\(\int \frac {x^9}{(a+b x^4) \sqrt {c+d x^4}} \, dx\) [240]

Optimal result

Integrand size = 24, antiderivative size = 123 \[ \int \frac {x^9}{\left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx=\frac {x^2 \sqrt {c+d x^4}}{4 b d}+\frac {a^{3/2} \arctan \left (\frac {\sqrt {b c-a d} x^2}{\sqrt {a} \sqrt {c+d x^4}}\right )}{2 b^2 \sqrt {b c-a d}}-\frac {(b c+2 a d) \text {arctanh}\left (\frac {\sqrt {d} x^2}{\sqrt {c+d x^4}}\right )}{4 b^2 d^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.73 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.11 \[ \int \frac {x^9}{\left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx=\frac {\frac {b x^2 \sqrt {c+d x^4}}{d}+\frac {2 a^{3/2} \arctan \left (\frac {a \sqrt {d}+b x^2 \left (\sqrt {d} x^2+\sqrt {c+d x^4}\right )}{\sqrt {a} \sqrt {b c-a d}}\right )}{\sqrt {b c-a d}}-\frac {(b c+2 a d) \log \left (\sqrt {d} x^2+\sqrt {c+d x^4}\right )}{d^{3/2}}}{4 b^2} \] Input:

Integrate[x^9/((a + b*x^4)*Sqrt[c + d*x^4]),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.09, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {965, 381, 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.

\(\displaystyle \int \frac {x^9}{\left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx\)

\(\Big \downarrow \) 965

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

\(\Big \downarrow \) 381

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

\(\Big \downarrow \) 398

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 218

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

Input:

Int[x^9/((a + b*x^4)*Sqrt[c + d*x^4]),x]
 

Output:

((x^2*Sqrt[c + d*x^4])/(2*b*d) - ((-2*a^(3/2)*d*ArcTan[(Sqrt[b*c - a*d]*x^ 
2)/(Sqrt[a]*Sqrt[c + d*x^4])])/(b*Sqrt[b*c - a*d]) + ((b*c + 2*a*d)*ArcTan 
h[(Sqrt[d]*x^2)/Sqrt[c + d*x^4]])/(b*Sqrt[d]))/(2*b*d))/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 381

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ 
), x_Symbol] :> Simp[e^3*(e*x)^(m - 3)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q 
+ 1)/(b*d*(m + 2*(p + q) + 1))), x] - Simp[e^4/(b*d*(m + 2*(p + q) + 1)) 
Int[(e*x)^(m - 4)*(a + b*x^2)^p*(c + d*x^2)^q*Simp[a*c*(m - 3) + (a*d*(m + 
2*q - 1) + b*c*(m + 2*p - 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, p, q 
}, x] && NeQ[b*c - a*d, 0] && 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 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 = 0.80 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.80

method	result	size
pseudoelliptic	\(-\frac {-\frac {2 a^{2} \operatorname {arctanh}\left (\frac {a \sqrt {d \,x^{4}+c}}{x^{2} \sqrt {a \left (a d -c b \right )}}\right )}{\sqrt {a \left (a d -c b \right )}}-\frac {\sqrt {d \,x^{4}+c}\, b \,x^{2}}{d}+\frac {\left (2 a d +c b \right ) \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{4}+c}}{x^{2} \sqrt {d}}\right )}{d^{\frac {3}{2}}}}{4 b^{2}}\)	\(98\)
risch	\(\frac {x^{2} \sqrt {d \,x^{4}+c}}{4 b d}-\frac {\frac {\left (2 a d +c b \right ) \ln \left (\sqrt {d}\, x^{2}+\sqrt {d \,x^{4}+c}\right )}{2 b \sqrt {d}}-\frac {2 a^{2} d \left (-\frac {\ln \left (\frac {-\frac {2 \left (a d -c b \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -c b}{b}}\, \sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{x^{2}-\frac {\sqrt {-a b}}{b}}\right )}{4 \sqrt {-a b}\, \sqrt {-\frac {a d -c b}{b}}}+\frac {\ln \left (\frac {-\frac {2 \left (a d -c b \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -c b}{b}}\, \sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{x^{2}+\frac {\sqrt {-a b}}{b}}\right )}{4 \sqrt {-a b}\, \sqrt {-\frac {a d -c b}{b}}}\right )}{b}}{2 b d}\)	\(395\)
default	\(\frac {\frac {x^{2} \sqrt {d \,x^{4}+c}}{4 d}-\frac {c \ln \left (\sqrt {d}\, x^{2}+\sqrt {d \,x^{4}+c}\right )}{4 d^{\frac {3}{2}}}}{b}+\frac {a^{2} \left (-\frac {\ln \left (\frac {-\frac {2 \left (a d -c b \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -c b}{b}}\, \sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{x^{2}-\frac {\sqrt {-a b}}{b}}\right )}{4 \sqrt {-a b}\, \sqrt {-\frac {a d -c b}{b}}}+\frac {\ln \left (\frac {-\frac {2 \left (a d -c b \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -c b}{b}}\, \sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{x^{2}+\frac {\sqrt {-a b}}{b}}\right )}{4 \sqrt {-a b}\, \sqrt {-\frac {a d -c b}{b}}}\right )}{b^{2}}-\frac {a \ln \left (\sqrt {d}\, x^{2}+\sqrt {d \,x^{4}+c}\right )}{2 b^{2} \sqrt {d}}\)	\(403\)
elliptic	\(\frac {x^{2} \sqrt {d \,x^{4}+c}}{4 b d}-\frac {c \ln \left (\sqrt {d}\, x^{2}+\sqrt {d \,x^{4}+c}\right )}{4 b \,d^{\frac {3}{2}}}-\frac {a \ln \left (\sqrt {d}\, x^{2}+\sqrt {d \,x^{4}+c}\right )}{2 b^{2} \sqrt {d}}-\frac {a^{2} \ln \left (\frac {-\frac {2 \left (a d -c b \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -c b}{b}}\, \sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{x^{2}-\frac {\sqrt {-a b}}{b}}\right )}{4 b^{2} \sqrt {-a b}\, \sqrt {-\frac {a d -c b}{b}}}+\frac {a^{2} \ln \left (\frac {-\frac {2 \left (a d -c b \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -c b}{b}}\, \sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{x^{2}+\frac {\sqrt {-a b}}{b}}\right )}{4 b^{2} \sqrt {-a b}\, \sqrt {-\frac {a d -c b}{b}}}\)	\(408\)

