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

Optimal. Leaf size=65 \[ -\frac{2 \sqrt{x}}{b^2 \sqrt{b x+2}}+\frac{2 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{5/2}}-\frac{2 x^{3/2}}{3 b (b x+2)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0144139, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {47, 54, 215} \[ -\frac{2 \sqrt{x}}{b^2 \sqrt{b x+2}}+\frac{2 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{5/2}}-\frac{2 x^{3/2}}{3 b (b x+2)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[x^(3/2)/(2 + b*x)^(5/2),x]

[Out]

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

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Rule 54

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[2/Sqrt[b], Subst[Int[1/Sqrt[b*c -
a*d + d*x^2], x], x, Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d}, x] && GtQ[b*c - a*d, 0] && GtQ[b, 0]

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rubi steps

\begin{align*} \int \frac{x^{3/2}}{(2+b x)^{5/2}} \, dx &=-\frac{2 x^{3/2}}{3 b (2+b x)^{3/2}}+\frac{\int \frac{\sqrt{x}}{(2+b x)^{3/2}} \, dx}{b}\\ &=-\frac{2 x^{3/2}}{3 b (2+b x)^{3/2}}-\frac{2 \sqrt{x}}{b^2 \sqrt{2+b x}}+\frac{\int \frac{1}{\sqrt{x} \sqrt{2+b x}} \, dx}{b^2}\\ &=-\frac{2 x^{3/2}}{3 b (2+b x)^{3/2}}-\frac{2 \sqrt{x}}{b^2 \sqrt{2+b x}}+\frac{2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{2+b x^2}} \, dx,x,\sqrt{x}\right )}{b^2}\\ &=-\frac{2 x^{3/2}}{3 b (2+b x)^{3/2}}-\frac{2 \sqrt{x}}{b^2 \sqrt{2+b x}}+\frac{2 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.0793311, size = 52, normalized size = 0.8 \[ \frac{2 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{5/2}}-\frac{4 \sqrt{x} (2 b x+3)}{3 b^2 (b x+2)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(3/2)/(2 + b*x)^(5/2),x]

[Out]

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

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Maple [A]  time = 0.029, size = 55, normalized size = 0.9 \begin{align*}{\frac{4}{3\,\sqrt{\pi }} \left ( -{\frac{\sqrt{\pi }\sqrt{2} \left ( 10\,bx+15 \right ) }{20}\sqrt{b}\sqrt{x} \left ({\frac{bx}{2}}+1 \right ) ^{-{\frac{3}{2}}}}+{\frac{3\,\sqrt{\pi }}{2}{\it Arcsinh} \left ({\frac{\sqrt{2}}{2}\sqrt{b}\sqrt{x}} \right ) } \right ){b}^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(3/2)/(b*x+2)^(5/2),x)

[Out]

4/3/b^(5/2)/Pi^(1/2)*(-1/20*Pi^(1/2)*x^(1/2)*2^(1/2)*b^(1/2)*(10*b*x+15)/(1/2*b*x+1)^(3/2)+3/2*Pi^(1/2)*arcsin
h(1/2*b^(1/2)*x^(1/2)*2^(1/2)))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.65766, size = 421, normalized size = 6.48 \begin{align*} \left [\frac{3 \,{\left (b^{2} x^{2} + 4 \, b x + 4\right )} \sqrt{b} \log \left (b x + \sqrt{b x + 2} \sqrt{b} \sqrt{x} + 1\right ) - 4 \,{\left (2 \, b^{2} x + 3 \, b\right )} \sqrt{b x + 2} \sqrt{x}}{3 \,{\left (b^{5} x^{2} + 4 \, b^{4} x + 4 \, b^{3}\right )}}, -\frac{2 \,{\left (3 \,{\left (b^{2} x^{2} + 4 \, b x + 4\right )} \sqrt{-b} \arctan \left (\frac{\sqrt{b x + 2} \sqrt{-b}}{b \sqrt{x}}\right ) + 2 \,{\left (2 \, b^{2} x + 3 \, b\right )} \sqrt{b x + 2} \sqrt{x}\right )}}{3 \,{\left (b^{5} x^{2} + 4 \, b^{4} x + 4 \, b^{3}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/3*(3*(b^2*x^2 + 4*b*x + 4)*sqrt(b)*log(b*x + sqrt(b*x + 2)*sqrt(b)*sqrt(x) + 1) - 4*(2*b^2*x + 3*b)*sqrt(b*
x + 2)*sqrt(x))/(b^5*x^2 + 4*b^4*x + 4*b^3), -2/3*(3*(b^2*x^2 + 4*b*x + 4)*sqrt(-b)*arctan(sqrt(b*x + 2)*sqrt(
-b)/(b*sqrt(x))) + 2*(2*b^2*x + 3*b)*sqrt(b*x + 2)*sqrt(x))/(b^5*x^2 + 4*b^4*x + 4*b^3)]

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Sympy [B]  time = 5.86438, size = 257, normalized size = 3.95 \begin{align*} - \frac{8 b^{\frac{11}{2}} x^{8}}{3 b^{\frac{15}{2}} x^{\frac{15}{2}} \sqrt{b x + 2} + 6 b^{\frac{13}{2}} x^{\frac{13}{2}} \sqrt{b x + 2}} - \frac{12 b^{\frac{9}{2}} x^{7}}{3 b^{\frac{15}{2}} x^{\frac{15}{2}} \sqrt{b x + 2} + 6 b^{\frac{13}{2}} x^{\frac{13}{2}} \sqrt{b x + 2}} + \frac{6 b^{5} x^{\frac{15}{2}} \sqrt{b x + 2} \operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{3 b^{\frac{15}{2}} x^{\frac{15}{2}} \sqrt{b x + 2} + 6 b^{\frac{13}{2}} x^{\frac{13}{2}} \sqrt{b x + 2}} + \frac{12 b^{4} x^{\frac{13}{2}} \sqrt{b x + 2} \operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{3 b^{\frac{15}{2}} x^{\frac{15}{2}} \sqrt{b x + 2} + 6 b^{\frac{13}{2}} x^{\frac{13}{2}} \sqrt{b x + 2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-8*b**(11/2)*x**8/(3*b**(15/2)*x**(15/2)*sqrt(b*x + 2) + 6*b**(13/2)*x**(13/2)*sqrt(b*x + 2)) - 12*b**(9/2)*x*
*7/(3*b**(15/2)*x**(15/2)*sqrt(b*x + 2) + 6*b**(13/2)*x**(13/2)*sqrt(b*x + 2)) + 6*b**5*x**(15/2)*sqrt(b*x + 2
)*asinh(sqrt(2)*sqrt(b)*sqrt(x)/2)/(3*b**(15/2)*x**(15/2)*sqrt(b*x + 2) + 6*b**(13/2)*x**(13/2)*sqrt(b*x + 2))
+ 12*b**4*x**(13/2)*sqrt(b*x + 2)*asinh(sqrt(2)*sqrt(b)*sqrt(x)/2)/(3*b**(15/2)*x**(15/2)*sqrt(b*x + 2) + 6*b
**(13/2)*x**(13/2)*sqrt(b*x + 2))

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out