Integrand size = 26, antiderivative size = 233 \[ \int \frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{x^3} \, dx=-\frac {b \left (12 a c-7 b^2 d\right ) \left (b d+2 c \sqrt {\frac {d}{x}}\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{64 c^4}+\frac {\left (32 a c-35 b^2 d+42 b c \sqrt {\frac {d}{x}}\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{120 c^3}-\frac {2 \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{5 c x}-\frac {b \sqrt {d} \left (12 a c-7 b^2 d\right ) \left (4 a c-b^2 d\right ) \text {arctanh}\left (\frac {b d+2 c \sqrt {\frac {d}{x}}}{2 \sqrt {c} \sqrt {d} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}\right )}{128 c^{9/2}} \]
-1/128*b*(-7*b^2*d+12*a*c)*(-b^2*d+4*a*c)*arctanh(1/2*(b*d+2*c*(d/x)^(1/2) )/c^(1/2)/d^(1/2)/(a+c/x+b*(d/x)^(1/2))^(1/2))*d^(1/2)/c^(9/2)-2/5*(a+c/x+ b*(d/x)^(1/2))^(3/2)/c/x+1/120*(a+c/x+b*(d/x)^(1/2))^(3/2)*(32*a*c-35*b^2* d+42*b*c*(d/x)^(1/2))/c^3-1/64*b*(-7*b^2*d+12*a*c)*(b*d+2*c*(d/x)^(1/2))*( a+c/x+b*(d/x)^(1/2))^(1/2)/c^4
Time = 1.21 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.05 \[ \int \frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{x^3} \, dx=\frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} \left (\frac {2 \sqrt {c} \left (-384 c^4-16 c^3 \left (8 a+3 b \sqrt {\frac {d}{x}}\right ) x+105 b^4 d^2 x^2-10 b^2 c d \left (46 a+7 b \sqrt {\frac {d}{x}}\right ) x^2+8 c^2 x \left (7 b^2 d+32 a^2 x+29 a b \sqrt {\frac {d}{x}} x\right )\right )}{x^2}+\frac {15 b d \left (48 a^2 c^2-40 a b^2 c d+7 b^4 d^2\right ) \log \left (b d+2 c \sqrt {\frac {d}{x}}-2 \sqrt {c} \sqrt {\frac {d \left (c+a x+b \sqrt {\frac {d}{x}} x\right )}{x}}\right )}{\sqrt {\frac {d \left (c+\left (a+b \sqrt {\frac {d}{x}}\right ) x\right )}{x}}}\right )}{1920 c^{9/2}} \]
(Sqrt[a + b*Sqrt[d/x] + c/x]*((2*Sqrt[c]*(-384*c^4 - 16*c^3*(8*a + 3*b*Sqr t[d/x])*x + 105*b^4*d^2*x^2 - 10*b^2*c*d*(46*a + 7*b*Sqrt[d/x])*x^2 + 8*c^ 2*x*(7*b^2*d + 32*a^2*x + 29*a*b*Sqrt[d/x]*x)))/x^2 + (15*b*d*(48*a^2*c^2 - 40*a*b^2*c*d + 7*b^4*d^2)*Log[b*d + 2*c*Sqrt[d/x] - 2*Sqrt[c]*Sqrt[(d*(c + a*x + b*Sqrt[d/x]*x))/x]])/Sqrt[(d*(c + (a + b*Sqrt[d/x])*x))/x]))/(192 0*c^(9/2))
Time = 0.42 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.95, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2066, 1693, 1166, 27, 1225, 1087, 1092, 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.
| \(\displaystyle \int \frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{x^3} \, dx\) |
| \(\Big \downarrow \) 2066 |
| \(\displaystyle -\frac {\int \frac {d \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{x}d\frac {d}{x}}{d^2}\) |
| \(\Big \downarrow \) 1693 |
| \(\displaystyle -\frac {2 \int \frac {d^3 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{x^3}d\sqrt {\frac {d}{x}}}{d^2}\) |
| \(\Big \downarrow \) 1166 |
| \(\displaystyle -\frac {2 \left (\frac {d \int -\frac {1}{2} \sqrt {a+\frac {b d}{x}+\frac {c d}{x^2}} \left (4 a+\frac {7 b d}{x}\right ) \sqrt {\frac {d}{x}}d\sqrt {\frac {d}{x}}}{5 c}+\frac {d^3 \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{5 c x^2}\right )}{d^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 \left (\frac {d^3 \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{5 c x^2}-\frac {d \int \sqrt {a+\frac {b d}{x}+\frac {c d}{x^2}} \left (4 a+\frac {7 b d}{x}\right ) \sqrt {\frac {d}{x}}d\sqrt {\frac {d}{x}}}{10 c}\right )}{d^2}\) |
| \(\Big \downarrow \) 1225 |
| \(\displaystyle -\frac {2 \left (\frac {d^3 \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{5 c x^2}-\frac {d \left (-\frac {5 b d \left (12 a c-7 b^2 d\right ) \int \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}d\sqrt {\frac {d}{x}}}{16 c^2}-\frac {d \left (d \left (35 b^2-\frac {32 a c}{d}\right )-42 b c \sqrt {\frac {d}{x}}\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{24 c^2}\right )}{10 c}\right )}{d^2}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle -\frac {2 \left (\frac {d^3 \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{5 c x^2}-\frac {d \left (-\frac {5 b d \left (12 a c-7 b^2 d\right ) \left (\frac {\left (4 a c-b^2 d\right ) \int \frac {1}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}d\sqrt {\frac {d}{x}}}{8 c}+\frac {\left (b d+2 c \sqrt {\frac {d}{x}}\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{4 c}\right )}{16 c^2}-\frac {d \left (d \left (35 b^2-\frac {32 a c}{d}\right )-42 b c \sqrt {\frac {d}{x}}\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{24 c^2}\right )}{10 c}\right )}{d^2}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle -\frac {2 \left (\frac {d^3 \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{5 c x^2}-\frac {d \left (-\frac {5 b d \left (12 a c-7 b^2 d\right ) \left (\frac {\left (4 a c-b^2 d\right ) \int \frac {1}{\frac {4 c}{d}-\frac {d^2}{x^2}}d\frac {2 \sqrt {\frac {d}{x}} c+b d}{d \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}}{4 c}+\frac {\left (b d+2 c \sqrt {\frac {d}{x}}\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{4 c}\right )}{16 c^2}-\frac {d \left (d \left (35 b^2-\frac {32 a c}{d}\right )-42 b c \sqrt {\frac {d}{x}}\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{24 c^2}\right )}{10 c}\right )}{d^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {2 \left (\frac {d^3 \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{5 c x^2}-\frac {d \left (-\frac {5 b d \left (12 a c-7 b^2 d\right ) \left (\frac {\sqrt {d} \left (4 a c-b^2 d\right ) \text {arctanh}\left (\frac {d^{3/2}}{2 \sqrt {c} x}\right )}{8 c^{3/2}}+\frac {\left (b d+2 c \sqrt {\frac {d}{x}}\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{4 c}\right )}{16 c^2}-\frac {d \left (d \left (35 b^2-\frac {32 a c}{d}\right )-42 b c \sqrt {\frac {d}{x}}\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{24 c^2}\right )}{10 c}\right )}{d^2}\) |
