Integrand size = 16, antiderivative size = 204 \[ \int \left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{5/2} \, dx=-\frac {5}{24} \left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{3/2} \left (7 b+\frac {6 c}{x}\right )-\frac {5 \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} \left (b \left (b^2+44 a c\right )+\frac {2 c \left (b^2+12 a c\right )}{x}\right )}{64 c}+\left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{5/2} x+\frac {5}{2} a^{3/2} b \text {arctanh}\left (\frac {2 a+\frac {b}{x}}{2 \sqrt {a} \sqrt {a+\frac {c}{x^2}+\frac {b}{x}}}\right )+\frac {5 \left (b^4-24 a b^2 c-48 a^2 c^2\right ) \text {arctanh}\left (\frac {b+\frac {2 c}{x}}{2 \sqrt {c} \sqrt {a+\frac {c}{x^2}+\frac {b}{x}}}\right )}{128 c^{3/2}} \] Output:
-5/24*(a+c/x^2+b/x)^(3/2)*(7*b+6*c/x)-5/64*(a+c/x^2+b/x)^(1/2)*(b*(44*a*c+ b^2)+2*c*(12*a*c+b^2)/x)/c+(a+c/x^2+b/x)^(5/2)*x+5/2*a^(3/2)*b*arctanh(1/2 *(2*a+b/x)/a^(1/2)/(a+c/x^2+b/x)^(1/2))+5/128*(-48*a^2*c^2-24*a*b^2*c+b^4) *arctanh(1/2*(b+2*c/x)/c^(1/2)/(a+c/x^2+b/x)^(1/2))/c^(3/2)
Time = 1.86 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.02 \[ \int \left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{5/2} \, dx=-\frac {\sqrt {a+\frac {c+b x}{x^2}} \left (15 \left (b^4-24 a b^2 c-48 a^2 c^2\right ) x^4 \text {arctanh}\left (\frac {\sqrt {a} x-\sqrt {c+x (b+a x)}}{\sqrt {c}}\right )+\sqrt {c} \left (\sqrt {c+x (b+a x)} \left (48 c^3+15 b^3 x^3+8 c^2 x (17 b+27 a x)+2 c x^2 \left (59 b^2+278 a b x-96 a^2 x^2\right )\right )+480 a^{3/2} b c x^4 \log \left (b+2 a x-2 \sqrt {a} \sqrt {c+x (b+a x)}\right )\right )\right )}{192 c^{3/2} x^3 \sqrt {c+x (b+a x)}} \] Input:
Integrate[(a + c/x^2 + b/x)^(5/2),x]
Output:
-1/192*(Sqrt[a + (c + b*x)/x^2]*(15*(b^4 - 24*a*b^2*c - 48*a^2*c^2)*x^4*Ar cTanh[(Sqrt[a]*x - Sqrt[c + x*(b + a*x)])/Sqrt[c]] + Sqrt[c]*(Sqrt[c + x*( b + a*x)]*(48*c^3 + 15*b^3*x^3 + 8*c^2*x*(17*b + 27*a*x) + 2*c*x^2*(59*b^2 + 278*a*b*x - 96*a^2*x^2)) + 480*a^(3/2)*b*c*x^4*Log[b + 2*a*x - 2*Sqrt[a ]*Sqrt[c + x*(b + a*x)]])))/(c^(3/2)*x^3*Sqrt[c + x*(b + a*x)])
Time = 0.48 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.07, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {1681, 1161, 1231, 25, 27, 1231, 27, 1269, 1092, 219, 1154, 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.
