\(\int \frac {\sqrt {c+e x+d x^2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{x^6} \, dx\) [79]

Optimal result

Integrand size = 40, antiderivative size = 368 \[ \int \frac {\sqrt {c+e x+d x^2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{x^6} \, dx=\frac {e \left (16 b c^2-12 a c d+7 a e^2\right ) (2 c+e x) \sqrt {c+e x+d x^2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{128 c^4 x^2 \left (a+b x^2\right )}-\frac {a \left (c+e x+d x^2\right )^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{5 c x^5 \left (a+b x^2\right )}+\frac {7 a e \left (c+e x+d x^2\right )^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{40 c^2 x^4 \left (a+b x^2\right )}-\frac {\left (80 b c^2-32 a c d+35 a e^2\right ) \left (c+e x+d x^2\right )^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{240 c^3 x^3 \left (a+b x^2\right )}+\frac {e \left (4 c d-e^2\right ) \left (16 b c^2-12 a c d+7 a e^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4} \text {arctanh}\left (\frac {2 c+e x}{2 \sqrt {c} \sqrt {c+e x+d x^2}}\right )}{256 c^{9/2} \left (a+b x^2\right )} \] Output:

1/128*e*(-12*a*c*d+7*a*e^2+16*b*c^2)*(e*x+2*c)*(d*x^2+e*x+c)^(1/2)*((b*x^2 
+a)^2)^(1/2)/c^4/x^2/(b*x^2+a)-1/5*a*(d*x^2+e*x+c)^(3/2)*((b*x^2+a)^2)^(1/ 
2)/c/x^5/(b*x^2+a)+7/40*a*e*(d*x^2+e*x+c)^(3/2)*((b*x^2+a)^2)^(1/2)/c^2/x^ 
4/(b*x^2+a)-1/240*(-32*a*c*d+35*a*e^2+80*b*c^2)*(d*x^2+e*x+c)^(3/2)*((b*x^ 
2+a)^2)^(1/2)/c^3/x^3/(b*x^2+a)+1/256*e*(4*c*d-e^2)*(-12*a*c*d+7*a*e^2+16* 
b*c^2)*((b*x^2+a)^2)^(1/2)*arctanh(1/2*(e*x+2*c)/c^(1/2)/(d*x^2+e*x+c)^(1/ 
2))/c^(9/2)/(b*x^2+a)
 

Mathematica [A] (verified)

Time = 2.28 (sec) , antiderivative size = 270, normalized size of antiderivative = 0.73 \[ \int \frac {\sqrt {c+e x+d x^2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{x^6} \, dx=-\frac {\sqrt {\left (a+b x^2\right )^2} \left (\sqrt {c} \sqrt {c+x (e+d x)} \left (80 b c^2 x^2 \left (8 c^2-3 e^2 x^2+2 c x (e+4 d x)\right )+a \left (384 c^4-105 e^4 x^4+16 c^3 x (3 e+8 d x)+10 c e^2 x^3 (7 e+46 d x)-8 c^2 x^2 \left (7 e^2+29 d e x+32 d^2 x^2\right )\right )\right )+15 e \left (64 b c^3 d+40 a c d e^2-7 a e^4\right ) x^5 \text {arctanh}\left (\frac {\sqrt {d} x-\sqrt {c+x (e+d x)}}{\sqrt {c}}\right )+240 c^2 e \left (3 a d^2+b e^2\right ) x^5 \text {arctanh}\left (\frac {-\sqrt {d} x+\sqrt {c+x (e+d x)}}{\sqrt {c}}\right )\right )}{1920 c^{9/2} x^5 \left (a+b x^2\right )} \] Input:

Integrate[(Sqrt[c + e*x + d*x^2]*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/x^6,x]
 

Output:

-1/1920*(Sqrt[(a + b*x^2)^2]*(Sqrt[c]*Sqrt[c + x*(e + d*x)]*(80*b*c^2*x^2* 
(8*c^2 - 3*e^2*x^2 + 2*c*x*(e + 4*d*x)) + a*(384*c^4 - 105*e^4*x^4 + 16*c^ 
3*x*(3*e + 8*d*x) + 10*c*e^2*x^3*(7*e + 46*d*x) - 8*c^2*x^2*(7*e^2 + 29*d* 
e*x + 32*d^2*x^2))) + 15*e*(64*b*c^3*d + 40*a*c*d*e^2 - 7*a*e^4)*x^5*ArcTa 
nh[(Sqrt[d]*x - Sqrt[c + x*(e + d*x)])/Sqrt[c]] + 240*c^2*e*(3*a*d^2 + b*e 
^2)*x^5*ArcTanh[(-(Sqrt[d]*x) + Sqrt[c + x*(e + d*x)])/Sqrt[c]]))/(c^(9/2) 
*x^5*(a + b*x^2))
 

