Integrand size = 20, antiderivative size = 216 \[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^{13}} \, dx=\frac {\left (b^2-4 a c\right ) \left (7 b^2-4 a c\right ) \left (2 a+b x^2\right ) \sqrt {a+b x^2+c x^4}}{1024 a^4 x^4}-\frac {\left (7 b^2-4 a c\right ) \left (2 a+b x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{384 a^3 x^8}-\frac {\left (a+b x^2+c x^4\right )^{5/2}}{12 a x^{12}}+\frac {7 b \left (a+b x^2+c x^4\right )^{5/2}}{120 a^2 x^{10}}-\frac {\left (b^2-4 a c\right )^2 \left (7 b^2-4 a c\right ) \text {arctanh}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{2048 a^{9/2}} \] Output:
1/1024*(-4*a*c+b^2)*(-4*a*c+7*b^2)*(b*x^2+2*a)*(c*x^4+b*x^2+a)^(1/2)/a^4/x ^4-1/384*(-4*a*c+7*b^2)*(b*x^2+2*a)*(c*x^4+b*x^2+a)^(3/2)/a^3/x^8-1/12*(c* x^4+b*x^2+a)^(5/2)/a/x^12+7/120*b*(c*x^4+b*x^2+a)^(5/2)/a^2/x^10-1/2048*(- 4*a*c+b^2)^2*(-4*a*c+7*b^2)*arctanh(1/2*(b*x^2+2*a)/a^(1/2)/(c*x^4+b*x^2+a )^(1/2))/a^(9/2)
Time = 1.35 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.93 \[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^{13}} \, dx=\frac {-\frac {\sqrt {a} \sqrt {a+b x^2+c x^4} \left (1280 a^5-105 b^5 x^{10}+10 a b^3 x^8 \left (7 b+76 c x^2\right )+64 a^4 \left (26 b x^2+35 c x^4\right )+48 a^3 x^4 \left (b^2+6 b c x^2+10 c^2 x^4\right )-8 a^2 b x^6 \left (7 b^2+54 b c x^2+162 c^2 x^4\right )\right )}{x^{12}}+15 \left (b^2-4 a c\right )^2 \left (7 b^2-4 a c\right ) \text {arctanh}\left (\frac {\sqrt {c} x^2-\sqrt {a+b x^2+c x^4}}{\sqrt {a}}\right )}{15360 a^{9/2}} \] Input:
Integrate[(a + b*x^2 + c*x^4)^(3/2)/x^13,x]
Output:
(-((Sqrt[a]*Sqrt[a + b*x^2 + c*x^4]*(1280*a^5 - 105*b^5*x^10 + 10*a*b^3*x^ 8*(7*b + 76*c*x^2) + 64*a^4*(26*b*x^2 + 35*c*x^4) + 48*a^3*x^4*(b^2 + 6*b* c*x^2 + 10*c^2*x^4) - 8*a^2*b*x^6*(7*b^2 + 54*b*c*x^2 + 162*c^2*x^4)))/x^1 2) + 15*(b^2 - 4*a*c)^2*(7*b^2 - 4*a*c)*ArcTanh[(Sqrt[c]*x^2 - Sqrt[a + b* x^2 + c*x^4])/Sqrt[a]])/(15360*a^(9/2))
Time = 0.66 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.03, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {1434, 1167, 27, 1228, 1152, 1152, 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.
