3.1294 \(\int \frac{(A+B x) \left (a+c x^2\right )}{(d+e x)^5} \, dx\)

Optimal. Leaf size=106 \[ -\frac{a B e^2-2 A c d e+3 B c d^2}{3 e^4 (d+e x)^3}+\frac{\left (a e^2+c d^2\right ) (B d-A e)}{4 e^4 (d+e x)^4}+\frac{c (3 B d-A e)}{2 e^4 (d+e x)^2}-\frac{B c}{e^4 (d+e x)} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.172787, antiderivative size = 106, 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 \[ -\frac{a B e^2-2 A c d e+3 B c d^2}{3 e^4 (d+e x)^3}+\frac{\left (a e^2+c d^2\right ) (B d-A e)}{4 e^4 (d+e x)^4}+\frac{c (3 B d-A e)}{2 e^4 (d+e x)^2}-\frac{B c}{e^4 (d+e x)} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 28.4649, size = 99, normalized size = 0.93 \[ - \frac{B c}{e^{4} \left (d + e x\right )} - \frac{c \left (A e - 3 B d\right )}{2 e^{4} \left (d + e x\right )^{2}} - \frac{- 2 A c d e + B a e^{2} + 3 B c d^{2}}{3 e^{4} \left (d + e x\right )^{3}} - \frac{\left (A e - B d\right ) \left (a e^{2} + c d^{2}\right )}{4 e^{4} \left (d + e x\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0795001, size = 87, normalized size = 0.82 \[ -\frac{3 a A e^3+a B e^2 (d+4 e x)+A c e \left (d^2+4 d e x+6 e^2 x^2\right )+3 B c \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )}{12 e^4 (d+e x)^4} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.009, size = 110, normalized size = 1. \[ -{\frac{aA{e}^{3}+Ac{d}^{2}e-aBd{e}^{2}-Bc{d}^{3}}{4\,{e}^{4} \left ( ex+d \right ) ^{4}}}-{\frac{-2\,Acde+aB{e}^{2}+3\,Bc{d}^{2}}{3\,{e}^{4} \left ( ex+d \right ) ^{3}}}-{\frac{c \left ( Ae-3\,Bd \right ) }{2\,{e}^{4} \left ( ex+d \right ) ^{2}}}-{\frac{Bc}{{e}^{4} \left ( ex+d \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 0.707898, size = 178, normalized size = 1.68 \[ -\frac{12 \, B c e^{3} x^{3} + 3 \, B c d^{3} + A c d^{2} e + B a d e^{2} + 3 \, A a e^{3} + 6 \,{\left (3 \, B c d e^{2} + A c e^{3}\right )} x^{2} + 4 \,{\left (3 \, B c d^{2} e + A c d e^{2} + B a e^{3}\right )} x}{12 \,{\left (e^{8} x^{4} + 4 \, d e^{7} x^{3} + 6 \, d^{2} e^{6} x^{2} + 4 \, d^{3} e^{5} x + d^{4} e^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)*(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*c*d^2*e + B*a*d*e^2 + 3*A*a*e^3 + 6*(3*B*c
*d*e^2 + A*c*e^3)*x^2 + 4*(3*B*c*d^2*e + A*c*d*e^2 + B*a*e^3)*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)

_______________________________________________________________________________________

Fricas [A]  time = 0.270358, size = 178, normalized size = 1.68 \[ -\frac{12 \, B c e^{3} x^{3} + 3 \, B c d^{3} + A c d^{2} e + B a d e^{2} + 3 \, A a e^{3} + 6 \,{\left (3 \, B c d e^{2} + A c e^{3}\right )} x^{2} + 4 \,{\left (3 \, B c d^{2} e + A c d e^{2} + B a e^{3}\right )} x}{12 \,{\left (e^{8} x^{4} + 4 \, d e^{7} x^{3} + 6 \, d^{2} e^{6} x^{2} + 4 \, d^{3} e^{5} x + d^{4} e^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)*(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*c*d^2*e + B*a*d*e^2 + 3*A*a*e^3 + 6*(3*B*c
*d*e^2 + A*c*e^3)*x^2 + 4*(3*B*c*d^2*e + A*c*d*e^2 + B*a*e^3)*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)

_______________________________________________________________________________________

Sympy [A]  time = 21.3971, size = 148, normalized size = 1.4 \[ - \frac{3 A a e^{3} + A c d^{2} e + B a d e^{2} + 3 B c d^{3} + 12 B c e^{3} x^{3} + x^{2} \left (6 A c e^{3} + 18 B c d e^{2}\right ) + x \left (4 A c d e^{2} + 4 B a e^{3} + 12 B c d^{2} e\right )}{12 d^{4} e^{4} + 48 d^{3} e^{5} x + 72 d^{2} e^{6} x^{2} + 48 d e^{7} x^{3} + 12 e^{8} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(3*A*a*e**3 + A*c*d**2*e + B*a*d*e**2 + 3*B*c*d**3 + 12*B*c*e**3*x**3 + x**2*(6
*A*c*e**3 + 18*B*c*d*e**2) + x*(4*A*c*d*e**2 + 4*B*a*e**3 + 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)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.286195, size = 212, normalized size = 2. \[ -\frac{1}{12} \,{\left (\frac{12 \, B c e^{8}}{x e + d} - \frac{18 \, B c d e^{8}}{{\left (x e + d\right )}^{2}} + \frac{12 \, B c d^{2} e^{8}}{{\left (x e + d\right )}^{3}} - \frac{3 \, B c d^{3} e^{8}}{{\left (x e + d\right )}^{4}} + \frac{6 \, A c e^{9}}{{\left (x e + d\right )}^{2}} - \frac{8 \, A c d e^{9}}{{\left (x e + d\right )}^{3}} + \frac{3 \, A c d^{2} e^{9}}{{\left (x e + d\right )}^{4}} + \frac{4 \, B a e^{10}}{{\left (x e + d\right )}^{3}} - \frac{3 \, B a d e^{10}}{{\left (x e + d\right )}^{4}} + \frac{3 \, A a e^{11}}{{\left (x e + d\right )}^{4}}\right )} e^{\left (-12\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)*(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*A*c*e^9/(x*e + d)^2 - 8*A*c*d*e^9/(x*e + d)^
3 + 3*A*c*d^2*e^9/(x*e + d)^4 + 4*B*a*e^10/(x*e + d)^3 - 3*B*a*d*e^10/(x*e + d)^
4 + 3*A*a*e^11/(x*e + d)^4)*e^(-12)