3.1670 \(\int \frac{\sqrt{a^2+2 a b x+b^2 x^2}}{(d+e x)^{5/2}} \, dx\)

Optimal. Leaf size=94 \[ \frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{3 e^2 (a+b x) (d+e x)^{3/2}}-\frac{2 b \sqrt{a^2+2 a b x+b^2 x^2}}{e^2 (a+b x) \sqrt{d+e x}} \]

[Out]

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

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Rubi [A]  time = 0.115679, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{3 e^2 (a+b x) (d+e x)^{3/2}}-\frac{2 b \sqrt{a^2+2 a b x+b^2 x^2}}{e^2 (a+b x) \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 12.7989, size = 78, normalized size = 0.83 \[ - \frac{2 \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{e \left (d + e x\right )^{\frac{3}{2}}} + \frac{4 \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{3 e^{2} \left (a + b x\right ) \left (d + e x\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0419418, size = 47, normalized size = 0.5 \[ -\frac{2 \sqrt{(a+b x)^2} (a e+2 b d+3 b e x)}{3 e^2 (a+b x) (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.004, size = 42, normalized size = 0.5 \[ -{\frac{6\,bex+2\,ae+4\,bd}{3\, \left ( bx+a \right ){e}^{2}}\sqrt{ \left ( bx+a \right ) ^{2}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.75572, size = 47, normalized size = 0.5 \[ -\frac{2 \,{\left (3 \, b e x + 2 \, b d + a e\right )}}{3 \,{\left (e^{3} x + d e^{2}\right )} \sqrt{e x + d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.207087, size = 47, normalized size = 0.5 \[ -\frac{2 \,{\left (3 \, b e x + 2 \, b d + a e\right )}}{3 \,{\left (e^{3} x + d e^{2}\right )} \sqrt{e x + d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.213652, size = 65, normalized size = 0.69 \[ -\frac{2 \,{\left (3 \,{\left (x e + d\right )} b{\rm sign}\left (b x + a\right ) - b d{\rm sign}\left (b x + a\right ) + a e{\rm sign}\left (b x + a\right )\right )} e^{\left (-2\right )}}{3 \,{\left (x e + d\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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