| \(\displaystyle \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^{13}} \, dx\) |
| \(\Big \downarrow \) 1434 |
| \(\displaystyle \frac {1}{2} \int \frac {\left (c x^4+b x^2+a\right )^{3/2}}{x^{14}}dx^2\) |
| \(\Big \downarrow \) 1167 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\int \frac {\left (2 c x^2+7 b\right ) \left (c x^4+b x^2+a\right )^{3/2}}{2 x^{12}}dx^2}{6 a}-\frac {\left (a+b x^2+c x^4\right )^{5/2}}{6 a x^{12}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\int \frac {\left (2 c x^2+7 b\right ) \left (c x^4+b x^2+a\right )^{3/2}}{x^{12}}dx^2}{12 a}-\frac {\left (a+b x^2+c x^4\right )^{5/2}}{6 a x^{12}}\right )\) |
| \(\Big \downarrow \) 1228 |
| \(\displaystyle \frac {1}{2} \left (-\frac {-\frac {\left (7 b^2-4 a c\right ) \int \frac {\left (c x^4+b x^2+a\right )^{3/2}}{x^{10}}dx^2}{2 a}-\frac {7 b \left (a+b x^2+c x^4\right )^{5/2}}{5 a x^{10}}}{12 a}-\frac {\left (a+b x^2+c x^4\right )^{5/2}}{6 a x^{12}}\right )\) |
| \(\Big \downarrow \) 1152 |
| \(\displaystyle \frac {1}{2} \left (-\frac {-\frac {\left (7 b^2-4 a c\right ) \left (-\frac {3 \left (b^2-4 a c\right ) \int \frac {\sqrt {c x^4+b x^2+a}}{x^6}dx^2}{16 a}-\frac {\left (2 a+b x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{8 a x^8}\right )}{2 a}-\frac {7 b \left (a+b x^2+c x^4\right )^{5/2}}{5 a x^{10}}}{12 a}-\frac {\left (a+b x^2+c x^4\right )^{5/2}}{6 a x^{12}}\right )\) |
| \(\Big \downarrow \) 1152 |
| \(\displaystyle \frac {1}{2} \left (-\frac {-\frac {\left (7 b^2-4 a c\right ) \left (-\frac {3 \left (b^2-4 a c\right ) \left (-\frac {\left (b^2-4 a c\right ) \int \frac {1}{x^2 \sqrt {c x^4+b x^2+a}}dx^2}{8 a}-\frac {\left (2 a+b x^2\right ) \sqrt {a+b x^2+c x^4}}{4 a x^4}\right )}{16 a}-\frac {\left (2 a+b x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{8 a x^8}\right )}{2 a}-\frac {7 b \left (a+b x^2+c x^4\right )^{5/2}}{5 a x^{10}}}{12 a}-\frac {\left (a+b x^2+c x^4\right )^{5/2}}{6 a x^{12}}\right )\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {1}{2} \left (-\frac {-\frac {\left (7 b^2-4 a c\right ) \left (-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\left (b^2-4 a c\right ) \int \frac {1}{4 a-x^4}d\frac {b x^2+2 a}{\sqrt {c x^4+b x^2+a}}}{4 a}-\frac {\left (2 a+b x^2\right ) \sqrt {a+b x^2+c x^4}}{4 a x^4}\right )}{16 a}-\frac {\left (2 a+b x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{8 a x^8}\right )}{2 a}-\frac {7 b \left (a+b x^2+c x^4\right )^{5/2}}{5 a x^{10}}}{12 a}-\frac {\left (a+b x^2+c x^4\right )^{5/2}}{6 a x^{12}}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{2} \left (-\frac {-\frac {\left (7 b^2-4 a c\right ) \left (-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{8 a^{3/2}}-\frac {\left (2 a+b x^2\right ) \sqrt {a+b x^2+c x^4}}{4 a x^4}\right )}{16 a}-\frac {\left (2 a+b x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{8 a x^8}\right )}{2 a}-\frac {7 b \left (a+b x^2+c x^4\right )^{5/2}}{5 a x^{10}}}{12 a}-\frac {\left (a+b x^2+c x^4\right )^{5/2}}{6 a x^{12}}\right )\) |
Input:
Int[(a + b*x^2 + c*x^4)^(3/2)/x^13,x]
Output:
(-1/6*(a + b*x^2 + c*x^4)^(5/2)/(a*x^12) - ((-7*b*(a + b*x^2 + c*x^4)^(5/2 ))/(5*a*x^10) - ((7*b^2 - 4*a*c)*(-1/8*((2*a + b*x^2)*(a + b*x^2 + c*x^4)^ (3/2))/(a*x^8) - (3*(b^2 - 4*a*c)*(-1/4*((2*a + b*x^2)*Sqrt[a + b*x^2 + c* x^4])/(a*x^4) + ((b^2 - 4*a*c)*ArcTanh[(2*a + b*x^2)/(2*Sqrt[a]*Sqrt[a + b *x^2 + c*x^4])])/(8*a^(3/2))))/(16*a)))/(2*a))/(12*a))/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[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(-(d + e*x)^(m + 1))*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b *x + c*x^2)^p/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[p*((b^2 - 4*a *c)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))) Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[m + 2*p + 2, 0] && GtQ[p, 0]
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[e*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/((m + 1)*(c*d ^2 - b*d*e + a*e^2))), x] + Simp[1/((m + 1)*(c*d^2 - b*d*e + a*e^2)) Int[ (d + e*x)^(m + 1)*Simp[c*d*(m + 1) - b*e*(m + p + 2) - c*e*(m + 2*p + 3)*x, x]*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[m , -1] && ((LtQ[m, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]) || (SumSimp lerQ[m, 1] && IntegerQ[p]) || ILtQ[Simplify[m + 2*p + 3], 0])
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Simp[(b*(e *f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^ (m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x ] && EqQ[Simplify[m + 2*p + 3], 0]
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp [1/2 Subst[Int[x^((m - 1)/2)*(a + b*x + c*x^2)^p, x], x, x^2], x] /; Free Q[{a, b, c, p}, x] && IntegerQ[(m - 1)/2]
Time = 0.32 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.88
| method | result | size |
| pseudoelliptic | \(\frac {\left (a c -\frac {b^{2}}{4}\right )^{2} \left (a c -\frac {7 b^{2}}{4}\right ) x^{12} \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )-\frac {52 \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (-\frac {7 \left (\frac {162}{7} c^{2} x^{4}+\frac {54}{7} b c \,x^{2}+b^{2}\right ) b \,x^{6} a^{\frac {5}{2}}}{208}+\frac {3 x^{4} \left (10 c^{2} x^{4}+6 b c \,x^{2}+b^{2}\right ) a^{\frac {7}{2}}}{104}+\frac {35 \left (\frac {76 c \,x^{2}}{7}+b \right ) b^{3} x^{8} a^{\frac {3}{2}}}{832}+x^{2} \left (\frac {35 c \,x^{2}}{26}+b \right ) a^{\frac {9}{2}}-\frac {105 \sqrt {a}\, b^{5} x^{10}}{1664}+\frac {10 a^{\frac {11}{2}}}{13}\right )}{15}}{32 a^{\frac {9}{2}} x^{12}}\) | \(189\) |
| risch | \(-\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (-1296 a^{2} b \,c^{2} x^{10}+760 a \,b^{3} c \,x^{10}-105 x^{10} b^{5}+480 a^{3} c^{2} x^{8}-432 a^{2} b^{2} c \,x^{8}+70 a \,x^{8} b^{4}+288 a^{3} b c \,x^{6}-56 a^{2} x^{6} b^{3}+2240 a^{4} c \,x^{4}+48 a^{3} x^{4} b^{2}+1664 x^{2} a^{4} b +1280 a^{5}\right )}{15360 x^{12} a^{4}}+\frac {\left (64 a^{3} c^{3}-144 a^{2} b^{2} c^{2}+60 a \,b^{4} c -7 b^{6}\right ) \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{2048 a^{\frac {9}{2}}}\) | \(212\) |
