\(\int \frac {x^{15}}{(a+b x^4)^2 \sqrt {c+d x^4}} \, dx\) [252]

Optimal result

Integrand size = 24, antiderivative size = 154 \[ \int \frac {x^{15}}{\left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx=-\frac {(b c+2 a d) \sqrt {c+d x^4}}{2 b^3 d^2}+\frac {a^3 \sqrt {c+d x^4}}{4 b^3 (b c-a d) \left (a+b x^4\right )}+\frac {\left (c+d x^4\right )^{3/2}}{6 b^2 d^2}-\frac {a^2 (6 b c-5 a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^4}}{\sqrt {b c-a d}}\right )}{4 b^{7/2} (b c-a d)^{3/2}} \] Output:

-1/2*(2*a*d+b*c)*(d*x^4+c)^(1/2)/b^3/d^2+1/4*a^3*(d*x^4+c)^(1/2)/b^3/(-a*d 
+b*c)/(b*x^4+a)+1/6*(d*x^4+c)^(3/2)/b^2/d^2-1/4*a^2*(-5*a*d+6*b*c)*arctanh 
(b^(1/2)*(d*x^4+c)^(1/2)/(-a*d+b*c)^(1/2))/b^(7/2)/(-a*d+b*c)^(3/2)
 

Mathematica [A] (verified)

Time = 0.47 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.14 \[ \int \frac {x^{15}}{\left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx=-\frac {\sqrt {c+d x^4} \left (-15 a^3 d^2+2 a^2 b d \left (4 c-5 d x^4\right )+2 b^3 c x^4 \left (2 c-d x^4\right )+2 a b^2 \left (2 c^2+3 c d x^4+d^2 x^8\right )\right )}{12 b^3 d^2 (b c-a d) \left (a+b x^4\right )}+\frac {a^2 (-6 b c+5 a d) \arctan \left (\frac {\sqrt {b} \sqrt {c+d x^4}}{\sqrt {-b c+a d}}\right )}{4 b^{7/2} (-b c+a d)^{3/2}} \] Input:

Integrate[x^15/((a + b*x^4)^2*Sqrt[c + d*x^4]),x]
 

Output:

-1/12*(Sqrt[c + d*x^4]*(-15*a^3*d^2 + 2*a^2*b*d*(4*c - 5*d*x^4) + 2*b^3*c* 
x^4*(2*c - d*x^4) + 2*a*b^2*(2*c^2 + 3*c*d*x^4 + d^2*x^8)))/(b^3*d^2*(b*c 
- a*d)*(a + b*x^4)) + (a^2*(-6*b*c + 5*a*d)*ArcTan[(Sqrt[b]*Sqrt[c + d*x^4 
])/Sqrt[-(b*c) + a*d]])/(4*b^(7/2)*(-(b*c) + a*d)^(3/2))
 

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.18, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {948, 109, 27, 164, 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.

\(\displaystyle \int \frac {x^{15}}{\left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx\)

\(\Big \downarrow \) 948

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

\(\Big \downarrow \) 109

\(\displaystyle \frac {1}{4} \left (\frac {a x^8 \sqrt {c+d x^4}}{b \left (a+b x^4\right ) (b c-a d)}-\frac {\int \frac {x^4 \left (4 a c-(2 b c-5 a d) x^4\right )}{2 \left (b x^4+a\right ) \sqrt {d x^4+c}}dx^4}{b (b c-a d)}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{4} \left (\frac {a x^8 \sqrt {c+d x^4}}{b \left (a+b x^4\right ) (b c-a d)}-\frac {\int \frac {x^4 \left (4 a c-(2 b c-5 a d) x^4\right )}{\left (b x^4+a\right ) \sqrt {d x^4+c}}dx^4}{2 b (b c-a d)}\right )\)

