3.136 \(\int \frac {(a+b x)^5 (A+B x)}{x^{12}} \, dx\)

Optimal. Leaf size=117 \[ -\frac {a^5 A}{11 x^{11}}-\frac {a^4 (a B+5 A b)}{10 x^{10}}-\frac {5 a^3 b (a B+2 A b)}{9 x^9}-\frac {5 a^2 b^2 (a B+A b)}{4 x^8}-\frac {b^4 (5 a B+A b)}{6 x^6}-\frac {5 a b^3 (2 a B+A b)}{7 x^7}-\frac {b^5 B}{5 x^5} \]

[Out]

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

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Rubi [A]  time = 0.06, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {76} \[ -\frac {5 a^2 b^2 (a B+A b)}{4 x^8}-\frac {a^4 (a B+5 A b)}{10 x^{10}}-\frac {5 a^3 b (a B+2 A b)}{9 x^9}-\frac {a^5 A}{11 x^{11}}-\frac {5 a b^3 (2 a B+A b)}{7 x^7}-\frac {b^4 (5 a B+A b)}{6 x^6}-\frac {b^5 B}{5 x^5} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^5*(A + B*x))/x^12,x]

[Out]

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

Rule 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])

Rubi steps

\begin {align*} \int \frac {(a+b x)^5 (A+B x)}{x^{12}} \, dx &=\int \left (\frac {a^5 A}{x^{12}}+\frac {a^4 (5 A b+a B)}{x^{11}}+\frac {5 a^3 b (2 A b+a B)}{x^{10}}+\frac {10 a^2 b^2 (A b+a B)}{x^9}+\frac {5 a b^3 (A b+2 a B)}{x^8}+\frac {b^4 (A b+5 a B)}{x^7}+\frac {b^5 B}{x^6}\right ) \, dx\\ &=-\frac {a^5 A}{11 x^{11}}-\frac {a^4 (5 A b+a B)}{10 x^{10}}-\frac {5 a^3 b (2 A b+a B)}{9 x^9}-\frac {5 a^2 b^2 (A b+a B)}{4 x^8}-\frac {5 a b^3 (A b+2 a B)}{7 x^7}-\frac {b^4 (A b+5 a B)}{6 x^6}-\frac {b^5 B}{5 x^5}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 107, normalized size = 0.91 \[ -\frac {126 a^5 (10 A+11 B x)+770 a^4 b x (9 A+10 B x)+1925 a^3 b^2 x^2 (8 A+9 B x)+2475 a^2 b^3 x^3 (7 A+8 B x)+1650 a b^4 x^4 (6 A+7 B x)+462 b^5 x^5 (5 A+6 B x)}{13860 x^{11}} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^5*(A + B*x))/x^12,x]

[Out]

-1/13860*(462*b^5*x^5*(5*A + 6*B*x) + 1650*a*b^4*x^4*(6*A + 7*B*x) + 2475*a^2*b^3*x^3*(7*A + 8*B*x) + 1925*a^3
*b^2*x^2*(8*A + 9*B*x) + 770*a^4*b*x*(9*A + 10*B*x) + 126*a^5*(10*A + 11*B*x))/x^11

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fricas [A]  time = 0.71, size = 119, normalized size = 1.02 \[ -\frac {2772 \, B b^{5} x^{6} + 1260 \, A a^{5} + 2310 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 9900 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 17325 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 7700 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 1386 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x}{13860 \, x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5*(B*x+A)/x^12,x, algorithm="fricas")

[Out]

-1/13860*(2772*B*b^5*x^6 + 1260*A*a^5 + 2310*(5*B*a*b^4 + A*b^5)*x^5 + 9900*(2*B*a^2*b^3 + A*a*b^4)*x^4 + 1732
5*(B*a^3*b^2 + A*a^2*b^3)*x^3 + 7700*(B*a^4*b + 2*A*a^3*b^2)*x^2 + 1386*(B*a^5 + 5*A*a^4*b)*x)/x^11

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giac [A]  time = 1.25, size = 123, normalized size = 1.05 \[ -\frac {2772 \, B b^{5} x^{6} + 11550 \, B a b^{4} x^{5} + 2310 \, A b^{5} x^{5} + 19800 \, B a^{2} b^{3} x^{4} + 9900 \, A a b^{4} x^{4} + 17325 \, B a^{3} b^{2} x^{3} + 17325 \, A a^{2} b^{3} x^{3} + 7700 \, B a^{4} b x^{2} + 15400 \, A a^{3} b^{2} x^{2} + 1386 \, B a^{5} x + 6930 \, A a^{4} b x + 1260 \, A a^{5}}{13860 \, x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5*(B*x+A)/x^12,x, algorithm="giac")

