3.491 \(\int \sqrt{x} \sqrt{a+b x} \, dx\)

Optimal. Leaf size=74 \[ -\frac{a^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{4 b^{3/2}}+\frac{1}{2} x^{3/2} \sqrt{a+b x}+\frac{a \sqrt{x} \sqrt{a+b x}}{4 b} \]

[Out]

(a*Sqrt[x]*Sqrt[a + b*x])/(4*b) + (x^(3/2)*Sqrt[a + b*x])/2 - (a^2*ArcTanh[(Sqrt
[b]*Sqrt[x])/Sqrt[a + b*x]])/(4*b^(3/2))

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Rubi [A]  time = 0.0554198, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{a^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{4 b^{3/2}}+\frac{1}{2} x^{3/2} \sqrt{a+b x}+\frac{a \sqrt{x} \sqrt{a+b x}}{4 b} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x]*Sqrt[a + b*x],x]

[Out]

(a*Sqrt[x]*Sqrt[a + b*x])/(4*b) + (x^(3/2)*Sqrt[a + b*x])/2 - (a^2*ArcTanh[(Sqrt
[b]*Sqrt[x])/Sqrt[a + b*x]])/(4*b^(3/2))

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Rubi in Sympy [A]  time = 7.64583, size = 65, normalized size = 0.88 \[ - \frac{a^{2} \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a + b x}} \right )}}{4 b^{\frac{3}{2}}} - \frac{a \sqrt{x} \sqrt{a + b x}}{4 b} + \frac{\sqrt{x} \left (a + b x\right )^{\frac{3}{2}}}{2 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**2*atanh(sqrt(b)*sqrt(x)/sqrt(a + b*x))/(4*b**(3/2)) - a*sqrt(x)*sqrt(a + b*x
)/(4*b) + sqrt(x)*(a + b*x)**(3/2)/(2*b)

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Mathematica [A]  time = 0.0353239, size = 65, normalized size = 0.88 \[ \frac{\sqrt{b} \sqrt{x} \sqrt{a+b x} (a+2 b x)-a^2 \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )}{4 b^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[x]*Sqrt[a + b*x],x]

[Out]

(Sqrt[b]*Sqrt[x]*Sqrt[a + b*x]*(a + 2*b*x) - a^2*Log[b*Sqrt[x] + Sqrt[b]*Sqrt[a
+ b*x]])/(4*b^(3/2))

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Maple [A]  time = 0.007, size = 81, normalized size = 1.1 \[{\frac{1}{2}{x}^{{\frac{3}{2}}}\sqrt{bx+a}}+{\frac{a}{4\,b}\sqrt{x}\sqrt{bx+a}}-{\frac{{a}^{2}}{8}\sqrt{x \left ( bx+a \right ) }\ln \left ({1 \left ({\frac{a}{2}}+bx \right ){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+ax} \right ){b}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{bx+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(1/2)*(b*x+a)^(1/2),x)

[Out]

1/2*x^(3/2)*(b*x+a)^(1/2)+1/4*a*x^(1/2)*(b*x+a)^(1/2)/b-1/8*a^2/b^(3/2)*(x*(b*x+
a))^(1/2)/x^(1/2)/(b*x+a)^(1/2)*ln((1/2*a+b*x)/b^(1/2)+(b*x^2+a*x)^(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(sqrt(b*x + a)*sqrt(x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.220622, size = 1, normalized size = 0.01 \[ \left [\frac{a^{2} \log \left (-2 \, \sqrt{b x + a} b \sqrt{x} +{\left (2 \, b x + a\right )} \sqrt{b}\right ) + 2 \,{\left (2 \, b x + a\right )} \sqrt{b x + a} \sqrt{b} \sqrt{x}}{8 \, b^{\frac{3}{2}}}, -\frac{a^{2} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right ) -{\left (2 \, b x + a\right )} \sqrt{b x + a} \sqrt{-b} \sqrt{x}}{4 \, \sqrt{-b} b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/8*(a^2*log(-2*sqrt(b*x + a)*b*sqrt(x) + (2*b*x + a)*sqrt(b)) + 2*(2*b*x + a)*
sqrt(b*x + a)*sqrt(b)*sqrt(x))/b^(3/2), -1/4*(a^2*arctan(sqrt(b*x + a)*sqrt(-b)/
(b*sqrt(x))) - (2*b*x + a)*sqrt(b*x + a)*sqrt(-b)*sqrt(x))/(sqrt(-b)*b)]

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Sympy [A]  time = 11.7171, size = 97, normalized size = 1.31 \[ \frac{a^{\frac{3}{2}} \sqrt{x}}{4 b \sqrt{1 + \frac{b x}{a}}} + \frac{3 \sqrt{a} x^{\frac{3}{2}}}{4 \sqrt{1 + \frac{b x}{a}}} - \frac{a^{2} \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{4 b^{\frac{3}{2}}} + \frac{b x^{\frac{5}{2}}}{2 \sqrt{a} \sqrt{1 + \frac{b x}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**(3/2)*sqrt(x)/(4*b*sqrt(1 + b*x/a)) + 3*sqrt(a)*x**(3/2)/(4*sqrt(1 + b*x/a))
- a**2*asinh(sqrt(b)*sqrt(x)/sqrt(a))/(4*b**(3/2)) + b*x**(5/2)/(2*sqrt(a)*sqrt(
1 + b*x/a))

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GIAC/XCAS [A]  time = 12.2479, size = 4, normalized size = 0.05 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x