3.761 \(\int \frac{A+B x}{\sqrt{x} \left (a^2+2 a b x+b^2 x^2\right )} \, dx\)

Optimal. Leaf size=63 \[ \frac{(a B+A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{3/2} b^{3/2}}+\frac{\sqrt{x} (A b-a B)}{a b (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.0775568, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138 \[ \frac{(a B+A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{3/2} b^{3/2}}+\frac{\sqrt{x} (A b-a B)}{a b (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 20.8778, size = 53, normalized size = 0.84 \[ \frac{\sqrt{x} \left (A b - B a\right )}{a b \left (a + b x\right )} + \frac{\left (A b + B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{a^{\frac{3}{2}} b^{\frac{3}{2}}} \]

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)/x**(1/2),x)

[Out]

sqrt(x)*(A*b - B*a)/(a*b*(a + b*x)) + (A*b + B*a)*atan(sqrt(b)*sqrt(x)/sqrt(a))/
(a**(3/2)*b**(3/2))

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Mathematica [A]  time = 0.064859, size = 64, normalized size = 1.02 \[ \frac{(a B+A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{3/2} b^{3/2}}-\frac{\sqrt{x} (a B-A b)}{a b (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.02, size = 69, normalized size = 1.1 \[{\frac{Ab-Ba}{ab \left ( bx+a \right ) }\sqrt{x}}+{\frac{A}{a}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{B}{b}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

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)/x^(1/2),x)

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.297203, size = 1, normalized size = 0.02 \[ \left [-\frac{2 \,{\left (B a - A b\right )} \sqrt{-a b} \sqrt{x} -{\left (B a^{2} + A a b +{\left (B a b + A b^{2}\right )} x\right )} \log \left (\frac{2 \, a b \sqrt{x} + \sqrt{-a b}{\left (b x - a\right )}}{b x + a}\right )}{2 \,{\left (a b^{2} x + a^{2} b\right )} \sqrt{-a b}}, -\frac{{\left (B a - A b\right )} \sqrt{a b} \sqrt{x} +{\left (B a^{2} + A a b +{\left (B a b + A b^{2}\right )} x\right )} \arctan \left (\frac{a}{\sqrt{a b} \sqrt{x}}\right )}{{\left (a b^{2} x + a^{2} b\right )} \sqrt{a b}}\right ] \]

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)*sqrt(x)),x, algorithm="fricas")

[Out]

[-1/2*(2*(B*a - A*b)*sqrt(-a*b)*sqrt(x) - (B*a^2 + A*a*b + (B*a*b + A*b^2)*x)*lo
g((2*a*b*sqrt(x) + sqrt(-a*b)*(b*x - a))/(b*x + a)))/((a*b^2*x + a^2*b)*sqrt(-a*
b)), -((B*a - A*b)*sqrt(a*b)*sqrt(x) + (B*a^2 + A*a*b + (B*a*b + A*b^2)*x)*arcta
n(a/(sqrt(a*b)*sqrt(x))))/((a*b^2*x + a^2*b)*sqrt(a*b))]

