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]

-1/12*A*(b*x+a)^11/a/x^12+1/132*(A*b-12*B*a)*(b*x+a)^11/a^2/x^11

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Rubi [A]  time = 0.01, antiderivative size = 44, normalized size of antiderivative = 1.00, 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 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]

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]))))

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*}

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Mathematica [B]  time = 0.07, size = 199, normalized size = 4.52 \[ -\frac {a^{10} (11 A+12 B x)+12 a^9 b x (10 A+11 B x)+66 a^8 b^2 x^2 (9 A+10 B x)+220 a^7 b^3 x^3 (8 A+9 B x)+495 a^6 b^4 x^4 (7 A+8 B x)+792 a^5 b^5 x^5 (6 A+7 B x)+924 a^4 b^6 x^6 (5 A+6 B x)+792 a^3 b^7 x^7 (4 A+5 B x)+495 a^2 b^8 x^8 (3 A+4 B x)+220 a b^9 x^9 (2 A+3 B x)+66 b^{10} x^{10} (A+2 B x)}{132 x^{12}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/132*(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) + 220*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))/x^12

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fricas [B]  time = 0.82, size = 243, normalized size = 5.52 \[ -\frac {132 \, B b^{10} x^{11} + 11 \, A a^{10} + 66 \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 220 \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 495 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 792 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 924 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 792 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 495 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 220 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 66 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 12 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{132 \, x^{12}} \]

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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giac [B]  time = 0.99, size = 243, normalized size = 5.52 \[ -\frac {132 \, B b^{10} x^{11} + 660 \, B a b^{9} x^{10} + 66 \, A b^{10} x^{10} + 1980 \, B a^{2} b^{8} x^{9} + 440 \, A a b^{9} x^{9} + 3960 \, B a^{3} b^{7} x^{8} + 1485 \, A a^{2} b^{8} x^{8} + 5544 \, B a^{4} b^{6} x^{7} + 3168 \, A a^{3} b^{7} x^{7} + 5544 \, B a^{5} b^{5} x^{6} + 4620 \, A a^{4} b^{6} x^{6} + 3960 \, B a^{6} b^{4} x^{5} + 4752 \, A a^{5} b^{5} x^{5} + 1980 \, B a^{7} b^{3} x^{4} + 3465 \, A a^{6} b^{4} x^{4} + 660 \, B a^{8} b^{2} x^{3} + 1760 \, A a^{7} b^{3} x^{3} + 132 \, B a^{9} b x^{2} + 594 \, A a^{8} b^{2} x^{2} + 12 \, B a^{10} x + 120 \, A a^{9} b x + 11 \, A a^{10}}{132 \, x^{12}} \]

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

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maple [B]  time = 0.01, size = 208, normalized size = 4.73 \[ -\frac {B \,b^{10}}{x}-\frac {\left (A b +10 B a \right ) b^{9}}{2 x^{2}}-\frac {5 \left (2 A b +9 B a \right ) a \,b^{8}}{3 x^{3}}-\frac {15 \left (3 A b +8 B a \right ) a^{2} b^{7}}{4 x^{4}}-\frac {6 \left (4 A b +7 B a \right ) a^{3} b^{6}}{x^{5}}-\frac {7 \left (5 A b +6 B a \right ) a^{4} b^{5}}{x^{6}}-\frac {6 \left (6 A b +5 B a \right ) a^{5} b^{4}}{x^{7}}-\frac {15 \left (7 A b +4 B a \right ) a^{6} b^{3}}{4 x^{8}}-\frac {5 \left (8 A b +3 B a \right ) a^{7} b^{2}}{3 x^{9}}-\frac {A \,a^{10}}{12 x^{12}}-\frac {\left (9 A b +2 B a \right ) a^{8} b}{2 x^{10}}-\frac {\left (10 A b +B a \right ) a^{9}}{11 x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 1.10, size = 243, normalized size = 5.52 \[ -\frac {132 \, B b^{10} x^{11} + 11 \, A a^{10} + 66 \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 220 \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 495 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 792 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 924 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 792 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 495 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 220 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 66 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 12 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{132 \, x^{12}} \]

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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mupad [B]  time = 0.15, size = 233, normalized size = 5.30 \[ -\frac {x\,\left (\frac {B\,a^{10}}{11}+\frac {10\,A\,b\,a^9}{11}\right )+\frac {A\,a^{10}}{12}+x^2\,\left (B\,a^9\,b+\frac {9\,A\,a^8\,b^2}{2}\right )+x^9\,\left (15\,B\,a^2\,b^8+\frac {10\,A\,a\,b^9}{3}\right )+x^{10}\,\left (\frac {A\,b^{10}}{2}+5\,B\,a\,b^9\right )+x^3\,\left (5\,B\,a^8\,b^2+\frac {40\,A\,a^7\,b^3}{3}\right )+x^5\,\left (30\,B\,a^6\,b^4+36\,A\,a^5\,b^5\right )+x^7\,\left (42\,B\,a^4\,b^6+24\,A\,a^3\,b^7\right )+x^6\,\left (42\,B\,a^5\,b^5+35\,A\,a^4\,b^6\right )+x^8\,\left (30\,B\,a^3\,b^7+\frac {45\,A\,a^2\,b^8}{4}\right )+x^4\,\left (15\,B\,a^7\,b^3+\frac {105\,A\,a^6\,b^4}{4}\right )+B\,b^{10}\,x^{11}}{x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x)^10)/x^13,x)

[Out]

-(x*((B*a^10)/11 + (10*A*a^9*b)/11) + (A*a^10)/12 + x^2*((9*A*a^8*b^2)/2 + B*a^9*b) + x^9*(15*B*a^2*b^8 + (10*
A*a*b^9)/3) + x^10*((A*b^10)/2 + 5*B*a*b^9) + x^3*((40*A*a^7*b^3)/3 + 5*B*a^8*b^2) + x^5*(36*A*a^5*b^5 + 30*B*
a^6*b^4) + x^7*(24*A*a^3*b^7 + 42*B*a^4*b^6) + x^6*(35*A*a^4*b^6 + 42*B*a^5*b^5) + x^8*((45*A*a^2*b^8)/4 + 30*
B*a^3*b^7) + x^4*((105*A*a^6*b^4)/4 + 15*B*a^7*b^3) + B*b^10*x^11)/x^12

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sympy [B]  time = 27.59, size = 260, normalized size = 5.91 \[ \frac {- 11 A a^{10} - 132 B b^{10} x^{11} + x^{10} \left (- 66 A b^{10} - 660 B a b^{9}\right ) + x^{9} \left (- 440 A a b^{9} - 1980 B a^{2} b^{8}\right ) + x^{8} \left (- 1485 A a^{2} b^{8} - 3960 B a^{3} b^{7}\right ) + x^{7} \left (- 3168 A a^{3} b^{7} - 5544 B a^{4} b^{6}\right ) + x^{6} \left (- 4620 A a^{4} b^{6} - 5544 B a^{5} b^{5}\right ) + x^{5} \left (- 4752 A a^{5} b^{5} - 3960 B a^{6} b^{4}\right ) + x^{4} \left (- 3465 A a^{6} b^{4} - 1980 B a^{7} b^{3}\right ) + x^{3} \left (- 1760 A a^{7} b^{3} - 660 B a^{8} b^{2}\right ) + x^{2} \left (- 594 A a^{8} b^{2} - 132 B a^{9} b\right ) + x \left (- 120 A a^{9} b - 12 B a^{10}\right )}{132 x^{12}} \]

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 - 12*B*a**10))/(132*x**12)

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