\(\int \frac {x^{13}}{(a+b x^4)^2 \sqrt {c+d x^4}} \, dx\) [258]

Optimal result

Integrand size = 24, antiderivative size = 191 \[ \int \frac {x^{13}}{\left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx=\frac {(b c-2 a d) x^2 \sqrt {c+d x^4}}{4 b^2 d (b c-a d)}+\frac {a x^6 \sqrt {c+d x^4}}{4 b (b c-a d) \left (a+b x^4\right )}+\frac {a^{3/2} (5 b c-4 a d) \arctan \left (\frac {\sqrt {b c-a d} x^2}{\sqrt {a} \sqrt {c+d x^4}}\right )}{4 b^3 (b c-a d)^{3/2}}-\frac {(b c+4 a d) \text {arctanh}\left (\frac {\sqrt {d} x^2}{\sqrt {c+d x^4}}\right )}{4 b^3 d^{3/2}} \] Output:

1/4*(-2*a*d+b*c)*x^2*(d*x^4+c)^(1/2)/b^2/d/(-a*d+b*c)+1/4*a*x^6*(d*x^4+c)^ 
(1/2)/b/(-a*d+b*c)/(b*x^4+a)+1/4*a^(3/2)*(-4*a*d+5*b*c)*arctan((-a*d+b*c)^ 
(1/2)*x^2/a^(1/2)/(d*x^4+c)^(1/2))/b^3/(-a*d+b*c)^(3/2)-1/4*(4*a*d+b*c)*ar 
ctanh(d^(1/2)*x^2/(d*x^4+c)^(1/2))/b^3/d^(3/2)
 

Mathematica [A] (verified)

Time = 2.03 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.99 \[ \int \frac {x^{13}}{\left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx=\frac {\frac {b x^2 \sqrt {c+d x^4} \left (-2 a^2 d+b^2 c x^4+a b \left (c-d x^4\right )\right )}{d (b c-a d) \left (a+b x^4\right )}+\frac {a^{3/2} (5 b c-4 a d) \arctan \left (\frac {a \sqrt {d}+b x^2 \left (\sqrt {d} x^2+\sqrt {c+d x^4}\right )}{\sqrt {a} \sqrt {b c-a d}}\right )}{(b c-a d)^{3/2}}-\frac {(b c+4 a d) \log \left (\sqrt {d} x^2+\sqrt {c+d x^4}\right )}{d^{3/2}}}{4 b^3} \] Input:

Integrate[x^13/((a + b*x^4)^2*Sqrt[c + d*x^4]),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.73 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.12, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {965, 372, 444, 27, 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.

\(\displaystyle \int \frac {x^{13}}{\left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx\)

\(\Big \downarrow \) 965

\(\displaystyle \frac {1}{2} \int \frac {x^{12}}{\left (b x^4+a\right )^2 \sqrt {d x^4+c}}dx^2\)

\(\Big \downarrow \) 372

\(\displaystyle \frac {1}{2} \left (\frac {a x^6 \sqrt {c+d x^4}}{2 b \left (a+b x^4\right ) (b c-a d)}-\frac {\int \frac {x^4 \left (3 a c-2 (b c-2 a d) x^4\right )}{\left (b x^4+a\right ) \sqrt {d x^4+c}}dx^2}{2 b (b c-a d)}\right )\)

\(\Big \downarrow \) 444

\(\displaystyle \frac {1}{2} \left (\frac {a x^6 \sqrt {c+d x^4}}{2 b \left (a+b x^4\right ) (b c-a d)}-\frac {-\frac {\int -\frac {2 \left ((b c-a d) (b c+4 a d) x^4+a c (b c-2 a d)\right )}{\left (b x^4+a\right ) \sqrt {d x^4+c}}dx^2}{2 b d}-\frac {x^2 \sqrt {c+d x^4} (b c-2 a d)}{b d}}{2 b (b c-a d)}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} \left (\frac {a x^6 \sqrt {c+d x^4}}{2 b \left (a+b x^4\right ) (b c-a d)}-\frac {\frac {\int \frac {(b c-a d) (b c+4 a d) x^4+a c (b c-2 a d)}{\left (b x^4+a\right ) \sqrt {d x^4+c}}dx^2}{b d}-\frac {x^2 \sqrt {c+d x^4} (b c-2 a d)}{b d}}{2 b (b c-a d)}\right )\)

