3.160 \(\int \frac{(a+b x)^{10} (A+B x)}{x^{13}} \, dx\)

Optimal. Leaf size=44 \[ \frac{(a+b x)^{11} (A b-12 a B)}{132 a^2 x^{11}}-\frac{A (a+b x)^{11}}{12 a x^{12}} \]

[Out]

-(A*(a + b*x)^11)/(12*a*x^12) + ((A*b - 12*a*B)*(a + b*x)^11)/(132*a^2*x^11)

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Rubi [A]  time = 0.0111734, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {78, 37} \[ \frac{(a+b x)^{11} (A b-12 a B)}{132 a^2 x^{11}}-\frac{A (a+b x)^{11}}{12 a x^{12}} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^10*(A + B*x))/x^13,x]

[Out]

-(A*(a + b*x)^11)/(12*a*x^12) + ((A*b - 12*a*B)*(a + b*x)^11)/(132*a^2*x^11)

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)^{10} (A+B x)}{x^{13}} \, dx &=-\frac{A (a+b x)^{11}}{12 a x^{12}}+\frac{(-A b+12 a B) \int \frac{(a+b x)^{10}}{x^{12}} \, dx}{12 a}\\ &=-\frac{A (a+b x)^{11}}{12 a x^{12}}+\frac{(A b-12 a B) (a+b x)^{11}}{132 a^2 x^{11}}\\ \end{align*}

Mathematica [B]  time = 0.0529479, size = 199, normalized size = 4.52 \[ -\frac{66 a^8 b^2 x^2 (9 A+10 B x)+220 a^7 b^3 x^3 (8 A+9 B x)+495 a^6 b^4 x^4 (7 A+8 B x)+792 a^5 b^5 x^5 (6 A+7 B x)+924 a^4 b^6 x^6 (5 A+6 B x)+792 a^3 b^7 x^7 (4 A+5 B x)+495 a^2 b^8 x^8 (3 A+4 B x)+12 a^9 b x (10 A+11 B x)+a^{10} (11 A+12 B x)+220 a b^9 x^9 (2 A+3 B x)+66 b^{10} x^{10} (A+2 B x)}{132 x^{12}} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^10*(A + B*x))/x^13,x]

[Out]

-(66*b^10*x^10*(A + 2*B*x) + 220*a*b^9*x^9*(2*A + 3*B*x) + 495*a^2*b^8*x^8*(3*A + 4*B*x) + 792*a^3*b^7*x^7*(4*
A + 5*B*x) + 924*a^4*b^6*x^6*(5*A + 6*B*x) + 792*a^5*b^5*x^5*(6*A + 7*B*x) + 495*a^6*b^4*x^4*(7*A + 8*B*x) + 2
20*a^7*b^3*x^3*(8*A + 9*B*x) + 66*a^8*b^2*x^2*(9*A + 10*B*x) + 12*a^9*b*x*(10*A + 11*B*x) + a^10*(11*A + 12*B*
x))/(132*x^12)

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Maple [B]  time = 0.008, size = 208, normalized size = 4.7 \begin{align*} -{\frac{5\,a{b}^{8} \left ( 2\,Ab+9\,Ba \right ) }{3\,{x}^{3}}}-6\,{\frac{{a}^{3}{b}^{6} \left ( 4\,Ab+7\,Ba \right ) }{{x}^{5}}}-{\frac{A{a}^{10}}{12\,{x}^{12}}}-{\frac{{a}^{9} \left ( 10\,Ab+Ba \right ) }{11\,{x}^{11}}}-{\frac{15\,{a}^{2}{b}^{7} \left ( 3\,Ab+8\,Ba \right ) }{4\,{x}^{4}}}-{\frac{15\,{a}^{6}{b}^{3} \left ( 7\,Ab+4\,Ba \right ) }{4\,{x}^{8}}}-{\frac{{b}^{9} \left ( Ab+10\,Ba \right ) }{2\,{x}^{2}}}-7\,{\frac{{a}^{4}{b}^{5} \left ( 5\,Ab+6\,Ba \right ) }{{x}^{6}}}-6\,{\frac{{a}^{5}{b}^{4} \left ( 6\,Ab+5\,Ba \right ) }{{x}^{7}}}-{\frac{B{b}^{10}}{x}}-{\frac{5\,{a}^{7}{b}^{2} \left ( 8\,Ab+3\,Ba \right ) }{3\,{x}^{9}}}-{\frac{{a}^{8}b \left ( 9\,Ab+2\,Ba \right ) }{2\,{x}^{10}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^10*(B*x+A)/x^13,x)

