3.2023 \(\int \frac{a+b x}{(d+e x)^4 \sqrt{a^2+2 a b x+b^2 x^2}} \, dx\)

Optimal. Leaf size=39 \[ -\frac{a+b x}{3 e \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^3} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.112966, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{a+b x}{3 e \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)/((d + e*x)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 16.6926, size = 36, normalized size = 0.92 \[ - \frac{a + b x}{3 e \left (d + e x\right )^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)/(e*x+d)**4/((b*x+a)**2)**(1/2),x)

[Out]

-(a + b*x)/(3*e*(d + e*x)**3*sqrt(a**2 + 2*a*b*x + b**2*x**2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0216398, size = 30, normalized size = 0.77 \[ -\frac{a+b x}{3 e \sqrt{(a+b x)^2} (d+e x)^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)/((d + e*x)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.005, size = 27, normalized size = 0.7 \[ -{\frac{bx+a}{3\,e \left ( ex+d \right ) ^{3}}{\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)/(e*x+d)^4/((b*x+a)^2)^(1/2),x)

[Out]

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

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)/(sqrt((b*x + a)^2)*(e*x + d)^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.275833, size = 47, normalized size = 1.21 \[ -\frac{1}{3 \,{\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)/(sqrt((b*x + a)^2)*(e*x + d)^4),x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 1.60376, size = 37, normalized size = 0.95 \[ - \frac{1}{3 d^{3} e + 9 d^{2} e^{2} x + 9 d e^{3} x^{2} + 3 e^{4} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)/(e*x+d)**4/((b*x+a)**2)**(1/2),x)

[Out]

-1/(3*d**3*e + 9*d**2*e**2*x + 9*d*e**3*x**2 + 3*e**4*x**3)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.286615, size = 24, normalized size = 0.62 \[ -\frac{e^{\left (-1\right )}{\rm sign}\left (b x + a\right )}{3 \,{\left (x e + d\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)/(sqrt((b*x + a)^2)*(e*x + d)^4),x, algorithm="giac")

[Out]

-1/3*e^(-1)*sign(b*x + a)/(x*e + d)^3