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Sympy [A]  time = 39.6527, size = 532, normalized size = 8.44 \[ A \left (\begin{cases} \frac{\tilde{\infty }}{x^{\frac{3}{2}}} & \text{for}\: a = 0 \wedge b = 0 \\\frac{2 \sqrt{x}}{a^{2}} & \text{for}\: b = 0 \\- \frac{2}{3 b^{2} x^{\frac{3}{2}}} & \text{for}\: a = 0 \\\frac{2 i \sqrt{a} b \sqrt{x} \sqrt{\frac{1}{b}}}{2 i a^{\frac{5}{2}} b \sqrt{\frac{1}{b}} + 2 i a^{\frac{3}{2}} b^{2} x \sqrt{\frac{1}{b}}} + \frac{a \log{\left (- i \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{2 i a^{\frac{5}{2}} b \sqrt{\frac{1}{b}} + 2 i a^{\frac{3}{2}} b^{2} x \sqrt{\frac{1}{b}}} - \frac{a \log{\left (i \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{2 i a^{\frac{5}{2}} b \sqrt{\frac{1}{b}} + 2 i a^{\frac{3}{2}} b^{2} x \sqrt{\frac{1}{b}}} + \frac{b x \log{\left (- i \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{2 i a^{\frac{5}{2}} b \sqrt{\frac{1}{b}} + 2 i a^{\frac{3}{2}} b^{2} x \sqrt{\frac{1}{b}}} - \frac{b x \log{\left (i \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{2 i a^{\frac{5}{2}} b \sqrt{\frac{1}{b}} + 2 i a^{\frac{3}{2}} b^{2} x \sqrt{\frac{1}{b}}} & \text{otherwise} \end{cases}\right ) - \frac{2 B a \sqrt{x}}{2 a^{2} b + 2 a b^{2} x} + \frac{B a \sqrt{- \frac{1}{a^{3} b}} \log{\left (- a^{2} \sqrt{- \frac{1}{a^{3} b}} + \sqrt{x} \right )}}{2 b} - \frac{B a \sqrt{- \frac{1}{a^{3} b}} \log{\left (a^{2} \sqrt{- \frac{1}{a^{3} b}} + \sqrt{x} \right )}}{2 b} + \frac{2 B \left (\begin{cases} \frac{\operatorname{atan}{\left (\frac{\sqrt{x}}{\sqrt{\frac{a}{b}}} \right )}}{b \sqrt{\frac{a}{b}}} & \text{for}\: \frac{a}{b} > 0 \\- \frac{\operatorname{acoth}{\left (\frac{\sqrt{x}}{\sqrt{- \frac{a}{b}}} \right )}}{b \sqrt{- \frac{a}{b}}} & \text{for}\: x > - \frac{a}{b} \wedge \frac{a}{b} < 0 \\- \frac{\operatorname{atanh}{\left (\frac{\sqrt{x}}{\sqrt{- \frac{a}{b}}} \right )}}{b \sqrt{- \frac{a}{b}}} & \text{for}\: x < - \frac{a}{b} \wedge \frac{a}{b} < 0 \end{cases}\right )}{b} \]

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)/x**(1/2),x)

[Out]

A*Piecewise((zoo/x**(3/2), Eq(a, 0) & Eq(b, 0)), (2*sqrt(x)/a**2, Eq(b, 0)), (-2
/(3*b**2*x**(3/2)), Eq(a, 0)), (2*I*sqrt(a)*b*sqrt(x)*sqrt(1/b)/(2*I*a**(5/2)*b*
sqrt(1/b) + 2*I*a**(3/2)*b**2*x*sqrt(1/b)) + a*log(-I*sqrt(a)*sqrt(1/b) + sqrt(x
))/(2*I*a**(5/2)*b*sqrt(1/b) + 2*I*a**(3/2)*b**2*x*sqrt(1/b)) - a*log(I*sqrt(a)*
sqrt(1/b) + sqrt(x))/(2*I*a**(5/2)*b*sqrt(1/b) + 2*I*a**(3/2)*b**2*x*sqrt(1/b))
+ b*x*log(-I*sqrt(a)*sqrt(1/b) + sqrt(x))/(2*I*a**(5/2)*b*sqrt(1/b) + 2*I*a**(3/
2)*b**2*x*sqrt(1/b)) - b*x*log(I*sqrt(a)*sqrt(1/b) + sqrt(x))/(2*I*a**(5/2)*b*sq
rt(1/b) + 2*I*a**(3/2)*b**2*x*sqrt(1/b)), True)) - 2*B*a*sqrt(x)/(2*a**2*b + 2*a
*b**2*x) + B*a*sqrt(-1/(a**3*b))*log(-a**2*sqrt(-1/(a**3*b)) + sqrt(x))/(2*b) -
B*a*sqrt(-1/(a**3*b))*log(a**2*sqrt(-1/(a**3*b)) + sqrt(x))/(2*b) + 2*B*Piecewis
e((atan(sqrt(x)/sqrt(a/b))/(b*sqrt(a/b)), a/b > 0), (-acoth(sqrt(x)/sqrt(-a/b))/
(b*sqrt(-a/b)), (a/b < 0) & (x > -a/b)), (-atanh(sqrt(x)/sqrt(-a/b))/(b*sqrt(-a/
b)), (a/b < 0) & (x < -a/b)))/b

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GIAC/XCAS [A]  time = 0.272159, size = 81, normalized size = 1.29 \[ \frac{{\left (B a + A b\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} a b} - \frac{B a \sqrt{x} - A b \sqrt{x}}{{\left (b x + a\right )} a b} \]

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)*sqrt(x)),x, algorithm="giac")

[Out]

(B*a + A*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a*b) - (B*a*sqrt(x) - A*b*sqr
t(x))/((b*x + a)*a*b)