\(\Big \downarrow \) 398

\(\displaystyle \frac {1}{2} \left (\frac {a x^6 \sqrt {c+d x^4}}{2 b \left (a+b x^4\right ) (b c-a d)}-\frac {\frac {\frac {(b c-a d) (4 a d+b c) \int \frac {1}{\sqrt {d x^4+c}}dx^2}{b}-\frac {a^2 d (5 b c-4 a d) \int \frac {1}{\left (b x^4+a\right ) \sqrt {d x^4+c}}dx^2}{b}}{b d}-\frac {x^2 \sqrt {c+d x^4} (b c-2 a d)}{b d}}{2 b (b c-a d)}\right )\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {1}{2} \left (\frac {a x^6 \sqrt {c+d x^4}}{2 b \left (a+b x^4\right ) (b c-a d)}-\frac {\frac {\frac {(b c-a d) (4 a d+b c) \int \frac {1}{1-d x^4}d\frac {x^2}{\sqrt {d x^4+c}}}{b}-\frac {a^2 d (5 b c-4 a d) \int \frac {1}{\left (b x^4+a\right ) \sqrt {d x^4+c}}dx^2}{b}}{b d}-\frac {x^2 \sqrt {c+d x^4} (b c-2 a d)}{b d}}{2 b (b c-a d)}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{2} \left (\frac {a x^6 \sqrt {c+d x^4}}{2 b \left (a+b x^4\right ) (b c-a d)}-\frac {\frac {\frac {(b c-a d) (4 a d+b c) \text {arctanh}\left (\frac {\sqrt {d} x^2}{\sqrt {c+d x^4}}\right )}{b \sqrt {d}}-\frac {a^2 d (5 b c-4 a d) \int \frac {1}{\left (b x^4+a\right ) \sqrt {d x^4+c}}dx^2}{b}}{b d}-\frac {x^2 \sqrt {c+d x^4} (b c-2 a d)}{b d}}{2 b (b c-a d)}\right )\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {1}{2} \left (\frac {a x^6 \sqrt {c+d x^4}}{2 b \left (a+b x^4\right ) (b c-a d)}-\frac {\frac {\frac {(b c-a d) (4 a d+b c) \text {arctanh}\left (\frac {\sqrt {d} x^2}{\sqrt {c+d x^4}}\right )}{b \sqrt {d}}-\frac {a^2 d (5 b c-4 a d) \int \frac {1}{a-(a d-b c) x^4}d\frac {x^2}{\sqrt {d x^4+c}}}{b}}{b d}-\frac {x^2 \sqrt {c+d x^4} (b c-2 a d)}{b d}}{2 b (b c-a d)}\right )\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {1}{2} \left (\frac {a x^6 \sqrt {c+d x^4}}{2 b \left (a+b x^4\right ) (b c-a d)}-\frac {\frac {\frac {(b c-a d) (4 a d+b c) \text {arctanh}\left (\frac {\sqrt {d} x^2}{\sqrt {c+d x^4}}\right )}{b \sqrt {d}}-\frac {a^{3/2} d (5 b c-4 a d) \arctan \left (\frac {x^2 \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^4}}\right )}{b \sqrt {b c-a d}}}{b d}-\frac {x^2 \sqrt {c+d x^4} (b c-2 a d)}{b d}}{2 b (b c-a d)}\right )\)

Input:

Int[x^13/((a + b*x^4)^2*Sqrt[c + d*x^4]),x]
 

Output:

