Integrand size = 19, antiderivative size = 172 \[ \int \frac {x}{\sqrt {a+b \sqrt {\frac {c}{x}}}} \, dx=-\frac {7 b c^2 \sqrt {a+b \sqrt {\frac {c}{x}}}}{12 a^2 \left (\frac {c}{x}\right )^{3/2}}-\frac {35 b^3 c^2 \sqrt {a+b \sqrt {\frac {c}{x}}}}{32 a^4 \sqrt {\frac {c}{x}}}+\frac {35 b^2 c \sqrt {a+b \sqrt {\frac {c}{x}}} x}{48 a^3}+\frac {\sqrt {a+b \sqrt {\frac {c}{x}}} x^2}{2 a}+\frac {35 b^4 c^2 \text {arctanh}\left (\frac {\sqrt {a+b \sqrt {\frac {c}{x}}}}{\sqrt {a}}\right )}{32 a^{9/2}} \]
35/32*b^4*c^2*arctanh((a+b*(c/x)^(1/2))^(1/2)/a^(1/2))/a^(9/2)-7/12*b*c^2* (a+b*(c/x)^(1/2))^(1/2)/a^2/(c/x)^(3/2)+35/48*b^2*c*x*(a+b*(c/x)^(1/2))^(1 /2)/a^3+1/2*x^2*(a+b*(c/x)^(1/2))^(1/2)/a-35/32*b^3*c^2*(a+b*(c/x)^(1/2))^ (1/2)/a^4/(c/x)^(1/2)
Time = 0.15 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.65 \[ \int \frac {x}{\sqrt {a+b \sqrt {\frac {c}{x}}}} \, dx=\frac {\sqrt {a+b \sqrt {\frac {c}{x}}} \left (48 a^3-56 a^2 b \sqrt {\frac {c}{x}}-105 b^3 \left (\frac {c}{x}\right )^{3/2}+\frac {70 a b^2 c}{x}\right ) x^2}{96 a^4}+\frac {35 b^4 c^2 \text {arctanh}\left (\frac {\sqrt {a+b \sqrt {\frac {c}{x}}}}{\sqrt {a}}\right )}{32 a^{9/2}} \]
(Sqrt[a + b*Sqrt[c/x]]*(48*a^3 - 56*a^2*b*Sqrt[c/x] - 105*b^3*(c/x)^(3/2) + (70*a*b^2*c)/x)*x^2)/(96*a^4) + (35*b^4*c^2*ArcTanh[Sqrt[a + b*Sqrt[c/x] ]/Sqrt[a]])/(32*a^(9/2))
Time = 0.28 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.13, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {893, 798, 52, 52, 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.
| \(\displaystyle \int \frac {x}{\sqrt {a+b \sqrt {\frac {c}{x}}}} \, dx\) |
| \(\Big \downarrow \) 893 |
| \(\displaystyle \int \frac {x}{\sqrt {a+\frac {b \sqrt {c}}{\sqrt {x}}}}dx\) |
| \(\Big \downarrow \) 798 |
| \(\displaystyle -2 \int \frac {x^{5/2}}{\sqrt {a+\frac {b \sqrt {c}}{\sqrt {x}}}}d\frac {1}{\sqrt {x}}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle -2 \left (-\frac {7 b \sqrt {c} \int \frac {x^2}{\sqrt {a+\frac {b \sqrt {c}}{\sqrt {x}}}}d\frac {1}{\sqrt {x}}}{8 a}-\frac {x^2 \sqrt {a+\frac {b \sqrt {c}}{\sqrt {x}}}}{4 a}\right )\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle -2 \left (-\frac {7 b \sqrt {c} \left (-\frac {5 b \sqrt {c} \int \frac {x^{3/2}}{\sqrt {a+\frac {b \sqrt {c}}{\sqrt {x}}}}d\frac {1}{\sqrt {x}}}{6 a}-\frac {x^{3/2} \sqrt {a+\frac {b \sqrt {c}}{\sqrt {x}}}}{3 a}\right )}{8 a}-\frac {x^2 \sqrt {a+\frac {b \sqrt {c}}{\sqrt {x}}}}{4 a}\right )\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle -2 \left (-\frac {7 b \sqrt {c} \left (-\frac {5 b \sqrt {c} \left (-\frac {3 b \sqrt {c} \int \frac {x}{\sqrt {a+\frac {b \sqrt {c}}{\sqrt {x}}}}d\frac {1}{\sqrt {x}}}{4 a}-\frac {x \sqrt {a+\frac {b \sqrt {c}}{\sqrt {x}}}}{2 