Rubi [A] (verified)

Time = 0.91 (sec) , antiderivative size = 250, normalized size of antiderivative = 0.68, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1384, 27, 2181, 27, 1237, 27, 1228, 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.

\(\displaystyle \int \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \sqrt {c+d x^2+e x}}{x^6} \, dx\)

\(\Big \downarrow \) 1384

\(\displaystyle \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int \frac {b \left (b x^2+a\right ) \sqrt {d x^2+e x+c}}{x^6}dx}{b \left (a+b x^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int \frac {\left (b x^2+a\right ) \sqrt {d x^2+e x+c}}{x^6}dx}{a+b x^2}\)

\(\Big \downarrow \) 2181

\(\displaystyle \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \left (-\frac {\int \frac {(7 a e-2 (5 b c-2 a d) x) \sqrt {d x^2+e x+c}}{2 x^5}dx}{5 c}-\frac {a \left (c+d x^2+e x\right )^{3/2}}{5 c x^5}\right )}{a+b x^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \left (-\frac {\int \frac {(7 a e-2 (5 b c-2 a d) x) \sqrt {d x^2+e x+c}}{x^5}dx}{10 c}-\frac {a \left (c+d x^2+e x\right )^{3/2}}{5 c x^5}\right )}{a+b x^2}\)

\(\Big \downarrow \) 1237

\(\displaystyle \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \left (-\frac {-\frac {\int \frac {\left (80 b c^2-32 a d c+35 a e^2+14 a d e x\right ) \sqrt {d x^2+e x+c}}{2 x^4}dx}{4 c}-\frac {7 a e \left (c+d x^2+e x\right )^{3/2}}{4 c x^4}}{10 c}-\frac {a \left (c+d x^2+e x\right )^{3/2}}{5 c x^5}\right )}{a+b x^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \left (-\frac {-\frac {\int \frac {\left (80 b c^2-32 a d c+35 a e^2+14 a d e x\right ) \sqrt {d x^2+e x+c}}{x^4}dx}{8 c}-\frac {7 a e \left (c+d x^2+e x\right )^{3/2}}{4 c x^4}}{10 c}-\frac {a \left (c+d x^2+e x\right )^{3/2}}{5 c x^5}\right )}{a+b x^2}\)

\(\Big \downarrow \) 1228

\(\displaystyle \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \left (-\frac {-\frac {-\frac {5 e \left (-12 a c d+7 a e^2+16 b c^2\right ) \int \frac {\sqrt {d x^2+e x+c}}{x^3}dx}{2 c}-\frac {\left (c+d x^2+e x\right )^{3/2} \left (-32 a c d+35 a e^2+80 b c^2\right )}{3 c x^3}}{8 c}-\frac {7 a e \left (c+d x^2+e x\right )^{3/2}}{4 c x^4}}{10 c}-\frac {a \left (c+d x^2+e x\right )^{3/2}}{5 c x^5}\right )}{a+b x^2}\)

