Integrand size = 15, antiderivative size = 70 \[ \int \frac {1}{x^{5/2} (-a+b x)^2} \, dx=-\frac {5}{3 a^2 x^{3/2}}-\frac {5 b}{a^3 \sqrt {x}}+\frac {1}{a x^{3/2} (a-b x)}+\frac {5 b^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{7/2}} \]
-5/3/a^2/x^(3/2)+1/a/x^(3/2)/(-b*x+a)+5*b^(3/2)*arctanh(b^(1/2)*x^(1/2)/a^ (1/2))/a^(7/2)-5*b/a^3/x^(1/2)
Time = 0.09 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.99 \[ \int \frac {1}{x^{5/2} (-a+b x)^2} \, dx=\frac {-2 a^2-10 a b x+15 b^2 x^2}{3 a^3 x^{3/2} (a-b x)}+\frac {5 b^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{7/2}} \]
(-2*a^2 - 10*a*b*x + 15*b^2*x^2)/(3*a^3*x^(3/2)*(a - b*x)) + (5*b^(3/2)*Ar cTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/a^(7/2)
Time = 0.18 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.19, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {52, 25, 61, 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.
\(\displaystyle \int \frac {1}{x^{5/2} (b x-a)^2} \, dx\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {1}{a x^{3/2} (a-b x)}-\frac {5 \int -\frac {1}{x^{5/2} (a-b x)}dx}{2 a}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {5 \int \frac {1}{x^{5/2} (a-b x)}dx}{2 a}+\frac {1}{a x^{3/2} (a-b x)}\) |
\(\Big \downarrow \) 61 |
\(\displaystyle \frac {5 \left (\frac {b \int \frac {1}{x^{3/2} (a-b x)}dx}{a}-\frac {2}{3 a x^{3/2}}\right )}{2 a}+\frac {1}{a x^{3/2} (a-b x)}\) |
\(\Big \downarrow \) 61 |
\(\displaystyle \frac {5 \left (\frac {b \left (\frac {b \int \frac {1}{\sqrt {x} (a-b x)}dx}{a}-\frac {2}{a \sqrt {x}}\right )}{a}-\frac {2}{3 a x^{3/2}}\right )}{2 a}+\frac {1}{a x^{3/2} (a-b x)}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {5 \left (\frac {b \left (\frac {2 b \int \frac {1}{a-b x}d\sqrt {x}}{a}-\frac {2}{a \sqrt {x}}\right )}{a}-\frac {2}{3 a x^{3/2}}\right )}{2 a}+\frac {1}{a x^{3/2} (a-b x)}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {5 \left (\frac {b \left (\frac {2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {2}{a \sqrt {x}}\right )}{a}-\frac {2}{3 a x^{3/2}}\right )}{2 a}+\frac {1}{a x^{3/2} (a-b x)}\) |
1/(a*x^(3/2)*(a - b*x)) + (5*(-2/(3*a*x^(3/2)) + (b*(-2/(a*Sqrt[x]) + (2*S qrt[b]*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/a^(3/2)))/a))/(2*a)
3.5.81.3.1 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 = 56, normalized size of antiderivative = 0.80
method | result | size |
risch | \(-\frac {2 \left (6 b x +a \right )}{3 a^{3} x^{\frac {3}{2}}}-\frac {b^{2} \left (\frac {\sqrt {x}}{b x -a}-\frac {5 \,\operatorname {arctanh}\left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b}}\right )}{a^{3}}\) | \(56\) |
derivativedivides | \(\frac {2 b^{2} \left (\frac {\sqrt {x}}{-2 b x +2 a}+\frac {5 \,\operatorname {arctanh}\left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{3}}-\frac {2}{3 a^{2} x^{\frac {3}{2}}}-\frac {4 b}{a^{3} \sqrt {x}}\) | \(59\) |
default | \(\frac {2 b^{2} \left (\frac {\sqrt {x}}{-2 b x +2 a}+\frac {5 \,\operatorname {arctanh}\left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{3}}-\frac {2}{3 a^{2} x^{\frac {3}{2}}}-\frac {4 b}{a^{3} \sqrt {x}}\) | \(59\) |
-2/3*(6*b*x+a)/a^3/x^(3/2)-1/a^3*b^2*(x^(1/2)/(b*x-a)-5/(a*b)^(1/2)*arctan h(b*x^(1/2)/(a*b)^(1/2)))
Time = 0.24 (sec) , antiderivative size = 187, normalized size of antiderivative = 2.67 \[ \int \frac {1}{x^{5/2} (-a+b x)^2} \, dx=\left [\frac {15 \, {\left (b^{2} x^{3} - a b x^{2}\right )} \sqrt {\frac {b}{a}} \log \left (\frac {b x + 2 \, a \sqrt {x} \sqrt {\frac {b}{a}} + a}{b x - a}\right ) - 2 \, {\left (15 \, b^{2} x^{2} - 10 \, a b x - 2 \, a^{2}\right )} \sqrt {x}}{6 \, {\left (a^{3} b x^{3} - a^{4} x^{2}\right )}}, -\frac {15 \, {\left (b^{2} x^{3} - a b x^{2}\right )} \sqrt {-\frac {b}{a}} \arctan \left (\frac {a \sqrt {-\frac {b}{a}}}{b \sqrt {x}}\right ) + {\left (15 \, b^{2} x^{2} - 10 \, a b x - 2 \, a^{2}\right )} \sqrt {x}}{3 \, {\left (a^{3} b x^{3} - a^{4} x^{2}\right )}}\right ] \]
[1/6*(15*(b^2*x^3 - a*b*x^2)*sqrt(b/a)*log((b*x + 2*a*sqrt(x)*sqrt(b/a) + a)/(b*x - a)) - 2*(15*b^2*x^2 - 10*a*b*x - 2*a^2)*sqrt(x))/(a^3*b*x^3 - a^ 4*x^2), -1/3*(15*(b^2*x^3 - a*b*x^2)*sqrt(-b/a)*arctan(a*sqrt(-b/a)/(b*sqr t(x))) + (15*b^2*x^2 - 10*a*b*x - 2*a^2)*sqrt(x))/(a^3*b*x^3 - a^4*x^2)]
Leaf count of result is larger than twice the leaf count of optimal. 416 vs. \(2 (65) = 130\).