| default | \(\frac {15 b^{4} c \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{512 a^{\frac {7}{2}}}-\frac {19 b^{3} c \sqrt {c \,x^{4}+b \,x^{2}+a}}{384 a^{3} x^{2}}+\frac {9 b^{2} c \sqrt {c \,x^{4}+b \,x^{2}+a}}{320 a^{2} x^{4}}-\frac {9 b^{2} c^{2} \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{128 a^{\frac {5}{2}}}-\frac {3 b c \sqrt {c \,x^{4}+b \,x^{2}+a}}{160 a \,x^{6}}+\frac {27 b \,c^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{320 a^{2} x^{2}}-\frac {b^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{320 a \,x^{8}}+\frac {7 b^{3} \sqrt {c \,x^{4}+b \,x^{2}+a}}{1920 a^{2} x^{6}}-\frac {7 b^{4} \sqrt {c \,x^{4}+b \,x^{2}+a}}{1536 a^{3} x^{4}}+\frac {7 b^{5} \sqrt {c \,x^{4}+b \,x^{2}+a}}{1024 a^{4} x^{2}}-\frac {7 b^{6} \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{2048 a^{\frac {9}{2}}}-\frac {c^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{32 a \,x^{4}}+\frac {c^{3} \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{32 a^{\frac {3}{2}}}-\frac {a \sqrt {c \,x^{4}+b \,x^{2}+a}}{12 x^{12}}-\frac {13 b \sqrt {c \,x^{4}+b \,x^{2}+a}}{120 x^{10}}-\frac {7 c \sqrt {c \,x^{4}+b \,x^{2}+a}}{48 x^{8}}\) | \(457\) |
| elliptic | \(\frac {15 b^{4} c \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{512 a^{\frac {7}{2}}}-\frac {19 b^{3} c \sqrt {c \,x^{4}+b \,x^{2}+a}}{384 a^{3} x^{2}}+\frac {9 b^{2} c \sqrt {c \,x^{4}+b \,x^{2}+a}}{320 a^{2} x^{4}}-\frac {9 b^{2} c^{2} \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{128 a^{\frac {5}{2}}}-\frac {3 b c \sqrt {c \,x^{4}+b \,x^{2}+a}}{160 a \,x^{6}}+\frac {27 b \,c^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{320 a^{2} x^{2}}-\frac {b^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{320 a \,x^{8}}+\frac {7 b^{3} \sqrt {c \,x^{4}+b \,x^{2}+a}}{1920 a^{2} x^{6}}-\frac {7 b^{4} \sqrt {c \,x^{4}+b \,x^{2}+a}}{1536 a^{3} x^{4}}+\frac {7 b^{5} \sqrt {c \,x^{4}+b \,x^{2}+a}}{1024 a^{4} x^{2}}-\frac {7 b^{6} \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{2048 a^{\frac {9}{2}}}-\frac {c^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{32 a \,x^{4}}+\frac {c^{3} \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{32 a^{\frac {3}{2}}}-\frac {a \sqrt {c \,x^{4}+b \,x^{2}+a}}{12 x^{12}}-\frac {13 b \sqrt {c \,x^{4}+b \,x^{2}+a}}{120 x^{10}}-\frac {7 c \sqrt {c \,x^{4}+b \,x^{2}+a}}{48 x^{8}}\) | \(457\) |
Input:
int((c*x^4+b*x^2+a)^(3/2)/x^13,x,method=_RETURNVERBOSE)
Output:
1/32/a^(9/2)*((a*c-1/4*b^2)^2*(a*c-7/4*b^2)*x^12*ln((2*a+b*x^2+2*a^(1/2)*( c*x^4+b*x^2+a)^(1/2))/x^2)-52/15*(c*x^4+b*x^2+a)^(1/2)*(-7/208*(162/7*c^2* x^4+54/7*b*c*x^2+b^2)*b*x^6*a^(5/2)+3/104*x^4*(10*c^2*x^4+6*b*c*x^2+b^2)*a ^(7/2)+35/832*(76/7*c*x^2+b)*b^3*x^8*a^(3/2)+x^2*(35/26*c*x^2+b)*a^(9/2)-1 05/1664*a^(1/2)*b^5*x^10+10/13*a^(11/2)))/x^12
Time = 0.25 (sec) , antiderivative size = 473, normalized size of antiderivative = 2.19 \[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^{13}} \, dx=\left [-\frac {15 \, {\left (7 \, b^{6} - 60 \, a b^{4} c + 144 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt {a} x^{12} \log \left (-\frac {{\left (b^{2} + 4 \, a c\right )} x^{4} + 8 \, a b x^{2} + 4 \, \sqrt {c x^{4} + b x^{2} + a} {\left (b x^{2} + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{4}}\right ) - 4 \, {\left ({\left (105 \, a b^{5} - 760 \, a^{2} b^{3} c + 1296 \, a^{3} b c^{2}\right )} x^{10} - 2 \, {\left (35 \, a^{2} b^{4} - 216 \, a^{3} b^{2} c + 240 \, a^{4} c^{2}\right )} x^{8} - 1664 \, a^{5} b x^{2} + 8 \, {\left (7 \, a^{3} b^{3} - 36 \, a^{4} b c\right )} x^{6} - 1280 \, a^{6} - 16 \, {\left (3 \, a^{4} b^{2} + 140 \, a^{5} c\right )} x^{4}\right )} \sqrt {c x^{4} + b x^{2} + a}}{61440 \, a^{5} x^{12}}, \frac {15 \, {\left (7 \, b^{6} - 60 \, a b^{4} c + 144 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt {-a} x^{12} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2} + a} {\left (b x^{2} + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{4} + a b x^{2} + a^{2}\right )}}\right ) + 2 \, {\left ({\left (105 \, a b^{5} - 760 \, a^{2} b^{3} c + 1296 \, a^{3} b c^{2}\right )} x^{10} - 2 \, {\left (35 \, a^{2} b^{4} - 216 \, a^{3} b^{2} c + 240 \, a^{4} c^{2}\right )} x^{8} - 1664 \, a^{5} b x^{2} + 8 \, {\left (7 \, a^{3} b^{3} - 36 \, a^{4} b c\right )} x^{6} - 1280 \, a^{6} - 16 \, {\left (3 \, a^{4} b^{2} + 140 \, a^{5} c\right )} x^{4}\right )} \sqrt {c x^{4} + b x^{2} + a}}{30720 \, a^{5} x^{12}}\right ] \] Input:
integrate((c*x^4+b*x^2+a)^(3/2)/x^13,x, algorithm="fricas")
Output:
[-1/61440*(15*(7*b^6 - 60*a*b^4*c + 144*a^2*b^2*c^2 - 64*a^3*c^3)*sqrt(a)* x^12*log(-((b^2 + 4*a*c)*x^4 + 8*a*b*x^2 + 4*sqrt(c*x^4 + b*x^2 + a)*(b*x^ 2 + 2*a)*sqrt(a) + 8*a^2)/x^4) - 4*((105*a*b^5 - 760*a^2*b^3*c + 1296*a^3* b*c^2)*x^10 - 2*(35*a^2*b^4 - 216*a^3*b^2*c + 240*a^4*c^2)*x^8 - 1664*a^5* b*x^2 + 8*(7*a^3*b^3 - 36*a^4*b*c)*x^6 - 1280*a^6 - 16*(3*a^4*b^2 + 140*a^ 5*c)*x^4)*sqrt(c*x^4 + b*x^2 + a))/(a^5*x^12), 1/30720*(15*(7*b^6 - 60*a*b ^4*c + 144*a^2*b^2*c^2 - 64*a^3*c^3)*sqrt(-a)*x^12*arctan(1/2*sqrt(c*x^4 + b*x^2 + a)*(b*x^2 + 2*a)*sqrt(-a)/(a*c*x^4 + a*b*x^2 + a^2)) + 2*((105*a* b^5 - 760*a^2*b^3*c + 1296*a^3*b*c^2)*x^10 - 2*(35*a^2*b^4 - 216*a^3*b^2*c + 240*a^4*c^2)*x^8 - 1664*a^5*b*x^2 + 8*(7*a^3*b^3 - 36*a^4*b*c)*x^6 - 12 80*a^6 - 16*(3*a^4*b^2 + 140*a^5*c)*x^4)*sqrt(c*x^4 + b*x^2 + a))/(a^5*x^1 2)]
\[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^{13}} \, dx=\int \frac {\left (a + b x^{2} + c x^{4}\right )^{\frac {3}{2}}}{x^{13}}\, dx \] Input:
integrate((c*x**4+b*x**2+a)**(3/2)/x**13,x)
Output:
Integral((a + b*x**2 + c*x**4)**(3/2)/x**13, x)
Exception generated. \[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^{13}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((c*x^4+b*x^2+a)^(3/2)/x^13,x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for more deta
Leaf count of result is larger than twice the leaf count of optimal. 1235 vs. \(2 (190) = 380\).
