3.1001 \(\int \frac{(a+b x) (A+B x)}{(d+e x)^3} \, dx\)

Optimal. Leaf size=70 \[ -\frac{(b d-a e) (B d-A e)}{2 e^3 (d+e x)^2}+\frac{-a B e-A b e+2 b B d}{e^3 (d+e x)}+\frac{b B \log (d+e x)}{e^3} \]

[Out]

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

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Rubi [A]  time = 0.11974, antiderivative size = 70, 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 \[ -\frac{(b d-a e) (B d-A e)}{2 e^3 (d+e x)^2}+\frac{-a B e-A b e+2 b B d}{e^3 (d+e x)}+\frac{b B \log (d+e x)}{e^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 19.1757, size = 63, normalized size = 0.9 \[ \frac{B b \log{\left (d + e x \right )}}{e^{3}} - \frac{A b e + B a e - 2 B b d}{e^{3} \left (d + e x\right )} - \frac{\left (A e - B d\right ) \left (a e - b d\right )}{2 e^{3} \left (d + e x\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0574929, size = 72, normalized size = 1.03 \[ \frac{-a e (A e+B (d+2 e x))+b (B d (3 d+4 e x)-A e (d+2 e x))+2 b B (d+e x)^2 \log (d+e x)}{2 e^3 (d+e x)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 118, normalized size = 1.7 \[{\frac{Bb\ln \left ( ex+d \right ) }{{e}^{3}}}-{\frac{Ab}{{e}^{2} \left ( ex+d \right ) }}-{\frac{Ba}{{e}^{2} \left ( ex+d \right ) }}+2\,{\frac{Bbd}{{e}^{3} \left ( ex+d \right ) }}-{\frac{Aa}{2\,e \left ( ex+d \right ) ^{2}}}+{\frac{Abd}{2\,{e}^{2} \left ( ex+d \right ) ^{2}}}+{\frac{Bad}{2\,{e}^{2} \left ( ex+d \right ) ^{2}}}-{\frac{bB{d}^{2}}{2\,{e}^{3} \left ( ex+d \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.34577, size = 117, normalized size = 1.67 \[ \frac{3 \, B b d^{2} - A a e^{2} -{\left (B a + A b\right )} d e + 2 \,{\left (2 \, B b d e -{\left (B a + A b\right )} e^{2}\right )} x}{2 \,{\left (e^{5} x^{2} + 2 \, d e^{4} x + d^{2} e^{3}\right )}} + \frac{B b \log \left (e x + d\right )}{e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.205933, size = 142, normalized size = 2.03 \[ \frac{3 \, B b d^{2} - A a e^{2} -{\left (B a + A b\right )} d e + 2 \,{\left (2 \, B b d e -{\left (B a + A b\right )} e^{2}\right )} x + 2 \,{\left (B b e^{2} x^{2} + 2 \, B b d e x + B b d^{2}\right )} \log \left (e x + d\right )}{2 \,{\left (e^{5} x^{2} + 2 \, d e^{4} x + d^{2} e^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 4.40619, size = 94, normalized size = 1.34 \[ \frac{B b \log{\left (d + e x \right )}}{e^{3}} - \frac{A a e^{2} + A b d e + B a d e - 3 B b d^{2} + x \left (2 A b e^{2} + 2 B a e^{2} - 4 B b d e\right )}{2 d^{2} e^{3} + 4 d e^{4} x + 2 e^{5} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.219398, size = 107, normalized size = 1.53 \[ B b e^{\left (-3\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) + \frac{{\left (2 \,{\left (2 \, B b d - B a e - A b e\right )} x +{\left (3 \, B b d^{2} - B a d e - A b d e - A a e^{2}\right )} e^{\left (-1\right )}\right )} e^{\left (-2\right )}}{2 \,{\left (x e + d\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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