Integrand size = 21, antiderivative size = 97 \[ \int \frac {1}{x \sqrt {a+b \sqrt {c+d x}}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {a-b \sqrt {c}}}\right )}{\sqrt {a-b \sqrt {c}}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {a+b \sqrt {c}}}\right )}{\sqrt {a+b \sqrt {c}}} \]
-2*arctanh((a+b*(d*x+c)^(1/2))^(1/2)/(a-b*c^(1/2))^(1/2))/(a-b*c^(1/2))^(1 /2)-2*arctanh((a+b*(d*x+c)^(1/2))^(1/2)/(a+b*c^(1/2))^(1/2))/(a+b*c^(1/2)) ^(1/2)
Time = 0.16 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.08 \[ \int \frac {1}{x \sqrt {a+b \sqrt {c+d x}}} \, dx=\frac {2 \arctan \left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {-a-b \sqrt {c}}}\right )}{\sqrt {-a-b \sqrt {c}}}+\frac {2 \arctan \left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {-a+b \sqrt {c}}}\right )}{\sqrt {-a+b \sqrt {c}}} \]
(2*ArcTan[Sqrt[a + b*Sqrt[c + d*x]]/Sqrt[-a - b*Sqrt[c]]])/Sqrt[-a - b*Sqr t[c]] + (2*ArcTan[Sqrt[a + b*Sqrt[c + d*x]]/Sqrt[-a + b*Sqrt[c]]])/Sqrt[-a + b*Sqrt[c]]
Time = 0.31 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.15, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {896, 25, 1732, 561, 27, 1480, 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.
\(\displaystyle \int \frac {1}{x \sqrt {a+b \sqrt {c+d x}}} \, dx\) |
\(\Big \downarrow \) 896 |
\(\displaystyle \int \frac {1}{d x \sqrt {a+b \sqrt {c+d x}}}d(c+d x)\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int -\frac {1}{d x \sqrt {a+b \sqrt {c+d x}}}d(c+d x)\) |
\(\Big \downarrow \) 1732 |
\(\displaystyle -2 \int -\frac {\sqrt {c+d x}}{d x \sqrt {a+b \sqrt {c+d x}}}d\sqrt {c+d x}\) |
\(\Big \downarrow \) 561 |
\(\displaystyle -\frac {4 \int \frac {a-c-d x}{b \left (\frac {a^2}{b^2}-\frac {2 (c+d x) a}{b^2}+\frac {(c+d x)^2}{b^2}-c\right )}d\sqrt {a+b \sqrt {c+d x}}}{b}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {4 \int \frac {a-c-d x}{\frac {a^2}{b^2}-\frac {2 (c+d x) a}{b^2}+\frac {(c+d x)^2}{b^2}-c}d\sqrt {a+b \sqrt {c+d x}}}{b^2}\) |
\(\Big \downarrow \) 1480 |
\(\displaystyle -\frac {4 \left (-\frac {1}{2} \int \frac {1}{\frac {c+d x}{b^2}-\frac {a-b \sqrt {c}}{b^2}}d\sqrt {a+b \sqrt {c+d x}}-\frac {1}{2} \int \frac {1}{\frac {c+d x}{b^2}-\frac {a+b \sqrt {c}}{b^2}}d\sqrt {a+b \sqrt {c+d x}}\right )}{b^2}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -\frac {4 \left (\frac {b^2 \text {arctanh}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {a-b \sqrt {c}}}\right )}{2 \sqrt {a-b \sqrt {c}}}+\frac {b^2 \text {arctanh}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {a+b \sqrt {c}}}\right )}{2 \sqrt {a+b \sqrt {c}}}\right )}{b^2}\) |
(-4*((b^2*ArcTanh[Sqrt[a + b*Sqrt[c + d*x]]/Sqrt[a - b*Sqrt[c]]])/(2*Sqrt[ a - b*Sqrt[c]]) + (b^2*ArcTanh[Sqrt[a + b*Sqrt[c + d*x]]/Sqrt[a + b*Sqrt[c ]]])/(2*Sqrt[a + b*Sqrt[c]])))/b^2
3.7.50.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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]
Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> With[{k = Denominator[n]}, Simp[k/d Subst[Int[x^(k*(n + 1) - 1)*(-c /d + x^k/d)^m*Simp[(b*c^2 + a*d^2)/d^2 - 2*b*c*(x^k/d^2) + b*(x^(2*k)/d^2), x]^p, x], x, (c + d*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, m, p}, x] && Frac tionQ[n] && IntegerQ[p] && IntegerQ[m]
Int[((a_) + (b_.)*(v_)^(n_))^(p_.)*(x_)^(m_.), x_Symbol] :> With[{c = Coeff icient[v, x, 0], d = Coefficient[v, x, 1]}, Simp[1/d^(m + 1) Subst[Int[Si mplifyIntegrand[(x - c)^m*(a + b*x^n)^p, x], x], x, v], x] /; NeQ[c, 0]] /; FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && IntegerQ[m]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q)) Int[1/( b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q)) Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
Int[((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symb ol] :> With[{g = Denominator[n]}, Simp[g Subst[Int[x^(g - 1)*(d + e*x^(g* n))^q*(a + c*x^(2*g*n))^p, x], x, x^(1/g)], x]] /; FreeQ[{a, c, d, e, p, q} , x] && EqQ[n2, 2*n] && FractionQ[n]
Time = 0.28 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.95
method | result | size |
derivativedivides | \(\frac {2 \arctan \left (\frac {\sqrt {a +b \sqrt {d x +c}}}{\sqrt {-\sqrt {b^{2} c}-a}}\right )}{\sqrt {-\sqrt {b^{2} c}-a}}+\frac {2 \arctan \left (\frac {\sqrt {a +b \sqrt {d x +c}}}{\sqrt {\sqrt {b^{2} c}-a}}\right )}{\sqrt {\sqrt {b^{2} c}-a}}\) | \(92\) |
default | \(\frac {2 \arctan \left (\frac {\sqrt {a +b \sqrt {d x +c}}}{\sqrt {-\sqrt {b^{2} c}-a}}\right )}{\sqrt {-\sqrt {b^{2} c}-a}}+\frac {2 \arctan \left (\frac {\sqrt {a +b \sqrt {d x +c}}}{\sqrt {\sqrt {b^{2} c}-a}}\right )}{\sqrt {\sqrt {b^{2} c}-a}}\) | \(92\) |
2/(-(b^2*c)^(1/2)-a)^(1/2)*arctan((a+b*(d*x+c)^(1/2))^(1/2)/(-(b^2*c)^(1/2 )-a)^(1/2))+2/((b^2*c)^(1/2)-a)^(1/2)*arctan((a+b*(d*x+c)^(1/2))^(1/2)/((b ^2*c)^(1/2)-a)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 743 vs. \(2 (73) = 146\).
Time = 0.33 (sec) , antiderivative size = 743, normalized size of antiderivative = 7.66 \[ \int \frac {1}{x \sqrt {a+b \sqrt {c+d x}}} \, dx=\sqrt {-\frac {{\left (b^{2} c - a^{2}\right )} \sqrt {\frac {b^{2} c}{b^{4} c^{2} - 2 \, a^{2} b^{2} c + a^{4}}} + a}{b^{2} c - a^{2}}} \log \left (4 \, {\left ({\left (b^{2} c - a^{2}\right )} \sqrt {\frac {b^{2} c}{b^{4} c^{2} - 2 \, a^{2} b^{2} c + a^{4}}} - a\right )} \sqrt {-\frac {{\left (b^{2} c - a^{2}\right )} \sqrt {\frac {b^{2} c}{b^{4} c^{2} - 2 \, a^{2} b^{2} c + a^{4}}} + a}{b^{2} c - a^{2}}} + 4 \, \sqrt {\sqrt {d x + c} b + a}\right ) - \sqrt {-\frac {{\left (b^{2} c - a^{2}\right )} \sqrt {\frac {b^{2} c}{b^{4} c^{2} - 2 \, a^{2} b^{2} c + a^{4}}} + a}{b^{2} c - a^{2}}} \log \left (-4 \, {\left ({\left (b^{2} c - a^{2}\right )} \sqrt {\frac {b^{2} c}{b^{4} c^{2} - 2 \, a^{2} b^{2} c + a^{4}}} - a\right )} \sqrt {-\frac {{\left (b^{2} c - a^{2}\right )} \sqrt {\frac {b^{2} c}{b^{4} c^{2} - 2 \, a^{2} b^{2} c + a^{4}}} + a}{b^{2} c - a^{2}}} + 4 \, \sqrt {\sqrt {d x + c} b + a}\right ) - \sqrt {\frac {{\left (b^{2} c - a^{2}\right )} \sqrt {\frac {b^{2} c}{b^{4} c^{2} - 2 \, a^{2} b^{2} c + a^{4}}} - a}{b^{2} c - a^{2}}} \log \left (4 \, {\left ({\left (b^{2} c - a^{2}\right )} \sqrt {\frac {b^{2} c}{b^{4} c^{2} - 2 \, a^{2} b^{2} c + a^{4}}} + a\right )} \sqrt {\frac {{\left (b^{2} c - a^{2}\right )} \sqrt {\frac {b^{2} c}{b^{4} c^{2} - 2 \, a^{2} b^{2} c + a^{4}}} - a}{b^{2} c - a^{2}}} + 4 \, \sqrt {\sqrt {d x + c} b + a}\right ) + \sqrt {\frac {{\left (b^{2} c - a^{2}\right )} \sqrt {\frac {b^{2} c}{b^{4} c^{2} - 2 \, a^{2} b^{2} c + a^{4}}} - a}{b^{2} c - a^{2}}} \log \left (-4 \, {\left ({\left (b^{2} c - a^{2}\right )} \sqrt {\frac {b^{2} c}{b^{4} c^{2} - 2 \, a^{2} b^{2} c + a^{4}}} + a\right )} \sqrt {\frac {{\left (b^{2} c - a^{2}\right )} \sqrt {\frac {b^{2} c}{b^{4} c^{2} - 2 \, a^{2} b^{2} c + a^{4}}} - a}{b^{2} c - a^{2}}} + 4 \, \sqrt {\sqrt {d x + c} b + a}\right ) \]
sqrt(-((b^2*c - a^2)*sqrt(b^2*c/(b^4*c^2 - 2*a^2*b^2*c + a^4)) + a)/(b^2*c - a^2))*log(4*((b^2*c - a^2)*sqrt(b^2*c/(b^4*c^2 - 2*a^2*b^2*c + a^4)) - a)*sqrt(-((b^2*c - a^2)*sqrt(b^2*c/(b^4*c^2 - 2*a^2*b^2*c + a^4)) + a)/(b^ 2*c - a^2)) + 4*sqrt(sqrt(d*x + c)*b + a)) - sqrt(-((b^2*c - a^2)*sqrt(b^2 *c/(b^4*c^2 - 2*a^2*b^2*c + a^4)) + a)/(b^2*c - a^2))*log(-4*((b^2*c - a^2 )*sqrt(b^2*c/(b^4*c^2 - 2*a^2*b^2*c + a^4)) - a)*sqrt(-((b^2*c - a^2)*sqrt (b^2*c/(b^4*c^2 - 2*a^2*b^2*c + a^4)) + a)/(b^2*c - a^2)) + 4*sqrt(sqrt(d* x + c)*b + a)) - sqrt(((b^2*c - a^2)*sqrt(b^2*c/(b^4*c^2 - 2*a^2*b^2*c + a ^4)) - a)/(b^2*c - a^2))*log(4*((b^2*c - a^2)*sqrt(b^2*c/(b^4*c^2 - 2*a^2* b^2*c + a^4)) + a)*sqrt(((b^2*c - a^2)*sqrt(b^2*c/(b^4*c^2 - 2*a^2*b^2*c + a^4)) - a)/(b^2*c - a^2)) + 4*sqrt(sqrt(d*x + c)*b + a)) + sqrt(((b^2*c - a^2)*sqrt(b^2*c/(b^4*c^2 - 2*a^2*b^2*c + a^4)) - a)/(b^2*c - a^2))*log(-4 *((b^2*c - a^2)*sqrt(b^2*c/(b^4*c^2 - 2*a^2*b^2*c + a^4)) + a)*sqrt(((b^2* c - a^2)*sqrt(b^2*c/(b^4*c^2 - 2*a^2*b^2*c + a^4)) - a)/(b^2*c - a^2)) + 4 *sqrt(sqrt(d*x + c)*b + a))
\[ \int \frac {1}{x \sqrt {a+b \sqrt {c+d x}}} \, dx=\int \frac {1}{x \sqrt {a + b \sqrt {c + d x}}}\, dx \]
\[ \int \frac {1}{x \sqrt {a+b \sqrt {c+d x}}} \, dx=\int { \frac {1}{\sqrt {\sqrt {d x + c} b + a} x} \,d x } \]
Time = 0.38 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.44 \[ \int \frac {1}{x \sqrt {a+b \sqrt {c+d x}}} \, dx=\frac {2 \, {\left (\frac {{\left (b^{2} \sqrt {c} {\left | b \right |} + a b^{2}\right )} \arctan \left (\frac {\sqrt {\sqrt {d x + c} b + a}}{\sqrt {-a + \sqrt {b^{2} c}}}\right )}{{\left (b \sqrt {c} + a\right )} \sqrt {b \sqrt {c} - a}} + \frac {{\left (b^{2} \sqrt {c} {\left | b \right |} - a b^{2}\right )} \arctan \left (\frac {\sqrt {\sqrt {d x + c} b + a}}{\sqrt {-a - \sqrt {b^{2} c}}}\right )}{{\left (b \sqrt {c} - a\right )} \sqrt {-b \sqrt {c} - a}}\right )}}{b^{2}} \]
2*((b^2*sqrt(c)*abs(b) + a*b^2)*arctan(sqrt(sqrt(d*x + c)*b + a)/sqrt(-a + sqrt(b^2*c)))/((b*sqrt(c) + a)*sqrt(b*sqrt(c) - a)) + (b^2*sqrt(c)*abs(b) - a*b^2)*arctan(sqrt(sqrt(d*x + c)*b + a)/sqrt(-a - sqrt(b^2*c)))/((b*sqr t(c) - a)*sqrt(-b*sqrt(c) - a)))/b^2
Timed out. \[ \int \frac {1}{x \sqrt {a+b \sqrt {c+d x}}} \, dx=\int \frac {1}{x\,\sqrt {a+b\,\sqrt {c+d\,x}}} \,d x \]