3.555 \(\int \frac{(A+B x) (a^2+2 a b x+b^2 x^2)^3}{x^{11}} \, dx\)

Optimal. Leaf size=143 \[ -\frac{5 a^3 b^2 (3 a B+4 A b)}{7 x^7}-\frac{5 a^2 b^3 (4 a B+3 A b)}{6 x^6}-\frac{a^5 (a B+6 A b)}{9 x^9}-\frac{3 a^4 b (2 a B+5 A b)}{8 x^8}-\frac{a^6 A}{10 x^{10}}-\frac{3 a b^4 (5 a B+2 A b)}{5 x^5}-\frac{b^5 (6 a B+A b)}{4 x^4}-\frac{b^6 B}{3 x^3} \]

[Out]

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

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Rubi [A]  time = 0.0764113, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {27, 76} \[ -\frac{5 a^3 b^2 (3 a B+4 A b)}{7 x^7}-\frac{5 a^2 b^3 (4 a B+3 A b)}{6 x^6}-\frac{a^5 (a B+6 A b)}{9 x^9}-\frac{3 a^4 b (2 a B+5 A b)}{8 x^8}-\frac{a^6 A}{10 x^{10}}-\frac{3 a b^4 (5 a B+2 A b)}{5 x^5}-\frac{b^5 (6 a B+A b)}{4 x^4}-\frac{b^6 B}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/x^11,x]

[Out]

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

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

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

Mathematica [A]  time = 0.034262, size = 126, normalized size = 0.88 \[ -\frac{675 a^4 b^2 x^2 (7 A+8 B x)+1200 a^3 b^3 x^3 (6 A+7 B x)+1260 a^2 b^4 x^4 (5 A+6 B x)+210 a^5 b x (8 A+9 B x)+28 a^6 (9 A+10 B x)+756 a b^5 x^5 (4 A+5 B x)+210 b^6 x^6 (3 A+4 B x)}{2520 x^{10}} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/x^11,x]

[Out]

-(210*b^6*x^6*(3*A + 4*B*x) + 756*a*b^5*x^5*(4*A + 5*B*x) + 1260*a^2*b^4*x^4*(5*A + 6*B*x) + 1200*a^3*b^3*x^3*
(6*A + 7*B*x) + 675*a^4*b^2*x^2*(7*A + 8*B*x) + 210*a^5*b*x*(8*A + 9*B*x) + 28*a^6*(9*A + 10*B*x))/(2520*x^10)

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Maple [A]  time = 0.007, size = 128, normalized size = 0.9 \begin{align*} -{\frac{A{a}^{6}}{10\,{x}^{10}}}-{\frac{{a}^{5} \left ( 6\,Ab+aB \right ) }{9\,{x}^{9}}}-{\frac{3\,{a}^{4}b \left ( 5\,Ab+2\,aB \right ) }{8\,{x}^{8}}}-{\frac{5\,{a}^{3}{b}^{2} \left ( 4\,Ab+3\,aB \right ) }{7\,{x}^{7}}}-{\frac{5\,{a}^{2}{b}^{3} \left ( 3\,Ab+4\,aB \right ) }{6\,{x}^{6}}}-{\frac{3\,a{b}^{4} \left ( 2\,Ab+5\,aB \right ) }{5\,{x}^{5}}}-{\frac{{b}^{5} \left ( Ab+6\,aB \right ) }{4\,{x}^{4}}}-{\frac{B{b}^{6}}{3\,{x}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3/x^11,x)

[Out]

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

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Maxima [A]  time = 0.990964, size = 198, normalized size = 1.38 \begin{align*} -\frac{840 \, B b^{6} x^{7} + 252 \, A a^{6} + 630 \,{\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 1512 \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 2100 \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 1800 \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 945 \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 280 \,{\left (B a^{6} + 6 \, A a^{5} b\right )} x}{2520 \, x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3/x^11,x, algorithm="maxima")

[Out]

