3.814 \(\int \frac{A+B x}{x^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}} \, dx\)

Optimal. Leaf size=99 \[ -\frac{2 A (a+b x)}{a \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 (a+b x) (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{3/2} \sqrt{b} \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

(-2*A*(a + b*x))/(a*Sqrt[x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (2*(A*b - a*B)*(a +
b*x)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(a^(3/2)*Sqrt[b]*Sqrt[a^2 + 2*a*b*x + b
^2*x^2])

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Rubi [A]  time = 0.187452, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129 \[ -\frac{2 A (a+b x)}{a \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 (a+b x) (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{3/2} \sqrt{b} \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*A*(a + b*x))/(a*Sqrt[x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (2*(A*b - a*B)*(a +
b*x)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(a^(3/2)*Sqrt[b]*Sqrt[a^2 + 2*a*b*x + b
^2*x^2])

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Rubi in Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RecursionError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 0.0702718, size = 81, normalized size = 0.82 \[ \frac{2 (a+b x) (a B-A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{3/2} \sqrt{b} \sqrt{(a+b x)^2}}-\frac{2 A (a+b x)}{a \sqrt{x} \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*A*(a + b*x))/(a*Sqrt[x]*Sqrt[(a + b*x)^2]) + (2*(-(A*b) + a*B)*(a + b*x)*Arc
Tan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(a^(3/2)*Sqrt[b]*Sqrt[(a + b*x)^2])

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Maple [A]  time = 0.011, size = 71, normalized size = 0.7 \[ -2\,{\frac{bx+a}{\sqrt{ \left ( bx+a \right ) ^{2}}a\sqrt{ab}\sqrt{x}} \left ( A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) \sqrt{x}b-B\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) \sqrt{x}a+A\sqrt{ab} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(b*x+a)*(A*arctan(x^(1/2)*b/(a*b)^(1/2))*x^(1/2)*b-B*arctan(x^(1/2)*b/(a*b)^(
1/2))*x^(1/2)*a+A*(a*b)^(1/2))/((b*x+a)^2)^(1/2)/a/(a*b)^(1/2)/x^(1/2)

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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((B*x + A)/(sqrt((b*x + a)^2)*x^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.280073, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (B a - A b\right )} \sqrt{x} \log \left (-\frac{2 \, a b \sqrt{x} - \sqrt{-a b}{\left (b x - a\right )}}{b x + a}\right ) + 2 \, \sqrt{-a b} A}{\sqrt{-a b} a \sqrt{x}}, -\frac{2 \,{\left ({\left (B a - A b\right )} \sqrt{x} \arctan \left (\frac{a}{\sqrt{a b} \sqrt{x}}\right ) + \sqrt{a b} A\right )}}{\sqrt{a b} a \sqrt{x}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-((B*a - A*b)*sqrt(x)*log(-(2*a*b*sqrt(x) - sqrt(-a*b)*(b*x - a))/(b*x + a)) +
2*sqrt(-a*b)*A)/(sqrt(-a*b)*a*sqrt(x)), -2*((B*a - A*b)*sqrt(x)*arctan(a/(sqrt(a
*b)*sqrt(x))) + sqrt(a*b)*A)/(sqrt(a*b)*a*sqrt(x))]

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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)/x**(3/2)/((b*x+a)**2)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.276885, size = 77, normalized size = 0.78 \[ \frac{2 \,{\left (B a{\rm sign}\left (b x + a\right ) - A b{\rm sign}\left (b x + a\right )\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} a} - \frac{2 \, A{\rm sign}\left (b x + a\right )}{a \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(B*a*sign(b*x + a) - A*b*sign(b*x + a))*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)
*a) - 2*A*sign(b*x + a)/(a*sqrt(x))