3.1113 \(\int \frac{A+B x}{(a+b x)^2 (d+e x)^3} \, dx\)

Optimal. Leaf size=157 \[ -\frac{b (A b-a B)}{(a+b x) (b d-a e)^3}+\frac{a B e-2 A b e+b B d}{(d+e x) (b d-a e)^3}+\frac{B d-A e}{2 (d+e x)^2 (b d-a e)^2}+\frac{b \log (a+b x) (2 a B e-3 A b e+b B d)}{(b d-a e)^4}-\frac{b \log (d+e x) (2 a B e-3 A b e+b B d)}{(b d-a e)^4} \]

[Out]

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

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Rubi [A]  time = 0.333449, antiderivative size = 157, 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 \[ -\frac{b (A b-a B)}{(a+b x) (b d-a e)^3}+\frac{a B e-2 A b e+b B d}{(d+e x) (b d-a e)^3}+\frac{B d-A e}{2 (d+e x)^2 (b d-a e)^2}+\frac{b \log (a+b x) (2 a B e-3 A b e+b B d)}{(b d-a e)^4}-\frac{b \log (d+e x) (2 a B e-3 A b e+b B d)}{(b d-a e)^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 57.6403, size = 146, normalized size = 0.93 \[ - \frac{b \left (3 A b e - 2 B a e - B b d\right ) \log{\left (a + b x \right )}}{\left (a e - b d\right )^{4}} + \frac{b \left (3 A b e - 2 B a e - B b d\right ) \log{\left (d + e x \right )}}{\left (a e - b d\right )^{4}} + \frac{b \left (A b - B a\right )}{\left (a + b x\right ) \left (a e - b d\right )^{3}} + \frac{2 A b e - B a e - B b d}{\left (d + e x\right ) \left (a e - b d\right )^{3}} - \frac{A e - B d}{2 \left (d + e x\right )^{2} \left (a e - b d\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.169474, size = 146, normalized size = 0.93 \[ \frac{\frac{(b d-a e)^2 (B d-A e)}{(d+e x)^2}-\frac{2 b (A b-a B) (b d-a e)}{a+b x}+\frac{2 (b d-a e) (a B e-2 A b e+b B d)}{d+e x}+2 b \log (a+b x) (2 a B e-3 A b e+b B d)-2 b \log (d+e x) (2 a B e-3 A b e+b B d)}{2 (b d-a e)^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.022, size = 289, normalized size = 1.8 \[ -{\frac{Ae}{2\, \left ( ae-bd \right ) ^{2} \left ( ex+d \right ) ^{2}}}+{\frac{Bd}{2\, \left ( ae-bd \right ) ^{2} \left ( ex+d \right ) ^{2}}}+3\,{\frac{{b}^{2}\ln \left ( ex+d \right ) Ae}{ \left ( ae-bd \right ) ^{4}}}-2\,{\frac{b\ln \left ( ex+d \right ) Bae}{ \left ( ae-bd \right ) ^{4}}}-{\frac{{b}^{2}\ln \left ( ex+d \right ) Bd}{ \left ( ae-bd \right ) ^{4}}}+2\,{\frac{Abe}{ \left ( ae-bd \right ) ^{3} \left ( ex+d \right ) }}-{\frac{Bae}{ \left ( ae-bd \right ) ^{3} \left ( ex+d \right ) }}-{\frac{Bbd}{ \left ( ae-bd \right ) ^{3} \left ( ex+d \right ) }}-3\,{\frac{{b}^{2}\ln \left ( bx+a \right ) Ae}{ \left ( ae-bd \right ) ^{4}}}+2\,{\frac{b\ln \left ( bx+a \right ) Bae}{ \left ( ae-bd \right ) ^{4}}}+{\frac{{b}^{2}\ln \left ( bx+a \right ) Bd}{ \left ( ae-bd \right ) ^{4}}}+{\frac{{b}^{2}A}{ \left ( ae-bd \right ) ^{3} \left ( bx+a \right ) }}-{\frac{Bba}{ \left ( ae-bd \right ) ^{3} \left ( bx+a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.3946, size = 647, normalized size = 4.12 \[ \frac{{\left (B b^{2} d +{\left (2 \, B a b - 3 \, A b^{2}\right )} e\right )} \log \left (b x + a\right )}{b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}} - \frac{{\left (B b^{2} d +{\left (2 \, B a b - 3 \, A b^{2}\right )} e\right )} \log \left (e x + d\right )}{b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}} + \frac{A a^{2} e^{2} +{\left (5 \, B a b - 2 \, A b^{2}\right )} d^{2} +{\left (B a^{2} - 5 \, A a b\right )} d e + 2 \,{\left (B b^{2} d e +{\left (2 \, B a b - 3 \, A b^{2}\right )} e^{2}\right )} x^{2} +{\left (3 \, B b^{2} d^{2} +{\left (7 \, B a b - 9 \, A b^{2}\right )} d e +{\left (2 \, B a^{2} - 3 \, A a b\right )} e^{2}\right )} x}{2 \,{\left (a b^{3} d^{5} - 3 \, a^{2} b^{2} d^{4} e + 3 \, a^{3} b d^{3} e^{2} - a^{4} d^{2} e^{3} +{\left (b^{4} d^{3} e^{2} - 3 \, a b^{3} d^{2} e^{3} + 3 \, a^{2} b^{2} d e^{4} - a^{3} b e^{5}\right )} x^{3} +{\left (2 \, b^{4} d^{4} e - 5 \, a b^{3} d^{3} e^{2} + 3 \, a^{2} b^{2} d^{2} e^{3} + a^{3} b d e^{4} - a^{4} e^{5}\right )} x^{2} +{\left (b^{4} d^{5} - a b^{3} d^{4} e - 3 \, a^{2} b^{2} d^{3} e^{2} + 5 \, a^{3} b d^{2} e^{3} - 2 \, a^{4} d e^{4}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(B*b^2*d + (2*B*a*b - 3*A*b^2)*e)*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) - (B*b^2*d + (2*B*a*b - 3*A*b^2)*e)*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*(A*a^2*e^2 + (5*B*a*b - 2*A*b^2)*d^2 + (B*a^2 - 5*A*a*b)*d*e + 2*(B*b^2*d*
e + (2*B*a*b - 3*A*b^2)*e^2)*x^2 + (3*B*b^2*d^2 + (7*B*a*b - 9*A*b^2)*d*e + (2*B
*a^2 - 3*A*a*b)*e^2)*x)/(a*b^3*d^5 - 3*a^2*b^2*d^4*e + 3*a^3*b*d^3*e^2 - a^4*d^2
*e^3 + (b^4*d^3*e^2 - 3*a*b^3*d^2*e^3 + 3*a^2*b^2*d*e^4 - a^3*b*e^5)*x^3 + (2*b^
4*d^4*e - 5*a*b^3*d^3*e^2 + 3*a^2*b^2*d^2*e^3 + a^3*b*d*e^4 - a^4*e^5)*x^2 + (b^
4*d^5 - a*b^3*d^4*e - 3*a^2*b^2*d^3*e^2 + 5*a^3*b*d^2*e^3 - 2*a^4*d*e^4)*x)

