Integrand size = 20, antiderivative size = 155 \[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x} \, dx=\frac {\left (b^2+8 a c+2 b c x^2\right ) \sqrt {a+b x^2+c x^4}}{16 c}+\frac {1}{6} \left (a+b x^2+c x^4\right )^{3/2}-\frac {1}{2} a^{3/2} \text {arctanh}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )-\frac {b \left (b^2-12 a c\right ) \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{32 c^{3/2}} \]
1/6*(c*x^4+b*x^2+a)^(3/2)-1/2*a^(3/2)*arctanh(1/2*(b*x^2+2*a)/a^(1/2)/(c*x ^4+b*x^2+a)^(1/2))-1/32*b*(-12*a*c+b^2)*arctanh(1/2*(2*c*x^2+b)/c^(1/2)/(c *x^4+b*x^2+a)^(1/2))/c^(3/2)+1/16*(2*b*c*x^2+8*a*c+b^2)*(c*x^4+b*x^2+a)^(1 /2)/c
Time = 0.54 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.95 \[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x} \, dx=\frac {\sqrt {a+b x^2+c x^4} \left (3 b^2+32 a c+14 b c x^2+8 c^2 x^4\right )}{48 c}+\frac {\left (-b^3+12 a b c\right ) \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{32 c^{3/2}}+a^{3/2} \text {arctanh}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}-\frac {\sqrt {a+b x^2+c x^4}}{\sqrt {a}}\right ) \]
(Sqrt[a + b*x^2 + c*x^4]*(3*b^2 + 32*a*c + 14*b*c*x^2 + 8*c^2*x^4))/(48*c) + ((-b^3 + 12*a*b*c)*ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c* x^4])])/(32*c^(3/2)) + a^(3/2)*ArcTanh[(Sqrt[c]*x^2)/Sqrt[a] - Sqrt[a + b* x^2 + c*x^4]/Sqrt[a]]
Time = 0.37 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.09, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1434, 1162, 25, 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.
| \(\displaystyle \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x} \, dx\) |
| \(\Big \downarrow \) 1434 |
| \(\displaystyle \frac {1}{2} \int \frac {\left (c x^4+b x^2+a\right )^{3/2}}{x^2}dx^2\) |
| \(\Big \downarrow \) 1162 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} \left (a+b x^2+c x^4\right )^{3/2}-\frac {1}{2} \int -\frac {\left (b x^2+2 a\right ) \sqrt {c x^4+b x^2+a}}{x^2}dx^2\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{2} \int \frac {\left (b x^2+2 a\right ) \sqrt {c x^4+b x^2+a}}{x^2}dx^2+\frac {1}{3} \left (a+b x^2+c x^4\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (\frac {\left (8 a c+b^2+2 b c x^2\right ) \sqrt {a+b x^2+c x^4}}{4 c}-\frac {\int -\frac {16 a^2 c-b \left (b^2-12 a c\right ) x^2}{2 x^2 \sqrt {c x^4+b x^2+a}}dx^2}{4 c}\right )+\frac {1}{3} \left (a+b x^2+c x^4\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (\frac {\int \frac {16 a^2 c-b \left (b^2-12 a c\right ) x^2}{x^2 \sqrt {c x^4+b x^2+a}}dx^2}{8 c}+\frac {\sqrt {a+b x^2+c x^4} \left (8 a c+b^2+2 b c x^2\right )}{4 c}\right )+\frac {1}{3} \left (a+b x^2+c x^4\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (\frac {16 a^2 c \int \frac {1}{x^2 \sqrt {c x^4+b x^2+a}}dx^2-b \left (b^2-12 a c\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx^2}{8 c}+\frac {\sqrt {a+b x^2+c x^4} \left (8 a c+b^2+2 b c x^2\right )}{4 c}\right )+\frac {1}{3} \left (a+b x^2+c x^4\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (\frac {16 a^2 c \int \frac {1}{x^2 \sqrt {c x^4+b x^2+a}}dx^2-2 b \left (b^2-12 a c\right ) \int \frac {1}{4 c-x^4}d\frac {2 c x^2+b}{\sqrt {c x^4+b x^2+a}}}{8 c}+\frac {\sqrt {a+b x^2+c x^4} \left (8 a c+b^2+2 b c x^2\right )}{4 c}\right )+\frac {1}{3} \left (a+b x^2+c x^4\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (\frac {16 a^2 c \int \frac {1}{x^2 \sqrt {c x^4+b x^2+a}}dx^2-\frac {b \left (b^2-12 a c\right ) \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{\sqrt {c}}}{8 c}+\frac {\sqrt {a+b x^2+c x^4} \left (8 a c+b^2+2 b c x^2\right )}{4 c}\right )+\frac {1}{3} \left (a+b x^2+c x^4\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (\frac {-32 a^2 c \int \frac {1}{4 a-x^4}d\frac {b x^2+2 a}{\sqrt {c x^4+b x^2+a}}-\frac {b \left (b^2-12 a c\right ) \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{\sqrt {c}}}{8 c}+\frac {\sqrt {a+b x^2+c x^4} \left (8 a c+b^2+2 b c x^2\right )}{4 c}\right )+\frac {1}{3} \left (a+b x^2+c x^4\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (\frac {-16 a^{3/2} c \text {arctanh}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )-\frac {b \left (b^2-12 a c\right ) \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{\sqrt {c}}}{8 c}+\frac {\sqrt {a+b x^2+c x^4} \left (8 a c+b^2+2 b c x^2\right )}{4 c}\right )+\frac {1}{3} \left (a+b x^2+c x^4\right )^{3/2}\right )\) |
((a + b*x^2 + c*x^4)^(3/2)/3 + (((b^2 + 8*a*c + 2*b*c*x^2)*Sqrt[a + b*x^2 + c*x^4])/(4*c) + (-16*a^(3/2)*c*ArcTanh[(2*a + b*x^2)/(2*Sqrt[a]*Sqrt[a + b*x^2 + c*x^4])] - (b*(b^2 - 12*a*c)*ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*Sqr t[a + b*x^2 + c*x^4])])/Sqrt[c])/(8*c))/2)/2
3.10.40.3.1 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[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 + 2*p + 1))), x ] - Simp[p/(e*(m + 2*p + 1)) Int[(d + e*x)^m*Simp[b*d - 2*a*e + (2*c*d - b*e)*x, x]*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x ] && GtQ[p, 0] && NeQ[m + 2*p + 1, 0] && ( !RationalQ[m] || LtQ[m, 1]) && !ILtQ[m + 2*p, 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[(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 = 0.13 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.24
| method | result | size |
| default | \(\frac {c \,x^{4} \sqrt {c \,x^{4}+b \,x^{2}+a}}{6}+\frac {7 b \,x^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{24}+\frac {b^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{16 c}-\frac {b^{3} \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{32 c^{\frac {3}{2}}}+\frac {3 a b \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{8 \sqrt {c}}+\frac {2 a \sqrt {c \,x^{4}+b \,x^{2}+a}}{3}-\frac {a^{\frac {3}{2}} \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{2}\) | \(192\) |
| elliptic | \(\frac {c \,x^{4} \sqrt {c \,x^{4}+b \,x^{2}+a}}{6}+\frac {7 b \,x^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{24}+\frac {b^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{16 c}-\frac {b^{3} \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{32 c^{\frac {3}{2}}}+\frac {3 a b \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{8 \sqrt {c}}+\frac {2 a \sqrt {c \,x^{4}+b \,x^{2}+a}}{3}-\frac {a^{\frac {3}{2}} \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{2}\) | \(192\) |
| pseudoelliptic | \(\frac {16 c^{\frac {5}{2}} x^{4} \sqrt {c \,x^{4}+b \,x^{2}+a}+28 b \,c^{\frac {3}{2}} x^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}-48 \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right ) c^{\frac {3}{2}} a^{\frac {3}{2}}+64 a \,c^{\frac {3}{2}} \sqrt {c \,x^{4}+b \,x^{2}+a}+6 b^{2} \sqrt {c}\, \sqrt {c \,x^{4}+b \,x^{2}+a}+36 \ln \left (\frac {2 c \,x^{2}+2 \sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {c}+b}{\sqrt {c}}\right ) a b c -3 \ln \left (\frac {2 c \,x^{2}+2 \sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {c}+b}{\sqrt {c}}\right ) b^{3}-36 \ln \left (2\right ) a b c +3 \ln \left (2\right ) b^{3}}{96 c^{\frac {3}{2}}}\) | \(223\) |
1/6*c*x^4*(c*x^4+b*x^2+a)^(1/2)+7/24*b*x^2*(c*x^4+b*x^2+a)^(1/2)+1/16/c*b^ 2*(c*x^4+b*x^2+a)^(1/2)-1/32/c^(3/2)*b^3*ln((1/2*b+c*x^2)/c^(1/2)+(c*x^4+b *x^2+a)^(1/2))+3/8*a*b*ln((1/2*b+c*x^2)/c^(1/2)+(c*x^4+b*x^2+a)^(1/2))/c^( 1/2)+2/3*a*(c*x^4+b*x^2+a)^(1/2)-1/2*a^(3/2)*ln((2*a+b*x^2+2*a^(1/2)*(c*x^ 4+b*x^2+a)^(1/2))/x^2)
