3.1120 \(\int \frac{(A+B x) (d+e x)}{(a+b x)^3} \, dx\)

Optimal. Leaf size=69 \[ -\frac{(A b-a B) (b d-a e)}{2 b^3 (a+b x)^2}-\frac{-2 a B e+A b e+b B d}{b^3 (a+b x)}+\frac{B e \log (a+b x)}{b^3} \]

[Out]

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

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Rubi [A]  time = 0.127622, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ -\frac{(A b-a B) (b d-a e)}{2 b^3 (a+b x)^2}-\frac{-2 a B e+A b e+b B d}{b^3 (a+b x)}+\frac{B e \log (a+b x)}{b^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 19.163, size = 63, normalized size = 0.91 \[ \frac{B e \log{\left (a + b x \right )}}{b^{3}} - \frac{A b e - 2 B a e + B b d}{b^{3} \left (a + b x\right )} + \frac{\left (A b - B a\right ) \left (a e - b d\right )}{2 b^{3} \left (a + b x\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0532225, size = 75, normalized size = 1.09 \[ \frac{B \left (3 a^2 e-a b d+4 a b e x-2 b^2 d x\right )-A b (a e+b d+2 b e x)+2 B e (a+b x)^2 \log (a+b x)}{2 b^3 (a+b x)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 118, normalized size = 1.7 \[{\frac{Be\ln \left ( bx+a \right ) }{{b}^{3}}}-{\frac{Ae}{ \left ( bx+a \right ){b}^{2}}}+2\,{\frac{Bae}{ \left ( bx+a \right ){b}^{3}}}-{\frac{Bd}{ \left ( bx+a \right ){b}^{2}}}+{\frac{Aae}{2\,{b}^{2} \left ( bx+a \right ) ^{2}}}-{\frac{Ad}{2\,b \left ( bx+a \right ) ^{2}}}-{\frac{B{a}^{2}e}{2\,{b}^{3} \left ( bx+a \right ) ^{2}}}+{\frac{Bad}{2\,{b}^{2} \left ( bx+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.34739, size = 124, normalized size = 1.8 \[ -\frac{{\left (B a b + A b^{2}\right )} d -{\left (3 \, B a^{2} - A a b\right )} e + 2 \,{\left (B b^{2} d -{\left (2 \, B a b - A b^{2}\right )} e\right )} x}{2 \,{\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}} + \frac{B e \log \left (b x + a\right )}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.215834, size = 149, normalized size = 2.16 \[ -\frac{{\left (B a b + A b^{2}\right )} d -{\left (3 \, B a^{2} - A a b\right )} e + 2 \,{\left (B b^{2} d -{\left (2 \, B a b - A b^{2}\right )} e\right )} x - 2 \,{\left (B b^{2} e x^{2} + 2 \, B a b e x + B a^{2} e\right )} \log \left (b x + a\right )}{2 \,{\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 4.35124, size = 94, normalized size = 1.36 \[ \frac{B e \log{\left (a + b x \right )}}{b^{3}} + \frac{- A a b e - A b^{2} d + 3 B a^{2} e - B a b d + x \left (- 2 A b^{2} e + 4 B a b e - 2 B b^{2} d\right )}{2 a^{2} b^{3} + 4 a b^{4} x + 2 b^{5} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.232129, size = 104, normalized size = 1.51 \[ \frac{B e{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{3}} - \frac{2 \,{\left (B b d - 2 \, B a e + A b e\right )} x + \frac{B a b d + A b^{2} d - 3 \, B a^{2} e + A a b e}{b}}{2 \,{\left (b x + a\right )}^{2} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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