\(\int \frac {\sqrt {c+d x^4}}{x^3 (a+b x^4)} \, dx\) [226]

Optimal result

Integrand size = 24, antiderivative size = 76 \[ \int \frac {\sqrt {c+d x^4}}{x^3 \left (a+b x^4\right )} \, dx=-\frac {\sqrt {c+d x^4}}{2 a x^2}-\frac {\sqrt {b c-a d} \arctan \left (\frac {\sqrt {b c-a d} x^2}{\sqrt {a} \sqrt {c+d x^4}}\right )}{2 a^{3/2}} \] Output:

-1/2*(d*x^4+c)^(1/2)/a/x^2-1/2*(-a*d+b*c)^(1/2)*arctan((-a*d+b*c)^(1/2)*x^ 
2/a^(1/2)/(d*x^4+c)^(1/2))/a^(3/2)
 

Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.26 \[ \int \frac {\sqrt {c+d x^4}}{x^3 \left (a+b x^4\right )} \, dx=-\frac {\sqrt {c+d x^4}}{2 a x^2}-\frac {\sqrt {b c-a d} \arctan \left (\frac {a \sqrt {d}+b x^2 \left (\sqrt {d} x^2+\sqrt {c+d x^4}\right )}{\sqrt {a} \sqrt {b c-a d}}\right )}{2 a^{3/2}} \] Input:

Integrate[Sqrt[c + d*x^4]/(x^3*(a + b*x^4)),x]
 

Output:

-1/2*Sqrt[c + d*x^4]/(a*x^2) - (Sqrt[b*c - a*d]*ArcTan[(a*Sqrt[d] + b*x^2* 
(Sqrt[d]*x^2 + Sqrt[c + d*x^4]))/(Sqrt[a]*Sqrt[b*c - a*d])])/(2*a^(3/2))
 

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {965, 377, 25, 27, 291, 218}

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.

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

\(\Big \downarrow \) 965

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

\(\Big \downarrow \) 377

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 218

\(\displaystyle \frac {1}{2} \left (-\frac {\sqrt {b c-a d} \arctan \left (\frac {x^2 \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^4}}\right )}{a^{3/2}}-\frac {\sqrt {c+d x^4}}{a x^2}\right )\)

Input:

Int[Sqrt[c + d*x^4]/(x^3*(a + b*x^4)),x]
 

Output:

(-(Sqrt[c + d*x^4]/(a*x^2)) - (Sqrt[b*c - a*d]*ArcTan[(Sqrt[b*c - a*d]*x^2 
)/(Sqrt[a]*Sqrt[c + d*x^4])])/a^(3/2))/2
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 218

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

rule 291

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst 
[Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, 
d}, x] && NeQ[b*c - a*d, 0]
 

rule 377

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_) 
, x_Symbol] :> Simp[(e*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(a*e*( 
m + 1))), x] - Simp[1/(a*e^2*(m + 1))   Int[(e*x)^(m + 2)*(a + b*x^2)^p*(c 
+ d*x^2)^(q - 1)*Simp[b*c*(m + 1) + 2*(b*c*(p + 1) + a*d*q) + d*(b*(m + 1) 
+ 2*b*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b 
*c - a*d, 0] && LtQ[0, q, 1] && LtQ[m, -1] && IntBinomialQ[a, b, c, d, e, m 
, 2, p, q, x]
 

rule 965

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

Maple [A] (verified)

Time = 0.59 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.91

method	result	size
pseudoelliptic	\(\frac {-\frac {\sqrt {d \,x^{4}+c}}{x^{2}}+\frac {\left (a d -c b \right ) \operatorname {arctanh}\left (\frac {a \sqrt {d \,x^{4}+c}}{x^{2} \sqrt {a \left (a d -c b \right )}}\right )}{\sqrt {a \left (a d -c b \right )}}}{2 a}\)	\(69\)
risch	\(-\frac {\sqrt {d \,x^{4}+c}}{2 a \,x^{2}}+\frac {\left (a d -c b \right ) \left (-\frac {\ln \left (\frac {-\frac {2 \left (a d -c b \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -c b}{b}}\, \sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{x^{2}-\frac {\sqrt {-a b}}{b}}\right )}{4 \sqrt {-a b}\, \sqrt {-\frac {a d -c b}{b}}}+\frac {\ln \left (\frac {-\frac {2 \left (a d -c b \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -c b}{b}}\, \sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{x^{2}+\frac {\sqrt {-a b}}{b}}\right )}{4 \sqrt {-a b}\, \sqrt {-\frac {a d -c b}{b}}}\right )}{a}\)	\(352\)
default	\(\frac {-\frac {\left (d \,x^{4}+c \right )^{\frac {3}{2}}}{2 c \,x^{2}}+\frac {d \,x^{2} \sqrt {d \,x^{4}+c}}{2 c}+\frac {\sqrt {d}\, \ln \left (\sqrt {d}\, x^{2}+\sqrt {d \,x^{4}+c}\right )}{2}}{a}-\frac {b \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 \sqrt {-a 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 \sqrt {-a b}}\right )}{a}\)	\(759\)
elliptic	\(\frac {-\frac {\left (d \,x^{4}+c \right )^{\frac {3}{2}}}{c \,x^{2}}+\frac {2 d \left (\frac {x^{2} \sqrt {d \,x^{4}+c}}{2}+\frac {c \ln \left (\sqrt {d}\, x^{2}+\sqrt {d \,x^{4}+c}\right )}{2 \sqrt {d}}\right )}{c}}{2 a}-\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 \sqrt {-a b}\, a}+\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 \sqrt {-a b}\, a}\)	\(765\)

Input:

int((d*x^4+c)^(1/2)/x^3/(b*x^4+a),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 281, normalized size of antiderivative = 3.70 \[ \int \frac {\sqrt {c+d x^4}}{x^3 \left (a+b x^4\right )} \, dx=\left [\frac {x^{2} \sqrt {-\frac {b c - a d}{a}} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{8} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{4} + a^{2} c^{2} - 4 \, {\left ({\left (a b c - 2 \, a^{2} d\right )} x^{6} - a^{2} c x^{2}\right )} \sqrt {d x^{4} + c} \sqrt {-\frac {b c - a d}{a}}}{b^{2} x^{8} + 2 \, a b x^{4} + a^{2}}\right ) - 4 \, \sqrt {d x^{4} + c}}{8 \, a x^{2}}, -\frac {x^{2} \sqrt {\frac {b c - a d}{a}} \arctan \left (\frac {{\left ({\left (b c - 2 \, a d\right )} x^{4} - a c\right )} \sqrt {d x^{4} + c} \sqrt {\frac {b c - a d}{a}}}{2 \, {\left ({\left (b c d - a d^{2}\right )} x^{6} + {\left (b c^{2} - a c d\right )} x^{2}\right )}}\right ) + 2 \, \sqrt {d x^{4} + c}}{4 \, a x^{2}}\right ] \] Input:

integrate((d*x^4+c)^(1/2)/x^3/(b*x^4+a),x, algorithm="fricas")
 

Output:

[1/8*(x^2*sqrt(-(b*c - a*d)/a)*log(((b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*x^8 
- 2*(3*a*b*c^2 - 4*a^2*c*d)*x^4 + a^2*c^2 - 4*((a*b*c - 2*a^2*d)*x^6 - a^2 
*c*x^2)*sqrt(d*x^4 + c)*sqrt(-(b*c - a*d)/a))/(b^2*x^8 + 2*a*b*x^4 + a^2)) 
 - 4*sqrt(d*x^4 + c))/(a*x^2), -1/4*(x^2*sqrt((b*c - a*d)/a)*arctan(1/2*(( 
b*c - 2*a*d)*x^4 - a*c)*sqrt(d*x^4 + c)*sqrt((b*c - a*d)/a)/((b*c*d - a*d^ 
2)*x^6 + (b*c^2 - a*c*d)*x^2)) + 2*sqrt(d*x^4 + c))/(a*x^2)]
 

Sympy [F]

\[ \int \frac {\sqrt {c+d x^4}}{x^3 \left (a+b x^4\right )} \, dx=\int \frac {\sqrt {c + d x^{4}}}{x^{3} \left (a + b x^{4}\right )}\, dx \] Input:

integrate((d*x**4+c)**(1/2)/x**3/(b*x**4+a),x)
 

Output:

Integral(sqrt(c + d*x**4)/(x**3*(a + b*x**4)), x)
 

Maxima [F]

\[ \int \frac {\sqrt {c+d x^4}}{x^3 \left (a+b x^4\right )} \, dx=\int { \frac {\sqrt {d x^{4} + c}}{{\left (b x^{4} + a\right )} x^{3}} \,d x } \] Input:

integrate((d*x^4+c)^(1/2)/x^3/(b*x^4+a),x, algorithm="maxima")
 

Output:

integrate(sqrt(d*x^4 + c)/((b*x^4 + a)*x^3), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 121 vs. \(2 (60) = 120\).

Time = 0.27 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.59 \[ \int \frac {\sqrt {c+d x^4}}{x^3 \left (a+b x^4\right )} \, dx=\frac {{\left (b c \sqrt {d} - a d^{\frac {3}{2}}\right )} \arctan \left (\frac {{\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{2 \, \sqrt {a b c d - a^{2} d^{2}} a} + \frac {c \sqrt {d}}{{\left ({\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{2} - c\right )} a} \] Input:

integrate((d*x^4+c)^(1/2)/x^3/(b*x^4+a),x, algorithm="giac")
 

Output:

1/2*(b*c*sqrt(d) - a*d^(3/2))*arctan(1/2*((sqrt(d)*x^2 - sqrt(d*x^4 + c))^ 
2*b - b*c + 2*a*d)/sqrt(a*b*c*d - a^2*d^2))/(sqrt(a*b*c*d - a^2*d^2)*a) + 
c*sqrt(d)/(((sqrt(d)*x^2 - sqrt(d*x^4 + c))^2 - c)*a)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {c+d x^4}}{x^3 \left (a+b x^4\right )} \, dx=\int \frac {\sqrt {d\,x^4+c}}{x^3\,\left (b\,x^4+a\right )} \,d x \] Input:

int((c + d*x^4)^(1/2)/(x^3*(a + b*x^4)),x)
 

Output:

int((c + d*x^4)^(1/2)/(x^3*(a + b*x^4)), x)
 

Reduce [B] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 455, normalized size of antiderivative = 5.99 \[ \int \frac {\sqrt {c+d x^4}}{x^3 \left (a+b x^4\right )} \, dx=\frac {\sqrt {d}\, \sqrt {a}\, \sqrt {d \,x^{4}+c}\, \sqrt {a d -b c}\, \mathrm {log}\left (-\sqrt {2 \sqrt {d}\, \sqrt {a}\, \sqrt {a d -b c}-2 a d +b c}+\sqrt {b}\, \sqrt {d \,x^{4}+c}+\sqrt {d}\, \sqrt {b}\, x^{2}\right ) x^{2}+\sqrt {d}\, \sqrt {a}\, \sqrt {d \,x^{4}+c}\, \sqrt {a d -b c}\, \mathrm {log}\left (\sqrt {2 \sqrt {d}\, \sqrt {a}\, \sqrt {a d -b c}-2 a d +b c}+\sqrt {b}\, \sqrt {d \,x^{4}+c}+\sqrt {d}\, \sqrt {b}\, x^{2}\right ) x^{2}-\sqrt {d}\, \sqrt {a}\, \sqrt {d \,x^{4}+c}\, \sqrt {a d -b c}\, \mathrm {log}\left (2 \sqrt {d}\, \sqrt {a}\, \sqrt {a d -b c}+2 \sqrt {d}\, \sqrt {d \,x^{4}+c}\, b \,x^{2}+2 a d +2 b d \,x^{4}\right ) x^{2}+\sqrt {a}\, \sqrt {a d -b c}\, \mathrm {log}\left (-\sqrt {2 \sqrt {d}\, \sqrt {a}\, \sqrt {a d -b c}-2 a d +b c}+\sqrt {b}\, \sqrt {d \,x^{4}+c}+\sqrt {d}\, \sqrt {b}\, x^{2}\right ) d \,x^{4}+\sqrt {a}\, \sqrt {a d -b c}\, \mathrm {log}\left (\sqrt {2 \sqrt {d}\, \sqrt {a}\, \sqrt {a d -b c}-2 a d +b c}+\sqrt {b}\, \sqrt {d \,x^{4}+c}+\sqrt {d}\, \sqrt {b}\, x^{2}\right ) d \,x^{4}-\sqrt {a}\, \sqrt {a d -b c}\, \mathrm {log}\left (2 \sqrt {d}\, \sqrt {a}\, \sqrt {a d -b c}+2 \sqrt {d}\, \sqrt {d \,x^{4}+c}\, b \,x^{2}+2 a d +2 b d \,x^{4}\right ) d \,x^{4}-4 \sqrt {d \,x^{4}+c}\, a d \,x^{2}-2 \sqrt {d}\, a c -4 \sqrt {d}\, a d \,x^{4}}{4 a^{2} x^{2} \left (\sqrt {d}\, \sqrt {d \,x^{4}+c}+d \,x^{2}\right )} \] Input:

int((d*x^4+c)^(1/2)/x^3/(b*x^4+a),x)
 

Output:

(sqrt(d)*sqrt(a)*sqrt(c + d*x**4)*sqrt(a*d - b*c)*log( - sqrt(2*sqrt(d)*sq 
rt(a)*sqrt(a*d - b*c) - 2*a*d + b*c) + sqrt(b)*sqrt(c + d*x**4) + sqrt(d)* 
sqrt(b)*x**2)*x**2 + sqrt(d)*sqrt(a)*sqrt(c + d*x**4)*sqrt(a*d - b*c)*log( 
sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) - 2*a*d + b*c) + sqrt(b)*sqrt(c + d 
*x**4) + sqrt(d)*sqrt(b)*x**2)*x**2 - sqrt(d)*sqrt(a)*sqrt(c + d*x**4)*sqr 
t(a*d - b*c)*log(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) + 2*sqrt(d)*sqrt(c + d* 
x**4)*b*x**2 + 2*a*d + 2*b*d*x**4)*x**2 + sqrt(a)*sqrt(a*d - b*c)*log( - s 
qrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) - 2*a*d + b*c) + sqrt(b)*sqrt(c + d* 
x**4) + sqrt(d)*sqrt(b)*x**2)*d*x**4 + sqrt(a)*sqrt(a*d - b*c)*log(sqrt(2* 
sqrt(d)*sqrt(a)*sqrt(a*d - b*c) - 2*a*d + b*c) + sqrt(b)*sqrt(c + d*x**4) 
+ sqrt(d)*sqrt(b)*x**2)*d*x**4 - sqrt(a)*sqrt(a*d - b*c)*log(2*sqrt(d)*sqr 
t(a)*sqrt(a*d - b*c) + 2*sqrt(d)*sqrt(c + d*x**4)*b*x**2 + 2*a*d + 2*b*d*x 
**4)*d*x**4 - 4*sqrt(c + d*x**4)*a*d*x**2 - 2*sqrt(d)*a*c - 4*sqrt(d)*a*d* 
x**4)/(4*a**2*x**2*(sqrt(d)*sqrt(c + d*x**4) + d*x**2))