3.1103 \(\int \frac{A+B x}{(a+b x) (d+e x)^4} \, dx\)

Optimal. Leaf size=146 \[ \frac{b^2 (A b-a B) \log (a+b x)}{(b d-a e)^4}-\frac{b^2 (A b-a B) \log (d+e x)}{(b d-a e)^4}+\frac{b (A b-a B)}{(d+e x) (b d-a e)^3}+\frac{A b-a B}{2 (d+e x)^2 (b d-a e)^2}-\frac{B d-A e}{3 e (d+e x)^3 (b d-a e)} \]

[Out]

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

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Rubi [A]  time = 0.263098, antiderivative size = 146, 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 \[ \frac{b^2 (A b-a B) \log (a+b x)}{(b d-a e)^4}-\frac{b^2 (A b-a B) \log (d+e x)}{(b d-a e)^4}+\frac{b (A b-a B)}{(d+e x) (b d-a e)^3}+\frac{A b-a B}{2 (d+e x)^2 (b d-a e)^2}-\frac{B d-A e}{3 e (d+e x)^3 (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/((a + b*x)*(d + e*x)^4),x]

[Out]

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

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Rubi in Sympy [A]  time = 43.4842, size = 119, normalized size = 0.82 \[ \frac{b^{2} \left (A b - B a\right ) \log{\left (a + b x \right )}}{\left (a e - b d\right )^{4}} - \frac{b^{2} \left (A b - B a\right ) \log{\left (d + e x \right )}}{\left (a e - b d\right )^{4}} - \frac{b \left (A b - B a\right )}{\left (d + e x\right ) \left (a e - b d\right )^{3}} + \frac{A b - B a}{2 \left (d + e x\right )^{2} \left (a e - b d\right )^{2}} - \frac{A e - B d}{3 e \left (d + e x\right )^{3} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/(b*x+a)/(e*x+d)**4,x)

[Out]

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

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Mathematica [A]  time = 0.35529, size = 145, normalized size = 0.99 \[ \frac{b^2 (A b-a B) \log (a+b x)}{(b d-a e)^4}+\frac{b^2 (a B-A b) \log (d+e x)}{(b d-a e)^4}+\frac{b (A b-a B)}{(d+e x) (b d-a e)^3}+\frac{A b-a B}{2 (d+e x)^2 (b d-a e)^2}+\frac{B d-A e}{3 e (d+e x)^3 (a e-b d)} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/((a + b*x)*(d + e*x)^4),x]

[Out]

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

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Maple [A]  time = 0.018, size = 220, normalized size = 1.5 \[ -{\frac{A}{ \left ( 3\,ae-3\,bd \right ) \left ( ex+d \right ) ^{3}}}+{\frac{Bd}{ \left ( 3\,ae-3\,bd \right ) e \left ( ex+d \right ) ^{3}}}-{\frac{{b}^{2}A}{ \left ( ae-bd \right ) ^{3} \left ( ex+d \right ) }}+{\frac{Bba}{ \left ( ae-bd \right ) ^{3} \left ( ex+d \right ) }}+{\frac{Ab}{2\, \left ( ae-bd \right ) ^{2} \left ( ex+d \right ) ^{2}}}-{\frac{Ba}{2\, \left ( ae-bd \right ) ^{2} \left ( ex+d \right ) ^{2}}}-{\frac{{b}^{3}\ln \left ( ex+d \right ) A}{ \left ( ae-bd \right ) ^{4}}}+{\frac{{b}^{2}\ln \left ( ex+d \right ) Ba}{ \left ( ae-bd \right ) ^{4}}}+{\frac{{b}^{3}\ln \left ( bx+a \right ) A}{ \left ( ae-bd \right ) ^{4}}}-{\frac{{b}^{2}\ln \left ( bx+a \right ) Ba}{ \left ( ae-bd \right ) ^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/(b*x+a)/(e*x+d)^4,x)

[Out]

