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

Optimal. Leaf size=67 \[ -\frac{2 \sqrt{x}}{b^2 \sqrt{2-b x}}+\frac{2 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{5/2}}+\frac{2 x^{3/2}}{3 b (2-b x)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0145247, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {47, 54, 216} \[ -\frac{2 \sqrt{x}}{b^2 \sqrt{2-b x}}+\frac{2 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{5/2}}+\frac{2 x^{3/2}}{3 b (2-b x)^{3/2}} \]

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*ArcSin[(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 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[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 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.0547212, size = 53, normalized size = 0.79 \[ \frac{4 \sqrt{x} (2 b x-3)}{3 b^2 (2-b x)^{3/2}}+\frac{2 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{5/2}} \]

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*ArcSin[(Sqrt[b]*Sqrt[x])/Sqrt[2]])/b^(5/2)

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Maple [A]  time = 0.032, size = 73, normalized size = 1.1 \begin{align*} -{\frac{4}{3\,\sqrt{\pi }b} \left ( -{\frac{\sqrt{\pi }\sqrt{2} \left ( -10\,bx+15 \right ) }{20\,{b}^{2}}\sqrt{x} \left ( -b \right ) ^{{\frac{5}{2}}} \left ( -{\frac{bx}{2}}+1 \right ) ^{-{\frac{3}{2}}}}+{\frac{3\,\sqrt{\pi }}{2} \left ( -b \right ) ^{{\frac{5}{2}}}\arcsin \left ({\frac{\sqrt{2}}{2}\sqrt{b}\sqrt{x}} \right ){b}^{-{\frac{5}{2}}}} \right ) \left ( -b \right ) ^{-{\frac{3}{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)^(3/2)/Pi^(1/2)/b*(-1/20*Pi^(1/2)*x^(1/2)*2^(1/2)*(-b)^(5/2)*(-10*b*x+15)/b^2/(-1/2*b*x+1)^(3/2)+3/2*
Pi^(1/2)*(-b)^(5/2)/b^(5/2)*arcsin(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 = 1.66662, size = 427, normalized size = 6.37 \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} \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)*sq
rt(-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)*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 = 5.82932, size = 649, normalized size = 9.69 \begin{align*} \begin{cases} \frac{8 i 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 i 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 i b^{5} x^{\frac{15}{2}} \sqrt{b x - 2} \operatorname{acosh}{\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{3 \pi b^{5} x^{\frac{15}{2}} \sqrt{b x - 2}}{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 i b^{4} x^{\frac{13}{2}} \sqrt{b x - 2} \operatorname{acosh}{\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{6 \pi b^{4} x^{\frac{13}{2}} \sqrt{b x - 2}}{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}} & \text{for}\: \frac{\left |{b x}\right |}{2} > 1 \\- \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{asin}{\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{asin}{\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}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((8*I*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
*I*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*I*b**5*x**(15
/2)*sqrt(b*x - 2)*acosh(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)) + 3*pi*b**5*x**(15/2)*sqrt(b*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*I*b**4*x**(13/2)*sqrt(b*x - 2)*acosh(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)) - 6*pi*b**4*x**(13/2)*sqrt(b*x - 2)/(3*b**(15/2)*x**(15/
2)*sqrt(b*x - 2) - 6*b**(13/2)*x**(13/2)*sqrt(b*x - 2)), Abs(b*x)/2 > 1), (-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)*asin(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)*asin(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)), True))

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Giac [B]  time = 18.0308, size = 244, normalized size = 3.64 \begin{align*} -\frac{{\left (\frac{3 \, \sqrt{-b} \log \left ({\left (\sqrt{-b x + 2} \sqrt{-b} - \sqrt{{\left (b x - 2\right )} b + 2 \, b}\right )}^{2}\right )}{b} - \frac{16 \,{\left (3 \,{\left (\sqrt{-b x + 2} \sqrt{-b} - \sqrt{{\left (b x - 2\right )} b + 2 \, b}\right )}^{4} \sqrt{-b} - 6 \,{\left (\sqrt{-b x + 2} \sqrt{-b} - \sqrt{{\left (b x - 2\right )} b + 2 \, b}\right )}^{2} \sqrt{-b} b + 8 \, \sqrt{-b} b^{2}\right )}}{{\left ({\left (\sqrt{-b x + 2} \sqrt{-b} - \sqrt{{\left (b x - 2\right )} b + 2 \, b}\right )}^{2} - 2 \, b\right )}^{3}}\right )}{\left | b \right |}}{3 \, b^{3}} \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]

-1/3*(3*sqrt(-b)*log((sqrt(-b*x + 2)*sqrt(-b) - sqrt((b*x - 2)*b + 2*b))^2)/b - 16*(3*(sqrt(-b*x + 2)*sqrt(-b)
- sqrt((b*x - 2)*b + 2*b))^4*sqrt(-b) - 6*(sqrt(-b*x + 2)*sqrt(-b) - sqrt((b*x - 2)*b + 2*b))^2*sqrt(-b)*b +
8*sqrt(-b)*b^2)/((sqrt(-b*x + 2)*sqrt(-b) - sqrt((b*x - 2)*b + 2*b))^2 - 2*b)^3)*abs(b)/b^3