Integrand size = 26, antiderivative size = 371 \[ \int \frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{x^4} \, dx=\frac {b \left (80 a^2 c^2-120 a b^2 c d+33 b^4 d^2\right ) \left (b d+2 c \sqrt {\frac {d}{x}}\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{512 c^6}-\frac {\left (1024 a^2 c^2-3276 a b^2 c d+1155 b^4 d^2+18 b c \left (148 a c-77 b^2 d\right ) \sqrt {\frac {d}{x}}\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{6720 c^5}+\frac {11 b \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2} \left (\frac {d}{x}\right )^{3/2}}{42 c^2 d}-\frac {2 \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{7 c x^2}+\frac {\left (32 a c-33 b^2 d\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{140 c^3 x}+\frac {b \sqrt {d} \left (4 a c-b^2 d\right ) \left (80 a^2 c^2-120 a b^2 c d+33 b^4 d^2\right ) \text {arctanh}\left (\frac {b d+2 c \sqrt {\frac {d}{x}}}{2 \sqrt {c} \sqrt {d} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}\right )}{1024 c^{13/2}} \]
1/1024*b*(-b^2*d+4*a*c)*(33*b^4*d^2-120*a*b^2*c*d+80*a^2*c^2)*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^(13/2)+11/42*b*(d/x)^(3/2)*(a+c/x+b*(d/x)^(1/2))^(3/2)/c^2/d-2/7*(a+c/x +b*(d/x)^(1/2))^(3/2)/c/x^2+1/140*(-33*b^2*d+32*a*c)*(a+c/x+b*(d/x)^(1/2)) ^(3/2)/c^3/x-1/6720*(a+c/x+b*(d/x)^(1/2))^(3/2)*(1024*a^2*c^2-3276*a*b^2*c *d+1155*b^4*d^2+18*b*c*(-77*b^2*d+148*a*c)*(d/x)^(1/2))/c^5+1/512*b*(33*b^ 4*d^2-120*a*b^2*c*d+80*a^2*c^2)*(b*d+2*c*(d/x)^(1/2))*(a+c/x+b*(d/x)^(1/2) )^(1/2)/c^6
Time = 2.03 (sec) , antiderivative size = 353, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{x^4} \, dx=\frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} \left (-\frac {2 \sqrt {c} \left (15360 c^6+256 c^5 \left (12 a+5 b \sqrt {\frac {d}{x}}\right ) x-3465 b^6 d^3 x^3+210 b^4 c d^2 \left (104 a+11 b \sqrt {\frac {d}{x}}\right ) x^3-168 b^2 c^2 d x^2 \left (11 b^2 d+206 a^2 x+72 a b \sqrt {\frac {d}{x}} x\right )-64 c^4 x \left (22 b^2 d+64 a^2 x+79 a b \sqrt {\frac {d}{x}} x\right )+16 c^3 x^2 \left (486 a b^2 d+99 b^3 d \sqrt {\frac {d}{x}}+512 a^3 x+794 a^2 b \sqrt {\frac {d}{x}} x\right )\right )}{x^3}+\frac {105 b d \left (-320 a^3 c^3+560 a^2 b^2 c^2 d-252 a b^4 c d^2+33 b^6 d^3\right ) \log \left (b d+2 c \sqrt {\frac {d}{x}}-2 \sqrt {c} \sqrt {\frac {d \left (c+a x+b \sqrt {\frac {d}{x}} x\right )}{x}}\right )}{\sqrt {\frac {d \left (c+\left (a+b \sqrt {\frac {d}{x}}\right ) x\right )}{x}}}\right )}{107520 c^{13/2}} \]
(Sqrt[a + b*Sqrt[d/x] + c/x]*((-2*Sqrt[c]*(15360*c^6 + 256*c^5*(12*a + 5*b *Sqrt[d/x])*x - 3465*b^6*d^3*x^3 + 210*b^4*c*d^2*(104*a + 11*b*Sqrt[d/x])* x^3 - 168*b^2*c^2*d*x^2*(11*b^2*d + 206*a^2*x + 72*a*b*Sqrt[d/x]*x) - 64*c ^4*x*(22*b^2*d + 64*a^2*x + 79*a*b*Sqrt[d/x]*x) + 16*c^3*x^2*(486*a*b^2*d + 99*b^3*d*Sqrt[d/x] + 512*a^3*x + 794*a^2*b*Sqrt[d/x]*x)))/x^3 + (105*b*d *(-320*a^3*c^3 + 560*a^2*b^2*c^2*d - 252*a*b^4*c*d^2 + 33*b^6*d^3)*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]))/(107520*c^(13/2))
Time = 0.70 (sec) , antiderivative size = 359, normalized size of antiderivative = 0.97, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {2066, 1693, 1166, 27, 1236, 27, 1236, 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.
