\(\int \frac {1}{x (-a+b x)^{5/2}} \, dx\) [365]
Optimal result
Integrand size = 15, antiderivative size = 60 \[
\int \frac {1}{x (-a+b x)^{5/2}} \, dx=-\frac {2}{3 a (-a+b x)^{3/2}}+\frac {2}{a^2 \sqrt {-a+b x}}+\frac {2 \arctan \left (\frac {\sqrt {-a+b x}}{\sqrt {a}}\right )}{a^{5/2}}
\]
[Out]
-2/3/a/(b*x-a)^(3/2)+2*arctan((b*x-a)^(1/2)/a^(1/2))/a^(5/2)+2/a^2/(b*x-a)^(1/2)
Rubi [A] (verified)
Time = 0.01 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number
of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {53, 65, 211}
\[
\int \frac {1}{x (-a+b x)^{5/2}} \, dx=\frac {2 \arctan \left (\frac {\sqrt {b x-a}}{\sqrt {a}}\right )}{a^{5/2}}+\frac {2}{a^2 \sqrt {b x-a}}-\frac {2}{3 a (b x-a)^{3/2}}
\]
[In]
Int[1/(x*(-a + b*x)^(5/2)),x]
[Out]
-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)
Rule 53
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[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]
Rule 65
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 211
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]
&& PosQ[a/b]
Rubi steps \begin{align*}
\text {integral}& = -\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*}
Mathematica [A] (verified)
Time = 0.06 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.87
\[
\int \frac {1}{x (-a+b x)^{5/2}} \, dx=-\frac {8 a-6 b x}{3 a^2 (-a+b x)^{3/2}}+\frac {2 \arctan \left (\frac {\sqrt {-a+b x}}{\sqrt {a}}\right )}{a^{5/2}}
\]
[In]
Integrate[1/(x*(-a + b*x)^(5/2)),x]
[Out]
-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)
Maple [A] (verified)
Time = 0.09 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.82
| | |
| 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 |
\(-\frac {2}{3 a \left (b x -a \right )^{\frac {3}{2}}}+\frac {2 \arctan \left (\frac {\sqrt {b x -a}}{\sqrt {a}}\right )}{a^{\frac {5}{2}}}+\frac {2}{a^{2} \sqrt {b x -a}}\) |
\(49\) |
| pseudoelliptic |
\(-\frac {2 \left (\sqrt {b x -a}\, \left (-b x +a \right ) \arctan \left (\frac {\sqrt {b x -a}}{\sqrt {a}}\right )-\sqrt {a}\, b x +\frac {4 a^{\frac {3}{2}}}{3}\right )}{a^{\frac {5}{2}} \left (b x -a \right )^{\frac {3}{2}}}\) |
\(58\) |
| | |
|
|
|
|
[In]
int(1/x/(b*x-a)^(5/2),x,method=_RETURNVERBOSE)
[Out]
-2/3/a/(b*x-a)^(3/2)+2*arctan((b*x-a)^(1/2)/a^(1/2))/a^(5/2)+2/a^2/(b*x-a)^(1/2)
Fricas [A] (verification not implemented)
none
Time = 0.24 (sec) , antiderivative size = 182, normalized size of antiderivative = 3.03
\[
\int \frac {1}{x (-a+b x)^{5/2}} \, dx=\left [-\frac {3 \, {\left (b^{2} x^{2} - 2 \, a b x + a^{2}\right )} \sqrt {-a} \log \left (\frac {b x - 2 \, \sqrt {b x - a} \sqrt {-a} - 2 \, a}{x}\right ) - 2 \, {\left (3 \, a b x - 4 \, a^{2}\right )} \sqrt {b x - a}}{3 \, {\left (a^{3} b^{2} x^{2} - 2 \, a^{4} b x + a^{5}\right )}}, \frac {2 \, {\left (3 \, {\left (b^{2} x^{2} - 2 \, a b x + a^{2}\right )} \sqrt {a} \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right ) + {\left (3 \, a b x - 4 \, a^{2}\right )} \sqrt {b x - a}\right )}}{3 \, {\left (a^{3} b^{2} x^{2} - 2 \, a^{4} b x + a^{5}\right )}}\right ]
\]
[In]
integrate(1/x/(b*x-a)^(5/2),x, algorithm="fricas")
[Out]
[-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
)/sqrt(a)) + (3*a*b*x - 4*a^2)*sqrt(b*x - a))/(a^3*b^2*x^2 - 2*a^4*b*x + a^5)]
Sympy [C] (verification not implemented)
Result contains complex when optimal does not.
Time = 142.31 (sec) , antiderivative size = 1950, normalized size of antiderivative = 32.50
\[
\int \frac {1}{x (-a+b x)^{5/2}} \, dx=\text {Too large to display}
\]
[In]
integrate(1/x/(b*x-a)**(5/2),x)
[Out]
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))
Maxima [A] (verification not implemented)
none
Time = 0.29 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.70
\[
\int \frac {1}{x (-a+b x)^{5/2}} \, dx=\frac {2 \, \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right )}{a^{\frac {5}{2}}} + \frac {2 \, {\left (3 \, b x - 4 \, a\right )}}{3 \, {\left (b x - a\right )}^{\frac {3}{2}} a^{2}}
\]
[In]
integrate(1/x/(b*x-a)^(5/2),x, algorithm="maxima")
[Out]
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)
Giac [A] (verification not implemented)
none
Time = 0.32 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.70
\[
\int \frac {1}{x (-a+b x)^{5/2}} \, dx=\frac {2 \, \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right )}{a^{\frac {5}{2}}} + \frac {2 \, {\left (3 \, b x - 4 \, a\right )}}{3 \, {\left (b x - a\right )}^{\frac {3}{2}} a^{2}}
\]
[In]
integrate(1/x/(b*x-a)^(5/2),x, algorithm="giac")
[Out]
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)
Mupad [B] (verification not implemented)
Time = 0.11 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.80
\[
\int \frac {1}{x (-a+b x)^{5/2}} \, dx=\frac {2\,\mathrm {atan}\left (\frac {\sqrt {b\,x-a}}{\sqrt {a}}\right )}{a^{5/2}}-\frac {\frac {2\,\left (a-b\,x\right )}{a^2}+\frac {2}{3\,a}}{{\left (b\,x-a\right )}^{3/2}}
\]
[In]
int(1/(x*(b*x - a)^(5/2)),x)
[Out]
(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)