\(\Big \downarrow \) 164

\(\displaystyle \frac {1}{4} \left (\frac {a x^8 \sqrt {c+d x^4}}{b \left (a+b x^4\right ) (b c-a d)}-\frac {\frac {2 \sqrt {c+d x^4} \left (-15 a^2 d^2-b d x^4 (2 b c-5 a d)+8 a b c d+4 b^2 c^2\right )}{3 b^2 d^2}-\frac {a^2 (6 b c-5 a d) \int \frac {1}{\left (b x^4+a\right ) \sqrt {d x^4+c}}dx^4}{b^2}}{2 b (b c-a d)}\right )\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {1}{4} \left (\frac {a x^8 \sqrt {c+d x^4}}{b \left (a+b x^4\right ) (b c-a d)}-\frac {\frac {2 \sqrt {c+d x^4} \left (-15 a^2 d^2-b d x^4 (2 b c-5 a d)+8 a b c d+4 b^2 c^2\right )}{3 b^2 d^2}-\frac {2 a^2 (6 b c-5 a d) \int \frac {1}{\frac {b x^8}{d}+a-\frac {b c}{d}}d\sqrt {d x^4+c}}{b^2 d}}{2 b (b c-a d)}\right )\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {1}{4} \left (\frac {a x^8 \sqrt {c+d x^4}}{b \left (a+b x^4\right ) (b c-a d)}-\frac {\frac {2 a^2 (6 b c-5 a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^4}}{\sqrt {b c-a d}}\right )}{b^{5/2} \sqrt {b c-a d}}+\frac {2 \sqrt {c+d x^4} \left (-15 a^2 d^2-b d x^4 (2 b c-5 a d)+8 a b c d+4 b^2 c^2\right )}{3 b^2 d^2}}{2 b (b c-a d)}\right )\)

Input:

Int[x^15/((a + b*x^4)^2*Sqrt[c + d*x^4]),x]
 

Output:

((a*x^8*Sqrt[c + d*x^4])/(b*(b*c - a*d)*(a + b*x^4)) - ((2*Sqrt[c + d*x^4] 
*(4*b^2*c^2 + 8*a*b*c*d - 15*a^2*d^2 - b*d*(2*b*c - 5*a*d)*x^4))/(3*b^2*d^ 
2) + (2*a^2*(6*b*c - 5*a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^4])/Sqrt[b*c - a 
*d]])/(b^(5/2)*Sqrt[b*c - a*d]))/(2*b*(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 109

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

rule 164

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_ 
))*((g_.) + (h_.)*(x_)), x_] :> Simp[(-(a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - 
 b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x))*(a + b*x)^(m + 1)*(( 
c + d*x)^(n + 1)/(b^2*d^2*(m + n + 2)*(m + n + 3))), x] + Simp[(a^2*d^2*f*h 
*(n + 1)*(n + 2) + a*b*d*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 
3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + 
d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3))   Int[( 
a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] 
&& NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]
 

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 = 0.37 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.11