| \(\displaystyle \int \frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{x^4} \, dx\) |
| \(\Big \downarrow \) 2066 |
| \(\displaystyle -\frac {\int \frac {d^2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{x^2}d\frac {d}{x}}{d^3}\) |
| \(\Big \downarrow \) 1693 |
| \(\displaystyle -\frac {2 \int \frac {d^5 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{x^5}d\sqrt {\frac {d}{x}}}{d^3}\) |
| \(\Big \downarrow \) 1166 |
| \(\displaystyle -\frac {2 \left (\frac {d \int -\frac {d^3 \left (8 a+11 b \sqrt {\frac {d}{x}}\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{2 x^3}d\sqrt {\frac {d}{x}}}{7 c}+\frac {d^5 \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{7 c x^4}\right )}{d^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 \left (\frac {d^5 \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{7 c x^4}-\frac {d \int \frac {d^3 \left (8 a+11 b \sqrt {\frac {d}{x}}\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{x^3}d\sqrt {\frac {d}{x}}}{14 c}\right )}{d^3}\) |
| \(\Big \downarrow \) 1236 |
| \(\displaystyle -\frac {2 \left (\frac {d^5 \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{7 c x^4}-\frac {d \left (\frac {d \int -\frac {3 d \left (22 a b d-\left (32 a c-33 b^2 d\right ) \sqrt {\frac {d}{x}}\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{2 x^2}d\sqrt {\frac {d}{x}}}{6 c}+\frac {11 b d^4 \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{6 c x^3}\right )}{14 c}\right )}{d^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 \left (\frac {d^5 \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{7 c x^4}-\frac {d \left (\frac {11 b d^4 \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{6 c x^3}-\frac {\int \frac {d^2 \left (22 a b d-\left (32 a c-33 b^2 d\right ) \sqrt {\frac {d}{x}}\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{x^2}d\sqrt {\frac {d}{x}}}{4 c}\right )}{14 c}\right )}{d^3}\) |
| \(\Big \downarrow \) 1236 |
| \(\displaystyle -\frac {2 \left (\frac {d^5 \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{7 c x^4}-\frac {d \left (\frac {11 b d^4 \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{6 c x^3}-\frac {\frac {d \int \frac {1}{2} \sqrt {a+\frac {b d}{x}+\frac {c d}{x^2}} \left (\frac {3 b d \left (148 a c-77 b^2 d\right )}{x}+4 a \left (32 a c-33 b^2 d\right )\right ) \sqrt {\frac {d}{x}}d\sqrt {\frac {d}{x}}}{5 c}-\frac {d^3 \left (32 a c-33 b^2 d\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{5 c x^2}}{4 c}\right )}{14 c}\right )}{d^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 \left (\frac {d^5 \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{7 c x^4}-\frac {d \left (\frac {11 b d^4 \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{6 c x^3}-\frac {\frac {d \int \sqrt {a+\frac {b d}{x}+\frac {c d}{x^2}} \left (\frac {3 b d \left (148 a c-77 b^2 d\right )}{x}+4 a \left (32 a c-33 b^2 d\right )\right ) \sqrt {\frac {d}{x}}d\sqrt {\frac {d}{x}}}{10 c}-\frac {d^3 \left (32 a c-33 b^2 d\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{5 c x^2}}{4 c}\right )}{14 c}\right )}{d^3}\) |
| \(\Big \downarrow \) 1225 |
| \(\displaystyle -\frac {2 \left (\frac {d^5 \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{7 c x^4}-\frac {d \left (\frac {11 b d^4 \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{6 c x^3}-\frac {\frac {d \left (-\frac {35 b d \left (80 a^2 c^2-120 a b^2 c d+33 b^4 d^2\right ) \int \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}d\sqrt {\frac {d}{x}}}{16 c^2}-\frac {d \left (d \left (-\frac {1024 a^2 c^2}{d}+3276 a b^2 c-1155 b^4 d\right )-18 b c \sqrt {\frac {d}{x}} \left (148 a c-77 b^2 d\right )\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{24 c^2}\right )}{10 c}-\frac {d^3 \left (32 a c-33 b^2 d\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{5 c x^2}}{4 c}\right )}{14 c}\right )}{d^3}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle -\frac {2 \left (\frac {d^5 \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{7 c x^4}-\frac {d \left (\frac {11 b d^4 \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{6 c x^3}-\frac {\frac {d \left (-\frac {35 b d \left (80 a^2 c^2-120 a b^2 c d+33 b^4 d^2\right ) \left (\frac {\left (4 a c-b^2 d\right ) \int \frac {1}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}d\sqrt {\frac {d}{x}}}{8 c}+\frac {\left (b d+2 c \sqrt {\frac {d}{x}}\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{4 c}\right )}{16 c^2}-\frac {d \left (d \left (-\frac {1024 a^2 c^2}{d}+3276 a b^2 c-1155 b^4 d\right )-18 b c \sqrt {\frac {d}{x}} \left (148 a c-77 b^2 d\right )\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{24 c^2}\right )}{10 c}-\frac {d^3 \left (32 a c-33 b^2 d\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{5 c x^2}}{4 c}\right )}{14 c}\right )}{d^3}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle -\frac {2 \left (\frac {d^5 \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{7 c x^4}-\frac {d \left (\frac {11 b d^4 \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{6 c x^3}-\frac {\frac {d \left (-\frac {35 b d \left (80 a^2 c^2-120 a b^2 c d+33 b^4 d^2\right ) \left (\frac {\left (4 a c-b^2 d\right ) \int \frac {1}{\frac {4 c}{d}-\frac {d^2}{x^2}}d\frac {2 \sqrt {\frac {d}{x}} c+b d}{d \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}}{4 c}+\frac {\left (b d+2 c \sqrt {\frac {d}{x}}\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{4 c}\right )}{16 c^2}-\frac {d \left (d \left (-\frac {1024 a^2 c^2}{d}+3276 a b^2 c-1155 b^4 d\right )-18 b c \sqrt {\frac {d}{x}} \left (148 a c-77 b^2 d\right )\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{24 c^2}\right )}{10 c}-\frac {d^3 \left (32 a c-33 b^2 d\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{5 c x^2}}{4 c}\right )}{14 c}\right )}{d^3}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {2 \left (\frac {d^5 \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{7 c x^4}-\frac {d \left (\frac {11 b d^4 \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{6 c x^3}-\frac {\frac {d \left (-\frac {35 b d \left (80 a^2 c^2-120 a b^2 c d+33 b^4 d^2\right ) \left (\frac {\sqrt {d} \left (4 a c-b^2 d\right ) \text {arctanh}\left (\frac {d^{3/2}}{2 \sqrt {c} x}\right )}{8 c^{3/2}}+\frac {\left (b d+2 c \sqrt {\frac {d}{x}}\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{4 c}\right )}{16 c^2}-\frac {d \left (d \left (-\frac {1024 a^2 c^2}{d}+3276 a b^2 c-1155 b^4 d\right )-18 b c \sqrt {\frac {d}{x}} \left (148 a c-77 b^2 d\right )\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{24 c^2}\right )}{10 c}-\frac {d^3 \left (32 a c-33 b^2 d\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{5 c x^2}}{4 c}\right )}{14 c}\right )}{d^3}\) |
(-2*((d^5*(a + b*Sqrt[d/x] + (c*d)/x^2)^(3/2))/(7*c*x^4) - (d*((11*b*d^4*( a + b*Sqrt[d/x] + (c*d)/x^2)^(3/2))/(6*c*x^3) - (-1/5*(d^3*(32*a*c - 33*b^ 2*d)*(a + b*Sqrt[d/x] + (c*d)/x^2)^(3/2))/(c*x^2) + (d*(-1/24*(d*(d*(3276* a*b^2*c - (1024*a^2*c^2)/d - 1155*b^4*d) - 18*b*c*(148*a*c - 77*b^2*d)*Sqr t[d/x])*(a + b*Sqrt[d/x] + (c*d)/x^2)^(3/2))/c^2 - (35*b*d*(80*a^2*c^2 - 1 20*a*b^2*c*d + 33*b^4*d^2)*(((b*d + 2*c*Sqrt[d/x])*Sqrt[a + b*Sqrt[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))/(4*c)))/(14*c)))/d^3
3.31.59.3.1 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[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 2))), x] + Simp[1/(c*(m + 2*p + 2)) Int[(d + e*x)^(m - 1 )*(a + b*x + c*x^2)^p*Simp[m*(c*d*f - a*e*g) + d*(2*c*f - b*g)*(p + 1) + (m *(c*e*f + c*d*g - b*e*g) + e*(p + 1)*(2*c*f - b*g))*x, x], x], x] /; FreeQ[ {a, b, c, d, e, f, g, p}, x] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (Intege rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) && !(IGtQ[m, 0] && EqQ[f, 0])
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. \(978\) vs. \(2(317)=634\).
