Integrand size = 15, antiderivative size = 72 \[ \int \frac {x^{3/2}}{(-a+b x)^3} \, dx=-\frac {x^{3/2}}{2 b (a-b x)^2}+\frac {3 \sqrt {x}}{4 b^2 (a-b x)}-\frac {3 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 \sqrt {a} b^{5/2}} \]
-1/2*x^(3/2)/b/(-b*x+a)^2-3/4*arctanh(b^(1/2)*x^(1/2)/a^(1/2))/b^(5/2)/a^( 1/2)+3/4*x^(1/2)/b^2/(-b*x+a)
Time = 0.12 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.83 \[ \int \frac {x^{3/2}}{(-a+b x)^3} \, dx=\frac {\sqrt {x} (3 a-5 b x)}{4 b^2 (a-b x)^2}-\frac {3 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 \sqrt {a} b^{5/2}} \]
(Sqrt[x]*(3*a - 5*b*x))/(4*b^2*(a - b*x)^2) - (3*ArcTanh[(Sqrt[b]*Sqrt[x]) /Sqrt[a]])/(4*Sqrt[a]*b^(5/2))
Time = 0.17 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.04, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {51, 51, 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.
\(\displaystyle \int \frac {x^{3/2}}{(b x-a)^3} \, dx\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {3 \int \frac {\sqrt {x}}{(a-b x)^2}dx}{4 b}-\frac {x^{3/2}}{2 b (a-b x)^2}\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {3 \left (\frac {\sqrt {x}}{b (a-b x)}-\frac {\int \frac {1}{\sqrt {x} (a-b x)}dx}{2 b}\right )}{4 b}-\frac {x^{3/2}}{2 b (a-b x)^2}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {3 \left (\frac {\sqrt {x}}{b (a-b x)}-\frac {\int \frac {1}{a-b x}d\sqrt {x}}{b}\right )}{4 b}-\frac {x^{3/2}}{2 b (a-b x)^2}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {3 \left (\frac {\sqrt {x}}{b (a-b x)}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2}}\right )}{4 b}-\frac {x^{3/2}}{2 b (a-b x)^2}\) |
-1/2*x^(3/2)/(b*(a - b*x)^2) + (3*(Sqrt[x]/(b*(a - b*x)) - ArcTanh[(Sqrt[b ]*Sqrt[x])/Sqrt[a]]/(Sqrt[a]*b^(3/2))))/(4*b)
3.5.84.3.1 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/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x ] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
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 = 0.08 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.71
method | result | size |
derivativedivides | \(-\frac {2 \left (\frac {5 x^{\frac {3}{2}}}{8 b}-\frac {3 a \sqrt {x}}{8 b^{2}}\right )}{\left (-b x +a \right )^{2}}-\frac {3 \,\operatorname {arctanh}\left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 b^{2} \sqrt {a b}}\) | \(51\) |
default | \(-\frac {2 \left (\frac {5 x^{\frac {3}{2}}}{8 b}-\frac {3 a \sqrt {x}}{8 b^{2}}\right )}{\left (-b x +a \right )^{2}}-\frac {3 \,\operatorname {arctanh}\left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 b^{2} \sqrt {a b}}\) | \(51\) |
-2*(5/8*x^(3/2)/b-3/8*a*x^(1/2)/b^2)/(-b*x+a)^2-3/4/b^2/(a*b)^(1/2)*arctan h(b*x^(1/2)/(a*b)^(1/2))
Time = 0.24 (sec) , antiderivative size = 186, normalized size of antiderivative = 2.58 \[ \int \frac {x^{3/2}}{(-a+b x)^3} \, dx=\left [\frac {3 \, {\left (b^{2} x^{2} - 2 \, a b x + a^{2}\right )} \sqrt {a b} \log \left (\frac {b x + a - 2 \, \sqrt {a b} \sqrt {x}}{b x - a}\right ) - 2 \, {\left (5 \, a b^{2} x - 3 \, a^{2} b\right )} \sqrt {x}}{8 \, {\left (a b^{5} x^{2} - 2 \, a^{2} b^{4} x + a^{3} b^{3}\right )}}, \frac {3 \, {\left (b^{2} x^{2} - 2 \, a b x + a^{2}\right )} \sqrt {-a b} \arctan \left (\frac {\sqrt {-a b}}{b \sqrt {x}}\right ) - {\left (5 \, a b^{2} x - 3 \, a^{2} b\right )} \sqrt {x}}{4 \, {\left (a b^{5} x^{2} - 2 \, a^{2} b^{4} x + a^{3} b^{3}\right )}}\right ] \]
[1/8*(3*(b^2*x^2 - 2*a*b*x + a^2)*sqrt(a*b)*log((b*x + a - 2*sqrt(a*b)*sqr t(x))/(b*x - a)) - 2*(5*a*b^2*x - 3*a^2*b)*sqrt(x))/(a*b^5*x^2 - 2*a^2*b^4 *x + a^3*b^3), 1/4*(3*(b^2*x^2 - 2*a*b*x + a^2)*sqrt(-a*b)*arctan(sqrt(-a* b)/(b*sqrt(x))) - (5*a*b^2*x - 3*a^2*b)*sqrt(x))/(a*b^5*x^2 - 2*a^2*b^4*x + a^3*b^3)]
Leaf count of result is larger than twice the leaf count of optimal. 552 vs. \(2 (61) = 122\).
