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

Optimal. Leaf size=80 \[ \frac{3 a^2 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )}{4 b^{5/2}}-\frac{3 a \sqrt{x} \sqrt{a-b x}}{4 b^2}-\frac{x^{3/2} \sqrt{a-b x}}{2 b} \]

[Out]

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

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Rubi [A]  time = 0.0228779, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {50, 63, 217, 203} \[ \frac{3 a^2 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )}{4 b^{5/2}}-\frac{3 a \sqrt{x} \sqrt{a-b x}}{4 b^2}-\frac{x^{3/2} \sqrt{a-b x}}{2 b} \]

Antiderivative was successfully verified.

[In]

Int[x^(3/2)/Sqrt[a - b*x],x]

[Out]

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

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 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 217

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

Rule 203

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

Rubi steps

\begin{align*} \int \frac{x^{3/2}}{\sqrt{a-b x}} \, dx &=-\frac{x^{3/2} \sqrt{a-b x}}{2 b}+\frac{(3 a) \int \frac{\sqrt{x}}{\sqrt{a-b x}} \, dx}{4 b}\\ &=-\frac{3 a \sqrt{x} \sqrt{a-b x}}{4 b^2}-\frac{x^{3/2} \sqrt{a-b x}}{2 b}+\frac{\left (3 a^2\right ) \int \frac{1}{\sqrt{x} \sqrt{a-b x}} \, dx}{8 b^2}\\ &=-\frac{3 a \sqrt{x} \sqrt{a-b x}}{4 b^2}-\frac{x^{3/2} \sqrt{a-b x}}{2 b}+\frac{\left (3 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a-b x^2}} \, dx,x,\sqrt{x}\right )}{4 b^2}\\ &=-\frac{3 a \sqrt{x} \sqrt{a-b x}}{4 b^2}-\frac{x^{3/2} \sqrt{a-b x}}{2 b}+\frac{\left (3 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{1+b x^2} \, dx,x,\frac{\sqrt{x}}{\sqrt{a-b x}}\right )}{4 b^2}\\ &=-\frac{3 a \sqrt{x} \sqrt{a-b x}}{4 b^2}-\frac{x^{3/2} \sqrt{a-b x}}{2 b}+\frac{3 a^2 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )}{4 b^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.050758, size = 86, normalized size = 1.08 \[ \frac{\sqrt{b} \sqrt{x} \left (-3 a^2+a b x+2 b^2 x^2\right )+3 a^{5/2} \sqrt{1-\frac{b x}{a}} \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 b^{5/2} \sqrt{a-b x}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(3/2)/Sqrt[a - b*x],x]

[Out]

(Sqrt[b]*Sqrt[x]*(-3*a^2 + a*b*x + 2*b^2*x^2) + 3*a^(5/2)*Sqrt[1 - (b*x)/a]*ArcSin[(Sqrt[b]*Sqrt[x])/Sqrt[a]])
/(4*b^(5/2)*Sqrt[a - b*x])

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Maple [A]  time = 0.004, size = 89, normalized size = 1.1 \begin{align*} -{\frac{1}{2\,b}{x}^{{\frac{3}{2}}}\sqrt{-bx+a}}-{\frac{3\,a}{4\,{b}^{2}}\sqrt{x}\sqrt{-bx+a}}+{\frac{3\,{a}^{2}}{8}\sqrt{x \left ( -bx+a \right ) }\arctan \left ({\sqrt{b} \left ( x-{\frac{a}{2\,b}} \right ){\frac{1}{\sqrt{-b{x}^{2}+ax}}}} \right ){b}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{-bx+a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.08532, size = 321, normalized size = 4.01 \begin{align*} \left [-\frac{3 \, a^{2} \sqrt{-b} \log \left (-2 \, b x + 2 \, \sqrt{-b x + a} \sqrt{-b} \sqrt{x} + a\right ) + 2 \,{\left (2 \, b^{2} x + 3 \, a b\right )} \sqrt{-b x + a} \sqrt{x}}{8 \, b^{3}}, -\frac{3 \, a^{2} \sqrt{b} \arctan \left (\frac{\sqrt{-b x + a}}{\sqrt{b} \sqrt{x}}\right ) +{\left (2 \, b^{2} x + 3 \, a b\right )} \sqrt{-b x + a} \sqrt{x}}{4 \, b^{3}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/8*(3*a^2*sqrt(-b)*log(-2*b*x + 2*sqrt(-b*x + a)*sqrt(-b)*sqrt(x) + a) + 2*(2*b^2*x + 3*a*b)*sqrt(-b*x + a)
*sqrt(x))/b^3, -1/4*(3*a^2*sqrt(b)*arctan(sqrt(-b*x + a)/(sqrt(b)*sqrt(x))) + (2*b^2*x + 3*a*b)*sqrt(-b*x + a)
*sqrt(x))/b^3]

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Sympy [A]  time = 5.28609, size = 216, normalized size = 2.7 \begin{align*} \begin{cases} \frac{3 i a^{\frac{3}{2}} \sqrt{x}}{4 b^{2} \sqrt{-1 + \frac{b x}{a}}} - \frac{i \sqrt{a} x^{\frac{3}{2}}}{4 b \sqrt{-1 + \frac{b x}{a}}} - \frac{3 i a^{2} \operatorname{acosh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{4 b^{\frac{5}{2}}} - \frac{i x^{\frac{5}{2}}}{2 \sqrt{a} \sqrt{-1 + \frac{b x}{a}}} & \text{for}\: \frac{\left |{b x}\right |}{\left |{a}\right |} > 1 \\- \frac{3 a^{\frac{3}{2}} \sqrt{x}}{4 b^{2} \sqrt{1 - \frac{b x}{a}}} + \frac{\sqrt{a} x^{\frac{3}{2}}}{4 b \sqrt{1 - \frac{b x}{a}}} + \frac{3 a^{2} \operatorname{asin}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{4 b^{\frac{5}{2}}} + \frac{x^{\frac{5}{2}}}{2 \sqrt{a} \sqrt{1 - \frac{b x}{a}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((3*I*a**(3/2)*sqrt(x)/(4*b**2*sqrt(-1 + b*x/a)) - I*sqrt(a)*x**(3/2)/(4*b*sqrt(-1 + b*x/a)) - 3*I*a*
*2*acosh(sqrt(b)*sqrt(x)/sqrt(a))/(4*b**(5/2)) - I*x**(5/2)/(2*sqrt(a)*sqrt(-1 + b*x/a)), Abs(b*x)/Abs(a) > 1)
, (-3*a**(3/2)*sqrt(x)/(4*b**2*sqrt(1 - b*x/a)) + sqrt(a)*x**(3/2)/(4*b*sqrt(1 - b*x/a)) + 3*a**2*asin(sqrt(b)
*sqrt(x)/sqrt(a))/(4*b**(5/2)) + x**(5/2)/(2*sqrt(a)*sqrt(1 - b*x/a)), True))

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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+a)^(1/2),x, algorithm="giac")

[Out]

Timed out