\(\Big \downarrow \) 1152

\(\displaystyle \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \left (-\frac {-\frac {-\frac {5 e \left (-12 a c d+7 a e^2+16 b c^2\right ) \left (\frac {\left (4 c d-e^2\right ) \int \frac {1}{x \sqrt {d x^2+e x+c}}dx}{8 c}-\frac {(2 c+e x) \sqrt {c+d x^2+e x}}{4 c x^2}\right )}{2 c}-\frac {\left (c+d x^2+e x\right )^{3/2} \left (-32 a c d+35 a e^2+80 b c^2\right )}{3 c x^3}}{8 c}-\frac {7 a e \left (c+d x^2+e x\right )^{3/2}}{4 c x^4}}{10 c}-\frac {a \left (c+d x^2+e x\right )^{3/2}}{5 c x^5}\right )}{a+b x^2}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \left (-\frac {-\frac {-\frac {5 e \left (-12 a c d+7 a e^2+16 b c^2\right ) \left (-\frac {\left (4 c d-e^2\right ) \int \frac {1}{4 c-\frac {(2 c+e x)^2}{d x^2+e x+c}}d\frac {2 c+e x}{\sqrt {d x^2+e x+c}}}{4 c}-\frac {(2 c+e x) \sqrt {c+d x^2+e x}}{4 c x^2}\right )}{2 c}-\frac {\left (c+d x^2+e x\right )^{3/2} \left (-32 a c d+35 a e^2+80 b c^2\right )}{3 c x^3}}{8 c}-\frac {7 a e \left (c+d x^2+e x\right )^{3/2}}{4 c x^4}}{10 c}-\frac {a \left (c+d x^2+e x\right )^{3/2}}{5 c x^5}\right )}{a+b x^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \left (-\frac {-\frac {-\frac {5 e \left (-12 a c d+7 a e^2+16 b c^2\right ) \left (-\frac {\left (4 c d-e^2\right ) \text {arctanh}\left (\frac {2 c+e x}{2 \sqrt {c} \sqrt {c+d x^2+e x}}\right )}{8 c^{3/2}}-\frac {(2 c+e x) \sqrt {c+d x^2+e x}}{4 c x^2}\right )}{2 c}-\frac {\left (c+d x^2+e x\right )^{3/2} \left (-32 a c d+35 a e^2+80 b c^2\right )}{3 c x^3}}{8 c}-\frac {7 a e \left (c+d x^2+e x\right )^{3/2}}{4 c x^4}}{10 c}-\frac {a \left (c+d x^2+e x\right )^{3/2}}{5 c x^5}\right )}{a+b x^2}\)

Input:

Int[(Sqrt[c + e*x + d*x^2]*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/x^6,x]
 

Output:

(Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]*(-1/5*(a*(c + e*x + d*x^2)^(3/2))/(c*x^5) 
 - ((-7*a*e*(c + e*x + d*x^2)^(3/2))/(4*c*x^4) - (-1/3*((80*b*c^2 - 32*a*c 
*d + 35*a*e^2)*(c + e*x + d*x^2)^(3/2))/(c*x^3) - (5*e*(16*b*c^2 - 12*a*c* 
d + 7*a*e^2)*(-1/4*((2*c + e*x)*Sqrt[c + e*x + d*x^2])/(c*x^2) - ((4*c*d - 
 e^2)*ArcTanh[(2*c + e*x)/(2*Sqrt[c]*Sqrt[c + e*x + d*x^2])])/(8*c^(3/2))) 
)/(2*c))/(8*c))/(10*c)))/(a + b*x^2)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

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])
 

rule 1152

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]
 

rule 1154

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]
 

rule 1228

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]
 

rule 1237

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)/((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)*(a + b*x + c*x^2)^p*Simp[ 
(c*d*f - f*b*e + a*e*g)*(m + 1) + b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m 
+ 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && LtQ[m, -1 
] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

rule 1384

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> S 
imp[(a + b*x^n + c*x^(2*n))^FracPart[p]/(c^IntPart[p]*(b/2 + c*x^n)^(2*Frac 
Part[p]))   Int[u*(b/2 + c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] 
&& EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2] && NeQ[u, x^(n 
- 1)] && NeQ[u, x^(2*n - 1)] &&  !(EqQ[p, 1/2] && EqQ[u, x^(-2*n - 1)])
 

rule 2181

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_ 
), x_Symbol] :> With[{Qx = PolynomialQuotient[Pq, d + e*x, x], R = Polynomi 
alRemainder[Pq, d + e*x, x]}, Simp[(e*R*(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)*(a + b*x + c*x^2)^p*ExpandToSum[(m 
+ 1)*(c*d^2 - b*d*e + a*e^2)*Qx + c*d*R*(m + 1) - b*e*R*(m + p + 2) - c*e*R 
*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, e, p}, x] && PolyQ[Pq, 
x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1]
 

Maple [A] (verified)

