3.616 \(\int \frac{x^{5/2}}{(2+b x)^{3/2}} \, dx\)

Optimal. Leaf size=86 \[ \frac{15 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{7/2}}-\frac{15 \sqrt{x} \sqrt{b x+2}}{2 b^3}+\frac{5 x^{3/2} \sqrt{b x+2}}{2 b^2}-\frac{2 x^{5/2}}{b \sqrt{b x+2}} \]

[Out]

(-2*x^(5/2))/(b*Sqrt[2 + b*x]) - (15*Sqrt[x]*Sqrt[2 + b*x])/(2*b^3) + (5*x^(3/2)
*Sqrt[2 + b*x])/(2*b^2) + (15*ArcSinh[(Sqrt[b]*Sqrt[x])/Sqrt[2]])/b^(7/2)

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Rubi [A]  time = 0.0673449, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{15 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{7/2}}-\frac{15 \sqrt{x} \sqrt{b x+2}}{2 b^3}+\frac{5 x^{3/2} \sqrt{b x+2}}{2 b^2}-\frac{2 x^{5/2}}{b \sqrt{b x+2}} \]

Antiderivative was successfully verified.

[In]  Int[x^(5/2)/(2 + b*x)^(3/2),x]

[Out]

(-2*x^(5/2))/(b*Sqrt[2 + b*x]) - (15*Sqrt[x]*Sqrt[2 + b*x])/(2*b^3) + (5*x^(3/2)
*Sqrt[2 + b*x])/(2*b^2) + (15*ArcSinh[(Sqrt[b]*Sqrt[x])/Sqrt[2]])/b^(7/2)

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Rubi in Sympy [A]  time = 10.4864, size = 82, normalized size = 0.95 \[ - \frac{2 x^{\frac{5}{2}}}{b \sqrt{b x + 2}} + \frac{5 x^{\frac{3}{2}} \sqrt{b x + 2}}{2 b^{2}} - \frac{15 \sqrt{x} \sqrt{b x + 2}}{2 b^{3}} + \frac{15 \operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(5/2)/(b*x+2)**(3/2),x)

[Out]

-2*x**(5/2)/(b*sqrt(b*x + 2)) + 5*x**(3/2)*sqrt(b*x + 2)/(2*b**2) - 15*sqrt(x)*s
qrt(b*x + 2)/(2*b**3) + 15*asinh(sqrt(2)*sqrt(b)*sqrt(x)/2)/b**(7/2)

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Mathematica [A]  time = 0.0801026, size = 59, normalized size = 0.69 \[ \frac{15 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{7/2}}+\frac{\sqrt{x} \left (b^2 x^2-5 b x-30\right )}{2 b^3 \sqrt{b x+2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(5/2)/(2 + b*x)^(3/2),x]

[Out]

(Sqrt[x]*(-30 - 5*b*x + b^2*x^2))/(2*b^3*Sqrt[2 + b*x]) + (15*ArcSinh[(Sqrt[b]*S
qrt[x])/Sqrt[2]])/b^(7/2)

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Maple [A]  time = 0.036, size = 106, normalized size = 1.2 \[{\frac{bx-7}{2\,{b}^{3}}\sqrt{x}\sqrt{bx+2}}+{1 \left ({\frac{15}{2}\ln \left ({(bx+1){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+2\,x} \right ){b}^{-{\frac{7}{2}}}}-8\,{\frac{1}{{b}^{4}}\sqrt{b \left ( x+2\,{b}^{-1} \right ) ^{2}-2\,x-4\,{b}^{-1}} \left ( x+2\,{b}^{-1} \right ) ^{-1}} \right ) \sqrt{x \left ( bx+2 \right ) }{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{bx+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(5/2)/(b*x+2)^(3/2),x)

[Out]

1/2*(b*x-7)*x^(1/2)*(b*x+2)^(1/2)/b^3+(15/2/b^(7/2)*ln((b*x+1)/b^(1/2)+(b*x^2+2*
x)^(1/2))-8/b^4/(x+2/b)*(b*(x+2/b)^2-2*x-4/b)^(1/2))*(x*(b*x+2))^(1/2)/x^(1/2)/(
b*x+2)^(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(x^(5/2)/(b*x + 2)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.229675, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, \sqrt{b x + 2} \sqrt{x} \log \left (\sqrt{b x + 2} b \sqrt{x} +{\left (b x + 1\right )} \sqrt{b}\right ) +{\left (b^{2} x^{3} - 5 \, b x^{2} - 30 \, x\right )} \sqrt{b}}{2 \, \sqrt{b x + 2} b^{\frac{7}{2}} \sqrt{x}}, \frac{30 \, \sqrt{b x + 2} \sqrt{x} \arctan \left (\frac{\sqrt{b x + 2} \sqrt{-b}}{b \sqrt{x}}\right ) +{\left (b^{2} x^{3} - 5 \, b x^{2} - 30 \, x\right )} \sqrt{-b}}{2 \, \sqrt{b x + 2} \sqrt{-b} b^{3} \sqrt{x}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(5/2)/(b*x + 2)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 80.6611, size = 80, normalized size = 0.93 \[ \frac{x^{\frac{5}{2}}}{2 b \sqrt{b x + 2}} - \frac{5 x^{\frac{3}{2}}}{2 b^{2} \sqrt{b x + 2}} - \frac{15 \sqrt{x}}{b^{3} \sqrt{b x + 2}} + \frac{15 \operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(5/2)/(b*x+2)**(3/2),x)

[Out]

x**(5/2)/(2*b*sqrt(b*x + 2)) - 5*x**(3/2)/(2*b**2*sqrt(b*x + 2)) - 15*sqrt(x)/(b
**3*sqrt(b*x + 2)) + 15*asinh(sqrt(2)*sqrt(b)*sqrt(x)/2)/b**(7/2)

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GIAC/XCAS [A]  time = 0.227263, size = 161, normalized size = 1.87 \[ \frac{{\left (\sqrt{{\left (b x + 2\right )} b - 2 \, b} \sqrt{b x + 2}{\left (\frac{b x + 2}{b^{3}} - \frac{9}{b^{3}}\right )} - \frac{15 \,{\rm ln}\left ({\left (\sqrt{b x + 2} \sqrt{b} - \sqrt{{\left (b x + 2\right )} b - 2 \, b}\right )}^{2}\right )}{b^{\frac{5}{2}}} - \frac{64}{{\left ({\left (\sqrt{b x + 2} \sqrt{b} - \sqrt{{\left (b x + 2\right )} b - 2 \, b}\right )}^{2} + 2 \, b\right )} b^{\frac{3}{2}}}\right )}{\left | b \right |}}{2 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(5/2)/(b*x + 2)^(3/2),x, algorithm="giac")

[Out]

1/2*(sqrt((b*x + 2)*b - 2*b)*sqrt(b*x + 2)*((b*x + 2)/b^3 - 9/b^3) - 15*ln((sqrt
(b*x + 2)*sqrt(b) - sqrt((b*x + 2)*b - 2*b))^2)/b^(5/2) - 64/(((sqrt(b*x + 2)*sq
rt(b) - sqrt((b*x + 2)*b - 2*b))^2 + 2*b)*b^(3/2)))*abs(b)/b^2