3.1115 \(\int \frac{A+B x}{(a+b x)^2 (d+e x)^5} \, dx\)
Optimal. Leaf size=239 \[ -\frac{b^3 (A b-a B)}{(a+b x) (b d-a e)^5}+\frac{b^3 \log (a+b x) (4 a B e-5 A b e+b B d)}{(b d-a e)^6}-\frac{b^3 \log (d+e x) (4 a B e-5 A b e+b B d)}{(b d-a e)^6}+\frac{b^2 (3 a B e-4 A b e+b B d)}{(d+e x) (b d-a e)^5}+\frac{b (2 a B e-3 A b e+b B d)}{2 (d+e x)^2 (b d-a e)^4}+\frac{a B e-2 A b e+b B d}{3 (d+e x)^3 (b d-a e)^3}+\frac{B d-A e}{4 (d+e x)^4 (b d-a e)^2} \]
[Out]
-((b^3*(A*b - a*B))/((b*d - a*e)^5*(a + b*x))) + (B*d - A*e)/(4*(b*d - a*e)^2*(d
+ e*x)^4) + (b*B*d - 2*A*b*e + a*B*e)/(3*(b*d - a*e)^3*(d + e*x)^3) + (b*(b*B*d
- 3*A*b*e + 2*a*B*e))/(2*(b*d - a*e)^4*(d + e*x)^2) + (b^2*(b*B*d - 4*A*b*e + 3
*a*B*e))/((b*d - a*e)^5*(d + e*x)) + (b^3*(b*B*d - 5*A*b*e + 4*a*B*e)*Log[a + b*
x])/(b*d - a*e)^6 - (b^3*(b*B*d - 5*A*b*e + 4*a*B*e)*Log[d + e*x])/(b*d - a*e)^6
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Rubi [A] time = 0.626265, antiderivative size = 239, 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
\[ -\frac{b^3 (A b-a B)}{(a+b x) (b d-a e)^5}+\frac{b^3 \log (a+b x) (4 a B e-5 A b e+b B d)}{(b d-a e)^6}-\frac{b^3 \log (d+e x) (4 a B e-5 A b e+b B d)}{(b d-a e)^6}+\frac{b^2 (3 a B e-4 A b e+b B d)}{(d+e x) (b d-a e)^5}+\frac{b (2 a B e-3 A b e+b B d)}{2 (d+e x)^2 (b d-a e)^4}+\frac{a B e-2 A b e+b B d}{3 (d+e x)^3 (b d-a e)^3}+\frac{B d-A e}{4 (d+e x)^4 (b d-a e)^2} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)/((a + b*x)^2*(d + e*x)^5),x]
[Out]
-((b^3*(A*b - a*B))/((b*d - a*e)^5*(a + b*x))) + (B*d - A*e)/(4*(b*d - a*e)^2*(d
+ e*x)^4) + (b*B*d - 2*A*b*e + a*B*e)/(3*(b*d - a*e)^3*(d + e*x)^3) + (b*(b*B*d
- 3*A*b*e + 2*a*B*e))/(2*(b*d - a*e)^4*(d + e*x)^2) + (b^2*(b*B*d - 4*A*b*e + 3
*a*B*e))/((b*d - a*e)^5*(d + e*x)) + (b^3*(b*B*d - 5*A*b*e + 4*a*B*e)*Log[a + b*
x])/(b*d - a*e)^6 - (b^3*(b*B*d - 5*A*b*e + 4*a*B*e)*Log[d + e*x])/(b*d - a*e)^6
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)/(b*x+a)**2/(e*x+d)**5,x)
[Out]
Timed out
