\(\int \frac {\sqrt {c+e x+d x^2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{x^4} \, dx\) [77]

Optimal result

Integrand size = 40, antiderivative size = 294 \[ \int \frac {\sqrt {c+e x+d x^2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{x^4} \, dx=\frac {\left (2 a c e-\left (8 b c^2-a e^2\right ) x\right ) \sqrt {c+e x+d x^2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{8 c^2 x^2 \left (a+b x^2\right )}-\frac {a \left (c+e x+d x^2\right )^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 c x^3 \left (a+b x^2\right )}+\frac {b \sqrt {d} \sqrt {a^2+2 a b x^2+b^2 x^4} \text {arctanh}\left (\frac {e+2 d x}{2 \sqrt {d} \sqrt {c+e x+d x^2}}\right )}{a+b x^2}-\frac {e \left (8 b c^2-a \left (4 c d-e^2\right )\right ) \sqrt {a^2+2 a b x^2+b^2 x^4} \text {arctanh}\left (\frac {2 c+e x}{2 \sqrt {c} \sqrt {c+e x+d x^2}}\right )}{16 c^{5/2} \left (a+b x^2\right )} \] Output:

1/8*(2*a*c*e-(-a*e^2+8*b*c^2)*x)*(d*x^2+e*x+c)^(1/2)*((b*x^2+a)^2)^(1/2)/c 
^2/x^2/(b*x^2+a)-1/3*a*(d*x^2+e*x+c)^(3/2)*((b*x^2+a)^2)^(1/2)/c/x^3/(b*x^ 
2+a)+b*d^(1/2)*((b*x^2+a)^2)^(1/2)*arctanh(1/2*(2*d*x+e)/d^(1/2)/(d*x^2+e* 
x+c)^(1/2))/(b*x^2+a)-1/16*e*(8*b*c^2-a*(4*c*d-e^2))*((b*x^2+a)^2)^(1/2)*a 
rctanh(1/2*(e*x+2*c)/c^(1/2)/(d*x^2+e*x+c)^(1/2))/c^(5/2)/(b*x^2+a)
 

Mathematica [A] (verified)

Time = 1.19 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.63 \[ \int \frac {\sqrt {c+e x+d x^2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{x^4} \, dx=\frac {\sqrt {\left (a+b x^2\right )^2} \left (-3 e \left (8 b c^2+a \left (-4 c d+e^2\right )\right ) x^3 \text {arctanh}\left (\frac {-\sqrt {d} x+\sqrt {c+x (e+d x)}}{\sqrt {c}}\right )-\sqrt {c} \left (\sqrt {c+x (e+d x)} \left (24 b c^2 x^2+a \left (8 c^2-3 e^2 x^2+2 c x (e+4 d x)\right )\right )+24 b c^2 \sqrt {d} x^3 \log \left (e+2 d x-2 \sqrt {d} \sqrt {c+x (e+d x)}\right )\right )\right )}{24 c^{5/2} x^3 \left (a+b x^2\right )} \] Input:

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

Output:

(Sqrt[(a + b*x^2)^2]*(-3*e*(8*b*c^2 + a*(-4*c*d + e^2))*x^3*ArcTanh[(-(Sqr 
t[d]*x) + Sqrt[c + x*(e + d*x)])/Sqrt[c]] - Sqrt[c]*(Sqrt[c + x*(e + d*x)] 
*(24*b*c^2*x^2 + a*(8*c^2 - 3*e^2*x^2 + 2*c*x*(e + 4*d*x))) + 24*b*c^2*Sqr 
t[d]*x^3*Log[e + 2*d*x - 2*Sqrt[d]*Sqrt[c + x*(e + d*x)]])))/(24*c^(5/2)*x 
^3*(a + b*x^2))
 

Rubi [A] (verified)

Time = 0.89 (sec) , antiderivative size = 220, normalized size of antiderivative = 0.75, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.275, Rules used = {1384, 27, 2181, 27, 1229, 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.

