Integrand size = 15, antiderivative size = 57 \[ \int \frac {1}{x^{3/2} (-a+b x)^2} \, dx=-\frac {3}{a^2 \sqrt {x}}+\frac {1}{a \sqrt {x} (a-b x)}+\frac {3 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{5/2}} \]
Time = 0.07 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.96 \[ \int \frac {1}{x^{3/2} (-a+b x)^2} \, dx=\frac {-2 a+3 b x}{a^2 \sqrt {x} (a-b x)}+\frac {3 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{5/2}} \]
(-2*a + 3*b*x)/(a^2*Sqrt[x]*(a - b*x)) + (3*Sqrt[b]*ArcTanh[(Sqrt[b]*Sqrt[ x])/Sqrt[a]])/a^(5/2)
Time = 0.17 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.14, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {52, 25, 61, 73, 221}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {1}{x^{3/2} (b x-a)^2} \, dx\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {1}{a \sqrt {x} (a-b x)}-\frac {3 \int -\frac {1}{x^{3/2} (a-b x)}dx}{2 a}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {3 \int \frac {1}{x^{3/2} (a-b x)}dx}{2 a}+\frac {1}{a \sqrt {x} (a-b x)}\) |
\(\Big \downarrow \) 61 |
\(\displaystyle \frac {3 \left (\frac {b \int \frac {1}{\sqrt {x} (a-b x)}dx}{a}-\frac {2}{a \sqrt {x}}\right )}{2 a}+\frac {1}{a \sqrt {x} (a-b x)}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {3 \left (\frac {2 b \int \frac {1}{a-b x}d\sqrt {x}}{a}-\frac {2}{a \sqrt {x}}\right )}{2 a}+\frac {1}{a \sqrt {x} (a-b x)}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {3 \left (\frac {2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {2}{a \sqrt {x}}\right )}{2 a}+\frac {1}{a \sqrt {x} (a-b x)}\) |
1/(a*Sqrt[x]*(a - b*x)) + (3*(-2/(a*Sqrt[x]) + (2*Sqrt[b]*ArcTanh[(Sqrt[b] *Sqrt[x])/Sqrt[a]])/a^(3/2)))/(2*a)
3.5.80.3.1 Defintions of rubi rules used
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] - Simp[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] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
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] - Simp[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] && 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]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[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] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Time = 0.09 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.84
method | result | size |
derivativedivides | \(\frac {2 b \left (\frac {\sqrt {x}}{-2 b x +2 a}+\frac {3 \,\operatorname {arctanh}\left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{2}}-\frac {2}{a^{2} \sqrt {x}}\) | \(48\) |
default | \(\frac {2 b \left (\frac {\sqrt {x}}{-2 b x +2 a}+\frac {3 \,\operatorname {arctanh}\left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{2}}-\frac {2}{a^{2} \sqrt {x}}\) | \(48\) |
risch | \(-\frac {2}{a^{2} \sqrt {x}}-\frac {b \left (\frac {\sqrt {x}}{b x -a}-\frac {3 \,\operatorname {arctanh}\left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b}}\right )}{a^{2}}\) | \(48\) |
Time = 0.23 (sec) , antiderivative size = 151, normalized size of antiderivative = 2.65 \[ \int \frac {1}{x^{3/2} (-a+b x)^2} \, dx=\left [\frac {3 \, {\left (b x^{2} - a x\right )} \sqrt {\frac {b}{a}} \log \left (\frac {b x + 2 \, a \sqrt {x} \sqrt {\frac {b}{a}} + a}{b x - a}\right ) - 2 \, {\left (3 \, b x - 2 \, a\right )} \sqrt {x}}{2 \, {\left (a^{2} b x^{2} - a^{3} x\right )}}, -\frac {3 \, {\left (b x^{2} - a x\right )} \sqrt {-\frac {b}{a}} \arctan \left (\frac {a \sqrt {-\frac {b}{a}}}{b \sqrt {x}}\right ) + {\left (3 \, b x - 2 \, a\right )} \sqrt {x}}{a^{2} b x^{2} - a^{3} x}\right ] \]
[1/2*(3*(b*x^2 - a*x)*sqrt(b/a)*log((b*x + 2*a*sqrt(x)*sqrt(b/a) + a)/(b*x - a)) - 2*(3*b*x - 2*a)*sqrt(x))/(a^2*b*x^2 - a^3*x), -(3*(b*x^2 - a*x)*s qrt(-b/a)*arctan(a*sqrt(-b/a)/(b*sqrt(x))) + (3*b*x - 2*a)*sqrt(x))/(a^2*b *x^2 - a^3*x)]
Leaf count of result is larger than twice the leaf count of optimal. 354 vs. \(2 (51) = 102\).