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Mathematica [A] time = 0.308992, size = 225, normalized size = 0.94 \[ \frac{-\frac{12 b^3 (A b-a B) (b d-a e)}{a+b x}+12 b^3 \log (a+b x) (4 a B e-5 A b e+b B d)-12 b^3 \log (d+e x) (4 a B e-5 A b e+b B d)+\frac{12 b^2 (b d-a e) (3 a B e-4 A b e+b B d)}{d+e x}+\frac{3 (b d-a e)^4 (B d-A e)}{(d+e x)^4}+\frac{4 (b d-a e)^3 (a B e-2 A b e+b B d)}{(d+e x)^3}+\frac{6 b (b d-a e)^2 (2 a B e-3 A b e+b B d)}{(d+e x)^2}}{12 (b d-a e)^6} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)/((a + b*x)^2*(d + e*x)^5),x]
[Out]
((-12*b^3*(A*b - a*B)*(b*d - a*e))/(a + b*x) + (3*(b*d - a*e)^4*(B*d - A*e))/(d
+ e*x)^4 + (4*(b*d - a*e)^3*(b*B*d - 2*A*b*e + a*B*e))/(d + e*x)^3 + (6*b*(b*d -
a*e)^2*(b*B*d - 3*A*b*e + 2*a*B*e))/(d + e*x)^2 + (12*b^2*(b*d - a*e)*(b*B*d -
4*A*b*e + 3*a*B*e))/(d + e*x) + 12*b^3*(b*B*d - 5*A*b*e + 4*a*B*e)*Log[a + b*x]
- 12*b^3*(b*B*d - 5*A*b*e + 4*a*B*e)*Log[d + e*x])/(12*(b*d - a*e)^6)
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Maple [A] time = 0.029, size = 438, normalized size = 1.8 \[ -{\frac{Ae}{4\, \left ( ae-bd \right ) ^{2} \left ( ex+d \right ) ^{4}}}+{\frac{Bd}{4\, \left ( ae-bd \right ) ^{2} \left ( ex+d \right ) ^{4}}}-{\frac{3\,{b}^{2}Ae}{2\, \left ( ae-bd \right ) ^{4} \left ( ex+d \right ) ^{2}}}+{\frac{Bbae}{ \left ( ae-bd \right ) ^{4} \left ( ex+d \right ) ^{2}}}+{\frac{{b}^{2}Bd}{2\, \left ( ae-bd \right ) ^{4} \left ( ex+d \right ) ^{2}}}+5\,{\frac{{b}^{4}\ln \left ( ex+d \right ) Ae}{ \left ( ae-bd \right ) ^{6}}}-4\,{\frac{{b}^{3}\ln \left ( ex+d \right ) Bae}{ \left ( ae-bd \right ) ^{6}}}-{\frac{{b}^{4}\ln \left ( ex+d \right ) Bd}{ \left ( ae-bd \right ) ^{6}}}+{\frac{2\,Abe}{3\, \left ( ae-bd \right ) ^{3} \left ( ex+d \right ) ^{3}}}-{\frac{Bae}{3\, \left ( ae-bd \right ) ^{3} \left ( ex+d \right ) ^{3}}}-{\frac{Bbd}{3\, \left ( ae-bd \right ) ^{3} \left ( ex+d \right ) ^{3}}}+4\,{\frac{{b}^{3}Ae}{ \left ( ae-bd \right ) ^{5} \left ( ex+d \right ) }}-3\,{\frac{{b}^{2}Bae}{ \left ( ae-bd \right ) ^{5} \left ( ex+d \right ) }}-{\frac{{b}^{3}Bd}{ \left ( ae-bd \right ) ^{5} \left ( ex+d \right ) }}-5\,{\frac{{b}^{4}\ln \left ( bx+a \right ) Ae}{ \left ( ae-bd \right ) ^{6}}}+4\,{\frac{{b}^{3}\ln \left ( bx+a \right ) Bae}{ \left ( ae-bd \right ) ^{6}}}+{\frac{{b}^{4}\ln \left ( bx+a \right ) Bd}{ \left ( ae-bd \right ) ^{6}}}+{\frac{{b}^{4}A}{ \left ( ae-bd \right ) ^{5} \left ( bx+a \right ) }}-{\frac{Ba{b}^{3}}{ \left ( ae-bd \right ) ^{5} \left ( bx+a \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)/(b*x+a)^2/(e*x+d)^5,x)
[Out]
-1/4/(a*e-b*d)^2/(e*x+d)^4*A*e+1/4/(a*e-b*d)^2/(e*x+d)^4*B*d-3/2*b^2/(a*e-b*d)^4
/(e*x+d)^2*A*e+b/(a*e-b*d)^4/(e*x+d)^2*B*a*e+1/2*b^2/(a*e-b*d)^4/(e*x+d)^2*B*d+5
*b^4/(a*e-b*d)^6*ln(e*x+d)*A*e-4*b^3/(a*e-b*d)^6*ln(e*x+d)*B*a*e-b^4/(a*e-b*d)^6
*ln(e*x+d)*B*d+2/3/(a*e-b*d)^3/(e*x+d)^3*A*b*e-1/3/(a*e-b*d)^3/(e*x+d)^3*B*a*e-1
/3/(a*e-b*d)^3/(e*x+d)^3*B*b*d+4*b^3/(a*e-b*d)^5/(e*x+d)*A*e-3*b^2/(a*e-b*d)^5/(
e*x+d)*B*a*e-b^3/(a*e-b*d)^5/(e*x+d)*B*d-5*b^4/(a*e-b*d)^6*ln(b*x+a)*A*e+4*b^3/(
a*e-b*d)^6*ln(b*x+a)*B*a*e+b^4/(a*e-b*d)^6*ln(b*x+a)*B*d+b^4/(a*e-b*d)^5/(b*x+a)
*A-b^3/(a*e-b*d)^5/(b*x+a)*B*a
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Maxima [A] time = 1.46333, size = 1472, normalized size = 6.16 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((b*x + a)^2*(e*x + d)^5),x, algorithm="maxima")
[Out]
(B*b^4*d + (4*B*a*b^3 - 5*A*b^4)*e)*log(b*x + a)/(b^6*d^6 - 6*a*b^5*d^5*e + 15*a
^2*b^4*d^4*e^2 - 20*a^3*b^3*d^3*e^3 + 15*a^4*b^2*d^2*e^4 - 6*a^5*b*d*e^5 + a^6*e
^6) - (B*b^4*d + (4*B*a*b^3 - 5*A*b^4)*e)*log(e*x + d)/(b^6*d^6 - 6*a*b^5*d^5*e
+ 15*a^2*b^4*d^4*e^2 - 20*a^3*b^3*d^3*e^3 + 15*a^4*b^2*d^2*e^4 - 6*a^5*b*d*e^5 +
a^6*e^6) + 1/12*(3*A*a^4*e^4 + (37*B*a*b^3 - 12*A*b^4)*d^4 + (29*B*a^2*b^2 - 77
*A*a*b^3)*d^3*e - (7*B*a^3*b - 43*A*a^2*b^2)*d^2*e^2 + (B*a^4 - 17*A*a^3*b)*d*e^
3 + 12*(B*b^4*d*e^3 + (4*B*a*b^3 - 5*A*b^4)*e^4)*x^4 + 6*(7*B*b^4*d^2*e^2 + (29*
B*a*b^3 - 35*A*b^4)*d*e^3 + (4*B*a^2*b^2 - 5*A*a*b^3)*e^4)*x^3 + 2*(26*B*b^4*d^3
*e + 5*(23*B*a*b^3 - 26*A*b^4)*d^2*e^2 + (43*B*a^2*b^2 - 55*A*a*b^3)*d*e^3 - (4*
B*a^3*b - 5*A*a^2*b^2)*e^4)*x^2 + (25*B*b^4*d^4 + (129*B*a*b^3 - 125*A*b^4)*d^3*
e + (109*B*a^2*b^2 - 145*A*a*b^3)*d^2*e^2 - (27*B*a^3*b - 35*A*a^2*b^2)*d*e^3 +
(4*B*a^4 - 5*A*a^3*b)*e^4)*x)/(a*b^5*d^9 - 5*a^2*b^4*d^8*e + 10*a^3*b^3*d^7*e^2
- 10*a^4*b^2*d^6*e^3 + 5*a^5*b*d^5*e^4 - a^6*d^4*e^5 + (b^6*d^5*e^4 - 5*a*b^5*d^
4*e^5 + 10*a^2*b^4*d^3*e^6 - 10*a^3*b^3*d^2*e^7 + 5*a^4*b^2*d*e^8 - a^5*b*e^9)*x
^5 + (4*b^6*d^6*e^3 - 19*a*b^5*d^5*e^4 + 35*a^2*b^4*d^4*e^5 - 30*a^3*b^3*d^3*e^6
+ 10*a^4*b^2*d^2*e^7 + a^5*b*d*e^8 - a^6*e^9)*x^4 + 2*(3*b^6*d^7*e^2 - 13*a*b^5
*d^6*e^3 + 20*a^2*b^4*d^5*e^4 - 10*a^3*b^3*d^4*e^5 - 5*a^4*b^2*d^3*e^6 + 7*a^5*b
*d^2*e^7 - 2*a^6*d*e^8)*x^3 + 2*(2*b^6*d^8*e - 7*a*b^5*d^7*e^2 + 5*a^2*b^4*d^6*e
^3 + 10*a^3*b^3*d^5*e^4 - 20*a^4*b^2*d^4*e^5 + 13*a^5*b*d^3*e^6 - 3*a^6*d^2*e^7)
*x^2 + (b^6*d^9 - a*b^5*d^8*e - 10*a^2*b^4*d^7*e^2 + 30*a^3*b^3*d^6*e^3 - 35*a^4
*b^2*d^5*e^4 + 19*a^5*b*d^4*e^5 - 4*a^6*d^3*e^6)*x)
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Fricas [A] time = 0.249473, size = 2298, normalized size = 9.62 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((b*x + a)^2*(e*x + d)^5),x, algorithm="fricas")
[Out]
-1/12*(3*A*a^5*e^5 - (37*B*a*b^4 - 12*A*b^5)*d^5 + (8*B*a^2*b^3 + 65*A*a*b^4)*d^
4*e + 12*(3*B*a^3*b^2 - 10*A*a^2*b^3)*d^3*e^2 - 4*(2*B*a^4*b - 15*A*a^3*b^2)*d^2
*e^3 + (B*a^5 - 20*A*a^4*b)*d*e^4 - 12*(B*b^5*d^2*e^3 + (3*B*a*b^4 - 5*A*b^5)*d*
e^4 - (4*B*a^2*b^3 - 5*A*a*b^4)*e^5)*x^4 - 6*(7*B*b^5*d^3*e^2 + (22*B*a*b^4 - 35
*A*b^5)*d^2*e^3 - 5*(5*B*a^2*b^3 - 6*A*a*b^4)*d*e^4 - (4*B*a^3*b^2 - 5*A*a^2*b^3
)*e^5)*x^3 - 2*(26*B*b^5*d^4*e + (89*B*a*b^4 - 130*A*b^5)*d^3*e^2 - 3*(24*B*a^2*
b^3 - 25*A*a*b^4)*d^2*e^3 - (47*B*a^3*b^2 - 60*A*a^2*b^3)*d*e^4 + (4*B*a^4*b - 5
*A*a^3*b^2)*e^5)*x^2 - (25*B*b^5*d^5 + (104*B*a*b^4 - 125*A*b^5)*d^4*e - 20*(B*a
