3.537 \(\int \frac{(A+B x) (a^2+2 a b x+b^2 x^2)^2}{x^{10}} \, dx\)

Optimal. Leaf size=99 \[ -\frac{a^3 (a B+4 A b)}{8 x^8}-\frac{2 a^2 b (2 a B+3 A b)}{7 x^7}-\frac{a^4 A}{9 x^9}-\frac{a b^2 (3 a B+2 A b)}{3 x^6}-\frac{b^3 (4 a B+A b)}{5 x^5}-\frac{b^4 B}{4 x^4} \]

[Out]

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

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Rubi [A]  time = 0.0476277, antiderivative size = 99, 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} \[ -\frac{a^3 (a B+4 A b)}{8 x^8}-\frac{2 a^2 b (2 a B+3 A b)}{7 x^7}-\frac{a^4 A}{9 x^9}-\frac{a b^2 (3 a B+2 A b)}{3 x^6}-\frac{b^3 (4 a B+A b)}{5 x^5}-\frac{b^4 B}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/x^10,x]

[Out]

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

Mathematica [A]  time = 0.025782, size = 88, normalized size = 0.89 \[ -\frac{360 a^2 b^2 x^2 (6 A+7 B x)+180 a^3 b x (7 A+8 B x)+35 a^4 (8 A+9 B x)+336 a b^3 x^3 (5 A+6 B x)+126 b^4 x^4 (4 A+5 B x)}{2520 x^9} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/x^10,x]

[Out]

-(126*b^4*x^4*(4*A + 5*B*x) + 336*a*b^3*x^3*(5*A + 6*B*x) + 360*a^2*b^2*x^2*(6*A + 7*B*x) + 180*a^3*b*x*(7*A +
8*B*x) + 35*a^4*(8*A + 9*B*x))/(2520*x^9)

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Maple [A]  time = 0.006, size = 88, normalized size = 0.9 \begin{align*} -{\frac{A{a}^{4}}{9\,{x}^{9}}}-{\frac{{a}^{3} \left ( 4\,Ab+aB \right ) }{8\,{x}^{8}}}-{\frac{2\,{a}^{2}b \left ( 3\,Ab+2\,aB \right ) }{7\,{x}^{7}}}-{\frac{a{b}^{2} \left ( 2\,Ab+3\,aB \right ) }{3\,{x}^{6}}}-{\frac{{b}^{3} \left ( Ab+4\,aB \right ) }{5\,{x}^{5}}}-{\frac{{b}^{4}B}{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)^2/x^10,x)

[Out]

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

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Maxima [A]  time = 0.969219, size = 134, normalized size = 1.35 \begin{align*} -\frac{630 \, B b^{4} x^{5} + 280 \, A a^{4} + 504 \,{\left (4 \, B a b^{3} + A b^{4}\right )} x^{4} + 840 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{3} + 720 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2} + 315 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} x}{2520 \, x^{9}} \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)^2/x^10,x, algorithm="maxima")

[Out]

-1/2520*(630*B*b^4*x^5 + 280*A*a^4 + 504*(4*B*a*b^3 + A*b^4)*x^4 + 840*(3*B*a^2*b^2 + 2*A*a*b^3)*x^3 + 720*(2*
B*a^3*b + 3*A*a^2*b^2)*x^2 + 315*(B*a^4 + 4*A*a^3*b)*x)/x^9

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Fricas [A]  time = 1.24425, size = 231, normalized size = 2.33 \begin{align*} -\frac{630 \, B b^{4} x^{5} + 280 \, A a^{4} + 504 \,{\left (4 \, B a b^{3} + A b^{4}\right )} x^{4} + 840 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{3} + 720 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2} + 315 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} x}{2520 \, x^{9}} \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)^2/x^10,x, algorithm="fricas")

[Out]

-1/2520*(630*B*b^4*x^5 + 280*A*a^4 + 504*(4*B*a*b^3 + A*b^4)*x^4 + 840*(3*B*a^2*b^2 + 2*A*a*b^3)*x^3 + 720*(2*
B*a^3*b + 3*A*a^2*b^2)*x^2 + 315*(B*a^4 + 4*A*a^3*b)*x)/x^9

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Sympy [A]  time = 8.01726, size = 102, normalized size = 1.03 \begin{align*} - \frac{280 A a^{4} + 630 B b^{4} x^{5} + x^{4} \left (504 A b^{4} + 2016 B a b^{3}\right ) + x^{3} \left (1680 A a b^{3} + 2520 B a^{2} b^{2}\right ) + x^{2} \left (2160 A a^{2} b^{2} + 1440 B a^{3} b\right ) + x \left (1260 A a^{3} b + 315 B a^{4}\right )}{2520 x^{9}} \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)**2/x**10,x)

[Out]

-(280*A*a**4 + 630*B*b**4*x**5 + x**4*(504*A*b**4 + 2016*B*a*b**3) + x**3*(1680*A*a*b**3 + 2520*B*a**2*b**2) +
x**2*(2160*A*a**2*b**2 + 1440*B*a**3*b) + x*(1260*A*a**3*b + 315*B*a**4))/(2520*x**9)

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Giac [A]  time = 1.14042, size = 134, normalized size = 1.35 \begin{align*} -\frac{630 \, B b^{4} x^{5} + 2016 \, B a b^{3} x^{4} + 504 \, A b^{4} x^{4} + 2520 \, B a^{2} b^{2} x^{3} + 1680 \, A a b^{3} x^{3} + 1440 \, B a^{3} b x^{2} + 2160 \, A a^{2} b^{2} x^{2} + 315 \, B a^{4} x + 1260 \, A a^{3} b x + 280 \, A a^{4}}{2520 \, x^{9}} \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)^2/x^10,x, algorithm="giac")

[Out]

-1/2520*(630*B*b^4*x^5 + 2016*B*a*b^3*x^4 + 504*A*b^4*x^4 + 2520*B*a^2*b^2*x^3 + 1680*A*a*b^3*x^3 + 1440*B*a^3
*b*x^2 + 2160*A*a^2*b^2*x^2 + 315*B*a^4*x + 1260*A*a^3*b*x + 280*A*a^4)/x^9