\(\int \frac {x^5}{(a+b x^4) \sqrt {c+d x^4}} \, dx\) [241]

Optimal result

Integrand size = 24, antiderivative size = 91 \[ \int \frac {x^5}{\left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx=-\frac {\sqrt {a} \arctan \left (\frac {\sqrt {b c-a d} x^2}{\sqrt {a} \sqrt {c+d x^4}}\right )}{2 b \sqrt {b c-a d}}+\frac {\text {arctanh}\left (\frac {\sqrt {d} x^2}{\sqrt {c+d x^4}}\right )}{2 b \sqrt {d}} \] Output:

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

Mathematica [A] (verified)

Time = 0.38 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.19 \[ \int \frac {x^5}{\left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx=\frac {-\frac {\sqrt {a} \arctan \left (\frac {a \sqrt {d}+b x^2 \left (\sqrt {d} x^2+\sqrt {c+d x^4}\right )}{\sqrt {a} \sqrt {b c-a d}}\right )}{\sqrt {b c-a d}}+\frac {\log \left (\sqrt {d} x^2+\sqrt {c+d x^4}\right )}{\sqrt {d}}}{2 b} \] Input:

Integrate[x^5/((a + b*x^4)*Sqrt[c + d*x^4]),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.99, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {965, 385, 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.

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

\(\Big \downarrow \) 965

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

\(\Big \downarrow \) 385

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 218

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

Input:

Int[x^5/((a + b*x^4)*Sqrt[c + d*x^4]),x]
 

Output:

(-((Sqrt[a]*ArcTan[(Sqrt[b*c - a*d]*x^2)/(Sqrt[a]*Sqrt[c + d*x^4])])/(b*Sq 
rt[b*c - a*d])) + ArcTanh[(Sqrt[d]*x^2)/Sqrt[c + d*x^4]]/(b*Sqrt[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 385

Int[(((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_)^2)^(q_.))/((a_) + (b_.)*(x_)^2), 
x_Symbol] :> Simp[e^2/b   Int[(e*x)^(m - 2)*(c + d*x^2)^q, x], x] - Simp[a* 
(e^2/b)   Int[(e*x)^(m - 2)*((c + d*x^2)^q/(a + b*x^2)), x], x] /; FreeQ[{a 
, b, c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && LeQ[2, m, 3] && IntBinomial 
Q[a, b, c, d, e, m, 2, -1, q, 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 = 0.44 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.77

method	result	size
pseudoelliptic	\(-\frac {\frac {a \,\operatorname {arctanh}\left (\frac {a \sqrt {d \,x^{4}+c}}{x^{2} \sqrt {a \left (a d -c b \right )}}\right )}{\sqrt {a \left (a d -c b \right )}}-\frac {\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{4}+c}}{x^{2} \sqrt {d}}\right )}{\sqrt {d}}}{2 b}\)	\(70\)
default	\(\frac {\ln \left (\sqrt {d}\, x^{2}+\sqrt {d \,x^{4}+c}\right )}{2 b \sqrt {d}}-\frac {a \left (-\frac {\ln \left (\frac {-\frac {2 \left (a d -c b \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -c b}{b}}\, \sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{x^{2}-\frac {\sqrt {-a b}}{b}}\right )}{4 \sqrt {-a b}\, \sqrt {-\frac {a d -c b}{b}}}+\frac {\ln \left (\frac {-\frac {2 \left (a d -c b \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -c b}{b}}\, \sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{x^{2}+\frac {\sqrt {-a b}}{b}}\right )}{4 \sqrt {-a b}\, \sqrt {-\frac {a d -c b}{b}}}\right )}{b}\)	\(355\)
elliptic	\(\frac {\ln \left (\sqrt {d}\, x^{2}+\sqrt {d \,x^{4}+c}\right )}{2 b \sqrt {d}}+\frac {a \ln \left (\frac {-\frac {2 \left (a d -c b \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -c b}{b}}\, \sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{x^{2}-\frac {\sqrt {-a b}}{b}}\right )}{4 \sqrt {-a b}\, b \sqrt {-\frac {a d -c b}{b}}}-\frac {a \ln \left (\frac {-\frac {2 \left (a d -c b \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -c b}{b}}\, \sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{x^{2}+\frac {\sqrt {-a b}}{b}}\right )}{4 \sqrt {-a b}\, b \sqrt {-\frac {a d -c b}{b}}}\)	\(356\)

Input:

int(x^5/(b*x^4+a)/(d*x^4+c)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 638, normalized size of antiderivative = 7.01 \[ \int \frac {x^5}{\left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx=\left [\frac {d \sqrt {-\frac {a}{b c - a d}} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{8} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{4} + a^{2} c^{2} - 4 \, {\left ({\left (b^{2} c^{2} - 3 \, a b c d + 2 \, a^{2} d^{2}\right )} x^{6} - {\left (a b c^{2} - a^{2} c d\right )} x^{2}\right )} \sqrt {d x^{4} + c} \sqrt {-\frac {a}{b c - a d}}}{b^{2} x^{8} + 2 \, a b x^{4} + a^{2}}\right ) + 2 \, \sqrt {d} \log \left (-2 \, d x^{4} - 2 \, \sqrt {d x^{4} + c} \sqrt {d} x^{2} - c\right )}{8 \, b d}, \frac {d \sqrt {-\frac {a}{b c - a d}} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{8} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{4} + a^{2} c^{2} - 4 \, {\left ({\left (b^{2} c^{2} - 3 \, a b c d + 2 \, a^{2} d^{2}\right )} x^{6} - {\left (a b c^{2} - a^{2} c d\right )} x^{2}\right )} \sqrt {d x^{4} + c} \sqrt {-\frac {a}{b c - a d}}}{b^{2} x^{8} + 2 \, a b x^{4} + a^{2}}\right ) - 4 \, \sqrt {-d} \arctan \left (\frac {\sqrt {d x^{4} + c} \sqrt {-d}}{d x^{2}}\right )}{8 \, b d}, \frac {d \sqrt {\frac {a}{b c - a d}} \arctan \left (-\frac {{\left ({\left (b c - 2 \, a d\right )} x^{4} - a c\right )} \sqrt {d x^{4} + c} \sqrt {\frac {a}{b c - a d}}}{2 \, {\left (a d x^{6} + a c x^{2}\right )}}\right ) + \sqrt {d} \log \left (-2 \, d x^{4} - 2 \, \sqrt {d x^{4} + c} \sqrt {d} x^{2} - c\right )}{4 \, b d}, \frac {d \sqrt {\frac {a}{b c - a d}} \arctan \left (-\frac {{\left ({\left (b c - 2 \, a d\right )} x^{4} - a c\right )} \sqrt {d x^{4} + c} \sqrt {\frac {a}{b c - a d}}}{2 \, {\left (a d x^{6} + a c x^{2}\right )}}\right ) - 2 \, \sqrt {-d} \arctan \left (\frac {\sqrt {d x^{4} + c} \sqrt {-d}}{d x^{2}}\right )}{4 \, b d}\right ] \] Input:

integrate(x^5/(b*x^4+a)/(d*x^4+c)^(1/2),x, algorithm="fricas")
 

Output:

[1/8*(d*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*sqrt(d)*log(-2*d*x^4 - 2*sqrt(d*x^4 + c)* 
sqrt(d)*x^2 - c))/(b*d), 1/8*(d*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*sqrt(-d)*arctan(s 
qrt(d*x^4 + c)*sqrt(-d)/(d*x^2)))/(b*d), 1/4*(d*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)) + sqrt(d)*log(-2*d*x^4 - 2*sqrt(d*x^4 + c)*sqrt(d)*x^2 - c) 
)/(b*d), 1/4*(d*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)) - 2*sqrt(-d)*arct 
an(sqrt(d*x^4 + c)*sqrt(-d)/(d*x^2)))/(b*d)]
 

Sympy [F]

\[ \int \frac {x^5}{\left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx=\int \frac {x^{5}}{\left (a + b x^{4}\right ) \sqrt {c + d x^{4}}}\, dx \] Input:

integrate(x**5/(b*x**4+a)/(d*x**4+c)**(1/2),x)
 

Output:

Integral(x**5/((a + b*x**4)*sqrt(c + d*x**4)), x)
 

Maxima [F]

\[ \int \frac {x^5}{\left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx=\int { \frac {x^{5}}{{\left (b x^{4} + a\right )} \sqrt {d x^{4} + c}} \,d x } \] Input:

integrate(x^5/(b*x^4+a)/(d*x^4+c)^(1/2),x, algorithm="maxima")
 

Output:

integrate(x^5/((b*x^4 + a)*sqrt(d*x^4 + c)), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {x^5}{\left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x^5/(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. \[ \int \frac {x^5}{\left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx=\int \frac {x^5}{\left (b\,x^4+a\right )\,\sqrt {d\,x^4+c}} \,d x \] Input:

int(x^5/((a + b*x^4)*(c + d*x^4)^(1/2)),x)
 

Output:

int(x^5/((a + b*x^4)*(c + d*x^4)^(1/2)), x)
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 250, normalized size of antiderivative = 2.75 \[ \int \frac {x^5}{\left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx=\frac {-\sqrt {d}\, \sqrt {a}\, \sqrt {a d -b c}\, \mathrm {log}\left (-\sqrt {2 \sqrt {d}\, \sqrt {a}\, \sqrt {a d -b c}-2 a d +b c}+\sqrt {b}\, \sqrt {d \,x^{4}+c}+\sqrt {d}\, \sqrt {b}\, x^{2}\right )-\sqrt {d}\, \sqrt {a}\, \sqrt {a d -b c}\, \mathrm {log}\left (\sqrt {2 \sqrt {d}\, \sqrt {a}\, \sqrt {a d -b c}-2 a d +b c}+\sqrt {b}\, \sqrt {d \,x^{4}+c}+\sqrt {d}\, \sqrt {b}\, x^{2}\right )+\sqrt {d}\, \sqrt {a}\, \sqrt {a d -b c}\, \mathrm {log}\left (2 \sqrt {d}\, \sqrt {a}\, \sqrt {a d -b c}+2 \sqrt {d}\, \sqrt {d \,x^{4}+c}\, b \,x^{2}+2 a d +2 b d \,x^{4}\right )+2 \,\mathrm {log}\left (\frac {\sqrt {d \,x^{4}+c}+\sqrt {d}\, x^{2}}{\sqrt {c}}\right ) a d -2 \,\mathrm {log}\left (\frac {\sqrt {d \,x^{4}+c}+\sqrt {d}\, x^{2}}{\sqrt {c}}\right ) b c}{4 \sqrt {d}\, b \left (a d -b c \right )} \] Input:

int(x^5/(b*x^4+a)/(d*x^4+c)^(1/2),x)
 

Output:

( - sqrt(d)*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) 
- sqrt(d)*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) + sqr 
t(d)*sqrt(a)*sqrt(a*d - b*c)*log(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) + 2*sqr 
t(d)*sqrt(c + d*x**4)*b*x**2 + 2*a*d + 2*b*d*x**4) + 2*log((sqrt(c + d*x** 
4) + sqrt(d)*x**2)/sqrt(c))*a*d - 2*log((sqrt(c + d*x**4) + sqrt(d)*x**2)/ 
sqrt(c))*b*c)/(4*sqrt(d)*b*(a*d - b*c))