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

Optimal. Leaf size=68 \[ \frac{3 b^2 \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right )}{4 \sqrt{a}}-\frac{(b x-a)^{3/2}}{2 x^2}-\frac{3 b \sqrt{b x-a}}{4 x} \]

[Out]

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

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Rubi [A]  time = 0.0148347, antiderivative size = 68, 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, 63, 205} \[ \frac{3 b^2 \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right )}{4 \sqrt{a}}-\frac{(b x-a)^{3/2}}{2 x^2}-\frac{3 b \sqrt{b x-a}}{4 x} \]

Antiderivative was successfully verified.

[In]

Int[(-a + b*x)^(3/2)/x^3,x]

[Out]

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

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 205

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

Rubi steps

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

Mathematica [A]  time = 0.0521995, size = 72, normalized size = 1.06 \[ -\frac{2 a^2+3 b^2 x^2 \sqrt{1-\frac{b x}{a}} \tanh ^{-1}\left (\sqrt{1-\frac{b x}{a}}\right )-7 a b x+5 b^2 x^2}{4 x^2 \sqrt{b x-a}} \]

Antiderivative was successfully verified.

[In]

Integrate[(-a + b*x)^(3/2)/x^3,x]

[Out]

-(2*a^2 - 7*a*b*x + 5*b^2*x^2 + 3*b^2*x^2*Sqrt[1 - (b*x)/a]*ArcTanh[Sqrt[1 - (b*x)/a]])/(4*x^2*Sqrt[-a + b*x])

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Maple [A]  time = 0.01, size = 53, normalized size = 0.8 \begin{align*} -{\frac{5}{4\,{x}^{2}} \left ( bx-a \right ) ^{{\frac{3}{2}}}}-{\frac{3\,a}{4\,{x}^{2}}\sqrt{bx-a}}+{\frac{3\,{b}^{2}}{4}\arctan \left ({\sqrt{bx-a}{\frac{1}{\sqrt{a}}}} \right ){\frac{1}{\sqrt{a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.557, size = 294, normalized size = 4.32 \begin{align*} \left [-\frac{3 \, \sqrt{-a} b^{2} x^{2} \log \left (\frac{b x - 2 \, \sqrt{b x - a} \sqrt{-a} - 2 \, a}{x}\right ) + 2 \,{\left (5 \, a b x - 2 \, a^{2}\right )} \sqrt{b x - a}}{8 \, a x^{2}}, \frac{3 \, \sqrt{a} b^{2} x^{2} \arctan \left (\frac{\sqrt{b x - a}}{\sqrt{a}}\right ) -{\left (5 \, a b x - 2 \, a^{2}\right )} \sqrt{b x - a}}{4 \, a x^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/8*(3*sqrt(-a)*b^2*x^2*log((b*x - 2*sqrt(b*x - a)*sqrt(-a) - 2*a)/x) + 2*(5*a*b*x - 2*a^2)*sqrt(b*x - a))/(
a*x^2), 1/4*(3*sqrt(a)*b^2*x^2*arctan(sqrt(b*x - a)/sqrt(a)) - (5*a*b*x - 2*a^2)*sqrt(b*x - a))/(a*x^2)]

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Sympy [A]  time = 4.50158, size = 194, normalized size = 2.85 \begin{align*} \begin{cases} \frac{i a^{2}}{2 \sqrt{b} x^{\frac{5}{2}} \sqrt{\frac{a}{b x} - 1}} - \frac{7 i a \sqrt{b}}{4 x^{\frac{3}{2}} \sqrt{\frac{a}{b x} - 1}} + \frac{5 i b^{\frac{3}{2}}}{4 \sqrt{x} \sqrt{\frac{a}{b x} - 1}} + \frac{3 i b^{2} \operatorname{acosh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{4 \sqrt{a}} & \text{for}\: \frac{\left |{a}\right |}{\left |{b}\right | \left |{x}\right |} > 1 \\\frac{a \sqrt{b} \sqrt{- \frac{a}{b x} + 1}}{2 x^{\frac{3}{2}}} - \frac{5 b^{\frac{3}{2}} \sqrt{- \frac{a}{b x} + 1}}{4 \sqrt{x}} - \frac{3 b^{2} \operatorname{asin}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{4 \sqrt{a}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((I*a**2/(2*sqrt(b)*x**(5/2)*sqrt(a/(b*x) - 1)) - 7*I*a*sqrt(b)/(4*x**(3/2)*sqrt(a/(b*x) - 1)) + 5*I*
b**(3/2)/(4*sqrt(x)*sqrt(a/(b*x) - 1)) + 3*I*b**2*acosh(sqrt(a)/(sqrt(b)*sqrt(x)))/(4*sqrt(a)), Abs(a)/(Abs(b)
*Abs(x)) > 1), (a*sqrt(b)*sqrt(-a/(b*x) + 1)/(2*x**(3/2)) - 5*b**(3/2)*sqrt(-a/(b*x) + 1)/(4*sqrt(x)) - 3*b**2
*asin(sqrt(a)/(sqrt(b)*sqrt(x)))/(4*sqrt(a)), True))

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Giac [A]  time = 1.21901, size = 89, normalized size = 1.31 \begin{align*} \frac{\frac{3 \, b^{3} \arctan \left (\frac{\sqrt{b x - a}}{\sqrt{a}}\right )}{\sqrt{a}} - \frac{5 \,{\left (b x - a\right )}^{\frac{3}{2}} b^{3} + 3 \, \sqrt{b x - a} a b^{3}}{b^{2} x^{2}}}{4 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(3*b^3*arctan(sqrt(b*x - a)/sqrt(a))/sqrt(a) - (5*(b*x - a)^(3/2)*b^3 + 3*sqrt(b*x - a)*a*b^3)/(b^2*x^2))/
b