3.1541 \(\int \frac{\sqrt{a^2+2 a b x+b^2 x^2}}{(d+e x)^5} \, dx\)

Optimal. Leaf size=92 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{4 e^2 (a+b x) (d+e x)^4}-\frac{b \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^2 (a+b x) (d+e x)^3} \]

[Out]

((b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*e^2*(a + b*x)*(d + e*x)^4) - (b*S
qrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^2*(a + b*x)*(d + e*x)^3)

_______________________________________________________________________________________

Rubi [A]  time = 0.125727, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{4 e^2 (a+b x) (d+e x)^4}-\frac{b \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^2 (a+b x) (d+e x)^3} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a^2 + 2*a*b*x + b^2*x^2]/(d + e*x)^5,x]

[Out]

((b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*e^2*(a + b*x)*(d + e*x)^4) - (b*S
qrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^2*(a + b*x)*(d + e*x)^3)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 12.8726, size = 73, normalized size = 0.79 \[ - \frac{\sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{3 e \left (d + e x\right )^{4}} + \frac{\left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{12 e^{2} \left (a + b x\right ) \left (d + e x\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(((b*x+a)**2)**(1/2)/(e*x+d)**5,x)

[Out]

-sqrt(a**2 + 2*a*b*x + b**2*x**2)/(3*e*(d + e*x)**4) + (a*e - b*d)*sqrt(a**2 + 2
*a*b*x + b**2*x**2)/(12*e**2*(a + b*x)*(d + e*x)**4)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0348337, size = 45, normalized size = 0.49 \[ -\frac{\sqrt{(a+b x)^2} (3 a e+b (d+4 e x))}{12 e^2 (a+b x) (d+e x)^4} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[(a + b*x)^2]*(3*a*e + b*(d + 4*e*x)))/(12*e^2*(a + b*x)*(d + e*x)^4)

_______________________________________________________________________________________

Maple [A]  time = 0.005, size = 42, normalized size = 0.5 \[ -{\frac{4\,bex+3\,ae+bd}{12\,{e}^{2} \left ( ex+d \right ) ^{4} \left ( bx+a \right ) }\sqrt{ \left ( bx+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(((b*x+a)^2)^(1/2)/(e*x+d)^5,x)

[Out]

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

_______________________________________________________________________________________

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(sqrt((b*x + a)^2)/(e*x + d)^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.205153, size = 82, normalized size = 0.89 \[ -\frac{4 \, b e x + b d + 3 \, a e}{12 \,{\left (e^{6} x^{4} + 4 \, d e^{5} x^{3} + 6 \, d^{2} e^{4} x^{2} + 4 \, d^{3} e^{3} x + d^{4} e^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 2.38849, size = 65, normalized size = 0.71 \[ - \frac{3 a e + b d + 4 b e x}{12 d^{4} e^{2} + 48 d^{3} e^{3} x + 72 d^{2} e^{4} x^{2} + 48 d e^{5} x^{3} + 12 e^{6} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((b*x+a)**2)**(1/2)/(e*x+d)**5,x)

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.211104, size = 61, normalized size = 0.66 \[ -\frac{{\left (4 \, b x e{\rm sign}\left (b x + a\right ) + b d{\rm sign}\left (b x + a\right ) + 3 \, a e{\rm sign}\left (b x + a\right )\right )} e^{\left (-2\right )}}{12 \,{\left (x e + d\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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