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

Optimal. Leaf size=74 \[ 5 a^{3/2} b \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right )-\frac{(b x-a)^{5/2}}{x}+\frac{5}{3} b (b x-a)^{3/2}-5 a b \sqrt{b x-a} \]

[Out]

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

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Rubi [A]  time = 0.0197625, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {47, 50, 63, 205} \[ 5 a^{3/2} b \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right )-\frac{(b x-a)^{5/2}}{x}+\frac{5}{3} b (b x-a)^{3/2}-5 a b \sqrt{b x-a} \]

Antiderivative was successfully verified.

[In]

Int[(-a + b*x)^(5/2)/x^2,x]

[Out]

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

Mathematica [C]  time = 0.0175202, size = 36, normalized size = 0.49 \[ \frac{2 b (b x-a)^{7/2} \, _2F_1\left (2,\frac{7}{2};\frac{9}{2};1-\frac{b x}{a}\right )}{7 a^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(-a + b*x)^(5/2)/x^2,x]

[Out]

(2*b*(-a + b*x)^(7/2)*Hypergeometric2F1[2, 7/2, 9/2, 1 - (b*x)/a])/(7*a^2)

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Maple [A]  time = 0.01, size = 64, normalized size = 0.9 \begin{align*}{\frac{2\,b}{3} \left ( bx-a \right ) ^{{\frac{3}{2}}}}-4\,ab\sqrt{bx-a}-{\frac{{a}^{2}}{x}\sqrt{bx-a}}+5\,{a}^{3/2}b\arctan \left ({\frac{\sqrt{bx-a}}{\sqrt{a}}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.5684, size = 306, normalized size = 4.14 \begin{align*} \left [\frac{15 \, \sqrt{-a} a b x \log \left (\frac{b x + 2 \, \sqrt{b x - a} \sqrt{-a} - 2 \, a}{x}\right ) + 2 \,{\left (2 \, b^{2} x^{2} - 14 \, a b x - 3 \, a^{2}\right )} \sqrt{b x - a}}{6 \, x}, \frac{15 \, a^{\frac{3}{2}} b x \arctan \left (\frac{\sqrt{b x - a}}{\sqrt{a}}\right ) +{\left (2 \, b^{2} x^{2} - 14 \, a b x - 3 \, a^{2}\right )} \sqrt{b x - a}}{3 \, x}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/6*(15*sqrt(-a)*a*b*x*log((b*x + 2*sqrt(b*x - a)*sqrt(-a) - 2*a)/x) + 2*(2*b^2*x^2 - 14*a*b*x - 3*a^2)*sqrt(
b*x - a))/x, 1/3*(15*a^(3/2)*b*x*arctan(sqrt(b*x - a)/sqrt(a)) + (2*b^2*x^2 - 14*a*b*x - 3*a^2)*sqrt(b*x - a))
/x]

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Sympy [C]  time = 4.85502, size = 246, normalized size = 3.32 \begin{align*} \begin{cases} - \frac{a^{\frac{5}{2}} \sqrt{-1 + \frac{b x}{a}}}{x} - \frac{14 a^{\frac{3}{2}} b \sqrt{-1 + \frac{b x}{a}}}{3} - \frac{5 i a^{\frac{3}{2}} b \log{\left (\frac{b x}{a} \right )}}{2} + 5 i a^{\frac{3}{2}} b \log{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )} - 5 a^{\frac{3}{2}} b \operatorname{asin}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )} + \frac{2 \sqrt{a} b^{2} x \sqrt{-1 + \frac{b x}{a}}}{3} & \text{for}\: \frac{\left |{b x}\right |}{\left |{a}\right |} > 1 \\- \frac{i a^{\frac{5}{2}} \sqrt{1 - \frac{b x}{a}}}{x} - \frac{14 i a^{\frac{3}{2}} b \sqrt{1 - \frac{b x}{a}}}{3} - \frac{5 i a^{\frac{3}{2}} b \log{\left (\frac{b x}{a} \right )}}{2} + 5 i a^{\frac{3}{2}} b \log{\left (\sqrt{1 - \frac{b x}{a}} + 1 \right )} + \frac{2 i \sqrt{a} b^{2} x \sqrt{1 - \frac{b x}{a}}}{3} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-a**(5/2)*sqrt(-1 + b*x/a)/x - 14*a**(3/2)*b*sqrt(-1 + b*x/a)/3 - 5*I*a**(3/2)*b*log(b*x/a)/2 + 5*I
*a**(3/2)*b*log(sqrt(b)*sqrt(x)/sqrt(a)) - 5*a**(3/2)*b*asin(sqrt(a)/(sqrt(b)*sqrt(x))) + 2*sqrt(a)*b**2*x*sqr
t(-1 + b*x/a)/3, Abs(b*x)/Abs(a) > 1), (-I*a**(5/2)*sqrt(1 - b*x/a)/x - 14*I*a**(3/2)*b*sqrt(1 - b*x/a)/3 - 5*
I*a**(3/2)*b*log(b*x/a)/2 + 5*I*a**(3/2)*b*log(sqrt(1 - b*x/a) + 1) + 2*I*sqrt(a)*b**2*x*sqrt(1 - b*x/a)/3, Tr
ue))

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Giac [A]  time = 1.14342, size = 101, normalized size = 1.36 \begin{align*} \frac{15 \, a^{\frac{3}{2}} b^{2} \arctan \left (\frac{\sqrt{b x - a}}{\sqrt{a}}\right ) + 2 \,{\left (b x - a\right )}^{\frac{3}{2}} b^{2} - 12 \, \sqrt{b x - a} a b^{2} - \frac{3 \, \sqrt{b x - a} a^{2} b}{x}}{3 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(15*a^(3/2)*b^2*arctan(sqrt(b*x - a)/sqrt(a)) + 2*(b*x - a)^(3/2)*b^2 - 12*sqrt(b*x - a)*a*b^2 - 3*sqrt(b*
x - a)*a^2*b/x)/b