3.1021 \(\int \frac{(a+b x) (A+B x)}{(d+e x)^6} \, dx\)

Optimal. Leaf size=77 \[ \frac{-a B e-A b e+2 b B d}{4 e^3 (d+e x)^4}-\frac{(b d-a e) (B d-A e)}{5 e^3 (d+e x)^5}-\frac{b B}{3 e^3 (d+e x)^3} \]

[Out]

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

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Rubi [A]  time = 0.0489671, antiderivative size = 77, 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} \[ \frac{-a B e-A b e+2 b B d}{4 e^3 (d+e x)^4}-\frac{(b d-a e) (B d-A e)}{5 e^3 (d+e x)^5}-\frac{b B}{3 e^3 (d+e x)^3} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(A + B*x))/(d + e*x)^6,x]

[Out]

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

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

Mathematica [A]  time = 0.0331288, size = 65, normalized size = 0.84 \[ -\frac{3 a e (4 A e+B (d+5 e x))+b \left (3 A e (d+5 e x)+2 B \left (d^2+5 d e x+10 e^2 x^2\right )\right )}{60 e^3 (d+e x)^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(3*a*e*(4*A*e + B*(d + 5*e*x)) + b*(3*A*e*(d + 5*e*x) + 2*B*(d^2 + 5*d*e*x + 10*e^2*x^2)))/(60*e^3*(d + e*x)^
5)

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Maple [A]  time = 0.005, size = 79, normalized size = 1. \begin{align*} -{\frac{Bb}{3\,{e}^{3} \left ( ex+d \right ) ^{3}}}-{\frac{aA{e}^{2}-Adbe-Bdae+bB{d}^{2}}{5\,{e}^{3} \left ( ex+d \right ) ^{5}}}-{\frac{Abe+Bae-2\,Bbd}{4\,{e}^{3} \left ( ex+d \right ) ^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.08327, size = 158, normalized size = 2.05 \begin{align*} -\frac{20 \, B b e^{2} x^{2} + 2 \, B b d^{2} + 12 \, A a e^{2} + 3 \,{\left (B a + A b\right )} d e + 5 \,{\left (2 \, B b d e + 3 \,{\left (B a + A b\right )} e^{2}\right )} x}{60 \,{\left (e^{8} x^{5} + 5 \, d e^{7} x^{4} + 10 \, d^{2} e^{6} x^{3} + 10 \, d^{3} e^{5} x^{2} + 5 \, d^{4} e^{4} x + d^{5} 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)^6,x, algorithm="maxima")

[Out]

-1/60*(20*B*b*e^2*x^2 + 2*B*b*d^2 + 12*A*a*e^2 + 3*(B*a + A*b)*d*e + 5*(2*B*b*d*e + 3*(B*a + A*b)*e^2)*x)/(e^8
*x^5 + 5*d*e^7*x^4 + 10*d^2*e^6*x^3 + 10*d^3*e^5*x^2 + 5*d^4*e^4*x + d^5*e^3)

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Fricas [A]  time = 1.855, size = 255, normalized size = 3.31 \begin{align*} -\frac{20 \, B b e^{2} x^{2} + 2 \, B b d^{2} + 12 \, A a e^{2} + 3 \,{\left (B a + A b\right )} d e + 5 \,{\left (2 \, B b d e + 3 \,{\left (B a + A b\right )} e^{2}\right )} x}{60 \,{\left (e^{8} x^{5} + 5 \, d e^{7} x^{4} + 10 \, d^{2} e^{6} x^{3} + 10 \, d^{3} e^{5} x^{2} + 5 \, d^{4} e^{4} x + d^{5} 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)^6,x, algorithm="fricas")

[Out]

-1/60*(20*B*b*e^2*x^2 + 2*B*b*d^2 + 12*A*a*e^2 + 3*(B*a + A*b)*d*e + 5*(2*B*b*d*e + 3*(B*a + A*b)*e^2)*x)/(e^8
*x^5 + 5*d*e^7*x^4 + 10*d^2*e^6*x^3 + 10*d^3*e^5*x^2 + 5*d^4*e^4*x + d^5*e^3)

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Sympy [A]  time = 5.12953, size = 134, normalized size = 1.74 \begin{align*} - \frac{12 A a e^{2} + 3 A b d e + 3 B a d e + 2 B b d^{2} + 20 B b e^{2} x^{2} + x \left (15 A b e^{2} + 15 B a e^{2} + 10 B b d e\right )}{60 d^{5} e^{3} + 300 d^{4} e^{4} x + 600 d^{3} e^{5} x^{2} + 600 d^{2} e^{6} x^{3} + 300 d e^{7} x^{4} + 60 e^{8} x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(12*A*a*e**2 + 3*A*b*d*e + 3*B*a*d*e + 2*B*b*d**2 + 20*B*b*e**2*x**2 + x*(15*A*b*e**2 + 15*B*a*e**2 + 10*B*b*
d*e))/(60*d**5*e**3 + 300*d**4*e**4*x + 600*d**3*e**5*x**2 + 600*d**2*e**6*x**3 + 300*d*e**7*x**4 + 60*e**8*x*
*5)

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Giac [A]  time = 2.05032, size = 96, normalized size = 1.25 \begin{align*} -\frac{{\left (20 \, B b x^{2} e^{2} + 10 \, B b d x e + 2 \, B b d^{2} + 15 \, B a x e^{2} + 15 \, A b x e^{2} + 3 \, B a d e + 3 \, A b d e + 12 \, A a e^{2}\right )} e^{\left (-3\right )}}{60 \,{\left (x e + d\right )}^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/60*(20*B*b*x^2*e^2 + 10*B*b*d*x*e + 2*B*b*d^2 + 15*B*a*x*e^2 + 15*A*b*x*e^2 + 3*B*a*d*e + 3*A*b*d*e + 12*A*
a*e^2)*e^(-3)/(x*e + d)^5