Time = 0.25 (sec) , antiderivative size = 979, normalized size of antiderivative = 2.64
| method | result | size |
| default | \(\frac {\sqrt {\frac {b \sqrt {\frac {d}{x}}\, x +a x +c}{x}}\, \left (22176 \left (b \sqrt {\frac {d}{x}}\, x +a x +c \right )^{\frac {3}{2}} \left (\frac {d}{x}\right )^{\frac {3}{2}} x^{3} b^{3} c^{3}-3465 \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 {7}{2}} x^{7} b^{7}+13860 \left (b \sqrt {\frac {d}{x}}\, x +a x +c \right )^{\frac {3}{2}} \left (\frac {d}{x}\right )^{\frac {5}{2}} x^{5} b^{5} c +28160 \left (b \sqrt {\frac {d}{x}}\, x +a x +c \right )^{\frac {3}{2}} \sqrt {\frac {d}{x}}\, x b \,c^{5}-16384 a^{2} \left (b \sqrt {\frac {d}{x}}\, x +a x +c \right )^{\frac {3}{2}} c^{4} x^{2}+24576 a \left (b \sqrt {\frac {d}{x}}\, x +a x +c \right )^{\frac {3}{2}} c^{5} x +6930 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, \left (\frac {d}{x}\right )^{\frac {7}{2}} x^{7} b^{7}-18480 \left (b \sqrt {\frac {d}{x}}\, x +a x +c \right )^{\frac {3}{2}} d^{2} x^{2} b^{4} c^{2}-25344 \left (b \sqrt {\frac {d}{x}}\, x +a x +c \right )^{\frac {3}{2}} d x \,b^{2} c^{4}+33600 a^{2} \left (b \sqrt {\frac {d}{x}}\, x +a x +c \right )^{\frac {3}{2}} \sqrt {\frac {d}{x}}\, x^{3} b \,c^{3}-30720 \left (b \sqrt {\frac {d}{x}}\, x +a x +c \right )^{\frac {3}{2}} c^{6}+67200 a^{2} \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, \left (\frac {d}{x}\right )^{\frac {3}{2}} x^{5} b^{3} c^{2}+6930 a \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, d^{3} x^{4} b^{6}-50400 a \left (b \sqrt {\frac {d}{x}}\, x +a x +c \right )^{\frac {3}{2}} \left (\frac {d}{x}\right )^{\frac {3}{2}} x^{4} b^{3} c^{2}-33600 a^{3} \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, \sqrt {\frac {d}{x}}\, x^{4} b \,c^{3}+25200 a \left (b \sqrt {\frac {d}{x}}\, x +a x +c \right )^{\frac {3}{2}} d^{2} x^{3} b^{4} c -16800 a^{2} \left (b \sqrt {\frac {d}{x}}\, x +a x +c \right )^{\frac {3}{2}} d \,x^{3} b^{2} c^{2}+52416 a \left (b \sqrt {\frac {d}{x}}\, x +a x +c \right )^{\frac {3}{2}} d \,x^{2} b^{2} c^{3}-25200 a^{2} \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, d^{2} x^{4} b^{4} c +16800 a^{3} \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, d \,x^{4} b^{2} c^{2}+33600 a^{3} c^{\frac {7}{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^{4} b -6930 \left (b \sqrt {\frac {d}{x}}\, x +a x +c \right )^{\frac {3}{2}} x^{3} d^{3} b^{6}-42624 a \left (b \sqrt {\frac {d}{x}}\, x +a x +c \right )^{\frac {3}{2}} \sqrt {\frac {d}{x}}\, x^{2} b \,c^{4}+26460 a \,c^{\frac {3}{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 ) \left (\frac {d}{x}\right )^{\frac {5}{2}} x^{6} b^{5}-58800 a^{2} c^{\frac {5}{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 ) \left (\frac {d}{x}\right )^{\frac {3}{2}} x^{5} b^{3}-39060 a \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, \left (\frac {d}{x}\right )^{\frac {5}{2}} x^{6} b^{5} c \right )}{107520 x^{3} \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, c^{7}}\) | \(979\) |
