3.1140 \(\int \frac{A+B x}{(a+b x)^3 (d+e x)^2} \, dx\)

Optimal. Leaf size=158 \[ -\frac{A b-a B}{2 (a+b x)^2 (b d-a e)^2}-\frac{a B e-2 A b e+b B d}{(a+b x) (b d-a e)^3}-\frac{e (B d-A e)}{(d+e x) (b d-a e)^3}-\frac{e \log (a+b x) (a B e-3 A b e+2 b B d)}{(b d-a e)^4}+\frac{e \log (d+e x) (a B e-3 A b e+2 b B d)}{(b d-a e)^4} \]

[Out]

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

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Rubi [A]  time = 0.158836, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {77} \[ -\frac{A b-a B}{2 (a+b x)^2 (b d-a e)^2}-\frac{a B e-2 A b e+b B d}{(a+b x) (b d-a e)^3}-\frac{e (B d-A e)}{(d+e x) (b d-a e)^3}-\frac{e \log (a+b x) (a B e-3 A b e+2 b B d)}{(b d-a e)^4}+\frac{e \log (d+e x) (a B e-3 A b e+2 b B d)}{(b d-a e)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.100739, size = 146, normalized size = 0.92 \[ \frac{\frac{(a B-A b) (b d-a e)^2}{(a+b x)^2}-\frac{2 (b d-a e) (a B e-2 A b e+b B d)}{a+b x}+\frac{2 e (b d-a e) (A e-B d)}{d+e x}-2 e \log (a+b x) (a B e-3 A b e+2 b B d)+2 e \log (d+e x) (a B e-3 A b e+2 b B d)}{2 (b d-a e)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.013, size = 287, normalized size = 1.8 \begin{align*} -{\frac{{e}^{2}A}{ \left ( ae-bd \right ) ^{3} \left ( ex+d \right ) }}+{\frac{eBd}{ \left ( ae-bd \right ) ^{3} \left ( ex+d \right ) }}-3\,{\frac{{e}^{2}\ln \left ( ex+d \right ) Ab}{ \left ( ae-bd \right ) ^{4}}}+{\frac{{e}^{2}\ln \left ( ex+d \right ) Ba}{ \left ( ae-bd \right ) ^{4}}}+2\,{\frac{e\ln \left ( ex+d \right ) Bbd}{ \left ( ae-bd \right ) ^{4}}}-2\,{\frac{Abe}{ \left ( ae-bd \right ) ^{3} \left ( bx+a \right ) }}+{\frac{Bae}{ \left ( ae-bd \right ) ^{3} \left ( bx+a \right ) }}+{\frac{Bbd}{ \left ( ae-bd \right ) ^{3} \left ( bx+a \right ) }}-{\frac{Ab}{2\, \left ( ae-bd \right ) ^{2} \left ( bx+a \right ) ^{2}}}+{\frac{Ba}{2\, \left ( ae-bd \right ) ^{2} \left ( bx+a \right ) ^{2}}}+3\,{\frac{{e}^{2}\ln \left ( bx+a \right ) Ab}{ \left ( ae-bd \right ) ^{4}}}-{\frac{{e}^{2}\ln \left ( bx+a \right ) Ba}{ \left ( ae-bd \right ) ^{4}}}-2\,{\frac{e\ln \left ( bx+a \right ) Bbd}{ \left ( ae-bd \right ) ^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.33684, size = 644, normalized size = 4.08 \begin{align*} -\frac{{\left (2 \, B b d e +{\left (B a - 3 \, A b\right )} e^{2}\right )} \log \left (b x + a\right )}{b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}} + \frac{{\left (2 \, B b d e +{\left (B a - 3 \, A b\right )} e^{2}\right )} \log \left (e x + d\right )}{b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}} + \frac{2 \, A a^{2} e^{2} -{\left (B a b + A b^{2}\right )} d^{2} - 5 \,{\left (B a^{2} - A a b\right )} d e - 2 \,{\left (2 \, B b^{2} d e +{\left (B a b - 3 \, A b^{2}\right )} e^{2}\right )} x^{2} -{\left (2 \, B b^{2} d^{2} +{\left (7 \, B a b - 3 \, A b^{2}\right )} d e + 3 \,{\left (B a^{2} - 3 \, A a b\right )} e^{2}\right )} x}{2 \,{\left (a^{2} b^{3} d^{4} - 3 \, a^{3} b^{2} d^{3} e + 3 \, a^{4} b d^{2} e^{2} - a^{5} d e^{3} +{\left (b^{5} d^{3} e - 3 \, a b^{4} d^{2} e^{2} + 3 \, a^{2} b^{3} d e^{3} - a^{3} b^{2} e^{4}\right )} x^{3} +{\left (b^{5} d^{4} - a b^{4} d^{3} e - 3 \, a^{2} b^{3} d^{2} e^{2} + 5 \, a^{3} b^{2} d e^{3} - 2 \, a^{4} b e^{4}\right )} x^{2} +{\left (2 \, a b^{4} d^{4} - 5 \, a^{2} b^{3} d^{3} e + 3 \, a^{3} b^{2} d^{2} e^{2} + a^{4} b d e^{3} - a^{5} e^{4}\right )} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(b*x+a)^3/(e*x+d)^2,x, algorithm="maxima")