Time = 0.33 (sec) , antiderivative size = 268, normalized size of antiderivative = 0.73

method	result	size
risch	\(-\frac {\sqrt {d \,x^{2}+e x +c}\, \left (-256 a \,c^{2} d^{2} x^{4}+460 a c d \,e^{2} x^{4}-105 a \,e^{4} x^{4}+640 b d \,x^{4} c^{3}-240 b \,c^{2} e^{2} x^{4}-232 a \,c^{2} d e \,x^{3}+70 a \,x^{3} e^{3} c +160 b \,c^{3} e \,x^{3}+128 a \,c^{3} d \,x^{2}-56 a \,c^{2} e^{2} x^{2}+640 b \,c^{4} x^{2}+48 a \,c^{3} e x +384 a \,c^{4}\right ) \sqrt {\left (b \,x^{2}+a \right )^{2}}}{1920 x^{5} c^{4} \left (b \,x^{2}+a \right )}-\frac {e \left (48 a \,c^{2} d^{2}-40 a c d \,e^{2}+7 e^{4} a -64 b \,c^{3} d +16 b \,c^{2} e^{2}\right ) \ln \left (\frac {2 c +e x +2 \sqrt {c}\, \sqrt {d \,x^{2}+e x +c}}{x}\right ) \sqrt {\left (b \,x^{2}+a \right )^{2}}}{256 c^{\frac {9}{2}} \left (b \,x^{2}+a \right )}\)	\(268\)
default	\(\frac {\sqrt {\left (b \,x^{2}+a \right )^{2}}\, \left (960 d \,c^{\frac {7}{2}} \ln \left (\frac {2 c +e x +2 \sqrt {c}\, \sqrt {d \,x^{2}+e x +c}}{x}\right ) b e \,x^{5}-720 d^{2} c^{\frac {5}{2}} \ln \left (\frac {2 c +e x +2 \sqrt {c}\, \sqrt {d \,x^{2}+e x +c}}{x}\right ) a e \,x^{5}-240 c^{\frac {5}{2}} \ln \left (\frac {2 c +e x +2 \sqrt {c}\, \sqrt {d \,x^{2}+e x +c}}{x}\right ) b \,e^{3} x^{5}+600 d \,c^{\frac {3}{2}} \ln \left (\frac {2 c +e x +2 \sqrt {c}\, \sqrt {d \,x^{2}+e x +c}}{x}\right ) a \,e^{3} x^{5}-105 \sqrt {c}\, \ln \left (\frac {2 c +e x +2 \sqrt {c}\, \sqrt {d \,x^{2}+e x +c}}{x}\right ) a \,e^{5} x^{5}-360 d^{2} \sqrt {d \,x^{2}+e x +c}\, a c \,e^{2} x^{6}+210 d \sqrt {d \,x^{2}+e x +c}\, a \,e^{4} x^{6}+480 d \sqrt {d \,x^{2}+e x +c}\, b \,c^{2} e^{2} x^{6}+360 d \left (d \,x^{2}+e x +c \right )^{\frac {3}{2}} a c \,e^{2} x^{4}-210 \left (d \,x^{2}+e x +c \right )^{\frac {3}{2}} a \,e^{4} x^{4}-480 \left (d \,x^{2}+e x +c \right )^{\frac {3}{2}} b \,c^{2} e^{2} x^{4}+720 d^{2} \sqrt {d \,x^{2}+e x +c}\, a \,c^{2} e \,x^{5}-780 d \sqrt {d \,x^{2}+e x +c}\, a c \,e^{3} x^{5}+210 \sqrt {d \,x^{2}+e x +c}\, a \,e^{5} x^{5}-960 d \sqrt {d \,x^{2}+e x +c}\, b \,c^{3} e \,x^{5}+480 \sqrt {d \,x^{2}+e x +c}\, b \,c^{2} e^{3} x^{5}-720 d \left (d \,x^{2}+e x +c \right )^{\frac {3}{2}} a \,c^{2} e \,x^{3}+420 \left (d \,x^{2}+e x +c \right )^{\frac {3}{2}} a c \,e^{3} x^{3}+960 \left (d \,x^{2}+e x +c \right )^{\frac {3}{2}} b \,c^{3} e \,x^{3}+512 d \left (d \,x^{2}+e x +c \right )^{\frac {3}{2}} a \,c^{3} x^{2}-560 \left (d \,x^{2}+e x +c \right )^{\frac {3}{2}} a \,c^{2} e^{2} x^{2}-1280 \left (d \,x^{2}+e x +c \right )^{\frac {3}{2}} b \,c^{4} x^{2}+672 \left (d \,x^{2}+e x +c \right )^{\frac {3}{2}} a \,c^{3} e x -768 \left (d \,x^{2}+e x +c \right )^{\frac {3}{2}} a \,c^{4}\right )}{3840 x^{5} c^{5} \left (b \,x^{2}+a \right )}\)	\(664\)

