\(\int \frac {x^{13}}{(a+b x^8)^2 \sqrt {c+d x^8}} \, dx\) [128]

Optimal result

Integrand size = 24, antiderivative size = 1177 \[ \int \frac {x^{13}}{\left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx =\text {Too large to display} \] Output:

1/8*d^(1/2)*x^2*(d*x^8+c)^(1/2)/b/(-a*d+b*c)/(c^(1/2)+d^(1/2)*x^4)-1/8*x^6 
*(d*x^8+c)^(1/2)/(-a*d+b*c)/(b*x^8+a)+1/32*(-a*d+3*b*c)*arctan((-a*d+b*c)^ 
(1/2)*x^2/(-a)^(1/4)/b^(1/4)/(d*x^8+c)^(1/2))/(-a)^(1/4)/b^(5/4)/(-a*d+b*c 
)^(3/2)-1/32*(-a*d+3*b*c)*arctanh((-a*d+b*c)^(1/2)*x^2/(-a)^(1/4)/b^(1/4)/ 
(d*x^8+c)^(1/2))/(-a)^(1/4)/b^(5/4)/(-a*d+b*c)^(3/2)-1/8*c^(1/4)*d^(1/4)*( 
c^(1/2)+d^(1/2)*x^4)*((d*x^8+c)/(c^(1/2)+d^(1/2)*x^4)^2)^(1/2)*EllipticE(s 
in(2*arctan(d^(1/4)*x^2/c^(1/4))),1/2*2^(1/2))/b/(-a*d+b*c)/(d*x^8+c)^(1/2 
)+1/16*c^(1/4)*d^(1/4)*(c^(1/2)+d^(1/2)*x^4)*((d*x^8+c)/(c^(1/2)+d^(1/2)*x 
^4)^2)^(1/2)*InverseJacobiAM(2*arctan(d^(1/4)*x^2/c^(1/4)),1/2*2^(1/2))/b/ 
(-a*d+b*c)/(d*x^8+c)^(1/2)-1/32*d^(1/4)*(-a*d+3*b*c)*(c^(1/2)+d^(1/2)*x^4) 
*((d*x^8+c)/(c^(1/2)+d^(1/2)*x^4)^2)^(1/2)*InverseJacobiAM(2*arctan(d^(1/4 
)*x^2/c^(1/4)),1/2*2^(1/2))/b^(3/2)/c^(1/4)/(b^(1/2)*c^(1/2)-(-a)^(1/2)*d^ 
(1/2))/(-a*d+b*c)/(d*x^8+c)^(1/2)-1/32*d^(1/4)*(-a*d+3*b*c)*(c^(1/2)+d^(1/ 
2)*x^4)*((d*x^8+c)/(c^(1/2)+d^(1/2)*x^4)^2)^(1/2)*InverseJacobiAM(2*arctan 
(d^(1/4)*x^2/c^(1/4)),1/2*2^(1/2))/b^(3/2)/c^(1/4)/(b^(1/2)*c^(1/2)+(-a)^( 
1/2)*d^(1/2))/(-a*d+b*c)/(d*x^8+c)^(1/2)+1/64*(b^(1/2)*c^(1/2)+(-a)^(1/2)* 
d^(1/2))*(-a*d+3*b*c)*(c^(1/2)+d^(1/2)*x^4)*((d*x^8+c)/(c^(1/2)+d^(1/2)*x^ 
4)^2)^(1/2)*EllipticPi(sin(2*arctan(d^(1/4)*x^2/c^(1/4))),-1/4*(b^(1/2)*c^ 
(1/2)-(-a)^(1/2)*d^(1/2))^2/(-a)^(1/2)/b^(1/2)/c^(1/2)/d^(1/2),1/2*2^(1/2) 
)/b^(3/2)/c^(1/4)/((-a)^(1/2)*b^(1/2)*c^(1/2)+a*d^(1/2))/d^(1/4)/(-a*d+...
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 10.12 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.14 \[ \int \frac {x^{13}}{\left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx=\frac {x^6 \left (-7 a \left (c+d x^8\right )+7 c \left (a+b x^8\right ) \sqrt {1+\frac {d x^8}{c}} \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},-\frac {d x^8}{c},-\frac {b x^8}{a}\right )+d x^8 \left (a+b x^8\right ) \sqrt {1+\frac {d x^8}{c}} \operatorname {AppellF1}\left (\frac {7}{4},\frac {1}{2},1,\frac {11}{4},-\frac {d x^8}{c},-\frac {b x^8}{a}\right )\right )}{56 a (b c-a d) \left (a+b x^8\right ) \sqrt {c+d x^8}} \] Input:

Integrate[x^13/((a + b*x^8)^2*Sqrt[c + d*x^8]),x]
 

Output:

(x^6*(-7*a*(c + d*x^8) + 7*c*(a + b*x^8)*Sqrt[1 + (d*x^8)/c]*AppellF1[3/4, 
 1/2, 1, 7/4, -((d*x^8)/c), -((b*x^8)/a)] + d*x^8*(a + b*x^8)*Sqrt[1 + (d* 
x^8)/c]*AppellF1[7/4, 1/2, 1, 11/4, -((d*x^8)/c), -((b*x^8)/a)]))/(56*a*(b 
*c - a*d)*(a + b*x^8)*Sqrt[c + d*x^8])
 

Rubi [A] (verified)

Time = 2.50 (sec) , antiderivative size = 1107, normalized size of antiderivative = 0.94, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {965, 971, 1054, 2009}

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.

\(\displaystyle \int \frac {x^{13}}{\left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx\)

\(\Big \downarrow \) 965

\(\displaystyle \frac {1}{2} \int \frac {x^{12}}{\left (b x^8+a\right )^2 \sqrt {d x^8+c}}dx^2\)

\(\Big \downarrow \) 971

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

\(\Big \downarrow \) 1054

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2} \left (\frac {-\frac {(3 b c-a d) \left (\sqrt {d} x^4+\sqrt {c}\right ) \sqrt {\frac {d x^8+c}{\left (\sqrt {d} x^4+\sqrt {c}\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right )^2}{4 \sqrt {-a} \sqrt {b} \sqrt {c} \sqrt {d}},2 \arctan \left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right ),\frac {1}{2}\right ) \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right )^2}{8 \sqrt {-a} b^{3/2} \sqrt [4]{c} \sqrt [4]{d} (b c+a d) \sqrt {d x^8+c}}+\frac {(3 b c-a d) \arctan \left (\frac {\sqrt {b c-a d} x^2}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {d x^8+c}}\right )}{4 \sqrt [4]{-a} b^{5/4} \sqrt {b c-a d}}-\frac {(3 b c-a d) \text {arctanh}\left (\frac {\sqrt {b c-a d} x^2}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {d x^8+c}}\right )}{4 \sqrt [4]{-a} b^{5/4} \sqrt {b c-a d}}-\frac {\sqrt [4]{c} \sqrt [4]{d} \left (\sqrt {d} x^4+\sqrt {c}\right ) \sqrt {\frac {d x^8+c}{\left (\sqrt {d} x^4+\sqrt {c}\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{b \sqrt {d x^8+c}}+\frac {\sqrt [4]{c} \sqrt [4]{d} \left (\sqrt {d} x^4+\sqrt {c}\right ) \sqrt {\frac {d x^8+c}{\left (\sqrt {d} x^4+\sqrt {c}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{2 b \sqrt {d x^8+c}}-\frac {\left (\sqrt {c}-\frac {\sqrt {-a} \sqrt {d}}{\sqrt {b}}\right ) \sqrt [4]{d} (3 b c-a d) \left (\sqrt {d} x^4+\sqrt {c}\right ) \sqrt {\frac {d x^8+c}{\left (\sqrt {d} x^4+\sqrt {c}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{4 b \sqrt [4]{c} (b c+a d) \sqrt {d x^8+c}}-\frac {\left (\sqrt {c}+\frac {\sqrt {-a} \sqrt {d}}{\sqrt {b}}\right ) \sqrt [4]{d} (3 b c-a d) \left (\sqrt {d} x^4+\sqrt {c}\right ) \sqrt {\frac {d x^8+c}{\left (\sqrt {d} x^4+\sqrt {c}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{4 b \sqrt [4]{c} (b c+a d) \sqrt {d x^8+c}}+\frac {\left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right )^2 (3 b c-a d) \left (\sqrt {d} x^4+\sqrt {c}\right ) \sqrt {\frac {d x^8+c}{\left (\sqrt {d} x^4+\sqrt {c}\right )^2}} \operatorname {EllipticPi}\left (-\frac {\sqrt {c} \left (\sqrt {b}-\frac {\sqrt {-a} \sqrt {d}}{\sqrt {c}}\right )^2}{4 \sqrt {-a} \sqrt {b} \sqrt {d}},2 \arctan \left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{8 \sqrt {-a} b^{3/2} \sqrt [4]{c} \sqrt [4]{d} (b c+a d) \sqrt {d x^8+c}}+\frac {\sqrt {d} x^2 \sqrt {d x^8+c}}{b \left (\sqrt {d} x^4+\sqrt {c}\right )}}{4 (b c-a d)}-\frac {x^6 \sqrt {d x^8+c}}{4 (b c-a d) \left (b x^8+a\right )}\right )\)

