Integrand size = 24, antiderivative size = 219 \[ \int \frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^5} \, dx=\frac {3 (3 b+2 c x) \sqrt {a x^2+b x^3+c x^4}}{4 x}-\frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^4}-\frac {3 \sqrt {a} b x \sqrt {a+b x+c x^2} \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {a x^2+b x^3+c x^4}}+\frac {3 \left (b^2+4 a c\right ) x \sqrt {a+b x+c x^2} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 \sqrt {c} \sqrt {a x^2+b x^3+c x^4}} \]
-(c*x^4+b*x^3+a*x^2)^(3/2)/x^4-3/2*b*x*arctanh(1/2*(b*x+2*a)/a^(1/2)/(c*x^ 2+b*x+a)^(1/2))*a^(1/2)*(c*x^2+b*x+a)^(1/2)/(c*x^4+b*x^3+a*x^2)^(1/2)+3/8* (4*a*c+b^2)*x*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))*(c*x^2+b* x+a)^(1/2)/c^(1/2)/(c*x^4+b*x^3+a*x^2)^(1/2)+3/4*(2*c*x+3*b)*(c*x^4+b*x^3+ a*x^2)^(1/2)/x
Time = 0.48 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.71 \[ \int \frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^5} \, dx=\frac {\sqrt {a+x (b+c x)} \left (2 \sqrt {c} \sqrt {a+x (b+c x)} (-4 a+x (5 b+2 c x))+24 \sqrt {a} b \sqrt {c} x \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )-3 \left (b^2+4 a c\right ) x \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )\right )}{8 \sqrt {c} \sqrt {x^2 (a+x (b+c x))}} \]
(Sqrt[a + x*(b + c*x)]*(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(-4*a + x*(5*b + 2 *c*x)) + 24*Sqrt[a]*b*Sqrt[c]*x*ArcTanh[(Sqrt[c]*x - Sqrt[a + x*(b + c*x)] )/Sqrt[a]] - 3*(b^2 + 4*a*c)*x*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[a + x*(b + c *x)]]))/(8*Sqrt[c]*Sqrt[x^2*(a + x*(b + c*x))])
Time = 0.45 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.86, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {1967, 1992, 27, 1980, 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.
\(\displaystyle \int \frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^5} \, dx\) |
\(\Big \downarrow \) 1967 |
\(\displaystyle \frac {3}{2} \int \frac {(b+2 c x) \sqrt {c x^4+b x^3+a x^2}}{x^2}dx-\frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^4}\) |
\(\Big \downarrow \) 1992 |
\(\displaystyle \frac {3}{2} \left (\frac {\int \frac {c \left (4 a b+\left (b^2+4 a c\right ) x\right )}{\sqrt {c x^4+b x^3+a x^2}}dx}{4 c}+\frac {(3 b+2 c x) \sqrt {a x^2+b x^3+c x^4}}{2 x}\right )-\frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^4}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {3}{2} \left (\frac {1}{4} \int \frac {4 a b+\left (b^2+4 a c\right ) x}{\sqrt {c x^4+b x^3+a x^2}}dx+\frac {(3 b+2 c x) \sqrt {a x^2+b x^3+c x^4}}{2 x}\right )-\frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^4}\) |
\(\Big \downarrow \) 1980 |
\(\displaystyle \frac {3}{2} \left (\frac {x \sqrt {a+b x+c x^2} \int \frac {4 a b+\left (b^2+4 a c\right ) x}{x \sqrt {c x^2+b x+a}}dx}{4 \sqrt {a x^2+b x^3+c x^4}}+\frac {(3 b+2 c x) \sqrt {a x^2+b x^3+c x^4}}{2 x}\right )-\frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^4}\) |
\(\Big \downarrow \) 1269 |
\(\displaystyle \frac {3}{2} \left (\frac {x \sqrt {a+b x+c x^2} \left (\left (4 a c+b^2\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx+4 a b \int \frac {1}{x \sqrt {c x^2+b x+a}}dx\right )}{4 \sqrt {a x^2+b x^3+c x^4}}+\frac {(3 b+2 c x) \sqrt {a x^2+b x^3+c x^4}}{2 x}\right )-\frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^4}\) |
\(\Big \downarrow \) 1092 |
\(\displaystyle \frac {3}{2} \left (\frac {x \sqrt {a+b x+c x^2} \left (2 \left (4 a c+b^2\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}+4 a b \int \frac {1}{x \sqrt {c x^2+b x+a}}dx\right )}{4 \sqrt {a x^2+b x^3+c x^4}}+\frac {(3 b+2 c x) \sqrt {a x^2+b x^3+c x^4}}{2 x}\right )-\frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^4}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {3}{2} \left (\frac {x \sqrt {a+b x+c x^2} \left (4 a b \int \frac {1}{x \sqrt {c x^2+b x+a}}dx+\frac {\left (4 a c+b^2\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{\sqrt {c}}\right )}{4 \sqrt {a x^2+b x^3+c x^4}}+\frac {(3 b+2 c x) \sqrt {a x^2+b x^3+c x^4}}{2 x}\right )-\frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^4}\) |
\(\Big \downarrow \) 1154 |
\(\displaystyle \frac {3}{2} \left (\frac {x \sqrt {a+b x+c x^2} \left (\frac {\left (4 a c+b^2\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{\sqrt {c}}-8 a b \int \frac {1}{4 a-\frac {(2 a+b x)^2}{c x^2+b x+a}}d\frac {2 a+b x}{\sqrt {c x^2+b x+a}}\right )}{4 \sqrt {a x^2+b x^3+c x^4}}+\frac {(3 b+2 c x) \sqrt {a x^2+b x^3+c x^4}}{2 x}\right )-\frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^4}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {3}{2} \left (\frac {x \sqrt {a+b x+c x^2} \left (\frac {\left (4 a c+b^2\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{\sqrt {c}}-4 \sqrt {a} b \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )\right )}{4 \sqrt {a x^2+b x^3+c x^4}}+\frac {(3 b+2 c x) \sqrt {a x^2+b x^3+c x^4}}{2 x}\right )-\frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^4}\) |
-((a*x^2 + b*x^3 + c*x^4)^(3/2)/x^4) + (3*(((3*b + 2*c*x)*Sqrt[a*x^2 + b*x ^3 + c*x^4])/(2*x) + (x*Sqrt[a + b*x + c*x^2]*(-4*Sqrt[a]*b*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])] + ((b^2 + 4*a*c)*ArcTanh[(b + 2*c *x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/Sqrt[c]))/(4*Sqrt[a*x^2 + b*x^3 + c*x^4])))/2
3.1.44.3.1 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_)*((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_.)*((b_.)*(x_)^(n_.) + (a_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(p_ ), x_Symbol] :> Simp[x^(m + 1)*((a*x^q + b*x^n + c*x^(2*n - q))^p/(m + p*q + 1)), x] - Simp[(n - q)*(p/(m + p*q + 1)) Int[x^(m + n)*(b + 2*c*x^(n - q))*(a*x^q + b*x^n + c*x^(2*n - q))^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && EqQ[r, 2*n - q] && PosQ[n - q] && !IntegerQ[p] && NeQ[b^2 - 4*a*c, 0] & & IGtQ[n, 0] && GtQ[p, 0] && RationalQ[m, q] && LeQ[m + p*q + 1, -(n - q) + 1] && NeQ[m + p*q + 1, 0]
Int[((A_) + (B_.)*(x_)^(j_.))/Sqrt[(b_.)*(x_)^(n_.) + (a_.)*(x_)^(q_.) + (c _.)*(x_)^(r_.)], x_Symbol] :> Simp[x^(q/2)*(Sqrt[a + b*x^(n - q) + c*x^(2*( n - q))]/Sqrt[a*x^q + b*x^n + c*x^(2*n - q)]) Int[(A + B*x^(n - q))/(x^(q /2)*Sqrt[a + b*x^(n - q) + c*x^(2*(n - q))]), x], x] /; FreeQ[{a, b, c, A, B, n, q}, x] && EqQ[j, n - q] && EqQ[r, 2*n - q] && PosQ[n - q] && EqQ[n, 3 ] && EqQ[q, 2]
Int[(x_)^(m_.)*((c_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.) + (a_.)*(x_)^(q_.))^(p_ .)*((A_) + (B_.)*(x_)^(r_.)), x_Symbol] :> Simp[x^(m + 1)*(b*B*(n - q)*p + A*c*(m + p*q + (n - q)*(2*p + 1) + 1) + B*c*(m + p*q + 2*(n - q)*p + 1)*x^( n - q))*((a*x^q + b*x^n + c*x^(2*n - q))^p/(c*(m + p*(2*n - q) + 1)*(m + p* q + (n - q)*(2*p + 1) + 1))), x] + Simp[(n - q)*(p/(c*(m + p*(2*n - q) + 1) *(m + p*q + (n - q)*(2*p + 1) + 1))) Int[x^(m + q)*Simp[2*a*A*c*(m + p*q + (n - q)*(2*p + 1) + 1) - a*b*B*(m + p*q + 1) + (2*a*B*c*(m + p*q + 2*(n - q)*p + 1) + A*b*c*(m + p*q + (n - q)*(2*p + 1) + 1) - b^2*B*(m + p*q + (n - q)*p + 1))*x^(n - q), x]*(a*x^q + b*x^n + c*x^(2*n - q))^(p - 1), x], x] /; FreeQ[{a, b, c, A, B}, x] && EqQ[r, n - q] && EqQ[j, 2*n - q] && !Integ erQ[p] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && GtQ[p, 0] && RationalQ[m, q] && GtQ[m + p*q, -(n - q) - 1] && NeQ[m + p*(2*n - q) + 1, 0] && NeQ[m + p* q + (n - q)*(2*p + 1) + 1, 0]
Time = 0.17 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.82
method | result | size |
pseudoelliptic | \(\frac {4 c^{\frac {3}{2}} x^{2} \sqrt {c \,x^{2}+b x +a}-12 \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x \sqrt {a}}\right ) b x \sqrt {a}\, \sqrt {c}+12 \ln \left (2\right ) b x \sqrt {a}\, \sqrt {c}+10 b \sqrt {c \,x^{2}+b x +a}\, x \sqrt {c}+12 \ln \left (2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}+2 c x +b \right ) a c x +3 \ln \left (2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}+2 c x +b \right ) b^{2} x -8 a \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}}{8 x \sqrt {c}}\) | \(180\) |
risch | \(-\frac {a \sqrt {x^{2} \left (c \,x^{2}+b x +a \right )}}{x^{2}}+\frac {\left (\frac {3 b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 \sqrt {c}}+\frac {3 a \sqrt {c}\, \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2}+\frac {c \sqrt {c \,x^{2}+b x +a}\, x}{2}+\frac {5 \sqrt {c \,x^{2}+b x +a}\, b}{4}-\frac {3 \sqrt {a}\, b \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{2}\right ) \sqrt {x^{2} \left (c \,x^{2}+b x +a \right )}}{x \sqrt {c \,x^{2}+b x +a}}\) | \(187\) |
default | \(\frac {\left (c \,x^{4}+b \,x^{3}+a \,x^{2}\right )^{\frac {3}{2}} \left (8 c^{\frac {5}{2}} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} x^{2}+12 c^{\frac {5}{2}} \sqrt {c \,x^{2}+b x +a}\, a \,x^{2}-12 c^{\frac {3}{2}} a^{\frac {3}{2}} \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) b x -8 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} c^{\frac {3}{2}}+8 c^{\frac {3}{2}} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} b x +18 c^{\frac {3}{2}} \sqrt {c \,x^{2}+b x +a}\, a b x +12 \ln \left (\frac {2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}+2 c x +b}{2 \sqrt {c}}\right ) a^{2} c^{2} x +3 c \ln \left (\frac {2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}+2 c x +b}{2 \sqrt {c}}\right ) a \,b^{2} x \right )}{8 x^{4} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a \,c^{\frac {3}{2}}}\) | \(254\) |
1/8*(4*c^(3/2)*x^2*(c*x^2+b*x+a)^(1/2)-12*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x +a)^(1/2))/x/a^(1/2))*b*x*a^(1/2)*c^(1/2)+12*ln(2)*b*x*a^(1/2)*c^(1/2)+10* b*(c*x^2+b*x+a)^(1/2)*x*c^(1/2)+12*ln(2*(c*x^2+b*x+a)^(1/2)*c^(1/2)+2*c*x+ b)*a*c*x+3*ln(2*(c*x^2+b*x+a)^(1/2)*c^(1/2)+2*c*x+b)*b^2*x-8*a*(c*x^2+b*x+ a)^(1/2)*c^(1/2))/x/c^(1/2)
Time = 0.34 (sec) , antiderivative size = 757, normalized size of antiderivative = 3.46 \[ \int \frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^5} \, dx=\left [\frac {12 \, \sqrt {a} b c x^{2} \log \left (-\frac {8 \, a b x^{2} + {\left (b^{2} + 4 \, a c\right )} x^{3} + 8 \, a^{2} x - 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (b x + 2 \, a\right )} \sqrt {a}}{x^{3}}\right ) + 3 \, {\left (b^{2} + 4 \, a c\right )} \sqrt {c} x^{2} \log \left (-\frac {8 \, c^{2} x^{3} + 8 \, b c x^{2} + 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c x + b\right )} \sqrt {c} + {\left (b^{2} + 4 \, a c\right )} x}{x}\right ) + 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c^{2} x^{2} + 5 \, b c x - 4 \, a c\right )}}{16 \, c x^{2}}, \frac {6 \, \sqrt {a} b c x^{2} \log \left (-\frac {8 \, a b x^{2} + {\left (b^{2} + 