Time = 22.12 (sec) , antiderivative size = 416, normalized size of antiderivative = 5.94 \[ \int \frac {1}{x^{5/2} (-a+b x)^2} \, dx=\begin {cases} \frac {\tilde {\infty }}{x^{\frac {7}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {2}{3 a^{2} x^{\frac {3}{2}}} & \text {for}\: b = 0 \\- \frac {2}{7 b^{2} x^{\frac {7}{2}}} & \text {for}\: a = 0 \\- \frac {4 a^{2} \sqrt {\frac {a}{b}}}{6 a^{4} x^{\frac {3}{2}} \sqrt {\frac {a}{b}} - 6 a^{3} b x^{\frac {5}{2}} \sqrt {\frac {a}{b}}} - \frac {15 a b x^{\frac {3}{2}} \log {\left (\sqrt {x} - \sqrt {\frac {a}{b}} \right )}}{6 a^{4} x^{\frac {3}{2}} \sqrt {\frac {a}{b}} - 6 a^{3} b x^{\frac {5}{2}} \sqrt {\frac {a}{b}}} + \frac {15 a b x^{\frac {3}{2}} \log {\left (\sqrt {x} + \sqrt {\frac {a}{b}} \right )}}{6 a^{4} x^{\frac {3}{2}} \sqrt {\frac {a}{b}} - 6 a^{3} b x^{\frac {5}{2}} \sqrt {\frac {a}{b}}} - \frac {20 a b x \sqrt {\frac {a}{b}}}{6 a^{4} x^{\frac {3}{2}} \sqrt {\frac {a}{b}} - 6 a^{3} b x^{\frac {5}{2}} \sqrt {\frac {a}{b}}} + \frac {15 b^{2} x^{\frac {5}{2}} \log {\left (\sqrt {x} - \sqrt {\frac {a}{b}} \right )}}{6 a^{4} x^{\frac {3}{2}} \sqrt {\frac {a}{b}} - 6 a^{3} b x^{\frac {5}{2}} \sqrt {\frac {a}{b}}} - \frac {15 b^{2} x^{\frac {5}{2}} \log {\left (\sqrt {x} + \sqrt {\frac {a}{b}} \right )}}{6 a^{4} x^{\frac {3}{2}} \sqrt {\frac {a}{b}} - 6 a^{3} b x^{\frac {5}{2}} \sqrt {\frac {a}{b}}} + \frac {30 b^{2} x^{2} \sqrt {\frac {a}{b}}}{6 a^{4} x^{\frac {3}{2}} \sqrt {\frac {a}{b}} - 6 a^{3} b x^{\frac {5}{2}} \sqrt {\frac {a}{b}}} & \text {otherwise} \end {cases} \]
Piecewise((zoo/x**(7/2), Eq(a, 0) & Eq(b, 0)), (-2/(3*a**2*x**(3/2)), Eq(b , 0)), (-2/(7*b**2*x**(7/2)), Eq(a, 0)), (-4*a**2*sqrt(a/b)/(6*a**4*x**(3/ 2)*sqrt(a/b) - 6*a**3*b*x**(5/2)*sqrt(a/b)) - 15*a*b*x**(3/2)*log(sqrt(x) - sqrt(a/b))/(6*a**4*x**(3/2)*sqrt(a/b) - 6*a**3*b*x**(5/2)*sqrt(a/b)) + 1 5*a*b*x**(3/2)*log(sqrt(x) + sqrt(a/b))/(6*a**4*x**(3/2)*sqrt(a/b) - 6*a** 3*b*x**(5/2)*sqrt(a/b)) - 20*a*b*x*sqrt(a/b)/(6*a**4*x**(3/2)*sqrt(a/b) - 6*a**3*b*x**(5/2)*sqrt(a/b)) + 15*b**2*x**(5/2)*log(sqrt(x) - sqrt(a/b))/( 6*a**4*x**(3/2)*sqrt(a/b) - 6*a**3*b*x**(5/2)*sqrt(a/b)) - 15*b**2*x**(5/2 )*log(sqrt(x) + sqrt(a/b))/(6*a**4*x**(3/2)*sqrt(a/b) - 6*a**3*b*x**(5/2)* sqrt(a/b)) + 30*b**2*x**2*sqrt(a/b)/(6*a**4*x**(3/2)*sqrt(a/b) - 6*a**3*b* x**(5/2)*sqrt(a/b)), True))