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

\(\Big \downarrow \) 1384

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2181

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1229

\(\displaystyle \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \left (-\frac {-\frac {\int \frac {16 b d x c^2+e \left (8 b c^2-a \left (4 c d-e^2\right )\right )}{2 x \sqrt {d x^2+e x+c}}dx}{4 c}-\frac {\sqrt {c+d x^2+e x} \left (2 a c e-x \left (8 b c^2-a e^2\right )\right )}{4 c x^2}}{2 c}-\frac {a \left (c+d x^2+e x\right )^{3/2}}{3 c x^3}\right )}{a+b x^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \left (-\frac {-\frac {\int \frac {16 b d x c^2+e \left (8 b c^2-a \left (4 c d-e^2\right )\right )}{x \sqrt {d x^2+e x+c}}dx}{8 c}-\frac {\sqrt {c+d x^2+e x} \left (2 a c e-x \left (8 b c^2-a e^2\right )\right )}{4 c x^2}}{2 c}-\frac {a \left (c+d x^2+e x\right )^{3/2}}{3 c x^3}\right )}{a+b x^2}\)

\(\Big \downarrow \) 1269

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

\(\Big \downarrow \) 1092

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

\(\Big \downarrow \) 219

\(\displaystyle \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \left (-\frac {-\frac {e \left (8 b c^2-a \left (4 c d-e^2\right )\right ) \int \frac {1}{x \sqrt {d x^2+e x+c}}dx+16 b c^2 \sqrt {d} \text {arctanh}\left (\frac {2 d x+e}{2 \sqrt {d} \sqrt {c+d x^2+e x}}\right )}{8 c}-\frac {\sqrt {c+d x^2+e x} \left (2 a c e-x \left (8 b c^2-a e^2\right )\right )}{4 c x^2}}{2 c}-\frac {a \left (c+d x^2+e x\right )^{3/2}}{3 c x^3}\right )}{a+b x^2}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \left (-\frac {-\frac {16 b c^2 \sqrt {d} \text {arctanh}\left (\frac {2 d x+e}{2 \sqrt {d} \sqrt {c+d x^2+e x}}\right )-2 e \left (8 b c^2-a \left (4 c d-e^2\right )\right ) \int \frac {1}{4 c-\frac {(2 c+e x)^2}{d x^2+e x+c}}d\frac {2 c+e x}{\sqrt {d x^2+e x+c}}}{8 c}-\frac {\sqrt {c+d x^2+e x} \left (2 a c e-x \left (8 b c^2-a e^2\right )\right )}{4 c x^2}}{2 c}-\frac {a \left (c+d x^2+e x\right )^{3/2}}{3 c x^3}\right )}{a+b x^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \left (-\frac {-\frac {16 b c^2 \sqrt {d} \text {arctanh}\left (\frac {2 d x+e}{2 \sqrt {d} \sqrt {c+d x^2+e x}}\right )-\frac {e \left (8 b c^2-a \left (4 c d-e^2\right )\right ) \text {arctanh}\left (\frac {2 c+e x}{2 \sqrt {c} \sqrt {c+d x^2+e x}}\right )}{\sqrt {c}}}{8 c}-\frac {\sqrt {c+d x^2+e x} \left (2 a c e-x \left (8 b c^2-a e^2\right )\right )}{4 c x^2}}{2 c}-\frac {a \left (c+d x^2+e x\right )^{3/2}}{3 c x^3}\right )}{a+b x^2}\)

Input:

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

Output:

(Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]*(-1/3*(a*(c + e*x + d*x^2)^(3/2))/(c*x^3) 
 - (-1/4*((2*a*c*e - (8*b*c^2 - a*e^2)*x)*Sqrt[c + e*x + d*x^2])/(c*x^2) - 
 (16*b*c^2*Sqrt[d]*ArcTanh[(e + 2*d*x)/(2*Sqrt[d]*Sqrt[c + e*x + d*x^2])] 
- (e*(8*b*c^2 - a*(4*c*d - e^2))*ArcTanh[(2*c + e*x)/(2*Sqrt[c]*Sqrt[c + e 
*x + d*x^2])])/Sqrt[c])/(8*c))/(2*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 1092

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]
 

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 1229

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*((a + b*x + c*x^2 
)^p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)))*((d*g - e*f*(m + 2))*(c* 
d^2 - b*d*e + a*e^2) - d*p*(2*c*d - b*e)*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 
- b*d*e + a*e^2) + p*(2*c*d - b*e)*(e*f - d*g))*x), x] - Simp[p/(e^2*(m + 1 
)*(m + 2)*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2 
)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) + b^2*e*(d*g*(p + 1) - e*f*(m + 
p + 2)) + b*(a*e^2*g*(m + 1) - c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2))) - c 
*(2*c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2)) - e*(2*a*e*g*(m + 1) - b*(d*g*( 
m - 2*p) + e*f*(m + 2*p + 2))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g 
}, x] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] &&  !ILtQ[m + 2*p + 3, 
0]
 

rule 1269

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]
 