((a*x^6*Sqrt[c + d*x^4])/(2*b*(b*c - a*d)*(a + b*x^4)) - (-(((b*c - 2*a*d) 
*x^2*Sqrt[c + d*x^4])/(b*d)) + (-((a^(3/2)*d*(5*b*c - 4*a*d)*ArcTan[(Sqrt[ 
b*c - a*d]*x^2)/(Sqrt[a]*Sqrt[c + d*x^4])])/(b*Sqrt[b*c - a*d])) + ((b*c - 
 a*d)*(b*c + 4*a*d)*ArcTanh[(Sqrt[d]*x^2)/Sqrt[c + d*x^4]])/(b*Sqrt[d]))/( 
b*d))/(2*b*(b*c - a*d)))/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 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 372

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ 
), x_Symbol] :> Simp[(-a)*e^3*(e*x)^(m - 3)*(a + b*x^2)^(p + 1)*((c + d*x^2 
)^(q + 1)/(2*b*(b*c - a*d)*(p + 1))), x] + Simp[e^4/(2*b*(b*c - a*d)*(p + 1 
))   Int[(e*x)^(m - 4)*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[a*c*(m - 3) + 
 (a*d*(m + 2*q - 1) + 2*b*c*(p + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, 
e, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[m, 3] && 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 444

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

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 = 1.74 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.75

method	result	size
pseudoelliptic	\(-\frac {\frac {a^{2} \left (-\frac {b \sqrt {d \,x^{4}+c}\, x^{2}}{b \,x^{4}+a}-\frac {\left (4 a d -5 c b \right ) \operatorname {arctanh}\left (\frac {a \sqrt {d \,x^{4}+c}}{x^{2} \sqrt {a \left (a d -c b \right )}}\right )}{\sqrt {a \left (a d -c b \right )}}\right )}{a d -c b}-\frac {\sqrt {d \,x^{4}+c}\, b \,x^{2}}{d}+\frac {\left (4 a d +c b \right ) \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{4}+c}}{x^{2} \sqrt {d}}\right )}{d^{\frac {3}{2}}}}{4 b^{3}}\)	\(143\)
risch	\(\frac {x^{2} \sqrt {d \,x^{4}+c}}{4 b^{2} d}-\frac {a \ln \left (\sqrt {d}\, x^{2}+\sqrt {d \,x^{4}+c}\right )}{b^{3} \sqrt {d}}-\frac {\ln \left (\sqrt {d}\, x^{2}+\sqrt {d \,x^{4}+c}\right ) c}{4 b^{2} d^{\frac {3}{2}}}-\frac {5 a^{2} \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} \sqrt {-a b}\, \sqrt {-\frac {a d -c b}{b}}}+\frac {5 a^{2} \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} \sqrt {-a b}\, \sqrt {-\frac {a d -c b}{b}}}+\frac {a^{2} \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 {d \,a^{2} \sqrt {-a b}\, \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^{4} \left (a d -c b \right ) \sqrt {-\frac {a d -c b}{b}}}+\frac {a^{2} \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 {d \,a^{2} \sqrt {-a b}\, \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^{4} \left (a d -c b \right ) \sqrt {-\frac {a d -c b}{b}}}\)	\(953\)
elliptic	\(\frac {x^{2} \sqrt {d \,x^{4}+c}}{4 b^{2} d}-\frac {a \ln \left (\sqrt {d}\, x^{2}+\sqrt {d \,x^{4}+c}\right )}{b^{3} \sqrt {d}}-\frac {\ln \left (\sqrt {d}\, x^{2}+\sqrt {d \,x^{4}+c}\right ) c}{4 b^{2} d^{\frac {3}{2}}}-\frac {5 a^{2} \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} \sqrt {-a b}\, \sqrt {-\frac {a d -c b}{b}}}+\frac {5 a^{2} \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} \sqrt {-a b}\, \sqrt {-\frac {a d -c b}{b}}}+\frac {a^{2} \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 {d \,a^{2} \sqrt {-a b}\, \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^{4} \left (a d -c b \right ) \sqrt {-\frac {a d -c b}{b}}}+\frac {a^{2} \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 {d \,a^{2} \sqrt {-a b}\, \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^{4} \left (a d -c b \right ) \sqrt {-\frac {a d -c b}{b}}}\)	\(953\)
default	\(\text {Expression too large to display}\)	\(1278\)

