\(\int \frac {\sqrt {c+d x^4}}{x^{11} (a+b x^4)} \, dx\) [228]

Optimal result

Integrand size = 24, antiderivative size = 159 \[ \int \frac {\sqrt {c+d x^4}}{x^{11} \left (a+b x^4\right )} \, dx=-\frac {\sqrt {c+d x^4}}{10 a x^{10}}+\frac {(5 b c-a d) \sqrt {c+d x^4}}{30 a^2 c x^6}-\frac {\left (15 b^2 c^2-5 a b c d-2 a^2 d^2\right ) \sqrt {c+d x^4}}{30 a^3 c^2 x^2}-\frac {b^2 \sqrt {b c-a d} \arctan \left (\frac {\sqrt {b c-a d} x^2}{\sqrt {a} \sqrt {c+d x^4}}\right )}{2 a^{7/2}} \] Output:

-1/10*(d*x^4+c)^(1/2)/a/x^10+1/30*(-a*d+5*b*c)*(d*x^4+c)^(1/2)/a^2/c/x^6-1 
/30*(-2*a^2*d^2-5*a*b*c*d+15*b^2*c^2)*(d*x^4+c)^(1/2)/a^3/c^2/x^2-1/2*b^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))/a^(7 
/2)
 

Mathematica [A] (verified)

Time = 0.83 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.96 \[ \int \frac {\sqrt {c+d x^4}}{x^{11} \left (a+b x^4\right )} \, dx=-\frac {\sqrt {c+d x^4} \left (15 b^2 c^2 x^8-5 a b c x^4 \left (c+d x^4\right )+a^2 \left (3 c^2+c d x^4-2 d^2 x^8\right )\right )}{30 a^3 c^2 x^{10}}-\frac {b^2 \sqrt {b c-a d} \arctan \left (\frac {a \sqrt {d}+b x^2 \left (\sqrt {d} x^2+\sqrt {c+d x^4}\right )}{\sqrt {a} \sqrt {b c-a d}}\right )}{2 a^{7/2}} \] Input:

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

Output:

-1/30*(Sqrt[c + d*x^4]*(15*b^2*c^2*x^8 - 5*a*b*c*x^4*(c + d*x^4) + a^2*(3* 
c^2 + c*d*x^4 - 2*d^2*x^8)))/(a^3*c^2*x^10) - (b^2*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])])/(2*a^(7/2))
 

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.09, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {965, 377, 25, 445, 445, 27, 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.

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

\(\Big \downarrow \) 965

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

\(\Big \downarrow \) 377

\(\displaystyle \frac {1}{2} \left (\frac {\int -\frac {4 b d x^4+5 b c-a d}{x^8 \left (b x^4+a\right ) \sqrt {d x^4+c}}dx^2}{5 a}-\frac {\sqrt {c+d x^4}}{5 a x^{10}}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{2} \left (-\frac {\int \frac {4 b d x^4+5 b c-a d}{x^8 \left (b x^4+a\right ) \sqrt {d x^4+c}}dx^2}{5 a}-\frac {\sqrt {c+d x^4}}{5 a x^{10}}\right )\)

\(\Big \downarrow \) 445

\(\displaystyle \frac {1}{2} \left (-\frac {-\frac {\int \frac {2 b d (5 b c-a d) x^4+15 b^2 c^2-2 a^2 d^2-5 a b c d}{x^4 \left (b x^4+a\right ) \sqrt {d x^4+c}}dx^2}{3 a c}-\frac {\sqrt {c+d x^4} (5 b c-a d)}{3 a c x^6}}{5 a}-\frac {\sqrt {c+d x^4}}{5 a x^{10}}\right )\)

\(\Big \downarrow \) 445

\(\displaystyle \frac {1}{2} \left (-\frac {-\frac {-\frac {\int \frac {15 b^2 c^2 (b c-a d)}{\left (b x^4+a\right ) \sqrt {d x^4+c}}dx^2}{a c}-\frac {\sqrt {c+d x^4} \left (\frac {15 b^2 c}{a}-\frac {2 a d^2}{c}-5 b d\right )}{x^2}}{3 a c}-\frac {\sqrt {c+d x^4} (5 b c-a d)}{3 a c x^6}}{5 a}-\frac {\sqrt {c+d x^4}}{5 a x^{10}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} \left (-\frac {-\frac {-\frac {15 b^2 c (b c-a d) \int \frac {1}{\left (b x^4+a\right ) \sqrt {d x^4+c}}dx^2}{a}-\frac {\sqrt {c+d x^4} \left (\frac {15 b^2 c}{a}-\frac {2 a d^2}{c}-5 b d\right )}{x^2}}{3 a c}-\frac {\sqrt {c+d x^4} (5 b c-a d)}{3 a c x^6}}{5 a}-\frac {\sqrt {c+d x^4}}{5 a x^{10}}\right )\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {1}{2} \left (-\frac {-\frac {-\frac {15 b^2 c (b c-a d) \int \frac {1}{a-(a d-b c) x^4}d\frac {x^2}{\sqrt {d x^4+c}}}{a}-\frac {\sqrt {c+d x^4} \left (\frac {15 b^2 c}{a}-\frac {2 a d^2}{c}-5 b d\right )}{x^2}}{3 a c}-\frac {\sqrt {c+d x^4} (5 b c-a d)}{3 a c x^6}}{5 a}-\frac {\sqrt {c+d x^4}}{5 a x^{10}}\right )\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {1}{2} \left (-\frac {-\frac {-\frac {15 b^2 c \sqrt {b c-a d} \arctan \left (\frac {x^2 \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^4}}\right )}{a^{3/2}}-\frac {\sqrt {c+d x^4} \left (\frac {15 b^2 c}{a}-\frac {2 a d^2}{c}-5 b d\right )}{x^2}}{3 a c}-\frac {\sqrt {c+d x^4} (5 b c-a d)}{3 a c x^6}}{5 a}-\frac {\sqrt {c+d x^4}}{5 a x^{10}}\right )\)

Input:

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

Output:

(-1/5*Sqrt[c + d*x^4]/(a*x^10) - (-1/3*((5*b*c - a*d)*Sqrt[c + d*x^4])/(a* 
c*x^6) - (-((((15*b^2*c)/a - 5*b*d - (2*a*d^2)/c)*Sqrt[c + d*x^4])/x^2) - 
(15*b^2*c*Sqrt[b*c - a*d]*ArcTan[(Sqrt[b*c - a*d]*x^2)/(Sqrt[a]*Sqrt[c + d 
*x^4])])/a^(3/2))/(3*a*c))/(5*a))/2
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 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 377

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

rule 445

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_ 
.)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^2)^(p 
+ 1)*((c + d*x^2)^(q + 1)/(a*c*g*(m + 1))), x] + Simp[1/(a*c*g^2*(m + 1)) 
 Int[(g*x)^(m + 2)*(a + b*x^2)^p*(c + d*x^2)^q*Simp[a*f*c*(m + 1) - e*(b*c 
+ a*d)*(m + 2 + 1) - e*2*(b*c*p + a*d*q) - b*e*d*(m + 2*(p + q + 2) + 1)*x^ 
2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && LtQ[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 = 1.69 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.87

method	result	size
pseudoelliptic	\(-\frac {\left (\left (-\frac {2 d \,x^{4}}{3}+c \right ) \left (d \,x^{4}+c \right ) a^{2}-\frac {5 a \left (d \,x^{4}+c \right ) b c \,x^{4}}{3}+5 b^{2} c^{2} x^{8}\right ) \sqrt {a \left (a d -c b \right )}\, \sqrt {d \,x^{4}+c}-5 b^{2} c^{2} x^{10} \left (a d -c b \right ) \operatorname {arctanh}\left (\frac {a \sqrt {d \,x^{4}+c}}{x^{2} \sqrt {a \left (a d -c b \right )}}\right )}{10 \sqrt {a \left (a d -c b \right )}\, x^{10} a^{3} c^{2}}\)	\(139\)
risch	\(-\frac {\sqrt {d \,x^{4}+c}\, \left (-2 a^{2} d^{2} x^{8}-5 a b c d \,x^{8}+15 b^{2} c^{2} x^{8}+a^{2} c d \,x^{4}-5 a b \,c^{2} x^{4}+3 a^{2} c^{2}\right )}{30 a^{3} x^{10} c^{2}}+\frac {b^{2} \left (a d -c b \right ) \left (-\frac {\ln \left (\frac {-\frac {2 \left (a d -c b \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -c b}{b}}\, \sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{x^{2}-\frac {\sqrt {-a b}}{b}}\right )}{4 \sqrt {-a b}\, \sqrt {-\frac {a d -c b}{b}}}+\frac {\ln \left (\frac {-\frac {2 \left (a d -c b \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -c b}{b}}\, \sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{x^{2}+\frac {\sqrt {-a b}}{b}}\right )}{4 \sqrt {-a b}\, \sqrt {-\frac {a d -c b}{b}}}\right )}{a^{3}}\)	\(417\)
default	\(-\frac {\left (d \,x^{4}+c \right )^{\frac {3}{2}} \left (-2 d \,x^{4}+3 c \right )}{30 a \,x^{10} c^{2}}+\frac {b^{2} \left (-\frac {\left (d \,x^{4}+c \right )^{\frac {3}{2}}}{2 c \,x^{2}}+\frac {d \,x^{2} \sqrt {d \,x^{4}+c}}{2 c}+\frac {\sqrt {d}\, \ln \left (\sqrt {d}\, x^{2}+\sqrt {d \,x^{4}+c}\right )}{2}\right )}{a^{3}}+\frac {b \left (d \,x^{4}+c \right )^{\frac {3}{2}}}{6 a^{2} x^{6} c}-\frac {b^{3} \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 )}{a^{3}}\)	\(815\)
elliptic	\(\frac {-\frac {\left (d \,x^{4}+c \right )^{\frac {3}{2}}}{5 c \,x^{10}}+\frac {2 d \left (d \,x^{4}+c \right )^{\frac {3}{2}}}{15 c^{2} x^{6}}}{2 a}+\frac {b^{2} \left (-\frac {\left (d \,x^{4}+c \right )^{\frac {3}{2}}}{c \,x^{2}}+\frac {2 d \left (\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}}\right )}{c}\right )}{2 a^{3}}+\frac {b \left (d \,x^{4}+c \right )^{\frac {3}{2}}}{6 a^{2} x^{6} c}-\frac {b^{3} \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 a^{3} \sqrt {-a b}}+\frac {b^{3} \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 a^{3} \sqrt {-a b}}\)	\(834\)

Input:

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

Output:

-1/10/(a*(a*d-b*c))^(1/2)*(((-2/3*d*x^4+c)*(d*x^4+c)*a^2-5/3*a*(d*x^4+c)*b 
*c*x^4+5*b^2*c^2*x^8)*(a*(a*d-b*c))^(1/2)*(d*x^4+c)^(1/2)-5*b^2*c^2*x^10*( 
a*d-b*c)*arctanh(a*(d*x^4+c)^(1/2)/x^2/(a*(a*d-b*c))^(1/2)))/x^10/a^3/c^2
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 413, normalized size of antiderivative = 2.60 \[ \int \frac {\sqrt {c+d x^4}}{x^{11} \left (a+b x^4\right )} \, dx=\left [\frac {15 \, b^{2} c^{2} x^{10} \sqrt {-\frac {b c - a d}{a}} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{8} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{4} + a^{2} c^{2} - 4 \, {\left ({\left (a b c - 2 \, a^{2} d\right )} x^{6} - a^{2} c x^{2}\right )} \sqrt {d x^{4} + c} \sqrt {-\frac {b c - a d}{a}}}{b^{2} x^{8} + 2 \, a b x^{4} + a^{2}}\right ) - 4 \, {\left ({\left (15 \, b^{2} c^{2} - 5 \, a b c d - 2 \, a^{2} d^{2}\right )} x^{8} - {\left (5 \, a b c^{2} - a^{2} c d\right )} x^{4} + 3 \, a^{2} c^{2}\right )} \sqrt {d x^{4} + c}}{120 \, a^{3} c^{2} x^{10}}, -\frac {15 \, b^{2} c^{2} x^{10} \sqrt {\frac {b c - a d}{a}} \arctan \left (\frac {{\left ({\left (b c - 2 \, a d\right )} x^{4} - a c\right )} \sqrt {d x^{4} + c} \sqrt {\frac {b c - a d}{a}}}{2 \, {\left ({\left (b c d - a d^{2}\right )} x^{6} + {\left (b c^{2} - a c d\right )} x^{2}\right )}}\right ) + 2 \, {\left ({\left (15 \, b^{2} c^{2} - 5 \, a b c d - 2 \, a^{2} d^{2}\right )} x^{8} - {\left (5 \, a b c^{2} - a^{2} c d\right )} x^{4} + 3 \, a^{2} c^{2}\right )} \sqrt {d x^{4} + c}}{60 \, a^{3} c^{2} x^{10}}\right ] \] Input:

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

Output:

[1/120*(15*b^2*c^2*x^10*sqrt(-(b*c - a*d)/a)*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*((a*b*c - 2*a^ 
2*d)*x^6 - a^2*c*x^2)*sqrt(d*x^4 + c)*sqrt(-(b*c - a*d)/a))/(b^2*x^8 + 2*a 
*b*x^4 + a^2)) - 4*((15*b^2*c^2 - 5*a*b*c*d - 2*a^2*d^2)*x^8 - (5*a*b*c^2 
- a^2*c*d)*x^4 + 3*a^2*c^2)*sqrt(d*x^4 + c))/(a^3*c^2*x^10), -1/60*(15*b^2 
*c^2*x^10*sqrt((b*c - a*d)/a)*arctan(1/2*((b*c - 2*a*d)*x^4 - a*c)*sqrt(d* 
x^4 + c)*sqrt((b*c - a*d)/a)/((b*c*d - a*d^2)*x^6 + (b*c^2 - a*c*d)*x^2)) 
+ 2*((15*b^2*c^2 - 5*a*b*c*d - 2*a^2*d^2)*x^8 - (5*a*b*c^2 - a^2*c*d)*x^4 
+ 3*a^2*c^2)*sqrt(d*x^4 + c))/(a^3*c^2*x^10)]
 

Sympy [F]

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

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

Output:

Integral(sqrt(c + d*x**4)/(x**11*(a + b*x**4)), x)
 

Maxima [F]

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

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

Output:

integrate(sqrt(d*x^4 + c)/((b*x^4 + a)*x^11), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 492 vs. \(2 (135) = 270\).

Time = 0.34 (sec) , antiderivative size = 492, normalized size of antiderivative = 3.09 \[ \int \frac {\sqrt {c+d x^4}}{x^{11} \left (a+b x^4\right )} \, dx=\frac {{\left (b^{3} c \sqrt {d} - a b^{2} d^{\frac {3}{2}}\right )} \arctan \left (\frac {{\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{2 \, \sqrt {a b c d - a^{2} d^{2}} a^{3}} + \frac {15 \, {\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{8} b^{2} c \sqrt {d} - 15 \, {\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{8} a b d^{\frac {3}{2}} - 60 \, {\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{6} b^{2} c^{2} \sqrt {d} + 30 \, {\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{6} a b c d^{\frac {3}{2}} + 30 \, {\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{6} a^{2} d^{\frac {5}{2}} + 90 \, {\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{4} b^{2} c^{3} \sqrt {d} - 20 \, {\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{4} a b c^{2} d^{\frac {3}{2}} + 10 \, {\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{4} a^{2} c d^{\frac {5}{2}} - 60 \, {\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{2} b^{2} c^{4} \sqrt {d} + 10 \, {\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{2} a b c^{3} d^{\frac {3}{2}} + 10 \, {\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{2} a^{2} c^{2} d^{\frac {5}{2}} + 15 \, b^{2} c^{5} \sqrt {d} - 5 \, a b c^{4} d^{\frac {3}{2}} - 2 \, a^{2} c^{3} d^{\frac {5}{2}}}{15 \, {\left ({\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{2} - c\right )}^{5} a^{3}} \] Input:

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

Output:

1/2*(b^3*c*sqrt(d) - a*b^2*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))/(sqrt(a*b*c*d - a^2*d^2) 
*a^3) + 1/15*(15*(sqrt(d)*x^2 - sqrt(d*x^4 + c))^8*b^2*c*sqrt(d) - 15*(sqr 
t(d)*x^2 - sqrt(d*x^4 + c))^8*a*b*d^(3/2) - 60*(sqrt(d)*x^2 - sqrt(d*x^4 + 
 c))^6*b^2*c^2*sqrt(d) + 30*(sqrt(d)*x^2 - sqrt(d*x^4 + c))^6*a*b*c*d^(3/2 
) + 30*(sqrt(d)*x^2 - sqrt(d*x^4 + c))^6*a^2*d^(5/2) + 90*(sqrt(d)*x^2 - s 
qrt(d*x^4 + c))^4*b^2*c^3*sqrt(d) - 20*(sqrt(d)*x^2 - sqrt(d*x^4 + c))^4*a 
*b*c^2*d^(3/2) + 10*(sqrt(d)*x^2 - sqrt(d*x^4 + c))^4*a^2*c*d^(5/2) - 60*( 
sqrt(d)*x^2 - sqrt(d*x^4 + c))^2*b^2*c^4*sqrt(d) + 10*(sqrt(d)*x^2 - sqrt( 
d*x^4 + c))^2*a*b*c^3*d^(3/2) + 10*(sqrt(d)*x^2 - sqrt(d*x^4 + c))^2*a^2*c 
^2*d^(5/2) + 15*b^2*c^5*sqrt(d) - 5*a*b*c^4*d^(3/2) - 2*a^2*c^3*d^(5/2))/( 
((sqrt(d)*x^2 - sqrt(d*x^4 + c))^2 - c)^5*a^3)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.51 (sec) , antiderivative size = 1815, normalized size of antiderivative = 11.42 \[ \int \frac {\sqrt {c+d x^4}}{x^{11} \left (a+b x^4\right )} \, dx =\text {Too large to display} \] Input:

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

Output:

(15*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)*b**2*c**3*x**10 + 180*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)*b**2*c**2*d*x**14 + 24 
0*sqrt(d)*sqrt(a)*sqrt(c + d*x**4)*sqrt(a*d - b*c)*log( - sqrt(2*sqrt(d)*s 
qrt(a)*sqrt(a*d - b*c) - 2*a*d + b*c) + sqrt(b)*sqrt(c + d*x**4) + sqrt(d) 
*sqrt(b)*x**2)*b**2*c*d**2*x**18 + 15*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) + s 
qrt(b)*sqrt(c + d*x**4) + sqrt(d)*sqrt(b)*x**2)*b**2*c**3*x**10 + 180*sqrt 
(d)*sqrt(a)*sqrt(c + d*x**4)*sqrt(a*d - b*c)*log(sqrt(2*sqrt(d)*sqrt(a)*sq 
rt(a*d - b*c) - 2*a*d + b*c) + sqrt(b)*sqrt(c + d*x**4) + sqrt(d)*sqrt(b)* 
x**2)*b**2*c**2*d*x**14 + 240*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)*s 
qrt(c + d*x**4) + sqrt(d)*sqrt(b)*x**2)*b**2*c*d**2*x**18 - 15*sqrt(d)*sqr 
t(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)*b**2*c**3*x**1 
0 - 180*sqrt(d)*sqrt(a)*sqrt(c + d*x**4)*sqrt(a*d - b*c)*log(2*sqrt(d)*sqr 
t(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)*b**2*c**2*d*x**14 - 240*sqrt(d)*sqrt(a)*sqrt(c + d*x**4)*sqrt(a*d ...