3.318 \(\int \frac{A+B x}{\sqrt{x} (a+b x)} \, dx\)

Optimal. Leaf size=49 \[ \frac{2 (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{\sqrt{a} b^{3/2}}+\frac{2 B \sqrt{x}}{b} \]

[Out]

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

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Rubi [A]  time = 0.0599834, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{2 (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{\sqrt{a} b^{3/2}}+\frac{2 B \sqrt{x}}{b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 7.96058, size = 44, normalized size = 0.9 \[ \frac{2 B \sqrt{x}}{b} + \frac{2 \left (A b - B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{\sqrt{a} b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0555414, size = 49, normalized size = 1. \[ \frac{2 B \sqrt{x}}{b}-\frac{2 (a B-A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{\sqrt{a} b^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.009, size = 53, normalized size = 1.1 \[ 2\,{\frac{B\sqrt{x}}{b}}+2\,{\frac{A}{\sqrt{ab}}\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ) }-2\,{\frac{Ba}{b\sqrt{ab}}\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/(b*x+a)/x^(1/2),x)

[Out]

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

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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*x + a)*sqrt(x)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.221423, size = 1, normalized size = 0.02 \[ \left [\frac{2 \, \sqrt{-a b} B \sqrt{x} -{\left (B a - A b\right )} \log \left (\frac{2 \, a b \sqrt{x} + \sqrt{-a b}{\left (b x - a\right )}}{b x + a}\right )}{\sqrt{-a b} b}, \frac{2 \,{\left (\sqrt{a b} B \sqrt{x} +{\left (B a - A b\right )} \arctan \left (\frac{a}{\sqrt{a b} \sqrt{x}}\right )\right )}}{\sqrt{a b} b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 13.7342, size = 202, normalized size = 4.12 \[ A \left (\begin{cases} \frac{\tilde{\infty }}{\sqrt{x}} & \text{for}\: a = 0 \wedge b = 0 \\\frac{2 \sqrt{x}}{a} & \text{for}\: b = 0 \\- \frac{2}{b \sqrt{x}} & \text{for}\: a = 0 \\- \frac{i \log{\left (- i \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{\sqrt{a} b \sqrt{\frac{1}{b}}} + \frac{i \log{\left (i \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{\sqrt{a} b \sqrt{\frac{1}{b}}} & \text{otherwise} \end{cases}\right ) - \frac{2 B a \left (\begin{cases} \frac{\operatorname{atan}{\left (\frac{\sqrt{x}}{\sqrt{\frac{a}{b}}} \right )}}{b \sqrt{\frac{a}{b}}} & \text{for}\: \frac{a}{b} > 0 \\- \frac{\operatorname{acoth}{\left (\frac{\sqrt{x}}{\sqrt{- \frac{a}{b}}} \right )}}{b \sqrt{- \frac{a}{b}}} & \text{for}\: x > - \frac{a}{b} \wedge \frac{a}{b} < 0 \\- \frac{\operatorname{atanh}{\left (\frac{\sqrt{x}}{\sqrt{- \frac{a}{b}}} \right )}}{b \sqrt{- \frac{a}{b}}} & \text{for}\: x < - \frac{a}{b} \wedge \frac{a}{b} < 0 \end{cases}\right )}{b} + \frac{2 B \sqrt{x}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*Piecewise((zoo/sqrt(x), Eq(a, 0) & Eq(b, 0)), (2*sqrt(x)/a, Eq(b, 0)), (-2/(b*
sqrt(x)), Eq(a, 0)), (-I*log(-I*sqrt(a)*sqrt(1/b) + sqrt(x))/(sqrt(a)*b*sqrt(1/b
)) + I*log(I*sqrt(a)*sqrt(1/b) + sqrt(x))/(sqrt(a)*b*sqrt(1/b)), True)) - 2*B*a*
Piecewise((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 + 2*B*sqrt(x)/b

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GIAC/XCAS [A]  time = 0.229671, size = 53, normalized size = 1.08 \[ \frac{2 \, B \sqrt{x}}{b} - \frac{2 \,{\left (B a - A b\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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