Input:

int(x^13/(b*x^4+a)^2/(d*x^4+c)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.56 (sec) , antiderivative size = 1391, normalized size of antiderivative = 7.28 \[ \int \frac {x^{13}}{\left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx=\text {Too large to display} \] Input:

integrate(x^13/(b*x^4+a)^2/(d*x^4+c)^(1/2),x, algorithm="fricas")
 

Output:

[1/16*(2*(a*b^2*c^2 + 3*a^2*b*c*d - 4*a^3*d^2 + (b^3*c^2 + 3*a*b^2*c*d - 4 
*a^2*b*d^2)*x^4)*sqrt(d)*log(-2*d*x^4 + 2*sqrt(d*x^4 + c)*sqrt(d)*x^2 - c) 
 + (5*a^2*b*c*d^2 - 4*a^3*d^3 + (5*a*b^2*c*d^2 - 4*a^2*b*d^3)*x^4)*sqrt(-a 
/(b*c - a*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^2*c^2 - 3*a*b*c*d + 2*a^2*d^2)*x^6 - (a*b 
*c^2 - a^2*c*d)*x^2)*sqrt(d*x^4 + c)*sqrt(-a/(b*c - a*d)))/(b^2*x^8 + 2*a* 
b*x^4 + a^2)) + 4*((b^3*c*d - a*b^2*d^2)*x^6 + (a*b^2*c*d - 2*a^2*b*d^2)*x 
^2)*sqrt(d*x^4 + c))/(a*b^4*c*d^2 - a^2*b^3*d^3 + (b^5*c*d^2 - a*b^4*d^3)* 
x^4), 1/16*(4*(a*b^2*c^2 + 3*a^2*b*c*d - 4*a^3*d^2 + (b^3*c^2 + 3*a*b^2*c* 
d - 4*a^2*b*d^2)*x^4)*sqrt(-d)*arctan(sqrt(d*x^4 + c)*sqrt(-d)/(d*x^2)) + 
(5*a^2*b*c*d^2 - 4*a^3*d^3 + (5*a*b^2*c*d^2 - 4*a^2*b*d^3)*x^4)*sqrt(-a/(b 
*c - a*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^2*c^2 - 3*a*b*c*d + 2*a^2*d^2)*x^6 - (a*b*c^ 
2 - a^2*c*d)*x^2)*sqrt(d*x^4 + c)*sqrt(-a/(b*c - a*d)))/(b^2*x^8 + 2*a*b*x 
^4 + a^2)) + 4*((b^3*c*d - a*b^2*d^2)*x^6 + (a*b^2*c*d - 2*a^2*b*d^2)*x^2) 
*sqrt(d*x^4 + c))/(a*b^4*c*d^2 - a^2*b^3*d^3 + (b^5*c*d^2 - a*b^4*d^3)*x^4 
), -1/8*((5*a^2*b*c*d^2 - 4*a^3*d^3 + (5*a*b^2*c*d^2 - 4*a^2*b*d^3)*x^4)*s 
qrt(a/(b*c - a*d))*arctan(-1/2*((b*c - 2*a*d)*x^4 - a*c)*sqrt(d*x^4 + c)*s 
qrt(a/(b*c - a*d))/(a*d*x^6 + a*c*x^2)) - (a*b^2*c^2 + 3*a^2*b*c*d - 4*a^3 
*d^2 + (b^3*c^2 + 3*a*b^2*c*d - 4*a^2*b*d^2)*x^4)*sqrt(d)*log(-2*d*x^4 ...
 

Sympy [F]

\[ \int \frac {x^{13}}{\left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx=\int \frac {x^{13}}{\left (a + b x^{4}\right )^{2} \sqrt {c + d x^{4}}}\, dx \] Input:

integrate(x**13/(b*x**4+a)**2/(d*x**4+c)**(1/2),x)
 

Output:

Integral(x**13/((a + b*x**4)**2*sqrt(c + d*x**4)), x)
 

Maxima [F]

\[ \int \frac {x^{13}}{\left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx=\int { \frac {x^{13}}{{\left (b x^{4} + a\right )}^{2} \sqrt {d x^{4} + c}} \,d x } \] Input:

integrate(x^13/(b*x^4+a)^2/(d*x^4+c)^(1/2),x, algorithm="maxima")
 

Output:

integrate(x^13/((b*x^4 + a)^2*sqrt(d*x^4 + c)), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 333 vs. \(2 (163) = 326\).