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 = 0.31 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.66

method	result	size
risch	\(-\frac {\sqrt {d \,x^{2}+e x +c}\, \left (8 a c d \,x^{2}-3 a \,e^{2} x^{2}+24 b \,c^{2} x^{2}+2 a c e x +8 a \,c^{2}\right ) \sqrt {\left (b \,x^{2}+a \right )^{2}}}{24 x^{3} c^{2} \left (b \,x^{2}+a \right )}-\frac {\left (-16 b \,c^{2} \sqrt {d}\, \ln \left (\frac {\frac {e}{2}+d x}{\sqrt {d}}+\sqrt {d \,x^{2}+e x +c}\right )-\frac {e \left (4 a c d -a \,e^{2}-8 b \,c^{2}\right ) \ln \left (\frac {2 c +e x +2 \sqrt {c}\, \sqrt {d \,x^{2}+e x +c}}{x}\right )}{\sqrt {c}}\right ) \sqrt {\left (b \,x^{2}+a \right )^{2}}}{16 c^{2} \left (b \,x^{2}+a \right )}\)	\(194\)
default	\(\frac {\sqrt {\left (b \,x^{2}+a \right )^{2}}\, \left (12 d^{\frac {5}{2}} c^{\frac {3}{2}} \ln \left (\frac {2 c +e x +2 \sqrt {c}\, \sqrt {d \,x^{2}+e x +c}}{x}\right ) a e \,x^{3}-24 d^{\frac {3}{2}} c^{\frac {5}{2}} \ln \left (\frac {2 c +e x +2 \sqrt {c}\, \sqrt {d \,x^{2}+e x +c}}{x}\right ) b e \,x^{3}+6 d^{\frac {5}{2}} \sqrt {d \,x^{2}+e x +c}\, a \,e^{2} x^{4}+48 d^{\frac {5}{2}} \sqrt {d \,x^{2}+e x +c}\, b \,c^{2} x^{4}-12 d^{\frac {5}{2}} \sqrt {d \,x^{2}+e x +c}\, a c e \,x^{3}-3 d^{\frac {3}{2}} \sqrt {c}\, \ln \left (\frac {2 c +e x +2 \sqrt {c}\, \sqrt {d \,x^{2}+e x +c}}{x}\right ) a \,e^{3} x^{3}-6 d^{\frac {3}{2}} \left (d \,x^{2}+e x +c \right )^{\frac {3}{2}} a \,e^{2} x^{2}-48 d^{\frac {3}{2}} \left (d \,x^{2}+e x +c \right )^{\frac {3}{2}} b \,c^{2} x^{2}+6 d^{\frac {3}{2}} \sqrt {d \,x^{2}+e x +c}\, a \,e^{3} x^{3}+48 d^{\frac {3}{2}} \sqrt {d \,x^{2}+e x +c}\, b \,c^{2} e \,x^{3}+48 \ln \left (\frac {2 \sqrt {d \,x^{2}+e x +c}\, \sqrt {d}+2 d x +e}{2 \sqrt {d}}\right ) b \,c^{3} d^{2} x^{3}+12 d^{\frac {3}{2}} \left (d \,x^{2}+e x +c \right )^{\frac {3}{2}} a c e x -16 d^{\frac {3}{2}} \left (d \,x^{2}+e x +c \right )^{\frac {3}{2}} a \,c^{2}\right )}{48 d^{\frac {3}{2}} x^{3} c^{3} \left (b \,x^{2}+a \right )}\)	\(412\)

Input:

int((d*x^2+e*x+c)^(1/2)*((b*x^2+a)^2)^(1/2)/x^4,x,method=_RETURNVERBOSE)
 

Output:

-1/24*(d*x^2+e*x+c)^(1/2)*(8*a*c*d*x^2-3*a*e^2*x^2+24*b*c^2*x^2+2*a*c*e*x+ 
8*a*c^2)/x^3/c^2*((b*x^2+a)^2)^(1/2)/(b*x^2+a)-1/16/c^2*(-16*b*c^2*d^(1/2) 
*ln((1/2*e+d*x)/d^(1/2)+(d*x^2+e*x+c)^(1/2))-e*(4*a*c*d-a*e^2-8*b*c^2)/c^( 
1/2)*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 = 0.41 (sec) , antiderivative size = 791, normalized size of antiderivative = 2.69 \[ \int \frac {\sqrt {c+e x+d x^2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{x^4} \, dx =\text {Too large to display} \] Input:

integrate((d*x^2+e*x+c)^(1/2)*((b*x^2+a)^2)^(1/2)/x^4,x, algorithm="fricas 
")
 

Output:

[1/96*(48*b*c^3*sqrt(d)*x^3*log(8*d^2*x^2 + 8*d*e*x + 4*sqrt(d*x^2 + e*x + 
 c)*(2*d*x + e)*sqrt(d) + 4*c*d + e^2) + 3*(a*e^3 + 4*(2*b*c^2 - a*c*d)*e) 
*sqrt(c)*x^3*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*(2*a*c^2*e*x + 8*a*c^3 + (24*b*c^3 + 8 
*a*c^2*d - 3*a*c*e^2)*x^2)*sqrt(d*x^2 + e*x + c))/(c^3*x^3), -1/96*(96*b*c 
^3*sqrt(-d)*x^3*arctan(1/2*sqrt(d*x^2 + e*x + c)*(2*d*x + e)*sqrt(-d)/(d^2 
*x^2 + d*e*x + c*d)) - 3*(a*e^3 + 4*(2*b*c^2 - a*c*d)*e)*sqrt(c)*x^3*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*(2*a*c^2*e*x + 8*a*c^3 + (24*b*c^3 + 8*a*c^2*d - 3*a*c*e 
^2)*x^2)*sqrt(d*x^2 + e*x + c))/(c^3*x^3), 1/48*(24*b*c^3*sqrt(d)*x^3*log( 
8*d^2*x^2 + 8*d*e*x + 4*sqrt(d*x^2 + e*x + c)*(2*d*x + e)*sqrt(d) + 4*c*d 
+ e^2) + 3*(a*e^3 + 4*(2*b*c^2 - a*c*d)*e)*sqrt(-c)*x^3*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*(2*a*c^2* 
e*x + 8*a*c^3 + (24*b*c^3 + 8*a*c^2*d - 3*a*c*e^2)*x^2)*sqrt(d*x^2 + e*x + 
 c))/(c^3*x^3), -1/48*(48*b*c^3*sqrt(-d)*x^3*arctan(1/2*sqrt(d*x^2 + e*x + 
 c)*(2*d*x + e)*sqrt(-d)/(d^2*x^2 + d*e*x + c*d)) - 3*(a*e^3 + 4*(2*b*c^2 
- a*c*d)*e)*sqrt(-c)*x^3*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*(2*a*c^2*e*x + 8*a*c^3 + (24*b*c^3 + 8*a 
*c^2*d - 3*a*c*e^2)*x^2)*sqrt(d*x^2 + e*x + c))/(c^3*x^3)]
 

Sympy [F(-1)]

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

integrate((d*x**2+e*x+c)**(1/2)*((b*x**2+a)**2)**(1/2)/x**4,x)
 

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((d*x^2+e*x+c)^(1/2)*((b*x^2+a)^2)^(1/2)/x^4,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. 683 vs. \(2 (224) = 448\).

Time = 0.18 (sec) , antiderivative size = 683, normalized size of antiderivative = 2.32 \[ \int \frac {\sqrt {c+e x+d x^2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{x^4} \, dx =\text {Too large to display} \] Input:

integrate((d*x^2+e*x+c)^(1/2)*((b*x^2+a)^2)^(1/2)/x^4,x, algorithm="giac")
 

Output:

-b*sqrt(d)*log(abs(-2*(sqrt(d)*x - sqrt(d*x^2 + e*x + c))*sqrt(d) - e))*sg 
n(b*x^2 + a) + 1/8*(8*b*c^2*e*sgn(b*x^2 + a) - 4*a*c*d*e*sgn(b*x^2 + a) + 
a*e^3*sgn(b*x^2 + a))*arctan(-(sqrt(d)*x - sqrt(d*x^2 + e*x + c))/sqrt(-c) 
)/(sqrt(-c)*c^2) + 1/24*(24*(sqrt(d)*x - sqrt(d*x^2 + e*x + c))^5*b*c^2*e* 
sgn(b*x^2 + a) + 12*(sqrt(d)*x - sqrt(d*x^2 + e*x + c))^5*a*c*d*e*sgn(b*x^ 
2 + a) - 3*(sqrt(d)*x - sqrt(d*x^2 + e*x + c))^5*a*e^3*sgn(b*x^2 + a) + 48 
*(sqrt(d)*x - sqrt(d*x^2 + e*x + c))^4*b*c^3*sqrt(d)*sgn(b*x^2 + a) + 48*( 
sqrt(d)*x - sqrt(d*x^2 + e*x + c))^4*a*c^2*d^(3/2)*sgn(b*x^2 + a) - 48*(sq 
rt(d)*x - sqrt(d*x^2 + e*x + c))^3*b*c^3*e*sgn(b*x^2 + a) + 48*(sqrt(d)*x 
- sqrt(d*x^2 + e*x + c))^3*a*c^2*d*e*sgn(b*x^2 + a) + 8*(sqrt(d)*x - sqrt( 
d*x^2 + e*x + c))^3*a*c*e^3*sgn(b*x^2 + a) - 96*(sqrt(d)*x - sqrt(d*x^2 + 
e*x + c))^2*b*c^4*sqrt(d)*sgn(b*x^2 + a) + 48*(sqrt(d)*x - sqrt(d*x^2 + e* 
x + c))^2*a*c^2*sqrt(d)*e^2*sgn(b*x^2 + a) + 24*(sqrt(d)*x - sqrt(d*x^2 + 
e*x + c))*b*c^4*e*sgn(b*x^2 + a) + 36*(sqrt(d)*x - sqrt(d*x^2 + e*x + c))* 
a*c^3*d*e*sgn(b*x^2 + a) + 3*(sqrt(d)*x - sqrt(d*x^2 + e*x + c))*a*c^2*e^3 
*sgn(b*x^2 + a) + 48*b*c^5*sqrt(d)*sgn(b*x^2 + a) + 16*a*c^4*d^(3/2)*sgn(b 
*x^2 + a))/(((sqrt(d)*x - sqrt(d*x^2 + e*x + c))^2 - c)^3*c^2)
 

