3.1669 \(\int \frac{\sqrt{a^2+2 a b x+b^2 x^2}}{(d+e x)^{3/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.115147, antiderivative size = 92, 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 \[ \frac{2 b \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x}}{e^2 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{e^2 (a+b x) \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 12.126, size = 76, normalized size = 0.83 \[ \frac{2 \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{e \sqrt{d + e x}} - \frac{4 \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{e^{2} \left (a + b x\right ) \sqrt{d + e x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(((b*x+a)**2)**(1/2)/(e*x+d)**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.0474541, size = 46, normalized size = 0.5 \[ -\frac{2 \sqrt{(a+b x)^2} (a e-b (2 d+e x))}{e^2 (a+b x) \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(((b*x+a)^2)^(1/2)/(e*x+d)^(3/2),x)

[Out]

-2/(e*x+d)^(1/2)*(-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 = 0.765362, size = 34, normalized size = 0.37 \[ \frac{2 \,{\left (b e x + 2 \, b d - a e\right )}}{\sqrt{e x + d} e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x + a)^2)/(e*x + d)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.205686, size = 34, normalized size = 0.37 \[ \frac{2 \,{\left (b e x + 2 \, b d - a e\right )}}{\sqrt{e x + d} e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x + a)^2)/(e*x + d)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{\left (a + b x\right )^{2}}}{\left (d + e x\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((b*x+a)**2)**(1/2)/(e*x+d)**(3/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.212497, size = 72, normalized size = 0.78 \[ 2 \, \sqrt{x e + d} b e^{\left (-2\right )}{\rm sign}\left (b x + a\right ) + \frac{2 \,{\left (b d{\rm sign}\left (b x + a\right ) - a e{\rm sign}\left (b x + a\right )\right )} e^{\left (-2\right )}}{\sqrt{x e + d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x + a)^2)/(e*x + d)^(3/2),x, algorithm="giac")

[Out]

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