(-2*((d^3*(a + b*Sqrt[d/x] + (c*d)/x^2)^(3/2))/(5*c*x^2) - (d*(-1/24*(d*(( 35*b^2 - (32*a*c)/d)*d - 42*b*c*Sqrt[d/x])*(a + b*Sqrt[d/x] + (c*d)/x^2)^( 3/2))/c^2 - (5*b*d*(12*a*c - 7*b^2*d)*(((b*d + 2*c*Sqrt[d/x])*Sqrt[a + b*S qrt[d/x] + (c*d)/x^2])/(4*c) + (Sqrt[d]*(4*a*c - b^2*d)*ArcTanh[d^(3/2)/(2 *Sqrt[c]*x)])/(8*c^(3/2))))/(16*c^2)))/(10*c)))/d^2
3.31.58.3.1 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[(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_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) *((a + b*x + c*x^2)^p/(2*c*(2*p + 1))), x] - Simp[p*((b^2 - 4*a*c)/(2*c*(2* p + 1))) Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[I nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a , b, c}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[e*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 1))), x] + Simp[1/(c*(m + 2*p + 1)) Int[(d + e*x)^(m - 2)*Simp[c*d^2*(m + 2*p + 1) - e*(a*e*(m - 1) + b*d*(p + 1)) + e*(2*c*d - b*e)*(m + p)*x, x]* (a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && If[Ration alQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadrat icQ[a, b, c, d, e, m, p, x]
Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*( x_)^2)^(p_), x_Symbol] :> Simp[(-(b*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 2*c*e*g*(p + 1)*x))*((a + b*x + c*x^2)^(p + 1)/(2*c^2*(p + 1)*(2*p + 3))), x] + Simp[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(2*c^2*(2*p + 3)) Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c , d, e, f, g, p}, x] && !LeQ[p, -1]
Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol ] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && IntegerQ [Simplify[(m + 1)/n]]
Int[(x_)^(m_.)*((a_) + (b_.)*((d_.)/(x_))^(n_) + (c_.)*(x_)^(n2_.))^(p_), x _Symbol] :> Simp[-d^(m + 1) Subst[Int[(a + b*x^n + (c/d^(2*n))*x^(2*n))^p /x^(m + 2), x], x, d/x], x] /; FreeQ[{a, b, c, d, n, p}, x] && EqQ[n2, -2*n ] && IntegerQ[2*n] && IntegerQ[m]
Leaf count of result is larger than twice the leaf count of optimal. \(614\) vs. \(2(193)=386\).
Time = 0.25 (sec) , antiderivative size = 615, normalized size of antiderivative = 2.64
| method | result | size |
| default | \(-\frac {\sqrt {\frac {b \sqrt {\frac {d}{x}}\, x +a x +c}{x}}\, \left (105 \sqrt {c}\, \ln \left (\frac {2 c +b \sqrt {\frac {d}{x}}\, x +2 \sqrt {c}\, \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}}{\sqrt {x}}\right ) \left (\frac {d}{x}\right )^{\frac {5}{2}} x^{5} b^{5}-210 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, \left (\frac {d}{x}\right )^{\frac {5}{2}} x^{5} b^{5}-600 c^{\frac {3}{2}} a \ln \left (\frac {2 c +b \sqrt {\frac {d}{x}}\, x +2 \sqrt {c}\, \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}}{\sqrt {x}}\right ) \left (\frac {d}{x}\right )^{\frac {3}{2}} x^{4} b^{3}-210 a \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, d^{2} x^{3} b^{4}+720 c^{\frac {5}{2}} a^{2} \ln \left (\frac {2 c +b \sqrt {\frac {d}{x}}\, x +2 \sqrt {c}\, \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}}{\sqrt {x}}\right ) \sqrt {\frac {d}{x}}\, x^{3} b +780 a \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, \left (\frac {d}{x}\right )^{\frac {3}{2}} x^{4} b^{3} c +360 a^{2} \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, d \,x^{3} b^{2} c +210 \left (b \sqrt {\frac {d}{x}}\, x +a x +c \right )^{\frac {3}{2}} x^{2} d^{2} b^{4}-420 \left (b \sqrt {\frac {d}{x}}\, x +a x +c \right )^{\frac {3}{2}} x^{3} \left (\frac {d}{x}\right )^{\frac {3}{2}} b^{3} c -720 a^{2} \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, \sqrt {\frac {d}{x}}\, x^{3} b \,c^{2}-360 a \left (b \sqrt {\frac {d}{x}}\, x +a x +c \right )^{\frac {3}{2}} d \,x^{2} b^{2} c +720 a \left (b \sqrt {\frac {d}{x}}\, x +a x +c \right )^{\frac {3}{2}} \sqrt {\frac {d}{x}}\, x^{2} b \,c^{2}+560 \left (b \sqrt {\frac {d}{x}}\, x +a x +c \right )^{\frac {3}{2}} d x \,b^{2} c^{2}-512 a \left (b \sqrt {\frac {d}{x}}\, x +a x +c \right )^{\frac {3}{2}} c^{3} x -672 \left (b \sqrt {\frac {d}{x}}\, x +a x +c \right )^{\frac {3}{2}} \sqrt {\frac {d}{x}}\, x b \,c^{3}+768 \left (b \sqrt {\frac {d}{x}}\, x +a x +c \right )^{\frac {3}{2}} c^{4}\right )}{1920 x^{2} \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, c^{5}}\) | \(615\) |
-1/1920*((b*(d/x)^(1/2)*x+a*x+c)/x)^(1/2)/x^2*(105*c^(1/2)*ln((2*c+b*(d/x) ^(1/2)*x+2*c^(1/2)*(b*(d/x)^(1/2)*x+a*x+c)^(1/2))/x^(1/2))*(d/x)^(5/2)*x^5 *b^5-210*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*(d/x)^(5/2)*x^5*b^5-600*c^(3/2)*a*l n((2*c+b*(d/x)^(1/2)*x+2*c^(1/2)*(b*(d/x)^(1/2)*x+a*x+c)^(1/2))/x^(1/2))*( d/x)^(3/2)*x^4*b^3-210*a*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*d^2*x^3*b^4+720*c^( 5/2)*a^2*ln((2*c+b*(d/x)^(1/2)*x+2*c^(1/2)*(b*(d/x)^(1/2)*x+a*x+c)^(1/2))/ x^(1/2))*(d/x)^(1/2)*x^3*b+780*a*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*(d/x)^(3/2) *x^4*b^3*c+360*a^2*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*d*x^3*b^2*c+210*(b*(d/x)^ (1/2)*x+a*x+c)^(3/2)*x^2*d^2*b^4-420*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*x^3*(d/ x)^(3/2)*b^3*c-720*a^2*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*(d/x)^(1/2)*x^3*b*c^2 -360*a*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*d*x^2*b^2*c+720*a*(b*(d/x)^(1/2)*x+a* x+c)^(3/2)*(d/x)^(1/2)*x^2*b*c^2+560*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*d*x*b^2 *c^2-512*a*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*c^3*x-672*(b*(d/x)^(1/2)*x+a*x+c) ^(3/2)*(d/x)^(1/2)*x*b*c^3+768*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*c^4)/(b*(d/x) ^(1/2)*x+a*x+c)^(1/2)/c^5
Timed out. \[ \int \frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{x^3} \, dx=\text {Timed out} \]
\[ \int \frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{x^3} \, dx=\int \frac {\sqrt {a + b \sqrt {\frac {d}{x}} + \frac {c}{x}}}{x^{3}}\, dx \]
\[ \int \frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{x^3} \, dx=\int { \frac {\sqrt {b \sqrt {\frac {d}{x}} + a + \frac {c}{x}}}{x^{3}} \,d x } \]
\[ \int \frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{x^3} \, dx=\int { \frac {\sqrt {b \sqrt {\frac {d}{x}} + a + \frac {c}{x}}}{x^{3}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{x^3} \, dx=\int \frac {\sqrt {a+\frac {c}{x}+b\,\sqrt {\frac {d}{x}}}}{x^3} \,d x \]