method	result	size
pseudoelliptic	\(-\frac {5 \left (-\left (a d -\frac {6 c b}{5}\right ) d^{2} \left (b \,x^{4}+a \right ) a^{2} \arctan \left (\frac {\sqrt {d \,x^{4}+c}\, b}{\sqrt {\left (a d -c b \right ) b}}\right )+\left (-\frac {4 \left (-\frac {d \,x^{4}}{2}+c \right ) x^{4} c \,b^{3}}{15}-\frac {4 a \left (\frac {d \,x^{4}}{2}+c \right ) \left (d \,x^{4}+c \right ) b^{2}}{15}-\frac {8 d \left (-\frac {5 d \,x^{4}}{4}+c \right ) a^{2} b}{15}+a^{3} d^{2}\right ) \sqrt {d \,x^{4}+c}\, \sqrt {\left (a d -c b \right ) b}\right )}{4 \sqrt {\left (a d -c b \right ) b}\, d^{2} b^{3} \left (a d -c b \right ) \left (b \,x^{4}+a \right )}\)	\(171\)
risch	\(-\frac {\left (-d b \,x^{4}+6 a d +2 c b \right ) \sqrt {d \,x^{4}+c}}{6 d^{2} b^{3}}-\frac {3 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 )}{4 b^{4} \sqrt {-\frac {a d -c b}{b}}}-\frac {3 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 )}{4 b^{4} \sqrt {-\frac {a d -c b}{b}}}+\frac {a^{2} \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^{4} \left (a d -c b \right ) \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}+\frac {a^{3} 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^{4} \left (a d -c b \right ) \sqrt {-\frac {a d -c b}{b}}}-\frac {a^{2} \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^{4} \left (a d -c b \right ) \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}+\frac {a^{3} 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^{4} \left (a d -c b \right ) \sqrt {-\frac {a d -c b}{b}}}\)	\(900\)
default	\(-\frac {\sqrt {d \,x^{4}+c}\, \left (-d \,x^{4}+2 c \right )}{6 b^{2} d^{2}}-\frac {a \sqrt {d \,x^{4}+c}}{b^{3} d}+\frac {3 a^{2} \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^{3}}-\frac {a^{3} \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^{3}}\)	\(918\)
elliptic	\(\frac {x^{4} \sqrt {d \,x^{4}+c}}{6 b^{2} d}-\frac {c \sqrt {d \,x^{4}+c}}{3 b^{2} d^{2}}-\frac {a \sqrt {d \,x^{4}+c}}{b^{3} d}-\frac {3 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 )}{4 b^{4} \sqrt {-\frac {a d -c b}{b}}}-\frac {3 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 )}{4 b^{4} \sqrt {-\frac {a d -c b}{b}}}+\frac {a^{2} \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^{4} \left (a d -c b \right ) \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}+\frac {a^{3} 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^{4} \left (a d -c b \right ) \sqrt {-\frac {a d -c b}{b}}}-\frac {a^{2} \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^{4} \left (a d -c b \right ) \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}+\frac {a^{3} 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^{4} \left (a d -c b \right ) \sqrt {-\frac {a d -c b}{b}}}\)	\(923\)

Input:

int(x^15/(b*x^4+a)^2/(d*x^4+c)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-5/4*(-(a*d-6/5*c*b)*d^2*(b*x^4+a)*a^2*arctan((d*x^4+c)^(1/2)*b/((a*d-b*c) 
*b)^(1/2))+(-4/15*(-1/2*d*x^4+c)*x^4*c*b^3-4/15*a*(1/2*d*x^4+c)*(d*x^4+c)* 
b^2-8/15*d*(-5/4*d*x^4+c)*a^2*b+a^3*d^2)*(d*x^4+c)^(1/2)*((a*d-b*c)*b)^(1/ 
2))/((a*d-b*c)*b)^(1/2)/d^2/b^3/(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. 303 vs. \(2 (130) = 260\).