Time = 12.42 (sec) , antiderivative size = 354, normalized size of antiderivative = 6.21 \[ \int \frac {1}{x^{3/2} (-a+b x)^2} \, dx=\begin {cases} \frac {\tilde {\infty }}{x^{\frac {5}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {2}{a^{2} \sqrt {x}} & \text {for}\: b = 0 \\- \frac {2}{5 b^{2} x^{\frac {5}{2}}} & \text {for}\: a = 0 \\- \frac {3 a \sqrt {x} \log {\left (\sqrt {x} - \sqrt {\frac {a}{b}} \right )}}{2 a^{3} \sqrt {x} \sqrt {\frac {a}{b}} - 2 a^{2} b x^{\frac {3}{2}} \sqrt {\frac {a}{b}}} + \frac {3 a \sqrt {x} \log {\left (\sqrt {x} + \sqrt {\frac {a}{b}} \right )}}{2 a^{3} \sqrt {x} \sqrt {\frac {a}{b}} - 2 a^{2} b x^{\frac {3}{2}} \sqrt {\frac {a}{b}}} - \frac {4 a \sqrt {\frac {a}{b}}}{2 a^{3} \sqrt {x} \sqrt {\frac {a}{b}} - 2 a^{2} b x^{\frac {3}{2}} \sqrt {\frac {a}{b}}} + \frac {3 b x^{\frac {3}{2}} \log {\left (\sqrt {x} - \sqrt {\frac {a}{b}} \right )}}{2 a^{3} \sqrt {x} \sqrt {\frac {a}{b}} - 2 a^{2} b x^{\frac {3}{2}} \sqrt {\frac {a}{b}}} - \frac {3 b x^{\frac {3}{2}} \log {\left (\sqrt {x} + \sqrt {\frac {a}{b}} \right )}}{2 a^{3} \sqrt {x} \sqrt {\frac {a}{b}} - 2 a^{2} b x^{\frac {3}{2}} \sqrt {\frac {a}{b}}} + \frac {6 b x \sqrt {\frac {a}{b}}}{2 a^{3} \sqrt {x} \sqrt {\frac {a}{b}} - 2 a^{2} b x^{\frac {3}{2}} \sqrt {\frac {a}{b}}} & \text {otherwise} \end {cases} \]
Piecewise((zoo/x**(5/2), Eq(a, 0) & Eq(b, 0)), (-2/(a**2*sqrt(x)), Eq(b, 0 )), (-2/(5*b**2*x**(5/2)), Eq(a, 0)), (-3*a*sqrt(x)*log(sqrt(x) - sqrt(a/b ))/(2*a**3*sqrt(x)*sqrt(a/b) - 2*a**2*b*x**(3/2)*sqrt(a/b)) + 3*a*sqrt(x)* log(sqrt(x) + sqrt(a/b))/(2*a**3*sqrt(x)*sqrt(a/b) - 2*a**2*b*x**(3/2)*sqr t(a/b)) - 4*a*sqrt(a/b)/(2*a**3*sqrt(x)*sqrt(a/b) - 2*a**2*b*x**(3/2)*sqrt (a/b)) + 3*b*x**(3/2)*log(sqrt(x) - sqrt(a/b))/(2*a**3*sqrt(x)*sqrt(a/b) - 2*a**2*b*x**(3/2)*sqrt(a/b)) - 3*b*x**(3/2)*log(sqrt(x) + sqrt(a/b))/(2*a **3*sqrt(x)*sqrt(a/b) - 2*a**2*b*x**(3/2)*sqrt(a/b)) + 6*b*x*sqrt(a/b)/(2* a**3*sqrt(x)*sqrt(a/b) - 2*a**2*b*x**(3/2)*sqrt(a/b)), True))
