3.117 \(\int \frac{(a+b x)^3 (A+B x)}{x^9} \, dx\)

Optimal. Leaf size=75 \[ -\frac{a^2 (a B+3 A b)}{7 x^7}-\frac{a^3 A}{8 x^8}-\frac{b^2 (3 a B+A b)}{5 x^5}-\frac{a b (a B+A b)}{2 x^6}-\frac{b^3 B}{4 x^4} \]

[Out]

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

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Rubi [A]  time = 0.0339948, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {76} \[ -\frac{a^2 (a B+3 A b)}{7 x^7}-\frac{a^3 A}{8 x^8}-\frac{b^2 (3 a B+A b)}{5 x^5}-\frac{a b (a B+A b)}{2 x^6}-\frac{b^3 B}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^3*(A + B*x))/x^9,x]

[Out]

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

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

Mathematica [A]  time = 0.0195082, size = 69, normalized size = 0.92 \[ -\frac{20 a^2 b x (6 A+7 B x)+5 a^3 (7 A+8 B x)+28 a b^2 x^2 (5 A+6 B x)+14 b^3 x^3 (4 A+5 B x)}{280 x^8} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^3*(A + B*x))/x^9,x]

[Out]

-(14*b^3*x^3*(4*A + 5*B*x) + 28*a*b^2*x^2*(5*A + 6*B*x) + 20*a^2*b*x*(6*A + 7*B*x) + 5*a^3*(7*A + 8*B*x))/(280
*x^8)

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Maple [A]  time = 0.004, size = 66, normalized size = 0.9 \begin{align*} -{\frac{A{a}^{3}}{8\,{x}^{8}}}-{\frac{{a}^{2} \left ( 3\,Ab+Ba \right ) }{7\,{x}^{7}}}-{\frac{ab \left ( Ab+Ba \right ) }{2\,{x}^{6}}}-{\frac{{b}^{2} \left ( Ab+3\,Ba \right ) }{5\,{x}^{5}}}-{\frac{B{b}^{3}}{4\,{x}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^3*(B*x+A)/x^9,x)

[Out]

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

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Maxima [A]  time = 1.02713, size = 99, normalized size = 1.32 \begin{align*} -\frac{70 \, B b^{3} x^{4} + 35 \, A a^{3} + 56 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 140 \,{\left (B a^{2} b + A a b^{2}\right )} x^{2} + 40 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x}{280 \, x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)/x^9,x, algorithm="maxima")

[Out]

-1/280*(70*B*b^3*x^4 + 35*A*a^3 + 56*(3*B*a*b^2 + A*b^3)*x^3 + 140*(B*a^2*b + A*a*b^2)*x^2 + 40*(B*a^3 + 3*A*a
^2*b)*x)/x^8

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Fricas [A]  time = 1.90523, size = 167, normalized size = 2.23 \begin{align*} -\frac{70 \, B b^{3} x^{4} + 35 \, A a^{3} + 56 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 140 \,{\left (B a^{2} b + A a b^{2}\right )} x^{2} + 40 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x}{280 \, x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)/x^9,x, algorithm="fricas")

[Out]

-1/280*(70*B*b^3*x^4 + 35*A*a^3 + 56*(3*B*a*b^2 + A*b^3)*x^3 + 140*(B*a^2*b + A*a*b^2)*x^2 + 40*(B*a^3 + 3*A*a
^2*b)*x)/x^8

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Sympy [A]  time = 3.05672, size = 78, normalized size = 1.04 \begin{align*} - \frac{35 A a^{3} + 70 B b^{3} x^{4} + x^{3} \left (56 A b^{3} + 168 B a b^{2}\right ) + x^{2} \left (140 A a b^{2} + 140 B a^{2} b\right ) + x \left (120 A a^{2} b + 40 B a^{3}\right )}{280 x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**3*(B*x+A)/x**9,x)

[Out]

-(35*A*a**3 + 70*B*b**3*x**4 + x**3*(56*A*b**3 + 168*B*a*b**2) + x**2*(140*A*a*b**2 + 140*B*a**2*b) + x*(120*A
*a**2*b + 40*B*a**3))/(280*x**8)

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Giac [A]  time = 1.21269, size = 101, normalized size = 1.35 \begin{align*} -\frac{70 \, B b^{3} x^{4} + 168 \, B a b^{2} x^{3} + 56 \, A b^{3} x^{3} + 140 \, B a^{2} b x^{2} + 140 \, A a b^{2} x^{2} + 40 \, B a^{3} x + 120 \, A a^{2} b x + 35 \, A a^{3}}{280 \, x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)/x^9,x, algorithm="giac")

[Out]

-1/280*(70*B*b^3*x^4 + 168*B*a*b^2*x^3 + 56*A*b^3*x^3 + 140*B*a^2*b*x^2 + 140*A*a*b^2*x^2 + 40*B*a^3*x + 120*A
*a^2*b*x + 35*A*a^3)/x^8