3.2029 \(\int \frac{a+b x}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=25 \[ -\frac{1}{b \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0142047, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038 \[ -\frac{1}{b \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 6.48289, size = 24, normalized size = 0.96 \[ - \frac{1}{b \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(b*sqrt(a**2 + 2*a*b*x + b**2*x**2))

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Mathematica [A]  time = 0.0101201, size = 16, normalized size = 0.64 \[ -\frac{1}{b \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

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

[Out]

-(1/(b*Sqrt[(a + b*x)^2]))

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Maple [A]  time = 0.005, size = 22, normalized size = 0.9 \[ -{\frac{ \left ( bx+a \right ) ^{2}}{b} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.720352, size = 31, normalized size = 1.24 \[ -\frac{1}{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(sqrt(b^2*x^2 + 2*a*b*x + a^2)*b)

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Fricas [A]  time = 0.276268, size = 18, normalized size = 0.72 \[ -\frac{1}{b^{2} x + a b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(b^2*x + a*b)

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Sympy [A]  time = 2.65489, size = 34, normalized size = 1.36 \[ \begin{cases} - \frac{1}{b \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} & \text{for}\: b \neq 0 \\\frac{a x}{\left (a^{2}\right )^{\frac{3}{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-1/(b*sqrt(a**2 + 2*a*b*x + b**2*x**2)), Ne(b, 0)), (a*x/(a**2)**(3/2
), True))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \mathit{undef} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

undef