3.505 \(\int x^{5/2} \sqrt{2+b x} \, dx\)

Optimal. Leaf size=108 \[ -\frac{5 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{4 b^{7/2}}+\frac{5 \sqrt{x} \sqrt{b x+2}}{8 b^3}-\frac{5 x^{3/2} \sqrt{b x+2}}{24 b^2}+\frac{1}{4} x^{7/2} \sqrt{b x+2}+\frac{x^{5/2} \sqrt{b x+2}}{12 b} \]

[Out]

(5*Sqrt[x]*Sqrt[2 + b*x])/(8*b^3) - (5*x^(3/2)*Sqrt[2 + b*x])/(24*b^2) + (x^(5/2
)*Sqrt[2 + b*x])/(12*b) + (x^(7/2)*Sqrt[2 + b*x])/4 - (5*ArcSinh[(Sqrt[b]*Sqrt[x
])/Sqrt[2]])/(4*b^(7/2))

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Rubi [A]  time = 0.0909795, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{5 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{4 b^{7/2}}+\frac{5 \sqrt{x} \sqrt{b x+2}}{8 b^3}-\frac{5 x^{3/2} \sqrt{b x+2}}{24 b^2}+\frac{1}{4} x^{7/2} \sqrt{b x+2}+\frac{x^{5/2} \sqrt{b x+2}}{12 b} \]

Antiderivative was successfully verified.

[In]  Int[x^(5/2)*Sqrt[2 + b*x],x]

[Out]

(5*Sqrt[x]*Sqrt[2 + b*x])/(8*b^3) - (5*x^(3/2)*Sqrt[2 + b*x])/(24*b^2) + (x^(5/2
)*Sqrt[2 + b*x])/(12*b) + (x^(7/2)*Sqrt[2 + b*x])/4 - (5*ArcSinh[(Sqrt[b]*Sqrt[x
])/Sqrt[2]])/(4*b^(7/2))

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Rubi in Sympy [A]  time = 11.9085, size = 104, normalized size = 0.96 \[ \frac{x^{\frac{5}{2}} \left (b x + 2\right )^{\frac{3}{2}}}{4 b} - \frac{5 x^{\frac{3}{2}} \left (b x + 2\right )^{\frac{3}{2}}}{12 b^{2}} + \frac{5 \sqrt{x} \left (b x + 2\right )^{\frac{3}{2}}}{8 b^{3}} - \frac{5 \sqrt{x} \sqrt{b x + 2}}{8 b^{3}} - \frac{5 \operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{4 b^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(5/2)*(b*x+2)**(1/2),x)

[Out]

x**(5/2)*(b*x + 2)**(3/2)/(4*b) - 5*x**(3/2)*(b*x + 2)**(3/2)/(12*b**2) + 5*sqrt
(x)*(b*x + 2)**(3/2)/(8*b**3) - 5*sqrt(x)*sqrt(b*x + 2)/(8*b**3) - 5*asinh(sqrt(
2)*sqrt(b)*sqrt(x)/2)/(4*b**(7/2))

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Mathematica [A]  time = 0.0738889, size = 70, normalized size = 0.65 \[ \frac{\sqrt{x} \sqrt{b x+2} \left (6 b^3 x^3+2 b^2 x^2-5 b x+15\right )}{24 b^3}-\frac{5 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{4 b^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(5/2)*Sqrt[2 + b*x],x]

[Out]

(Sqrt[x]*Sqrt[2 + b*x]*(15 - 5*b*x + 2*b^2*x^2 + 6*b^3*x^3))/(24*b^3) - (5*ArcSi
nh[(Sqrt[b]*Sqrt[x])/Sqrt[2]])/(4*b^(7/2))