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

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Maxima [A]  time = 1.38131, size = 599, normalized size = 4.1 \[ -\frac{{\left (B a b^{2} - A b^{3}\right )} \log \left (b x + a\right )}{b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}} + \frac{{\left (B a b^{2} - A b^{3}\right )} \log \left (e x + d\right )}{b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}} - \frac{2 \, B b^{2} d^{3} - 2 \, A a^{2} e^{3} + 6 \,{\left (B a b - A b^{2}\right )} e^{3} x^{2} +{\left (5 \, B a b - 11 \, A b^{2}\right )} d^{2} e -{\left (B a^{2} - 7 \, A a b\right )} d e^{2} + 3 \,{\left (5 \,{\left (B a b - A b^{2}\right )} d e^{2} -{\left (B a^{2} - A a b\right )} e^{3}\right )} x}{6 \,{\left (b^{3} d^{6} e - 3 \, a b^{2} d^{5} e^{2} + 3 \, a^{2} b d^{4} e^{3} - a^{3} d^{3} e^{4} +{\left (b^{3} d^{3} e^{4} - 3 \, a b^{2} d^{2} e^{5} + 3 \, a^{2} b d e^{6} - a^{3} e^{7}\right )} x^{3} + 3 \,{\left (b^{3} d^{4} e^{3} - 3 \, a b^{2} d^{3} e^{4} + 3 \, a^{2} b d^{2} e^{5} - a^{3} d e^{6}\right )} x^{2} + 3 \,{\left (b^{3} d^{5} e^{2} - 3 \, a b^{2} d^{4} e^{3} + 3 \, a^{2} b d^{3} e^{4} - a^{3} d^{2} e^{5}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((b*x + a)*(e*x + d)^4),x, algorithm="maxima")

[Out]

-(B*a*b^2 - A*b^3)*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*a*b^2 - A*b^3)*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) - 1/6*(2*B*b^2*d^3 - 2*A*a^2*e^
3 + 6*(B*a*b - A*b^2)*e^3*x^2 + (5*B*a*b - 11*A*b^2)*d^2*e - (B*a^2 - 7*A*a*b)*d
*e^2 + 3*(5*(B*a*b - A*b^2)*d*e^2 - (B*a^2 - A*a*b)*e^3)*x)/(b^3*d^6*e - 3*a*b^2
*d^5*e^2 + 3*a^2*b*d^4*e^3 - a^3*d^3*e^4 + (b^3*d^3*e^4 - 3*a*b^2*d^2*e^5 + 3*a^
2*b*d*e^6 - a^3*e^7)*x^3 + 3*(b^3*d^4*e^3 - 3*a*b^2*d^3*e^4 + 3*a^2*b*d^2*e^5 -
a^3*d*e^6)*x^2 + 3*(b^3*d^5*e^2 - 3*a*b^2*d^4*e^3 + 3*a^2*b*d^3*e^4 - a^3*d^2*e^
5)*x)

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Fricas [A]  time = 0.219307, size = 821, normalized size = 5.62 \[ -\frac{2 \, B b^{3} d^{4} + 2 \, A a^{3} e^{4} +{\left (3 \, B a b^{2} - 11 \, A b^{3}\right )} d^{3} e - 6 \,{\left (B a^{2} b - 3 \, A a b^{2}\right )} d^{2} e^{2} +{\left (B a^{3} - 9 \, A a^{2} b\right )} d e^{3} + 6 \,{\left ({\left (B a b^{2} - A b^{3}\right )} d e^{3} -{\left (B a^{2} b - A a b^{2}\right )} e^{4}\right )} x^{2} + 3 \,{\left (5 \,{\left (B a b^{2} - A b^{3}\right )} d^{2} e^{2} - 6 \,{\left (B a^{2} b - A a b^{2}\right )} d e^{3} +{\left (B a^{3} - A a^{2} b\right )} e^{4}\right )} x + 6 \,{\left ({\left (B a b^{2} - A b^{3}\right )} e^{4} x^{3} + 3 \,{\left (B a b^{2} - A b^{3}\right )} d e^{3} x^{2} + 3 \,{\left (B a b^{2} - A b^{3}\right )} d^{2} e^{2} x +{\left (B a b^{2} - A b^{3}\right )} d^{3} e\right )} \log \left (b x + a\right ) - 6 \,{\left ({\left (B a b^{2} - A b^{3}\right )} e^{4} x^{3} + 3 \,{\left (B a b^{2} - A b^{3}\right )} d e^{3} x^{2} + 3 \,{\left (B a b^{2} - A b^{3}\right )} d^{2} e^{2} x +{\left (B a b^{2} - A b^{3}\right )} d^{3} e\right )} \log \left (e x + d\right )}{6 \,{\left (b^{4} d^{7} e - 4 \, a b^{3} d^{6} e^{2} + 6 \, a^{2} b^{2} d^{5} e^{3} - 4 \, a^{3} b d^{4} e^{4} + a^{4} d^{3} e^{5} +{\left (b^{4} d^{4} e^{4} - 4 \, a b^{3} d^{3} e^{5} + 6 \, a^{2} b^{2} d^{2} e^{6} - 4 \, a^{3} b d e^{7} + a^{4} e^{8}\right )} x^{3} + 3 \,{\left (b^{4} d^{5} e^{3} - 4 \, a b^{3} d^{4} e^{4} + 6 \, a^{2} b^{2} d^{3} e^{5} - 4 \, a^{3} b d^{2} e^{6} + a^{4} d e^{7}\right )} x^{2} + 3 \,{\left (b^{4} d^{6} e^{2} - 4 \, a b^{3} d^{5} e^{3} + 6 \, a^{2} b^{2} d^{4} e^{4} - 4 \, a^{3} b d^{3} e^{5} + a^{4} d^{2} e^{6}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((b*x + a)*(e*x + d)^4),x, algorithm="fricas")

[Out]

-1/6*(2*B*b^3*d^4 + 2*A*a^3*e^4 + (3*B*a*b^2 - 11*A*b^3)*d^3*e - 6*(B*a^2*b - 3*
A*a*b^2)*d^2*e^2 + (B*a^3 - 9*A*a^2*b)*d*e^3 + 6*((B*a*b^2 - A*b^3)*d*e^3 - (B*a
^2*b - A*a*b^2)*e^4)*x^2 + 3*(5*(B*a*b^2 - A*b^3)*d^2*e^2 - 6*(B*a^2*b - A*a*b^2
)*d*e^3 + (B*a^3 - A*a^2*b)*e^4)*x + 6*((B*a*b^2 - A*b^3)*e^4*x^3 + 3*(B*a*b^2 -
A*b^3)*d*e^3*x^2 + 3*(B*a*b^2 - A*b^3)*d^2*e^2*x + (B*a*b^2 - A*b^3)*d^3*e)*log
(b*x + a) - 6*((B*a*b^2 - A*b^3)*e^4*x^3 + 3*(B*a*b^2 - A*b^3)*d*e^3*x^2 + 3*(B*
a*b^2 - A*b^3)*d^2*e^2*x + (B*a*b^2 - A*b^3)*d^3*e)*log(e*x + d))/(b^4*d^7*e - 4
*a*b^3*d^6*e^2 + 6*a^2*b^2*d^5*e^3 - 4*a^3*b*d^4*e^4 + a^4*d^3*e^5 + (b^4*d^4*e^
4 - 4*a*b^3*d^3*e^5 + 6*a^2*b^2*d^2*e^6 - 4*a^3*b*d*e^7 + a^4*e^8)*x^3 + 3*(b^4*
d^5*e^3 - 4*a*b^3*d^4*e^4 + 6*a^2*b^2*d^3*e^5 - 4*a^3*b*d^2*e^6 + a^4*d*e^7)*x^2
+ 3*(b^4*d^6*e^2 - 4*a*b^3*d^5*e^3 + 6*a^2*b^2*d^4*e^4 - 4*a^3*b*d^3*e^5 + a^4*
d^2*e^6)*x)

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Sympy [A]  time = 11.9238, size = 818, normalized size = 5.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)/(b*x+a)/(e*x+d)**4,x)

[Out]

b**2*(-A*b + B*a)*log(x + (-A*a*b**3*e - A*b**4*d + B*a**2*b**2*e + B*a*b**3*d -
a**5*b**2*e**5*(-A*b + B*a)/(a*e - b*d)**4 + 5*a**4*b**3*d*e**4*(-A*b + B*a)/(a
*e - b*d)**4 - 10*a**3*b**4*d**2*e**3*(-A*b + B*a)/(a*e - b*d)**4 + 10*a**2*b**5
*d**3*e**2*(-A*b + B*a)/(a*e - b*d)**4 - 5*a*b**6*d**4*e*(-A*b + B*a)/(a*e - b*d
)**4 + b**7*d**5*(-A*b + B*a)/(a*e - b*d)**4)/(-2*A*b**4*e + 2*B*a*b**3*e))/(a*e
- b*d)**4 - b**2*(-A*b + B*a)*log(x + (-A*a*b**3*e - A*b**4*d + B*a**2*b**2*e +
B*a*b**3*d + a**5*b**2*e**5*(-A*b + B*a)/(a*e - b*d)**4 - 5*a**4*b**3*d*e**4*(-
A*b + B*a)/(a*e - b*d)**4 + 10*a**3*b**4*d**2*e**3*(-A*b + B*a)/(a*e - b*d)**4 -
10*a**2*b**5*d**3*e**2*(-A*b + B*a)/(a*e - b*d)**4 + 5*a*b**6*d**4*e*(-A*b + B*
a)/(a*e - b*d)**4 - b**7*d**5*(-A*b + B*a)/(a*e - b*d)**4)/(-2*A*b**4*e + 2*B*a*
b**3*e))/(a*e - b*d)**4 + (-2*A*a**2*e**3 + 7*A*a*b*d*e**2 - 11*A*b**2*d**2*e -
B*a**2*d*e**2 + 5*B*a*b*d**2*e + 2*B*b**2*d**3 + x**2*(-6*A*b**2*e**3 + 6*B*a*b*
e**3) + x*(3*A*a*b*e**3 - 15*A*b**2*d*e**2 - 3*B*a**2*e**3 + 15*B*a*b*d*e**2))/(
6*a**3*d**3*e**4 - 18*a**2*b*d**4*e**3 + 18*a*b**2*d**5*e**2 - 6*b**3*d**6*e + x
**3*(6*a**3*e**7 - 18*a**2*b*d*e**6 + 18*a*b**2*d**2*e**5 - 6*b**3*d**3*e**4) +
x**2*(18*a**3*d*e**6 - 54*a**2*b*d**2*e**5 + 54*a*b**2*d**3*e**4 - 18*b**3*d**4*
e**3) + x*(18*a**3*d**2*e**5 - 54*a**2*b*d**3*e**4 + 54*a*b**2*d**4*e**3 - 18*b*
*3*d**5*e**2))

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GIAC/XCAS [A]  time = 0.215111, size = 489, normalized size = 3.35 \[ -\frac{{\left (B a b^{3} - A b^{4}\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{5} d^{4} - 4 \, a b^{4} d^{3} e + 6 \, a^{2} b^{3} d^{2} e^{2} - 4 \, a^{3} b^{2} d e^{3} + a^{4} b e^{4}} + \frac{{\left (B a b^{2} e - A b^{3} e\right )}{\rm ln}\left ({\left | x e + d \right |}\right )}{b^{4} d^{4} e - 4 \, a b^{3} d^{3} e^{2} + 6 \, a^{2} b^{2} d^{2} e^{3} - 4 \, a^{3} b d e^{4} + a^{4} e^{5}} - \frac{{\left (2 \, B b^{3} d^{4} + 3 \, B a b^{2} d^{3} e - 11 \, A b^{3} d^{3} e - 6 \, B a^{2} b d^{2} e^{2} + 18 \, A a b^{2} d^{2} e^{2} + B a^{3} d e^{3} - 9 \, A a^{2} b d e^{3} + 2 \, A a^{3} e^{4} + 6 \,{\left (B a b^{2} d e^{3} - A b^{3} d e^{3} - B a^{2} b e^{4} + A a b^{2} e^{4}\right )} x^{2} + 3 \,{\left (5 \, B a b^{2} d^{2} e^{2} - 5 \, A b^{3} d^{2} e^{2} - 6 \, B a^{2} b d e^{3} + 6 \, A a b^{2} d e^{3} + B a^{3} e^{4} - A a^{2} b e^{4}\right )} x\right )} e^{\left (-1\right )}}{6 \,{\left (b d - a e\right )}^{4}{\left (x e + d\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((b*x + a)*(e*x + d)^4),x, algorithm="giac")

[Out]

-(B*a*b^3 - A*b^4)*ln(abs(b*x + a))/(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^2*e - A*b^3*e)*ln(abs(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) - 1/6*(2*B*
b^3*d^4 + 3*B*a*b^2*d^3*e - 11*A*b^3*d^3*e - 6*B*a^2*b*d^2*e^2 + 18*A*a*b^2*d^2*
e^2 + B*a^3*d*e^3 - 9*A*a^2*b*d*e^3 + 2*A*a^3*e^4 + 6*(B*a*b^2*d*e^3 - A*b^3*d*e
^3 - B*a^2*b*e^4 + A*a*b^2*e^4)*x^2 + 3*(5*B*a*b^2*d^2*e^2 - 5*A*b^3*d^2*e^2 - 6
*B*a^2*b*d*e^3 + 6*A*a*b^2*d*e^3 + B*a^3*e^4 - A*a^2*b*e^4)*x)*e^(-1)/((b*d - a*
e)^4*(x*e + d)^3)