Input:

int((d*x^2+e*x+c)^(1/2)*((b*x^2+a)^2)^(1/2)/x^6,x,method=_RETURNVERBOSE)
 

Output:

-1/1920*(d*x^2+e*x+c)^(1/2)*(-256*a*c^2*d^2*x^4+460*a*c*d*e^2*x^4-105*a*e^ 
4*x^4+640*b*c^3*d*x^4-240*b*c^2*e^2*x^4-232*a*c^2*d*e*x^3+70*a*c*e^3*x^3+1 
60*b*c^3*e*x^3+128*a*c^3*d*x^2-56*a*c^2*e^2*x^2+640*b*c^4*x^2+48*a*c^3*e*x 
+384*a*c^4)/x^5/c^4*((b*x^2+a)^2)^(1/2)/(b*x^2+a)-1/256*e*(48*a*c^2*d^2-40 
*a*c*d*e^2+7*a*e^4-64*b*c^3*d+16*b*c^2*e^2)/c^(9/2)*ln((2*c+e*x+2*c^(1/2)* 
(d*x^2+e*x+c)^(1/2))/x)*((b*x^2+a)^2)^(1/2)/(b*x^2+a)
 

Fricas [A] (verification not implemented)

Time = 0.89 (sec) , antiderivative size = 492, normalized size of antiderivative = 1.34 \[ \int \frac {\sqrt {c+e x+d x^2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{x^6} \, dx=\left [\frac {15 \, {\left (7 \, a e^{5} + 8 \, {\left (2 \, b c^{2} - 5 \, a c d\right )} e^{3} - 16 \, {\left (4 \, b c^{3} d - 3 \, a c^{2} d^{2}\right )} e\right )} \sqrt {c} x^{5} \log \left (\frac {8 \, c e x + {\left (4 \, c d + e^{2}\right )} x^{2} - 4 \, \sqrt {d x^{2} + e x + c} {\left (e x + 2 \, c\right )} \sqrt {c} + 8 \, c^{2}}{x^{2}}\right ) - 4 \, {\left (48 \, a c^{4} e x + 384 \, a c^{5} + {\left (640 \, b c^{4} d - 256 \, a c^{3} d^{2} - 105 \, a c e^{4} - 20 \, {\left (12 \, b c^{3} - 23 \, a c^{2} d\right )} e^{2}\right )} x^{4} + 2 \, {\left (35 \, a c^{2} e^{3} + 4 \, {\left (20 \, b c^{4} - 29 \, a c^{3} d\right )} e\right )} x^{3} + 8 \, {\left (80 \, b c^{5} + 16 \, a c^{4} d - 7 \, a c^{3} e^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + e x + c}}{7680 \, c^{5} x^{5}}, \frac {15 \, {\left (7 \, a e^{5} + 8 \, {\left (2 \, b c^{2} - 5 \, a c d\right )} e^{3} - 16 \, {\left (4 \, b c^{3} d - 3 \, a c^{2} d^{2}\right )} e\right )} \sqrt {-c} x^{5} \arctan \left (\frac {\sqrt {d x^{2} + e x + c} {\left (e x + 2 \, c\right )} \sqrt {-c}}{2 \, {\left (c d x^{2} + c e x + c^{2}\right )}}\right ) - 2 \, {\left (48 \, a c^{4} e x + 384 \, a c^{5} + {\left (640 \, b c^{4} d - 256 \, a c^{3} d^{2} - 105 \, a c e^{4} - 20 \, {\left (12 \, b c^{3} - 23 \, a c^{2} d\right )} e^{2}\right )} x^{4} + 2 \, {\left (35 \, a c^{2} e^{3} + 4 \, {\left (20 \, b c^{4} - 29 \, a c^{3} d\right )} e\right )} x^{3} + 8 \, {\left (80 \, b c^{5} + 16 \, a c^{4} d - 7 \, a c^{3} e^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + e x + c}}{3840 \, c^{5} x^{5}}\right ] \] Input:

integrate((d*x^2+e*x+c)^(1/2)*((b*x^2+a)^2)^(1/2)/x^6,x, algorithm="fricas 
")
 

