\(\int \frac {x^{11}}{(a+b x^4)^2 \sqrt {c+d x^4}} \, dx\) [253]

Optimal result

Integrand size = 24, antiderivative size = 123 \[ \int \frac {x^{11}}{\left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx=\frac {\sqrt {c+d x^4}}{2 b^2 d}-\frac {a^2 \sqrt {c+d x^4}}{4 b^2 (b c-a d) \left (a+b x^4\right )}+\frac {a (4 b c-3 a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^4}}{\sqrt {b c-a d}}\right )}{4 b^{5/2} (b c-a d)^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.06 \[ \int \frac {x^{11}}{\left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx=\frac {\frac {\sqrt {b} \sqrt {c+d x^4} \left (-3 a^2 d+2 b^2 c x^4+2 a b \left (c-d x^4\right )\right )}{d (b c-a d) \left (a+b x^4\right )}+\frac {a (4 b c-3 a d) \arctan \left (\frac {\sqrt {b} \sqrt {c+d x^4}}{\sqrt {-b c+a d}}\right )}{(-b c+a d)^{3/2}}}{4 b^{5/2}} \] Input:

Integrate[x^11/((a + b*x^4)^2*Sqrt[c + d*x^4]),x]
 

Output:

((Sqrt[b]*Sqrt[c + d*x^4]*(-3*a^2*d + 2*b^2*c*x^4 + 2*a*b*(c - d*x^4)))/(d 
*(b*c - a*d)*(a + b*x^4)) + (a*(4*b*c - 3*a*d)*ArcTan[(Sqrt[b]*Sqrt[c + d* 
x^4])/Sqrt[-(b*c) + a*d]])/(-(b*c) + a*d)^(3/2))/(4*b^(5/2))
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.17, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {948, 100, 27, 90, 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.

\(\displaystyle \int \frac {x^{11}}{\left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx\)

\(\Big \downarrow \) 948

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

\(\Big \downarrow \) 100

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 90

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

Int[x^11/((a + b*x^4)^2*Sqrt[c + d*x^4]),x]
 

Output:

(-((a^2*Sqrt[c + d*x^4])/(b^2*(b*c - a*d)*(a + b*x^4))) - ((-4*(b*c - a*d) 
*Sqrt[c + d*x^4])/d - (2*a*(4*b*c - 3*a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^4 
])/Sqrt[b*c - a*d]])/(Sqrt[b]*Sqrt[b*c - a*d]))/(2*b^2*(b*c - a*d)))/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 90

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

rule 100

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( 
p_), x_] :> Simp[(b*c - a*d)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d 
*e - c*f)*(n + 1))), x] - Simp[1/(d^2*(d*e - c*f)*(n + 1))   Int[(c + d*x)^ 
(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*( 
p + 1)) - 2*a*b*d*(d*e*(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x 
, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ[n, -1] || (EqQ[n 
 + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))
 

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 = 0.26 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.08

