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

Optimal. Leaf size=50 \[ \frac{2 B \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{\sqrt{b}}-\frac{2 A \sqrt{a+b x}}{a \sqrt{x}} \]

[Out]

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

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Rubi [A]  time = 0.0561823, antiderivative size = 50, 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 \[ \frac{2 B \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{\sqrt{b}}-\frac{2 A \sqrt{a+b x}}{a \sqrt{x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 5.45555, size = 46, normalized size = 0.92 \[ - \frac{2 A \sqrt{a + b x}}{a \sqrt{x}} + \frac{2 B \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{b} \sqrt{x}} \right )}}{\sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0487142, size = 53, normalized size = 1.06 \[ \frac{2 B \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )}{\sqrt{b}}-\frac{2 A \sqrt{a+b x}}{a \sqrt{x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.023, size = 73, normalized size = 1.5 \[{\frac{1}{a} \left ( B\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a \right ){\frac{1}{\sqrt{b}}}} \right ) xa-2\,A\sqrt{x \left ( bx+a \right ) }\sqrt{b} \right ) \sqrt{bx+a}{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{x \left ( bx+a \right ) }}}{\frac{1}{\sqrt{b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.237359, size = 1, normalized size = 0.02 \[ \left [\frac{B a x \log \left (2 \, \sqrt{b x + a} b \sqrt{x} +{\left (2 \, b x + a\right )} \sqrt{b}\right ) - 2 \, \sqrt{b x + a} A \sqrt{b} \sqrt{x}}{a \sqrt{b} x}, \frac{2 \,{\left (B a x \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right ) - \sqrt{b x + a} A \sqrt{-b} \sqrt{x}\right )}}{a \sqrt{-b} x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 19.2886, size = 44, normalized size = 0.88 \[ - \frac{2 A \sqrt{b} \sqrt{\frac{a}{b x} + 1}}{a} + \frac{2 B \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{\sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 12.743, size = 4, normalized size = 0.08 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x