Output:

[1/7680*(15*(7*a*e^5 + 8*(2*b*c^2 - 5*a*c*d)*e^3 - 16*(4*b*c^3*d - 3*a*c^2 
*d^2)*e)*sqrt(c)*x^5*log((8*c*e*x + (4*c*d + e^2)*x^2 - 4*sqrt(d*x^2 + e*x 
 + c)*(e*x + 2*c)*sqrt(c) + 8*c^2)/x^2) - 4*(48*a*c^4*e*x + 384*a*c^5 + (6 
40*b*c^4*d - 256*a*c^3*d^2 - 105*a*c*e^4 - 20*(12*b*c^3 - 23*a*c^2*d)*e^2) 
*x^4 + 2*(35*a*c^2*e^3 + 4*(20*b*c^4 - 29*a*c^3*d)*e)*x^3 + 8*(80*b*c^5 + 
16*a*c^4*d - 7*a*c^3*e^2)*x^2)*sqrt(d*x^2 + e*x + c))/(c^5*x^5), 1/3840*(1 
5*(7*a*e^5 + 8*(2*b*c^2 - 5*a*c*d)*e^3 - 16*(4*b*c^3*d - 3*a*c^2*d^2)*e)*s 
qrt(-c)*x^5*arctan(1/2*sqrt(d*x^2 + e*x + c)*(e*x + 2*c)*sqrt(-c)/(c*d*x^2 
 + c*e*x + c^2)) - 2*(48*a*c^4*e*x + 384*a*c^5 + (640*b*c^4*d - 256*a*c^3* 
d^2 - 105*a*c*e^4 - 20*(12*b*c^3 - 23*a*c^2*d)*e^2)*x^4 + 2*(35*a*c^2*e^3 
+ 4*(20*b*c^4 - 29*a*c^3*d)*e)*x^3 + 8*(80*b*c^5 + 16*a*c^4*d - 7*a*c^3*e^ 
2)*x^2)*sqrt(d*x^2 + e*x + c))/(c^5*x^5)]
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\sqrt {c+e x+d x^2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{x^6} \, dx=\text {Timed out} \] Input:

integrate((d*x**2+e*x+c)**(1/2)*((b*x**2+a)**2)**(1/2)/x**6,x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sqrt {c+e x+d x^2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{x^6} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((d*x^2+e*x+c)^(1/2)*((b*x^2+a)^2)^(1/2)/x^6,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(e^2-4*c*d>0)', see `assume?` for 
 more deta
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1697 vs. \(2 (287) = 574\).

Time = 0.17 (sec) , antiderivative size = 1697, normalized size of antiderivative = 4.61 \[ \int \frac {\sqrt {c+e x+d x^2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{x^6} \, dx=\text {Too large to display} \] Input:

integrate((d*x^2+e*x+c)^(1/2)*((b*x^2+a)^2)^(1/2)/x^6,x, algorithm="giac")
 

Output:

-1/128*(64*b*c^3*d*e*sgn(b*x^2 + a) - 48*a*c^2*d^2*e*sgn(b*x^2 + a) - 16*b 
*c^2*e^3*sgn(b*x^2 + a) + 40*a*c*d*e^3*sgn(b*x^2 + a) - 7*a*e^5*sgn(b*x^2 
+ a))*arctan(-(sqrt(d)*x - sqrt(d*x^2 + e*x + c))/sqrt(-c))/(sqrt(-c)*c^4) 
 + 1/1920*(960*(sqrt(d)*x - sqrt(d*x^2 + e*x + c))^9*b*c^3*d*e*sgn(b*x^2 + 
 a) - 720*(sqrt(d)*x - sqrt(d*x^2 + e*x + c))^9*a*c^2*d^2*e*sgn(b*x^2 + a) 
 - 240*(sqrt(d)*x - sqrt(d*x^2 + e*x + c))^9*b*c^2*e^3*sgn(b*x^2 + a) + 60 
0*(sqrt(d)*x - sqrt(d*x^2 + e*x + c))^9*a*c*d*e^3*sgn(b*x^2 + a) - 105*(sq 
rt(d)*x - sqrt(d*x^2 + e*x + c))^9*a*e^5*sgn(b*x^2 + a) + 3840*(sqrt(d)*x 
- sqrt(d*x^2 + e*x + c))^8*b*c^4*d^(3/2)*sgn(b*x^2 + a) + 1920*(sqrt(d)*x 
- sqrt(d*x^2 + e*x + c))^7*b*c^4*d*e*sgn(b*x^2 + a) + 3360*(sqrt(d)*x - sq 
rt(d*x^2 + e*x + c))^7*a*c^3*d^2*e*sgn(b*x^2 + a) + 1120*(sqrt(d)*x - sqrt 
(d*x^2 + e*x + c))^7*b*c^3*e^3*sgn(b*x^2 + a) - 2800*(sqrt(d)*x - sqrt(d*x 
^2 + e*x + c))^7*a*c^2*d*e^3*sgn(b*x^2 + a) + 490*(sqrt(d)*x - sqrt(d*x^2 
+ e*x + c))^7*a*c*e^5*sgn(b*x^2 + a) - 7680*(sqrt(d)*x - sqrt(d*x^2 + e*x 
+ c))^6*b*c^5*d^(3/2)*sgn(b*x^2 + a) + 7680*(sqrt(d)*x - sqrt(d*x^2 + e*x 
+ c))^6*a*c^4*d^(5/2)*sgn(b*x^2 + a) + 3840*(sqrt(d)*x - sqrt(d*x^2 + e*x 
+ c))^6*b*c^4*sqrt(d)*e^2*sgn(b*x^2 + a) - 3840*(sqrt(d)*x - sqrt(d*x^2 + 
e*x + c))^5*b*c^5*d*e*sgn(b*x^2 + a) + 15360*(sqrt(d)*x - sqrt(d*x^2 + e*x 
 + c))^5*a*c^4*d^2*e*sgn(b*x^2 + a) - 1280*(sqrt(d)*x - sqrt(d*x^2 + e*x + 
 c))^5*b*c^4*e^3*sgn(b*x^2 + a) + 5120*(sqrt(d)*x - sqrt(d*x^2 + e*x + ...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {c+e x+d x^2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{x^6} \, dx=\int \frac {\sqrt {{\left (b\,x^2+a\right )}^2}\,\sqrt {d\,x^2+e\,x+c}}{x^6} \,d x \] Input:

int((((a + b*x^2)^2)^(1/2)*(c + e*x + d*x^2)^(1/2))/x^6,x)
 

Output:

int((((a + b*x^2)^2)^(1/2)*(c + e*x + d*x^2)^(1/2))/x^6, x)
 

Reduce [B] (verification not implemented)