Time = 0.28 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.21 \[ \int \frac {1}{x^{3/2} (-a+b x)^2} \, dx=-\frac {3 \, b x - 2 \, a}{a^{2} b x^{\frac {3}{2}} - a^{3} \sqrt {x}} - \frac {3 \, b \log \left (\frac {b \sqrt {x} - \sqrt {a b}}{b \sqrt {x} + \sqrt {a b}}\right )}{2 \, \sqrt {a b} a^{2}} \]
-(3*b*x - 2*a)/(a^2*b*x^(3/2) - a^3*sqrt(x)) - 3/2*b*log((b*sqrt(x) - sqrt (a*b))/(b*sqrt(x) + sqrt(a*b)))/(sqrt(a*b)*a^2)
Time = 0.30 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.91 \[ \int \frac {1}{x^{3/2} (-a+b x)^2} \, dx=-\frac {3 \, b \arctan \left (\frac {b \sqrt {x}}{\sqrt {-a b}}\right )}{\sqrt {-a b} a^{2}} - \frac {3 \, b x - 2 \, a}{{\left (b x^{\frac {3}{2}} - a \sqrt {x}\right )} a^{2}} \]
-3*b*arctan(b*sqrt(x)/sqrt(-a*b))/(sqrt(-a*b)*a^2) - (3*b*x - 2*a)/((b*x^( 3/2) - a*sqrt(x))*a^2)
Time = 0.09 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.86 \[ \int \frac {1}{x^{3/2} (-a+b x)^2} \, dx=\frac {3\,\sqrt {b}\,\mathrm {atanh}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )}{a^{5/2}}-\frac {\frac {2}{a}-\frac {3\,b\,x}{a^2}}{a\,\sqrt {x}-b\,x^{3/2}} \]
(3*b^(1/2)*atanh((b^(1/2)*x^(1/2))/a^(1/2)))/a^(5/2) - (2/a - (3*b*x)/a^2) /(a*x^(1/2) - b*x^(3/2))
Time = 0.00 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.96 \[ \int \frac {1}{x^{3/2} (-a+b x)^2} \, dx=\frac {-3 \sqrt {x}\, \sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (-\sqrt {b}\, \sqrt {a}+\sqrt {x}\, b \right ) a +3 \sqrt {x}\, \sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (-\sqrt {b}\, \sqrt {a}+\sqrt {x}\, b \right ) b x +3 \sqrt {x}\, \sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {b}\, \sqrt {a}+\sqrt {x}\, b \right ) a -3 \sqrt {x}\, \sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {b}\, \sqrt {a}+\sqrt {x}\, b \right ) b x -4 a^{2}+6 a b x}{2 \sqrt {x}\, a^{3} \left (-b x +a \right )} \]
( - 3*sqrt(x)*sqrt(b)*sqrt(a)*log( - sqrt(b)*sqrt(a) + sqrt(x)*b)*a + 3*sq rt(x)*sqrt(b)*sqrt(a)*log( - sqrt(b)*sqrt(a) + sqrt(x)*b)*b*x + 3*sqrt(x)* sqrt(b)*sqrt(a)*log(sqrt(b)*sqrt(a) + sqrt(x)*b)*a - 3*sqrt(x)*sqrt(b)*sqr t(a)*log(sqrt(b)*sqrt(a) + sqrt(x)*b)*b*x - 4*a**2 + 6*a*b*x)/(2*sqrt(x)*a **3*(a - b*x))