Integrand size = 15, antiderivative size = 72 \[ \int \frac {1}{\sqrt {x} (-a+b x)^3} \, dx=-\frac {\sqrt {x}}{2 a (a-b x)^2}-\frac {3 \sqrt {x}}{4 a^2 (a-b x)}-\frac {3 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{5/2} \sqrt {b}} \] Output:
-1/2*x^(1/2)/a/(-b*x+a)^2-3/4*x^(1/2)/a^2/(-b*x+a)-3/4*arctanh(b^(1/2)*x^( 1/2)/a^(1/2))/a^(5/2)/b^(1/2)
Time = 0.07 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.83 \[ \int \frac {1}{\sqrt {x} (-a+b x)^3} \, dx=\frac {\sqrt {x} (-5 a+3 b x)}{4 a^2 (a-b x)^2}-\frac {3 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{5/2} \sqrt {b}} \] Input:
Integrate[1/(Sqrt[x]*(-a + b*x)^3),x]
Output:
(Sqrt[x]*(-5*a + 3*b*x))/(4*a^2*(a - b*x)^2) - (3*ArcTanh[(Sqrt[b]*Sqrt[x] )/Sqrt[a]])/(4*a^(5/2)*Sqrt[b])
Time = 0.15 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.03, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {52, 52, 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.
\(\displaystyle \int \frac {1}{\sqrt {x} (b x-a)^3} \, dx\) |
\(\Big \downarrow \) 52 |
\(\displaystyle -\frac {3 \int \frac {1}{\sqrt {x} (a-b x)^2}dx}{4 a}-\frac {\sqrt {x}}{2 a (a-b x)^2}\) |
\(\Big \downarrow \) 52 |
\(\displaystyle -\frac {3 \left (\frac {\int \frac {1}{\sqrt {x} (a-b x)}dx}{2 a}+\frac {\sqrt {x}}{a (a-b x)}\right )}{4 a}-\frac {\sqrt {x}}{2 a (a-b x)^2}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle -\frac {3 \left (\frac {\int \frac {1}{a-b x}d\sqrt {x}}{a}+\frac {\sqrt {x}}{a (a-b x)}\right )}{4 a}-\frac {\sqrt {x}}{2 a (a-b x)^2}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -\frac {3 \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{3/2} \sqrt {b}}+\frac {\sqrt {x}}{a (a-b x)}\right )}{4 a}-\frac {\sqrt {x}}{2 a (a-b x)^2}\) |
Input:
Int[1/(Sqrt[x]*(-a + b*x)^3),x]
Output:
-1/2*Sqrt[x]/(a*(a - b*x)^2) - (3*(Sqrt[x]/(a*(a - b*x)) + ArcTanh[(Sqrt[b ]*Sqrt[x])/Sqrt[a]]/(a^(3/2)*Sqrt[b])))/(4*a)
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] :> 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 = 61, normalized size of antiderivative = 0.85
method | result | size |
derivativedivides | \(-\frac {\sqrt {x}}{2 a \left (-b x +a \right )^{2}}-\frac {3 \left (\frac {\sqrt {x}}{2 a \left (-b x +a \right )}+\frac {\operatorname {arctanh}\left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{2 a \sqrt {a b}}\right )}{2 a}\) | \(61\) |