1/107520*((b*(d/x)^(1/2)*x+a*x+c)/x)^(1/2)/x^3*(22176*(b*(d/x)^(1/2)*x+a*x +c)^(3/2)*(d/x)^(3/2)*x^3*b^3*c^3-3465*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)^(7/2)*x^7*b^7+13860*( b*(d/x)^(1/2)*x+a*x+c)^(3/2)*(d/x)^(5/2)*x^5*b^5*c+28160*(b*(d/x)^(1/2)*x+ a*x+c)^(3/2)*(d/x)^(1/2)*x*b*c^5-16384*a^2*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*c ^4*x^2+24576*a*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*c^5*x+6930*(b*(d/x)^(1/2)*x+a *x+c)^(1/2)*(d/x)^(7/2)*x^7*b^7-18480*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*d^2*x^ 2*b^4*c^2-25344*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*d*x*b^2*c^4+33600*a^2*(b*(d/ x)^(1/2)*x+a*x+c)^(3/2)*(d/x)^(1/2)*x^3*b*c^3-30720*(b*(d/x)^(1/2)*x+a*x+c )^(3/2)*c^6+67200*a^2*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*(d/x)^(3/2)*x^5*b^3*c^ 2+6930*a*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*d^3*x^4*b^6-50400*a*(b*(d/x)^(1/2)* x+a*x+c)^(3/2)*(d/x)^(3/2)*x^4*b^3*c^2-33600*a^3*(b*(d/x)^(1/2)*x+a*x+c)^( 1/2)*(d/x)^(1/2)*x^4*b*c^3+25200*a*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*d^2*x^3*b ^4*c-16800*a^2*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*d*x^3*b^2*c^2+52416*a*(b*(d/x )^(1/2)*x+a*x+c)^(3/2)*d*x^2*b^2*c^3-25200*a^2*(b*(d/x)^(1/2)*x+a*x+c)^(1/ 2)*d^2*x^4*b^4*c+16800*a^3*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*d*x^4*b^2*c^2+336 00*a^3*c^(7/2)*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^4*b-6930*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*x^3*d^ 3*b^6-42624*a*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*(d/x)^(1/2)*x^2*b*c^4+26460*a* c^(3/2)*ln((2*c+b*(d/x)^(1/2)*x+2*c^(1/2)*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)...
Timed out. \[ \int \frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{x^4} \, dx=\text {Timed out} \]
\[ \int \frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{x^4} \, dx=\int \frac {\sqrt {a + b \sqrt {\frac {d}{x}} + \frac {c}{x}}}{x^{4}}\, dx \]
\[ \int \frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{x^4} \, dx=\int { \frac {\sqrt {b \sqrt {\frac {d}{x}} + a + \frac {c}{x}}}{x^{4}} \,d x } \]
\[ \int \frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{x^4} \, dx=\int { \frac {\sqrt {b \sqrt {\frac {d}{x}} + a + \frac {c}{x}}}{x^{4}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{x^4} \, dx=\int \frac {\sqrt {a+\frac {c}{x}+b\,\sqrt {\frac {d}{x}}}}{x^4} \,d x \]