[Out]

-(2*B*b*d*e + (B*a - 3*A*b)*e^2)*log(b*x + a)/(b^4*d^4 - 4*a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 - 4*a^3*b*d*e^3 + a
^4*e^4) + (2*B*b*d*e + (B*a - 3*A*b)*e^2)*log(e*x + d)/(b^4*d^4 - 4*a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 - 4*a^3*b*
d*e^3 + a^4*e^4) + 1/2*(2*A*a^2*e^2 - (B*a*b + A*b^2)*d^2 - 5*(B*a^2 - A*a*b)*d*e - 2*(2*B*b^2*d*e + (B*a*b -
3*A*b^2)*e^2)*x^2 - (2*B*b^2*d^2 + (7*B*a*b - 3*A*b^2)*d*e + 3*(B*a^2 - 3*A*a*b)*e^2)*x)/(a^2*b^3*d^4 - 3*a^3*
b^2*d^3*e + 3*a^4*b*d^2*e^2 - a^5*d*e^3 + (b^5*d^3*e - 3*a*b^4*d^2*e^2 + 3*a^2*b^3*d*e^3 - a^3*b^2*e^4)*x^3 +
(b^5*d^4 - a*b^4*d^3*e - 3*a^2*b^3*d^2*e^2 + 5*a^3*b^2*d*e^3 - 2*a^4*b*e^4)*x^2 + (2*a*b^4*d^4 - 5*a^2*b^3*d^3
*e + 3*a^3*b^2*d^2*e^2 + a^4*b*d*e^3 - a^5*e^4)*x)

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Fricas [B]  time = 1.79527, size = 1652, normalized size = 10.46 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(b*x+a)^3/(e*x+d)^2,x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 4.09984, size = 1066, normalized size = 6.75 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(b*x+a)**3/(e*x+d)**2,x)

[Out]

