Integrand size = 32, antiderivative size = 114 \[ \int \frac {1}{x \sqrt {c+d x} \sqrt {b c^2-b d^2 x^2}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c} \sqrt {c+d x}}{\sqrt {b c^2-b d^2 x^2}}\right )}{\sqrt {b} c^{3/2}}+\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {c} \sqrt {c+d x}}{\sqrt {b c^2-b d^2 x^2}}\right )}{\sqrt {b} c^{3/2}} \] Output:
-2*arctanh(b^(1/2)*c^(1/2)*(d*x+c)^(1/2)/(-b*d^2*x^2+b*c^2)^(1/2))/b^(1/2) /c^(3/2)+2^(1/2)*arctanh(2^(1/2)*b^(1/2)*c^(1/2)*(d*x+c)^(1/2)/(-b*d^2*x^2 +b*c^2)^(1/2))/b^(1/2)/c^(3/2)
Time = 0.24 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.04 \[ \int \frac {1}{x \sqrt {c+d x} \sqrt {b c^2-b d^2 x^2}} \, dx=\frac {\sqrt {c^2-d^2 x^2} \left (-2 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {c+d x}}{\sqrt {c^2-d^2 x^2}}\right )+\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {c+d x}}{\sqrt {c^2-d^2 x^2}}\right )\right )}{c^{3/2} \sqrt {b \left (c^2-d^2 x^2\right )}} \] Input:
Integrate[1/(x*Sqrt[c + d*x]*Sqrt[b*c^2 - b*d^2*x^2]),x]
Output:
(Sqrt[c^2 - d^2*x^2]*(-2*ArcTanh[(Sqrt[c]*Sqrt[c + d*x])/Sqrt[c^2 - d^2*x^ 2]] + Sqrt[2]*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[c + d*x])/Sqrt[c^2 - d^2*x^2]] ))/(c^(3/2)*Sqrt[b*(c^2 - d^2*x^2)])
Time = 0.44 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.11, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {586, 97, 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.
\(\displaystyle \int \frac {1}{x \sqrt {c+d x} \sqrt {b c^2-b d^2 x^2}} \, dx\) |
\(\Big \downarrow \) 586 |
\(\displaystyle \frac {\sqrt {c+d x} \sqrt {b c-b d x} \int \frac {1}{x (c+d x) \sqrt {b c-b d x}}dx}{\sqrt {b c^2-b d^2 x^2}}\) |
\(\Big \downarrow \) 97 |
\(\displaystyle \frac {\sqrt {c+d x} \sqrt {b c-b d x} \left (\frac {\int \frac {1}{x \sqrt {b c-b d x}}dx}{c}-\frac {d \int \frac {1}{(c+d x) \sqrt {b c-b d x}}dx}{c}\right )}{\sqrt {b c^2-b d^2 x^2}}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {\sqrt {c+d x} \sqrt {b c-b d x} \left (\frac {2 \int \frac {1}{2 c-\frac {b c-b d x}{b}}d\sqrt {b c-b d x}}{b c}-\frac {2 \int \frac {1}{\frac {c}{d}-\frac {b c-b d x}{b d}}d\sqrt {b c-b d x}}{b c d}\right )}{\sqrt {b c^2-b d^2 x^2}}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {\sqrt {c+d x} \sqrt {b c-b d x} \left (\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {b c-b d x}}{\sqrt {2} \sqrt {b} \sqrt {c}}\right )}{\sqrt {b} c^{3/2}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {b c-b d x}}{\sqrt {b} \sqrt {c}}\right )}{\sqrt {b} c^{3/2}}\right )}{\sqrt {b c^2-b d^2 x^2}}\) |
Input:
Int[1/(x*Sqrt[c + d*x]*Sqrt[b*c^2 - b*d^2*x^2]),x]
Output:
(Sqrt[c + d*x]*Sqrt[b*c - b*d*x]*((-2*ArcTanh[Sqrt[b*c - b*d*x]/(Sqrt[b]*S qrt[c])])/(Sqrt[b]*c^(3/2)) + (Sqrt[2]*ArcTanh[Sqrt[b*c - b*d*x]/(Sqrt[2]* Sqrt[b]*Sqrt[c])])/(Sqrt[b]*c^(3/2))))/Sqrt[b*c^2 - b*d^2*x^2]
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[((e_.) + (f_.)*(x_))^(p_)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_] :> Simp[b/(b*c - a*d) Int[(e + f*x)^p/(a + b*x), x], x] - Simp[d/(b*c - a*d) Int[(e + f*x)^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && !IntegerQ[p]
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]
Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_) , x_Symbol] :> Simp[(a + b*x^2)^FracPart[p]/((c + d*x)^FracPart[p]*(a/c + ( b*x)/d)^FracPart[p]) Int[(e*x)^m*(c + d*x)^(n + p)*(a/c + (b/d)*x)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && EqQ[b*c^2 + a*d^2, 0]
Time = 0.24 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.77
method | result | size |
default | \(\frac {\sqrt {b \left (-d^{2} x^{2}+c^{2}\right )}\, \left (\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {\left (-d x +c \right ) b}\, \sqrt {2}}{2 \sqrt {b c}}\right )-2 \,\operatorname {arctanh}\left (\frac {\sqrt {\left (-d x +c \right ) b}}{\sqrt {b c}}\right )\right )}{\sqrt {d x +c}\, \sqrt {\left (-d x +c \right ) b}\, c \sqrt {b c}}\) | \(88\) |
Input:
int(1/x/(d*x+c)^(1/2)/(-b*d^2*x^2+b*c^2)^(1/2),x,method=_RETURNVERBOSE)
Output:
(b*(-d^2*x^2+c^2))^(1/2)*(2^(1/2)*arctanh(1/2*((-d*x+c)*b)^(1/2)*2^(1/2)/( b*c)^(1/2))-2*arctanh(((-d*x+c)*b)^(1/2)/(b*c)^(1/2)))/(d*x+c)^(1/2)/((-d* x+c)*b)^(1/2)/c/(b*c)^(1/2)
Time = 0.15 (sec) , antiderivative size = 308, normalized size of antiderivative = 2.70 \[ \int \frac {1}{x \sqrt {c+d x} \sqrt {b c^2-b d^2 x^2}} \, dx=\left [\frac {\sqrt {2} b c \sqrt {\frac {1}{b c}} \log \left (-\frac {d^{2} x^{2} - 2 \, c d x - 2 \, \sqrt {2} \sqrt {-b d^{2} x^{2} + b c^{2}} \sqrt {d x + c} c \sqrt {\frac {1}{b c}} - 3 \, c^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) + 2 \, \sqrt {b c} \log \left (-\frac {b d^{2} x^{2} - b c d x - 2 \, b c^{2} + 2 \, \sqrt {-b d^{2} x^{2} + b c^{2}} \sqrt {b c} \sqrt {d x + c}}{d x^{2} + c x}\right )}{2 \, b c^{2}}, \frac {\sqrt {2} b c \sqrt {-\frac {1}{b c}} \arctan \left (\frac {\sqrt {2} \sqrt {-b d^{2} x^{2} + b c^{2}} \sqrt {d x + c} c \sqrt {-\frac {1}{b c}}}{d^{2} x^{2} - c^{2}}\right ) + 2 \, \sqrt {-b c} \arctan \left (\frac {\sqrt {-b d^{2} x^{2} + b c^{2}} \sqrt {-b c} \sqrt {d x + c}}{b c d x + b c^{2}}\right )}{b c^{2}}\right ] \] Input:
integrate(1/x/(d*x+c)^(1/2)/(-b*d^2*x^2+b*c^2)^(1/2),x, algorithm="fricas" )
Output:
[1/2*(sqrt(2)*b*c*sqrt(1/(b*c))*log(-(d^2*x^2 - 2*c*d*x - 2*sqrt(2)*sqrt(- b*d^2*x^2 + b*c^2)*sqrt(d*x + c)*c*sqrt(1/(b*c)) - 3*c^2)/(d^2*x^2 + 2*c*d *x + c^2)) + 2*sqrt(b*c)*log(-(b*d^2*x^2 - b*c*d*x - 2*b*c^2 + 2*sqrt(-b*d ^2*x^2 + b*c^2)*sqrt(b*c)*sqrt(d*x + c))/(d*x^2 + c*x)))/(b*c^2), (sqrt(2) *b*c*sqrt(-1/(b*c))*arctan(sqrt(2)*sqrt(-b*d^2*x^2 + b*c^2)*sqrt(d*x + c)* c*sqrt(-1/(b*c))/(d^2*x^2 - c^2)) + 2*sqrt(-b*c)*arctan(sqrt(-b*d^2*x^2 + b*c^2)*sqrt(-b*c)*sqrt(d*x + c)/(b*c*d*x + b*c^2)))/(b*c^2)]
\[ \int \frac {1}{x \sqrt {c+d x} \sqrt {b c^2-b d^2 x^2}} \, dx=\int \frac {1}{x \sqrt {- b \left (- c + d x\right ) \left (c + d x\right )} \sqrt {c + d x}}\, dx \] Input:
integrate(1/x/(d*x+c)**(1/2)/(-b*d**2*x**2+b*c**2)**(1/2),x)
Output:
Integral(1/(x*sqrt(-b*(-c + d*x)*(c + d*x))*sqrt(c + d*x)), x)
\[ \int \frac {1}{x \sqrt {c+d x} \sqrt {b c^2-b d^2 x^2}} \, dx=\int { \frac {1}{\sqrt {-b d^{2} x^{2} + b c^{2}} \sqrt {d x + c} x} \,d x } \] Input:
integrate(1/x/(d*x+c)^(1/2)/(-b*d^2*x^2+b*c^2)^(1/2),x, algorithm="maxima" )
Output:
integrate(1/(sqrt(-b*d^2*x^2 + b*c^2)*sqrt(d*x + c)*x), x)
Time = 0.12 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.67 \[ \int \frac {1}{x \sqrt {c+d x} \sqrt {b c^2-b d^2 x^2}} \, dx=-\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {-{\left (d x + c\right )} b + 2 \, b c}}{2 \, \sqrt {-b c}}\right )}{\sqrt {-b c} c} + \frac {2 \, \arctan \left (\frac {\sqrt {-{\left (d x + c\right )} b + 2 \, b c}}{\sqrt {-b c}}\right )}{\sqrt {-b c} c} \] Input:
integrate(1/x/(d*x+c)^(1/2)/(-b*d^2*x^2+b*c^2)^(1/2),x, algorithm="giac")
Output:
-sqrt(2)*arctan(1/2*sqrt(2)*sqrt(-(d*x + c)*b + 2*b*c)/sqrt(-b*c))/(sqrt(- b*c)*c) + 2*arctan(sqrt(-(d*x + c)*b + 2*b*c)/sqrt(-b*c))/(sqrt(-b*c)*c)
Timed out. \[ \int \frac {1}{x \sqrt {c+d x} \sqrt {b c^2-b d^2 x^2}} \, dx=\int \frac {1}{x\,\sqrt {b\,c^2-b\,d^2\,x^2}\,\sqrt {c+d\,x}} \,d x \] Input:
int(1/(x*(b*c^2 - b*d^2*x^2)^(1/2)*(c + d*x)^(1/2)),x)
Output:
int(1/(x*(b*c^2 - b*d^2*x^2)^(1/2)*(c + d*x)^(1/2)), x)
Time = 0.21 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.68 \[ \int \frac {1}{x \sqrt {c+d x} \sqrt {b c^2-b d^2 x^2}} \, dx=\frac {\sqrt {c}\, \sqrt {b}\, \left (-\sqrt {2}\, \mathrm {log}\left (\sqrt {-d x +c}-\sqrt {c}\, \sqrt {2}\right )+\sqrt {2}\, \mathrm {log}\left (\sqrt {-d x +c}+\sqrt {c}\, \sqrt {2}\right )+2 \,\mathrm {log}\left (\sqrt {-d x +c}-\sqrt {c}\right )-2 \,\mathrm {log}\left (\sqrt {-d x +c}+\sqrt {c}\right )\right )}{2 b \,c^{2}} \] Input:
int(1/x/(d*x+c)^(1/2)/(-b*d^2*x^2+b*c^2)^(1/2),x)
Output:
(sqrt(c)*sqrt(b)*( - sqrt(2)*log(sqrt(c - d*x) - sqrt(c)*sqrt(2)) + sqrt(2 )*log(sqrt(c - d*x) + sqrt(c)*sqrt(2)) + 2*log(sqrt(c - d*x) - sqrt(c)) - 2*log(sqrt(c - d*x) + sqrt(c))))/(2*b*c**2)