4 \, a c\right )} x^{3} + 8 \, a^{2} x - 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (b x + 2 \, a\right )} \sqrt {a}}{x^{3}}\right ) - 3 \, {\left (b^{2} + 4 \, a c\right )} \sqrt {-c} x^{2} \arctan \left (\frac {\sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{3} + b c x^{2} + a c x\right )}}\right ) + 2 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c^{2} x^{2} + 5 \, b c x - 4 \, a c\right )}}{8 \, c x^{2}}, \frac {24 \, \sqrt {-a} b c x^{2} \arctan \left (\frac {\sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (b x + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{3} + a b x^{2} + a^{2} x\right )}}\right ) + 3 \, {\left (b^{2} + 4 \, a c\right )} \sqrt {c} x^{2} \log \left (-\frac {8 \, c^{2} x^{3} + 8 \, b c x^{2} + 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c x + b\right )} \sqrt {c} + {\left (b^{2} + 4 \, a c\right )} x}{x}\right ) + 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c^{2} x^{2} + 5 \, b c x - 4 \, a c\right )}}{16 \, c x^{2}}, \frac {12 \, \sqrt {-a} b c x^{2} \arctan \left (\frac {\sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (b x + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{3} + a b x^{2} + a^{2} x\right )}}\right ) - 3 \, {\left (b^{2} + 4 \, a c\right )} \sqrt {-c} x^{2} \arctan \left (\frac {\sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{3} + b c x^{2} + a c x\right )}}\right ) + 2 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c^{2} x^{2} + 5 \, b c x - 4 \, a c\right )}}{8 \, c x^{2}}\right ] \]
[1/16*(12*sqrt(a)*b*c*x^2*log(-(8*a*b*x^2 + (b^2 + 4*a*c)*x^3 + 8*a^2*x - 4*sqrt(c*x^4 + b*x^3 + a*x^2)*(b*x + 2*a)*sqrt(a))/x^3) + 3*(b^2 + 4*a*c)* sqrt(c)*x^2*log(-(8*c^2*x^3 + 8*b*c*x^2 + 4*sqrt(c*x^4 + b*x^3 + a*x^2)*(2 *c*x + b)*sqrt(c) + (b^2 + 4*a*c)*x)/x) + 4*sqrt(c*x^4 + b*x^3 + a*x^2)*(2 *c^2*x^2 + 5*b*c*x - 4*a*c))/(c*x^2), 1/8*(6*sqrt(a)*b*c*x^2*log(-(8*a*b*x ^2 + (b^2 + 4*a*c)*x^3 + 8*a^2*x - 4*sqrt(c*x^4 + b*x^3 + a*x^2)*(b*x + 2* a)*sqrt(a))/x^3) - 3*(b^2 + 4*a*c)*sqrt(-c)*x^2*arctan(1/2*sqrt(c*x^4 + b* x^3 + a*x^2)*(2*c*x + b)*sqrt(-c)/(c^2*x^3 + b*c*x^2 + a*c*x)) + 2*sqrt(c* x^4 + b*x^3 + a*x^2)*(2*c^2*x^2 + 5*b*c*x - 4*a*c))/(c*x^2), 1/16*(24*sqrt (-a)*b*c*x^2*arctan(1/2*sqrt(c*x^4 + b*x^3 + a*x^2)*(b*x + 2*a)*sqrt(-a)/( a*c*x^3 + a*b*x^2 + a^2*x)) + 3*(b^2 + 4*a*c)*sqrt(c)*x^2*log(-(8*c^2*x^3 + 8*b*c*x^2 + 4*sqrt(c*x^4 + b*x^3 + a*x^2)*(2*c*x + b)*sqrt(c) + (b^2 + 4 *a*c)*x)/x) + 4*sqrt(c*x^4 + b*x^3 + a*x^2)*(2*c^2*x^2 + 5*b*c*x - 4*a*c)) /(c*x^2), 1/8*(12*sqrt(-a)*b*c*x^2*arctan(1/2*sqrt(c*x^4 + b*x^3 + a*x^2)* (b*x + 2*a)*sqrt(-a)/(a*c*x^3 + a*b*x^2 + a^2*x)) - 3*(b^2 + 4*a*c)*sqrt(- c)*x^2*arctan(1/2*sqrt(c*x^4 + b*x^3 + a*x^2)*(2*c*x + b)*sqrt(-c)/(c^2*x^ 3 + b*c*x^2 + a*c*x)) + 2*sqrt(c*x^4 + b*x^3 + a*x^2)*(2*c^2*x^2 + 5*b*c*x - 4*a*c))/(c*x^2)]
\[ \int \frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^5} \, dx=\int \frac {\left (x^{2} \left (a + b x + c x^{2}\right )\right )^{\frac {3}{2}}}{x^{5}}\, dx \]
\[ \int \frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^5} \, dx=\int { \frac {{\left (c x^{4} + b x^{3} + a x^{2}\right )}^{\frac {3}{2}}}{x^{5}} \,d x } \]
Exception generated. \[ \int \frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^5} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Not invertible Error: Bad Argument Value
Timed out. \[ \int \frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^5} \, dx=\int \frac {{\left (c\,x^4+b\,x^3+a\,x^2\right )}^{3/2}}{x^5} \,d x \]