Integrand size = 22, antiderivative size = 76 \[ \int \frac {\sqrt {c+d x}}{x^2 \sqrt {a+b x}} \, dx=-\frac {\sqrt {a+b x} \sqrt {c+d x}}{a x}+\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2} \sqrt {c}} \] Output:
-(b*x+a)^(1/2)*(d*x+c)^(1/2)/a/x+(-a*d+b*c)*arctanh(c^(1/2)*(b*x+a)^(1/2)/ a^(1/2)/(d*x+c)^(1/2))/a^(3/2)/c^(1/2)
Time = 10.04 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.01 \[ \int \frac {\sqrt {c+d x}}{x^2 \sqrt {a+b x}} \, dx=-\frac {\sqrt {a+b x} \sqrt {c+d x}}{a x}-\frac {(-b c+a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2} \sqrt {c}} \] Input:
Integrate[Sqrt[c + d*x]/(x^2*Sqrt[a + b*x]),x]
Output:
-((Sqrt[a + b*x]*Sqrt[c + d*x])/(a*x)) - ((-(b*c) + a*d)*ArcTanh[(Sqrt[c]* Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(a^(3/2)*Sqrt[c])
Time = 0.17 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {105, 104, 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.
\(\displaystyle \int \frac {\sqrt {c+d x}}{x^2 \sqrt {a+b x}} \, dx\) |
\(\Big \downarrow \) 105 |
\(\displaystyle -\frac {(b c-a d) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx}{2 a}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{a x}\) |
\(\Big \downarrow \) 104 |
\(\displaystyle -\frac {(b c-a d) \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{a}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{a x}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2} \sqrt {c}}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{a x}\) |
Input:
Int[Sqrt[c + d*x]/(x^2*Sqrt[a + b*x]),x]
Output:
-((Sqrt[a + b*x]*Sqrt[c + d*x])/(a*x)) + ((b*c - a*d)*ArcTanh[(Sqrt[c]*Sqr t[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(a^(3/2)*Sqrt[c])
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Simp[n*((d*e - c*f)/((m + 1)*(b*e - a*f))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] || !SumSimplerQ[p, 1]) && NeQ[m, -1]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(146\) vs. \(2(60)=120\).
Time = 0.22 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.93
method | result | size |
default | \(-\frac {\sqrt {x d +c}\, \sqrt {b x +a}\, \left (\ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) a d x -\ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}\right )}{2 a \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, x \sqrt {a c}}\) | \(147\) |
Input:
int((d*x+c)^(1/2)/x^2/(b*x+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/2*(d*x+c)^(1/2)*(b*x+a)^(1/2)/a*(ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a) *(d*x+c))^(1/2)+2*a*c)/x)*a*d*x-ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d* x+c))^(1/2)+2*a*c)/x)*b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/((b*x+a )*(d*x+c))^(1/2)/x/(a*c)^(1/2)
Time = 0.14 (sec) , antiderivative size = 252, normalized size of antiderivative = 3.32 \[ \int \frac {\sqrt {c+d x}}{x^2 \sqrt {a+b x}} \, dx=\left [-\frac {\sqrt {a c} {\left (b c - a d\right )} x \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) + 4 \, \sqrt {b x + a} \sqrt {d x + c} a c}{4 \, a^{2} c x}, -\frac {\sqrt {-a c} {\left (b c - a d\right )} x \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) + 2 \, \sqrt {b x + a} \sqrt {d x + c} a c}{2 \, a^{2} c x}\right ] \] Input:
integrate((d*x+c)^(1/2)/x^2/(b*x+a)^(1/2),x, algorithm="fricas")
Output:
[-1/4*(sqrt(a*c)*(b*c - a*d)*x*log((8*a^2*c^2 + (b^2*c^2 + 6*a*b*c*d + a^2 *d^2)*x^2 - 4*(2*a*c + (b*c + a*d)*x)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c ) + 8*(a*b*c^2 + a^2*c*d)*x)/x^2) + 4*sqrt(b*x + a)*sqrt(d*x + c)*a*c)/(a^ 2*c*x), -1/2*(sqrt(-a*c)*(b*c - a*d)*x*arctan(1/2*(2*a*c + (b*c + a*d)*x)* sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x + c)/(a*b*c*d*x^2 + a^2*c^2 + (a*b*c^2 + a^2*c*d)*x)) + 2*sqrt(b*x + a)*sqrt(d*x + c)*a*c)/(a^2*c*x)]
\[ \int \frac {\sqrt {c+d x}}{x^2 \sqrt {a+b x}} \, dx=\int \frac {\sqrt {c + d x}}{x^{2} \sqrt {a + b x}}\, dx \] Input:
integrate((d*x+c)**(1/2)/x**2/(b*x+a)**(1/2),x)
Output:
Integral(sqrt(c + d*x)/(x**2*sqrt(a + b*x)), x)
Exception generated. \[ \int \frac {\sqrt {c+d x}}{x^2 \sqrt {a+b x}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((d*x+c)^(1/2)/x^2/(b*x+a)^(1/2),x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for m ore detail
Leaf count of result is larger than twice the leaf count of optimal. 409 vs. \(2 (60) = 120\).