^2*b^3 + A*a*b^4)*d^3*e^2 - 4*(34*B*a^3*b^2 - 45*A*a^2*b^3)*d^2*e^3 + (31*B*a^4*
b - 40*A*a^3*b^2)*d*e^4 - (4*B*a^5 - 5*A*a^4*b)*e^5)*x - 12*(B*a*b^4*d^5 + (4*B*
a^2*b^3 - 5*A*a*b^4)*d^4*e + (B*b^5*d*e^4 + (4*B*a*b^4 - 5*A*b^5)*e^5)*x^5 + (4*
B*b^5*d^2*e^3 + (17*B*a*b^4 - 20*A*b^5)*d*e^4 + (4*B*a^2*b^3 - 5*A*a*b^4)*e^5)*x
^4 + 2*(3*B*b^5*d^3*e^2 + (14*B*a*b^4 - 15*A*b^5)*d^2*e^3 + 2*(4*B*a^2*b^3 - 5*A
*a*b^4)*d*e^4)*x^3 + 2*(2*B*b^5*d^4*e + (11*B*a*b^4 - 10*A*b^5)*d^3*e^2 + 3*(4*B
*a^2*b^3 - 5*A*a*b^4)*d^2*e^3)*x^2 + (B*b^5*d^5 + (8*B*a*b^4 - 5*A*b^5)*d^4*e +
4*(4*B*a^2*b^3 - 5*A*a*b^4)*d^3*e^2)*x)*log(b*x + a) + 12*(B*a*b^4*d^5 + (4*B*a^
2*b^3 - 5*A*a*b^4)*d^4*e + (B*b^5*d*e^4 + (4*B*a*b^4 - 5*A*b^5)*e^5)*x^5 + (4*B*
b^5*d^2*e^3 + (17*B*a*b^4 - 20*A*b^5)*d*e^4 + (4*B*a^2*b^3 - 5*A*a*b^4)*e^5)*x^4
+ 2*(3*B*b^5*d^3*e^2 + (14*B*a*b^4 - 15*A*b^5)*d^2*e^3 + 2*(4*B*a^2*b^3 - 5*A*a
*b^4)*d*e^4)*x^3 + 2*(2*B*b^5*d^4*e + (11*B*a*b^4 - 10*A*b^5)*d^3*e^2 + 3*(4*B*a
^2*b^3 - 5*A*a*b^4)*d^2*e^3)*x^2 + (B*b^5*d^5 + (8*B*a*b^4 - 5*A*b^5)*d^4*e + 4*
(4*B*a^2*b^3 - 5*A*a*b^4)*d^3*e^2)*x)*log(e*x + d))/(a*b^6*d^10 - 6*a^2*b^5*d^9*
e + 15*a^3*b^4*d^8*e^2 - 20*a^4*b^3*d^7*e^3 + 15*a^5*b^2*d^6*e^4 - 6*a^6*b*d^5*e
^5 + a^7*d^4*e^6 + (b^7*d^6*e^4 - 6*a*b^6*d^5*e^5 + 15*a^2*b^5*d^4*e^6 - 20*a^3*
b^4*d^3*e^7 + 15*a^4*b^3*d^2*e^8 - 6*a^5*b^2*d*e^9 + a^6*b*e^10)*x^5 + (4*b^7*d^
7*e^3 - 23*a*b^6*d^6*e^4 + 54*a^2*b^5*d^5*e^5 - 65*a^3*b^4*d^4*e^6 + 40*a^4*b^3*
d^3*e^7 - 9*a^5*b^2*d^2*e^8 - 2*a^6*b*d*e^9 + a^7*e^10)*x^4 + 2*(3*b^7*d^8*e^2 -
16*a*b^6*d^7*e^3 + 33*a^2*b^5*d^6*e^4 - 30*a^3*b^4*d^5*e^5 + 5*a^4*b^3*d^4*e^6
+ 12*a^5*b^2*d^3*e^7 - 9*a^6*b*d^2*e^8 + 2*a^7*d*e^9)*x^3 + 2*(2*b^7*d^9*e - 9*a
*b^6*d^8*e^2 + 12*a^2*b^5*d^7*e^3 + 5*a^3*b^4*d^6*e^4 - 30*a^4*b^3*d^5*e^5 + 33*
a^5*b^2*d^4*e^6 - 16*a^6*b*d^3*e^7 + 3*a^7*d^2*e^8)*x^2 + (b^7*d^10 - 2*a*b^6*d^
9*e - 9*a^2*b^5*d^8*e^2 + 40*a^3*b^4*d^7*e^3 - 65*a^4*b^3*d^6*e^4 + 54*a^5*b^2*d
^5*e^5 - 23*a^6*b*d^4*e^6 + 4*a^7*d^3*e^7)*x)
