3.1018 \(\int \frac{(a+b x) (A+B x)}{(d+e x)^3} \, dx\)

Optimal. Leaf size=70 \[ -\frac{(b d-a e) (B d-A e)}{2 e^3 (d+e x)^2}+\frac{-a B e-A b e+2 b B d}{e^3 (d+e x)}+\frac{b B \log (d+e x)}{e^3} \]

[Out]

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

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Rubi [A]  time = 0.0499388, antiderivative size = 70, 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} \[ -\frac{(b d-a e) (B d-A e)}{2 e^3 (d+e x)^2}+\frac{-a B e-A b e+2 b B d}{e^3 (d+e x)}+\frac{b B \log (d+e x)}{e^3} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(A + B*x))/(d + e*x)^3,x]

[Out]

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

Mathematica [A]  time = 0.0362283, size = 72, normalized size = 1.03 \[ \frac{-a e (A e+B (d+2 e x))+b (B d (3 d+4 e x)-A e (d+2 e x))+2 b B (d+e x)^2 \log (d+e x)}{2 e^3 (d+e x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.005, size = 118, normalized size = 1.7 \begin{align*} -{\frac{Aa}{2\,e \left ( ex+d \right ) ^{2}}}+{\frac{Adb}{2\,{e}^{2} \left ( ex+d \right ) ^{2}}}+{\frac{Bda}{2\,{e}^{2} \left ( ex+d \right ) ^{2}}}-{\frac{bB{d}^{2}}{2\,{e}^{3} \left ( ex+d \right ) ^{2}}}-{\frac{Ab}{{e}^{2} \left ( ex+d \right ) }}-{\frac{Ba}{{e}^{2} \left ( ex+d \right ) }}+2\,{\frac{Bbd}{{e}^{3} \left ( ex+d \right ) }}+{\frac{Bb\ln \left ( ex+d \right ) }{{e}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.01484, size = 117, normalized size = 1.67 \begin{align*} \frac{3 \, B b d^{2} - A a e^{2} -{\left (B a + A b\right )} d e + 2 \,{\left (2 \, B b d e -{\left (B a + A b\right )} e^{2}\right )} x}{2 \,{\left (e^{5} x^{2} + 2 \, d e^{4} x + d^{2} e^{3}\right )}} + \frac{B b \log \left (e x + d\right )}{e^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.85916, size = 227, normalized size = 3.24 \begin{align*} \frac{3 \, B b d^{2} - A a e^{2} -{\left (B a + A b\right )} d e + 2 \,{\left (2 \, B b d e -{\left (B a + A b\right )} e^{2}\right )} x + 2 \,{\left (B b e^{2} x^{2} + 2 \, B b d e x + B b d^{2}\right )} \log \left (e x + d\right )}{2 \,{\left (e^{5} x^{2} + 2 \, d e^{4} x + d^{2} 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)^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.36353, size = 94, normalized size = 1.34 \begin{align*} \frac{B b \log{\left (d + e x \right )}}{e^{3}} - \frac{A a e^{2} + A b d e + B a d e - 3 B b d^{2} + x \left (2 A b e^{2} + 2 B a e^{2} - 4 B b d e\right )}{2 d^{2} e^{3} + 4 d e^{4} x + 2 e^{5} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 2.26373, size = 107, normalized size = 1.53 \begin{align*} B b e^{\left (-3\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{{\left (2 \,{\left (2 \, B b d - B a e - A b e\right )} x +{\left (3 \, B b d^{2} - B a d e - A b d e - A a e^{2}\right )} e^{\left (-1\right )}\right )} e^{\left (-2\right )}}{2 \,{\left (x e + d\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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