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

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

[Out]

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

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Rubi [A]  time = 0.0215647, antiderivative size = 67, 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 = {47, 63, 217, 203} \[ 2 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )-\frac{2 (a-b x)^{3/2}}{3 x^{3/2}}+\frac{2 b \sqrt{a-b x}}{\sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Int[(a - b*x)^(3/2)/x^(5/2),x]

[Out]

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

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

Mathematica [C]  time = 0.0101356, size = 49, normalized size = 0.73 \[ -\frac{2 a \sqrt{a-b x} \, _2F_1\left (-\frac{3}{2},-\frac{3}{2};-\frac{1}{2};\frac{b x}{a}\right )}{3 x^{3/2} \sqrt{1-\frac{b x}{a}}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a - b*x)^(3/2)/x^(5/2),x]

[Out]

(-2*a*Sqrt[a - b*x]*Hypergeometric2F1[-3/2, -3/2, -1/2, (b*x)/a])/(3*x^(3/2)*Sqrt[1 - (b*x)/a])

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Maple [A]  time = 0.016, size = 71, normalized size = 1.1 \begin{align*} -{\frac{-8\,bx+2\,a}{3}\sqrt{-bx+a}{x}^{-{\frac{3}{2}}}}+{{b}^{{\frac{3}{2}}}\arctan \left ({\sqrt{b} \left ( x-{\frac{a}{2\,b}} \right ){\frac{1}{\sqrt{-b{x}^{2}+ax}}}} \right ) \sqrt{x \left ( -bx+a \right ) }{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{-bx+a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.73592, size = 306, normalized size = 4.57 \begin{align*} \left [\frac{3 \, \sqrt{-b} b x^{2} \log \left (-2 \, b x - 2 \, \sqrt{-b x + a} \sqrt{-b} \sqrt{x} + a\right ) + 2 \,{\left (4 \, b x - a\right )} \sqrt{-b x + a} \sqrt{x}}{3 \, x^{2}}, -\frac{2 \,{\left (3 \, b^{\frac{3}{2}} x^{2} \arctan \left (\frac{\sqrt{-b x + a}}{\sqrt{b} \sqrt{x}}\right ) -{\left (4 \, b x - a\right )} \sqrt{-b x + a} \sqrt{x}\right )}}{3 \, x^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C]  time = 4.43924, size = 190, normalized size = 2.84 \begin{align*} \begin{cases} - \frac{2 a \sqrt{b} \sqrt{\frac{a}{b x} - 1}}{3 x} + \frac{8 b^{\frac{3}{2}} \sqrt{\frac{a}{b x} - 1}}{3} - 2 i b^{\frac{3}{2}} \log{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )} + i b^{\frac{3}{2}} \log{\left (\frac{a}{b x} \right )} + 2 b^{\frac{3}{2}} \operatorname{asin}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )} & \text{for}\: \frac{\left |{a}\right |}{\left |{b}\right | \left |{x}\right |} > 1 \\- \frac{2 i a \sqrt{b} \sqrt{- \frac{a}{b x} + 1}}{3 x} + \frac{8 i b^{\frac{3}{2}} \sqrt{- \frac{a}{b x} + 1}}{3} + i b^{\frac{3}{2}} \log{\left (\frac{a}{b x} \right )} - 2 i b^{\frac{3}{2}} \log{\left (\sqrt{- \frac{a}{b x} + 1} + 1 \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-2*a*sqrt(b)*sqrt(a/(b*x) - 1)/(3*x) + 8*b**(3/2)*sqrt(a/(b*x) - 1)/3 - 2*I*b**(3/2)*log(sqrt(a)/(s
qrt(b)*sqrt(x))) + I*b**(3/2)*log(a/(b*x)) + 2*b**(3/2)*asin(sqrt(b)*sqrt(x)/sqrt(a)), Abs(a)/(Abs(b)*Abs(x))
> 1), (-2*I*a*sqrt(b)*sqrt(-a/(b*x) + 1)/(3*x) + 8*I*b**(3/2)*sqrt(-a/(b*x) + 1)/3 + I*b**(3/2)*log(a/(b*x)) -
2*I*b**(3/2)*log(sqrt(-a/(b*x) + 1) + 1), 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((-b*x+a)^(3/2)/x^(5/2),x, algorithm="giac")

[Out]

Timed out