3.100 \(\int \frac{(a+b x)^2 (A+B x)}{x^5} \, dx\)

Optimal. Leaf size=44 \[ \frac{(a+b x)^3 (A b-4 a B)}{12 a^2 x^3}-\frac{A (a+b x)^3}{4 a x^4} \]

[Out]

-(A*(a + b*x)^3)/(4*a*x^4) + ((A*b - 4*a*B)*(a + b*x)^3)/(12*a^2*x^3)

________________________________________________________________________________________

Rubi [A]  time = 0.0107101, 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} \[ \frac{(a+b x)^3 (A b-4 a B)}{12 a^2 x^3}-\frac{A (a+b x)^3}{4 a x^4} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^2*(A + B*x))/x^5,x]

[Out]

-(A*(a + b*x)^3)/(4*a*x^4) + ((A*b - 4*a*B)*(a + b*x)^3)/(12*a^2*x^3)

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)^2 (A+B x)}{x^5} \, dx &=-\frac{A (a+b x)^3}{4 a x^4}+\frac{(-A b+4 a B) \int \frac{(a+b x)^2}{x^4} \, dx}{4 a}\\ &=-\frac{A (a+b x)^3}{4 a x^4}+\frac{(A b-4 a B) (a+b x)^3}{12 a^2 x^3}\\ \end{align*}

Mathematica [A]  time = 0.0148921, size = 47, normalized size = 1.07 \[ -\frac{a^2 (3 A+4 B x)+4 a b x (2 A+3 B x)+6 b^2 x^2 (A+2 B x)}{12 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^2*(A + B*x))/x^5,x]

[Out]

-(6*b^2*x^2*(A + 2*B*x) + 4*a*b*x*(2*A + 3*B*x) + a^2*(3*A + 4*B*x))/(12*x^4)

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 48, normalized size = 1.1 \begin{align*} -{\frac{a \left ( 2\,Ab+Ba \right ) }{3\,{x}^{3}}}-{\frac{A{a}^{2}}{4\,{x}^{4}}}-{\frac{b \left ( Ab+2\,Ba \right ) }{2\,{x}^{2}}}-{\frac{B{b}^{2}}{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^2*(B*x+A)/x^5,x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.00285, size = 69, normalized size = 1.57 \begin{align*} -\frac{12 \, B b^{2} x^{3} + 3 \, A a^{2} + 6 \,{\left (2 \, B a b + A b^{2}\right )} x^{2} + 4 \,{\left (B a^{2} + 2 \, A a b\right )} x}{12 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*(B*x+A)/x^5,x, algorithm="maxima")

[Out]

-1/12*(12*B*b^2*x^3 + 3*A*a^2 + 6*(2*B*a*b + A*b^2)*x^2 + 4*(B*a^2 + 2*A*a*b)*x)/x^4

________________________________________________________________________________________

Fricas [A]  time = 1.83281, size = 116, normalized size = 2.64 \begin{align*} -\frac{12 \, B b^{2} x^{3} + 3 \, A a^{2} + 6 \,{\left (2 \, B a b + A b^{2}\right )} x^{2} + 4 \,{\left (B a^{2} + 2 \, A a b\right )} x}{12 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*(B*x+A)/x^5,x, algorithm="fricas")

[Out]

-1/12*(12*B*b^2*x^3 + 3*A*a^2 + 6*(2*B*a*b + A*b^2)*x^2 + 4*(B*a^2 + 2*A*a*b)*x)/x^4

________________________________________________________________________________________

Sympy [A]  time = 0.896727, size = 54, normalized size = 1.23 \begin{align*} - \frac{3 A a^{2} + 12 B b^{2} x^{3} + x^{2} \left (6 A b^{2} + 12 B a b\right ) + x \left (8 A a b + 4 B a^{2}\right )}{12 x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**2*(B*x+A)/x**5,x)

[Out]

-(3*A*a**2 + 12*B*b**2*x**3 + x**2*(6*A*b**2 + 12*B*a*b) + x*(8*A*a*b + 4*B*a**2))/(12*x**4)

________________________________________________________________________________________

Giac [A]  time = 1.18372, size = 69, normalized size = 1.57 \begin{align*} -\frac{12 \, B b^{2} x^{3} + 12 \, B a b x^{2} + 6 \, A b^{2} x^{2} + 4 \, B a^{2} x + 8 \, A a b x + 3 \, A a^{2}}{12 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*(B*x+A)/x^5,x, algorithm="giac")

[Out]

-1/12*(12*B*b^2*x^3 + 12*B*a*b*x^2 + 6*A*b^2*x^2 + 4*B*a^2*x + 8*A*a*b*x + 3*A*a^2)/x^4