Input:

Int[x^13/((a + b*x^8)^2*Sqrt[c + d*x^8]),x]
 

Output:

(-1/4*(x^6*Sqrt[c + d*x^8])/((b*c - a*d)*(a + b*x^8)) + ((Sqrt[d]*x^2*Sqrt 
[c + d*x^8])/(b*(Sqrt[c] + Sqrt[d]*x^4)) + ((3*b*c - a*d)*ArcTan[(Sqrt[b*c 
 - a*d]*x^2)/((-a)^(1/4)*b^(1/4)*Sqrt[c + d*x^8])])/(4*(-a)^(1/4)*b^(5/4)* 
Sqrt[b*c - a*d]) - ((3*b*c - a*d)*ArcTanh[(Sqrt[b*c - a*d]*x^2)/((-a)^(1/4 
)*b^(1/4)*Sqrt[c + d*x^8])])/(4*(-a)^(1/4)*b^(5/4)*Sqrt[b*c - a*d]) - (c^( 
1/4)*d^(1/4)*(Sqrt[c] + Sqrt[d]*x^4)*Sqrt[(c + d*x^8)/(Sqrt[c] + Sqrt[d]*x 
^4)^2]*EllipticE[2*ArcTan[(d^(1/4)*x^2)/c^(1/4)], 1/2])/(b*Sqrt[c + d*x^8] 
) + (c^(1/4)*d^(1/4)*(Sqrt[c] + Sqrt[d]*x^4)*Sqrt[(c + d*x^8)/(Sqrt[c] + S 
qrt[d]*x^4)^2]*EllipticF[2*ArcTan[(d^(1/4)*x^2)/c^(1/4)], 1/2])/(2*b*Sqrt[ 
c + d*x^8]) - ((Sqrt[c] - (Sqrt[-a]*Sqrt[d])/Sqrt[b])*d^(1/4)*(3*b*c - a*d 
)*(Sqrt[c] + Sqrt[d]*x^4)*Sqrt[(c + d*x^8)/(Sqrt[c] + Sqrt[d]*x^4)^2]*Elli 
pticF[2*ArcTan[(d^(1/4)*x^2)/c^(1/4)], 1/2])/(4*b*c^(1/4)*(b*c + a*d)*Sqrt 
[c + d*x^8]) - ((Sqrt[c] + (Sqrt[-a]*Sqrt[d])/Sqrt[b])*d^(1/4)*(3*b*c - a* 
d)*(Sqrt[c] + Sqrt[d]*x^4)*Sqrt[(c + d*x^8)/(Sqrt[c] + Sqrt[d]*x^4)^2]*Ell 
ipticF[2*ArcTan[(d^(1/4)*x^2)/c^(1/4)], 1/2])/(4*b*c^(1/4)*(b*c + a*d)*Sqr 
t[c + d*x^8]) - ((Sqrt[b]*Sqrt[c] - Sqrt[-a]*Sqrt[d])^2*(3*b*c - a*d)*(Sqr 
t[c] + Sqrt[d]*x^4)*Sqrt[(c + d*x^8)/(Sqrt[c] + Sqrt[d]*x^4)^2]*EllipticPi 
[(Sqrt[b]*Sqrt[c] + Sqrt[-a]*Sqrt[d])^2/(4*Sqrt[-a]*Sqrt[b]*Sqrt[c]*Sqrt[d 
]), 2*ArcTan[(d^(1/4)*x^2)/c^(1/4)], 1/2])/(8*Sqrt[-a]*b^(3/2)*c^(1/4)*d^( 
1/4)*(b*c + a*d)*Sqrt[c + d*x^8]) + ((Sqrt[b]*Sqrt[c] + Sqrt[-a]*Sqrt[d...
 

