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

Optimal. Leaf size=123 \[ \frac{3 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{4 b^{5/2}}-\frac{3 \sqrt{x} \sqrt{b x+2}}{8 b^2}+\frac{1}{5} x^{5/2} (b x+2)^{5/2}+\frac{1}{4} x^{5/2} (b x+2)^{3/2}+\frac{1}{4} x^{5/2} \sqrt{b x+2}+\frac{x^{3/2} \sqrt{b x+2}}{8 b} \]

[Out]

(-3*Sqrt[x]*Sqrt[2 + b*x])/(8*b^2) + (x^(3/2)*Sqrt[2 + b*x])/(8*b) + (x^(5/2)*Sq
rt[2 + b*x])/4 + (x^(5/2)*(2 + b*x)^(3/2))/4 + (x^(5/2)*(2 + b*x)^(5/2))/5 + (3*
ArcSinh[(Sqrt[b]*Sqrt[x])/Sqrt[2]])/(4*b^(5/2))

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Rubi [A]  time = 0.0872171, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{3 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{4 b^{5/2}}-\frac{3 \sqrt{x} \sqrt{b x+2}}{8 b^2}+\frac{1}{5} x^{5/2} (b x+2)^{5/2}+\frac{1}{4} x^{5/2} (b x+2)^{3/2}+\frac{1}{4} x^{5/2} \sqrt{b x+2}+\frac{x^{3/2} \sqrt{b x+2}}{8 b} \]

Antiderivative was successfully verified.

[In]  Int[x^(3/2)*(2 + b*x)^(5/2),x]

[Out]

(-3*Sqrt[x]*Sqrt[2 + b*x])/(8*b^2) + (x^(3/2)*Sqrt[2 + b*x])/(8*b) + (x^(5/2)*Sq
rt[2 + b*x])/4 + (x^(5/2)*(2 + b*x)^(3/2))/4 + (x^(5/2)*(2 + b*x)^(5/2))/5 + (3*
ArcSinh[(Sqrt[b]*Sqrt[x])/Sqrt[2]])/(4*b^(5/2))

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Rubi in Sympy [A]  time = 15.0231, size = 121, normalized size = 0.98 \[ \frac{x^{\frac{3}{2}} \left (b x + 2\right )^{\frac{7}{2}}}{5 b} - \frac{3 \sqrt{x} \left (b x + 2\right )^{\frac{7}{2}}}{20 b^{2}} + \frac{\sqrt{x} \left (b x + 2\right )^{\frac{5}{2}}}{20 b^{2}} + \frac{\sqrt{x} \left (b x + 2\right )^{\frac{3}{2}}}{8 b^{2}} + \frac{3 \sqrt{x} \sqrt{b x + 2}}{8 b^{2}} + \frac{3 \operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{4 b^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**(3/2)*(b*x + 2)**(7/2)/(5*b) - 3*sqrt(x)*(b*x + 2)**(7/2)/(20*b**2) + sqrt(x)
*(b*x + 2)**(5/2)/(20*b**2) + sqrt(x)*(b*x + 2)**(3/2)/(8*b**2) + 3*sqrt(x)*sqrt
(b*x + 2)/(8*b**2) + 3*asinh(sqrt(2)*sqrt(b)*sqrt(x)/2)/(4*b**(5/2))

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Mathematica [A]  time = 0.079967, size = 78, normalized size = 0.63 \[ \frac{3 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{4 b^{5/2}}+\frac{\sqrt{x} \sqrt{b x+2} \left (8 b^4 x^4+42 b^3 x^3+62 b^2 x^2+5 b x-15\right )}{40 b^2} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x]*Sqrt[2 + b*x]*(-15 + 5*b*x + 62*b^2*x^2 + 42*b^3*x^3 + 8*b^4*x^4))/(40*
b^2) + (3*ArcSinh[(Sqrt[b]*Sqrt[x])/Sqrt[2]])/(4*b^(5/2))

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Maple [A]  time = 0.008, size = 123, normalized size = 1. \[{\frac{1}{5\,b}{x}^{{\frac{3}{2}}} \left ( bx+2 \right ) ^{{\frac{7}{2}}}}-{\frac{3}{20\,{b}^{2}}\sqrt{x} \left ( bx+2 \right ) ^{{\frac{7}{2}}}}+{\frac{1}{20\,{b}^{2}} \left ( bx+2 \right ) ^{{\frac{5}{2}}}\sqrt{x}}+{\frac{1}{8\,{b}^{2}} \left ( bx+2 \right ) ^{{\frac{3}{2}}}\sqrt{x}}+{\frac{3}{8\,{b}^{2}}\sqrt{x}\sqrt{bx+2}}+{\frac{3}{8}\sqrt{x \left ( bx+2 \right ) }\ln \left ({(bx+1){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+2\,x} \right ){b}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{bx+2}}}{\frac{1}{\sqrt{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(3/2)*(b*x+2)^(5/2),x)

[Out]

1/5/b*x^(3/2)*(b*x+2)^(7/2)-3/20/b^2*x^(1/2)*(b*x+2)^(7/2)+1/20/b^2*x^(1/2)*(b*x
+2)^(5/2)+1/8/b^2*x^(1/2)*(b*x+2)^(3/2)+3/8*x^(1/2)*(b*x+2)^(1/2)/b^2+3/8/b^(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))

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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)*x^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.230263, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (8 \, b^{4} x^{4} + 42 \, b^{3} x^{3} + 62 \, b^{2} x^{2} + 5 \, b x - 15\right )} \sqrt{b x + 2} \sqrt{b} \sqrt{x} + 15 \, \log \left (\sqrt{b x + 2} b \sqrt{x} +{\left (b x + 1\right )} \sqrt{b}\right )}{40 \, b^{\frac{5}{2}}}, \frac{{\left (8 \, b^{4} x^{4} + 42 \, b^{3} x^{3} + 62 \, b^{2} x^{2} + 5 \, b x - 15\right )} \sqrt{b x + 2} \sqrt{-b} \sqrt{x} + 30 \, \arctan \left (\frac{\sqrt{b x + 2} \sqrt{-b}}{b \sqrt{x}}\right )}{40 \, \sqrt{-b} b^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/40*((8*b^4*x^4 + 42*b^3*x^3 + 62*b^2*x^2 + 5*b*x - 15)*sqrt(b*x + 2)*sqrt(b)*
sqrt(x) + 15*log(sqrt(b*x + 2)*b*sqrt(x) + (b*x + 1)*sqrt(b)))/b^(5/2), 1/40*((8
*b^4*x^4 + 42*b^3*x^3 + 62*b^2*x^2 + 5*b*x - 15)*sqrt(b*x + 2)*sqrt(-b)*sqrt(x)
+ 30*arctan(sqrt(b*x + 2)*sqrt(-b)/(b*sqrt(x))))/(sqrt(-b)*b^2)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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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)*x^(3/2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError