3.1003 \(\int \frac{(a+b x) (A+B x)}{(d+e x)^5} \, dx\)

Optimal. Leaf size=77 \[ \frac{-a B e-A b e+2 b B d}{3 e^3 (d+e x)^3}-\frac{(b d-a e) (B d-A e)}{4 e^3 (d+e x)^4}-\frac{b B}{2 e^3 (d+e x)^2} \]

[Out]

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

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Rubi [A]  time = 0.128088, antiderivative size = 77, 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 \[ \frac{-a B e-A b e+2 b B d}{3 e^3 (d+e x)^3}-\frac{(b d-a e) (B d-A e)}{4 e^3 (d+e x)^4}-\frac{b B}{2 e^3 (d+e x)^2} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)*(A + B*x))/(d + e*x)^5,x]

[Out]

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

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Rubi in Sympy [A]  time = 19.9748, size = 70, normalized size = 0.91 \[ - \frac{B b}{2 e^{3} \left (d + e x\right )^{2}} - \frac{A b e + B a e - 2 B b d}{3 e^{3} \left (d + e x\right )^{3}} - \frac{\left (A e - B d\right ) \left (a e - b d\right )}{4 e^{3} \left (d + e x\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)*(B*x+A)/(e*x+d)**5,x)

[Out]

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

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Mathematica [A]  time = 0.0545648, size = 62, normalized size = 0.81 \[ -\frac{a e (3 A e+B (d+4 e x))+b \left (A e (d+4 e x)+B \left (d^2+4 d e x+6 e^2 x^2\right )\right )}{12 e^3 (d+e x)^4} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)*(A + B*x))/(d + e*x)^5,x]

[Out]

-(a*e*(3*A*e + B*(d + 4*e*x)) + b*(A*e*(d + 4*e*x) + B*(d^2 + 4*d*e*x + 6*e^2*x^
2)))/(12*e^3*(d + e*x)^4)

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Maple [A]  time = 0.007, size = 79, normalized size = 1. \[ -{\frac{Abe+Bae-2\,Bbd}{3\,{e}^{3} \left ( ex+d \right ) ^{3}}}-{\frac{Bb}{2\,{e}^{3} \left ( ex+d \right ) ^{2}}}-{\frac{aA{e}^{2}-Abde-Bade+bB{d}^{2}}{4\,{e}^{3} \left ( ex+d \right ) ^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(B*x+A)/(e*x+d)^5,x)

[Out]

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

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Maxima [A]  time = 1.33743, size = 138, normalized size = 1.79 \[ -\frac{6 \, B b e^{2} x^{2} + B b d^{2} + 3 \, A a e^{2} +{\left (B a + A b\right )} d e + 4 \,{\left (B b d e +{\left (B a + A b\right )} e^{2}\right )} x}{12 \,{\left (e^{7} x^{4} + 4 \, d e^{6} x^{3} + 6 \, d^{2} e^{5} x^{2} + 4 \, d^{3} e^{4} x + d^{4} e^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)/(e*x + d)^5,x, algorithm="maxima")

[Out]

-1/12*(6*B*b*e^2*x^2 + B*b*d^2 + 3*A*a*e^2 + (B*a + A*b)*d*e + 4*(B*b*d*e + (B*a
+ A*b)*e^2)*x)/(e^7*x^4 + 4*d*e^6*x^3 + 6*d^2*e^5*x^2 + 4*d^3*e^4*x + d^4*e^3)

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Fricas [A]  time = 0.20064, size = 138, normalized size = 1.79 \[ -\frac{6 \, B b e^{2} x^{2} + B b d^{2} + 3 \, A a e^{2} +{\left (B a + A b\right )} d e + 4 \,{\left (B b d e +{\left (B a + A b\right )} e^{2}\right )} x}{12 \,{\left (e^{7} x^{4} + 4 \, d e^{6} x^{3} + 6 \, d^{2} e^{5} x^{2} + 4 \, d^{3} e^{4} x + d^{4} e^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)/(e*x + d)^5,x, algorithm="fricas")

[Out]

-1/12*(6*B*b*e^2*x^2 + B*b*d^2 + 3*A*a*e^2 + (B*a + A*b)*d*e + 4*(B*b*d*e + (B*a
+ A*b)*e^2)*x)/(e^7*x^4 + 4*d*e^6*x^3 + 6*d^2*e^5*x^2 + 4*d^3*e^4*x + d^4*e^3)

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Sympy [A]  time = 11.9947, size = 117, normalized size = 1.52 \[ - \frac{3 A a e^{2} + A b d e + B a d e + B b d^{2} + 6 B b e^{2} x^{2} + x \left (4 A b e^{2} + 4 B a e^{2} + 4 B b d e\right )}{12 d^{4} e^{3} + 48 d^{3} e^{4} x + 72 d^{2} e^{5} x^{2} + 48 d e^{6} x^{3} + 12 e^{7} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)*(B*x+A)/(e*x+d)**5,x)

[Out]

-(3*A*a*e**2 + A*b*d*e + B*a*d*e + B*b*d**2 + 6*B*b*e**2*x**2 + x*(4*A*b*e**2 +
4*B*a*e**2 + 4*B*b*d*e))/(12*d**4*e**3 + 48*d**3*e**4*x + 72*d**2*e**5*x**2 + 48
*d*e**6*x**3 + 12*e**7*x**4)

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GIAC/XCAS [A]  time = 0.219088, size = 166, normalized size = 2.16 \[ -\frac{1}{12} \,{\left (\frac{6 \, B b e}{{\left (x e + d\right )}^{2}} - \frac{8 \, B b d e}{{\left (x e + d\right )}^{3}} + \frac{3 \, B b d^{2} e}{{\left (x e + d\right )}^{4}} + \frac{4 \, B a e^{2}}{{\left (x e + d\right )}^{3}} + \frac{4 \, A b e^{2}}{{\left (x e + d\right )}^{3}} - \frac{3 \, B a d e^{2}}{{\left (x e + d\right )}^{4}} - \frac{3 \, A b d e^{2}}{{\left (x e + d\right )}^{4}} + \frac{3 \, A a e^{3}}{{\left (x e + d\right )}^{4}}\right )} e^{\left (-4\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)/(e*x + d)^5,x, algorithm="giac")

[Out]

-1/12*(6*B*b*e/(x*e + d)^2 - 8*B*b*d*e/(x*e + d)^3 + 3*B*b*d^2*e/(x*e + d)^4 + 4
*B*a*e^2/(x*e + d)^3 + 4*A*b*e^2/(x*e + d)^3 - 3*B*a*d*e^2/(x*e + d)^4 - 3*A*b*d
*e^2/(x*e + d)^4 + 3*A*a*e^3/(x*e + d)^4)*e^(-4)