[Out]

-5/3*a*b^8*(2*A*b+9*B*a)/x^3-6*a^3*b^6*(4*A*b+7*B*a)/x^5-1/12*A*a^10/x^12-1/11*a^9*(10*A*b+B*a)/x^11-15/4*a^2*
b^7*(3*A*b+8*B*a)/x^4-15/4*a^6*b^3*(7*A*b+4*B*a)/x^8-1/2*b^9*(A*b+10*B*a)/x^2-7*a^4*b^5*(5*A*b+6*B*a)/x^6-6*a^
5*b^4*(6*A*b+5*B*a)/x^7-B*b^10/x-5/3*a^7*b^2*(8*A*b+3*B*a)/x^9-1/2*a^8*b*(9*A*b+2*B*a)/x^10

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Maxima [B]  time = 1.02908, size = 328, normalized size = 7.45 \begin{align*} -\frac{132 \, B b^{10} x^{11} + 11 \, A a^{10} + 66 \,{\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 220 \,{\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 495 \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 792 \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 924 \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 792 \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 495 \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 220 \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 66 \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 12 \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x}{132 \, x^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/x^13,x, algorithm="maxima")

[Out]

-1/132*(132*B*b^10*x^11 + 11*A*a^10 + 66*(10*B*a*b^9 + A*b^10)*x^10 + 220*(9*B*a^2*b^8 + 2*A*a*b^9)*x^9 + 495*
(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^8 + 792*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 + 924*(6*B*a^5*b^5 + 5*A*a^4*b^6)*x^6 +
792*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 495*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 + 220*(3*B*a^8*b^2 + 8*A*a^7*b^3)*x^
3 + 66*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 12*(B*a^10 + 10*A*a^9*b)*x)/x^12

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Fricas [B]  time = 1.46961, size = 544, normalized size = 12.36 \begin{align*} -\frac{132 \, B b^{10} x^{11} + 11 \, A a^{10} + 66 \,{\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 220 \,{\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 495 \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 792 \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 924 \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 792 \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 495 \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 220 \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 66 \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 12 \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x}{132 \, x^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/x^13,x, algorithm="fricas")

[Out]

-1/132*(132*B*b^10*x^11 + 11*A*a^10 + 66*(10*B*a*b^9 + A*b^10)*x^10 + 220*(9*B*a^2*b^8 + 2*A*a*b^9)*x^9 + 495*
(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^8 + 792*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 + 924*(6*B*a^5*b^5 + 5*A*a^4*b^6)*x^6 +
792*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 495*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 + 220*(3*B*a^8*b^2 + 8*A*a^7*b^3)*x^
3 + 66*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 12*(B*a^10 + 10*A*a^9*b)*x)/x^12