Time = 0.33 (sec) , antiderivative size = 727, normalized size of antiderivative = 4.69 \[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x} \, dx=\left [\frac {48 \, a^{\frac {3}{2}} c^{2} \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 ) - 3 \, {\left (b^{3} - 12 \, a b c\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{4} - 8 \, b c x^{2} - b^{2} - 4 \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} + b\right )} \sqrt {c} - 4 \, a c\right ) + 4 \, {\left (8 \, c^{3} x^{4} + 14 \, b c^{2} x^{2} + 3 \, b^{2} c + 32 \, a c^{2}\right )} \sqrt {c x^{4} + b x^{2} + a}}{192 \, c^{2}}, \frac {24 \, a^{\frac {3}{2}} c^{2} \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 ) + 3 \, {\left (b^{3} - 12 \, a b c\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{4} + b c x^{2} + a c\right )}}\right ) + 2 \, {\left (8 \, c^{3} x^{4} + 14 \, b c^{2} x^{2} + 3 \, b^{2} c + 32 \, a c^{2}\right )} \sqrt {c x^{4} + b x^{2} + a}}{96 \, c^{2}}, \frac {96 \, \sqrt {-a} a c^{2} \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 ) - 3 \, {\left (b^{3} - 12 \, a b c\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{4} - 8 \, b c x^{2} - b^{2} - 4 \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} + b\right )} \sqrt {c} - 4 \, a c\right ) + 4 \, {\left (8 \, c^{3} x^{4} + 14 \, b c^{2} x^{2} + 3 \, b^{2} c + 32 \, a c^{2}\right )} \sqrt {c x^{4} + b x^{2} + a}}{192 \, c^{2}}, \frac {48 \, \sqrt {-a} a c^{2} \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 ) + 3 \, {\left (b^{3} - 12 \, a b c\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{4} + b c x^{2} + a c\right )}}\right ) + 2 \, {\left (8 \, c^{3} x^{4} + 14 \, b c^{2} x^{2} + 3 \, b^{2} c + 32 \, a c^{2}\right )} \sqrt {c x^{4} + b x^{2} + a}}{96 \, c^{2}}\right ] \]
[1/192*(48*a^(3/2)*c^2*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) - 3*(b^3 - 12*a*b*c)*sqrt (c)*log(-8*c^2*x^4 - 8*b*c*x^2 - b^2 - 4*sqrt(c*x^4 + b*x^2 + a)*(2*c*x^2 + b)*sqrt(c) - 4*a*c) + 4*(8*c^3*x^4 + 14*b*c^2*x^2 + 3*b^2*c + 32*a*c^2)* sqrt(c*x^4 + b*x^2 + a))/c^2, 1/96*(24*a^(3/2)*c^2*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) + 3*(b^3 - 12*a*b*c)*sqrt(-c)*arctan(1/2*sqrt(c*x^4 + b*x^2 + a)*(2*c*x ^2 + b)*sqrt(-c)/(c^2*x^4 + b*c*x^2 + a*c)) + 2*(8*c^3*x^4 + 14*b*c^2*x^2 + 3*b^2*c + 32*a*c^2)*sqrt(c*x^4 + b*x^2 + a))/c^2, 1/192*(96*sqrt(-a)*a*c ^2*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)) - 3*(b^3 - 12*a*b*c)*sqrt(c)*log(-8*c^2*x^4 - 8*b*c*x^2 - b^ 2 - 4*sqrt(c*x^4 + b*x^2 + a)*(2*c*x^2 + b)*sqrt(c) - 4*a*c) + 4*(8*c^3*x^ 4 + 14*b*c^2*x^2 + 3*b^2*c + 32*a*c^2)*sqrt(c*x^4 + b*x^2 + a))/c^2, 1/96* (48*sqrt(-a)*a*c^2*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)) + 3*(b^3 - 12*a*b*c)*sqrt(-c)*arctan(1/2*sqr t(c*x^4 + b*x^2 + a)*(2*c*x^2 + b)*sqrt(-c)/(c^2*x^4 + b*c*x^2 + a*c)) + 2 *(8*c^3*x^4 + 14*b*c^2*x^2 + 3*b^2*c + 32*a*c^2)*sqrt(c*x^4 + b*x^2 + a))/ c^2]
\[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x} \, dx=\int \frac {\left (a + b x^{2} + c x^{4}\right )^{\frac {3}{2}}}{x}\, dx \]
Exception generated. \[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x} \, dx=\text {Exception raised: ValueError} \]
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
Exception generated. \[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
Timed out. \[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x} \, dx=\int \frac {{\left (c\,x^4+b\,x^2+a\right )}^{3/2}}{x} \,d x \]