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Maple [A]  time = 0.011, size = 108, normalized size = 1. \[{\frac{1}{4\,b}{x}^{{\frac{5}{2}}} \left ( bx+2 \right ) ^{{\frac{3}{2}}}}-{\frac{5}{12\,{b}^{2}}{x}^{{\frac{3}{2}}} \left ( bx+2 \right ) ^{{\frac{3}{2}}}}+{\frac{5}{8\,{b}^{3}} \left ( bx+2 \right ) ^{{\frac{3}{2}}}\sqrt{x}}-{\frac{5}{8\,{b}^{3}}\sqrt{x}\sqrt{bx+2}}-{\frac{5}{8}\sqrt{x \left ( bx+2 \right ) }\ln \left ({(bx+1){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+2\,x} \right ){b}^{-{\frac{7}{2}}}{\frac{1}{\sqrt{bx+2}}}{\frac{1}{\sqrt{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(5/2)*(b*x+2)^(1/2),x)

[Out]

1/4/b*x^(5/2)*(b*x+2)^(3/2)-5/12/b^2*x^(3/2)*(b*x+2)^(3/2)+5/8/b^3*x^(1/2)*(b*x+
2)^(3/2)-5/8*x^(1/2)*(b*x+2)^(1/2)/b^3-5/8/b^(7/2)*(x*(b*x+2))^(1/2)/(b*x+2)^(1/
2)/x^(1/2)*ln((b*x+1)/b^(1/2)+(b*x^2+2*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 + 2)*x^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.227665, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (6 \, b^{3} x^{3} + 2 \, b^{2} x^{2} - 5 \, b x + 15\right )} \sqrt{b x + 2} \sqrt{b} \sqrt{x} + 15 \, \log \left (-\sqrt{b x + 2} b \sqrt{x} +{\left (b x + 1\right )} \sqrt{b}\right )}{24 \, b^{\frac{7}{2}}}, \frac{{\left (6 \, b^{3} x^{3} + 2 \, b^{2} x^{2} - 5 \, b x + 15\right )} \sqrt{b x + 2} \sqrt{-b} \sqrt{x} - 30 \, \arctan \left (\frac{\sqrt{b x + 2} \sqrt{-b}}{b \sqrt{x}}\right )}{24 \, \sqrt{-b} b^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x + 2)*x^(5/2),x, algorithm="fricas")

[Out]

[1/24*((6*b^3*x^3 + 2*b^2*x^2 - 5*b*x + 15)*sqrt(b*x + 2)*sqrt(b)*sqrt(x) + 15*l
og(-sqrt(b*x + 2)*b*sqrt(x) + (b*x + 1)*sqrt(b)))/b^(7/2), 1/24*((6*b^3*x^3 + 2*
b^2*x^2 - 5*b*x + 15)*sqrt(b*x + 2)*sqrt(-b)*sqrt(x) - 30*arctan(sqrt(b*x + 2)*s
qrt(-b)/(b*sqrt(x))))/(sqrt(-b)*b^3)]

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Sympy [A]  time = 101.496, size = 117, normalized size = 1.08 \[ \frac{b x^{\frac{9}{2}}}{4 \sqrt{b x + 2}} + \frac{7 x^{\frac{7}{2}}}{12 \sqrt{b x + 2}} - \frac{x^{\frac{5}{2}}}{24 b \sqrt{b x + 2}} + \frac{5 x^{\frac{3}{2}}}{24 b^{2} \sqrt{b x + 2}} + \frac{5 \sqrt{x}}{4 b^{3} \sqrt{b x + 2}} - \frac{5 \operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{4 b^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(5/2)*(b*x+2)**(1/2),x)

[Out]

b*x**(9/2)/(4*sqrt(b*x + 2)) + 7*x**(7/2)/(12*sqrt(b*x + 2)) - x**(5/2)/(24*b*sq
rt(b*x + 2)) + 5*x**(3/2)/(24*b**2*sqrt(b*x + 2)) + 5*sqrt(x)/(4*b**3*sqrt(b*x +
2)) - 5*asinh(sqrt(2)*sqrt(b)*sqrt(x)/2)/(4*b**(7/2))

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x + 2)*x^(5/2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError