∫ 1____ x(−a+bx)5∕2 dx [365]

Optimal. Leaf size=60 \[ -\frac {2}{3 a (-a+b x)^{3/2}}+\frac {2}{a^2 \sqrt {-a+b x}}+\frac {2 \tan ^{-1}\left (\frac {\sqrt {-a+b x}}{\sqrt {a}}\right )}{a^{5/2}} \]

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Rubi [A]
time = 0.01, antiderivative size = 60, 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 = {53, 65, 211} \begin {gather*} \frac {2 \tan ^{-1}\left (\frac {\sqrt {b x-a}}{\sqrt {a}}\right )}{a^{5/2}}+\frac {2}{a^2 \sqrt {b x-a}}-\frac {2}{3 a (b x-a)^{3/2}} \end {gather*}

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

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1

)/((b*c - a*d)*(m + 1))), x] - Dist[d*((m + n + 2)/((b*c - a*d)*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^n,

x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[

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,

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

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

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Mathematica [A]
time = 0.05, size = 52, normalized size = 0.87 \begin {gather*} -\frac {8 a-6 b x}{3 a^2 (-a+b x)^{3/2}}+\frac {2 \tan ^{-1}\left (\frac {\sqrt {-a+b x}}{\sqrt {a}}\right )}{a^{5/2}} \end {gather*}

-1/3*(8*a - 6*b*x)/(a^2*(-a + b*x)^(3/2)) + (2*ArcTan[Sqrt[-a + b*x]/Sqrt[a]])/a^(5/2)

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 9.95, size = 1070, normalized size = 17.83

Piecewise[{{(-8 a ^ 3 Sqrt[(-a + b x) / a] + 3 a ^ 3 (-2 ArcSin[Sqrt[a] / (Sqrt[b] Sqrt[x])] - I Log[b x / a]

+ 2 I Log[Sqrt[b] Sqrt[x] / Sqrt[a]]) + a ^ 2 b x (-18 I Log[Sqrt[b] Sqrt[x] / Sqrt[a]] + 9 I Log[b x / a] + 1

4 Sqrt[(-a + b x) / a] + 18 ArcSin[Sqrt[a] / (Sqrt[b] Sqrt[x])]) + 3 a b ^ 2 x ^ 2 (-6 ArcSin[Sqrt[a] / (Sqrt[

b] Sqrt[x])] - 2 Sqrt[(-a + b x) / a] - 3 I Log[b x / a] + 6 I Log[Sqrt[b] Sqrt[x] / Sqrt[a]]) + 3 b ^ 3 x ^ 3

(-2 I Log[Sqrt[b] Sqrt[x] / Sqrt[a]] + I Log[b x / a] + 2 ArcSin[Sqrt[a] / (Sqrt[b] Sqrt[x])])) / (3 a ^ (5 /

2) (a ^ 3 - 3 a ^ 2 b x + 3 a b ^ 2 x ^ 2 - b ^ 3 x ^ 3)), Abs[b x / a] > 1}}, -6 I a ^ 7 Log[1 + Sqrt[1 - b

x / a]] / (-3 a ^ (19 / 2) + 9 a ^ (17 / 2) b x - 9 a ^ (15 / 2) b ^ 2 x ^ 2 + 3 a ^ (13 / 2) b ^ 3 x ^ 3) + I

3 a ^ 7 Log[b x / a] / (-3 a ^ (19 / 2) + 9 a ^ (17 / 2) b x - 9 a ^ (15 / 2) b ^ 2 x ^ 2 + 3 a ^ (13 / 2) b

^ 3 x ^ 3) + I 8 a ^ 7 Sqrt[1 - b x / a] / (-3 a ^ (19 / 2) + 9 a ^ (17 / 2) b x - 9 a ^ (15 / 2) b ^ 2 x ^ 2

+ 3 a ^ (13 / 2) b ^ 3 x ^ 3) + 3 Pi a ^ 7 / (-3 a ^ (19 / 2) + 9 a ^ (17 / 2) b x - 9 a ^ (15 / 2) b ^ 2 x ^

2 + 3 a ^ (13 / 2) b ^ 3 x ^ 3) - 14 I a ^ 6 b x Sqrt[1 - b x / a] / (-3 a ^ (19 / 2) + 9 a ^ (17 / 2) b x - 9

a ^ (15 / 2) b ^ 2 x ^ 2 + 3 a ^ (13 / 2) b ^ 3 x ^ 3) - 9 I a ^ 6 b x Log[b x / a] / (-3 a ^ (19 / 2) + 9 a

^ (17 / 2) b x - 9 a ^ (15 / 2) b ^ 2 x ^ 2 + 3 a ^ (13 / 2) b ^ 3 x ^ 3) + I 18 a ^ 6 b x Log[1 + Sqrt[1 - b

x / a]] / (-3 a ^ (19 / 2) + 9 a ^ (17 / 2) b x - 9 a ^ (15 / 2) b ^ 2 x ^ 2 + 3 a ^ (13 / 2) b ^ 3 x ^ 3) - 9

Pi a ^ 6 b x / (-3 a ^ (19 / 2) + 9 a ^ (17 / 2) b x - 9 a ^ (15 / 2) b ^ 2 x ^ 2 + 3 a ^ (13 / 2) b ^ 3 x ^

3) - 18 I a ^ 5 b ^ 2 x ^ 2 Log[1 + Sqrt[1 - b x / a]] / (-3 a ^ (19 / 2) + 9 a ^ (17 / 2) b x - 9 a ^ (15 / 2

) b ^ 2 x ^ 2 + 3 a ^ (13 / 2) b ^ 3 x ^ 3) + I 6 a ^ 5 b ^ 2 x ^ 2 Sqrt[1 - b x / a] / (-3 a ^ (19 / 2) + 9 a

^ (17 / 2) b x - 9 a ^ (15 / 2) b ^ 2 x ^ 2 + 3 a ^ (13 / 2) b ^ 3 x ^ 3) + I 9 a ^ 5 b ^ 2 x ^ 2 Log[b x / a

] / (-3 a ^ (19 / 2) + 9 a ^ (17 / 2) b x - 9 a ^ (15 / 2) b ^ 2 x ^ 2 + 3 a ^ (13 / 2) b ^ 3 x ^ 3) + 9 Pi a

^ 5 b ^ 2 x ^ 2 / (-3 a ^ (19 / 2) + 9 a ^ (17 / 2) b x - 9 a ^ (15 / 2) b ^ 2 x ^ 2 + 3 a ^ (13 / 2) b ^ 3 x

^ 3) - 3 I a ^ 4 b ^ 3 x ^ 3 Log[b x / a] / (-3 a ^ (19 / 2) + 9 a ^ (17 / 2) b x - 9 a ^ (15 / 2) b ^ 2 x ^ 2

+ 3 a ^ (13 / 2) b ^ 3 x ^ 3) + I 6 a ^ 4 b ^ 3 x ^ 3 Log[1 + Sqrt[1 - b x / a]] / (-3 a ^ (19 / 2) + 9 a ^ (

17 / 2) b x - 9 a ^ (15 / 2) b ^ 2 x ^ 2 + 3 a ^ (13 / 2) b ^ 3 x ^ 3) - 3 Pi a ^ 4 b ^ 3 x ^ 3 / (-3 a ^ (19

/ 2) + 9 a ^ (17 / 2) b x - 9 a ^ (15 / 2) b ^ 2 x ^ 2 + 3 a ^ (13 / 2) b ^ 3 x ^ 3)]

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method	result	size

derivativedivides	\(-\frac {2}{3 a \left (b x -a \right )^{\frac {3}{2}}}+\frac {2 \arctan \left (\frac {\sqrt {b x -a}}{\sqrt {a}}\right )}{a^{\frac {5}{2}}}+\frac {2}{a^{2} \sqrt {b x -a}}\)	\(49\)
default	\(-\frac {2}{3 a \left (b x -a \right )^{\frac {3}{2}}}+\frac {2 \arctan \left (\frac {\sqrt {b x -a}}{\sqrt {a}}\right )}{a^{\frac {5}{2}}}+\frac {2}{a^{2} \sqrt {b x -a}}\)	\(49\)

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Maxima [A]
time = 0.38, size = 42, normalized size = 0.70 \begin {gather*} \frac {2 \, \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right )}{a^{\frac {5}{2}}} + \frac {2 \, {\left (3 \, b x - 4 \, a\right )}}{3 \, {\left (b x - a\right )}^{\frac {3}{2}} a^{2}} \end {gather*}

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

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Fricas [A]
time = 0.32, size = 182, normalized size = 3.03 \begin {gather*} \left [-\frac {3 \, {\left (b^{2} x^{2} - 2 \, a b x + a^{2}\right )} \sqrt {-a} \log \left (\frac {b x - 2 \, \sqrt {b x - a} \sqrt {-a} - 2 \, a}{x}\right ) - 2 \, {\left (3 \, a b x - 4 \, a^{2}\right )} \sqrt {b x - a}}{3 \, {\left (a^{3} b^{2} x^{2} - 2 \, a^{4} b x + a^{5}\right )}}, \frac {2 \, {\left (3 \, {\left (b^{2} x^{2} - 2 \, a b x + a^{2}\right )} \sqrt {a} \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right ) + {\left (3 \, a b x - 4 \, a^{2}\right )} \sqrt {b x - a}\right )}}{3 \, {\left (a^{3} b^{2} x^{2} - 2 \, a^{4} b x + a^{5}\right )}}\right ] \end {gather*}

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

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

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Sympy [C] Result contains complex when optimal does not.
time = 104.42, size = 1950, normalized size = 32.50

Piecewise((8*a**7*sqrt(-1 + b*x/a)/(-3*a**(19/2) + 9*a**(17/2)*b*x - 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*

x**3) + 3*I*a**7*log(b*x/a)/(-3*a**(19/2) + 9*a**(17/2)*b*x - 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) -

6*I*a**7*log(sqrt(b)*sqrt(x)/sqrt(a))/(-3*a**(19/2) + 9*a**(17/2)*b*x - 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b

**3*x**3) + 6*a**7*asin(sqrt(a)/(sqrt(b)*sqrt(x)))/(-3*a**(19/2) + 9*a**(17/2)*b*x - 9*a**(15/2)*b**2*x**2 + 3

*a**(13/2)*b**3*x**3) - 14*a**6*b*x*sqrt(-1 + b*x/a)/(-3*a**(19/2) + 9*a**(17/2)*b*x - 9*a**(15/2)*b**2*x**2 +

3*a**(13/2)*b**3*x**3) - 9*I*a**6*b*x*log(b*x/a)/(-3*a**(19/2) + 9*a**(17/2)*b*x - 9*a**(15/2)*b**2*x**2 + 3*

a**(13/2)*b**3*x**3) + 18*I*a**6*b*x*log(sqrt(b)*sqrt(x)/sqrt(a))/(-3*a**(19/2) + 9*a**(17/2)*b*x - 9*a**(15/2

)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) - 18*a**6*b*x*asin(sqrt(a)/(sqrt(b)*sqrt(x)))/(-3*a**(19/2) + 9*a**(17/2)

*b*x - 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) + 6*a**5*b**2*x**2*sqrt(-1 + b*x/a)/(-3*a**(19/2) + 9*a*

*(17/2)*b*x - 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) + 9*I*a**5*b**2*x**2*log(b*x/a)/(-3*a**(19/2) + 9

*a**(17/2)*b*x - 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) - 18*I*a**5*b**2*x**2*log(sqrt(b)*sqrt(x)/sqrt

(a))/(-3*a**(19/2) + 9*a**(17/2)*b*x - 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) + 18*a**5*b**2*x**2*asin

(sqrt(a)/(sqrt(b)*sqrt(x)))/(-3*a**(19/2) + 9*a**(17/2)*b*x - 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) -

3*I*a**4*b**3*x**3*log(b*x/a)/(-3*a**(19/2) + 9*a**(17/2)*b*x - 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3

) + 6*I*a**4*b**3*x**3*log(sqrt(b)*sqrt(x)/sqrt(a))/(-3*a**(19/2) + 9*a**(17/2)*b*x - 9*a**(15/2)*b**2*x**2 +

3*a**(13/2)*b**3*x**3) - 6*a**4*b**3*x**3*asin(sqrt(a)/(sqrt(b)*sqrt(x)))/(-3*a**(19/2) + 9*a**(17/2)*b*x - 9*

a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3), Abs(b*x/a) > 1), (8*I*a**7*sqrt(1 - b*x/a)/(-3*a**(19/2) + 9*a**

(17/2)*b*x - 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) + 3*I*a**7*log(b*x/a)/(-3*a**(19/2) + 9*a**(17/2)*

b*x - 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) - 6*I*a**7*log(sqrt(1 - b*x/a) + 1)/(-3*a**(19/2) + 9*a**

(17/2)*b*x - 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) + 3*pi*a**7/(-3*a**(19/2) + 9*a**(17/2)*b*x - 9*a*

*(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) - 14*I*a**6*b*x*sqrt(1 - b*x/a)/(-3*a**(19/2) + 9*a**(17/2)*b*x - 9

*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) - 9*I*a**6*b*x*log(b*x/a)/(-3*a**(19/2) + 9*a**(17/2)*b*x - 9*a*

*(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) + 18*I*a**6*b*x*log(sqrt(1 - b*x/a) + 1)/(-3*a**(19/2) + 9*a**(17/2

)*b*x - 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) - 9*pi*a**6*b*x/(-3*a**(19/2) + 9*a**(17/2)*b*x - 9*a**

(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) + 6*I*a**5*b**2*x**2*sqrt(1 - b*x/a)/(-3*a**(19/2) + 9*a**(17/2)*b*x

- 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) + 9*I*a**5*b**2*x**2*log(b*x/a)/(-3*a**(19/2) + 9*a**(17/2)*

b*x - 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) - 18*I*a**5*b**2*x**2*log(sqrt(1 - b*x/a) + 1)/(-3*a**(19

/2) + 9*a**(17/2)*b*x - 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) + 9*pi*a**5*b**2*x**2/(-3*a**(19/2) + 9

*a**(17/2)*b*x - 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) - 3*I*a**4*b**3*x**3*log(b*x/a)/(-3*a**(19/2)

+ 9*a**(17/2)*b*x - 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) + 6*I*a**4*b**3*x**3*log(sqrt(1 - b*x/a) +

1)/(-3*a**(19/2) + 9*a**(17/2)*b*x - 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) - 3*pi*a**4*b**3*x**3/(-3*

a**(19/2) + 9*a**(17/2)*b*x - 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3), True))

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Giac [A]
time = 0.00, size = 68, normalized size = 1.13 \begin {gather*} 2 \left (\frac {3 \left (-a+b x\right )-a}{3 a^{2} \sqrt {-a+b x} \left (-a+b x\right )}+\frac {\arctan \left (\frac {\sqrt {-a+b x}}{\sqrt {a}}\right )}{a^{2} \sqrt {a}}\right ) \end {gather*}

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

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Mupad [B]
time = 0.09, size = 48, normalized size = 0.80 \begin {gather*} \frac {2\,\mathrm {atan}\left (\frac {\sqrt {b\,x-a}}{\sqrt {a}}\right )}{a^{5/2}}-\frac {\frac {2\,\left (a-b\,x\right )}{a^2}+\frac {2}{3\,a}}{{\left (b\,x-a\right )}^{3/2}} \end {gather*}

(2*atan((b*x - a)^(1/2)/a^(1/2)))/a^(5/2) - ((2*(a - b*x))/a^2 + 2/(3*a))/(b*x - a)^(3/2)

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3.4.65 \(\int \frac {1}{x (-a+b x)^{5/2}} \, dx\) [365]