\(\int \frac {\sqrt {c+e x+d x^2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{x} \, dx\) [74]

Optimal result

Integrand size = 40, antiderivative size = 286 \[ \int \frac {\sqrt {c+e x+d x^2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{x} \, dx=\frac {\left (8 a d^2-b e^2-2 b d e x\right ) \sqrt {c+e x+d x^2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{8 d^2 \left (a+b x^2\right )}+\frac {b \left (c+e x+d x^2\right )^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 d \left (a+b x^2\right )}+\frac {e \left (8 a d^2-b \left (4 c d-e^2\right )\right ) \sqrt {a^2+2 a b x^2+b^2 x^4} \text {arctanh}\left (\frac {e+2 d x}{2 \sqrt {d} \sqrt {c+e x+d x^2}}\right )}{16 d^{5/2} \left (a+b x^2\right )}-\frac {a \sqrt {c} \sqrt {a^2+2 a b x^2+b^2 x^4} \text {arctanh}\left (\frac {2 c+e x}{2 \sqrt {c} \sqrt {c+e x+d x^2}}\right )}{a+b x^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.77 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.61 \[ \int \frac {\sqrt {c+e x+d x^2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{x} \, dx=\frac {\sqrt {\left (a+b x^2\right )^2} \left (2 \sqrt {d} \sqrt {c+x (e+d x)} \left (24 a d^2+b \left (8 c d-3 e^2+2 d e x+8 d^2 x^2\right )\right )+96 a \sqrt {c} d^{5/2} \text {arctanh}\left (\frac {\sqrt {d} x-\sqrt {c+x (e+d x)}}{\sqrt {c}}\right )-3 e \left (8 a d^2+b \left (-4 c d+e^2\right )\right ) \log \left (e+2 d x-2 \sqrt {d} \sqrt {c+x (e+d x)}\right )\right )}{48 d^{5/2} \left (a+b x^2\right )} \] Input:

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

Output:

(Sqrt[(a + b*x^2)^2]*(2*Sqrt[d]*Sqrt[c + x*(e + d*x)]*(24*a*d^2 + b*(8*c*d 
 - 3*e^2 + 2*d*e*x + 8*d^2*x^2)) + 96*a*Sqrt[c]*d^(5/2)*ArcTanh[(Sqrt[d]*x 
 - Sqrt[c + x*(e + d*x)])/Sqrt[c]] - 3*e*(8*a*d^2 + b*(-4*c*d + e^2))*Log[ 
e + 2*d*x - 2*Sqrt[d]*Sqrt[c + x*(e + d*x)]]))/(48*d^(5/2)*(a + b*x^2))
 

Rubi [A] (verified)

Time = 0.88 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.73, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.275, Rules used = {1384, 27, 2184, 27, 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.

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

\(\Big \downarrow \) 1384

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2184

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1231

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1269

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

\(\Big \downarrow \) 1092

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 1154

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

(Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]*((b*(c + e*x + d*x^2)^(3/2))/(3*d) + (((8 
*a*d^2 - b*e^2 - 2*b*d*e*x)*Sqrt[c + e*x + d*x^2])/(4*d) + ((e*(8*a*d^2 - 
b*(4*c*d - e^2))*ArcTanh[(e + 2*d*x)/(2*Sqrt[d]*Sqrt[c + e*x + d*x^2])])/S 
qrt[d] - 16*a*Sqrt[c]*d^2*ArcTanh[(2*c + e*x)/(2*Sqrt[c]*Sqrt[c + e*x + d* 
x^2])])/(8*d))/(2*d)))/(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 1231

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])
 

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 2184

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p 
_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, S 
imp[f*(d + e*x)^(m + q - 1)*((a + b*x + c*x^2)^(p + 1)/(c*e^(q - 1)*(m + q 
+ 2*p + 1))), x] + Simp[1/(c*e^q*(m + q + 2*p + 1))   Int[(d + e*x)^m*(a + 
b*x + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*f*(m + q + 2*p + 
1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(b*d*e*(p + 1) + a*e^2*(m + q - 1) - c 
*d^2*(m + q + 2*p + 1) - e*(2*c*d - b*e)*(m + q + p)*x), x], x], x] /; GtQ[ 
q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, d, e, m, p}, x] && Pol 
yQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] &&  !(IGt 
Q[m, 0] && RationalQ[a, b, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
 

Maple [A] (verified)

Time = 0.32 (sec) , antiderivative size = 251, normalized size of antiderivative = 0.88

method	result	size
default	\(-\frac {\sqrt {\left (b \,x^{2}+a \right )^{2}}\, \left (48 \sqrt {c}\, d^{\frac {7}{2}} \ln \left (\frac {2 c +e x +2 \sqrt {c}\, \sqrt {d \,x^{2}+e x +c}}{x}\right ) a -16 d^{\frac {5}{2}} \left (d \,x^{2}+e x +c \right )^{\frac {3}{2}} b +12 d^{\frac {5}{2}} \sqrt {d \,x^{2}+e x +c}\, b e x -24 \ln \left (\frac {2 \sqrt {d \,x^{2}+e x +c}\, \sqrt {d}+2 d x +e}{2 \sqrt {d}}\right ) d^{3} a e -48 d^{\frac {7}{2}} \sqrt {d \,x^{2}+e x +c}\, a +6 d^{\frac {3}{2}} \sqrt {d \,x^{2}+e x +c}\, b \,e^{2}+12 \ln \left (\frac {2 \sqrt {d \,x^{2}+e x +c}\, \sqrt {d}+2 d x +e}{2 \sqrt {d}}\right ) b c \,d^{2} e -3 \ln \left (\frac {2 \sqrt {d \,x^{2}+e x +c}\, \sqrt {d}+2 d x +e}{2 \sqrt {d}}\right ) b d \,e^{3}\right )}{48 \left (b \,x^{2}+a \right ) d^{\frac {7}{2}}}\)	\(251\)

