3.1540 \(\int \frac{\sqrt{a^2+2 a b x+b^2 x^2}}{(d+e x)^4} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.126945, 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 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{3 e^2 (a+b x) (d+e x)^3}-\frac{b \sqrt{a^2+2 a b x+b^2 x^2}}{2 e^2 (a+b x) (d+e x)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 12.6675, size = 73, normalized size = 0.79 \[ - \frac{\sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{2 e \left (d + e x\right )^{3}} + \frac{\left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{6 e^{2} \left (a + b x\right ) \left (d + e x\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(a**2 + 2*a*b*x + b**2*x**2)/(2*e*(d + e*x)**3) + (a*e - b*d)*sqrt(a**2 + 2
*a*b*x + b**2*x**2)/(6*e**2*(a + b*x)*(d + e*x)**3)

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Mathematica [A]  time = 0.0405863, size = 45, normalized size = 0.49 \[ -\frac{\sqrt{(a+b x)^2} (2 a e+b (d+3 e x))}{6 e^2 (a+b x) (d+e x)^3} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[(a + b*x)^2]*(2*a*e + b*(d + 3*e*x)))/(6*e^2*(a + b*x)*(d + e*x)^3)

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Maple [A]  time = 0.004, size = 42, normalized size = 0.5 \[ -{\frac{3\,bex+2\,ae+bd}{6\,{e}^{2} \left ( ex+d \right ) ^{3} \left ( bx+a \right ) }\sqrt{ \left ( bx+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.204741, size = 68, normalized size = 0.74 \[ -\frac{3 \, b e x + b d + 2 \, a e}{6 \,{\left (e^{5} x^{3} + 3 \, d e^{4} x^{2} + 3 \, d^{2} e^{3} x + d^{3} e^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.0343, size = 53, normalized size = 0.58 \[ - \frac{2 a e + b d + 3 b e x}{6 d^{3} e^{2} + 18 d^{2} e^{3} x + 18 d e^{4} x^{2} + 6 e^{5} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(2*a*e + b*d + 3*b*e*x)/(6*d**3*e**2 + 18*d**2*e**3*x + 18*d*e**4*x**2 + 6*e**5
*x**3)

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GIAC/XCAS [A]  time = 0.21189, size = 61, normalized size = 0.66 \[ -\frac{{\left (3 \, b x e{\rm sign}\left (b x + a\right ) + b d{\rm sign}\left (b x + a\right ) + 2 \, a e{\rm sign}\left (b x + a\right )\right )} e^{\left (-2\right )}}{6 \,{\left (x e + d\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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