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

Optimal. Leaf size=79 \[ \frac{1}{3} \sqrt{x} (b x+2)^{5/2}+\frac{5}{6} \sqrt{x} (b x+2)^{3/2}+\frac{5}{2} \sqrt{x} \sqrt{b x+2}+\frac{5 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{\sqrt{b}} \]

[Out]

(5*Sqrt[x]*Sqrt[2 + b*x])/2 + (5*Sqrt[x]*(2 + b*x)^(3/2))/6 + (Sqrt[x]*(2 + b*x)
^(5/2))/3 + (5*ArcSinh[(Sqrt[b]*Sqrt[x])/Sqrt[2]])/Sqrt[b]

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Rubi [A]  time = 0.0538055, antiderivative size = 79, 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 \[ \frac{1}{3} \sqrt{x} (b x+2)^{5/2}+\frac{5}{6} \sqrt{x} (b x+2)^{3/2}+\frac{5}{2} \sqrt{x} \sqrt{b x+2}+\frac{5 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{\sqrt{b}} \]

Antiderivative was successfully verified.

[In]  Int[(2 + b*x)^(5/2)/Sqrt[x],x]

[Out]

(5*Sqrt[x]*Sqrt[2 + b*x])/2 + (5*Sqrt[x]*(2 + b*x)^(3/2))/6 + (Sqrt[x]*(2 + b*x)
^(5/2))/3 + (5*ArcSinh[(Sqrt[b]*Sqrt[x])/Sqrt[2]])/Sqrt[b]

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Rubi in Sympy [A]  time = 8.12388, size = 73, normalized size = 0.92 \[ \frac{\sqrt{x} \left (b x + 2\right )^{\frac{5}{2}}}{3} + \frac{5 \sqrt{x} \left (b x + 2\right )^{\frac{3}{2}}}{6} + \frac{5 \sqrt{x} \sqrt{b x + 2}}{2} + \frac{5 \operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{\sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(x)*(b*x + 2)**(5/2)/3 + 5*sqrt(x)*(b*x + 2)**(3/2)/6 + 5*sqrt(x)*sqrt(b*x +
2)/2 + 5*asinh(sqrt(2)*sqrt(b)*sqrt(x)/2)/sqrt(b)

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Mathematica [A]  time = 0.0540694, size = 57, normalized size = 0.72 \[ \frac{1}{6} \sqrt{x} \sqrt{b x+2} \left (2 b^2 x^2+13 b x+33\right )+\frac{5 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{\sqrt{b}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x]*Sqrt[2 + b*x]*(33 + 13*b*x + 2*b^2*x^2))/6 + (5*ArcSinh[(Sqrt[b]*Sqrt[x
])/Sqrt[2]])/Sqrt[b]

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Maple [A]  time = 0.007, size = 84, normalized size = 1.1 \[{\frac{1}{3} \left ( bx+2 \right ) ^{{\frac{5}{2}}}\sqrt{x}}+{\frac{5}{6} \left ( bx+2 \right ) ^{{\frac{3}{2}}}\sqrt{x}}+{\frac{5}{2}\sqrt{x}\sqrt{bx+2}}+{\frac{5}{2}\sqrt{x \left ( bx+2 \right ) }\ln \left ({(bx+1){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+2\,x} \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{bx+2}}}{\frac{1}{\sqrt{b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+2)^(5/2)/x^(1/2),x)

[Out]

1/3*(b*x+2)^(5/2)*x^(1/2)+5/6*(b*x+2)^(3/2)*x^(1/2)+5/2*x^(1/2)*(b*x+2)^(1/2)+5/
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))/
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 + 2)^(5/2)/sqrt(x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.251911, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (2 \, b^{2} x^{2} + 13 \, b x + 33\right )} \sqrt{b x + 2} \sqrt{b} \sqrt{x} + 15 \, \log \left (\sqrt{b x + 2} b \sqrt{x} +{\left (b x + 1\right )} \sqrt{b}\right )}{6 \, \sqrt{b}}, \frac{{\left (2 \, b^{2} x^{2} + 13 \, b x + 33\right )} \sqrt{b x + 2} \sqrt{-b} \sqrt{x} + 30 \, \arctan \left (\frac{\sqrt{b x + 2} \sqrt{-b}}{b \sqrt{x}}\right )}{6 \, \sqrt{-b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/6*((2*b^2*x^2 + 13*b*x + 33)*sqrt(b*x + 2)*sqrt(b)*sqrt(x) + 15*log(sqrt(b*x
+ 2)*b*sqrt(x) + (b*x + 1)*sqrt(b)))/sqrt(b), 1/6*((2*b^2*x^2 + 13*b*x + 33)*sqr
t(b*x + 2)*sqrt(-b)*sqrt(x) + 30*arctan(sqrt(b*x + 2)*sqrt(-b)/(b*sqrt(x))))/sqr
t(-b)]

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Sympy [A]  time = 59.0891, size = 97, normalized size = 1.23 \[ \frac{b^{3} x^{\frac{7}{2}}}{3 \sqrt{b x + 2}} + \frac{17 b^{2} x^{\frac{5}{2}}}{6 \sqrt{b x + 2}} + \frac{59 b x^{\frac{3}{2}}}{6 \sqrt{b x + 2}} + \frac{11 \sqrt{x}}{\sqrt{b x + 2}} + \frac{5 \operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{\sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b**3*x**(7/2)/(3*sqrt(b*x + 2)) + 17*b**2*x**(5/2)/(6*sqrt(b*x + 2)) + 59*b*x**(
3/2)/(6*sqrt(b*x + 2)) + 11*sqrt(x)/sqrt(b*x + 2) + 5*asinh(sqrt(2)*sqrt(b)*sqrt
(x)/2)/sqrt(b)

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

[Out]

Exception raised: NotImplementedError