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

Optimal. Leaf size=60 \[ \frac{2 \sqrt{x} (A b-a B)}{a b \sqrt{a+b x}}+\frac{2 B \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{b^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0619321, antiderivative size = 60, 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 \[ \frac{2 \sqrt{x} (A b-a B)}{a b \sqrt{a+b x}}+\frac{2 B \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{b^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 6.4396, size = 53, normalized size = 0.88 \[ \frac{2 B \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a + b x}} \right )}}{b^{\frac{3}{2}}} + \frac{2 \sqrt{x} \left (A b - B a\right )}{a b \sqrt{a + b x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0810571, size = 63, normalized size = 1.05 \[ \frac{2 B \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )}{b^{3/2}}-\frac{2 \sqrt{x} (a B-A b)}{a b \sqrt{a+b x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.025, size = 121, normalized size = 2. \[{\frac{1}{a} \left ( B\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a \right ){\frac{1}{\sqrt{b}}}} \right ) xab+2\,A{b}^{3/2}\sqrt{x \left ( bx+a \right ) }+B\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a \right ){\frac{1}{\sqrt{b}}}} \right ){a}^{2}-2\,Ba\sqrt{b}\sqrt{x \left ( bx+a \right ) } \right ) \sqrt{x}{b}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{x \left ( bx+a \right ) }}}{\frac{1}{\sqrt{bx+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.24141, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{b x + a} B a \sqrt{x} \log \left (2 \, \sqrt{b x + a} b \sqrt{x} +{\left (2 \, b x + a\right )} \sqrt{b}\right ) - 2 \,{\left (B a - A b\right )} \sqrt{b} x}{\sqrt{b x + a} a b^{\frac{3}{2}} \sqrt{x}}, \frac{2 \,{\left (\sqrt{b x + a} B a \sqrt{x} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right ) -{\left (B a - A b\right )} \sqrt{-b} x\right )}}{\sqrt{b x + a} a \sqrt{-b} b \sqrt{x}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 37.2562, size = 68, normalized size = 1.13 \[ \frac{2 A}{a \sqrt{b} \sqrt{\frac{a}{b x} + 1}} + B \left (\frac{2 \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{b^{\frac{3}{2}}} - \frac{2 \sqrt{x}}{\sqrt{a} b \sqrt{1 + \frac{b x}{a}}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.234905, size = 131, normalized size = 2.18 \[ -\frac{B{\rm ln}\left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2}\right )}{\sqrt{b}{\left | b \right |}} - \frac{4 \,{\left (B a \sqrt{b} - A b^{\frac{3}{2}}\right )}}{{\left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-B*ln((sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2)/(sqrt(b)*abs(b)) - 4*
(B*a*sqrt(b) - A*b^(3/2))/(((sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2
+ a*b)*abs(b))