Time = 0.20 (sec) , antiderivative size = 1235, normalized size of antiderivative = 5.72 \[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^{13}} \, dx=\text {Too large to display} \] Input:
integrate((c*x^4+b*x^2+a)^(3/2)/x^13,x, algorithm="giac")
Output:
1/1024*(7*b^6 - 60*a*b^4*c + 144*a^2*b^2*c^2 - 64*a^3*c^3)*arctan(-(sqrt(c )*x^2 - sqrt(c*x^4 + b*x^2 + a))/sqrt(-a))/(sqrt(-a)*a^4) - 1/15360*(105*( sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))^11*b^6 - 900*(sqrt(c)*x^2 - sqrt(c* x^4 + b*x^2 + a))^11*a*b^4*c + 2160*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a) )^11*a^2*b^2*c^2 - 960*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))^11*a^3*c^3 - 595*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))^9*a*b^6 + 5100*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))^9*a^2*b^4*c - 12240*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))^9*a^3*b^2*c^2 - 15040*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a)) ^9*a^4*c^3 - 76800*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))^8*a^4*b*c^(5/2) + 1386*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))^7*a^2*b^6 - 11880*(sqrt(c) *x^2 - sqrt(c*x^4 + b*x^2 + a))^7*a^3*b^4*c - 97440*(sqrt(c)*x^2 - sqrt(c* x^4 + b*x^2 + a))^7*a^4*b^2*c^2 - 24960*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))^7*a^5*c^3 - 112640*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))^6*a^4*b^3 *c^(3/2) - 61440*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))^6*a^5*b*c^(5/2) - 1686*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))^5*a^3*b^6 - 42600*(sqrt(c)*x ^2 - sqrt(c*x^4 + b*x^2 + a))^5*a^4*b^4*c - 128160*(sqrt(c)*x^2 - sqrt(c*x ^4 + b*x^2 + a))^5*a^5*b^2*c^2 - 24960*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))^5*a^6*c^3 - 15360*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))^4*a^4*b^5*s qrt(c) - 61440*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))^4*a^5*b^3*c^(3/2) - 92160*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))^4*a^6*b*c^(5/2) - 595*(s...
Timed out. \[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^{13}} \, dx=\int \frac {{\left (c\,x^4+b\,x^2+a\right )}^{3/2}}{x^{13}} \,d x \] Input:
int((a + b*x^2 + c*x^4)^(3/2)/x^13,x)
Output:
int((a + b*x^2 + c*x^4)^(3/2)/x^13, x)
\[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^{13}} \, dx=\frac {-960 \sqrt {c \,x^{4}+b \,x^{2}+a}\, a^{3} b c +400 \sqrt {c \,x^{4}+b \,x^{2}+a}\, a^{2} b^{3}-1248 \sqrt {c \,x^{4}+b \,x^{2}+a}\, a^{2} b^{2} c \,x^{2}-1680 \sqrt {c \,x^{4}+b \,x^{2}+a}\, a^{2} b \,c^{2} x^{4}-160 \sqrt {c \,x^{4}+b \,x^{2}+a}\, a^{2} c^{3} x^{6}+520 \sqrt {c \,x^{4}+b \,x^{2}+a}\, a \,b^{4} x^{2}+664 \sqrt {c \,x^{4}+b \,x^{2}+a}\, a \,b^{3} c \,x^{4}+144 \sqrt {c \,x^{4}+b \,x^{2}+a}\, a \,b^{2} c^{2} x^{6}-160 \sqrt {c \,x^{4}+b \,x^{2}+a}\, a b \,c^{3} x^{8}+320 \sqrt {c \,x^{4}+b \,x^{2}+a}\, a \,c^{4} x^{10}+15 \sqrt {c \,x^{4}+b \,x^{2}+a}\, b^{5} x^{4}-18 \sqrt {c \,x^{4}+b \,x^{2}+a}\, b^{4} c \,x^{6}+24 \sqrt {c \,x^{4}+b \,x^{2}+a}\, b^{3} c^{2} x^{8}-48 \sqrt {c \,x^{4}+b \,x^{2}+a}\, b^{2} c^{3} x^{10}-11520 \left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{12 a \,c^{2} x^{11}-5 b^{2} c \,x^{11}+12 a b c \,x^{9}-5 b^{3} x^{9}+12 a^{2} c \,x^{7}-5 a \,b^{2} x^{7}}d x \right ) a^{4} c^{4} x^{12}+30720 \left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{12 a \,c^{2} x^{11}-5 b^{2} c \,x^{11}+12 a b c \,x^{9}-5 b^{3} x^{9}+12 a^{2} c \,x^{7}-5 a \,b^{2} x^{7}}d x \right ) a^{3} b^{2} c^{3} x^{12}-21600 \left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{12 a \,c^{2} x^{11}-5 b^{2} c \,x^{11}+12 a b c \,x^{9}-5 b^{3} x^{9}+12 a^{2} c \,x^{7}-5 a \,b^{2} x^{7}}d x \right ) a^{2} b^{4} c^{2} x^{12}+5760 \left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{12 a \,c^{2} x^{11}-5 b^{2} c \,x^{11}+12 a b c \,x^{9}-5 b^{3} x^{9}+12 a^{2} c \,x^{7}-5 a \,b^{2} x^{7}}d x \right ) a \,b^{6} c \,x^{12}-525 \left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{12 a \,c^{2} x^{11}-5 b^{2} c \,x^{11}+12 a b c \,x^{9}-5 b^{3} x^{9}+12 a^{2} c \,x^{7}-5 a \,b^{2} x^{7}}d x \right ) b^{8} x^{12}}{960 a b \,x^{12} \left (12 a c -5 b^{2}\right )} \] Input:
int((c*x^4+b*x^2+a)^(3/2)/x^13,x)
Output:
( - 960*sqrt(a + b*x**2 + c*x**4)*a**3*b*c + 400*sqrt(a + b*x**2 + c*x**4) *a**2*b**3 - 1248*sqrt(a + b*x**2 + c*x**4)*a**2*b**2*c*x**2 - 1680*sqrt(a + b*x**2 + c*x**4)*a**2*b*c**2*x**4 - 160*sqrt(a + b*x**2 + c*x**4)*a**2* c**3*x**6 + 520*sqrt(a + b*x**2 + c*x**4)*a*b**4*x**2 + 664*sqrt(a + b*x** 2 + c*x**4)*a*b**3*c*x**4 + 144*sqrt(a + b*x**2 + c*x**4)*a*b**2*c**2*x**6 - 160*sqrt(a + b*x**2 + c*x**4)*a*b*c**3*x**8 + 320*sqrt(a + b*x**2 + c*x **4)*a*c**4*x**10 + 15*sqrt(a + b*x**2 + c*x**4)*b**5*x**4 - 18*sqrt(a + b *x**2 + c*x**4)*b**4*c*x**6 + 24*sqrt(a + b*x**2 + c*x**4)*b**3*c**2*x**8 - 48*sqrt(a + b*x**2 + c*x**4)*b**2*c**3*x**10 - 11520*int(sqrt(a + b*x**2 + c*x**4)/(12*a**2*c*x**7 - 5*a*b**2*x**7 + 12*a*b*c*x**9 + 12*a*c**2*x** 11 - 5*b**3*x**9 - 5*b**2*c*x**11),x)*a**4*c**4*x**12 + 30720*int(sqrt(a + b*x**2 + c*x**4)/(12*a**2*c*x**7 - 5*a*b**2*x**7 + 12*a*b*c*x**9 + 12*a*c **2*x**11 - 5*b**3*x**9 - 5*b**2*c*x**11),x)*a**3*b**2*c**3*x**12 - 21600* int(sqrt(a + b*x**2 + c*x**4)/(12*a**2*c*x**7 - 5*a*b**2*x**7 + 12*a*b*c*x **9 + 12*a*c**2*x**11 - 5*b**3*x**9 - 5*b**2*c*x**11),x)*a**2*b**4*c**2*x* *12 + 5760*int(sqrt(a + b*x**2 + c*x**4)/(12*a**2*c*x**7 - 5*a*b**2*x**7 + 12*a*b*c*x**9 + 12*a*c**2*x**11 - 5*b**3*x**9 - 5*b**2*c*x**11),x)*a*b**6 *c*x**12 - 525*int(sqrt(a + b*x**2 + c*x**4)/(12*a**2*c*x**7 - 5*a*b**2*x* *7 + 12*a*b*c*x**9 + 12*a*c**2*x**11 - 5*b**3*x**9 - 5*b**2*c*x**11),x)*b* *8*x**12)/(960*a*b*x**12*(12*a*c - 5*b**2))