3.47 \(\int \frac{(a+b x) (a c-b c x)^6}{x^{11}} \, dx\)

Optimal. Leaf size=89 \[ -\frac{13 b^3 c^6 (a-b x)^7}{2520 a^3 x^7}-\frac{13 b^2 c^6 (a-b x)^7}{360 a^2 x^8}-\frac{13 b c^6 (a-b x)^7}{90 a x^9}-\frac{c^6 (a-b x)^7}{10 x^{10}} \]

[Out]

-(c^6*(a - b*x)^7)/(10*x^10) - (13*b*c^6*(a - b*x)^7)/(90*a*x^9) - (13*b^2*c^6*(a - b*x)^7)/(360*a^2*x^8) - (1
3*b^3*c^6*(a - b*x)^7)/(2520*a^3*x^7)

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Rubi [A]  time = 0.0241516, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {78, 45, 37} \[ -\frac{13 b^3 c^6 (a-b x)^7}{2520 a^3 x^7}-\frac{13 b^2 c^6 (a-b x)^7}{360 a^2 x^8}-\frac{13 b c^6 (a-b x)^7}{90 a x^9}-\frac{c^6 (a-b x)^7}{10 x^{10}} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(a*c - b*c*x)^6)/x^11,x]

[Out]

-(c^6*(a - b*x)^7)/(10*x^10) - (13*b*c^6*(a - b*x)^7)/(90*a*x^9) - (13*b^2*c^6*(a - b*x)^7)/(360*a^2*x^8) - (1
3*b^3*c^6*(a - b*x)^7)/(2520*a^3*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^{11}} \, dx &=-\frac{c^6 (a-b x)^7}{10 x^{10}}+\frac{1}{10} (13 b) \int \frac{(a c-b c x)^6}{x^{10}} \, dx\\ &=-\frac{c^6 (a-b x)^7}{10 x^{10}}-\frac{13 b c^6 (a-b x)^7}{90 a x^9}+\frac{\left (13 b^2\right ) \int \frac{(a c-b c x)^6}{x^9} \, dx}{45 a}\\ &=-\frac{c^6 (a-b x)^7}{10 x^{10}}-\frac{13 b c^6 (a-b x)^7}{90 a x^9}-\frac{13 b^2 c^6 (a-b x)^7}{360 a^2 x^8}+\frac{\left (13 b^3\right ) \int \frac{(a c-b c x)^6}{x^8} \, dx}{360 a^2}\\ &=-\frac{c^6 (a-b x)^7}{10 x^{10}}-\frac{13 b c^6 (a-b x)^7}{90 a x^9}-\frac{13 b^2 c^6 (a-b x)^7}{360 a^2 x^8}-\frac{13 b^3 c^6 (a-b x)^7}{2520 a^3 x^7}\\ \end{align*}

Mathematica [A]  time = 0.0076477, size = 119, normalized size = 1.34 \[ -\frac{9 a^5 b^2 c^6}{8 x^8}+\frac{5 a^4 b^3 c^6}{7 x^7}+\frac{5 a^3 b^4 c^6}{6 x^6}-\frac{9 a^2 b^5 c^6}{5 x^5}+\frac{5 a^6 b c^6}{9 x^9}-\frac{a^7 c^6}{10 x^{10}}+\frac{5 a b^6 c^6}{4 x^4}-\frac{b^7 c^6}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(a*c - b*c*x)^6)/x^11,x]

[Out]

-(a^7*c^6)/(10*x^10) + (5*a^6*b*c^6)/(9*x^9) - (9*a^5*b^2*c^6)/(8*x^8) + (5*a^4*b^3*c^6)/(7*x^7) + (5*a^3*b^4*
c^6)/(6*x^6) - (9*a^2*b^5*c^6)/(5*x^5) + (5*a*b^6*c^6)/(4*x^4) - (b^7*c^6)/(3*x^3)

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Maple [A]  time = 0.004, size = 84, normalized size = 0.9 \begin{align*}{c}^{6} \left ( -{\frac{{b}^{7}}{3\,{x}^{3}}}-{\frac{9\,{a}^{2}{b}^{5}}{5\,{x}^{5}}}+{\frac{5\,a{b}^{6}}{4\,{x}^{4}}}-{\frac{9\,{a}^{5}{b}^{2}}{8\,{x}^{8}}}+{\frac{5\,{a}^{3}{b}^{4}}{6\,{x}^{6}}}+{\frac{5\,{a}^{4}{b}^{3}}{7\,{x}^{7}}}+{\frac{5\,{a}^{6}b}{9\,{x}^{9}}}-{\frac{{a}^{7}}{10\,{x}^{10}}} \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^11,x)

