3.556 \(\int \frac{(A+B x) (a^2+2 a b x+b^2 x^2)^3}{x^{12}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0716047, 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} \[ -\frac{5 a^3 b^2 (3 a B+4 A b)}{8 x^8}-\frac{5 a^2 b^3 (4 a B+3 A b)}{7 x^7}-\frac{a^5 (a B+6 A b)}{10 x^{10}}-\frac{a^4 b (2 a B+5 A b)}{3 x^9}-\frac{a^6 A}{11 x^{11}}-\frac{a b^4 (5 a B+2 A b)}{2 x^6}-\frac{b^5 (6 a B+A b)}{5 x^5}-\frac{b^6 B}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/x^12,x]

[Out]

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

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

Mathematica [A]  time = 0.0337319, size = 126, normalized size = 0.88 \[ -\frac{1925 a^4 b^2 x^2 (8 A+9 B x)+3300 a^3 b^3 x^3 (7 A+8 B x)+3300 a^2 b^4 x^4 (6 A+7 B x)+616 a^5 b x (9 A+10 B x)+84 a^6 (10 A+11 B x)+1848 a b^5 x^5 (5 A+6 B x)+462 b^6 x^6 (4 A+5 B x)}{9240 x^{11}} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/x^12,x]

[Out]

-(462*b^6*x^6*(4*A + 5*B*x) + 1848*a*b^5*x^5*(5*A + 6*B*x) + 3300*a^2*b^4*x^4*(6*A + 7*B*x) + 3300*a^3*b^3*x^3
*(7*A + 8*B*x) + 1925*a^4*b^2*x^2*(8*A + 9*B*x) + 616*a^5*b*x*(9*A + 10*B*x) + 84*a^6*(10*A + 11*B*x))/(9240*x
^11)

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Maple [A]  time = 0.008, size = 128, normalized size = 0.9 \begin{align*} -{\frac{A{a}^{6}}{11\,{x}^{11}}}-{\frac{{a}^{5} \left ( 6\,Ab+aB \right ) }{10\,{x}^{10}}}-{\frac{{a}^{4}b \left ( 5\,Ab+2\,aB \right ) }{3\,{x}^{9}}}-{\frac{5\,{a}^{3}{b}^{2} \left ( 4\,Ab+3\,aB \right ) }{8\,{x}^{8}}}-{\frac{5\,{a}^{2}{b}^{3} \left ( 3\,Ab+4\,aB \right ) }{7\,{x}^{7}}}-{\frac{a{b}^{4} \left ( 2\,Ab+5\,aB \right ) }{2\,{x}^{6}}}-{\frac{{b}^{5} \left ( Ab+6\,aB \right ) }{5\,{x}^{5}}}-{\frac{B{b}^{6}}{4\,{x}^{4}}} \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^12,x)

[Out]

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

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Maxima [A]  time = 0.980805, size = 198, normalized size = 1.38 \begin{align*} -\frac{2310 \, B b^{6} x^{7} + 840 \, A a^{6} + 1848 \,{\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 4620 \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 6600 \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 5775 \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 3080 \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 924 \,{\left (B a^{6} + 6 \, A a^{5} b\right )} x}{9240 \, x^{11}} \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^12,x, algorithm="maxima")

[Out]

-1/9240*(2310*B*b^6*x^7 + 840*A*a^6 + 1848*(6*B*a*b^5 + A*b^6)*x^6 + 4620*(5*B*a^2*b^4 + 2*A*a*b^5)*x^5 + 6600
*(4*B*a^3*b^3 + 3*A*a^2*b^4)*x^4 + 5775*(3*B*a^4*b^2 + 4*A*a^3*b^3)*x^3 + 3080*(2*B*a^5*b + 5*A*a^4*b^2)*x^2 +
924*(B*a^6 + 6*A*a^5*b)*x)/x^11

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Fricas [A]  time = 1.19436, size = 343, normalized size = 2.4 \begin{align*} -\frac{2310 \, B b^{6} x^{7} + 840 \, A a^{6} + 1848 \,{\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 4620 \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 6600 \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 5775 \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 3080 \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 924 \,{\left (B a^{6} + 6 \, A a^{5} b\right )} x}{9240 \, x^{11}} \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^12,x, algorithm="fricas")

[Out]

-1/9240*(2310*B*b^6*x^7 + 840*A*a^6 + 1848*(6*B*a*b^5 + A*b^6)*x^6 + 4620*(5*B*a^2*b^4 + 2*A*a*b^5)*x^5 + 6600
*(4*B*a^3*b^3 + 3*A*a^2*b^4)*x^4 + 5775*(3*B*a^4*b^2 + 4*A*a^3*b^3)*x^3 + 3080*(2*B*a^5*b + 5*A*a^4*b^2)*x^2 +
924*(B*a^6 + 6*A*a^5*b)*x)/x^11

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Sympy [A]  time = 33.9449, size = 150, normalized size = 1.05 \begin{align*} - \frac{840 A a^{6} + 2310 B b^{6} x^{7} + x^{6} \left (1848 A b^{6} + 11088 B a b^{5}\right ) + x^{5} \left (9240 A a b^{5} + 23100 B a^{2} b^{4}\right ) + x^{4} \left (19800 A a^{2} b^{4} + 26400 B a^{3} b^{3}\right ) + x^{3} \left (23100 A a^{3} b^{3} + 17325 B a^{4} b^{2}\right ) + x^{2} \left (15400 A a^{4} b^{2} + 6160 B a^{5} b\right ) + x \left (5544 A a^{5} b + 924 B a^{6}\right )}{9240 x^{11}} \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**12,x)

[Out]

-(840*A*a**6 + 2310*B*b**6*x**7 + x**6*(1848*A*b**6 + 11088*B*a*b**5) + x**5*(9240*A*a*b**5 + 23100*B*a**2*b**
4) + x**4*(19800*A*a**2*b**4 + 26400*B*a**3*b**3) + x**3*(23100*A*a**3*b**3 + 17325*B*a**4*b**2) + x**2*(15400
*A*a**4*b**2 + 6160*B*a**5*b) + x*(5544*A*a**5*b + 924*B*a**6))/(9240*x**11)

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Giac [A]  time = 1.09211, size = 198, normalized size = 1.38 \begin{align*} -\frac{2310 \, B b^{6} x^{7} + 11088 \, B a b^{5} x^{6} + 1848 \, A b^{6} x^{6} + 23100 \, B a^{2} b^{4} x^{5} + 9240 \, A a b^{5} x^{5} + 26400 \, B a^{3} b^{3} x^{4} + 19800 \, A a^{2} b^{4} x^{4} + 17325 \, B a^{4} b^{2} x^{3} + 23100 \, A a^{3} b^{3} x^{3} + 6160 \, B a^{5} b x^{2} + 15400 \, A a^{4} b^{2} x^{2} + 924 \, B a^{6} x + 5544 \, A a^{5} b x + 840 \, A a^{6}}{9240 \, x^{11}} \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^12,x, algorithm="giac")

[Out]

-1/9240*(2310*B*b^6*x^7 + 11088*B*a*b^5*x^6 + 1848*A*b^6*x^6 + 23100*B*a^2*b^4*x^5 + 9240*A*a*b^5*x^5 + 26400*
B*a^3*b^3*x^4 + 19800*A*a^2*b^4*x^4 + 17325*B*a^4*b^2*x^3 + 23100*A*a^3*b^3*x^3 + 6160*B*a^5*b*x^2 + 15400*A*a
^4*b^2*x^2 + 924*B*a^6*x + 5544*A*a^5*b*x + 840*A*a^6)/x^11