Input:

int(x^9/(b*x^4+a)/(d*x^4+c)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-1/4/b^2*(-2*a^2/(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+(2*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 = 0.21 (sec) , antiderivative size = 745, normalized size of antiderivative = 6.06 \[ \int \frac {x^9}{\left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx=\left [\frac {2 \, \sqrt {d x^{4} + c} b d x^{2} + a d^{2} \sqrt {-\frac {a}{b c - a 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^{2} c^{2} - 3 \, a b c d + 2 \, a^{2} d^{2}\right )} x^{6} - {\left (a b c^{2} - a^{2} c d\right )} x^{2}\right )} \sqrt {d x^{4} + c} \sqrt {-\frac {a}{b c - a d}}}{b^{2} x^{8} + 2 \, a b x^{4} + a^{2}}\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^{2}}, \frac {2 \, \sqrt {d x^{4} + c} b d x^{2} + a d^{2} \sqrt {-\frac {a}{b c - a 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^{2} c^{2} - 3 \, a b c d + 2 \, a^{2} d^{2}\right )} x^{6} - {\left (a b c^{2} - a^{2} c d\right )} x^{2}\right )} \sqrt {d x^{4} + c} \sqrt {-\frac {a}{b c - a d}}}{b^{2} x^{8} + 2 \, a b x^{4} + a^{2}}\right ) + 2 \, {\left (b c + 2 \, a d\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {d x^{4} + c} \sqrt {-d}}{d x^{2}}\right )}{8 \, b^{2} d^{2}}, \frac {2 \, \sqrt {d x^{4} + c} b d x^{2} - 2 \, a d^{2} \sqrt {\frac {a}{b c - a d}} \arctan \left (-\frac {{\left ({\left (b c - 2 \, a d\right )} x^{4} - a c\right )} \sqrt {d x^{4} + c} \sqrt {\frac {a}{b c - a d}}}{2 \, {\left (a d x^{6} + a c 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^{2}}, \frac {\sqrt {d x^{4} + c} b d x^{2} - a d^{2} \sqrt {\frac {a}{b c - a d}} \arctan \left (-\frac {{\left ({\left (b c - 2 \, a d\right )} x^{4} - a c\right )} \sqrt {d x^{4} + c} \sqrt {\frac {a}{b c - a d}}}{2 \, {\left (a d x^{6} + a c 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^{2}}\right ] \] Input:

integrate(x^9/(b*x^4+a)/(d*x^4+c)^(1/2),x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int \frac {x^9}{\left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx=\int \frac {x^{9}}{\left (a + b x^{4}\right ) \sqrt {c + d x^{4}}}\, dx \] Input:

integrate(x**9/(b*x**4+a)/(d*x**4+c)**(1/2),x)
 

Output:

Integral(x**9/((a + b*x**4)*sqrt(c + d*x**4)), x)
 

Maxima [F]

\[ \int \frac {x^9}{\left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx=\int { \frac {x^{9}}{{\left (b x^{4} + a\right )} \sqrt {d x^{4} + c}} \,d x } \] Input:

integrate(x^9/(b*x^4+a)/(d*x^4+c)^(1/2),x, algorithm="maxima")
 

Output:

integrate(x^9/((b*x^4 + a)*sqrt(d*x^4 + c)), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {x^9}{\left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x^9/(b*x^4+a)/(d*x^4+c)^(1/2),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. \[ \int \frac {x^9}{\left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx=\int \frac {x^9}{\left (b\,x^4+a\right )\,\sqrt {d\,x^4+c}} \,d x \] Input:

int(x^9/((a + b*x^4)*(c + d*x^4)^(1/2)),x)
 

Output:

int(x^9/((a + b*x^4)*(c + d*x^4)^(1/2)), x)
 

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 1128, normalized size of antiderivative = 9.17 \[ \int \frac {x^9}{\left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx =\text {Too large to display} \] Input:

int(x^9/(b*x^4+a)/(d*x^4+c)^(1/2),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)*a*d**2*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)*a*d**2*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)*sqrt(c + d*x**4)*b*x**2 + 2*a*d + 2*b*d*x**4)*a*d**2*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)*a*c*d**2 + 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)*a*d**3*x**4 + sq 
rt(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)*a*c*d**2 + 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)*a*d**3*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)*a*c*d**2 - 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)*a*d**3*x**4 - 4*sqrt(c + d*x**4)*log((sqrt(c + d 
*x**4) + sqrt(d)*x**2)/sqrt(c))*a**2*d**3*x**2 + 2*sqrt(c + d*x**4)*log...