Input:

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

Output:

-1/48*((b*x^2+a)^2)^(1/2)*(48*c^(1/2)*d^(7/2)*ln((2*c+e*x+2*c^(1/2)*(d*x^2 
+e*x+c)^(1/2))/x)*a-16*d^(5/2)*(d*x^2+e*x+c)^(3/2)*b+12*d^(5/2)*(d*x^2+e*x 
+c)^(1/2)*b*e*x-24*ln(1/2*(2*(d*x^2+e*x+c)^(1/2)*d^(1/2)+2*d*x+e)/d^(1/2)) 
*d^3*a*e-48*d^(7/2)*(d*x^2+e*x+c)^(1/2)*a+6*d^(3/2)*(d*x^2+e*x+c)^(1/2)*b* 
e^2+12*ln(1/2*(2*(d*x^2+e*x+c)^(1/2)*d^(1/2)+2*d*x+e)/d^(1/2))*b*c*d^2*e-3 
*ln(1/2*(2*(d*x^2+e*x+c)^(1/2)*d^(1/2)+2*d*x+e)/d^(1/2))*b*d*e^3)/(b*x^2+a 
)/d^(7/2)
 

Fricas [A] (verification not implemented)

Time = 0.50 (sec) , antiderivative size = 743, normalized size of antiderivative = 2.60 \[ \int \frac {\sqrt {c+e x+d x^2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{x} \, dx =\text {Too large to display} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

Integral(sqrt(c + d*x**2 + e*x)*sqrt((a + b*x**2)**2)/x, x)
 

Maxima [F(-2)]

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

integrate((d*x^2+e*x+c)^(1/2)*((b*x^2+a)^2)^(1/2)/x,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>0)', see `assume?` for more de 
tails)Is e
 

Giac [F(-2)]

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

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Degree mismatch inside factorisatio 
n over extensionNot implemented, e.g. for multivariate mod/approx polynomi 
alsError:
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 1153, normalized size of antiderivative = 4.03 \[ \int \frac {\sqrt {c+e x+d x^2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{x} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 48*sqrt(c)*sqrt(4*sqrt(d)*sqrt(c)*e - 4*c*d - e**2)*atan((2*sqrt(d)*sq 
rt(c + d*x**2 + e*x) + 2*d*x + e)/sqrt(4*sqrt(d)*sqrt(c)*e - 4*c*d - e**2) 
)*a*d**3*e - 96*sqrt(d)*sqrt(4*sqrt(d)*sqrt(c)*e - 4*c*d - e**2)*atan((2*s 
qrt(d)*sqrt(c + d*x**2 + e*x) + 2*d*x + e)/sqrt(4*sqrt(d)*sqrt(c)*e - 4*c* 
d - e**2))*a*c*d**3 - 24*sqrt(c)*sqrt(4*sqrt(d)*sqrt(c)*e + 4*c*d + e**2)* 
log( - sqrt(4*sqrt(d)*sqrt(c)*e + 4*c*d + e**2) + 2*sqrt(d)*sqrt(c + d*x** 
2 + e*x) + 2*d*x + e)*a*d**3*e + 24*sqrt(c)*sqrt(4*sqrt(d)*sqrt(c)*e + 4*c 
*d + e**2)*log(sqrt(4*sqrt(d)*sqrt(c)*e + 4*c*d + e**2) + 2*sqrt(d)*sqrt(c 
 + d*x**2 + e*x) + 2*d*x + e)*a*d**3*e + 48*sqrt(d)*sqrt(4*sqrt(d)*sqrt(c) 
*e + 4*c*d + e**2)*log( - sqrt(4*sqrt(d)*sqrt(c)*e + 4*c*d + e**2) + 2*sqr 
t(d)*sqrt(c + d*x**2 + e*x) + 2*d*x + e)*a*c*d**3 - 48*sqrt(d)*sqrt(4*sqrt 
(d)*sqrt(c)*e + 4*c*d + e**2)*log(sqrt(4*sqrt(d)*sqrt(c)*e + 4*c*d + e**2) 
 + 2*sqrt(d)*sqrt(c + d*x**2 + e*x) + 2*d*x + e)*a*c*d**3 + 192*sqrt(c + d 
*x**2 + e*x)*a*c*d**4 - 48*sqrt(c + d*x**2 + e*x)*a*d**3*e**2 + 64*sqrt(c 
+ d*x**2 + e*x)*b*c**2*d**3 + 64*sqrt(c + d*x**2 + e*x)*b*c*d**4*x**2 + 16 
*sqrt(c + d*x**2 + e*x)*b*c*d**3*e*x - 40*sqrt(c + d*x**2 + e*x)*b*c*d**2* 
e**2 - 16*sqrt(c + d*x**2 + e*x)*b*d**3*e**2*x**2 - 4*sqrt(c + d*x**2 + e* 
x)*b*d**2*e**3*x + 6*sqrt(c + d*x**2 + e*x)*b*d*e**4 + 96*sqrt(c)*log( - s 
qrt(4*sqrt(d)*sqrt(c)*e + 4*c*d + e**2) + 2*sqrt(d)*sqrt(c + d*x**2 + e*x) 
 + 2*d*x + e)*a*c*d**4 - 24*sqrt(c)*log( - sqrt(4*sqrt(d)*sqrt(c)*e + 4...