3.133 \(\int \frac{(a+b x)^5 (A+B x)}{x^9} \, dx\)

Optimal. Leaf size=70 \[ -\frac{b (a+b x)^6 (A b-4 a B)}{168 a^3 x^6}+\frac{(a+b x)^6 (A b-4 a B)}{28 a^2 x^7}-\frac{A (a+b x)^6}{8 a x^8} \]

[Out]

-(A*(a + b*x)^6)/(8*a*x^8) + ((A*b - 4*a*B)*(a + b*x)^6)/(28*a^2*x^7) - (b*(A*b - 4*a*B)*(a + b*x)^6)/(168*a^3
*x^6)

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Rubi [A]  time = 0.0283154, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {78, 45, 37} \[ -\frac{b (a+b x)^6 (A b-4 a B)}{168 a^3 x^6}+\frac{(a+b x)^6 (A b-4 a B)}{28 a^2 x^7}-\frac{A (a+b x)^6}{8 a x^8} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^5*(A + B*x))/x^9,x]

[Out]

-(A*(a + b*x)^6)/(8*a*x^8) + ((A*b - 4*a*B)*(a + b*x)^6)/(28*a^2*x^7) - (b*(A*b - 4*a*B)*(a + b*x)^6)/(168*a^3
*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)^5 (A+B x)}{x^9} \, dx &=-\frac{A (a+b x)^6}{8 a x^8}+\frac{(-2 A b+8 a B) \int \frac{(a+b x)^5}{x^8} \, dx}{8 a}\\ &=-\frac{A (a+b x)^6}{8 a x^8}+\frac{(A b-4 a B) (a+b x)^6}{28 a^2 x^7}+\frac{(b (A b-4 a B)) \int \frac{(a+b x)^5}{x^7} \, dx}{28 a^2}\\ &=-\frac{A (a+b x)^6}{8 a x^8}+\frac{(A b-4 a B) (a+b x)^6}{28 a^2 x^7}-\frac{b (A b-4 a B) (a+b x)^6}{168 a^3 x^6}\\ \end{align*}

Mathematica [A]  time = 0.0279759, size = 107, normalized size = 1.53 \[ -\frac{56 a^3 b^2 x^2 (5 A+6 B x)+84 a^2 b^3 x^3 (4 A+5 B x)+20 a^4 b x (6 A+7 B x)+3 a^5 (7 A+8 B x)+70 a b^4 x^4 (3 A+4 B x)+28 b^5 x^5 (2 A+3 B x)}{168 x^8} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^5*(A + B*x))/x^9,x]

[Out]

-(28*b^5*x^5*(2*A + 3*B*x) + 70*a*b^4*x^4*(3*A + 4*B*x) + 84*a^2*b^3*x^3*(4*A + 5*B*x) + 56*a^3*b^2*x^2*(5*A +
6*B*x) + 20*a^4*b*x*(6*A + 7*B*x) + 3*a^5*(7*A + 8*B*x))/(168*x^8)

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Maple [A]  time = 0.007, size = 104, normalized size = 1.5 \begin{align*} -{\frac{{b}^{4} \left ( Ab+5\,Ba \right ) }{3\,{x}^{3}}}-2\,{\frac{{b}^{2}{a}^{2} \left ( Ab+Ba \right ) }{{x}^{5}}}-{\frac{5\,a{b}^{3} \left ( Ab+2\,Ba \right ) }{4\,{x}^{4}}}-{\frac{A{a}^{5}}{8\,{x}^{8}}}-{\frac{B{b}^{5}}{2\,{x}^{2}}}-{\frac{5\,{a}^{3}b \left ( 2\,Ab+Ba \right ) }{6\,{x}^{6}}}-{\frac{{a}^{4} \left ( 5\,Ab+Ba \right ) }{7\,{x}^{7}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^5*(B*x+A)/x^9,x)

[Out]

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

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Maxima [A]  time = 1.02494, size = 161, normalized size = 2.3 \begin{align*} -\frac{84 \, B b^{5} x^{6} + 21 \, A a^{5} + 56 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 210 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 336 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 140 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 24 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x}{168 \, x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/168*(84*B*b^5*x^6 + 21*A*a^5 + 56*(5*B*a*b^4 + A*b^5)*x^5 + 210*(2*B*a^2*b^3 + A*a*b^4)*x^4 + 336*(B*a^3*b^
2 + A*a^2*b^3)*x^3 + 140*(B*a^4*b + 2*A*a^3*b^2)*x^2 + 24*(B*a^5 + 5*A*a^4*b)*x)/x^8

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Fricas [A]  time = 1.72684, size = 265, normalized size = 3.79 \begin{align*} -\frac{84 \, B b^{5} x^{6} + 21 \, A a^{5} + 56 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 210 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 336 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 140 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 24 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x}{168 \, x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/168*(84*B*b^5*x^6 + 21*A*a^5 + 56*(5*B*a*b^4 + A*b^5)*x^5 + 210*(2*B*a^2*b^3 + A*a*b^4)*x^4 + 336*(B*a^3*b^
2 + A*a^2*b^3)*x^3 + 140*(B*a^4*b + 2*A*a^3*b^2)*x^2 + 24*(B*a^5 + 5*A*a^4*b)*x)/x^8

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Sympy [A]  time = 6.21498, size = 126, normalized size = 1.8 \begin{align*} - \frac{21 A a^{5} + 84 B b^{5} x^{6} + x^{5} \left (56 A b^{5} + 280 B a b^{4}\right ) + x^{4} \left (210 A a b^{4} + 420 B a^{2} b^{3}\right ) + x^{3} \left (336 A a^{2} b^{3} + 336 B a^{3} b^{2}\right ) + x^{2} \left (280 A a^{3} b^{2} + 140 B a^{4} b\right ) + x \left (120 A a^{4} b + 24 B a^{5}\right )}{168 x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**5*(B*x+A)/x**9,x)

[Out]

-(21*A*a**5 + 84*B*b**5*x**6 + x**5*(56*A*b**5 + 280*B*a*b**4) + x**4*(210*A*a*b**4 + 420*B*a**2*b**3) + x**3*
(336*A*a**2*b**3 + 336*B*a**3*b**2) + x**2*(280*A*a**3*b**2 + 140*B*a**4*b) + x*(120*A*a**4*b + 24*B*a**5))/(1
68*x**8)

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Giac [A]  time = 1.22015, size = 166, normalized size = 2.37 \begin{align*} -\frac{84 \, B b^{5} x^{6} + 280 \, B a b^{4} x^{5} + 56 \, A b^{5} x^{5} + 420 \, B a^{2} b^{3} x^{4} + 210 \, A a b^{4} x^{4} + 336 \, B a^{3} b^{2} x^{3} + 336 \, A a^{2} b^{3} x^{3} + 140 \, B a^{4} b x^{2} + 280 \, A a^{3} b^{2} x^{2} + 24 \, B a^{5} x + 120 \, A a^{4} b x + 21 \, A a^{5}}{168 \, x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/168*(84*B*b^5*x^6 + 280*B*a*b^4*x^5 + 56*A*b^5*x^5 + 420*B*a^2*b^3*x^4 + 210*A*a*b^4*x^4 + 336*B*a^3*b^2*x^
3 + 336*A*a^2*b^3*x^3 + 140*B*a^4*b*x^2 + 280*A*a^3*b^2*x^2 + 24*B*a^5*x + 120*A*a^4*b*x + 21*A*a^5)/x^8