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

Optimal. Leaf size=63 \[ \frac{1}{2} \sqrt{x} (2-b x)^{3/2}+\frac{3}{2} \sqrt{x} \sqrt{2-b x}+\frac{3 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{\sqrt{b}} \]

[Out]

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

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Rubi [A]  time = 0.011925, antiderivative size = 63, 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 = {50, 54, 216} \[ \frac{1}{2} \sqrt{x} (2-b x)^{3/2}+\frac{3}{2} \sqrt{x} \sqrt{2-b x}+\frac{3 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{\sqrt{b}} \]

Antiderivative was successfully verified.

[In]

Int[(2 - b*x)^(3/2)/Sqrt[x],x]

[Out]

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

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 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{(2-b x)^{3/2}}{\sqrt{x}} \, dx &=\frac{1}{2} \sqrt{x} (2-b x)^{3/2}+\frac{3}{2} \int \frac{\sqrt{2-b x}}{\sqrt{x}} \, dx\\ &=\frac{3}{2} \sqrt{x} \sqrt{2-b x}+\frac{1}{2} \sqrt{x} (2-b x)^{3/2}+\frac{3}{2} \int \frac{1}{\sqrt{x} \sqrt{2-b x}} \, dx\\ &=\frac{3}{2} \sqrt{x} \sqrt{2-b x}+\frac{1}{2} \sqrt{x} (2-b x)^{3/2}+3 \operatorname{Subst}\left (\int \frac{1}{\sqrt{2-b x^2}} \, dx,x,\sqrt{x}\right )\\ &=\frac{3}{2} \sqrt{x} \sqrt{2-b x}+\frac{1}{2} \sqrt{x} (2-b x)^{3/2}+\frac{3 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{\sqrt{b}}\\ \end{align*}

Mathematica [A]  time = 0.0275864, size = 49, normalized size = 0.78 \[ \frac{3 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{\sqrt{b}}-\frac{1}{2} \sqrt{x} \sqrt{2-b x} (b x-5) \]

Antiderivative was successfully verified.

[In]

Integrate[(2 - b*x)^(3/2)/Sqrt[x],x]

[Out]

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

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Maple [A]  time = 0.005, size = 78, normalized size = 1.2 \begin{align*}{\frac{1}{2} \left ( -bx+2 \right ) ^{{\frac{3}{2}}}\sqrt{x}}+{\frac{3}{2}\sqrt{x}\sqrt{-bx+2}}+{\frac{3}{2}\sqrt{ \left ( -bx+2 \right ) x}\arctan \left ({\sqrt{b} \left ( x-{b}^{-1} \right ){\frac{1}{\sqrt{-b{x}^{2}+2\,x}}}} \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{-bx+2}}}{\frac{1}{\sqrt{b}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-b*x+2)^(3/2)/x^(1/2),x)

[Out]

1/2*(-b*x+2)^(3/2)*x^(1/2)+3/2*x^(1/2)*(-b*x+2)^(1/2)+3/2*((-b*x+2)*x)^(1/2)/(-b*x+2)^(1/2)/x^(1/2)/b^(1/2)*ar
ctan(b^(1/2)*(x-1/b)/(-b*x^2+2*x)^(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((-b*x+2)^(3/2)/x^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.83868, size = 286, normalized size = 4.54 \begin{align*} \left [-\frac{{\left (b^{2} x - 5 \, b\right )} \sqrt{-b x + 2} \sqrt{x} + 3 \, \sqrt{-b} \log \left (-b x + \sqrt{-b x + 2} \sqrt{-b} \sqrt{x} + 1\right )}{2 \, b}, -\frac{{\left (b^{2} x - 5 \, b\right )} \sqrt{-b x + 2} \sqrt{x} + 6 \, \sqrt{b} \arctan \left (\frac{\sqrt{-b x + 2}}{\sqrt{b} \sqrt{x}}\right )}{2 \, b}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x+2)^(3/2)/x^(1/2),x, algorithm="fricas")

[Out]

[-1/2*((b^2*x - 5*b)*sqrt(-b*x + 2)*sqrt(x) + 3*sqrt(-b)*log(-b*x + sqrt(-b*x + 2)*sqrt(-b)*sqrt(x) + 1))/b, -
1/2*((b^2*x - 5*b)*sqrt(-b*x + 2)*sqrt(x) + 6*sqrt(b)*arctan(sqrt(-b*x + 2)/(sqrt(b)*sqrt(x))))/b]

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Sympy [A]  time = 3.24902, size = 167, normalized size = 2.65 \begin{align*} \begin{cases} - \frac{i b^{2} x^{\frac{5}{2}}}{2 \sqrt{b x - 2}} + \frac{7 i b x^{\frac{3}{2}}}{2 \sqrt{b x - 2}} - \frac{5 i \sqrt{x}}{\sqrt{b x - 2}} - \frac{3 i \operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{\sqrt{b}} & \text{for}\: \frac{\left |{b x}\right |}{2} > 1 \\\frac{b^{2} x^{\frac{5}{2}}}{2 \sqrt{- b x + 2}} - \frac{7 b x^{\frac{3}{2}}}{2 \sqrt{- b x + 2}} + \frac{5 \sqrt{x}}{\sqrt{- b x + 2}} + \frac{3 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{\sqrt{b}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x+2)**(3/2)/x**(1/2),x)

[Out]

Piecewise((-I*b**2*x**(5/2)/(2*sqrt(b*x - 2)) + 7*I*b*x**(3/2)/(2*sqrt(b*x - 2)) - 5*I*sqrt(x)/sqrt(b*x - 2) -
3*I*acosh(sqrt(2)*sqrt(b)*sqrt(x)/2)/sqrt(b), Abs(b*x)/2 > 1), (b**2*x**(5/2)/(2*sqrt(-b*x + 2)) - 7*b*x**(3/
2)/(2*sqrt(-b*x + 2)) + 5*sqrt(x)/sqrt(-b*x + 2) + 3*asin(sqrt(2)*sqrt(b)*sqrt(x)/2)/sqrt(b), True))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x+2)^(3/2)/x^(1/2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError