3.118 \(\int \frac{(a+b x)^3 (A+B x)}{x^{10}} \, dx\)

Optimal. Leaf size=75 \[ -\frac{a^2 (a B+3 A b)}{8 x^8}-\frac{a^3 A}{9 x^9}-\frac{b^2 (3 a B+A b)}{6 x^6}-\frac{3 a b (a B+A b)}{7 x^7}-\frac{b^3 B}{5 x^5} \]

[Out]

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

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Rubi [A]  time = 0.0329464, 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} \[ -\frac{a^2 (a B+3 A b)}{8 x^8}-\frac{a^3 A}{9 x^9}-\frac{b^2 (3 a B+A b)}{6 x^6}-\frac{3 a b (a B+A b)}{7 x^7}-\frac{b^3 B}{5 x^5} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^3*(A + B*x))/x^10,x]

[Out]

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

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

Mathematica [A]  time = 0.0198967, size = 69, normalized size = 0.92 \[ -\frac{135 a^2 b x (7 A+8 B x)+35 a^3 (8 A+9 B x)+180 a b^2 x^2 (6 A+7 B x)+84 b^3 x^3 (5 A+6 B x)}{2520 x^9} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^3*(A + B*x))/x^10,x]

[Out]

-(84*b^3*x^3*(5*A + 6*B*x) + 180*a*b^2*x^2*(6*A + 7*B*x) + 135*a^2*b*x*(7*A + 8*B*x) + 35*a^3*(8*A + 9*B*x))/(
2520*x^9)

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Maple [A]  time = 0.005, size = 66, normalized size = 0.9 \begin{align*} -{\frac{A{a}^{3}}{9\,{x}^{9}}}-{\frac{{a}^{2} \left ( 3\,Ab+Ba \right ) }{8\,{x}^{8}}}-{\frac{3\,ab \left ( Ab+Ba \right ) }{7\,{x}^{7}}}-{\frac{{b}^{2} \left ( Ab+3\,Ba \right ) }{6\,{x}^{6}}}-{\frac{B{b}^{3}}{5\,{x}^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^3*(B*x+A)/x^10,x)

[Out]

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

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Maxima [A]  time = 1.03978, size = 99, normalized size = 1.32 \begin{align*} -\frac{504 \, B b^{3} x^{4} + 280 \, A a^{3} + 420 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 1080 \,{\left (B a^{2} b + A a b^{2}\right )} x^{2} + 315 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x}{2520 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)/x^10,x, algorithm="maxima")

[Out]

-1/2520*(504*B*b^3*x^4 + 280*A*a^3 + 420*(3*B*a*b^2 + A*b^3)*x^3 + 1080*(B*a^2*b + A*a*b^2)*x^2 + 315*(B*a^3 +
3*A*a^2*b)*x)/x^9

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Fricas [A]  time = 2.11728, size = 176, normalized size = 2.35 \begin{align*} -\frac{504 \, B b^{3} x^{4} + 280 \, A a^{3} + 420 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 1080 \,{\left (B a^{2} b + A a b^{2}\right )} x^{2} + 315 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x}{2520 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)/x^10,x, algorithm="fricas")

[Out]

-1/2520*(504*B*b^3*x^4 + 280*A*a^3 + 420*(3*B*a*b^2 + A*b^3)*x^3 + 1080*(B*a^2*b + A*a*b^2)*x^2 + 315*(B*a^3 +
3*A*a^2*b)*x)/x^9

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Sympy [A]  time = 3.80808, size = 78, normalized size = 1.04 \begin{align*} - \frac{280 A a^{3} + 504 B b^{3} x^{4} + x^{3} \left (420 A b^{3} + 1260 B a b^{2}\right ) + x^{2} \left (1080 A a b^{2} + 1080 B a^{2} b\right ) + x \left (945 A a^{2} b + 315 B a^{3}\right )}{2520 x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**3*(B*x+A)/x**10,x)

[Out]

-(280*A*a**3 + 504*B*b**3*x**4 + x**3*(420*A*b**3 + 1260*B*a*b**2) + x**2*(1080*A*a*b**2 + 1080*B*a**2*b) + x*
(945*A*a**2*b + 315*B*a**3))/(2520*x**9)

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Giac [A]  time = 1.17353, size = 101, normalized size = 1.35 \begin{align*} -\frac{504 \, B b^{3} x^{4} + 1260 \, B a b^{2} x^{3} + 420 \, A b^{3} x^{3} + 1080 \, B a^{2} b x^{2} + 1080 \, A a b^{2} x^{2} + 315 \, B a^{3} x + 945 \, A a^{2} b x + 280 \, A a^{3}}{2520 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)/x^10,x, algorithm="giac")

[Out]

-1/2520*(504*B*b^3*x^4 + 1260*B*a*b^2*x^3 + 420*A*b^3*x^3 + 1080*B*a^2*b*x^2 + 1080*A*a*b^2*x^2 + 315*B*a^3*x
+ 945*A*a^2*b*x + 280*A*a^3)/x^9