method	result	size
pseudoelliptic	\(\frac {-\frac {3 d \left (a d -\frac {4 c b}{3}\right ) \left (b \,x^{4}+a \right ) a \arctan \left (\frac {\sqrt {d \,x^{4}+c}\, b}{\sqrt {\left (a d -c b \right ) b}}\right )}{4}+\frac {3 \left (-\frac {2 b^{2} c \,x^{4}}{3}-\frac {2 a \left (-d \,x^{4}+c \right ) b}{3}+a^{2} d \right ) \sqrt {d \,x^{4}+c}\, \sqrt {\left (a d -c b \right ) b}}{4}}{b^{2} \left (a d -c b \right ) d \left (b \,x^{4}+a \right ) \sqrt {\left (a d -c b \right ) b}}\)	\(133\)
risch	\(\frac {\sqrt {d \,x^{4}+c}}{2 b^{2} d}+\frac {a \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 )}{2 b^{3} \sqrt {-\frac {a d -c b}{b}}}+\frac {a \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 )}{2 b^{3} \sqrt {-\frac {a d -c b}{b}}}-\frac {a \sqrt {-a 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}}}{8 b^{3} \left (a d -c b \right ) \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}-\frac {a^{2} d \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 )}{8 b^{3} \left (a d -c b \right ) \sqrt {-\frac {a d -c b}{b}}}+\frac {a \sqrt {-a 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}}}{8 b^{3} \left (a d -c b \right ) \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}-\frac {a^{2} d \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 )}{8 b^{3} \left (a d -c b \right ) \sqrt {-\frac {a d -c b}{b}}}\)	\(876\)
elliptic	\(\frac {\sqrt {d \,x^{4}+c}}{2 b^{2} d}+\frac {a \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 )}{2 b^{3} \sqrt {-\frac {a d -c b}{b}}}+\frac {a \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 )}{2 b^{3} \sqrt {-\frac {a d -c b}{b}}}-\frac {a \sqrt {-a 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}}}{8 b^{3} \left (a d -c b \right ) \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}-\frac {a^{2} d \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 )}{8 b^{3} \left (a d -c b \right ) \sqrt {-\frac {a d -c b}{b}}}+\frac {a \sqrt {-a 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}}}{8 b^{3} \left (a d -c b \right ) \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}-\frac {a^{2} d \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 )}{8 b^{3} \left (a d -c b \right ) \sqrt {-\frac {a d -c b}{b}}}\)	\(876\)
default	\(\frac {\sqrt {d \,x^{4}+c}}{2 b^{2} d}+\frac {a^{2} \left (-\frac {\sqrt {-a 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}}}{8 a b \left (a d -c b \right ) \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}-\frac {d \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 )}{8 b \left (a d -c b \right ) \sqrt {-\frac {a d -c b}{b}}}+\frac {\sqrt {-a 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}}}{8 a b \left (a d -c b \right ) \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}-\frac {d \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 )}{8 b \left (a d -c b \right ) \sqrt {-\frac {a d -c b}{b}}}\right )}{b^{2}}-\frac {2 a \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 )}{b^{2}}\)	\(887\)

Input:

int(x^11/(b*x^4+a)^2/(d*x^4+c)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 230 vs. \(2 (103) = 206\).

Time = 0.12 (sec) , antiderivative size = 475, normalized size of antiderivative = 3.86 \[ \int \frac {x^{11}}{\left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx=\left [\frac {{\left (4 \, a^{2} b c d - 3 \, a^{3} d^{2} + {\left (4 \, a b^{2} c d - 3 \, a^{2} b d^{2}\right )} x^{4}\right )} \sqrt {b^{2} c - a b d} \log \left (\frac {b d x^{4} + 2 \, b c - a d + 2 \, \sqrt {d x^{4} + c} \sqrt {b^{2} c - a b d}}{b x^{4} + a}\right ) + 2 \, {\left (2 \, a b^{3} c^{2} - 5 \, a^{2} b^{2} c d + 3 \, a^{3} b d^{2} + 2 \, {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{4}\right )} \sqrt {d x^{4} + c}}{8 \, {\left (a b^{5} c^{2} d - 2 \, a^{2} b^{4} c d^{2} + a^{3} b^{3} d^{3} + {\left (b^{6} c^{2} d - 2 \, a b^{5} c d^{2} + a^{2} b^{4} d^{3}\right )} x^{4}\right )}}, -\frac {{\left (4 \, a^{2} b c d - 3 \, a^{3} d^{2} + {\left (4 \, a b^{2} c d - 3 \, a^{2} b d^{2}\right )} x^{4}\right )} \sqrt {-b^{2} c + a b d} \arctan \left (\frac {\sqrt {d x^{4} + c} \sqrt {-b^{2} c + a b d}}{b d x^{4} + b c}\right ) - {\left (2 \, a b^{3} c^{2} - 5 \, a^{2} b^{2} c d + 3 \, a^{3} b d^{2} + 2 \, {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{4}\right )} \sqrt {d x^{4} + c}}{4 \, {\left (a b^{5} c^{2} d - 2 \, a^{2} b^{4} c d^{2} + a^{3} b^{3} d^{3} + {\left (b^{6} c^{2} d - 2 \, a b^{5} c d^{2} + a^{2} b^{4} d^{3}\right )} x^{4}\right )}}\right ] \] Input:

integrate(x^11/(b*x^4+a)^2/(d*x^4+c)^(1/2),x, algorithm="fricas")
 