Time = 0.12 (sec) , antiderivative size = 622, normalized size of antiderivative = 4.04 \[ \int \frac {x^{15}}{\left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx=\left [\frac {3 \, {\left (6 \, a^{3} b c d^{2} - 5 \, a^{4} d^{3} + {\left (6 \, a^{2} b^{2} c d^{2} - 5 \, a^{3} b d^{3}\right )} x^{4}\right )} \sqrt {b^{2} c - a b d} \log \left (\frac {b d x^{4} + 2 \, b c - a d - 2 \, \sqrt {d x^{4} + c} \sqrt {b^{2} c - a b d}}{b x^{4} + a}\right ) + 2 \, {\left (2 \, {\left (b^{5} c^{2} d - 2 \, a b^{4} c d^{2} + a^{2} b^{3} d^{3}\right )} x^{8} - 4 \, a b^{4} c^{3} - 4 \, a^{2} b^{3} c^{2} d + 23 \, a^{3} b^{2} c d^{2} - 15 \, a^{4} b d^{3} - 2 \, {\left (2 \, b^{5} c^{3} + a b^{4} c^{2} d - 8 \, a^{2} b^{3} c d^{2} + 5 \, a^{3} b^{2} d^{3}\right )} x^{4}\right )} \sqrt {d x^{4} + c}}{24 \, {\left (a b^{6} c^{2} d^{2} - 2 \, a^{2} b^{5} c d^{3} + a^{3} b^{4} d^{4} + {\left (b^{7} c^{2} d^{2} - 2 \, a b^{6} c d^{3} + a^{2} b^{5} d^{4}\right )} x^{4}\right )}}, \frac {3 \, {\left (6 \, a^{3} b c d^{2} - 5 \, a^{4} d^{3} + {\left (6 \, a^{2} b^{2} c d^{2} - 5 \, a^{3} b d^{3}\right )} x^{4}\right )} \sqrt {-b^{2} c + a b d} \arctan \left (\frac {\sqrt {d x^{4} + c} \sqrt {-b^{2} c + a b d}}{b d x^{4} + b c}\right ) + {\left (2 \, {\left (b^{5} c^{2} d - 2 \, a b^{4} c d^{2} + a^{2} b^{3} d^{3}\right )} x^{8} - 4 \, a b^{4} c^{3} - 4 \, a^{2} b^{3} c^{2} d + 23 \, a^{3} b^{2} c d^{2} - 15 \, a^{4} b d^{3} - 2 \, {\left (2 \, b^{5} c^{3} + a b^{4} c^{2} d - 8 \, a^{2} b^{3} c d^{2} + 5 \, a^{3} b^{2} d^{3}\right )} x^{4}\right )} \sqrt {d x^{4} + c}}{12 \, {\left (a b^{6} c^{2} d^{2} - 2 \, a^{2} b^{5} c d^{3} + a^{3} b^{4} d^{4} + {\left (b^{7} c^{2} d^{2} - 2 \, a b^{6} c d^{3} + a^{2} b^{5} d^{4}\right )} x^{4}\right )}}\right ] \] Input:

integrate(x^15/(b*x^4+a)^2/(d*x^4+c)^(1/2),x, algorithm="fricas")
 

Output:

[1/24*(3*(6*a^3*b*c*d^2 - 5*a^4*d^3 + (6*a^2*b^2*c*d^2 - 5*a^3*b*d^3)*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*(b^5*c^2*d - 2*a*b^4*c*d^2 + a^2*b^3*d^ 
3)*x^8 - 4*a*b^4*c^3 - 4*a^2*b^3*c^2*d + 23*a^3*b^2*c*d^2 - 15*a^4*b*d^3 - 
 2*(2*b^5*c^3 + a*b^4*c^2*d - 8*a^2*b^3*c*d^2 + 5*a^3*b^2*d^3)*x^4)*sqrt(d 
*x^4 + c))/(a*b^6*c^2*d^2 - 2*a^2*b^5*c*d^3 + a^3*b^4*d^4 + (b^7*c^2*d^2 - 
 2*a*b^6*c*d^3 + a^2*b^5*d^4)*x^4), 1/12*(3*(6*a^3*b*c*d^2 - 5*a^4*d^3 + ( 
6*a^2*b^2*c*d^2 - 5*a^3*b*d^3)*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*(b^5*c^2*d - 2*a*b^4*c*d^ 
2 + a^2*b^3*d^3)*x^8 - 4*a*b^4*c^3 - 4*a^2*b^3*c^2*d + 23*a^3*b^2*c*d^2 - 
15*a^4*b*d^3 - 2*(2*b^5*c^3 + a*b^4*c^2*d - 8*a^2*b^3*c*d^2 + 5*a^3*b^2*d^ 
3)*x^4)*sqrt(d*x^4 + c))/(a*b^6*c^2*d^2 - 2*a^2*b^5*c*d^3 + a^3*b^4*d^4 + 
(b^7*c^2*d^2 - 2*a*b^6*c*d^3 + a^2*b^5*d^4)*x^4)]
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^{15}}{\left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx=\text {Timed out} \] Input:

integrate(x**15/(b*x**4+a)**2/(d*x**4+c)**(1/2),x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^{15}}{\left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(x^15/(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 = 0.13 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.17 \[ \int \frac {x^{15}}{\left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx=\frac {\sqrt {d x^{4} + c} a^{3} d}{4 \, {\left (b^{4} c - a b^{3} d\right )} {\left ({\left (d x^{4} + c\right )} b - b c + a d\right )}} + \frac {{\left (6 \, a^{2} b c - 5 \, a^{3} d\right )} \arctan \left (\frac {\sqrt {d x^{4} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{4 \, {\left (b^{4} c - a b^{3} d\right )} \sqrt {-b^{2} c + a b d}} + \frac {{\left (d x^{4} + c\right )}^{\frac {3}{2}} b^{4} d^{4} - 3 \, \sqrt {d x^{4} + c} b^{4} c d^{4} - 6 \, \sqrt {d x^{4} + c} a b^{3} d^{5}}{6 \, b^{6} d^{6}} \] Input:

integrate(x^15/(b*x^4+a)^2/(d*x^4+c)^(1/2),x, algorithm="giac")
 

Output:

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

Mupad [B] (verification not implemented)

Time = 4.48 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.21 \[ \int \frac {x^{15}}{\left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx=\frac {{\left (d\,x^4+c\right )}^{3/2}}{6\,b^2\,d^2}-\left (\frac {3\,c}{2\,b^2\,d^2}+\frac {a\,d-b\,c}{b^3\,d^2}\right )\,\sqrt {d\,x^4+c}+\frac {a^2\,\mathrm {atan}\left (\frac {a^2\,\sqrt {b}\,\sqrt {d\,x^4+c}\,\left (5\,a\,d-6\,b\,c\right )}{\sqrt {a\,d-b\,c}\,\left (5\,a^3\,d-6\,a^2\,b\,c\right )}\right )\,\left (5\,a\,d-6\,b\,c\right )}{4\,b^{7/2}\,{\left (a\,d-b\,c\right )}^{3/2}}-\frac {a^3\,d\,\sqrt {d\,x^4+c}}{2\,\left (a\,d-b\,c\right )\,\left (2\,b^4\,\left (d\,x^4+c\right )-2\,b^4\,c+2\,a\,b^3\,d\right )} \] Input:

int(x^15/((a + b*x^4)^2*(c + d*x^4)^(1/2)),x)
 

Output:

(c + d*x^4)^(3/2)/(6*b^2*d^2) - ((3*c)/(2*b^2*d^2) + (a*d - b*c)/(b^3*d^2) 
)*(c + d*x^4)^(1/2) + (a^2*atan((a^2*b^(1/2)*(c + d*x^4)^(1/2)*(5*a*d - 6* 
b*c))/((a*d - b*c)^(1/2)*(5*a^3*d - 6*a^2*b*c)))*(5*a*d - 6*b*c))/(4*b^(7/ 
2)*(a*d - b*c)^(3/2)) - (a^3*d*(c + d*x^4)^(1/2))/(2*(a*d - b*c)*(2*b^4*(c 
 + d*x^4) - 2*b^4*c + 2*a*b^3*d))
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 3265, normalized size of antiderivative = 21.20 \[ \int \frac {x^{15}}{\left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx =\text {Too large to display} \] Input:

int(x^15/(b*x^4+a)^2/(d*x^4+c)^(1/2),x)
 

Output:

(15*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**4*c**2*d**3 + 180*sqrt(b)*sqrt(c 
+ d*x**4)*sqrt(a*d - b*c)*atan((sqrt(d)*sqrt(b)*sqrt(c + d*x**4)*x**2 + sq 
rt(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**4*c*d**4*x**4 + 240*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**4*d**5*x**8 - 18*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*b*c**3*d**2 - 
 201*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*b*c**2*d**3*x**4 - 108*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*b*c*d**4*x**8 + 240*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*b*d**5*x**12 - 18*sqrt(b)*sqrt(c + d*x**4)*sqrt(a*d - ...