3.557 \(\int \frac{(A+B x) (a^2+2 a b x+b^2 x^2)^3}{x^{13}} \, dx\)

Optimal. Leaf size=143 \[ -\frac{5 a^3 b^2 (3 a B+4 A b)}{9 x^9}-\frac{5 a^2 b^3 (4 a B+3 A b)}{8 x^8}-\frac{a^5 (a B+6 A b)}{11 x^{11}}-\frac{3 a^4 b (2 a B+5 A b)}{10 x^{10}}-\frac{a^6 A}{12 x^{12}}-\frac{3 a b^4 (5 a B+2 A b)}{7 x^7}-\frac{b^5 (6 a B+A b)}{6 x^6}-\frac{b^6 B}{5 x^5} \]

[Out]

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

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Rubi [A]  time = 0.0688152, 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} \[ -\frac{5 a^3 b^2 (3 a B+4 A b)}{9 x^9}-\frac{5 a^2 b^3 (4 a B+3 A b)}{8 x^8}-\frac{a^5 (a B+6 A b)}{11 x^{11}}-\frac{3 a^4 b (2 a B+5 A b)}{10 x^{10}}-\frac{a^6 A}{12 x^{12}}-\frac{3 a b^4 (5 a B+2 A b)}{7 x^7}-\frac{b^5 (6 a B+A b)}{6 x^6}-\frac{b^6 B}{5 x^5} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/x^13,x]

[Out]

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

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^{13}} \, dx &=\int \frac{(a+b x)^6 (A+B x)}{x^{13}} \, dx\\ &=\int \left (\frac{a^6 A}{x^{13}}+\frac{a^5 (6 A b+a B)}{x^{12}}+\frac{3 a^4 b (5 A b+2 a B)}{x^{11}}+\frac{5 a^3 b^2 (4 A b+3 a B)}{x^{10}}+\frac{5 a^2 b^3 (3 A b+4 a B)}{x^9}+\frac{3 a b^4 (2 A b+5 a B)}{x^8}+\frac{b^5 (A b+6 a B)}{x^7}+\frac{b^6 B}{x^6}\right ) \, dx\\ &=-\frac{a^6 A}{12 x^{12}}-\frac{a^5 (6 A b+a B)}{11 x^{11}}-\frac{3 a^4 b (5 A b+2 a B)}{10 x^{10}}-\frac{5 a^3 b^2 (4 A b+3 a B)}{9 x^9}-\frac{5 a^2 b^3 (3 A b+4 a B)}{8 x^8}-\frac{3 a b^4 (2 A b+5 a B)}{7 x^7}-\frac{b^5 (A b+6 a B)}{6 x^6}-\frac{b^6 B}{5 x^5}\\ \end{align*}

Mathematica [A]  time = 0.0325521, size = 126, normalized size = 0.88 \[ -\frac{4620 a^4 b^2 x^2 (9 A+10 B x)+7700 a^3 b^3 x^3 (8 A+9 B x)+7425 a^2 b^4 x^4 (7 A+8 B x)+1512 a^5 b x (10 A+11 B x)+210 a^6 (11 A+12 B x)+3960 a b^5 x^5 (6 A+7 B x)+924 b^6 x^6 (5 A+6 B x)}{27720 x^{12}} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/x^13,x]

[Out]

-(924*b^6*x^6*(5*A + 6*B*x) + 3960*a*b^5*x^5*(6*A + 7*B*x) + 7425*a^2*b^4*x^4*(7*A + 8*B*x) + 7700*a^3*b^3*x^3
*(8*A + 9*B*x) + 4620*a^4*b^2*x^2*(9*A + 10*B*x) + 1512*a^5*b*x*(10*A + 11*B*x) + 210*a^6*(11*A + 12*B*x))/(27
720*x^12)

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Maple [A]  time = 0.007, size = 128, normalized size = 0.9 \begin{align*} -{\frac{A{a}^{6}}{12\,{x}^{12}}}-{\frac{{a}^{5} \left ( 6\,Ab+aB \right ) }{11\,{x}^{11}}}-{\frac{3\,{a}^{4}b \left ( 5\,Ab+2\,aB \right ) }{10\,{x}^{10}}}-{\frac{5\,{a}^{3}{b}^{2} \left ( 4\,Ab+3\,aB \right ) }{9\,{x}^{9}}}-{\frac{5\,{a}^{2}{b}^{3} \left ( 3\,Ab+4\,aB \right ) }{8\,{x}^{8}}}-{\frac{3\,a{b}^{4} \left ( 2\,Ab+5\,aB \right ) }{7\,{x}^{7}}}-{\frac{{b}^{5} \left ( Ab+6\,aB \right ) }{6\,{x}^{6}}}-{\frac{B{b}^{6}}{5\,{x}^{5}}} \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^13,x)

[Out]

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

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Maxima [A]  time = 1.02854, size = 198, normalized size = 1.38 \begin{align*} -\frac{5544 \, B b^{6} x^{7} + 2310 \, A a^{6} + 4620 \,{\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 11880 \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 17325 \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 15400 \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 8316 \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 2520 \,{\left (B a^{6} + 6 \, A a^{5} b\right )} x}{27720 \, x^{12}} \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^13,x, algorithm="maxima")

[Out]

-1/27720*(5544*B*b^6*x^7 + 2310*A*a^6 + 4620*(6*B*a*b^5 + A*b^6)*x^6 + 11880*(5*B*a^2*b^4 + 2*A*a*b^5)*x^5 + 1
7325*(4*B*a^3*b^3 + 3*A*a^2*b^4)*x^4 + 15400*(3*B*a^4*b^2 + 4*A*a^3*b^3)*x^3 + 8316*(2*B*a^5*b + 5*A*a^4*b^2)*
x^2 + 2520*(B*a^6 + 6*A*a^5*b)*x)/x^12

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Fricas [A]  time = 1.25401, size = 351, normalized size = 2.45 \begin{align*} -\frac{5544 \, B b^{6} x^{7} + 2310 \, A a^{6} + 4620 \,{\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 11880 \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 17325 \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 15400 \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 8316 \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 2520 \,{\left (B a^{6} + 6 \, A a^{5} b\right )} x}{27720 \, x^{12}} \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^13,x, algorithm="fricas")

[Out]

-1/27720*(5544*B*b^6*x^7 + 2310*A*a^6 + 4620*(6*B*a*b^5 + A*b^6)*x^6 + 11880*(5*B*a^2*b^4 + 2*A*a*b^5)*x^5 + 1
7325*(4*B*a^3*b^3 + 3*A*a^2*b^4)*x^4 + 15400*(3*B*a^4*b^2 + 4*A*a^3*b^3)*x^3 + 8316*(2*B*a^5*b + 5*A*a^4*b^2)*
x^2 + 2520*(B*a^6 + 6*A*a^5*b)*x)/x^12

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Sympy [A]  time = 43.4522, size = 150, normalized size = 1.05 \begin{align*} - \frac{2310 A a^{6} + 5544 B b^{6} x^{7} + x^{6} \left (4620 A b^{6} + 27720 B a b^{5}\right ) + x^{5} \left (23760 A a b^{5} + 59400 B a^{2} b^{4}\right ) + x^{4} \left (51975 A a^{2} b^{4} + 69300 B a^{3} b^{3}\right ) + x^{3} \left (61600 A a^{3} b^{3} + 46200 B a^{4} b^{2}\right ) + x^{2} \left (41580 A a^{4} b^{2} + 16632 B a^{5} b\right ) + x \left (15120 A a^{5} b + 2520 B a^{6}\right )}{27720 x^{12}} \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**13,x)

[Out]

-(2310*A*a**6 + 5544*B*b**6*x**7 + x**6*(4620*A*b**6 + 27720*B*a*b**5) + x**5*(23760*A*a*b**5 + 59400*B*a**2*b
**4) + x**4*(51975*A*a**2*b**4 + 69300*B*a**3*b**3) + x**3*(61600*A*a**3*b**3 + 46200*B*a**4*b**2) + x**2*(415
80*A*a**4*b**2 + 16632*B*a**5*b) + x*(15120*A*a**5*b + 2520*B*a**6))/(27720*x**12)

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Giac [A]  time = 1.12966, size = 198, normalized size = 1.38 \begin{align*} -\frac{5544 \, B b^{6} x^{7} + 27720 \, B a b^{5} x^{6} + 4620 \, A b^{6} x^{6} + 59400 \, B a^{2} b^{4} x^{5} + 23760 \, A a b^{5} x^{5} + 69300 \, B a^{3} b^{3} x^{4} + 51975 \, A a^{2} b^{4} x^{4} + 46200 \, B a^{4} b^{2} x^{3} + 61600 \, A a^{3} b^{3} x^{3} + 16632 \, B a^{5} b x^{2} + 41580 \, A a^{4} b^{2} x^{2} + 2520 \, B a^{6} x + 15120 \, A a^{5} b x + 2310 \, A a^{6}}{27720 \, x^{12}} \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^13,x, algorithm="giac")

[Out]

-1/27720*(5544*B*b^6*x^7 + 27720*B*a*b^5*x^6 + 4620*A*b^6*x^6 + 59400*B*a^2*b^4*x^5 + 23760*A*a*b^5*x^5 + 6930
0*B*a^3*b^3*x^4 + 51975*A*a^2*b^4*x^4 + 46200*B*a^4*b^2*x^3 + 61600*A*a^3*b^3*x^3 + 16632*B*a^5*b*x^2 + 41580*
A*a^4*b^2*x^2 + 2520*B*a^6*x + 15120*A*a^5*b*x + 2310*A*a^6)/x^12