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

Optimal. Leaf size=56 \[ \frac{(2 A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{b^{3/2}}+\frac{B \sqrt{x} \sqrt{a+b x}}{b} \]

[Out]

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

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Rubi [A]  time = 0.070188, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{(2 A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{b^{3/2}}+\frac{B \sqrt{x} \sqrt{a+b x}}{b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 6.00705, size = 51, normalized size = 0.91 \[ \frac{B \sqrt{x} \sqrt{a + b x}}{b} + \frac{2 \left (A b - \frac{B a}{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{b} \sqrt{x}} \right )}}{b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0542538, size = 59, normalized size = 1.05 \[ \frac{(2 A b-a B) \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )}{b^{3/2}}+\frac{B \sqrt{x} \sqrt{a+b x}}{b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.02, size = 101, normalized size = 1.8 \[{\frac{1}{2}\sqrt{x}\sqrt{bx+a} \left ( 2\,A\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) b-B\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a \right ){\frac{1}{\sqrt{b}}}} \right ) a+2\,B\sqrt{x \left ( bx+a \right ) }\sqrt{b} \right ){b}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{x \left ( bx+a \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*x^(1/2)*(b*x+a)^(1/2)/b^(3/2)*(2*A*ln(1/2*(2*(x*(b*x+a))^(1/2)*b^(1/2)+2*b*x
+a)/b^(1/2))*b-B*ln(1/2*(2*(x*(b*x+a))^(1/2)*b^(1/2)+2*b*x+a)/b^(1/2))*a+2*B*(x*
(b*x+a))^(1/2)*b^(1/2))/(x*(b*x+a))^(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)*sqrt(x)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.241709, size = 1, normalized size = 0.02 \[ \left [\frac{2 \, \sqrt{b x + a} B \sqrt{b} \sqrt{x} -{\left (B a - 2 \, A b\right )} \log \left (2 \, \sqrt{b x + a} b \sqrt{x} +{\left (2 \, b x + a\right )} \sqrt{b}\right )}{2 \, b^{\frac{3}{2}}}, \frac{\sqrt{b x + a} B \sqrt{-b} \sqrt{x} -{\left (B a - 2 \, A b\right )} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right )}{\sqrt{-b} b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*(2*sqrt(b*x + a)*B*sqrt(b)*sqrt(x) - (B*a - 2*A*b)*log(2*sqrt(b*x + a)*b*sq
rt(x) + (2*b*x + a)*sqrt(b)))/b^(3/2), (sqrt(b*x + a)*B*sqrt(-b)*sqrt(x) - (B*a
- 2*A*b)*arctan(sqrt(b*x + a)*sqrt(-b)/(b*sqrt(x))))/(sqrt(-b)*b)]

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Sympy [A]  time = 17.0874, size = 73, normalized size = 1.3 \[ \frac{2 A \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{\sqrt{b}} + \frac{B \sqrt{a} \sqrt{x} \sqrt{1 + \frac{b x}{a}}}{b} - \frac{B a \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*A*asinh(sqrt(b)*sqrt(x)/sqrt(a))/sqrt(b) + B*sqrt(a)*sqrt(x)*sqrt(1 + b*x/a)/b
- B*a*asinh(sqrt(b)*sqrt(x)/sqrt(a))/b**(3/2)

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GIAC/XCAS [A]  time = 12.746, size = 4, normalized size = 0.07 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x