3.1019 \(\int \frac{(a+b x) (A+B x)}{(d+e x)^4} \, dx\)

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

[Out]

-((b*d - a*e)*(B*d - A*e))/(3*e^3*(d + e*x)^3) + (2*b*B*d - A*b*e - a*B*e)/(2*e^3*(d + e*x)^2) - (b*B)/(e^3*(d
+ e*x))

________________________________________________________________________________________

Rubi [A]  time = 0.0490479, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {77} \[ \frac{-a B e-A b e+2 b B d}{2 e^3 (d+e x)^2}-\frac{(b d-a e) (B d-A e)}{3 e^3 (d+e x)^3}-\frac{b B}{e^3 (d+e x)} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(A + B*x))/(d + e*x)^4,x]

[Out]

-((b*d - a*e)*(B*d - A*e))/(3*e^3*(d + e*x)^3) + (2*b*B*d - A*b*e - a*B*e)/(2*e^3*(d + e*x)^2) - (b*B)/(e^3*(d
+ e*x))

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin{align*} \int \frac{(a+b x) (A+B x)}{(d+e x)^4} \, dx &=\int \left (\frac{(-b d+a e) (-B d+A e)}{e^2 (d+e x)^4}+\frac{-2 b B d+A b e+a B e}{e^2 (d+e x)^3}+\frac{b B}{e^2 (d+e x)^2}\right ) \, dx\\ &=-\frac{(b d-a e) (B d-A e)}{3 e^3 (d+e x)^3}+\frac{2 b B d-A b e-a B e}{2 e^3 (d+e x)^2}-\frac{b B}{e^3 (d+e x)}\\ \end{align*}

Mathematica [A]  time = 0.0312522, size = 63, normalized size = 0.84 \[ -\frac{a e (2 A e+B (d+3 e x))+b \left (A e (d+3 e x)+2 B \left (d^2+3 d e x+3 e^2 x^2\right )\right )}{6 e^3 (d+e x)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(A + B*x))/(d + e*x)^4,x]

[Out]

-(a*e*(2*A*e + B*(d + 3*e*x)) + b*(A*e*(d + 3*e*x) + 2*B*(d^2 + 3*d*e*x + 3*e^2*x^2)))/(6*e^3*(d + e*x)^3)

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 79, normalized size = 1.1 \begin{align*} -{\frac{aA{e}^{2}-Adbe-Bdae+bB{d}^{2}}{3\,{e}^{3} \left ( ex+d \right ) ^{3}}}-{\frac{Abe+Bae-2\,Bbd}{2\,{e}^{3} \left ( ex+d \right ) ^{2}}}-{\frac{Bb}{{e}^{3} \left ( ex+d \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(B*x+A)/(e*x+d)^4,x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.20904, size = 126, normalized size = 1.68 \begin{align*} -\frac{6 \, B b e^{2} x^{2} + 2 \, B b d^{2} + 2 \, A a e^{2} +{\left (B a + A b\right )} d e + 3 \,{\left (2 \, B b d e +{\left (B a + A b\right )} e^{2}\right )} x}{6 \,{\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(B*x+A)/(e*x+d)^4,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.81589, size = 200, normalized size = 2.67 \begin{align*} -\frac{6 \, B b e^{2} x^{2} + 2 \, B b d^{2} + 2 \, A a e^{2} +{\left (B a + A b\right )} d e + 3 \,{\left (2 \, B b d e +{\left (B a + A b\right )} e^{2}\right )} x}{6 \,{\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(B*x+A)/(e*x+d)^4,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 2.15033, size = 107, normalized size = 1.43 \begin{align*} - \frac{2 A a e^{2} + A b d e + B a d e + 2 B b d^{2} + 6 B b e^{2} x^{2} + x \left (3 A b e^{2} + 3 B a e^{2} + 6 B b d e\right )}{6 d^{3} e^{3} + 18 d^{2} e^{4} x + 18 d e^{5} x^{2} + 6 e^{6} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(B*x+A)/(e*x+d)**4,x)

[Out]

-(2*A*a*e**2 + A*b*d*e + B*a*d*e + 2*B*b*d**2 + 6*B*b*e**2*x**2 + x*(3*A*b*e**2 + 3*B*a*e**2 + 6*B*b*d*e))/(6*
d**3*e**3 + 18*d**2*e**4*x + 18*d*e**5*x**2 + 6*e**6*x**3)

________________________________________________________________________________________

Giac [A]  time = 1.9704, size = 93, normalized size = 1.24 \begin{align*} -\frac{{\left (6 \, B b x^{2} e^{2} + 6 \, B b d x e + 2 \, B b d^{2} + 3 \, B a x e^{2} + 3 \, A b x e^{2} + B a d e + A b d e + 2 \, A a e^{2}\right )} e^{\left (-3\right )}}{6 \,{\left (x e + d\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(B*x+A)/(e*x+d)^4,x, algorithm="giac")

[Out]

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