3.523 \(\int \sqrt{x} (a+b x)^{3/2} \, dx\)

Optimal. Leaf size=95 \[ -\frac{a^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{8 b^{3/2}}+\frac{a^2 \sqrt{x} \sqrt{a+b x}}{8 b}+\frac{1}{4} a x^{3/2} \sqrt{a+b x}+\frac{1}{3} x^{3/2} (a+b x)^{3/2} \]

[Out]

(a^2*Sqrt[x]*Sqrt[a + b*x])/(8*b) + (a*x^(3/2)*Sqrt[a + b*x])/4 + (x^(3/2)*(a +
b*x)^(3/2))/3 - (a^3*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a + b*x]])/(8*b^(3/2))

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Rubi [A]  time = 0.070938, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{a^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{8 b^{3/2}}+\frac{a^2 \sqrt{x} \sqrt{a+b x}}{8 b}+\frac{1}{4} a x^{3/2} \sqrt{a+b x}+\frac{1}{3} x^{3/2} (a+b x)^{3/2} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x]*(a + b*x)^(3/2),x]

[Out]

(a^2*Sqrt[x]*Sqrt[a + b*x])/(8*b) + (a*x^(3/2)*Sqrt[a + b*x])/4 + (x^(3/2)*(a +
b*x)^(3/2))/3 - (a^3*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a + b*x]])/(8*b^(3/2))

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Rubi in Sympy [A]  time = 11.472, size = 85, normalized size = 0.89 \[ - \frac{a^{3} \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{b} \sqrt{x}} \right )}}{8 b^{\frac{3}{2}}} - \frac{a^{2} \sqrt{x} \sqrt{a + b x}}{8 b} - \frac{a \sqrt{x} \left (a + b x\right )^{\frac{3}{2}}}{12 b} + \frac{\sqrt{x} \left (a + b x\right )^{\frac{5}{2}}}{3 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**3*atanh(sqrt(a + b*x)/(sqrt(b)*sqrt(x)))/(8*b**(3/2)) - a**2*sqrt(x)*sqrt(a
+ b*x)/(8*b) - a*sqrt(x)*(a + b*x)**(3/2)/(12*b) + sqrt(x)*(a + b*x)**(5/2)/(3*b
)

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Mathematica [A]  time = 0.0477117, size = 78, normalized size = 0.82 \[ \frac{\sqrt{b} \sqrt{x} \sqrt{a+b x} \left (3 a^2+14 a b x+8 b^2 x^2\right )-3 a^3 \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )}{24 b^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[b]*Sqrt[x]*Sqrt[a + b*x]*(3*a^2 + 14*a*b*x + 8*b^2*x^2) - 3*a^3*Log[b*Sqrt
[x] + Sqrt[b]*Sqrt[a + b*x]])/(24*b^(3/2))

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Maple [A]  time = 0.009, size = 96, normalized size = 1. \[{\frac{1}{3}{x}^{{\frac{3}{2}}} \left ( bx+a \right ) ^{{\frac{3}{2}}}}+{\frac{a}{4}{x}^{{\frac{3}{2}}}\sqrt{bx+a}}+{\frac{{a}^{2}}{8\,b}\sqrt{x}\sqrt{bx+a}}-{\frac{{a}^{3}}{16}\sqrt{x \left ( bx+a \right ) }\ln \left ({1 \left ({\frac{a}{2}}+bx \right ){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+ax} \right ){b}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{bx+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(3/2)*x^(1/2),x)

[Out]

1/3*x^(3/2)*(b*x+a)^(3/2)+1/4*a*x^(3/2)*(b*x+a)^(1/2)+1/8*a^2*x^(1/2)*(b*x+a)^(1
/2)/b-1/16*a^3/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((b*x + a)^(3/2)*sqrt(x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.221572, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, a^{3} \log \left (-2 \, \sqrt{b x + a} b \sqrt{x} +{\left (2 \, b x + a\right )} \sqrt{b}\right ) + 2 \,{\left (8 \, b^{2} x^{2} + 14 \, a b x + 3 \, a^{2}\right )} \sqrt{b x + a} \sqrt{b} \sqrt{x}}{48 \, b^{\frac{3}{2}}}, -\frac{3 \, a^{3} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right ) -{\left (8 \, b^{2} x^{2} + 14 \, a b x + 3 \, a^{2}\right )} \sqrt{b x + a} \sqrt{-b} \sqrt{x}}{24 \, \sqrt{-b} b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/48*(3*a^3*log(-2*sqrt(b*x + a)*b*sqrt(x) + (2*b*x + a)*sqrt(b)) + 2*(8*b^2*x^
2 + 14*a*b*x + 3*a^2)*sqrt(b*x + a)*sqrt(b)*sqrt(x))/b^(3/2), -1/24*(3*a^3*arcta
n(sqrt(b*x + a)*sqrt(-b)/(b*sqrt(x))) - (8*b^2*x^2 + 14*a*b*x + 3*a^2)*sqrt(b*x
+ a)*sqrt(-b)*sqrt(x))/(sqrt(-b)*b)]

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Sympy [A]  time = 24.9127, size = 124, normalized size = 1.31 \[ \frac{a^{\frac{5}{2}} \sqrt{x}}{8 b \sqrt{1 + \frac{b x}{a}}} + \frac{17 a^{\frac{3}{2}} x^{\frac{3}{2}}}{24 \sqrt{1 + \frac{b x}{a}}} + \frac{11 \sqrt{a} b x^{\frac{5}{2}}}{12 \sqrt{1 + \frac{b x}{a}}} - \frac{a^{3} \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{8 b^{\frac{3}{2}}} + \frac{b^{2} x^{\frac{7}{2}}}{3 \sqrt{a} \sqrt{1 + \frac{b x}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**(5/2)*sqrt(x)/(8*b*sqrt(1 + b*x/a)) + 17*a**(3/2)*x**(3/2)/(24*sqrt(1 + b*x/a
)) + 11*sqrt(a)*b*x**(5/2)/(12*sqrt(1 + b*x/a)) - a**3*asinh(sqrt(b)*sqrt(x)/sqr
t(a))/(8*b**(3/2)) + b**2*x**(7/2)/(3*sqrt(a)*sqrt(1 + b*x/a))

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GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out