a}\right )}{6 a}-\frac {x^{3/2} \sqrt {a+\frac {b \sqrt {c}}{\sqrt {x}}}}{3 a}\right )}{8 a}-\frac {x^2 \sqrt {a+\frac {b \sqrt {c}}{\sqrt {x}}}}{4 a}\right )\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle -2 \left (-\frac {7 b \sqrt {c} \left (-\frac {5 b \sqrt {c} \left (-\frac {3 b \sqrt {c} \left (-\frac {b \sqrt {c} \int \frac {\sqrt {x}}{\sqrt {a+\frac {b \sqrt {c}}{\sqrt {x}}}}d\frac {1}{\sqrt {x}}}{2 a}-\frac {\sqrt {x} \sqrt {a+\frac {b \sqrt {c}}{\sqrt {x}}}}{a}\right )}{4 a}-\frac {x \sqrt {a+\frac {b \sqrt {c}}{\sqrt {x}}}}{2 a}\right )}{6 a}-\frac {x^{3/2} \sqrt {a+\frac {b \sqrt {c}}{\sqrt {x}}}}{3 a}\right )}{8 a}-\frac {x^2 \sqrt {a+\frac {b \sqrt {c}}{\sqrt {x}}}}{4 a}\right )\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -2 \left (-\frac {7 b \sqrt {c} \left (-\frac {5 b \sqrt {c} \left (-\frac {3 b \sqrt {c} \left (-\frac {\int \frac {1}{\frac {1}{b \sqrt {c} x}-\frac {a}{b \sqrt {c}}}d\sqrt {a+\frac {b \sqrt {c}}{\sqrt {x}}}}{a}-\frac {\sqrt {x} \sqrt {a+\frac {b \sqrt {c}}{\sqrt {x}}}}{a}\right )}{4 a}-\frac {x \sqrt {a+\frac {b \sqrt {c}}{\sqrt {x}}}}{2 a}\right )}{6 a}-\frac {x^{3/2} \sqrt {a+\frac {b \sqrt {c}}{\sqrt {x}}}}{3 a}\right )}{8 a}-\frac {x^2 \sqrt {a+\frac {b \sqrt {c}}{\sqrt {x}}}}{4 a}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -2 \left (-\frac {7 b \sqrt {c} \left (-\frac {5 b \sqrt {c} \left (-\frac {3 b \sqrt {c} \left (\frac {b \sqrt {c} \text {arctanh}\left (\frac {\sqrt {a+\frac {b \sqrt {c}}{\sqrt {x}}}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {\sqrt {x} \sqrt {a+\frac {b \sqrt {c}}{\sqrt {x}}}}{a}\right )}{4 a}-\frac {x \sqrt {a+\frac {b \sqrt {c}}{\sqrt {x}}}}{2 a}\right )}{6 a}-\frac {x^{3/2} \sqrt {a+\frac {b \sqrt {c}}{\sqrt {x}}}}{3 a}\right )}{8 a}-\frac {x^2 \sqrt {a+\frac {b \sqrt {c}}{\sqrt {x}}}}{4 a}\right )\) |
-2*(-1/4*(Sqrt[a + (b*Sqrt[c])/Sqrt[x]]*x^2)/a - (7*b*Sqrt[c]*(-1/3*(Sqrt[ a + (b*Sqrt[c])/Sqrt[x]]*x^(3/2))/a - (5*b*Sqrt[c]*(-1/2*(Sqrt[a + (b*Sqrt [c])/Sqrt[x]]*x)/a - (3*b*Sqrt[c]*(-((Sqrt[a + (b*Sqrt[c])/Sqrt[x]]*Sqrt[x ])/a) + (b*Sqrt[c]*ArcTanh[Sqrt[a + (b*Sqrt[c])/Sqrt[x]]/Sqrt[a]])/a^(3/2) ))/(4*a)))/(6*a)))/(8*a))
3.30.88.3.1 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 + 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]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/n Subst [Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*((c_.)*(x_)^(q_))^(n_))^(p_.), x_Symbo l] :> With[{k = Denominator[n]}, Subst[Int[(d*x)^m*(a + b*c^n*x^(n*q))^p, x ], x^(1/k), (c*x^q)^(1/k)/(c^(1/k)*(x^(1/k))^(q - 1))]] /; FreeQ[{a, b, c, d, m, p, q}, x] && FractionQ[n]
Leaf count of result is larger than twice the leaf count of optimal. \(297\) vs. \(2(134)=268\).
