*c^6)/x^9