\(\int \frac {\sqrt {c+d x^4}}{x^5 (a+b x^4)} \, dx\) [223]

Optimal result

Integrand size = 24, antiderivative size = 115 \[ \int \frac {\sqrt {c+d x^4}}{x^5 \left (a+b x^4\right )} \, dx=-\frac {\sqrt {c+d x^4}}{4 a x^4}+\frac {(2 b c-a d) \text {arctanh}\left (\frac {\sqrt {c+d x^4}}{\sqrt {c}}\right )}{4 a^2 \sqrt {c}}-\frac {\sqrt {b} \sqrt {b c-a d} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^4}}{\sqrt {b c-a d}}\right )}{2 a^2} \] Output:

-1/4*(d*x^4+c)^(1/2)/a/x^4+1/4*(-a*d+2*b*c)*arctanh((d*x^4+c)^(1/2)/c^(1/2 
))/a^2/c^(1/2)-1/2*b^(1/2)*(-a*d+b*c)^(1/2)*arctanh(b^(1/2)*(d*x^4+c)^(1/2 
)/(-a*d+b*c)^(1/2))/a^2
 

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.93 \[ \int \frac {\sqrt {c+d x^4}}{x^5 \left (a+b x^4\right )} \, dx=\frac {-\frac {a \sqrt {c+d x^4}}{x^4}-2 \sqrt {b} \sqrt {-b c+a d} \arctan \left (\frac {\sqrt {b} \sqrt {c+d x^4}}{\sqrt {-b c+a d}}\right )+\frac {(2 b c-a d) \text {arctanh}\left (\frac {\sqrt {c+d x^4}}{\sqrt {c}}\right )}{\sqrt {c}}}{4 a^2} \] Input:

Integrate[Sqrt[c + d*x^4]/(x^5*(a + b*x^4)),x]
 

Output:

(-((a*Sqrt[c + d*x^4])/x^4) - 2*Sqrt[b]*Sqrt[-(b*c) + a*d]*ArcTan[(Sqrt[b] 
*Sqrt[c + d*x^4])/Sqrt[-(b*c) + a*d]] + ((2*b*c - a*d)*ArcTanh[Sqrt[c + d* 
x^4]/Sqrt[c]])/Sqrt[c])/(4*a^2)
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.05, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {948, 110, 27, 174, 73, 221}

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.

\(\displaystyle \int \frac {\sqrt {c+d x^4}}{x^5 \left (a+b x^4\right )} \, dx\)

\(\Big \downarrow \) 948

\(\displaystyle \frac {1}{4} \int \frac {\sqrt {d x^4+c}}{x^8 \left (b x^4+a\right )}dx^4\)

\(\Big \downarrow \) 110

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 174

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

\(\displaystyle \frac {1}{4} \left (-\frac {\frac {4 \sqrt {b} \sqrt {b c-a d} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^4}}{\sqrt {b c-a d}}\right )}{a}-\frac {2 (2 b c-a d) \text {arctanh}\left (\frac {\sqrt {c+d x^4}}{\sqrt {c}}\right )}{a \sqrt {c}}}{2 a}-\frac {\sqrt {c+d x^4}}{a x^4}\right )\)

Input:

Int[Sqrt[c + d*x^4]/(x^5*(a + b*x^4)),x]
 

Output:

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

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 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 110

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 
1)*(b*e - a*f))), x] - Simp[1/((m + 1)*(b*e - a*f))   Int[(a + b*x)^(m + 1) 
*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + 
p + 2)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && Gt 
Q[n, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, 
 m + n])
 

rule 174

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* 
((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d)   Int[(e + f*x)^ 
p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d)   Int[(e + f*x)^p/(c + d 
*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 948

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. 
), x_Symbol] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^ 
p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ 
[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]
 

Maple [A] (verified)

Time = 0.24 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.83

method	result	size
pseudoelliptic	\(\frac {-\frac {a \sqrt {d \,x^{4}+c}}{x^{4}}-\frac {\left (a d -2 c b \right ) \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{4}+c}}{\sqrt {c}}\right )}{\sqrt {c}}-\frac {2 \left (a d -c b \right ) b \arctan \left (\frac {\sqrt {d \,x^{4}+c}\, b}{\sqrt {\left (a d -c b \right ) b}}\right )}{\sqrt {\left (a d -c b \right ) b}}}{4 a^{2}}\)	\(96\)
risch	\(-\frac {\sqrt {d \,x^{4}+c}}{4 a \,x^{4}}-\frac {\frac {\left (a d -2 c b \right ) \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{4}+c}}{x^{2}}\right )}{2 a \sqrt {c}}+\frac {2 \left (a d -c b \right ) b \left (-\frac {\ln \left (\frac {-\frac {2 \left (a d -c b \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -c b}{b}}\, \sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{x^{2}-\frac {\sqrt {-a b}}{b}}\right )}{4 b \sqrt {-\frac {a d -c b}{b}}}-\frac {\ln \left (\frac {-\frac {2 \left (a d -c b \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -c b}{b}}\, \sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{x^{2}+\frac {\sqrt {-a b}}{b}}\right )}{4 b \sqrt {-\frac {a d -c b}{b}}}\right )}{a}}{2 a}\)	\(393\)
elliptic	\(\frac {-\frac {\left (d \,x^{4}+c \right )^{\frac {3}{2}}}{2 c \,x^{4}}+\frac {d \left (\sqrt {d \,x^{4}+c}-\sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{4}+c}}{x^{2}}\right )\right )}{2 c}}{2 a}-\frac {b \left (\sqrt {d \,x^{4}+c}-\sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{4}+c}}{x^{2}}\right )\right )}{2 a^{2}}+\frac {b \left (\sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}+\frac {\sqrt {d}\, \sqrt {-a b}\, \ln \left (\frac {\frac {d \sqrt {-a b}}{b}+\left (x^{2}-\frac {\sqrt {-a b}}{b}\right ) d}{\sqrt {d}}+\sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}\right )}{b}+\frac {\left (a d -c b \right ) \ln \left (\frac {-\frac {2 \left (a d -c b \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -c b}{b}}\, \sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{x^{2}-\frac {\sqrt {-a b}}{b}}\right )}{b \sqrt {-\frac {a d -c b}{b}}}\right )}{4 a^{2}}+\frac {b \left (\sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}-\frac {\sqrt {d}\, \sqrt {-a b}\, \ln \left (\frac {-\frac {d \sqrt {-a b}}{b}+\left (x^{2}+\frac {\sqrt {-a b}}{b}\right ) d}{\sqrt {d}}+\sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}\right )}{b}+\frac {\left (a d -c b \right ) \ln \left (\frac {-\frac {2 \left (a d -c b \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -c b}{b}}\, \sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{x^{2}+\frac {\sqrt {-a b}}{b}}\right )}{b \sqrt {-\frac {a d -c b}{b}}}\right )}{4 a^{2}}\)	\(796\)
default	\(\frac {-\frac {\left (d \,x^{4}+c \right )^{\frac {3}{2}}}{4 c \,x^{4}}-\frac {d \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{4}+c}}{x^{2}}\right )}{4 \sqrt {c}}+\frac {d \sqrt {d \,x^{4}+c}}{4 c}}{a}+\frac {b^{2} \left (\frac {\sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}+\frac {\sqrt {d}\, \sqrt {-a b}\, \ln \left (\frac {\frac {d \sqrt {-a b}}{b}+\left (x^{2}-\frac {\sqrt {-a b}}{b}\right ) d}{\sqrt {d}}+\sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}\right )}{b}+\frac {\left (a d -c b \right ) \ln \left (\frac {-\frac {2 \left (a d -c b \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -c b}{b}}\, \sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{x^{2}-\frac {\sqrt {-a b}}{b}}\right )}{b \sqrt {-\frac {a d -c b}{b}}}}{4 b}+\frac {\sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}-\frac {\sqrt {d}\, \sqrt {-a b}\, \ln \left (\frac {-\frac {d \sqrt {-a b}}{b}+\left (x^{2}+\frac {\sqrt {-a b}}{b}\right ) d}{\sqrt {d}}+\sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}\right )}{b}+\frac {\left (a d -c b \right ) \ln \left (\frac {-\frac {2 \left (a d -c b \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -c b}{b}}\, \sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{x^{2}+\frac {\sqrt {-a b}}{b}}\right )}{b \sqrt {-\frac {a d -c b}{b}}}}{4 b}\right )}{a^{2}}-\frac {b \left (\frac {\sqrt {d \,x^{4}+c}}{2}-\frac {\sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{4}+c}}{x^{2}}\right )}{2}\right )}{a^{2}}\)	\(803\)

Input:

int((d*x^4+c)^(1/2)/x^5/(b*x^4+a),x,method=_RETURNVERBOSE)
 

Output:

1/4/a^2*(-a*(d*x^4+c)^(1/2)/x^4-(a*d-2*b*c)/c^(1/2)*arctanh((d*x^4+c)^(1/2 
)/c^(1/2))-2*(a*d-b*c)*b/((a*d-b*c)*b)^(1/2)*arctan((d*x^4+c)^(1/2)*b/((a* 
d-b*c)*b)^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 507, normalized size of antiderivative = 4.41 \[ \int \frac {\sqrt {c+d x^4}}{x^5 \left (a+b x^4\right )} \, dx=\left [\frac {2 \, \sqrt {b^{2} c - a b d} c x^{4} \log \left (\frac {b d x^{4} + 2 \, b c - a d - 2 \, \sqrt {d x^{4} + c} \sqrt {b^{2} c - a b d}}{b x^{4} + a}\right ) - {\left (2 \, b c - a d\right )} \sqrt {c} x^{4} \log \left (\frac {d x^{4} - 2 \, \sqrt {d x^{4} + c} \sqrt {c} + 2 \, c}{x^{4}}\right ) - 2 \, \sqrt {d x^{4} + c} a c}{8 \, a^{2} c x^{4}}, \frac {4 \, \sqrt {-b^{2} c + a b d} c x^{4} \arctan \left (\frac {\sqrt {d x^{4} + c} \sqrt {-b^{2} c + a b d}}{b d x^{4} + b c}\right ) - {\left (2 \, b c - a d\right )} \sqrt {c} x^{4} \log \left (\frac {d x^{4} - 2 \, \sqrt {d x^{4} + c} \sqrt {c} + 2 \, c}{x^{4}}\right ) - 2 \, \sqrt {d x^{4} + c} a c}{8 \, a^{2} c x^{4}}, -\frac {{\left (2 \, b c - a d\right )} \sqrt {-c} x^{4} \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x^{4} + c}}\right ) - \sqrt {b^{2} c - a b d} c x^{4} \log \left (\frac {b d x^{4} + 2 \, b c - a d - 2 \, \sqrt {d x^{4} + c} \sqrt {b^{2} c - a b d}}{b x^{4} + a}\right ) + \sqrt {d x^{4} + c} a c}{4 \, a^{2} c x^{4}}, \frac {2 \, \sqrt {-b^{2} c + a b d} c x^{4} \arctan \left (\frac {\sqrt {d x^{4} + c} \sqrt {-b^{2} c + a b d}}{b d x^{4} + b c}\right ) - {\left (2 \, b c - a d\right )} \sqrt {-c} x^{4} \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x^{4} + c}}\right ) - \sqrt {d x^{4} + c} a c}{4 \, a^{2} c x^{4}}\right ] \] Input:

integrate((d*x^4+c)^(1/2)/x^5/(b*x^4+a),x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int \frac {\sqrt {c+d x^4}}{x^5 \left (a+b x^4\right )} \, dx=\int \frac {\sqrt {c + d x^{4}}}{x^{5} \left (a + b x^{4}\right )}\, dx \] Input:

integrate((d*x**4+c)**(1/2)/x**5/(b*x**4+a),x)
 

