3.116 \(\int \frac{(a+b x)^3 (A+B x)}{x^8} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0340929, 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} \[ -\frac{a^2 (a B+3 A b)}{6 x^6}-\frac{a^3 A}{7 x^7}-\frac{b^2 (3 a B+A b)}{4 x^4}-\frac{3 a b (a B+A b)}{5 x^5}-\frac{b^3 B}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^3*(A + B*x))/x^8,x]

[Out]

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

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

Mathematica [A]  time = 0.0194462, size = 69, normalized size = 0.92 \[ -\frac{42 a^2 b x (5 A+6 B x)+10 a^3 (6 A+7 B x)+63 a b^2 x^2 (4 A+5 B x)+35 b^3 x^3 (3 A+4 B x)}{420 x^7} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^3*(A + B*x))/x^8,x]

[Out]

-(35*b^3*x^3*(3*A + 4*B*x) + 63*a*b^2*x^2*(4*A + 5*B*x) + 42*a^2*b*x*(5*A + 6*B*x) + 10*a^3*(6*A + 7*B*x))/(42
0*x^7)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^3*(B*x+A)/x^8,x)

[Out]

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

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Maxima [A]  time = 0.992542, size = 99, normalized size = 1.32 \begin{align*} -\frac{140 \, B b^{3} x^{4} + 60 \, A a^{3} + 105 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 252 \,{\left (B a^{2} b + A a b^{2}\right )} x^{2} + 70 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x}{420 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)/x^8,x, algorithm="maxima")

[Out]

-1/420*(140*B*b^3*x^4 + 60*A*a^3 + 105*(3*B*a*b^2 + A*b^3)*x^3 + 252*(B*a^2*b + A*a*b^2)*x^2 + 70*(B*a^3 + 3*A
*a^2*b)*x)/x^7

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Fricas [A]  time = 1.87264, size = 170, normalized size = 2.27 \begin{align*} -\frac{140 \, B b^{3} x^{4} + 60 \, A a^{3} + 105 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 252 \,{\left (B a^{2} b + A a b^{2}\right )} x^{2} + 70 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x}{420 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)/x^8,x, algorithm="fricas")

[Out]

-1/420*(140*B*b^3*x^4 + 60*A*a^3 + 105*(3*B*a*b^2 + A*b^3)*x^3 + 252*(B*a^2*b + A*a*b^2)*x^2 + 70*(B*a^3 + 3*A
*a^2*b)*x)/x^7

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Sympy [A]  time = 2.51975, size = 78, normalized size = 1.04 \begin{align*} - \frac{60 A a^{3} + 140 B b^{3} x^{4} + x^{3} \left (105 A b^{3} + 315 B a b^{2}\right ) + x^{2} \left (252 A a b^{2} + 252 B a^{2} b\right ) + x \left (210 A a^{2} b + 70 B a^{3}\right )}{420 x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**3*(B*x+A)/x**8,x)

[Out]

-(60*A*a**3 + 140*B*b**3*x**4 + x**3*(105*A*b**3 + 315*B*a*b**2) + x**2*(252*A*a*b**2 + 252*B*a**2*b) + x*(210
*A*a**2*b + 70*B*a**3))/(420*x**7)

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Giac [A]  time = 1.21151, size = 101, normalized size = 1.35 \begin{align*} -\frac{140 \, B b^{3} x^{4} + 315 \, B a b^{2} x^{3} + 105 \, A b^{3} x^{3} + 252 \, B a^{2} b x^{2} + 252 \, A a b^{2} x^{2} + 70 \, B a^{3} x + 210 \, A a^{2} b x + 60 \, A a^{3}}{420 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)/x^8,x, algorithm="giac")

[Out]

-1/420*(140*B*b^3*x^4 + 315*B*a*b^2*x^3 + 105*A*b^3*x^3 + 252*B*a^2*b*x^2 + 252*A*a*b^2*x^2 + 70*B*a^3*x + 210
*A*a^2*b*x + 60*A*a^3)/x^7