3.1112 \(\int \frac{(A+B x) \left (b x+c x^2\right )}{(d+e x)^5} \, dx\)

Optimal. Leaf size=116 \[ \frac{-A c e-b B e+3 B c d}{2 e^4 (d+e x)^2}-\frac{B d (3 c d-2 b e)-A e (2 c d-b e)}{3 e^4 (d+e x)^3}+\frac{d (B d-A e) (c d-b e)}{4 e^4 (d+e x)^4}-\frac{B c}{e^4 (d+e x)} \]

[Out]

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

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Rubi [A]  time = 0.245107, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{-A c e-b B e+3 B c d}{2 e^4 (d+e x)^2}-\frac{B d (3 c d-2 b e)-A e (2 c d-b e)}{3 e^4 (d+e x)^3}+\frac{d (B d-A e) (c d-b e)}{4 e^4 (d+e x)^4}-\frac{B c}{e^4 (d+e x)} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(b*x + c*x^2))/(d + e*x)^5,x]

[Out]

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

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Rubi in Sympy [A]  time = 38.3363, size = 110, normalized size = 0.95 \[ - \frac{B c}{e^{4} \left (d + e x\right )} + \frac{d \left (A e - B d\right ) \left (b e - c d\right )}{4 e^{4} \left (d + e x\right )^{4}} - \frac{A c e + B b e - 3 B c d}{2 e^{4} \left (d + e x\right )^{2}} - \frac{A b e^{2} - 2 A c d e - 2 B b d e + 3 B c d^{2}}{3 e^{4} \left (d + e x\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+b*x)/(e*x+d)**5,x)

[Out]

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

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Mathematica [A]  time = 0.0844099, size = 101, normalized size = 0.87 \[ -\frac{A e \left (b e (d+4 e x)+c \left (d^2+4 d e x+6 e^2 x^2\right )\right )+B \left (b e \left (d^2+4 d e x+6 e^2 x^2\right )+3 c \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )\right )}{12 e^4 (d+e x)^4} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(b*x + c*x^2))/(d + e*x)^5,x]

[Out]

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

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Maple [A]  time = 0.009, size = 118, normalized size = 1. \[{\frac{d \left ( Ab{e}^{2}-Acde-Bbde+Bc{d}^{2} \right ) }{4\,{e}^{4} \left ( ex+d \right ) ^{4}}}-{\frac{Ab{e}^{2}-2\,Acde-2\,Bbde+3\,Bc{d}^{2}}{3\,{e}^{4} \left ( ex+d \right ) ^{3}}}-{\frac{Ace+bBe-3\,Bcd}{2\,{e}^{4} \left ( ex+d \right ) ^{2}}}-{\frac{Bc}{{e}^{4} \left ( ex+d \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x)/(e*x+d)^5,x)

[Out]

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

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Maxima [A]  time = 0.703519, size = 189, normalized size = 1.63 \[ -\frac{12 \, B c e^{3} x^{3} + 3 \, B c d^{3} + A b d e^{2} +{\left (B b + A c\right )} d^{2} e + 6 \,{\left (3 \, B c d e^{2} +{\left (B b + A c\right )} e^{3}\right )} x^{2} + 4 \,{\left (3 \, B c d^{2} e + A b e^{3} +{\left (B b + A c\right )} d e^{2}\right )} x}{12 \,{\left (e^{8} x^{4} + 4 \, d e^{7} x^{3} + 6 \, d^{2} e^{6} x^{2} + 4 \, d^{3} e^{5} x + d^{4} e^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)*(B*x + A)/(e*x + d)^5,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.356466, size = 189, normalized size = 1.63 \[ -\frac{12 \, B c e^{3} x^{3} + 3 \, B c d^{3} + A b d e^{2} +{\left (B b + A c\right )} d^{2} e + 6 \,{\left (3 \, B c d e^{2} +{\left (B b + A c\right )} e^{3}\right )} x^{2} + 4 \,{\left (3 \, B c d^{2} e + A b e^{3} +{\left (B b + A c\right )} d e^{2}\right )} x}{12 \,{\left (e^{8} x^{4} + 4 \, d e^{7} x^{3} + 6 \, d^{2} e^{6} x^{2} + 4 \, d^{3} e^{5} x + d^{4} e^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)*(B*x + A)/(e*x + d)^5,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 25.9418, size = 167, normalized size = 1.44 \[ - \frac{A b d e^{2} + A c d^{2} e + B b d^{2} e + 3 B c d^{3} + 12 B c e^{3} x^{3} + x^{2} \left (6 A c e^{3} + 6 B b e^{3} + 18 B c d e^{2}\right ) + x \left (4 A b e^{3} + 4 A c d e^{2} + 4 B b d e^{2} + 12 B c d^{2} e\right )}{12 d^{4} e^{4} + 48 d^{3} e^{5} x + 72 d^{2} e^{6} x^{2} + 48 d e^{7} x^{3} + 12 e^{8} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+b*x)/(e*x+d)**5,x)

[Out]

-(A*b*d*e**2 + A*c*d**2*e + B*b*d**2*e + 3*B*c*d**3 + 12*B*c*e**3*x**3 + x**2*(6
*A*c*e**3 + 6*B*b*e**3 + 18*B*c*d*e**2) + x*(4*A*b*e**3 + 4*A*c*d*e**2 + 4*B*b*d
*e**2 + 12*B*c*d**2*e))/(12*d**4*e**4 + 48*d**3*e**5*x + 72*d**2*e**6*x**2 + 48*
d*e**7*x**3 + 12*e**8*x**4)

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GIAC/XCAS [A]  time = 0.282563, size = 255, normalized size = 2.2 \[ -\frac{1}{12} \,{\left (\frac{12 \, B c e^{8}}{x e + d} - \frac{18 \, B c d e^{8}}{{\left (x e + d\right )}^{2}} + \frac{12 \, B c d^{2} e^{8}}{{\left (x e + d\right )}^{3}} - \frac{3 \, B c d^{3} e^{8}}{{\left (x e + d\right )}^{4}} + \frac{6 \, B b e^{9}}{{\left (x e + d\right )}^{2}} + \frac{6 \, A c e^{9}}{{\left (x e + d\right )}^{2}} - \frac{8 \, B b d e^{9}}{{\left (x e + d\right )}^{3}} - \frac{8 \, A c d e^{9}}{{\left (x e + d\right )}^{3}} + \frac{3 \, B b d^{2} e^{9}}{{\left (x e + d\right )}^{4}} + \frac{3 \, A c d^{2} e^{9}}{{\left (x e + d\right )}^{4}} + \frac{4 \, A b e^{10}}{{\left (x e + d\right )}^{3}} - \frac{3 \, A b d e^{10}}{{\left (x e + d\right )}^{4}}\right )} e^{\left (-12\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)*(B*x + A)/(e*x + d)^5,x, algorithm="giac")

[Out]

-1/12*(12*B*c*e^8/(x*e + d) - 18*B*c*d*e^8/(x*e + d)^2 + 12*B*c*d^2*e^8/(x*e + d
)^3 - 3*B*c*d^3*e^8/(x*e + d)^4 + 6*B*b*e^9/(x*e + d)^2 + 6*A*c*e^9/(x*e + d)^2
- 8*B*b*d*e^9/(x*e + d)^3 - 8*A*c*d*e^9/(x*e + d)^3 + 3*B*b*d^2*e^9/(x*e + d)^4
+ 3*A*c*d^2*e^9/(x*e + d)^4 + 4*A*b*e^10/(x*e + d)^3 - 3*A*b*d*e^10/(x*e + d)^4)
*e^(-12)