| \(\displaystyle \int \left (a+\frac {b}{x}+\frac {c}{x^2}\right )^{5/2} \, dx\) |
| \(\Big \downarrow \) 1681 |
| \(\displaystyle -\int \left (a+\frac {b}{x}+\frac {c}{x^2}\right )^{5/2} x^2d\frac {1}{x}\) |
| \(\Big \downarrow \) 1161 |
| \(\displaystyle x \left (a+\frac {b}{x}+\frac {c}{x^2}\right )^{5/2}-\frac {5}{2} \int \left (a+\frac {b}{x}+\frac {c}{x^2}\right )^{3/2} \left (b+\frac {2 c}{x}\right ) xd\frac {1}{x}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle x \left (a+\frac {b}{x}+\frac {c}{x^2}\right )^{5/2}-\frac {5}{2} \left (\frac {1}{12} \left (7 b+\frac {6 c}{x}\right ) \left (a+\frac {b}{x}+\frac {c}{x^2}\right )^{3/2}-\frac {\int -c \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \left (8 a b+\frac {b^2+12 a c}{x}\right ) xd\frac {1}{x}}{8 c}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle x \left (a+\frac {b}{x}+\frac {c}{x^2}\right )^{5/2}-\frac {5}{2} \left (\frac {\int c \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \left (8 a b+\frac {b^2+12 a c}{x}\right ) xd\frac {1}{x}}{8 c}+\frac {1}{12} \left (7 b+\frac {6 c}{x}\right ) \left (a+\frac {b}{x}+\frac {c}{x^2}\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle x \left (a+\frac {b}{x}+\frac {c}{x^2}\right )^{5/2}-\frac {5}{2} \left (\frac {1}{8} \int \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \left (8 a b+\frac {b^2+12 a c}{x}\right ) xd\frac {1}{x}+\frac {1}{12} \left (7 b+\frac {6 c}{x}\right ) \left (a+\frac {b}{x}+\frac {c}{x^2}\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle x \left (a+\frac {b}{x}+\frac {c}{x^2}\right )^{5/2}-\frac {5}{2} \left (\frac {1}{8} \left (\frac {\sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \left (\frac {2 c \left (12 a c+b^2\right )}{x}+b \left (44 a c+b^2\right )\right )}{4 c}-\frac {\int -\frac {\left (64 a^2 b c-\frac {b^4-24 a c b^2-48 a^2 c^2}{x}\right ) x}{2 \sqrt {a+\frac {b}{x}+\frac {c}{x^2}}}d\frac {1}{x}}{4 c}\right )+\frac {1}{12} \left (7 b+\frac {6 c}{x}\right ) \left (a+\frac {b}{x}+\frac {c}{x^2}\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle x \left (a+\frac {b}{x}+\frac {c}{x^2}\right )^{5/2}-\frac {5}{2} \left (\frac {1}{8} \left (\frac {\int \frac {\left (64 a^2 b c-\frac {b^4-24 a c b^2-48 a^2 c^2}{x}\right ) x}{\sqrt {a+\frac {b}{x}+\frac {c}{x^2}}}d\frac {1}{x}}{8 c}+\frac {\sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \left (\frac {2 c \left (12 a c+b^2\right )}{x}+b \left (44 a c+b^2\right )\right )}{4 c}\right )+\frac {1}{12} \left (7 b+\frac {6 c}{x}\right ) \left (a+\frac {b}{x}+\frac {c}{x^2}\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle x \left (a+\frac {b}{x}+\frac {c}{x^2}\right )^{5/2}-\frac {5}{2} \left (\frac {1}{8} \left (\frac {64 a^2 b c \int \frac {x}{\sqrt {a+\frac {b}{x}+\frac {c}{x^2}}}d\frac {1}{x}-\left (-48 a^2 c^2-24 a b^2 c+b^4\right ) \int \frac {1}{\sqrt {a+\frac {b}{x}+\frac {c}{x^2}}}d\frac {1}{x}}{8 c}+\frac {\sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \left (\frac {2 c \left (12 a c+b^2\right )}{x}+b \left (44 a c+b^2\right )\right )}{4 c}\right )+\frac {1}{12} \left (7 b+\frac {6 c}{x}\right ) \left (a+\frac {b}{x}+\frac {c}{x^2}\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle x \left (a+\frac {b}{x}+\frac {c}{x^2}\right )^{5/2}-\frac {5}{2} \left (\frac {1}{8} \left (\frac {64 a^2 b c \int \frac {x}{\sqrt {a+\frac {b}{x}+\frac {c}{x^2}}}d\frac {1}{x}-2 \left (-48 a^2 c^2-24 a b^2 c+b^4\right ) \int \frac {1}{4 c-\frac {1}{x^2}}d\frac {b+\frac {2 c}{x}}{\sqrt {a+\frac {b}{x}+\frac {c}{x^2}}}}{8 c}+\frac {\sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \left (\frac {2 c \left (12 a c+b^2\right )}{x}+b \left (44 a c+b^2\right )\right )}{4 c}\right )+\frac {1}{12} \left (7 b+\frac {6 c}{x}\right ) \left (a+\frac {b}{x}+\frac {c}{x^2}\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle x \left (a+\frac {b}{x}+\frac {c}{x^2}\right )^{5/2}-\frac {5}{2} \left (\frac {1}{8} \left (\frac {64 a^2 b c \int \frac {x}{\sqrt {a+\frac {b}{x}+\frac {c}{x^2}}}d\frac {1}{x}-\frac {\left (-48 a^2 c^2-24 a b^2 c+b^4\right ) \text {arctanh}\left (\frac {b+\frac {2 c}{x}}{2 \sqrt {c} \sqrt {a+\frac {b}{x}+\frac {c}{x^2}}}\right )}{\sqrt {c}}}{8 c}+\frac {\sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \left (\frac {2 c \left (12 a c+b^2\right )}{x}+b \left (44 a c+b^2\right )\right )}{4 c}\right )+\frac {1}{12} \left (7 b+\frac {6 c}{x}\right ) \left (a+\frac {b}{x}+\frac {c}{x^2}\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle x \left (a+\frac {b}{x}+\frac {c}{x^2}\right )^{5/2}-\frac {5}{2} \left (\frac {1}{8} \left (\frac {-128 a^2 b c \int \frac {1}{4 a-\frac {1}{x^2}}d\frac {2 a+\frac {b}{x}}{\sqrt {a+\frac {b}{x}+\frac {c}{x^2}}}-\frac {\left (-48 a^2 c^2-24 a b^2 c+b^4\right ) \text {arctanh}\left (\frac {b+\frac {2 c}{x}}{2 \sqrt {c} \sqrt {a+\frac {b}{x}+\frac {c}{x^2}}}\right )}{\sqrt {c}}}{8 c}+\frac {\sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \left (\frac {2 c \left (12 a c+b^2\right )}{x}+b \left (44 a c+b^2\right )\right )}{4 c}\right )+\frac {1}{12} \left (7 b+\frac {6 c}{x}\right ) \left (a+\frac {b}{x}+\frac {c}{x^2}\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle x \left (a+\frac {b}{x}+\frac {c}{x^2}\right )^{5/2}-\frac {5}{2} \left (\frac {1}{8} \left (\frac {-64 a^{3/2} b c \text {arctanh}\left (\frac {2 a+\frac {b}{x}}{2 \sqrt {a} \sqrt {a+\frac {b}{x}+\frac {c}{x^2}}}\right )-\frac {\left (-48 a^2 c^2-24 a b^2 c+b^4\right ) \text {arctanh}\left (\frac {b+\frac {2 c}{x}}{2 \sqrt {c} \sqrt {a+\frac {b}{x}+\frac {c}{x^2}}}\right )}{\sqrt {c}}}{8 c}+\frac {\sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \left (\frac {2 c \left (12 a c+b^2\right )}{x}+b \left (44 a c+b^2\right )\right )}{4 c}\right )+\frac {1}{12} \left (7 b+\frac {6 c}{x}\right ) \left (a+\frac {b}{x}+\frac {c}{x^2}\right )^{3/2}\right )\) |
Input:
Int[(a + c/x^2 + b/x)^(5/2),x]
Output:
(a + c/x^2 + b/x)^(5/2)*x - (5*(((a + c/x^2 + b/x)^(3/2)*(7*b + (6*c)/x))/ 12 + ((Sqrt[a + c/x^2 + b/x]*(b*(b^2 + 44*a*c) + (2*c*(b^2 + 12*a*c))/x))/ (4*c) + (-64*a^(3/2)*b*c*ArcTanh[(2*a + b/x)/(2*Sqrt[a]*Sqrt[a + c/x^2 + b /x])] - ((b^4 - 24*a*b^2*c - 48*a^2*c^2)*ArcTanh[(b + (2*c)/x)/(2*Sqrt[c]* Sqrt[a + c/x^2 + b/x])])/Sqrt[c])/(8*c))/8))/2
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[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[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym bol] :> Simp[-2 Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c , d, e}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(d + e*x)^(m + 1)*((a + b*x + c*x^2)^p/(e*(m + 1))), x] - Si mp[p/(e*(m + 1)) Int[(d + e*x)^(m + 1)*(b + 2*c*x)*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && GtQ[p, 0] && (IntegerQ[p] || LtQ[m, -1]) && NeQ[m, -1] && !ILtQ[m + 2*p + 1, 0] && IntQuadraticQ[a, b, c, d, e, m, p, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*((a + b*x + c*x^2)^p/ (c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] - Simp[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)) Int[(d + e*x)^m*(a + b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2* a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p - c*d - 2* c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c ^2*d^2*(1 + 2*p) - c*e*(b*d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x ] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] || !R ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) && !ILtQ[m + 2*p, 0] && (Integer Q[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + b*x + c*x^2)^ p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
Int[((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[ Int[(a + b/x^n + c/x^(2*n))^p/x^2, x], x, 1/x] /; FreeQ[{a, b, c, p}, x] && EqQ[n2, 2*n] && ILtQ[n, 0]
Time = 0.09 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.22
| method | result | size |