Time = 4.40 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.73
| method | result | size |
| default | \(\frac {\sqrt {a +b \sqrt {\frac {c}{x}}}\, \sqrt {x}\, \left (174 a^{\frac {3}{2}} \sqrt {a x +b \sqrt {\frac {c}{x}}\, x}\, \left (\frac {c}{x}\right )^{\frac {3}{2}} x^{\frac {3}{2}} b^{3}-384 a^{\frac {3}{2}} \sqrt {x \left (a +b \sqrt {\frac {c}{x}}\right )}\, \left (\frac {c}{x}\right )^{\frac {3}{2}} x^{\frac {3}{2}} b^{3}+348 c \,a^{\frac {5}{2}} \sqrt {a x +b \sqrt {\frac {c}{x}}\, x}\, \sqrt {x}\, b^{2}-87 c^{2} \ln \left (\frac {b \sqrt {\frac {c}{x}}\, \sqrt {x}+2 \sqrt {a x +b \sqrt {\frac {c}{x}}\, x}\, \sqrt {a}+2 a \sqrt {x}}{2 \sqrt {a}}\right ) a \,b^{4}+192 a \,c^{2} \ln \left (\frac {b \sqrt {\frac {c}{x}}\, \sqrt {x}+2 \sqrt {x \left (a +b \sqrt {\frac {c}{x}}\right )}\, \sqrt {a}+2 a \sqrt {x}}{2 \sqrt {a}}\right ) b^{4}+96 \sqrt {x}\, \left (a x +b \sqrt {\frac {c}{x}}\, x \right )^{\frac {3}{2}} a^{\frac {7}{2}}-208 a^{\frac {5}{2}} \left (a x +b \sqrt {\frac {c}{x}}\, x \right )^{\frac {3}{2}} \sqrt {\frac {c}{x}}\, \sqrt {x}\, b \right )}{192 \sqrt {x \left (a +b \sqrt {\frac {c}{x}}\right )}\, a^{\frac {11}{2}}}\) | \(298\) |
1/192*(a+b*(c/x)^(1/2))^(1/2)*x^(1/2)*(174*a^(3/2)*(a*x+b*(c/x)^(1/2)*x)^( 1/2)*(c/x)^(3/2)*x^(3/2)*b^3-384*a^(3/2)*(x*(a+b*(c/x)^(1/2)))^(1/2)*(c/x) ^(3/2)*x^(3/2)*b^3+348*c*a^(5/2)*(a*x+b*(c/x)^(1/2)*x)^(1/2)*x^(1/2)*b^2-8 7*c^2*ln(1/2*(b*(c/x)^(1/2)*x^(1/2)+2*(a*x+b*(c/x)^(1/2)*x)^(1/2)*a^(1/2)+ 2*a*x^(1/2))/a^(1/2))*a*b^4+192*a*c^2*ln(1/2*(b*(c/x)^(1/2)*x^(1/2)+2*(x*( a+b*(c/x)^(1/2)))^(1/2)*a^(1/2)+2*a*x^(1/2))/a^(1/2))*b^4+96*x^(1/2)*(a*x+ b*(c/x)^(1/2)*x)^(3/2)*a^(7/2)-208*a^(5/2)*(a*x+b*(c/x)^(1/2)*x)^(3/2)*(c/ x)^(1/2)*x^(1/2)*b)/(x*(a+b*(c/x)^(1/2)))^(1/2)/a^(11/2)
Time = 0.33 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.30 \[ \int \frac {x}{\sqrt {a+b \sqrt {\frac {c}{x}}}} \, dx=\left [\frac {105 \, \sqrt {a} b^{4} c^{2} \log \left (2 \, \sqrt {b \sqrt {\frac {c}{x}} + a} \sqrt {a} x \sqrt {\frac {c}{x}} + 2 \, a x \sqrt {\frac {c}{x}} + b c\right ) + 2 \, {\left (70 \, a^{2} b^{2} c x + 48 \, a^{4} x^{2} - 7 \, {\left (15 \, a b^{3} c x + 8 \, a^{3} b x^{2}\right )} \sqrt {\frac {c}{x}}\right )} \sqrt {b \sqrt {\frac {c}{x}} + a}}{192 \, a^{5}}, -\frac {105 \, \sqrt {-a} b^{4} c^{2} \arctan \left (\frac {\sqrt {b \sqrt {\frac {c}{x}} + a} \sqrt {-a}}{a}\right ) - {\left (70 \, a^{2} b^{2} c x + 48 \, a^{4} x^{2} - 7 \, {\left (15 \, a b^{3} c x + 8 \, a^{3} b x^{2}\right )} \sqrt {\frac {c}{x}}\right )} \sqrt {b \sqrt {\frac {c}{x}} + a}}{96 \, a^{5}}\right ] \]