e*(-3*A*b*e + B*a*e + 2*B*b*d)*log(x + (-3*A*a*b*e**3 - 3*A*b**2*d*e**2 + B*a**2*e**3 + 3*B*a*b*d*e**2 + 2*B*b
**2*d**2*e - a**5*e**6*(-3*A*b*e + B*a*e + 2*B*b*d)/(a*e - b*d)**4 + 5*a**4*b*d*e**5*(-3*A*b*e + B*a*e + 2*B*b
*d)/(a*e - b*d)**4 - 10*a**3*b**2*d**2*e**4*(-3*A*b*e + B*a*e + 2*B*b*d)/(a*e - b*d)**4 + 10*a**2*b**3*d**3*e*
*3*(-3*A*b*e + B*a*e + 2*B*b*d)/(a*e - b*d)**4 - 5*a*b**4*d**4*e**2*(-3*A*b*e + B*a*e + 2*B*b*d)/(a*e - b*d)**
4 + b**5*d**5*e*(-3*A*b*e + B*a*e + 2*B*b*d)/(a*e - b*d)**4)/(-6*A*b**2*e**3 + 2*B*a*b*e**3 + 4*B*b**2*d*e**2)
)/(a*e - b*d)**4 - e*(-3*A*b*e + B*a*e + 2*B*b*d)*log(x + (-3*A*a*b*e**3 - 3*A*b**2*d*e**2 + B*a**2*e**3 + 3*B
*a*b*d*e**2 + 2*B*b**2*d**2*e + a**5*e**6*(-3*A*b*e + B*a*e + 2*B*b*d)/(a*e - b*d)**4 - 5*a**4*b*d*e**5*(-3*A*
b*e + B*a*e + 2*B*b*d)/(a*e - b*d)**4 + 10*a**3*b**2*d**2*e**4*(-3*A*b*e + B*a*e + 2*B*b*d)/(a*e - b*d)**4 - 1
0*a**2*b**3*d**3*e**3*(-3*A*b*e + B*a*e + 2*B*b*d)/(a*e - b*d)**4 + 5*a*b**4*d**4*e**2*(-3*A*b*e + B*a*e + 2*B
*b*d)/(a*e - b*d)**4 - b**5*d**5*e*(-3*A*b*e + B*a*e + 2*B*b*d)/(a*e - b*d)**4)/(-6*A*b**2*e**3 + 2*B*a*b*e**3
+ 4*B*b**2*d*e**2))/(a*e - b*d)**4 + (-2*A*a**2*e**2 - 5*A*a*b*d*e + A*b**2*d**2 + 5*B*a**2*d*e + B*a*b*d**2
+ x**2*(-6*A*b**2*e**2 + 2*B*a*b*e**2 + 4*B*b**2*d*e) + x*(-9*A*a*b*e**2 - 3*A*b**2*d*e + 3*B*a**2*e**2 + 7*B*
a*b*d*e + 2*B*b**2*d**2))/(2*a**5*d*e**3 - 6*a**4*b*d**2*e**2 + 6*a**3*b**2*d**3*e - 2*a**2*b**3*d**4 + x**3*(
2*a**3*b**2*e**4 - 6*a**2*b**3*d*e**3 + 6*a*b**4*d**2*e**2 - 2*b**5*d**3*e) + x**2*(4*a**4*b*e**4 - 10*a**3*b*
*2*d*e**3 + 6*a**2*b**3*d**2*e**2 + 2*a*b**4*d**3*e - 2*b**5*d**4) + x*(2*a**5*e**4 - 2*a**4*b*d*e**3 - 6*a**3
*b**2*d**2*e**2 + 10*a**2*b**3*d**3*e - 4*a*b**4*d**4))

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Giac [A]  time = 2.44232, size = 402, normalized size = 2.54 \begin{align*} -\frac{{\left (2 \, B b d e^{2} + B a e^{3} - 3 \, A b e^{3}\right )} \log \left ({\left | -b + \frac{b d}{x e + d} - \frac{a e}{x e + d} \right |}\right )}{b^{4} d^{4} e - 4 \, a b^{3} d^{3} e^{2} + 6 \, a^{2} b^{2} d^{2} e^{3} - 4 \, a^{3} b d e^{4} + a^{4} e^{5}} - \frac{\frac{B d e^{4}}{x e + d} - \frac{A e^{5}}{x e + d}}{b^{3} d^{3} e^{3} - 3 \, a b^{2} d^{2} e^{4} + 3 \, a^{2} b d e^{5} - a^{3} e^{6}} - \frac{2 \, B b^{3} d e + 3 \, B a b^{2} e^{2} - 5 \, A b^{3} e^{2} - \frac{2 \,{\left (B b^{3} d^{2} e^{2} + B a b^{2} d e^{3} - 3 \, A b^{3} d e^{3} - 2 \, B a^{2} b e^{4} + 3 \, A a b^{2} e^{4}\right )} e^{\left (-1\right )}}{x e + d}}{2 \,{\left (b d - a e\right )}^{4}{\left (b - \frac{b d}{x e + d} + \frac{a e}{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)^3/(e*x+d)^2,x, algorithm="giac")

[Out]

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