[Out]

-1/13860*(2772*B*b^5*x^6 + 11550*B*a*b^4*x^5 + 2310*A*b^5*x^5 + 19800*B*a^2*b^3*x^4 + 9900*A*a*b^4*x^4 + 17325
*B*a^3*b^2*x^3 + 17325*A*a^2*b^3*x^3 + 7700*B*a^4*b*x^2 + 15400*A*a^3*b^2*x^2 + 1386*B*a^5*x + 6930*A*a^4*b*x
+ 1260*A*a^5)/x^11

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maple [A]  time = 0.01, size = 104, normalized size = 0.89 \[ -\frac {B \,b^{5}}{5 x^{5}}-\frac {\left (A b +5 B a \right ) b^{4}}{6 x^{6}}-\frac {5 \left (A b +2 B a \right ) a \,b^{3}}{7 x^{7}}-\frac {5 \left (A b +B a \right ) a^{2} b^{2}}{4 x^{8}}-\frac {A \,a^{5}}{11 x^{11}}-\frac {5 \left (2 A b +B a \right ) a^{3} b}{9 x^{9}}-\frac {\left (5 A b +B a \right ) a^{4}}{10 x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^5*(B*x+A)/x^12,x)

[Out]

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

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maxima [A]  time = 1.09, size = 119, normalized size = 1.02 \[ -\frac {2772 \, B b^{5} x^{6} + 1260 \, A a^{5} + 2310 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 9900 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 17325 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 7700 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 1386 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x}{13860 \, x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5*(B*x+A)/x^12,x, algorithm="maxima")

[Out]

-1/13860*(2772*B*b^5*x^6 + 1260*A*a^5 + 2310*(5*B*a*b^4 + A*b^5)*x^5 + 9900*(2*B*a^2*b^3 + A*a*b^4)*x^4 + 1732
5*(B*a^3*b^2 + A*a^2*b^3)*x^3 + 7700*(B*a^4*b + 2*A*a^3*b^2)*x^2 + 1386*(B*a^5 + 5*A*a^4*b)*x)/x^11

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mupad [B]  time = 0.32, size = 120, normalized size = 1.03 \[ -\frac {x\,\left (\frac {B\,a^5}{10}+\frac {A\,b\,a^4}{2}\right )+\frac {A\,a^5}{11}+x^4\,\left (\frac {10\,B\,a^2\,b^3}{7}+\frac {5\,A\,a\,b^4}{7}\right )+x^2\,\left (\frac {5\,B\,a^4\,b}{9}+\frac {10\,A\,a^3\,b^2}{9}\right )+x^5\,\left (\frac {A\,b^5}{6}+\frac {5\,B\,a\,b^4}{6}\right )+x^3\,\left (\frac {5\,B\,a^3\,b^2}{4}+\frac {5\,A\,a^2\,b^3}{4}\right )+\frac {B\,b^5\,x^6}{5}}{x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x)^5)/x^12,x)

[Out]

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

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sympy [A]  time = 11.28, size = 133, normalized size = 1.14 \[ \frac {- 1260 A a^{5} - 2772 B b^{5} x^{6} + x^{5} \left (- 2310 A b^{5} - 11550 B a b^{4}\right ) + x^{4} \left (- 9900 A a b^{4} - 19800 B a^{2} b^{3}\right ) + x^{3} \left (- 17325 A a^{2} b^{3} - 17325 B a^{3} b^{2}\right ) + x^{2} \left (- 15400 A a^{3} b^{2} - 7700 B a^{4} b\right ) + x \left (- 6930 A a^{4} b - 1386 B a^{5}\right )}{13860 x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**5*(B*x+A)/x**12,x)

[Out]

(-1260*A*a**5 - 2772*B*b**5*x**6 + x**5*(-2310*A*b**5 - 11550*B*a*b**4) + x**4*(-9900*A*a*b**4 - 19800*B*a**2*
b**3) + x**3*(-17325*A*a**2*b**3 - 17325*B*a**3*b**2) + x**2*(-15400*A*a**3*b**2 - 7700*B*a**4*b) + x*(-6930*A
*a**4*b - 1386*B*a**5))/(13860*x**11)

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