3.536 \(\int \frac{(A+B x) (a^2+2 a b x+b^2 x^2)^2}{x^9} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0462386, 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} \[ -\frac{a^3 (a B+4 A b)}{7 x^7}-\frac{a^2 b (2 a B+3 A b)}{3 x^6}-\frac{a^4 A}{8 x^8}-\frac{2 a b^2 (3 a B+2 A b)}{5 x^5}-\frac{b^3 (4 a B+A b)}{4 x^4}-\frac{b^4 B}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/x^9,x]

[Out]

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

Mathematica [A]  time = 0.0257552, size = 88, normalized size = 0.89 \[ -\frac{168 a^2 b^2 x^2 (5 A+6 B x)+80 a^3 b x (6 A+7 B x)+15 a^4 (7 A+8 B x)+168 a b^3 x^3 (4 A+5 B x)+70 b^4 x^4 (3 A+4 B x)}{840 x^8} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/x^9,x]

[Out]

-(70*b^4*x^4*(3*A + 4*B*x) + 168*a*b^3*x^3*(4*A + 5*B*x) + 168*a^2*b^2*x^2*(5*A + 6*B*x) + 80*a^3*b*x*(6*A + 7
*B*x) + 15*a^4*(7*A + 8*B*x))/(840*x^8)

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Maple [A]  time = 0.007, size = 88, normalized size = 0.9 \begin{align*} -{\frac{A{a}^{4}}{8\,{x}^{8}}}-{\frac{{a}^{3} \left ( 4\,Ab+aB \right ) }{7\,{x}^{7}}}-{\frac{{a}^{2}b \left ( 3\,Ab+2\,aB \right ) }{3\,{x}^{6}}}-{\frac{2\,a{b}^{2} \left ( 2\,Ab+3\,aB \right ) }{5\,{x}^{5}}}-{\frac{{b}^{3} \left ( Ab+4\,aB \right ) }{4\,{x}^{4}}}-{\frac{{b}^{4}B}{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)^2/x^9,x)

[Out]

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

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Maxima [A]  time = 0.994885, size = 134, normalized size = 1.35 \begin{align*} -\frac{280 \, B b^{4} x^{5} + 105 \, A a^{4} + 210 \,{\left (4 \, B a b^{3} + A b^{4}\right )} x^{4} + 336 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{3} + 280 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2} + 120 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} x}{840 \, x^{8}} \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^9,x, algorithm="maxima")

[Out]

-1/840*(280*B*b^4*x^5 + 105*A*a^4 + 210*(4*B*a*b^3 + A*b^4)*x^4 + 336*(3*B*a^2*b^2 + 2*A*a*b^3)*x^3 + 280*(2*B
*a^3*b + 3*A*a^2*b^2)*x^2 + 120*(B*a^4 + 4*A*a^3*b)*x)/x^8

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Fricas [A]  time = 1.28031, size = 230, normalized size = 2.32 \begin{align*} -\frac{280 \, B b^{4} x^{5} + 105 \, A a^{4} + 210 \,{\left (4 \, B a b^{3} + A b^{4}\right )} x^{4} + 336 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{3} + 280 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2} + 120 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} x}{840 \, x^{8}} \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^9,x, algorithm="fricas")

[Out]

-1/840*(280*B*b^4*x^5 + 105*A*a^4 + 210*(4*B*a*b^3 + A*b^4)*x^4 + 336*(3*B*a^2*b^2 + 2*A*a*b^3)*x^3 + 280*(2*B
*a^3*b + 3*A*a^2*b^2)*x^2 + 120*(B*a^4 + 4*A*a^3*b)*x)/x^8

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Sympy [A]  time = 4.65636, size = 102, normalized size = 1.03 \begin{align*} - \frac{105 A a^{4} + 280 B b^{4} x^{5} + x^{4} \left (210 A b^{4} + 840 B a b^{3}\right ) + x^{3} \left (672 A a b^{3} + 1008 B a^{2} b^{2}\right ) + x^{2} \left (840 A a^{2} b^{2} + 560 B a^{3} b\right ) + x \left (480 A a^{3} b + 120 B a^{4}\right )}{840 x^{8}} \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**9,x)

[Out]

-(105*A*a**4 + 280*B*b**4*x**5 + x**4*(210*A*b**4 + 840*B*a*b**3) + x**3*(672*A*a*b**3 + 1008*B*a**2*b**2) + x
**2*(840*A*a**2*b**2 + 560*B*a**3*b) + x*(480*A*a**3*b + 120*B*a**4))/(840*x**8)

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Giac [A]  time = 1.16342, size = 134, normalized size = 1.35 \begin{align*} -\frac{280 \, B b^{4} x^{5} + 840 \, B a b^{3} x^{4} + 210 \, A b^{4} x^{4} + 1008 \, B a^{2} b^{2} x^{3} + 672 \, A a b^{3} x^{3} + 560 \, B a^{3} b x^{2} + 840 \, A a^{2} b^{2} x^{2} + 120 \, B a^{4} x + 480 \, A a^{3} b x + 105 \, A a^{4}}{840 \, x^{8}} \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^9,x, algorithm="giac")

[Out]

-1/840*(280*B*b^4*x^5 + 840*B*a*b^3*x^4 + 210*A*b^4*x^4 + 1008*B*a^2*b^2*x^3 + 672*A*a*b^3*x^3 + 560*B*a^3*b*x
^2 + 840*A*a^2*b^2*x^2 + 120*B*a^4*x + 480*A*a^3*b*x + 105*A*a^4)/x^8