Time = 0.22 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.74 \[ \int \frac {x^{13}}{\left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx=-\frac {{\left (5 \, a^{2} b c \sqrt {d} - 4 \, a^{3} d^{\frac {3}{2}}\right )} \arctan \left (\frac {{\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{4 \, {\left (b^{4} c - a b^{3} d\right )} \sqrt {a b c d - a^{2} d^{2}}} + \frac {\sqrt {d x^{4} + c} x^{2}}{4 \, b^{2} d} + \frac {{\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{2} a^{2} b c d - 2 \, {\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{2} a^{3} d^{2} - a^{2} b c^{2} d}{2 \, {\left (b^{4} c \sqrt {d} - a b^{3} d^{\frac {3}{2}}\right )} {\left ({\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{4} b - 2 \, {\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{2} b c + 4 \, {\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{2} a d + b c^{2}\right )}} + \frac {{\left (b c + 4 \, a d\right )} \log \left ({\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{2}\right )}{8 \, b^{3} d^{\frac {3}{2}}} \] Input:

integrate(x^13/(b*x^4+a)^2/(d*x^4+c)^(1/2),x, algorithm="giac")
 

Output:

-1/4*(5*a^2*b*c*sqrt(d) - 4*a^3*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))/((b^4*c - a*b^3*d)* 
sqrt(a*b*c*d - a^2*d^2)) + 1/4*sqrt(d*x^4 + c)*x^2/(b^2*d) + 1/2*((sqrt(d) 
*x^2 - sqrt(d*x^4 + c))^2*a^2*b*c*d - 2*(sqrt(d)*x^2 - sqrt(d*x^4 + c))^2* 
a^3*d^2 - a^2*b*c^2*d)/((b^4*c*sqrt(d) - a*b^3*d^(3/2))*((sqrt(d)*x^2 - sq 
rt(d*x^4 + c))^4*b - 2*(sqrt(d)*x^2 - sqrt(d*x^4 + c))^2*b*c + 4*(sqrt(d)* 
x^2 - sqrt(d*x^4 + c))^2*a*d + b*c^2)) + 1/8*(b*c + 4*a*d)*log((sqrt(d)*x^ 
2 - sqrt(d*x^4 + c))^2)/(b^3*d^(3/2))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^{13}}{\left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx=\int \frac {x^{13}}{{\left (b\,x^4+a\right )}^2\,\sqrt {d\,x^4+c}} \,d x \] Input:

int(x^13/((a + b*x^4)^2*(c + d*x^4)^(1/2)),x)
 

Output:

int(x^13/((a + b*x^4)^2*(c + d*x^4)^(1/2)), x)
 

Reduce [B] (verification not implemented)

Time = 0.56 (sec) , antiderivative size = 5868, normalized size of antiderivative = 30.72 \[ \int \frac {x^{13}}{\left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx =\text {Too large to display} \] Input:

int(x^13/(b*x^4+a)^2/(d*x^4+c)^(1/2),x)
 

Output:

(16*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)*a**3*c*d**3*x**2 + 32*sqrt(d)*sqrt(a)*sqrt(c + d*x**4)*sq 
rt(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)*a**3*d**4*x**6 - 20*sq 
rt(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)*sqr 
t(b)*x**2)*a**2*b*c**2*d**2*x**2 - 24*sqrt(d)*sqrt(a)*sqrt(c + d*x**4)*sqr 
t(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)*a**2*b*c*d**3*x**6 + 32 
*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)*a**2*b*d**4*x**10 - 20*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)*a*b**2*c**2*d**2*x**6 - 
40*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)*a*b**2*c*d**3*x**10 + 16*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)*a**3*c*d**3*x**2 + 3...