Output:

Integral(sqrt(c + d*x**4)/(x**5*(a + b*x**4)), x)
 

Maxima [F]

\[ \int \frac {\sqrt {c+d x^4}}{x^5 \left (a+b x^4\right )} \, dx=\int { \frac {\sqrt {d x^{4} + c}}{{\left (b x^{4} + a\right )} x^{5}} \,d x } \] Input:

integrate((d*x^4+c)^(1/2)/x^5/(b*x^4+a),x, algorithm="maxima")
 

Output:

integrate(sqrt(d*x^4 + c)/((b*x^4 + a)*x^5), x)
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.93 \[ \int \frac {\sqrt {c+d x^4}}{x^5 \left (a+b x^4\right )} \, dx=\frac {{\left (b^{2} c - a b d\right )} \arctan \left (\frac {\sqrt {d x^{4} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{2 \, \sqrt {-b^{2} c + a b d} a^{2}} - \frac {{\left (2 \, b c - a d\right )} \arctan \left (\frac {\sqrt {d x^{4} + c}}{\sqrt {-c}}\right )}{4 \, a^{2} \sqrt {-c}} - \frac {\sqrt {d x^{4} + c}}{4 \, a x^{4}} \] Input:

integrate((d*x^4+c)^(1/2)/x^5/(b*x^4+a),x, algorithm="giac")
 

Output:

1/2*(b^2*c - a*b*d)*arctan(sqrt(d*x^4 + c)*b/sqrt(-b^2*c + a*b*d))/(sqrt(- 
b^2*c + a*b*d)*a^2) - 1/4*(2*b*c - a*d)*arctan(sqrt(d*x^4 + c)/sqrt(-c))/( 
a^2*sqrt(-c)) - 1/4*sqrt(d*x^4 + c)/(a*x^4)
 

Mupad [B] (verification not implemented)

Time = 4.14 (sec) , antiderivative size = 269, normalized size of antiderivative = 2.34 \[ \int \frac {\sqrt {c+d x^4}}{x^5 \left (a+b x^4\right )} \, dx=\frac {\mathrm {atanh}\left (\frac {b^3\,d^4\,\sqrt {d\,x^4+c}\,\sqrt {b^2\,c-a\,b\,d}}{16\,\left (\frac {a\,b^3\,d^5}{16}-\frac {b^4\,c\,d^4}{16}\right )}\right )\,\sqrt {b^2\,c-a\,b\,d}}{2\,a^2}-\frac {\sqrt {d\,x^4+c}}{4\,a\,x^4}-\frac {\mathrm {atanh}\left (\frac {b^4\,\sqrt {c}\,d^4\,\sqrt {d\,x^4+c}}{16\,\left (\frac {b^4\,c\,d^4}{16}-\frac {3\,a\,b^3\,d^5}{32}+\frac {a^2\,b^2\,d^6}{32\,c}\right )}-\frac {3\,b^3\,d^5\,\sqrt {d\,x^4+c}}{32\,\sqrt {c}\,\left (\frac {a\,b^2\,d^6}{32\,c}-\frac {3\,b^3\,d^5}{32}+\frac {b^4\,c\,d^4}{16\,a}\right )}+\frac {b^2\,d^6\,\sqrt {d\,x^4+c}}{32\,c^{3/2}\,\left (\frac {b^2\,d^6}{32\,c}-\frac {3\,b^3\,d^5}{32\,a}+\frac {b^4\,c\,d^4}{16\,a^2}\right )}\right )\,\left (a\,d-2\,b\,c\right )}{4\,a^2\,\sqrt {c}} \] Input:

int((c + d*x^4)^(1/2)/(x^5*(a + b*x^4)),x)
 

Output:

(atanh((b^3*d^4*(c + d*x^4)^(1/2)*(b^2*c - a*b*d)^(1/2))/(16*((a*b^3*d^5)/ 
16 - (b^4*c*d^4)/16)))*(b^2*c - a*b*d)^(1/2))/(2*a^2) - (c + d*x^4)^(1/2)/ 
(4*a*x^4) - (atanh((b^4*c^(1/2)*d^4*(c + d*x^4)^(1/2))/(16*((b^4*c*d^4)/16 
 - (3*a*b^3*d^5)/32 + (a^2*b^2*d^6)/(32*c))) - (3*b^3*d^5*(c + d*x^4)^(1/2 
))/(32*c^(1/2)*((a*b^2*d^6)/(32*c) - (3*b^3*d^5)/32 + (b^4*c*d^4)/(16*a))) 
 + (b^2*d^6*(c + d*x^4)^(1/2))/(32*c^(3/2)*((b^2*d^6)/(32*c) - (3*b^3*d^5) 
/(32*a) + (b^4*c*d^4)/(16*a^2))))*(a*d - 2*b*c))/(4*a^2*c^(1/2))
 

Reduce [B] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 2930, normalized size of antiderivative = 25.48 \[ \int \frac {\sqrt {c+d x^4}}{x^5 \left (a+b x^4\right )} \, dx =\text {Too large to display} \] Input:

int((d*x^4+c)^(1/2)/x^5/(b*x^4+a),x)
 

Output:

( - 4*sqrt(b)*sqrt(a)*sqrt(c + d*x**4)*sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b 
*c) + 2*a*d - b*c)*sqrt(a*d - b*c)*atan((sqrt(b)*sqrt(c + d*x**4) + sqrt(d 
)*sqrt(b)*x**2)/sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) + 2*a*d - b*c))*d*x 
**6 - 2*sqrt(d)*sqrt(b)*sqrt(a)*sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) + 2 
*a*d - b*c)*sqrt(a*d - b*c)*atan((sqrt(b)*sqrt(c + d*x**4) + sqrt(d)*sqrt( 
b)*x**2)/sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) + 2*a*d - b*c))*c*x**4 - 4 
*sqrt(d)*sqrt(b)*sqrt(a)*sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) + 2*a*d - 
b*c)*sqrt(a*d - b*c)*atan((sqrt(b)*sqrt(c + d*x**4) + sqrt(d)*sqrt(b)*x**2 
)/sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) + 2*a*d - b*c))*d*x**8 + 4*sqrt(d 
)*sqrt(b)*sqrt(c + d*x**4)*sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) + 2*a*d 
- b*c)*atan((sqrt(b)*sqrt(c + d*x**4) + sqrt(d)*sqrt(b)*x**2)/sqrt(2*sqrt( 
d)*sqrt(a)*sqrt(a*d - b*c) + 2*a*d - b*c))*a*d*x**6 - 4*sqrt(d)*sqrt(b)*sq 
rt(c + d*x**4)*sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) + 2*a*d - b*c)*atan( 
(sqrt(b)*sqrt(c + d*x**4) + sqrt(d)*sqrt(b)*x**2)/sqrt(2*sqrt(d)*sqrt(a)*s 
qrt(a*d - b*c) + 2*a*d - b*c))*b*c*x**6 + 2*sqrt(b)*sqrt(2*sqrt(d)*sqrt(a) 
*sqrt(a*d - b*c) + 2*a*d - b*c)*atan((sqrt(b)*sqrt(c + d*x**4) + sqrt(d)*s 
qrt(b)*x**2)/sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) + 2*a*d - b*c))*a*c*d* 
x**4 + 4*sqrt(b)*sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) + 2*a*d - b*c)*ata 
n((sqrt(b)*sqrt(c + d*x**4) + sqrt(d)*sqrt(b)*x**2)/sqrt(2*sqrt(d)*sqrt(a) 
*sqrt(a*d - b*c) + 2*a*d - b*c))*a*d**2*x**8 - 2*sqrt(b)*sqrt(2*sqrt(d)...