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Sympy [A] time = 33.4686, size = 1877, normalized size = 7.85 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)/(b*x+a)**2/(e*x+d)**5,x)
[Out]
-b**3*(-5*A*b*e + 4*B*a*e + B*b*d)*log(x + (-5*A*a*b**4*e**2 - 5*A*b**5*d*e + 4*
B*a**2*b**3*e**2 + 5*B*a*b**4*d*e + B*b**5*d**2 - a**7*b**3*e**7*(-5*A*b*e + 4*B
*a*e + B*b*d)/(a*e - b*d)**6 + 7*a**6*b**4*d*e**6*(-5*A*b*e + 4*B*a*e + B*b*d)/(
a*e - b*d)**6 - 21*a**5*b**5*d**2*e**5*(-5*A*b*e + 4*B*a*e + B*b*d)/(a*e - b*d)*
*6 + 35*a**4*b**6*d**3*e**4*(-5*A*b*e + 4*B*a*e + B*b*d)/(a*e - b*d)**6 - 35*a**
3*b**7*d**4*e**3*(-5*A*b*e + 4*B*a*e + B*b*d)/(a*e - b*d)**6 + 21*a**2*b**8*d**5
*e**2*(-5*A*b*e + 4*B*a*e + B*b*d)/(a*e - b*d)**6 - 7*a*b**9*d**6*e*(-5*A*b*e +
4*B*a*e + B*b*d)/(a*e - b*d)**6 + b**10*d**7*(-5*A*b*e + 4*B*a*e + B*b*d)/(a*e -
b*d)**6)/(-10*A*b**5*e**2 + 8*B*a*b**4*e**2 + 2*B*b**5*d*e))/(a*e - b*d)**6 + b
**3*(-5*A*b*e + 4*B*a*e + B*b*d)*log(x + (-5*A*a*b**4*e**2 - 5*A*b**5*d*e + 4*B*
a**2*b**3*e**2 + 5*B*a*b**4*d*e + B*b**5*d**2 + a**7*b**3*e**7*(-5*A*b*e + 4*B*a
*e + B*b*d)/(a*e - b*d)**6 - 7*a**6*b**4*d*e**6*(-5*A*b*e + 4*B*a*e + B*b*d)/(a*
e - b*d)**6 + 21*a**5*b**5*d**2*e**5*(-5*A*b*e + 4*B*a*e + B*b*d)/(a*e - b*d)**6
- 35*a**4*b**6*d**3*e**4*(-5*A*b*e + 4*B*a*e + B*b*d)/(a*e - b*d)**6 + 35*a**3*
b**7*d**4*e**3*(-5*A*b*e + 4*B*a*e + B*b*d)/(a*e - b*d)**6 - 21*a**2*b**8*d**5*e
**2*(-5*A*b*e + 4*B*a*e + B*b*d)/(a*e - b*d)**6 + 7*a*b**9*d**6*e*(-5*A*b*e + 4*
B*a*e + B*b*d)/(a*e - b*d)**6 - b**10*d**7*(-5*A*b*e + 4*B*a*e + B*b*d)/(a*e - b
*d)**6)/(-10*A*b**5*e**2 + 8*B*a*b**4*e**2 + 2*B*b**5*d*e))/(a*e - b*d)**6 - (3*
A*a**4*e**4 - 17*A*a**3*b*d*e**3 + 43*A*a**2*b**2*d**2*e**2 - 77*A*a*b**3*d**3*e
- 12*A*b**4*d**4 + B*a**4*d*e**3 - 7*B*a**3*b*d**2*e**2 + 29*B*a**2*b**2*d**3*e
+ 37*B*a*b**3*d**4 + x**4*(-60*A*b**4*e**4 + 48*B*a*b**3*e**4 + 12*B*b**4*d*e**
3) + x**3*(-30*A*a*b**3*e**4 - 210*A*b**4*d*e**3 + 24*B*a**2*b**2*e**4 + 174*B*a