Output:

[1/8*((4*a^2*b*c*d - 3*a^3*d^2 + (4*a*b^2*c*d - 3*a^2*b*d^2)*x^4)*sqrt(b^2 
*c - a*b*d)*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*(2*a*b^3*c^2 - 5*a^2*b^2*c*d + 3*a^3*b*d^2 + 2*(b^4 
*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2)*x^4)*sqrt(d*x^4 + c))/(a*b^5*c^2*d - 2*a 
^2*b^4*c*d^2 + a^3*b^3*d^3 + (b^6*c^2*d - 2*a*b^5*c*d^2 + a^2*b^4*d^3)*x^4 
), -1/4*((4*a^2*b*c*d - 3*a^3*d^2 + (4*a*b^2*c*d - 3*a^2*b*d^2)*x^4)*sqrt( 
-b^2*c + a*b*d)*arctan(sqrt(d*x^4 + c)*sqrt(-b^2*c + a*b*d)/(b*d*x^4 + b*c 
)) - (2*a*b^3*c^2 - 5*a^2*b^2*c*d + 3*a^3*b*d^2 + 2*(b^4*c^2 - 2*a*b^3*c*d 
 + a^2*b^2*d^2)*x^4)*sqrt(d*x^4 + c))/(a*b^5*c^2*d - 2*a^2*b^4*c*d^2 + a^3 
*b^3*d^3 + (b^6*c^2*d - 2*a*b^5*c*d^2 + a^2*b^4*d^3)*x^4)]
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^{11}}{\left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx=\text {Timed out} \] Input:

integrate(x**11/(b*x**4+a)**2/(d*x**4+c)**(1/2),x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^{11}}{\left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(x^11/(b*x^4+a)^2/(d*x^4+c)^(1/2),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(a*d-b*c>0)', see `assume?` for m 
ore detail
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.15 \[ \int \frac {x^{11}}{\left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx=-\frac {\frac {\sqrt {d x^{4} + c} a^{2} d^{3}}{{\left (b^{3} c - a b^{2} d\right )} {\left ({\left (d x^{4} + c\right )} b - b c + a d\right )}} + \frac {{\left (4 \, a b c d^{2} - 3 \, a^{2} d^{3}\right )} \arctan \left (\frac {\sqrt {d x^{4} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{{\left (b^{3} c - a b^{2} d\right )} \sqrt {-b^{2} c + a b d}} - \frac {2 \, \sqrt {d x^{4} + c} d}{b^{2}}}{4 \, d^{2}} \] Input:

integrate(x^11/(b*x^4+a)^2/(d*x^4+c)^(1/2),x, algorithm="giac")
 

Output:

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

Mupad [B] (verification not implemented)

Time = 4.47 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.17 \[ \int \frac {x^{11}}{\left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx=\frac {\sqrt {d\,x^4+c}}{2\,b^2\,d}-\frac {a\,\mathrm {atan}\left (\frac {a\,\sqrt {b}\,\sqrt {d\,x^4+c}\,\left (3\,a\,d-4\,b\,c\right )}{\left (3\,a^2\,d-4\,a\,b\,c\right )\,\sqrt {a\,d-b\,c}}\right )\,\left (3\,a\,d-4\,b\,c\right )}{4\,b^{5/2}\,{\left (a\,d-b\,c\right )}^{3/2}}+\frac {a^2\,d\,\sqrt {d\,x^4+c}}{2\,\left (a\,d-b\,c\right )\,\left (2\,b^3\,\left (d\,x^4+c\right )-2\,b^3\,c+2\,a\,b^2\,d\right )} \] Input:

int(x^11/((a + b*x^4)^2*(c + d*x^4)^(1/2)),x)
 

Output:

(c + d*x^4)^(1/2)/(2*b^2*d) - (a*atan((a*b^(1/2)*(c + d*x^4)^(1/2)*(3*a*d 
- 4*b*c))/((3*a^2*d - 4*a*b*c)*(a*d - b*c)^(1/2)))*(3*a*d - 4*b*c))/(4*b^( 
5/2)*(a*d - b*c)^(3/2)) + (a^2*d*(c + d*x^4)^(1/2))/(2*(a*d - b*c)*(2*b^3* 
(c + d*x^4) - 2*b^3*c + 2*a*b^2*d))
 

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 2072, normalized size of antiderivative = 16.85 \[ \int \frac {x^{11}}{\left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx =\text {Too large to display} \] Input:

int(x^11/(b*x^4+a)^2/(d*x^4+c)^(1/2),x)
 

Output:

( - 3*sqrt(b)*sqrt(c + d*x**4)*sqrt(a*d - b*c)*atan((sqrt(d)*sqrt(b)*sqrt( 
c + d*x**4)*x**2 + sqrt(b)*c + sqrt(b)*d*x**4)/(sqrt(c + d*x**4)*sqrt(a*d 
- b*c) + sqrt(d)*sqrt(a*d - b*c)*x**2))*a**3*c*d**2 - 12*sqrt(b)*sqrt(c + 
d*x**4)*sqrt(a*d - b*c)*atan((sqrt(d)*sqrt(b)*sqrt(c + d*x**4)*x**2 + sqrt 
(b)*c + sqrt(b)*d*x**4)/(sqrt(c + d*x**4)*sqrt(a*d - b*c) + sqrt(d)*sqrt(a 
*d - b*c)*x**2))*a**3*d**3*x**4 + 4*sqrt(b)*sqrt(c + d*x**4)*sqrt(a*d - b* 
c)*atan((sqrt(d)*sqrt(b)*sqrt(c + d*x**4)*x**2 + sqrt(b)*c + sqrt(b)*d*x** 
4)/(sqrt(c + d*x**4)*sqrt(a*d - b*c) + sqrt(d)*sqrt(a*d - b*c)*x**2))*a**2 
*b*c**2*d + 13*sqrt(b)*sqrt(c + d*x**4)*sqrt(a*d - b*c)*atan((sqrt(d)*sqrt 
(b)*sqrt(c + d*x**4)*x**2 + sqrt(b)*c + sqrt(b)*d*x**4)/(sqrt(c + d*x**4)* 
sqrt(a*d - b*c) + sqrt(d)*sqrt(a*d - b*c)*x**2))*a**2*b*c*d**2*x**4 - 12*s 
qrt(b)*sqrt(c + d*x**4)*sqrt(a*d - b*c)*atan((sqrt(d)*sqrt(b)*sqrt(c + d*x 
**4)*x**2 + sqrt(b)*c + sqrt(b)*d*x**4)/(sqrt(c + d*x**4)*sqrt(a*d - b*c) 
+ sqrt(d)*sqrt(a*d - b*c)*x**2))*a**2*b*d**3*x**8 + 4*sqrt(b)*sqrt(c + d*x 
**4)*sqrt(a*d - b*c)*atan((sqrt(d)*sqrt(b)*sqrt(c + d*x**4)*x**2 + sqrt(b) 
*c + sqrt(b)*d*x**4)/(sqrt(c + d*x**4)*sqrt(a*d - b*c) + sqrt(d)*sqrt(a*d 
- b*c)*x**2))*a*b**2*c**2*d*x**4 + 16*sqrt(b)*sqrt(c + d*x**4)*sqrt(a*d - 
b*c)*atan((sqrt(d)*sqrt(b)*sqrt(c + d*x**4)*x**2 + sqrt(b)*c + sqrt(b)*d*x 
**4)/(sqrt(c + d*x**4)*sqrt(a*d - b*c) + sqrt(d)*sqrt(a*d - b*c)*x**2))*a* 
b**2*c*d**2*x**8 - 9*sqrt(d)*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(d)*sqrt...