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Sympy [B]  time = 49.3216, size = 245, normalized size = 5.57 \begin{align*} - \frac{11 A a^{10} + 132 B b^{10} x^{11} + x^{10} \left (66 A b^{10} + 660 B a b^{9}\right ) + x^{9} \left (440 A a b^{9} + 1980 B a^{2} b^{8}\right ) + x^{8} \left (1485 A a^{2} b^{8} + 3960 B a^{3} b^{7}\right ) + x^{7} \left (3168 A a^{3} b^{7} + 5544 B a^{4} b^{6}\right ) + x^{6} \left (4620 A a^{4} b^{6} + 5544 B a^{5} b^{5}\right ) + x^{5} \left (4752 A a^{5} b^{5} + 3960 B a^{6} b^{4}\right ) + x^{4} \left (3465 A a^{6} b^{4} + 1980 B a^{7} b^{3}\right ) + x^{3} \left (1760 A a^{7} b^{3} + 660 B a^{8} b^{2}\right ) + x^{2} \left (594 A a^{8} b^{2} + 132 B a^{9} b\right ) + x \left (120 A a^{9} b + 12 B a^{10}\right )}{132 x^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**10*(B*x+A)/x**13,x)

[Out]

-(11*A*a**10 + 132*B*b**10*x**11 + x**10*(66*A*b**10 + 660*B*a*b**9) + x**9*(440*A*a*b**9 + 1980*B*a**2*b**8)
+ x**8*(1485*A*a**2*b**8 + 3960*B*a**3*b**7) + x**7*(3168*A*a**3*b**7 + 5544*B*a**4*b**6) + x**6*(4620*A*a**4*
b**6 + 5544*B*a**5*b**5) + x**5*(4752*A*a**5*b**5 + 3960*B*a**6*b**4) + x**4*(3465*A*a**6*b**4 + 1980*B*a**7*b
**3) + x**3*(1760*A*a**7*b**3 + 660*B*a**8*b**2) + x**2*(594*A*a**8*b**2 + 132*B*a**9*b) + x*(120*A*a**9*b + 1
2*B*a**10))/(132*x**12)

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Giac [B]  time = 1.2336, size = 328, normalized size = 7.45 \begin{align*} -\frac{132 \, B b^{10} x^{11} + 660 \, B a b^{9} x^{10} + 66 \, A b^{10} x^{10} + 1980 \, B a^{2} b^{8} x^{9} + 440 \, A a b^{9} x^{9} + 3960 \, B a^{3} b^{7} x^{8} + 1485 \, A a^{2} b^{8} x^{8} + 5544 \, B a^{4} b^{6} x^{7} + 3168 \, A a^{3} b^{7} x^{7} + 5544 \, B a^{5} b^{5} x^{6} + 4620 \, A a^{4} b^{6} x^{6} + 3960 \, B a^{6} b^{4} x^{5} + 4752 \, A a^{5} b^{5} x^{5} + 1980 \, B a^{7} b^{3} x^{4} + 3465 \, A a^{6} b^{4} x^{4} + 660 \, B a^{8} b^{2} x^{3} + 1760 \, A a^{7} b^{3} x^{3} + 132 \, B a^{9} b x^{2} + 594 \, A a^{8} b^{2} x^{2} + 12 \, B a^{10} x + 120 \, A a^{9} b x + 11 \, A a^{10}}{132 \, x^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/x^13,x, algorithm="giac")

[Out]

-1/132*(132*B*b^10*x^11 + 660*B*a*b^9*x^10 + 66*A*b^10*x^10 + 1980*B*a^2*b^8*x^9 + 440*A*a*b^9*x^9 + 3960*B*a^
3*b^7*x^8 + 1485*A*a^2*b^8*x^8 + 5544*B*a^4*b^6*x^7 + 3168*A*a^3*b^7*x^7 + 5544*B*a^5*b^5*x^6 + 4620*A*a^4*b^6
*x^6 + 3960*B*a^6*b^4*x^5 + 4752*A*a^5*b^5*x^5 + 1980*B*a^7*b^3*x^4 + 3465*A*a^6*b^4*x^4 + 660*B*a^8*b^2*x^3 +
1760*A*a^7*b^3*x^3 + 132*B*a^9*b*x^2 + 594*A*a^8*b^2*x^2 + 12*B*a^10*x + 120*A*a^9*b*x + 11*A*a^10)/x^12