| risch | \(-\frac {\left (556 a b c \,x^{3}+15 b^{3} x^{3}+216 a \,c^{2} x^{2}+118 b^{2} c \,x^{2}+136 b \,c^{2} x +48 c^{3}\right ) \sqrt {\frac {a \,x^{2}+b x +c}{x^{2}}}}{192 x^{3} c}+\frac {\left (-\frac {\left (240 a^{2} c^{2}+120 a \,b^{2} c -5 b^{4}\right ) \ln \left (\frac {2 c +b x +2 \sqrt {c}\, \sqrt {a \,x^{2}+b x +c}}{x}\right )}{\sqrt {c}}+384 a^{\frac {3}{2}} b c \ln \left (\frac {\frac {b}{2}+x a}{\sqrt {a}}+\sqrt {a \,x^{2}+b x +c}\right )+128 a^{3} c \left (\frac {\sqrt {a \,x^{2}+b x +c}}{a}-\frac {b \ln \left (\frac {\frac {b}{2}+x a}{\sqrt {a}}+\sqrt {a \,x^{2}+b x +c}\right )}{2 a^{\frac {3}{2}}}\right )\right ) \sqrt {\frac {a \,x^{2}+b x +c}{x^{2}}}\, x}{128 c \sqrt {a \,x^{2}+b x +c}}\) | \(249\) |
| default | \(\frac {\left (\frac {a \,x^{2}+b x +c}{x^{2}}\right )^{\frac {5}{2}} x \left (-360 \ln \left (\frac {2 c +b x +2 \sqrt {c}\, \sqrt {a \,x^{2}+b x +c}}{x}\right ) c^{\frac {7}{2}} a^{\frac {5}{2}} b^{2} x^{4}+660 a^{\frac {5}{2}} \sqrt {a \,x^{2}+b x +c}\, b^{2} c^{3} x^{4}+152 a^{\frac {7}{2}} \left (a \,x^{2}+b x +c \right )^{\frac {5}{2}} b c \,x^{5}-152 a^{\frac {5}{2}} \left (a \,x^{2}+b x +c \right )^{\frac {7}{2}} b c \,x^{3}+148 a^{\frac {5}{2}} \left (a \,x^{2}+b x +c \right )^{\frac {5}{2}} b^{2} c \,x^{4}+280 a^{\frac {7}{2}} \left (a \,x^{2}+b x +c \right )^{\frac {3}{2}} b \,c^{2} x^{5}-10 a^{\frac {5}{2}} \left (a \,x^{2}+b x +c \right )^{\frac {3}{2}} b^{3} c \,x^{5}-720 \ln \left (\frac {2 c +b x +2 \sqrt {c}\, \sqrt {a \,x^{2}+b x +c}}{x}\right ) c^{\frac {9}{2}} a^{\frac {7}{2}} x^{4}+15 \ln \left (\frac {2 c +b x +2 \sqrt {c}\, \sqrt {a \,x^{2}+b x +c}}{x}\right ) c^{\frac {5}{2}} a^{\frac {3}{2}} b^{4} x^{4}+260 a^{\frac {5}{2}} \left (a \,x^{2}+b x +c \right )^{\frac {3}{2}} b^{2} c^{2} x^{4}+600 a^{\frac {7}{2}} \sqrt {a \,x^{2}+b x +c}\, b \,c^{3} x^{5}-30 a^{\frac {5}{2}} \sqrt {a \,x^{2}+b x +c}\, b^{3} c^{2} x^{5}+960 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x +c}\, \sqrt {a}+2 x a +b}{2 \sqrt {a}}\right ) a^{3} b \,c^{4} x^{4}+6 a^{\frac {3}{2}} \left (a \,x^{2}+b x +c \right )^{\frac {7}{2}} b^{3} x^{3}-6 a^{\frac {3}{2}} \left (a \,x^{2}+b x +c \right )^{\frac {5}{2}} b^{4} x^{4}-6 a^{\frac {5}{2}} \left (a \,x^{2}+b x +c \right )^{\frac {5}{2}} b^{3} x^{5}+144 a^{\frac {7}{2}} \left (a \,x^{2}+b x +c \right )^{\frac {5}{2}} c^{2} x^{4}-144 a^{\frac {5}{2}} \left (a \,x^{2}+b x +c \right )^{\frac {7}{2}} c^{2} x^{2}+4 a^{\frac {3}{2}} \left (a \,x^{2}+b x +c \right )^{\frac {7}{2}} b^{2} c \,x^{2}+240 a^{\frac {7}{2}} \left (a \,x^{2}+b x +c \right )^{\frac {3}{2}} c^{3} x^{4}-10 a^{\frac {3}{2}} \left (a \,x^{2}+b x +c \right )^{\frac {3}{2}} b^{4} c \,x^{4}+16 a^{\frac {3}{2}} \left (a \,x^{2}+b x +c \right )^{\frac {7}{2}} b \,c^{2} x +720 a^{\frac {7}{2}} \sqrt {a \,x^{2}+b x +c}\, c^{4} x^{4}-30 a^{\frac {3}{2}} \sqrt {a \,x^{2}+b x +c}\, b^{4} c^{2} x^{4}-96 \left (a \,x^{2}+b x +c \right )^{\frac {7}{2}} c^{3} a^{\frac {3}{2}}\right )}{384 \left (a \,x^{2}+b x +c \right )^{\frac {5}{2}} c^{4} a^{\frac {3}{2}}}\) | \(701\) |
