3.459 \(\int \frac{a+c x^2}{(d+e x)^5} \, dx\)

Optimal. Leaf size=57 \[ -\frac{a e^2+c d^2}{4 e^3 (d+e x)^4}-\frac{c}{2 e^3 (d+e x)^2}+\frac{2 c d}{3 e^3 (d+e x)^3} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0846528, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ -\frac{a e^2+c d^2}{4 e^3 (d+e x)^4}-\frac{c}{2 e^3 (d+e x)^2}+\frac{2 c d}{3 e^3 (d+e x)^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + c*x^2)/(d + e*x)^5,x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 13.0632, size = 51, normalized size = 0.89 \[ \frac{2 c d}{3 e^{3} \left (d + e x\right )^{3}} - \frac{c}{2 e^{3} \left (d + e x\right )^{2}} - \frac{a e^{2} + c d^{2}}{4 e^{3} \left (d + e x\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0290433, size = 40, normalized size = 0.7 \[ -\frac{3 a e^2+c \left (d^2+4 d e x+6 e^2 x^2\right )}{12 e^3 (d+e x)^4} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.008, size = 52, normalized size = 0.9 \[ -{\frac{c}{2\,{e}^{3} \left ( ex+d \right ) ^{2}}}+{\frac{2\,cd}{3\,{e}^{3} \left ( ex+d \right ) ^{3}}}-{\frac{a{e}^{2}+c{d}^{2}}{4\,{e}^{3} \left ( ex+d \right ) ^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+a)/(e*x+d)^5,x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 0.707306, size = 101, normalized size = 1.77 \[ -\frac{6 \, c e^{2} x^{2} + 4 \, c d e x + c d^{2} + 3 \, a e^{2}}{12 \,{\left (e^{7} x^{4} + 4 \, d e^{6} x^{3} + 6 \, d^{2} e^{5} x^{2} + 4 \, d^{3} e^{4} x + d^{4} e^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/12*(6*c*e^2*x^2 + 4*c*d*e*x + c*d^2 + 3*a*e^2)/(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)

_______________________________________________________________________________________

Fricas [A]  time = 0.205127, size = 101, normalized size = 1.77 \[ -\frac{6 \, c e^{2} x^{2} + 4 \, c d e x + c d^{2} + 3 \, a e^{2}}{12 \,{\left (e^{7} x^{4} + 4 \, d e^{6} x^{3} + 6 \, d^{2} e^{5} x^{2} + 4 \, d^{3} e^{4} x + d^{4} e^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/12*(6*c*e^2*x^2 + 4*c*d*e*x + c*d^2 + 3*a*e^2)/(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)

_______________________________________________________________________________________

Sympy [A]  time = 2.80733, size = 80, normalized size = 1.4 \[ - \frac{3 a e^{2} + c d^{2} + 4 c d e x + 6 c e^{2} x^{2}}{12 d^{4} e^{3} + 48 d^{3} e^{4} x + 72 d^{2} e^{5} x^{2} + 48 d e^{6} x^{3} + 12 e^{7} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(3*a*e**2 + c*d**2 + 4*c*d*e*x + 6*c*e**2*x**2)/(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)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.211404, size = 82, normalized size = 1.44 \[ -\frac{1}{12} \,{\left (\frac{6 \, c e}{{\left (x e + d\right )}^{2}} - \frac{8 \, c d e}{{\left (x e + d\right )}^{3}} + \frac{3 \, c d^{2} e}{{\left (x e + d\right )}^{4}} + \frac{3 \, a e^{3}}{{\left (x e + d\right )}^{4}}\right )} e^{\left (-4\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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