Time = 15.79 (sec) , antiderivative size = 552, normalized size of antiderivative = 7.67 \[ \int \frac {x^{3/2}}{(-a+b x)^3} \, dx=\begin {cases} \frac {\tilde {\infty }}{\sqrt {x}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {2 x^{\frac {5}{2}}}{5 a^{3}} & \text {for}\: b = 0 \\- \frac {2}{b^{3} \sqrt {x}} & \text {for}\: a = 0 \\\frac {3 a^{2} \log {\left (\sqrt {x} - \sqrt {\frac {a}{b}} \right )}}{8 a^{2} b^{3} \sqrt {\frac {a}{b}} - 16 a b^{4} x \sqrt {\frac {a}{b}} + 8 b^{5} x^{2} \sqrt {\frac {a}{b}}} - \frac {3 a^{2} \log {\left (\sqrt {x} + \sqrt {\frac {a}{b}} \right )}}{8 a^{2} b^{3} \sqrt {\frac {a}{b}} - 16 a b^{4} x \sqrt {\frac {a}{b}} + 8 b^{5} x^{2} \sqrt {\frac {a}{b}}} + \frac {6 a b \sqrt {x} \sqrt {\frac {a}{b}}}{8 a^{2} b^{3} \sqrt {\frac {a}{b}} - 16 a b^{4} x \sqrt {\frac {a}{b}} + 8 b^{5} x^{2} \sqrt {\frac {a}{b}}} - \frac {6 a b x \log {\left (\sqrt {x} - \sqrt {\frac {a}{b}} \right )}}{8 a^{2} b^{3} \sqrt {\frac {a}{b}} - 16 a b^{4} x \sqrt {\frac {a}{b}} + 8 b^{5} x^{2} \sqrt {\frac {a}{b}}} + \frac {6 a b x \log {\left (\sqrt {x} + \sqrt {\frac {a}{b}} \right )}}{8 a^{2} b^{3} \sqrt {\frac {a}{b}} - 16 a b^{4} x \sqrt {\frac {a}{b}} + 8 b^{5} x^{2} \sqrt {\frac {a}{b}}} - \frac {10 b^{2} x^{\frac {3}{2}} \sqrt {\frac {a}{b}}}{8 a^{2} b^{3} \sqrt {\frac {a}{b}} - 16 a b^{4} x \sqrt {\frac {a}{b}} + 8 b^{5} x^{2} \sqrt {\frac {a}{b}}} + \frac {3 b^{2} x^{2} \log {\left (\sqrt {x} - \sqrt {\frac {a}{b}} \right )}}{8 a^{2} b^{3} \sqrt {\frac {a}{b}} - 16 a b^{4} x \sqrt {\frac {a}{b}} + 8 b^{5} x^{2} \sqrt {\frac {a}{b}}} - \frac {3 b^{2} x^{2} \log {\left (\sqrt {x} + \sqrt {\frac {a}{b}} \right )}}{8 a^{2} b^{3} \sqrt {\frac {a}{b}} - 16 a b^{4} x \sqrt {\frac {a}{b}} + 8 b^{5} x^{2} \sqrt {\frac {a}{b}}} & \text {otherwise} \end {cases} \]