Time = 0.38 (sec) , antiderivative size = 409, normalized size of antiderivative = 5.38 \[ \int \frac {\sqrt {c+d x}}{x^2 \sqrt {a+b x}} \, dx=\frac {{\left (\frac {{\left (\sqrt {b d} b^{2} c - \sqrt {b d} a b d\right )} \arctan \left (-\frac {b^{2} c + a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt {-a b c d} b}\right )}{\sqrt {-a b c d} a b} - \frac {2 \, {\left (\sqrt {b d} b^{4} c^{2} - 2 \, \sqrt {b d} a b^{3} c d + \sqrt {b d} a^{2} b^{2} d^{2} - \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{2} c - \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b d\right )}}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2} - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{2} c - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b d + {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4}\right )} a}\right )} {\left | b \right |}}{b} \] Input:
integrate((d*x+c)^(1/2)/x^2/(b*x+a)^(1/2),x, algorithm="giac")
Output:
((sqrt(b*d)*b^2*c - sqrt(b*d)*a*b*d)*arctan(-1/2*(b^2*c + a*b*d - (sqrt(b* d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)/(sqrt(-a*b*c*d) *b))/(sqrt(-a*b*c*d)*a*b) - 2*(sqrt(b*d)*b^4*c^2 - 2*sqrt(b*d)*a*b^3*c*d + sqrt(b*d)*a^2*b^2*d^2 - sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^2*c - sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sq rt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b*d)/((b^4*c^2 - 2*a*b^3*c*d + a^2* b^2*d^2 - 2*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d) )^2*b^2*c - 2*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b* d))^2*a*b*d + (sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b* d))^4)*a))*abs(b)/b
Time = 3.73 (sec) , antiderivative size = 439, normalized size of antiderivative = 5.78 \[ \int \frac {\sqrt {c+d x}}{x^2 \sqrt {a+b x}} \, dx=\frac {\frac {\left (\frac {c\,b^2}{4}+\frac {a\,d\,b}{4}\right )\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{a^{3/2}\,\sqrt {c}\,d\,\left (\sqrt {c+d\,x}-\sqrt {c}\right )}-\frac {b^2}{4\,a\,d}+\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2\,\left (\frac {a^2\,d^2}{4}-\frac {3\,a\,b\,c\,d}{4}+\frac {b^2\,c^2}{4}\right )}{a^2\,c\,d\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^2}}{\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^3}{{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^3}+\frac {b\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{d\,\left (\sqrt {c+d\,x}-\sqrt {c}\right )}-\frac {\left (a\,d+b\,c\right )\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{\sqrt {a}\,\sqrt {c}\,d\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^2}}-\frac {d\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{4\,a\,\left (\sqrt {c+d\,x}-\sqrt {c}\right )}-\frac {\ln \left (\frac {\left (\sqrt {c}\,\sqrt {a+b\,x}-\sqrt {a}\,\sqrt {c+d\,x}\right )\,\left (b\,\sqrt {c}-\frac {\sqrt {a}\,d\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{\sqrt {c+d\,x}-\sqrt {c}}\right )}{\sqrt {c+d\,x}-\sqrt {c}}\right )\,\left (\sqrt {a}\,b\,c^{3/2}-a^{3/2}\,\sqrt {c}\,d\right )}{2\,a^2\,c}+\frac {\ln \left (\frac {\sqrt {a+b\,x}-\sqrt {a}}{\sqrt {c+d\,x}-\sqrt {c}}\right )\,\left (\sqrt {a}\,b\,c^{3/2}-a^{3/2}\,\sqrt {c}\,d\right )}{2\,a^2\,c} \] Input:
int((c + d*x)^(1/2)/(x^2*(a + b*x)^(1/2)),x)
Output:
((((b^2*c)/4 + (a*b*d)/4)*((a + b*x)^(1/2) - a^(1/2)))/(a^(3/2)*c^(1/2)*d* ((c + d*x)^(1/2) - c^(1/2))) - b^2/(4*a*d) + (((a + b*x)^(1/2) - a^(1/2))^ 2*((a^2*d^2)/4 + (b^2*c^2)/4 - (3*a*b*c*d)/4))/(a^2*c*d*((c + d*x)^(1/2) - c^(1/2))^2))/(((a + b*x)^(1/2) - a^(1/2))^3/((c + d*x)^(1/2) - c^(1/2))^3 + (b*((a + b*x)^(1/2) - a^(1/2)))/(d*((c + d*x)^(1/2) - c^(1/2))) - ((a*d + b*c)*((a + b*x)^(1/2) - a^(1/2))^2)/(a^(1/2)*c^(1/2)*d*((c + d*x)^(1/2) - c^(1/2))^2)) - (d*((a + b*x)^(1/2) - a^(1/2)))/(4*a*((c + d*x)^(1/2) - c^(1/2))) - (log(((c^(1/2)*(a + b*x)^(1/2) - a^(1/2)*(c + d*x)^(1/2))*(b*c ^(1/2) - (a^(1/2)*d*((a + b*x)^(1/2) - a^(1/2)))/((c + d*x)^(1/2) - c^(1/2 ))))/((c + d*x)^(1/2) - c^(1/2)))*(a^(1/2)*b*c^(3/2) - a^(3/2)*c^(1/2)*d)) /(2*a^2*c) + (log(((a + b*x)^(1/2) - a^(1/2))/((c + d*x)^(1/2) - c^(1/2))) *(a^(1/2)*b*c^(3/2) - a^(3/2)*c^(1/2)*d))/(2*a^2*c)
Time = 0.22 (sec) , antiderivative size = 305, normalized size of antiderivative = 4.01 \[ \int \frac {\sqrt {c+d x}}{x^2 \sqrt {a+b x}} \, dx=\frac {-2 \sqrt {d x +c}\, \sqrt {b x +a}\, a c +\sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (-\sqrt {2 \sqrt {d}\, \sqrt {c}\, \sqrt {b}\, \sqrt {a}+a d +b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}\right ) a d x -\sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (-\sqrt {2 \sqrt {d}\, \sqrt {c}\, \sqrt {b}\, \sqrt {a}+a d +b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}\right ) b c x +\sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {2 \sqrt {d}\, \sqrt {c}\, \sqrt {b}\, \sqrt {a}+a d +b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}\right ) a d x -\sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {2 \sqrt {d}\, \sqrt {c}\, \sqrt {b}\, \sqrt {a}+a d +b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}\right ) b c x -\sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (2 \sqrt {d}\, \sqrt {b}\, \sqrt {d x +c}\, \sqrt {b x +a}+2 \sqrt {d}\, \sqrt {c}\, \sqrt {b}\, \sqrt {a}+2 b d x \right ) a d x +\sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (2 \sqrt {d}\, \sqrt {b}\, \sqrt {d x +c}\, \sqrt {b x +a}+2 \sqrt {d}\, \sqrt {c}\, \sqrt {b}\, \sqrt {a}+2 b d x \right ) b c x}{2 a^{2} c x} \] Input:
int((d*x+c)^(1/2)/x^2/(b*x+a)^(1/2),x)
Output:
( - 2*sqrt(c + d*x)*sqrt(a + b*x)*a*c + sqrt(c)*sqrt(a)*log( - sqrt(2*sqrt (d)*sqrt(c)*sqrt(b)*sqrt(a) + a*d + b*c) + sqrt(d)*sqrt(a + b*x) + sqrt(b) *sqrt(c + d*x))*a*d*x - sqrt(c)*sqrt(a)*log( - sqrt(2*sqrt(d)*sqrt(c)*sqrt (b)*sqrt(a) + a*d + b*c) + sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))* b*c*x + sqrt(c)*sqrt(a)*log(sqrt(2*sqrt(d)*sqrt(c)*sqrt(b)*sqrt(a) + a*d + b*c) + sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))*a*d*x - sqrt(c)*sqr t(a)*log(sqrt(2*sqrt(d)*sqrt(c)*sqrt(b)*sqrt(a) + a*d + b*c) + sqrt(d)*sqr t(a + b*x) + sqrt(b)*sqrt(c + d*x))*b*c*x - sqrt(c)*sqrt(a)*log(2*sqrt(d)* sqrt(b)*sqrt(c + d*x)*sqrt(a + b*x) + 2*sqrt(d)*sqrt(c)*sqrt(b)*sqrt(a) + 2*b*d*x)*a*d*x + sqrt(c)*sqrt(a)*log(2*sqrt(d)*sqrt(b)*sqrt(c + d*x)*sqrt( a + b*x) + 2*sqrt(d)*sqrt(c)*sqrt(b)*sqrt(a) + 2*b*d*x)*b*c*x)/(2*a**2*c*x )