3.1123 \(\int \frac{A+B x}{(a+b x)^3 (d+e x)^2} \, dx\)

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 = 0.343769, 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 \[ -\frac{A b-a B}{2 (a+b x)^2 (b d-a e)^2}-\frac{a B e-2 A b e+b B d}{(a+b x) (b d-a e)^3}-\frac{e (B d-A e)}{(d+e x) (b d-a e)^3}-\frac{e \log (a+b x) (a B e-3 A b e+2 b B d)}{(b d-a e)^4}+\frac{e \log (d+e x) (a B e-3 A b e+2 b B d)}{(b d-a e)^4} \]

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

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Rubi in Sympy [A]  time = 58.8292, size = 146, normalized size = 0.92 \[ \frac{e \left (3 A b e - B a e - 2 B b d\right ) \log{\left (a + b x \right )}}{\left (a e - b d\right )^{4}} - \frac{e \left (3 A b e - B a e - 2 B b d\right ) \log{\left (d + e x \right )}}{\left (a e - b d\right )^{4}} - \frac{e \left (A e - B d\right )}{\left (d + e x\right ) \left (a e - b d\right )^{3}} - \frac{2 A b e - B a e - B b d}{\left (a + b x\right ) \left (a e - b d\right )^{3}} - \frac{A b - B a}{2 \left (a + b x\right )^{2} \left (a e - b d\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_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(a + b*x)/(a*e - b*d)**4 - e*(3*A*b*e - B*a*e -
2*B*b*d)*log(d + e*x)/(a*e - b*d)**4 - e*(A*e - B*d)/((d + e*x)*(a*e - b*d)**3)
- (2*A*b*e - B*a*e - B*b*d)/((a + b*x)*(a*e - b*d)**3) - (A*b - B*a)/(2*(a + b*
x)**2*(a*e - b*d)**2)

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Mathematica [A]  time = 0.167255, 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 = 0.022, size = 287, normalized size = 1.8 \[ -{\frac{{e}^{2}A}{ \left ( ae-bd \right ) ^{3} \left ( ex+d \right ) }}+{\frac{eBd}{ \left ( ae-bd \right ) ^{3} \left ( ex+d \right ) }}-3\,{\frac{{e}^{2}\ln \left ( ex+d \right ) Ab}{ \left ( ae-bd \right ) ^{4}}}+{\frac{{e}^{2}\ln \left ( ex+d \right ) Ba}{ \left ( ae-bd \right ) ^{4}}}+2\,{\frac{e\ln \left ( ex+d \right ) Bbd}{ \left ( ae-bd \right ) ^{4}}}-2\,{\frac{Abe}{ \left ( ae-bd \right ) ^{3} \left ( bx+a \right ) }}+{\frac{Bae}{ \left ( ae-bd \right ) ^{3} \left ( bx+a \right ) }}+{\frac{Bbd}{ \left ( ae-bd \right ) ^{3} \left ( bx+a \right ) }}-{\frac{Ab}{2\, \left ( ae-bd \right ) ^{2} \left ( bx+a \right ) ^{2}}}+{\frac{Ba}{2\, \left ( ae-bd \right ) ^{2} \left ( bx+a \right ) ^{2}}}+3\,{\frac{{e}^{2}\ln \left ( bx+a \right ) Ab}{ \left ( ae-bd \right ) ^{4}}}-{\frac{{e}^{2}\ln \left ( bx+a \right ) Ba}{ \left ( ae-bd \right ) ^{4}}}-2\,{\frac{e\ln \left ( bx+a \right ) Bbd}{ \left ( ae-bd \right ) ^{4}}} \]

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 [A]  time = 1.36977, size = 644, normalized size = 4.08 \[ -\frac{{\left (2 \, B b d e +{\left (B a - 3 \, A b\right )} e^{2}\right )} \log \left (b x + a\right )}{b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}} + \frac{{\left (2 \, B b d e +{\left (B a - 3 \, A b\right )} e^{2}\right )} \log \left (e x + d\right )}{b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}} + \frac{2 \, A a^{2} e^{2} -{\left (B a b + A b^{2}\right )} d^{2} - 5 \,{\left (B a^{2} - A a b\right )} d e - 2 \,{\left (2 \, B b^{2} d e +{\left (B a b - 3 \, A b^{2}\right )} e^{2}\right )} x^{2} -{\left (2 \, B b^{2} d^{2} +{\left (7 \, B a b - 3 \, A b^{2}\right )} d e + 3 \,{\left (B a^{2} - 3 \, A a b\right )} e^{2}\right )} x}{2 \,{\left (a^{2} b^{3} d^{4} - 3 \, a^{3} b^{2} d^{3} e + 3 \, a^{4} b d^{2} e^{2} - a^{5} d e^{3} +{\left (b^{5} d^{3} e - 3 \, a b^{4} d^{2} e^{2} + 3 \, a^{2} b^{3} d e^{3} - a^{3} b^{2} e^{4}\right )} x^{3} +{\left (b^{5} d^{4} - a b^{4} d^{3} e - 3 \, a^{2} b^{3} d^{2} e^{2} + 5 \, a^{3} b^{2} d e^{3} - 2 \, a^{4} b e^{4}\right )} x^{2} +{\left (2 \, a b^{4} d^{4} - 5 \, a^{2} b^{3} d^{3} e + 3 \, a^{3} b^{2} d^{2} e^{2} + a^{4} b d e^{3} - a^{5} e^{4}\right )} x\right )}} \]

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 [A]  time = 0.23087, size = 1084, normalized size = 6.86 \[ \text{result too large to display} \]

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)*lo
g(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 [A]  time = 14.1017, size = 1066, normalized size = 6.75 \[ \text{result too large to display} \]

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/XCAS [A]  time = 0.233786, size = 402, normalized size = 2.54 \[ -\frac{{\left (2 \, B b d e^{2} + B a e^{3} - 3 \, A b e^{3}\right )}{\rm ln}\left ({\left | -b + \frac{b d}{x e + d} - \frac{a e}{x e + d} \right |}\right )}{b^{4} d^{4} e - 4 \, a b^{3} d^{3} e^{2} + 6 \, a^{2} b^{2} d^{2} e^{3} - 4 \, a^{3} b d e^{4} + a^{4} e^{5}} - \frac{\frac{B d e^{4}}{x e + d} - \frac{A e^{5}}{x e + d}}{b^{3} d^{3} e^{3} - 3 \, a b^{2} d^{2} e^{4} + 3 \, a^{2} b d e^{5} - a^{3} e^{6}} - \frac{2 \, B b^{3} d e + 3 \, B a b^{2} e^{2} - 5 \, A b^{3} e^{2} - \frac{2 \,{\left (B b^{3} d^{2} e^{2} + B a b^{2} d e^{3} - 3 \, A b^{3} d e^{3} - 2 \, B a^{2} b e^{4} + 3 \, A a b^{2} e^{4}\right )} e^{\left (-1\right )}}{x e + d}}{2 \,{\left (b d - a e\right )}^{4}{\left (b - \frac{b d}{x e + d} + \frac{a e}{x e + d}\right )}^{2}} \]

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)*ln(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)