Integrand size = 31, antiderivative size = 87 \[ \int \frac {\sqrt {c+\sqrt {b+a x}}}{d-\sqrt {b+a x}} \, dx=-\frac {4 \sqrt {c+\sqrt {b+a x}} \left (c+3 d+\sqrt {b+a x}\right )}{3 a}-\frac {4 \sqrt {-c-d} d \arctan \left (\frac {\sqrt {-c-d} \sqrt {c+\sqrt {b+a x}}}{c+d}\right )}{a} \]
-4/3*(c+(a*x+b)^(1/2))^(1/2)*(c+3*d+(a*x+b)^(1/2))/a-4*(-c-d)^(1/2)*d*arct an((-c-d)^(1/2)*(c+(a*x+b)^(1/2))^(1/2)/(c+d))/a
Time = 0.16 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.93 \[ \int \frac {\sqrt {c+\sqrt {b+a x}}}{d-\sqrt {b+a x}} \, dx=\frac {-4 \sqrt {c+\sqrt {b+a x}} \left (c+3 d+\sqrt {b+a x}\right )+12 \sqrt {-c-d} d \arctan \left (\frac {\sqrt {c+\sqrt {b+a x}}}{\sqrt {-c-d}}\right )}{3 a} \]
(-4*Sqrt[c + Sqrt[b + a*x]]*(c + 3*d + Sqrt[b + a*x]) + 12*Sqrt[-c - d]*d* ArcTan[Sqrt[c + Sqrt[b + a*x]]/Sqrt[-c - d]])/(3*a)
Time = 0.20 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.90, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {938, 900, 90, 60, 73, 219}
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.
\(\displaystyle \int \frac {\sqrt {\sqrt {a x+b}+c}}{d-\sqrt {a x+b}} \, dx\) |
\(\Big \downarrow \) 938 |
\(\displaystyle \frac {\int \frac {\sqrt {c+\sqrt {b+a x}}}{d-\sqrt {b+a x}}d(b+a x)}{a}\) |
\(\Big \downarrow \) 900 |
\(\displaystyle \frac {2 \int \frac {\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}}{-b+d-a x}d\sqrt {b+a x}}{a}\) |
\(\Big \downarrow \) 90 |
\(\displaystyle \frac {2 \left (d \int \frac {\sqrt {c+\sqrt {b+a x}}}{-b+d-a x}d\sqrt {b+a x}-\frac {2}{3} \left (\sqrt {a x+b}+c\right )^{3/2}\right )}{a}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {2 \left (d \left ((c+d) \int \frac {1}{(-b+d-a x) \sqrt {c+\sqrt {b+a x}}}d\sqrt {b+a x}-2 \sqrt {\sqrt {a x+b}+c}\right )-\frac {2}{3} \left (\sqrt {a x+b}+c\right )^{3/2}\right )}{a}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {2 \left (d \left (2 (c+d) \int \frac {1}{-b+c+d-a x}d\sqrt {c+\sqrt {b+a x}}-2 \sqrt {\sqrt {a x+b}+c}\right )-\frac {2}{3} \left (\sqrt {a x+b}+c\right )^{3/2}\right )}{a}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {2 \left (d \left (2 \sqrt {c+d} \text {arctanh}\left (\frac {\sqrt {\sqrt {a x+b}+c}}{\sqrt {c+d}}\right )-2 \sqrt {\sqrt {a x+b}+c}\right )-\frac {2}{3} \left (\sqrt {a x+b}+c\right )^{3/2}\right )}{a}\) |
(2*((-2*(c + Sqrt[b + a*x])^(3/2))/3 + d*(-2*Sqrt[c + Sqrt[b + a*x]] + 2*S qrt[c + d]*ArcTanh[Sqrt[c + Sqrt[b + a*x]]/Sqrt[c + d]])))/a
3.12.95.3.1 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 + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
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_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p + 2)) Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol ] :> With[{g = Denominator[n]}, Simp[g Subst[Int[x^(g - 1)*(a + b*x^(g*n) )^p*(c + d*x^(g*n))^q, x], x, x^(1/g)], x]] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && FractionQ[n]
Int[((a_.) + (b_.)*(u_)^(n_))^(p_.)*((c_.) + (d_.)*(u_)^(n_))^(q_.), x_Symb ol] :> Simp[1/Coefficient[u, x, 1] Subst[Int[(a + b*x^n)^p*(c + d*x^n)^q, x], x, u], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && LinearQ[u, x] && NeQ[u , x]
Time = 0.70 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.69
method | result | size |
derivativedivides | \(\frac {-\frac {4 \left (c +\sqrt {a x +b}\right )^{\frac {3}{2}}}{3}-4 d \sqrt {c +\sqrt {a x +b}}+4 d \sqrt {c +d}\, \operatorname {arctanh}\left (\frac {\sqrt {c +\sqrt {a x +b}}}{\sqrt {c +d}}\right )}{a}\) | \(60\) |
default | \(\frac {-\frac {4 \left (c +\sqrt {a x +b}\right )^{\frac {3}{2}}}{3}-4 d \sqrt {c +\sqrt {a x +b}}+4 d \sqrt {c +d}\, \operatorname {arctanh}\left (\frac {\sqrt {c +\sqrt {a x +b}}}{\sqrt {c +d}}\right )}{a}\) | \(60\) |
2/a*(-2/3*(c+(a*x+b)^(1/2))^(3/2)-2*d*(c+(a*x+b)^(1/2))^(1/2)+2*d*(c+d)^(1 /2)*arctanh((c+(a*x+b)^(1/2))^(1/2)/(c+d)^(1/2)))
Time = 0.26 (sec) , antiderivative size = 183, normalized size of antiderivative = 2.10 \[ \int \frac {\sqrt {c+\sqrt {b+a x}}}{d-\sqrt {b+a x}} \, dx=\left [\frac {2 \, {\left (3 \, \sqrt {c + d} d \log \left (-\frac {2 \, c d + d^{2} + a x + 2 \, \sqrt {a x + b} {\left (c + d\right )} + 2 \, {\left (\sqrt {c + d} d + \sqrt {a x + b} \sqrt {c + d}\right )} \sqrt {c + \sqrt {a x + b}} + b}{d^{2} - a x - b}\right ) - 2 \, {\left (c + 3 \, d + \sqrt {a x + b}\right )} \sqrt {c + \sqrt {a x + b}}\right )}}{3 \, a}, -\frac {4 \, {\left (3 \, \sqrt {-c - d} d \arctan \left (\frac {\sqrt {c + \sqrt {a x + b}} \sqrt {-c - d}}{c + d}\right ) + {\left (c + 3 \, d + \sqrt {a x + b}\right )} \sqrt {c + \sqrt {a x + b}}\right )}}{3 \, a}\right ] \]
[2/3*(3*sqrt(c + d)*d*log(-(2*c*d + d^2 + a*x + 2*sqrt(a*x + b)*(c + d) + 2*(sqrt(c + d)*d + sqrt(a*x + b)*sqrt(c + d))*sqrt(c + sqrt(a*x + b)) + b) /(d^2 - a*x - b)) - 2*(c + 3*d + sqrt(a*x + b))*sqrt(c + sqrt(a*x + b)))/a , -4/3*(3*sqrt(-c - d)*d*arctan(sqrt(c + sqrt(a*x + b))*sqrt(-c - d)/(c + d)) + (c + 3*d + sqrt(a*x + b))*sqrt(c + sqrt(a*x + b)))/a]
Time = 1.09 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.08 \[ \int \frac {\sqrt {c+\sqrt {b+a x}}}{d-\sqrt {b+a x}} \, dx=\begin {cases} \frac {2 \left (- 2 d \sqrt {c + \sqrt {a x + b}} - \frac {2 d \left (c + d\right ) \operatorname {atan}{\left (\frac {\sqrt {c + \sqrt {a x + b}}}{\sqrt {- c - d}} \right )}}{\sqrt {- c - d}} - \frac {2 \left (c + \sqrt {a x + b}\right )^{\frac {3}{2}}}{3}\right )}{a} & \text {for}\: a \neq 0 \\\frac {x \sqrt {\sqrt {b} + c}}{- \sqrt {b} + d} & \text {otherwise} \end {cases} \]
Piecewise((2*(-2*d*sqrt(c + sqrt(a*x + b)) - 2*d*(c + d)*atan(sqrt(c + sqr t(a*x + b))/sqrt(-c - d))/sqrt(-c - d) - 2*(c + sqrt(a*x + b))**(3/2)/3)/a , Ne(a, 0)), (x*sqrt(sqrt(b) + c)/(-sqrt(b) + d), True))
Time = 0.28 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.01 \[ \int \frac {\sqrt {c+\sqrt {b+a x}}}{d-\sqrt {b+a x}} \, dx=-\frac {2 \, {\left (2 \, {\left (c + \sqrt {a x + b}\right )}^{\frac {3}{2}} + 6 \, \sqrt {c + \sqrt {a x + b}} d + \frac {3 \, {\left (c d + d^{2}\right )} \log \left (-\frac {\sqrt {c + d} - \sqrt {c + \sqrt {a x + b}}}{\sqrt {c + d} + \sqrt {c + \sqrt {a x + b}}}\right )}{\sqrt {c + d}}\right )}}{3 \, a} \]
-2/3*(2*(c + sqrt(a*x + b))^(3/2) + 6*sqrt(c + sqrt(a*x + b))*d + 3*(c*d + d^2)*log(-(sqrt(c + d) - sqrt(c + sqrt(a*x + b)))/(sqrt(c + d) + sqrt(c + sqrt(a*x + b))))/sqrt(c + d))/a
Time = 0.28 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt {c+\sqrt {b+a x}}}{d-\sqrt {b+a x}} \, dx=-\frac {4 \, {\left (c d + d^{2}\right )} \arctan \left (\frac {\sqrt {c + \sqrt {a x + b}}}{\sqrt {-c - d}}\right )}{a \sqrt {-c - d}} - \frac {4 \, {\left (a^{2} {\left (c + \sqrt {a x + b}\right )}^{\frac {3}{2}} + 3 \, a^{2} \sqrt {c + \sqrt {a x + b}} d\right )}}{3 \, a^{3}} \]
-4*(c*d + d^2)*arctan(sqrt(c + sqrt(a*x + b))/sqrt(-c - d))/(a*sqrt(-c - d )) - 4/3*(a^2*(c + sqrt(a*x + b))^(3/2) + 3*a^2*sqrt(c + sqrt(a*x + b))*d) /a^3
Time = 6.06 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.72 \[ \int \frac {\sqrt {c+\sqrt {b+a x}}}{d-\sqrt {b+a x}} \, dx=\frac {4\,d\,\mathrm {atanh}\left (\frac {\sqrt {c+\sqrt {b+a\,x}}}{\sqrt {c+d}}\right )\,\sqrt {c+d}}{a}-\frac {4\,d\,\sqrt {c+\sqrt {b+a\,x}}}{a}-\frac {4\,{\left (c+\sqrt {b+a\,x}\right )}^{3/2}}{3\,a} \]