\(\int \frac {x^5 \sqrt {c+d x^4}}{a+b x^4} \, dx\) [224]

Optimal result

Integrand size = 24, antiderivative size = 120 \[ \int \frac {x^5 \sqrt {c+d x^4}}{a+b x^4} \, dx=\frac {x^2 \sqrt {c+d x^4}}{4 b}-\frac {\sqrt {a} \sqrt {b c-a d} \arctan \left (\frac {\sqrt {b c-a d} x^2}{\sqrt {a} \sqrt {c+d x^4}}\right )}{2 b^2}+\frac {(b c-2 a d) \text {arctanh}\left (\frac {\sqrt {d} x^2}{\sqrt {c+d x^4}}\right )}{4 b^2 \sqrt {d}} \] Output:

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

Mathematica [A] (verified)

Time = 0.86 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.18 \[ \int \frac {x^5 \sqrt {c+d x^4}}{a+b x^4} \, dx=\frac {b \sqrt {d} x^2 \sqrt {c+d x^4}-2 \sqrt {a} \sqrt {d} \sqrt {b c-a d} \arctan \left (\frac {a \sqrt {d}+b x^2 \left (\sqrt {d} x^2+\sqrt {c+d x^4}\right )}{\sqrt {a} \sqrt {b c-a d}}\right )+(b c-2 a d) \log \left (\sqrt {d} x^2+\sqrt {c+d x^4}\right )}{4 b^2 \sqrt {d}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.07, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {965, 380, 398, 224, 219, 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.

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

\(\Big \downarrow \) 965

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

\(\Big \downarrow \) 380

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

\(\Big \downarrow \) 398

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 218

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

Input:

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

Output:

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

Defintions of rubi rules used

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 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 224

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

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 380

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

rule 398

Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]) 
, x_Symbol] :> Simp[f/b   Int[1/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/ 
b   Int[1/((a + b*x^2)*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f} 
, 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 = 0.60 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.85

method	result	size
pseudoelliptic	\(-\frac {-\frac {2 a \left (a d -c b \right ) \operatorname {arctanh}\left (\frac {a \sqrt {d \,x^{4}+c}}{x^{2} \sqrt {a \left (a d -c b \right )}}\right )}{\sqrt {a \left (a d -c b \right )}}-\sqrt {d \,x^{4}+c}\, b \,x^{2}+\frac {\left (2 a d -c b \right ) \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{4}+c}}{x^{2} \sqrt {d}}\right )}{\sqrt {d}}}{4 b^{2}}\)	\(102\)
risch	\(\frac {x^{2} \sqrt {d \,x^{4}+c}}{4 b}-\frac {\frac {\left (2 a d -c b \right ) \ln \left (\sqrt {d}\, x^{2}+\sqrt {d \,x^{4}+c}\right )}{2 b \sqrt {d}}-\frac {2 a \left (a d -c b \right ) \left (-\frac {\ln \left (\frac {-\frac {2 \left (a d -c b \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -c b}{b}}\, \sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{x^{2}-\frac {\sqrt {-a b}}{b}}\right )}{4 \sqrt {-a b}\, \sqrt {-\frac {a d -c b}{b}}}+\frac {\ln \left (\frac {-\frac {2 \left (a d -c b \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -c b}{b}}\, \sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{x^{2}+\frac {\sqrt {-a b}}{b}}\right )}{4 \sqrt {-a b}\, \sqrt {-\frac {a d -c b}{b}}}\right )}{b}}{2 b}\)	\(395\)
default	\(\frac {\frac {x^{2} \sqrt {d \,x^{4}+c}}{4}+\frac {c \ln \left (\sqrt {d}\, x^{2}+\sqrt {d \,x^{4}+c}\right )}{4 \sqrt {d}}}{b}-\frac {a \left (\frac {\sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}+\frac {\sqrt {d}\, \sqrt {-a b}\, \ln \left (\frac {\frac {d \sqrt {-a b}}{b}+\left (x^{2}-\frac {\sqrt {-a b}}{b}\right ) d}{\sqrt {d}}+\sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}\right )}{b}+\frac {\left (a d -c b \right ) \ln \left (\frac {-\frac {2 \left (a d -c b \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -c b}{b}}\, \sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{x^{2}-\frac {\sqrt {-a b}}{b}}\right )}{b \sqrt {-\frac {a d -c b}{b}}}}{4 \sqrt {-a b}}-\frac {\sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}-\frac {\sqrt {d}\, \sqrt {-a b}\, \ln \left (\frac {-\frac {d \sqrt {-a b}}{b}+\left (x^{2}+\frac {\sqrt {-a b}}{b}\right ) d}{\sqrt {d}}+\sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}\right )}{b}+\frac {\left (a d -c b \right ) \ln \left (\frac {-\frac {2 \left (a d -c b \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -c b}{b}}\, \sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{x^{2}+\frac {\sqrt {-a b}}{b}}\right )}{b \sqrt {-\frac {a d -c b}{b}}}}{4 \sqrt {-a b}}\right )}{b}\)	\(739\)
elliptic	\(\frac {\frac {x^{2} \sqrt {d \,x^{4}+c}}{2}+\frac {c \ln \left (\sqrt {d}\, x^{2}+\sqrt {d \,x^{4}+c}\right )}{2 \sqrt {d}}}{2 b}-\frac {a \left (\sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}+\frac {\sqrt {d}\, \sqrt {-a b}\, \ln \left (\frac {\frac {d \sqrt {-a b}}{b}+\left (x^{2}-\frac {\sqrt {-a b}}{b}\right ) d}{\sqrt {d}}+\sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}\right )}{b}+\frac {\left (a d -c b \right ) \ln \left (\frac {-\frac {2 \left (a d -c b \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -c b}{b}}\, \sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{x^{2}-\frac {\sqrt {-a b}}{b}}\right )}{b \sqrt {-\frac {a d -c b}{b}}}\right )}{4 \sqrt {-a b}\, b}+\frac {a \left (\sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}-\frac {\sqrt {d}\, \sqrt {-a b}\, \ln \left (\frac {-\frac {d \sqrt {-a b}}{b}+\left (x^{2}+\frac {\sqrt {-a b}}{b}\right ) d}{\sqrt {d}}+\sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}\right )}{b}+\frac {\left (a d -c b \right ) \ln \left (\frac {-\frac {2 \left (a d -c b \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -c b}{b}}\, \sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{x^{2}+\frac {\sqrt {-a b}}{b}}\right )}{b \sqrt {-\frac {a d -c b}{b}}}\right )}{4 \sqrt {-a b}\, b}\)	\(741\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 720, normalized size of antiderivative = 6.00 \[ \int \frac {x^5 \sqrt {c+d x^4}}{a+b x^4} \, dx=\left [\frac {2 \, \sqrt {d x^{4} + c} b d x^{2} - {\left (b c - 2 \, a d\right )} \sqrt {d} \log \left (-2 \, d x^{4} + 2 \, \sqrt {d x^{4} + c} \sqrt {d} x^{2} - c\right ) + \sqrt {-a b c + a^{2} d} d \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 (b c - 2 \, a d\right )} x^{6} - a c x^{2}\right )} \sqrt {d x^{4} + c} \sqrt {-a b c + a^{2} d}}{b^{2} x^{8} + 2 \, a b x^{4} + a^{2}}\right )}{8 \, b^{2} d}, \frac {2 \, \sqrt {d x^{4} + c} b d x^{2} - 2 \, {\left (b c - 2 \, a d\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {d x^{4} + c} \sqrt {-d}}{d x^{2}}\right ) + \sqrt {-a b c + a^{2} d} d \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 (b c - 2 \, a d\right )} x^{6} - a c x^{2}\right )} \sqrt {d x^{4} + c} \sqrt {-a b c + a^{2} d}}{b^{2} x^{8} + 2 \, a b x^{4} + a^{2}}\right )}{8 \, b^{2} d}, \frac {2 \, \sqrt {d x^{4} + c} b d x^{2} - 2 \, \sqrt {a b c - a^{2} d} d \arctan \left (\frac {{\left ({\left (b c - 2 \, a d\right )} x^{4} - a c\right )} \sqrt {d x^{4} + c} \sqrt {a b c - a^{2} d}}{2 \, {\left ({\left (a b c d - a^{2} d^{2}\right )} x^{6} + {\left (a b c^{2} - a^{2} c d\right )} x^{2}\right )}}\right ) - {\left (b c - 2 \, a d\right )} \sqrt {d} \log \left (-2 \, d x^{4} + 2 \, \sqrt {d x^{4} + c} \sqrt {d} x^{2} - c\right )}{8 \, b^{2} d}, \frac {\sqrt {d x^{4} + c} b d x^{2} - \sqrt {a b c - a^{2} d} d \arctan \left (\frac {{\left ({\left (b c - 2 \, a d\right )} x^{4} - a c\right )} \sqrt {d x^{4} + c} \sqrt {a b c - a^{2} d}}{2 \, {\left ({\left (a b c d - a^{2} d^{2}\right )} x^{6} + {\left (a b c^{2} - a^{2} c d\right )} x^{2}\right )}}\right ) - {\left (b c - 2 \, a d\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {d x^{4} + c} \sqrt {-d}}{d x^{2}}\right )}{4 \, b^{2} d}\right ] \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {x^5 \sqrt {c+d x^4}}{a+b x^4} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E 
rror: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^5 \sqrt {c+d x^4}}{a+b x^4} \, dx=\int \frac {x^5\,\sqrt {d\,x^4+c}}{b\,x^4+a} \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 870, normalized size of antiderivative = 7.25 \[ \int \frac {x^5 \sqrt {c+d x^4}}{a+b x^4} \, dx =\text {Too large to display} \] Input:

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

Output:

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