Mupad [F(-1)]

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

int((((a + b*x^2)^2)^(1/2)*(c + e*x + d*x^2)^(1/2))/x^4,x)
 

Output:

int((((a + b*x^2)^2)^(1/2)*(c + e*x + d*x^2)^(1/2))/x^4, x)
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 288, normalized size of antiderivative = 0.98 \[ \int \frac {\sqrt {c+e x+d x^2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{x^4} \, dx=\frac {-16 \sqrt {d \,x^{2}+e x +c}\, a \,c^{3}-16 \sqrt {d \,x^{2}+e x +c}\, a \,c^{2} d \,x^{2}-4 \sqrt {d \,x^{2}+e x +c}\, a \,c^{2} e x +6 \sqrt {d \,x^{2}+e x +c}\, a c \,e^{2} x^{2}-48 \sqrt {d \,x^{2}+e x +c}\, b \,c^{3} x^{2}-12 \sqrt {c}\, \mathrm {log}\left (2 \sqrt {c}\, \sqrt {d \,x^{2}+e x +c}-2 c -e x \right ) a c d e \,x^{3}+3 \sqrt {c}\, \mathrm {log}\left (2 \sqrt {c}\, \sqrt {d \,x^{2}+e x +c}-2 c -e x \right ) a \,e^{3} x^{3}+24 \sqrt {c}\, \mathrm {log}\left (2 \sqrt {c}\, \sqrt {d \,x^{2}+e x +c}-2 c -e x \right ) b \,c^{2} e \,x^{3}+12 \sqrt {c}\, \mathrm {log}\left (x \right ) a c d e \,x^{3}-3 \sqrt {c}\, \mathrm {log}\left (x \right ) a \,e^{3} x^{3}-24 \sqrt {c}\, \mathrm {log}\left (x \right ) b \,c^{2} e \,x^{3}+48 \sqrt {d}\, \mathrm {log}\left (-2 \sqrt {d}\, \sqrt {d \,x^{2}+e x +c}-2 d x -e \right ) b \,c^{3} x^{3}}{48 c^{3} x^{3}} \] Input:

int((d*x^2+e*x+c)^(1/2)*((b*x^2+a)^2)^(1/2)/x^4,x)
 

Output:

( - 16*sqrt(c + d*x**2 + e*x)*a*c**3 - 16*sqrt(c + d*x**2 + e*x)*a*c**2*d* 
x**2 - 4*sqrt(c + d*x**2 + e*x)*a*c**2*e*x + 6*sqrt(c + d*x**2 + e*x)*a*c* 
e**2*x**2 - 48*sqrt(c + d*x**2 + e*x)*b*c**3*x**2 - 12*sqrt(c)*log(2*sqrt( 
c)*sqrt(c + d*x**2 + e*x) - 2*c - e*x)*a*c*d*e*x**3 + 3*sqrt(c)*log(2*sqrt 
(c)*sqrt(c + d*x**2 + e*x) - 2*c - e*x)*a*e**3*x**3 + 24*sqrt(c)*log(2*sqr 
t(c)*sqrt(c + d*x**2 + e*x) - 2*c - e*x)*b*c**2*e*x**3 + 12*sqrt(c)*log(x) 
*a*c*d*e*x**3 - 3*sqrt(c)*log(x)*a*e**3*x**3 - 24*sqrt(c)*log(x)*b*c**2*e* 
x**3 + 48*sqrt(d)*log( - 2*sqrt(d)*sqrt(c + d*x**2 + e*x) - 2*d*x - e)*b*c 
**3*x**3)/(48*c**3*x**3)