[Out]

c^6*(-1/3*b^7/x^3-9/5*a^2*b^5/x^5+5/4*a*b^6/x^4-9/8*a^5*b^2/x^8+5/6*a^3*b^4/x^6+5/7*a^4*b^3/x^7+5/9*a^6*b/x^9-
1/10*a^7/x^10)

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Maxima [A]  time = 1.05873, size = 139, normalized size = 1.56 \begin{align*} -\frac{840 \, b^{7} c^{6} x^{7} - 3150 \, a b^{6} c^{6} x^{6} + 4536 \, a^{2} b^{5} c^{6} x^{5} - 2100 \, a^{3} b^{4} c^{6} x^{4} - 1800 \, a^{4} b^{3} c^{6} x^{3} + 2835 \, a^{5} b^{2} c^{6} x^{2} - 1400 \, a^{6} b c^{6} x + 252 \, a^{7} c^{6}}{2520 \, x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^6/x^11,x, algorithm="maxima")

[Out]

-1/2520*(840*b^7*c^6*x^7 - 3150*a*b^6*c^6*x^6 + 4536*a^2*b^5*c^6*x^5 - 2100*a^3*b^4*c^6*x^4 - 1800*a^4*b^3*c^6
*x^3 + 2835*a^5*b^2*c^6*x^2 - 1400*a^6*b*c^6*x + 252*a^7*c^6)/x^10

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Fricas [A]  time = 1.97889, size = 240, normalized size = 2.7 \begin{align*} -\frac{840 \, b^{7} c^{6} x^{7} - 3150 \, a b^{6} c^{6} x^{6} + 4536 \, a^{2} b^{5} c^{6} x^{5} - 2100 \, a^{3} b^{4} c^{6} x^{4} - 1800 \, a^{4} b^{3} c^{6} x^{3} + 2835 \, a^{5} b^{2} c^{6} x^{2} - 1400 \, a^{6} b c^{6} x + 252 \, a^{7} c^{6}}{2520 \, x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^6/x^11,x, algorithm="fricas")

[Out]

-1/2520*(840*b^7*c^6*x^7 - 3150*a*b^6*c^6*x^6 + 4536*a^2*b^5*c^6*x^5 - 2100*a^3*b^4*c^6*x^4 - 1800*a^4*b^3*c^6
*x^3 + 2835*a^5*b^2*c^6*x^2 - 1400*a^6*b*c^6*x + 252*a^7*c^6)/x^10

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Sympy [A]  time = 0.990725, size = 112, normalized size = 1.26 \begin{align*} - \frac{252 a^{7} c^{6} - 1400 a^{6} b c^{6} x + 2835 a^{5} b^{2} c^{6} x^{2} - 1800 a^{4} b^{3} c^{6} x^{3} - 2100 a^{3} b^{4} c^{6} x^{4} + 4536 a^{2} b^{5} c^{6} x^{5} - 3150 a b^{6} c^{6} x^{6} + 840 b^{7} c^{6} x^{7}}{2520 x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)**6/x**11,x)

[Out]

-(252*a**7*c**6 - 1400*a**6*b*c**6*x + 2835*a**5*b**2*c**6*x**2 - 1800*a**4*b**3*c**6*x**3 - 2100*a**3*b**4*c*
*6*x**4 + 4536*a**2*b**5*c**6*x**5 - 3150*a*b**6*c**6*x**6 + 840*b**7*c**6*x**7)/(2520*x**10)

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Giac [A]  time = 1.163, size = 139, normalized size = 1.56 \begin{align*} -\frac{840 \, b^{7} c^{6} x^{7} - 3150 \, a b^{6} c^{6} x^{6} + 4536 \, a^{2} b^{5} c^{6} x^{5} - 2100 \, a^{3} b^{4} c^{6} x^{4} - 1800 \, a^{4} b^{3} c^{6} x^{3} + 2835 \, a^{5} b^{2} c^{6} x^{2} - 1400 \, a^{6} b c^{6} x + 252 \, a^{7} c^{6}}{2520 \, x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^6/x^11,x, algorithm="giac")

[Out]

-1/2520*(840*b^7*c^6*x^7 - 3150*a*b^6*c^6*x^6 + 4536*a^2*b^5*c^6*x^5 - 2100*a^3*b^4*c^6*x^4 - 1800*a^4*b^3*c^6
*x^3 + 2835*a^5*b^2*c^6*x^2 - 1400*a^6*b*c^6*x + 252*a^7*c^6)/x^10