*b**3*d*e**3 + 42*B*b**4*d**2*e**2) + x**2*(10*A*a**2*b**2*e**4 - 110*A*a*b**3*d
*e**3 - 260*A*b**4*d**2*e**2 - 8*B*a**3*b*e**4 + 86*B*a**2*b**2*d*e**3 + 230*B*a
*b**3*d**2*e**2 + 52*B*b**4*d**3*e) + x*(-5*A*a**3*b*e**4 + 35*A*a**2*b**2*d*e**
3 - 145*A*a*b**3*d**2*e**2 - 125*A*b**4*d**3*e + 4*B*a**4*e**4 - 27*B*a**3*b*d*e
**3 + 109*B*a**2*b**2*d**2*e**2 + 129*B*a*b**3*d**3*e + 25*B*b**4*d**4))/(12*a**
6*d**4*e**5 - 60*a**5*b*d**5*e**4 + 120*a**4*b**2*d**6*e**3 - 120*a**3*b**3*d**7
*e**2 + 60*a**2*b**4*d**8*e - 12*a*b**5*d**9 + x**5*(12*a**5*b*e**9 - 60*a**4*b*
*2*d*e**8 + 120*a**3*b**3*d**2*e**7 - 120*a**2*b**4*d**3*e**6 + 60*a*b**5*d**4*e
**5 - 12*b**6*d**5*e**4) + x**4*(12*a**6*e**9 - 12*a**5*b*d*e**8 - 120*a**4*b**2
*d**2*e**7 + 360*a**3*b**3*d**3*e**6 - 420*a**2*b**4*d**4*e**5 + 228*a*b**5*d**5
*e**4 - 48*b**6*d**6*e**3) + x**3*(48*a**6*d*e**8 - 168*a**5*b*d**2*e**7 + 120*a
**4*b**2*d**3*e**6 + 240*a**3*b**3*d**4*e**5 - 480*a**2*b**4*d**5*e**4 + 312*a*b
**5*d**6*e**3 - 72*b**6*d**7*e**2) + x**2*(72*a**6*d**2*e**7 - 312*a**5*b*d**3*e
**6 + 480*a**4*b**2*d**4*e**5 - 240*a**3*b**3*d**5*e**4 - 120*a**2*b**4*d**6*e**
3 + 168*a*b**5*d**7*e**2 - 48*b**6*d**8*e) + x*(48*a**6*d**3*e**6 - 228*a**5*b*d
**4*e**5 + 420*a**4*b**2*d**5*e**4 - 360*a**3*b**3*d**6*e**3 + 120*a**2*b**4*d**
7*e**2 + 12*a*b**5*d**8*e - 12*b**6*d**9))
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GIAC/XCAS [A] time = 0.248432, size = 770, normalized size = 3.22 \[ -\frac{{\left (B b^{5} d + 4 \, B a b^{4} e - 5 \, A b^{5} e\right )}{\rm ln}\left ({\left | -\frac{b d}{b x + a} + \frac{a e}{b x + a} - e \right |}\right )}{b^{7} d^{6} - 6 \, a b^{6} d^{5} e + 15 \, a^{2} b^{5} d^{4} e^{2} - 20 \, a^{3} b^{4} d^{3} e^{3} + 15 \, a^{4} b^{3} d^{2} e^{4} - 6 \, a^{5} b^{2} d e^{5} + a^{6} b e^{6}} + \frac{\frac{B a b^{8}}{b x + a} - \frac{A b^{9}}{b x + a}}{b^{10} d^{5} - 5 \, a b^{9} d^{4} e + 10 \, a^{2} b^{8} d^{3} e^{2} - 10 \, a^{3} b^{7} d^{2} e^{3} + 5 \, a^{4} b^{6} d e^{4} - a^{5} b^{5} e^{5}} - \frac{25 \, B b^{4} d e^{4} + 52 \, B a b^{3} e^{5} - 77 \, A b^{4} e^{5} + \frac{4 \,{\left (22 \, B b^{6} d^{2} e^{3} + 21 \, B a b^{5} d e^{4} - 65 \, A b^{6} d e^{4} - 43 \, B a^{2} b^{4} e^{5} + 65 \, A a b^{5} e^{5}\right )}}{{\left (b x + a\right )} b} + \frac{12 \,{\left (9 \, B b^{8} d^{3} e^{2} - 2 \, B a b^{7} d^{2} e^{3} - 25 \, A b^{8} d^{2} e^{3} - 23 \, B a^{2} b^{6} d e^{4} + 50 \, A a b^{7} d e^{4} + 16 \, B a^{3} b^{5} e^{5} - 25 \, A a^{2} b^{6} e^{5}\right )}}{{\left (b x + a\right )}^{2} b^{2}} + \frac{24 \,{\left (2 \, B b^{10} d^{4} e - 3 \, B a b^{9} d^{3} e^{2} - 5 \, A b^{10} d^{3} e^{2} - 3 \, B a^{2} b^{8} d^{2} e^{3} + 15 \, A a b^{9} d^{2} e^{3} + 7 \, B a^{3} b^{7} d e^{4} - 15 \, A a^{2} b^{8} d e^{4} - 3 \, B a^{4} b^{6} e^{5} + 5 \, A a^{3} b^{7} e^{5}\right )}}{{\left (b x + a\right )}^{3} b^{3}}}{12 \,{\left (b d - a e\right )}^{6}{\left (\frac{b d}{b x + a} - \frac{a e}{b x + a} + e\right )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((b*x + a)^2*(e*x + d)^5),x, algorithm="giac")
[Out]
-(B*b^5*d + 4*B*a*b^4*e - 5*A*b^5*e)*ln(abs(-b*d/(b*x + a) + a*e/(b*x + a) - e))
/(b^7*d^6 - 6*a*b^6*d^5*e + 15*a^2*b^5*d^4*e^2 - 20*a^3*b^4*d^3*e^3 + 15*a^4*b^3
*d^2*e^4 - 6*a^5*b^2*d*e^5 + a^6*b*e^6) + (B*a*b^8/(b*x + a) - A*b^9/(b*x + a))/
(b^10*d^5 - 5*a*b^9*d^4*e + 10*a^2*b^8*d^3*e^2 - 10*a^3*b^7*d^2*e^3 + 5*a^4*b^6*
d*e^4 - a^5*b^5*e^5) - 1/12*(25*B*b^4*d*e^4 + 52*B*a*b^3*e^5 - 77*A*b^4*e^5 + 4*
(22*B*b^6*d^2*e^3 + 21*B*a*b^5*d*e^4 - 65*A*b^6*d*e^4 - 43*B*a^2*b^4*e^5 + 65*A*
a*b^5*e^5)/((b*x + a)*b) + 12*(9*B*b^8*d^3*e^2 - 2*B*a*b^7*d^2*e^3 - 25*A*b^8*d^
2*e^3 - 23*B*a^2*b^6*d*e^4 + 50*A*a*b^7*d*e^4 + 16*B*a^3*b^5*e^5 - 25*A*a^2*b^6*
e^5)/((b*x + a)^2*b^2) + 24*(2*B*b^10*d^4*e - 3*B*a*b^9*d^3*e^2 - 5*A*b^10*d^3*e
^2 - 3*B*a^2*b^8*d^2*e^3 + 15*A*a*b^9*d^2*e^3 + 7*B*a^3*b^7*d*e^4 - 15*A*a^2*b^8
*d*e^4 - 3*B*a^4*b^6*e^5 + 5*A*a^3*b^7*e^5)/((b*x + a)^3*b^3))/((b*d - a*e)^6*(b
*d/(b*x + a) - a*e/(b*x + a) + e)^4)