Input:
int((a+c/x^2+b/x)^(5/2),x,method=_RETURNVERBOSE)
Output:
-1/192*(556*a*b*c*x^3+15*b^3*x^3+216*a*c^2*x^2+118*b^2*c*x^2+136*b*c^2*x+4 8*c^3)/x^3/c*((a*x^2+b*x+c)/x^2)^(1/2)+1/128/c*(-(240*a^2*c^2+120*a*b^2*c- 5*b^4)/c^(1/2)*ln((2*c+b*x+2*c^(1/2)*(a*x^2+b*x+c)^(1/2))/x)+384*a^(3/2)*b *c*ln((1/2*b+x*a)/a^(1/2)+(a*x^2+b*x+c)^(1/2))+128*a^3*c*(1/a*(a*x^2+b*x+c )^(1/2)-1/2*b/a^(3/2)*ln((1/2*b+x*a)/a^(1/2)+(a*x^2+b*x+c)^(1/2))))*((a*x^ 2+b*x+c)/x^2)^(1/2)*x/(a*x^2+b*x+c)^(1/2)
Time = 0.20 (sec) , antiderivative size = 959, normalized size of antiderivative = 4.70 \[ \int \left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{5/2} \, dx =\text {Too large to display} \] Input:
integrate((a+c/x^2+b/x)^(5/2),x, algorithm="fricas")
Output:
[1/768*(960*a^(3/2)*b*c^2*x^3*log(-8*a^2*x^2 - 8*a*b*x - b^2 - 4*a*c - 4*( 2*a*x^2 + b*x)*sqrt(a)*sqrt((a*x^2 + b*x + c)/x^2)) - 15*(b^4 - 24*a*b^2*c - 48*a^2*c^2)*sqrt(c)*x^3*log(-(8*b*c*x + (b^2 + 4*a*c)*x^2 + 8*c^2 - 4*( b*x^2 + 2*c*x)*sqrt(c)*sqrt((a*x^2 + b*x + c)/x^2))/x^2) + 4*(192*a^2*c^2* x^4 - 136*b*c^3*x - 48*c^4 - (15*b^3*c + 556*a*b*c^2)*x^3 - 2*(59*b^2*c^2 + 108*a*c^3)*x^2)*sqrt((a*x^2 + b*x + c)/x^2))/(c^2*x^3), -1/768*(1920*sqr t(-a)*a*b*c^2*x^3*arctan(1/2*(2*a*x^2 + b*x)*sqrt(-a)*sqrt((a*x^2 + b*x + c)/x^2)/(a^2*x^2 + a*b*x + a*c)) + 15*(b^4 - 24*a*b^2*c - 48*a^2*c^2)*sqrt (c)*x^3*log(-(8*b*c*x + (b^2 + 4*a*c)*x^2 + 8*c^2 - 4*(b*x^2 + 2*c*x)*sqrt (c)*sqrt((a*x^2 + b*x + c)/x^2))/x^2) - 4*(192*a^2*c^2*x^4 - 136*b*c^3*x - 48*c^4 - (15*b^3*c + 556*a*b*c^2)*x^3 - 2*(59*b^2*c^2 + 108*a*c^3)*x^2)*s qrt((a*x^2 + b*x + c)/x^2))/(c^2*x^3), 1/384*(480*a^(3/2)*b*c^2*x^3*log(-8 *a^2*x^2 - 8*a*b*x - b^2 - 4*a*c - 4*(2*a*x^2 + b*x)*sqrt(a)*sqrt((a*x^2 + b*x + c)/x^2)) - 15*(b^4 - 24*a*b^2*c - 48*a^2*c^2)*sqrt(-c)*x^3*arctan(1 /2*(b*x^2 + 2*c*x)*sqrt(-c)*sqrt((a*x^2 + b*x + c)/x^2)/(a*c*x^2 + b*c*x + c^2)) + 2*(192*a^2*c^2*x^4 - 136*b*c^3*x - 48*c^4 - (15*b^3*c + 556*a*b*c ^2)*x^3 - 2*(59*b^2*c^2 + 108*a*c^3)*x^2)*sqrt((a*x^2 + b*x + c)/x^2))/(c^ 2*x^3), -1/384*(960*sqrt(-a)*a*b*c^2*x^3*arctan(1/2*(2*a*x^2 + b*x)*sqrt(- a)*sqrt((a*x^2 + b*x + c)/x^2)/(a^2*x^2 + a*b*x + a*c)) + 15*(b^4 - 24*a*b ^2*c - 48*a^2*c^2)*sqrt(-c)*x^3*arctan(1/2*(b*x^2 + 2*c*x)*sqrt(-c)*sqr...
\[ \int \left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{5/2} \, dx=\int \left (a + \frac {b}{x} + \frac {c}{x^{2}}\right )^{\frac {5}{2}}\, dx \] Input:
integrate((a+c/x**2+b/x)**(5/2),x)
Output:
Integral((a + b/x + c/x**2)**(5/2), x)
\[ \int \left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{5/2} \, dx=\int { {\left (a + \frac {b}{x} + \frac {c}{x^{2}}\right )}^{\frac {5}{2}} \,d x } \] Input:
integrate((a+c/x^2+b/x)^(5/2),x, algorithm="maxima")
Output:
integrate((a + b/x + c/x^2)^(5/2), x)
Timed out. \[ \int \left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{5/2} \, dx=\text {Timed out} \] Input:
integrate((a+c/x^2+b/x)^(5/2),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{5/2} \, dx=\int {\left (a+\frac {b}{x}+\frac {c}{x^2}\right )}^{5/2} \,d x \] Input:
int((a + b/x + c/x^2)^(5/2),x)
Output:
int((a + b/x + c/x^2)^(5/2), x)
Time = 0.30 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.63 \[ \int \left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{5/2} \, dx=\frac {384 \sqrt {a \,x^{2}+b x +c}\, a^{2} c^{2} x^{4}-1112 \sqrt {a \,x^{2}+b x +c}\, a b \,c^{2} x^{3}-432 \sqrt {a \,x^{2}+b x +c}\, a \,c^{3} x^{2}-30 \sqrt {a \,x^{2}+b x +c}\, b^{3} c \,x^{3}-236 \sqrt {a \,x^{2}+b x +c}\, b^{2} c^{2} x^{2}-272 \sqrt {a \,x^{2}+b x +c}\, b \,c^{3} x -96 \sqrt {a \,x^{2}+b x +c}\, c^{4}+960 \sqrt {a}\, \mathrm {log}\left (-2 \sqrt {a}\, \sqrt {a \,x^{2}+b x +c}-2 a x -b \right ) a b \,c^{2} x^{4}+720 \sqrt {c}\, \mathrm {log}\left (2 \sqrt {c}\, \sqrt {a \,x^{2}+b x +c}-b x -2 c \right ) a^{2} c^{2} x^{4}+360 \sqrt {c}\, \mathrm {log}\left (2 \sqrt {c}\, \sqrt {a \,x^{2}+b x +c}-b x -2 c \right ) a \,b^{2} c \,x^{4}-15 \sqrt {c}\, \mathrm {log}\left (2 \sqrt {c}\, \sqrt {a \,x^{2}+b x +c}-b x -2 c \right ) b^{4} x^{4}-720 \sqrt {c}\, \mathrm {log}\left (x \right ) a^{2} c^{2} x^{4}-360 \sqrt {c}\, \mathrm {log}\left (x \right ) a \,b^{2} c \,x^{4}+15 \sqrt {c}\, \mathrm {log}\left (x \right ) b^{4} x^{4}}{384 c^{2} x^{4}} \] Input:
int((a+c/x^2+b/x)^(5/2),x)
Output:
(384*sqrt(a*x**2 + b*x + c)*a**2*c**2*x**4 - 1112*sqrt(a*x**2 + b*x + c)*a *b*c**2*x**3 - 432*sqrt(a*x**2 + b*x + c)*a*c**3*x**2 - 30*sqrt(a*x**2 + b *x + c)*b**3*c*x**3 - 236*sqrt(a*x**2 + b*x + c)*b**2*c**2*x**2 - 272*sqrt (a*x**2 + b*x + c)*b*c**3*x - 96*sqrt(a*x**2 + b*x + c)*c**4 + 960*sqrt(a) *log( - 2*sqrt(a)*sqrt(a*x**2 + b*x + c) - 2*a*x - b)*a*b*c**2*x**4 + 720* sqrt(c)*log(2*sqrt(c)*sqrt(a*x**2 + b*x + c) - b*x - 2*c)*a**2*c**2*x**4 + 360*sqrt(c)*log(2*sqrt(c)*sqrt(a*x**2 + b*x + c) - b*x - 2*c)*a*b**2*c*x* *4 - 15*sqrt(c)*log(2*sqrt(c)*sqrt(a*x**2 + b*x + c) - b*x - 2*c)*b**4*x** 4 - 720*sqrt(c)*log(x)*a**2*c**2*x**4 - 360*sqrt(c)*log(x)*a*b**2*c*x**4 + 15*sqrt(c)*log(x)*b**4*x**4)/(384*c**2*x**4)