Defintions of rubi rules used

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]
 

rule 971

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

rule 1054

Int[(((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n 
_)))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[(g*x)^m*(a 
+ b*x^n)^p*((e + f*x^n)/(c + d*x^n)), x], x] /; FreeQ[{a, b, c, d, e, f, g, 
 m, p}, x] && IGtQ[n, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [F]

\[\int \frac {x^{13}}{\left (b \,x^{8}+a \right )^{2} \sqrt {d \,x^{8}+c}}d x\]

Input:

int(x^13/(b*x^8+a)^2/(d*x^8+c)^(1/2),x)
 

Output:

int(x^13/(b*x^8+a)^2/(d*x^8+c)^(1/2),x)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {x^{13}}{\left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx=\text {Timed out} \] Input:

integrate(x^13/(b*x^8+a)^2/(d*x^8+c)^(1/2),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {x^{13}}{\left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx=\int \frac {x^{13}}{\left (a + b x^{8}\right )^{2} \sqrt {c + d x^{8}}}\, dx \] Input:

integrate(x**13/(b*x**8+a)**2/(d*x**8+c)**(1/2),x)
 

Output:

Integral(x**13/((a + b*x**8)**2*sqrt(c + d*x**8)), x)
 

Maxima [F]

\[ \int \frac {x^{13}}{\left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx=\int { \frac {x^{13}}{{\left (b x^{8} + a\right )}^{2} \sqrt {d x^{8} + c}} \,d x } \] Input:

integrate(x^13/(b*x^8+a)^2/(d*x^8+c)^(1/2),x, algorithm="maxima")
 

Output:

integrate(x^13/((b*x^8 + a)^2*sqrt(d*x^8 + c)), x)
 

Giac [F]

\[ \int \frac {x^{13}}{\left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx=\int { \frac {x^{13}}{{\left (b x^{8} + a\right )}^{2} \sqrt {d x^{8} + c}} \,d x } \] Input:

integrate(x^13/(b*x^8+a)^2/(d*x^8+c)^(1/2),x, algorithm="giac")
 

Output:

integrate(x^13/((b*x^8 + a)^2*sqrt(d*x^8 + c)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^{13}}{\left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx=\int \frac {x^{13}}{{\left (b\,x^8+a\right )}^2\,\sqrt {d\,x^8+c}} \,d x \] Input:

int(x^13/((a + b*x^8)^2*(c + d*x^8)^(1/2)),x)
 

Output:

int(x^13/((a + b*x^8)^2*(c + d*x^8)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {x^{13}}{\left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx=\int \frac {\sqrt {d \,x^{8}+c}\, x^{13}}{b^{2} d \,x^{24}+2 a b d \,x^{16}+b^{2} c \,x^{16}+a^{2} d \,x^{8}+2 a b c \,x^{8}+a^{2} c}d x \] Input:

int(x^13/(b*x^8+a)^2/(d*x^8+c)^(1/2),x)
 

Output:

int((sqrt(c + d*x**8)*x**13)/(a**2*c + a**2*d*x**8 + 2*a*b*c*x**8 + 2*a*b* 
d*x**16 + b**2*c*x**16 + b**2*d*x**24),x)