Piecewise((zoo/sqrt(x), Eq(a, 0) & Eq(b, 0)), (-2*x**(5/2)/(5*a**3), Eq(b, 0)), (-2/(b**3*sqrt(x)), Eq(a, 0)), (3*a**2*log(sqrt(x) - sqrt(a/b))/(8*a **2*b**3*sqrt(a/b) - 16*a*b**4*x*sqrt(a/b) + 8*b**5*x**2*sqrt(a/b)) - 3*a* *2*log(sqrt(x) + sqrt(a/b))/(8*a**2*b**3*sqrt(a/b) - 16*a*b**4*x*sqrt(a/b) + 8*b**5*x**2*sqrt(a/b)) + 6*a*b*sqrt(x)*sqrt(a/b)/(8*a**2*b**3*sqrt(a/b) - 16*a*b**4*x*sqrt(a/b) + 8*b**5*x**2*sqrt(a/b)) - 6*a*b*x*log(sqrt(x) - sqrt(a/b))/(8*a**2*b**3*sqrt(a/b) - 16*a*b**4*x*sqrt(a/b) + 8*b**5*x**2*sq rt(a/b)) + 6*a*b*x*log(sqrt(x) + sqrt(a/b))/(8*a**2*b**3*sqrt(a/b) - 16*a* b**4*x*sqrt(a/b) + 8*b**5*x**2*sqrt(a/b)) - 10*b**2*x**(3/2)*sqrt(a/b)/(8* a**2*b**3*sqrt(a/b) - 16*a*b**4*x*sqrt(a/b) + 8*b**5*x**2*sqrt(a/b)) + 3*b **2*x**2*log(sqrt(x) - sqrt(a/b))/(8*a**2*b**3*sqrt(a/b) - 16*a*b**4*x*sqr t(a/b) + 8*b**5*x**2*sqrt(a/b)) - 3*b**2*x**2*log(sqrt(x) + sqrt(a/b))/(8* a**2*b**3*sqrt(a/b) - 16*a*b**4*x*sqrt(a/b) + 8*b**5*x**2*sqrt(a/b)), True ))
Time = 0.29 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.08 \[ \int \frac {x^{3/2}}{(-a+b x)^3} \, dx=-\frac {5 \, b x^{\frac {3}{2}} - 3 \, a \sqrt {x}}{4 \, {\left (b^{4} x^{2} - 2 \, a b^{3} x + a^{2} b^{2}\right )}} + \frac {3 \, \log \left (\frac {b \sqrt {x} - \sqrt {a b}}{b \sqrt {x} + \sqrt {a b}}\right )}{8 \, \sqrt {a b} b^{2}} \]
-1/4*(5*b*x^(3/2) - 3*a*sqrt(x))/(b^4*x^2 - 2*a*b^3*x + a^2*b^2) + 3/8*log ((b*sqrt(x) - sqrt(a*b))/(b*sqrt(x) + sqrt(a*b)))/(sqrt(a*b)*b^2)
Time = 0.29 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.71 \[ \int \frac {x^{3/2}}{(-a+b x)^3} \, dx=\frac {3 \, \arctan \left (\frac {b \sqrt {x}}{\sqrt {-a b}}\right )}{4 \, \sqrt {-a b} b^{2}} - \frac {5 \, b x^{\frac {3}{2}} - 3 \, a \sqrt {x}}{4 \, {\left (b x - a\right )}^{2} b^{2}} \]
3/4*arctan(b*sqrt(x)/sqrt(-a*b))/(sqrt(-a*b)*b^2) - 1/4*(5*b*x^(3/2) - 3*a *sqrt(x))/((b*x - a)^2*b^2)
Time = 0.17 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.81 \[ \int \frac {x^{3/2}}{(-a+b x)^3} \, dx=-\frac {\frac {5\,x^{3/2}}{4\,b}-\frac {3\,a\,\sqrt {x}}{4\,b^2}}{a^2-2\,a\,b\,x+b^2\,x^2}-\frac {3\,\mathrm {atanh}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )}{4\,\sqrt {a}\,b^{5/2}} \]