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

Optimal. Leaf size=55 \[ 2 a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b x-a}}{\sqrt {a}}\right )-2 a \sqrt {b x-a}+\frac {2}{3} (b x-a)^{3/2} \]

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Rubi [A]  time = 0.01, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {50, 63, 205} \begin {gather*} 2 a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b x-a}}{\sqrt {a}}\right )-2 a \sqrt {b x-a}+\frac {2}{3} (b x-a)^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-a + b*x)^(3/2)/x,x]

[Out]

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

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

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Mathematica [A]  time = 0.02, size = 48, normalized size = 0.87 \begin {gather*} 2 a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b x-a}}{\sqrt {a}}\right )+\frac {2}{3} (b x-4 a) \sqrt {b x-a} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-a + b*x)^(3/2)/x,x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.03, size = 58, normalized size = 1.05 \begin {gather*} 2 a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b x-a}}{\sqrt {a}}\right )-\frac {2}{3} \left (3 a \sqrt {b x-a}-(b x-a)^{3/2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-a + b*x)^(3/2)/x,x]

[Out]

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

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fricas [A]  time = 0.91, size = 93, normalized size = 1.69 \begin {gather*} \left [\sqrt {-a} a \log \left (\frac {b x + 2 \, \sqrt {b x - a} \sqrt {-a} - 2 \, a}{x}\right ) + \frac {2}{3} \, \sqrt {b x - a} {\left (b x - 4 \, a\right )}, 2 \, a^{\frac {3}{2}} \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right ) + \frac {2}{3} \, \sqrt {b x - a} {\left (b x - 4 \, a\right )}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 1.06, size = 43, normalized size = 0.78 \begin {gather*} 2 \, a^{\frac {3}{2}} \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right ) + \frac {2}{3} \, {\left (b x - a\right )}^{\frac {3}{2}} - 2 \, \sqrt {b x - a} a \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.01, size = 44, normalized size = 0.80 \begin {gather*} 2 a^{\frac {3}{2}} \arctan \left (\frac {\sqrt {b x -a}}{\sqrt {a}}\right )-2 \sqrt {b x -a}\, a +\frac {2 \left (b x -a \right )^{\frac {3}{2}}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 3.02, size = 43, normalized size = 0.78 \begin {gather*} 2 \, a^{\frac {3}{2}} \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right ) + \frac {2}{3} \, {\left (b x - a\right )}^{\frac {3}{2}} - 2 \, \sqrt {b x - a} a \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.04, size = 43, normalized size = 0.78 \begin {gather*} 2\,a^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {b\,x-a}}{\sqrt {a}}\right )-2\,a\,\sqrt {b\,x-a}+\frac {2\,{\left (b\,x-a\right )}^{3/2}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x - a)^(3/2)/x,x)

[Out]

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

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sympy [C]  time = 2.46, size = 187, normalized size = 3.40 \begin {gather*} \begin {cases} - \frac {8 a^{\frac {3}{2}} \sqrt {-1 + \frac {b x}{a}}}{3} - i a^{\frac {3}{2}} \log {\left (\frac {b x}{a} \right )} + 2 i a^{\frac {3}{2}} \log {\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )} - 2 a^{\frac {3}{2}} \operatorname {asin}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )} + \frac {2 \sqrt {a} b x \sqrt {-1 + \frac {b x}{a}}}{3} & \text {for}\: \left |{\frac {b x}{a}}\right | > 1 \\- \frac {8 i a^{\frac {3}{2}} \sqrt {1 - \frac {b x}{a}}}{3} - i a^{\frac {3}{2}} \log {\left (\frac {b x}{a} \right )} + 2 i a^{\frac {3}{2}} \log {\left (\sqrt {1 - \frac {b x}{a}} + 1 \right )} + \frac {2 i \sqrt {a} b x \sqrt {1 - \frac {b x}{a}}}{3} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-8*a**(3/2)*sqrt(-1 + b*x/a)/3 - I*a**(3/2)*log(b*x/a) + 2*I*a**(3/2)*log(sqrt(b)*sqrt(x)/sqrt(a))
- 2*a**(3/2)*asin(sqrt(a)/(sqrt(b)*sqrt(x))) + 2*sqrt(a)*b*x*sqrt(-1 + b*x/a)/3, Abs(b*x/a) > 1), (-8*I*a**(3/
2)*sqrt(1 - b*x/a)/3 - I*a**(3/2)*log(b*x/a) + 2*I*a**(3/2)*log(sqrt(1 - b*x/a) + 1) + 2*I*sqrt(a)*b*x*sqrt(1
- b*x/a)/3, True))

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