Time = 0.29 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.17 \[ \int \frac {1}{x^{5/2} (-a+b x)^2} \, dx=-\frac {15 \, b^{2} x^{2} - 10 \, a b x - 2 \, a^{2}}{3 \, {\left (a^{3} b x^{\frac {5}{2}} - a^{4} x^{\frac {3}{2}}\right )}} - \frac {5 \, b^{2} \log \left (\frac {b \sqrt {x} - \sqrt {a b}}{b \sqrt {x} + \sqrt {a b}}\right )}{2 \, \sqrt {a b} a^{3}} \]
-1/3*(15*b^2*x^2 - 10*a*b*x - 2*a^2)/(a^3*b*x^(5/2) - a^4*x^(3/2)) - 5/2*b ^2*log((b*sqrt(x) - sqrt(a*b))/(b*sqrt(x) + sqrt(a*b)))/(sqrt(a*b)*a^3)
Time = 0.30 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.87 \[ \int \frac {1}{x^{5/2} (-a+b x)^2} \, dx=-\frac {5 \, b^{2} \arctan \left (\frac {b \sqrt {x}}{\sqrt {-a b}}\right )}{\sqrt {-a b} a^{3}} - \frac {b^{2} \sqrt {x}}{{\left (b x - a\right )} a^{3}} - \frac {2 \, {\left (6 \, b x + a\right )}}{3 \, a^{3} x^{\frac {3}{2}}} \]
-5*b^2*arctan(b*sqrt(x)/sqrt(-a*b))/(sqrt(-a*b)*a^3) - b^2*sqrt(x)/((b*x - a)*a^3) - 2/3*(6*b*x + a)/(a^3*x^(3/2))
Time = 0.16 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.86 \[ \int \frac {1}{x^{5/2} (-a+b x)^2} \, dx=\frac {5\,b^{3/2}\,\mathrm {atanh}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )}{a^{7/2}}-\frac {\frac {2}{3\,a}-\frac {5\,b^2\,x^2}{a^3}+\frac {10\,b\,x}{3\,a^2}}{a\,x^{3/2}-b\,x^{5/2}} \]
(5*b^(3/2)*atanh((b^(1/2)*x^(1/2))/a^(1/2)))/a^(7/2) - (2/(3*a) - (5*b^2*x ^2)/a^3 + (10*b*x)/(3*a^2))/(a*x^(3/2) - b*x^(5/2))
Time = 0.00 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.97 \[ \int \frac {1}{x^{5/2} (-a+b x)^2} \, dx=\frac {-15 \sqrt {x}\, \sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (-\sqrt {b}\, \sqrt {a}+\sqrt {x}\, b \right ) a b x +15 \sqrt {x}\, \sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (-\sqrt {b}\, \sqrt {a}+\sqrt {x}\, b \right ) b^{2} x^{2}+15 \sqrt {x}\, \sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {b}\, \sqrt {a}+\sqrt {x}\, b \right ) a b x -15 \sqrt {x}\, \sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {b}\, \sqrt {a}+\sqrt {x}\, b \right ) b^{2} x^{2}-4 a^{3}-20 a^{2} b x +30 a \,b^{2} x^{2}}{6 \sqrt {x}\, a^{4} x \left (-b x +a \right )} \]
( - 15*sqrt(x)*sqrt(b)*sqrt(a)*log( - sqrt(b)*sqrt(a) + sqrt(x)*b)*a*b*x + 15*sqrt(x)*sqrt(b)*sqrt(a)*log( - sqrt(b)*sqrt(a) + sqrt(x)*b)*b**2*x**2 + 15*sqrt(x)*sqrt(b)*sqrt(a)*log(sqrt(b)*sqrt(a) + sqrt(x)*b)*a*b*x - 15*s qrt(x)*sqrt(b)*sqrt(a)*log(sqrt(b)*sqrt(a) + sqrt(x)*b)*b**2*x**2 - 4*a**3 - 20*a**2*b*x + 30*a*b**2*x**2)/(6*sqrt(x)*a**4*x*(a - b*x))