default | \(-\frac {\sqrt {x}}{2 a \left (-b x +a \right )^{2}}-\frac {3 \left (\frac {\sqrt {x}}{2 a \left (-b x +a \right )}+\frac {\operatorname {arctanh}\left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{2 a \sqrt {a b}}\right )}{2 a}\) | \(61\) |
Input:
int(1/x^(1/2)/(b*x-a)^3,x,method=_RETURNVERBOSE)
Output:
-1/2*x^(1/2)/a/(-b*x+a)^2-3/2/a*(1/2*x^(1/2)/a/(-b*x+a)+1/2/a/(a*b)^(1/2)* arctanh(b*x^(1/2)/(a*b)^(1/2)))
Time = 0.09 (sec) , antiderivative size = 185, normalized size of antiderivative = 2.57 \[ \int \frac {1}{\sqrt {x} (-a+b x)^3} \, dx=\left [\frac {3 \, {\left (b^{2} x^{2} - 2 \, a b x + a^{2}\right )} \sqrt {a b} \log \left (\frac {b x + a - 2 \, \sqrt {a b} \sqrt {x}}{b x - a}\right ) + 2 \, {\left (3 \, a b^{2} x - 5 \, a^{2} b\right )} \sqrt {x}}{8 \, {\left (a^{3} b^{3} x^{2} - 2 \, a^{4} b^{2} x + a^{5} b\right )}}, \frac {3 \, {\left (b^{2} x^{2} - 2 \, a b x + a^{2}\right )} \sqrt {-a b} \arctan \left (\frac {\sqrt {-a b}}{b \sqrt {x}}\right ) + {\left (3 \, a b^{2} x - 5 \, a^{2} b\right )} \sqrt {x}}{4 \, {\left (a^{3} b^{3} x^{2} - 2 \, a^{4} b^{2} x + a^{5} b\right )}}\right ] \] Input:
integrate(1/x^(1/2)/(b*x-a)^3,x, algorithm="fricas")
Output:
[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*(3*a*b^2*x - 5*a^2*b)*sqrt(x))/(a^3*b^3*x^2 - 2*a^4*b ^2*x + a^5*b), 1/4*(3*(b^2*x^2 - 2*a*b*x + a^2)*sqrt(-a*b)*arctan(sqrt(-a* b)/(b*sqrt(x))) + (3*a*b^2*x - 5*a^2*b)*sqrt(x))/(a^3*b^3*x^2 - 2*a^4*b^2* x + a^5*b)]
Leaf count of result is larger than twice the leaf count of optimal. 580 vs. \(2 (63) = 126\).
Time = 7.57 (sec) , antiderivative size = 580, normalized size of antiderivative = 8.06 \[ \int \frac {1}{\sqrt {x} (-a+b x)^3} \, dx=\begin {cases} \frac {\tilde {\infty }}{x^{\frac {5}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {2 \sqrt {x}}{a^{3}} & \text {for}\: b = 0 \\- \frac {2}{5 b^{3} x^{\frac {5}{2}}} & \text {for}\: a = 0 \\\frac {3 a^{2} \log {\left (\sqrt {x} - \sqrt {\frac {a}{b}} \right )}}{8 a^{4} b \sqrt {\frac {a}{b}} - 16 a^{3} b^{2} x \sqrt {\frac {a}{b}} + 8 a^{2} b^{3} x^{2} \sqrt {\frac {a}{b}}} - \frac {3 a^{2} \log {\left (\sqrt {x} + \sqrt {\frac {a}{b}} \right )}}{8 a^{4} b \sqrt {\frac {a}{b}} - 16 a^{3} b^{2} x \sqrt {\frac {a}{b}} + 8 a^{2} b^{3} x^{2} \sqrt {\frac {a}{b}}} - \frac {10 a b \sqrt {x} \sqrt {\frac {a}{b}}}{8 a^{4} b \sqrt {\frac {a}{b}} - 16 a^{3} b^{2} x \sqrt {\frac {a}{b}} + 8 a^{2} b^{3} x^{2} \sqrt {\frac {a}{b}}} - \frac {6 a b x \log {\left (\sqrt {x} - \sqrt {\frac {a}{b}} \right )}}{8 a^{4} b \sqrt {\frac {a}{b}} - 16 a^{3} b^{2} x \sqrt {\frac {a}{b}} + 8 a^{2} b^{3} x^{2} \sqrt {\frac {a}{b}}} + \frac {6 a b x \log {\left (\sqrt {x} + \sqrt {\frac {a}{b}} \right )}}{8 a^{4} b \sqrt {\frac {a}{b}} - 16 a^{3} b^{2} x \sqrt {\frac {a}{b}} + 8 a^{2} b^{3} x^{2} \sqrt {\frac {a}{b}}} + \frac {6 b^{2} x^{\frac {3}{2}} \sqrt {\frac {a}{b}}}{8 a^{4} b \sqrt {\frac {a}{b}} - 16 a^{3} b^{2} x \sqrt {\frac {a}{b}} + 8 a^{2} b^{3} x^{2} \sqrt {\frac {a}{b}}} + \frac {3 b^{2} x^{2} \log {\left (\sqrt {x} - \sqrt {\frac {a}{b}} \right )}}{8 a^{4} b \sqrt {\frac {a}{b}} - 16 a^{3} b^{2} x \sqrt {\frac {a}{b}} + 8 a^{2} b^{3} x^{2} \sqrt {\frac {a}{b}}} - \frac {3 b^{2} x^{2} \log {\left (\sqrt {x} + \sqrt {\frac {a}{b}} \right )}}{8 a^{4} b \sqrt {\frac {a}{b}} - 16 a^{3} b^{2} x \sqrt {\frac {a}{b}} + 8 a^{2} b^{3} x^{2} \sqrt {\frac {a}{b}}} & \text {otherwise} \end {cases} \] Input:
integrate(1/x**(1/2)/(b*x-a)**3,x)
Output:
Piecewise((zoo/x**(5/2), Eq(a, 0) & Eq(b, 0)), (-2*sqrt(x)/a**3, Eq(b, 0)) , (-2/(5*b**3*x**(5/2)), Eq(a, 0)), (3*a**2*log(sqrt(x) - sqrt(a/b))/(8*a* *4*b*sqrt(a/b) - 16*a**3*b**2*x*sqrt(a/b) + 8*a**2*b**3*x**2*sqrt(a/b)) - 3*a**2*log(sqrt(x) + sqrt(a/b))/(8*a**4*b*sqrt(a/b) - 16*a**3*b**2*x*sqrt( a/b) + 8*a**2*b**3*x**2*sqrt(a/b)) - 10*a*b*sqrt(x)*sqrt(a/b)/(8*a**4*b*sq rt(a/b) - 16*a**3*b**2*x*sqrt(a/b) + 8*a**2*b**3*x**2*sqrt(a/b)) - 6*a*b*x *log(sqrt(x) - sqrt(a/b))/(8*a**4*b*sqrt(a/b) - 16*a**3*b**2*x*sqrt(a/b) + 8*a**2*b**3*x**2*sqrt(a/b)) + 6*a*b*x*log(sqrt(x) + sqrt(a/b))/(8*a**4*b* sqrt(a/b) - 16*a**3*b**2*x*sqrt(a/b) + 8*a**2*b**3*x**2*sqrt(a/b)) + 6*b** 2*x**(3/2)*sqrt(a/b)/(8*a**4*b*sqrt(a/b) - 16*a**3*b**2*x*sqrt(a/b) + 8*a* *2*b**3*x**2*sqrt(a/b)) + 3*b**2*x**2*log(sqrt(x) - sqrt(a/b))/(8*a**4*b*s qrt(a/b) - 16*a**3*b**2*x*sqrt(a/b) + 8*a**2*b**3*x**2*sqrt(a/b)) - 3*b**2 *x**2*log(sqrt(x) + sqrt(a/b))/(8*a**4*b*sqrt(a/b) - 16*a**3*b**2*x*sqrt(a /b) + 8*a**2*b**3*x**2*sqrt(a/b)), True))
Time = 0.13 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.07 \[ \int \frac {1}{\sqrt {x} (-a+b x)^3} \, dx=\frac {3 \, b x^{\frac {3}{2}} - 5 \, a \sqrt {x}}{4 \, {\left (a^{2} b^{2} x^{2} - 2 \, a^{3} b x + a^{4}\right )}} + \frac {3 \, \log \left (\frac {b \sqrt {x} - \sqrt {a b}}{b \sqrt {x} + \sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{2}} \] Input:
integrate(1/x^(1/2)/(b*x-a)^3,x, algorithm="maxima")
Output:
1/4*(3*b*x^(3/2) - 5*a*sqrt(x))/(a^2*b^2*x^2 - 2*a^3*b*x + a^4) + 3/8*log( (b*sqrt(x) - sqrt(a*b))/(b*sqrt(x) + sqrt(a*b)))/(sqrt(a*b)*a^2)
Time = 0.13 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.71 \[ \int \frac {1}{\sqrt {x} (-a+b x)^3} \, dx=\frac {3 \, \arctan \left (\frac {b \sqrt {x}}{\sqrt {-a b}}\right )}{4 \, \sqrt {-a b} a^{2}} + \frac {3 \, b x^{\frac {3}{2}} - 5 \, a \sqrt {x}}{4 \, {\left (b x - a\right )}^{2} a^{2}} \] Input:
integrate(1/x^(1/2)/(b*x-a)^3,x, algorithm="giac")
Output:
3/4*arctan(b*sqrt(x)/sqrt(-a*b))/(sqrt(-a*b)*a^2) + 1/4*(3*b*x^(3/2) - 5*a *sqrt(x))/((b*x - a)^2*a^2)
Time = 0.08 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.81 \[ \int \frac {1}{\sqrt {x} (-a+b x)^3} \, dx=-\frac {\frac {5\,\sqrt {x}}{4\,a}-\frac {3\,b\,x^{3/2}}{4\,a^2}}{a^2-2\,a\,b\,x+b^2\,x^2}-\frac {3\,\mathrm {atanh}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )}{4\,a^{5/2}\,\sqrt {b}} \] Input:
int(-1/(x^(1/2)*(a - b*x)^3),x)
Output:
- ((5*x^(1/2))/(4*a) - (3*b*x^(3/2))/(4*a^2))/(a^2 + b^2*x^2 - 2*a*b*x) - (3*atanh((b^(1/2)*x^(1/2))/a^(1/2)))/(4*a^(5/2)*b^(1/2))
Time = 0.16 (sec) , antiderivative size = 173, normalized size of antiderivative = 2.40 \[ \int \frac {1}{\sqrt {x} (-a+b x)^3} \, dx=\frac {3 \sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (-\sqrt {b}\, \sqrt {a}+\sqrt {x}\, b \right ) a^{2}-6 \sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (-\sqrt {b}\, \sqrt {a}+\sqrt {x}\, b \right ) a b x +3 \sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (-\sqrt {b}\, \sqrt {a}+\sqrt {x}\, b \right ) b^{2} x^{2}-3 \sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {b}\, \sqrt {a}+\sqrt {x}\, b \right ) a^{2}+6 \sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {b}\, \sqrt {a}+\sqrt {x}\, b \right ) a b x -3 \sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {b}\, \sqrt {a}+\sqrt {x}\, b \right ) b^{2} x^{2}-10 \sqrt {x}\, a^{2} b +6 \sqrt {x}\, a \,b^{2} x}{8 a^{3} b \left (b^{2} x^{2}-2 a b x +a^{2}\right )} \] Input:
int(1/x^(1/2)/(b*x-a)^3,x)
Output:
(3*sqrt(b)*sqrt(a)*log( - sqrt(b)*sqrt(a) + sqrt(x)*b)*a**2 - 6*sqrt(b)*sq rt(a)*log( - sqrt(b)*sqrt(a) + sqrt(x)*b)*a*b*x + 3*sqrt(b)*sqrt(a)*log( - sqrt(b)*sqrt(a) + sqrt(x)*b)*b**2*x**2 - 3*sqrt(b)*sqrt(a)*log(sqrt(b)*sq rt(a) + sqrt(x)*b)*a**2 + 6*sqrt(b)*sqrt(a)*log(sqrt(b)*sqrt(a) + sqrt(x)* b)*a*b*x - 3*sqrt(b)*sqrt(a)*log(sqrt(b)*sqrt(a) + sqrt(x)*b)*b**2*x**2 - 10*sqrt(x)*a**2*b + 6*sqrt(x)*a*b**2*x)/(8*a**3*b*(a**2 - 2*a*b*x + b**2*x **2))