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Fricas [A]  time = 0.244846, size = 1081, normalized size = 6.89 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(A*a^3*e^3 - (5*B*a*b^2 - 2*A*b^3)*d^3 + (4*B*a^2*b + 3*A*a*b^2)*d^2*e + (B
*a^3 - 6*A*a^2*b)*d*e^2 - 2*(B*b^3*d^2*e + (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 - (3*B*b^3*d^3 + (4*B*a*b^2 - 9*A*b^3)*d^2*e - (5*B*a^2*
b - 6*A*a*b^2)*d*e^2 - (2*B*a^3 - 3*A*a^2*b)*e^3)*x - 2*(B*a*b^2*d^3 + (2*B*a^2*
b - 3*A*a*b^2)*d^2*e + (B*b^3*d*e^2 + (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 - 6*A*b^3)*d*e^2 + (2*B*a^2*b - 3*A*a*b^2)*e^3)*x^2 + (B*b^3*
d^3 + (4*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)*x)*log(b*x
+ a) + 2*(B*a*b^2*d^3 + (2*B*a^2*b - 3*A*a*b^2)*d^2*e + (B*b^3*d*e^2 + (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 - 6*A*b^3)*d*e^2 + (2*B*a^2*
b - 3*A*a*b^2)*e^3)*x^2 + (B*b^3*d^3 + (4*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)*x)*log(e*x + d))/(a*b^4*d^6 - 4*a^2*b^3*d^5*e + 6*a^3*b^2*
d^4*e^2 - 4*a^4*b*d^3*e^3 + a^5*d^2*e^4 + (b^5*d^4*e^2 - 4*a*b^4*d^3*e^3 + 6*a^2
*b^3*d^2*e^4 - 4*a^3*b^2*d*e^5 + a^4*b*e^6)*x^3 + (2*b^5*d^5*e - 7*a*b^4*d^4*e^2
+ 8*a^2*b^3*d^3*e^3 - 2*a^3*b^2*d^2*e^4 - 2*a^4*b*d*e^5 + a^5*e^6)*x^2 + (b^5*d
^6 - 2*a*b^4*d^5*e - 2*a^2*b^3*d^4*e^2 + 8*a^3*b^2*d^3*e^3 - 7*a^4*b*d^2*e^4 + 2
*a^5*d*e^5)*x)

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Sympy [A]  time = 13.7414, size = 1066, normalized size = 6.79 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.239668, size = 413, normalized size = 2.63 \[ -\frac{{\left (B b^{3} d + 2 \, B a b^{2} e - 3 \, A b^{3} e\right )}{\rm ln}\left ({\left | -\frac{b d}{b x + a} + \frac{a e}{b x + a} - e \right |}\right )}{b^{5} d^{4} - 4 \, a b^{4} d^{3} e + 6 \, a^{2} b^{3} d^{2} e^{2} - 4 \, a^{3} b^{2} d e^{3} + a^{4} b e^{4}} + \frac{\frac{B a b^{4}}{b x + a} - \frac{A b^{5}}{b x + a}}{b^{6} d^{3} - 3 \, a b^{5} d^{2} e + 3 \, a^{2} b^{4} d e^{2} - a^{3} b^{3} e^{3}} - \frac{3 \, B b^{2} d e^{2} + 2 \, B a b e^{3} - 5 \, A b^{2} e^{3} + \frac{2 \,{\left (2 \, B b^{4} d^{2} e - B a b^{3} d e^{2} - 3 \, A b^{4} d e^{2} - B a^{2} b^{2} e^{3} + 3 \, A a b^{3} e^{3}\right )}}{{\left (b x + a\right )} b}}{2 \,{\left (b d - a e\right )}^{4}{\left (\frac{b d}{b x + a} - \frac{a e}{b x + a} + e\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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