3.114 \(\int \frac{(a+b x)^3 (A+B x)}{x^6} \, dx\)

Optimal. Leaf size=44 \[ \frac{(a+b x)^4 (A b-5 a B)}{20 a^2 x^4}-\frac{A (a+b x)^4}{5 a x^5} \]

[Out]

-(A*(a + b*x)^4)/(5*a*x^5) + ((A*b - 5*a*B)*(a + b*x)^4)/(20*a^2*x^4)

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Rubi [A]  time = 0.0106216, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {78, 37} \[ \frac{(a+b x)^4 (A b-5 a B)}{20 a^2 x^4}-\frac{A (a+b x)^4}{5 a x^5} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^3*(A + B*x))/x^6,x]

[Out]

-(A*(a + b*x)^4)/(5*a*x^5) + ((A*b - 5*a*B)*(a + b*x)^4)/(20*a^2*x^4)

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \frac{(a+b x)^3 (A+B x)}{x^6} \, dx &=-\frac{A (a+b x)^4}{5 a x^5}+\frac{(-A b+5 a B) \int \frac{(a+b x)^3}{x^5} \, dx}{5 a}\\ &=-\frac{A (a+b x)^4}{5 a x^5}+\frac{(A b-5 a B) (a+b x)^4}{20 a^2 x^4}\\ \end{align*}

Mathematica [A]  time = 0.019693, size = 66, normalized size = 1.5 \[ -\frac{5 a^2 b x (3 A+4 B x)+a^3 (4 A+5 B x)+10 a b^2 x^2 (2 A+3 B x)+10 b^3 x^3 (A+2 B x)}{20 x^5} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^3*(A + B*x))/x^6,x]

[Out]

-(10*b^3*x^3*(A + 2*B*x) + 10*a*b^2*x^2*(2*A + 3*B*x) + 5*a^2*b*x*(3*A + 4*B*x) + a^3*(4*A + 5*B*x))/(20*x^5)

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Maple [A]  time = 0.004, size = 66, normalized size = 1.5 \begin{align*} -{\frac{ab \left ( Ab+Ba \right ) }{{x}^{3}}}-{\frac{A{a}^{3}}{5\,{x}^{5}}}-{\frac{{a}^{2} \left ( 3\,Ab+Ba \right ) }{4\,{x}^{4}}}-{\frac{{b}^{2} \left ( Ab+3\,Ba \right ) }{2\,{x}^{2}}}-{\frac{B{b}^{3}}{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^3*(B*x+A)/x^6,x)

[Out]

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

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Maxima [A]  time = 1.02769, size = 99, normalized size = 2.25 \begin{align*} -\frac{20 \, B b^{3} x^{4} + 4 \, A a^{3} + 10 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 20 \,{\left (B a^{2} b + A a b^{2}\right )} x^{2} + 5 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x}{20 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)/x^6,x, algorithm="maxima")

[Out]

-1/20*(20*B*b^3*x^4 + 4*A*a^3 + 10*(3*B*a*b^2 + A*b^3)*x^3 + 20*(B*a^2*b + A*a*b^2)*x^2 + 5*(B*a^3 + 3*A*a^2*b
)*x)/x^5

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Fricas [A]  time = 1.58714, size = 162, normalized size = 3.68 \begin{align*} -\frac{20 \, B b^{3} x^{4} + 4 \, A a^{3} + 10 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 20 \,{\left (B a^{2} b + A a b^{2}\right )} x^{2} + 5 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x}{20 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)/x^6,x, algorithm="fricas")

[Out]

-1/20*(20*B*b^3*x^4 + 4*A*a^3 + 10*(3*B*a*b^2 + A*b^3)*x^3 + 20*(B*a^2*b + A*a*b^2)*x^2 + 5*(B*a^3 + 3*A*a^2*b
)*x)/x^5

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Sympy [B]  time = 1.62473, size = 78, normalized size = 1.77 \begin{align*} - \frac{4 A a^{3} + 20 B b^{3} x^{4} + x^{3} \left (10 A b^{3} + 30 B a b^{2}\right ) + x^{2} \left (20 A a b^{2} + 20 B a^{2} b\right ) + x \left (15 A a^{2} b + 5 B a^{3}\right )}{20 x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**3*(B*x+A)/x**6,x)

[Out]

-(4*A*a**3 + 20*B*b**3*x**4 + x**3*(10*A*b**3 + 30*B*a*b**2) + x**2*(20*A*a*b**2 + 20*B*a**2*b) + x*(15*A*a**2
*b + 5*B*a**3))/(20*x**5)

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Giac [A]  time = 1.17623, size = 101, normalized size = 2.3 \begin{align*} -\frac{20 \, B b^{3} x^{4} + 30 \, B a b^{2} x^{3} + 10 \, A b^{3} x^{3} + 20 \, B a^{2} b x^{2} + 20 \, A a b^{2} x^{2} + 5 \, B a^{3} x + 15 \, A a^{2} b x + 4 \, A a^{3}}{20 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)/x^6,x, algorithm="giac")

[Out]

-1/20*(20*B*b^3*x^4 + 30*B*a*b^2*x^3 + 10*A*b^3*x^3 + 20*B*a^2*b*x^2 + 20*A*a*b^2*x^2 + 5*B*a^3*x + 15*A*a^2*b
*x + 4*A*a^3)/x^5