[1/192*(105*sqrt(a)*b^4*c^2*log(2*sqrt(b*sqrt(c/x) + a)*sqrt(a)*x*sqrt(c/x ) + 2*a*x*sqrt(c/x) + b*c) + 2*(70*a^2*b^2*c*x + 48*a^4*x^2 - 7*(15*a*b^3* c*x + 8*a^3*b*x^2)*sqrt(c/x))*sqrt(b*sqrt(c/x) + a))/a^5, -1/96*(105*sqrt( -a)*b^4*c^2*arctan(sqrt(b*sqrt(c/x) + a)*sqrt(-a)/a) - (70*a^2*b^2*c*x + 4 8*a^4*x^2 - 7*(15*a*b^3*c*x + 8*a^3*b*x^2)*sqrt(c/x))*sqrt(b*sqrt(c/x) + a ))/a^5]
\[ \int \frac {x}{\sqrt {a+b \sqrt {\frac {c}{x}}}} \, dx=\int \frac {x}{\sqrt {a + b \sqrt {\frac {c}{x}}}}\, dx \]
Time = 0.30 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.23 \[ \int \frac {x}{\sqrt {a+b \sqrt {\frac {c}{x}}}} \, dx=-\frac {1}{192} \, c^{2} {\left (\frac {105 \, b^{4} \log \left (\frac {\sqrt {b \sqrt {\frac {c}{x}} + a} - \sqrt {a}}{\sqrt {b \sqrt {\frac {c}{x}} + a} + \sqrt {a}}\right )}{a^{\frac {9}{2}}} + \frac {2 \, {\left (105 \, {\left (b \sqrt {\frac {c}{x}} + a\right )}^{\frac {7}{2}} b^{4} - 385 \, {\left (b \sqrt {\frac {c}{x}} + a\right )}^{\frac {5}{2}} a b^{4} + 511 \, {\left (b \sqrt {\frac {c}{x}} + a\right )}^{\frac {3}{2}} a^{2} b^{4} - 279 \, \sqrt {b \sqrt {\frac {c}{x}} + a} a^{3} b^{4}\right )}}{{\left (b \sqrt {\frac {c}{x}} + a\right )}^{4} a^{4} - 4 \, {\left (b \sqrt {\frac {c}{x}} + a\right )}^{3} a^{5} + 6 \, {\left (b \sqrt {\frac {c}{x}} + a\right )}^{2} a^{6} - 4 \, {\left (b \sqrt {\frac {c}{x}} + a\right )} a^{7} + a^{8}}\right )} \]
-1/192*c^2*(105*b^4*log((sqrt(b*sqrt(c/x) + a) - sqrt(a))/(sqrt(b*sqrt(c/x ) + a) + sqrt(a)))/a^(9/2) + 2*(105*(b*sqrt(c/x) + a)^(7/2)*b^4 - 385*(b*s qrt(c/x) + a)^(5/2)*a*b^4 + 511*(b*sqrt(c/x) + a)^(3/2)*a^2*b^4 - 279*sqrt (b*sqrt(c/x) + a)*a^3*b^4)/((b*sqrt(c/x) + a)^4*a^4 - 4*(b*sqrt(c/x) + a)^ 3*a^5 + 6*(b*sqrt(c/x) + a)^2*a^6 - 4*(b*sqrt(c/x) + a)*a^7 + a^8))
Time = 0.35 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.02 \[ \int \frac {x}{\sqrt {a+b \sqrt {\frac {c}{x}}}} \, dx=\frac {\frac {105 \, b^{4} c^{4} \log \left (c^{2} {\left | b \right |}\right )}{\sqrt {a c} a^{4}} - \frac {105 \, b^{4} c^{4} \log \left ({\left | -b c^{2} - 2 \, \sqrt {a c} {\left (\sqrt {a c} \sqrt {c x} - \sqrt {a c^{2} x + \sqrt {c x} b c^{2}}\right )} \right |}\right )}{\sqrt {a c} a^{4}} - 2 \, \sqrt {a c^{2} x + \sqrt {c x} b c^{2}} {\left (2 \, \sqrt {c x} {\left (4 \, \sqrt {c x} {\left (\frac {7 \, b}{a^{2}} - \frac {6 \, \sqrt {c x}}{a c}\right )} - \frac {35 \, b^{2} c}{a^{3}}\right )} + \frac {105 \, b^{3} c^{2}}{a^{4}}\right )}}{192 \, c^{\frac {3}{2}} \mathrm {sgn}\left (x\right )} \]
1/192*(105*b^4*c^4*log(c^2*abs(b))/(sqrt(a*c)*a^4) - 105*b^4*c^4*log(abs(- b*c^2 - 2*sqrt(a*c)*(sqrt(a*c)*sqrt(c*x) - sqrt(a*c^2*x + sqrt(c*x)*b*c^2) )))/(sqrt(a*c)*a^4) - 2*sqrt(a*c^2*x + sqrt(c*x)*b*c^2)*(2*sqrt(c*x)*(4*sq rt(c*x)*(7*b/a^2 - 6*sqrt(c*x)/(a*c)) - 35*b^2*c/a^3) + 105*b^3*c^2/a^4))/ (c^(3/2)*sgn(x))
Timed out. \[ \int \frac {x}{\sqrt {a+b \sqrt {\frac {c}{x}}}} \, dx=\int \frac {x}{\sqrt {a+b\,\sqrt {\frac {c}{x}}}} \,d x \]