-1/2520*(840*B*b^6*x^7 + 252*A*a^6 + 630*(6*B*a*b^5 + A*b^6)*x^6 + 1512*(5*B*a^2*b^4 + 2*A*a*b^5)*x^5 + 2100*(
4*B*a^3*b^3 + 3*A*a^2*b^4)*x^4 + 1800*(3*B*a^4*b^2 + 4*A*a^3*b^3)*x^3 + 945*(2*B*a^5*b + 5*A*a^4*b^2)*x^2 + 28
0*(B*a^6 + 6*A*a^5*b)*x)/x^10

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Fricas [A]  time = 1.20178, size = 339, normalized size = 2.37 \begin{align*} -\frac{840 \, B b^{6} x^{7} + 252 \, A a^{6} + 630 \,{\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 1512 \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 2100 \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 1800 \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 945 \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 280 \,{\left (B a^{6} + 6 \, A a^{5} b\right )} x}{2520 \, x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3/x^11,x, algorithm="fricas")

[Out]

-1/2520*(840*B*b^6*x^7 + 252*A*a^6 + 630*(6*B*a*b^5 + A*b^6)*x^6 + 1512*(5*B*a^2*b^4 + 2*A*a*b^5)*x^5 + 2100*(
4*B*a^3*b^3 + 3*A*a^2*b^4)*x^4 + 1800*(3*B*a^4*b^2 + 4*A*a^3*b^3)*x^3 + 945*(2*B*a^5*b + 5*A*a^4*b^2)*x^2 + 28
0*(B*a^6 + 6*A*a^5*b)*x)/x^10

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Sympy [A]  time = 25.536, size = 150, normalized size = 1.05 \begin{align*} - \frac{252 A a^{6} + 840 B b^{6} x^{7} + x^{6} \left (630 A b^{6} + 3780 B a b^{5}\right ) + x^{5} \left (3024 A a b^{5} + 7560 B a^{2} b^{4}\right ) + x^{4} \left (6300 A a^{2} b^{4} + 8400 B a^{3} b^{3}\right ) + x^{3} \left (7200 A a^{3} b^{3} + 5400 B a^{4} b^{2}\right ) + x^{2} \left (4725 A a^{4} b^{2} + 1890 B a^{5} b\right ) + x \left (1680 A a^{5} b + 280 B a^{6}\right )}{2520 x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**3/x**11,x)

[Out]

-(252*A*a**6 + 840*B*b**6*x**7 + x**6*(630*A*b**6 + 3780*B*a*b**5) + x**5*(3024*A*a*b**5 + 7560*B*a**2*b**4) +
x**4*(6300*A*a**2*b**4 + 8400*B*a**3*b**3) + x**3*(7200*A*a**3*b**3 + 5400*B*a**4*b**2) + x**2*(4725*A*a**4*b
**2 + 1890*B*a**5*b) + x*(1680*A*a**5*b + 280*B*a**6))/(2520*x**10)

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Giac [A]  time = 1.14101, size = 198, normalized size = 1.38 \begin{align*} -\frac{840 \, B b^{6} x^{7} + 3780 \, B a b^{5} x^{6} + 630 \, A b^{6} x^{6} + 7560 \, B a^{2} b^{4} x^{5} + 3024 \, A a b^{5} x^{5} + 8400 \, B a^{3} b^{3} x^{4} + 6300 \, A a^{2} b^{4} x^{4} + 5400 \, B a^{4} b^{2} x^{3} + 7200 \, A a^{3} b^{3} x^{3} + 1890 \, B a^{5} b x^{2} + 4725 \, A a^{4} b^{2} x^{2} + 280 \, B a^{6} x + 1680 \, A a^{5} b x + 252 \, A a^{6}}{2520 \, x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3/x^11,x, algorithm="giac")

[Out]

-1/2520*(840*B*b^6*x^7 + 3780*B*a*b^5*x^6 + 630*A*b^6*x^6 + 7560*B*a^2*b^4*x^5 + 3024*A*a*b^5*x^5 + 8400*B*a^3
*b^3*x^4 + 6300*A*a^2*b^4*x^4 + 5400*B*a^4*b^2*x^3 + 7200*A*a^3*b^3*x^3 + 1890*B*a^5*b*x^2 + 4725*A*a^4*b^2*x^
2 + 280*B*a^6*x + 1680*A*a^5*b*x + 252*A*a^6)/x^10