Time = 0.48 (sec) , antiderivative size = 549, normalized size of antiderivative = 1.49 \[ \int \frac {\sqrt {c+e x+d x^2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{x^6} \, dx=\frac {-768 \sqrt {d \,x^{2}+e x +c}\, a \,c^{5}-256 \sqrt {d \,x^{2}+e x +c}\, a \,c^{4} d \,x^{2}-96 \sqrt {d \,x^{2}+e x +c}\, a \,c^{4} e x +512 \sqrt {d \,x^{2}+e x +c}\, a \,c^{3} d^{2} x^{4}+464 \sqrt {d \,x^{2}+e x +c}\, a \,c^{3} d e \,x^{3}+112 \sqrt {d \,x^{2}+e x +c}\, a \,c^{3} e^{2} x^{2}-920 \sqrt {d \,x^{2}+e x +c}\, a \,c^{2} d \,e^{2} x^{4}-140 \sqrt {d \,x^{2}+e x +c}\, a \,c^{2} e^{3} x^{3}+210 \sqrt {d \,x^{2}+e x +c}\, a c \,e^{4} x^{4}-1280 \sqrt {d \,x^{2}+e x +c}\, b \,c^{5} x^{2}-1280 \sqrt {d \,x^{2}+e x +c}\, b \,c^{4} d \,x^{4}-320 \sqrt {d \,x^{2}+e x +c}\, b \,c^{4} e \,x^{3}+480 \sqrt {d \,x^{2}+e x +c}\, b \,c^{3} e^{2} x^{4}+720 \sqrt {c}\, \mathrm {log}\left (2 \sqrt {c}\, \sqrt {d \,x^{2}+e x +c}-2 c -e x \right ) a \,c^{2} d^{2} e \,x^{5}-600 \sqrt {c}\, \mathrm {log}\left (2 \sqrt {c}\, \sqrt {d \,x^{2}+e x +c}-2 c -e x \right ) a c d \,e^{3} x^{5}+105 \sqrt {c}\, \mathrm {log}\left (2 \sqrt {c}\, \sqrt {d \,x^{2}+e x +c}-2 c -e x \right ) a \,e^{5} x^{5}-960 \sqrt {c}\, \mathrm {log}\left (2 \sqrt {c}\, \sqrt {d \,x^{2}+e x +c}-2 c -e x \right ) b \,c^{3} d e \,x^{5}+240 \sqrt {c}\, \mathrm {log}\left (2 \sqrt {c}\, \sqrt {d \,x^{2}+e x +c}-2 c -e x \right ) b \,c^{2} e^{3} x^{5}-720 \sqrt {c}\, \mathrm {log}\left (x \right ) a \,c^{2} d^{2} e \,x^{5}+600 \sqrt {c}\, \mathrm {log}\left (x \right ) a c d \,e^{3} x^{5}-105 \sqrt {c}\, \mathrm {log}\left (x \right ) a \,e^{5} x^{5}+960 \sqrt {c}\, \mathrm {log}\left (x \right ) b \,c^{3} d e \,x^{5}-240 \sqrt {c}\, \mathrm {log}\left (x \right ) b \,c^{2} e^{3} x^{5}}{3840 c^{5} x^{5}} \] Input:

int((d*x^2+e*x+c)^(1/2)*((b*x^2+a)^2)^(1/2)/x^6,x)
 

Output:

( - 768*sqrt(c + d*x**2 + e*x)*a*c**5 - 256*sqrt(c + d*x**2 + e*x)*a*c**4* 
d*x**2 - 96*sqrt(c + d*x**2 + e*x)*a*c**4*e*x + 512*sqrt(c + d*x**2 + e*x) 
*a*c**3*d**2*x**4 + 464*sqrt(c + d*x**2 + e*x)*a*c**3*d*e*x**3 + 112*sqrt( 
c + d*x**2 + e*x)*a*c**3*e**2*x**2 - 920*sqrt(c + d*x**2 + e*x)*a*c**2*d*e 
**2*x**4 - 140*sqrt(c + d*x**2 + e*x)*a*c**2*e**3*x**3 + 210*sqrt(c + d*x* 
*2 + e*x)*a*c*e**4*x**4 - 1280*sqrt(c + d*x**2 + e*x)*b*c**5*x**2 - 1280*s 
qrt(c + d*x**2 + e*x)*b*c**4*d*x**4 - 320*sqrt(c + d*x**2 + e*x)*b*c**4*e* 
x**3 + 480*sqrt(c + d*x**2 + e*x)*b*c**3*e**2*x**4 + 720*sqrt(c)*log(2*sqr 
t(c)*sqrt(c + d*x**2 + e*x) - 2*c - e*x)*a*c**2*d**2*e*x**5 - 600*sqrt(c)* 
log(2*sqrt(c)*sqrt(c + d*x**2 + e*x) - 2*c - e*x)*a*c*d*e**3*x**5 + 105*sq 
rt(c)*log(2*sqrt(c)*sqrt(c + d*x**2 + e*x) - 2*c - e*x)*a*e**5*x**5 - 960* 
sqrt(c)*log(2*sqrt(c)*sqrt(c + d*x**2 + e*x) - 2*c - e*x)*b*c**3*d*e*x**5 
+ 240*sqrt(c)*log(2*sqrt(c)*sqrt(c + d*x**2 + e*x) - 2*c - e*x)*b*c**2*e** 
3*x**5 - 720*sqrt(c)*log(x)*a*c**2*d**2*e*x**5 + 600*sqrt(c)*log(x)*a*c*d* 
e**3*x**5 - 105*sqrt(c)*log(x)*a*e**5*x**5 + 960*sqrt(c)*log(x)*b*c**3*d*e 
*x**5 - 240*sqrt(c)*log(x)*b*c**2*e**3*x**5)/(3840*c**5*x**5)