\(\int \frac {x^5}{(a+b x^8)^2 \sqrt {c+d x^8}} \, dx\) [129]

Optimal result

Integrand size = 24, antiderivative size = 1187 \[ \int \frac {x^5}{\left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx =\text {Too large to display} \] Output:

-1/8*d^(1/2)*x^2*(d*x^8+c)^(1/2)/a/(-a*d+b*c)/(c^(1/2)+d^(1/2)*x^4)+1/8*b* 
x^6*(d*x^8+c)^(1/2)/a/(-a*d+b*c)/(b*x^8+a)-1/32*(-3*a*d+b*c)*arctan((-a*d+ 
b*c)^(1/2)*x^2/(-a)^(1/4)/b^(1/4)/(d*x^8+c)^(1/2))/(-a)^(5/4)/b^(1/4)/(-a* 
d+b*c)^(3/2)+1/32*(-3*a*d+b*c)*arctanh((-a*d+b*c)^(1/2)*x^2/(-a)^(1/4)/b^( 
1/4)/(d*x^8+c)^(1/2))/(-a)^(5/4)/b^(1/4)/(-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)*Ellipt 
icE(sin(2*arctan(d^(1/4)*x^2/c^(1/4))),1/2*2^(1/2))/a/(-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 
))/a/(-a*d+b*c)/(d*x^8+c)^(1/2)-1/32*d^(1/4)*(-3*a*d+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))/a/b^(1/2)/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)*(-3*a*d+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))/a/b^(1/2)/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))*(-3*a*d+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))/a/b^(1/2)/c^(1/4)/((-a)^(1/2)*b^(1/2)*c^(1/2)+a*d^(1/2))/d^(...
 

Mathematica [C] (warning: unable to verify)

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

Time = 10.15 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.14 \[ \int \frac {x^5}{\left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx=\frac {x^6 \left (21 a b \left (c+d x^8\right )+7 (b c-4 a d) \left (a+b x^8\right ) \sqrt {1+\frac {d x^8}{c}} \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},-\frac {d x^8}{c},-\frac {b x^8}{a}\right )-3 b d x^8 \left (a+b x^8\right ) \sqrt {1+\frac {d x^8}{c}} \operatorname {AppellF1}\left (\frac {7}{4},\frac {1}{2},1,\frac {11}{4},-\frac {d x^8}{c},-\frac {b x^8}{a}\right )\right )}{168 a^2 (b c-a d) \left (a+b x^8\right ) \sqrt {c+d x^8}} \] Input:

Integrate[x^5/((a + b*x^8)^2*Sqrt[c + d*x^8]),x]
 

Output:

(x^6*(21*a*b*(c + d*x^8) + 7*(b*c - 4*a*d)*(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)] - 3*b*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)]))/(168*a^2*(b*c - a*d)*(a + b*x^8)*Sqrt[c + d*x^8])
 

Rubi [A] (verified)

Time = 2.23 (sec) , antiderivative size = 1093, normalized size of antiderivative = 0.92, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {965, 972, 25, 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.

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

\(\Big \downarrow \) 965

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

\(\Big \downarrow \) 972

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1054

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

\(\Big \downarrow \) 2009

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

Input:

Int[x^5/((a + b*x^8)^2*Sqrt[c + d*x^8]),x]
 

Output:

((b*x^6*Sqrt[c + d*x^8])/(4*a*(b*c - a*d)*(a + b*x^8)) + (-((Sqrt[d]*x^2*S 
qrt[c + d*x^8])/(Sqrt[c] + Sqrt[d]*x^4)) + ((b*c - 3*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^(1/4)* 
Sqrt[b*c - a*d]) - ((b*c - 3*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^(1/4)*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])/Sqrt[c + d*x^8] - 
(c^(1/4)*d^(1/4)*(Sqrt[c] + Sqrt[d]*x^4)*Sqrt[(c + d*x^8)/(Sqrt[c] + Sqrt[ 
d]*x^4)^2]*EllipticF[2*ArcTan[(d^(1/4)*x^2)/c^(1/4)], 1/2])/(2*Sqrt[c + d* 
x^8]) - ((Sqrt[c] - (Sqrt[-a]*Sqrt[d])/Sqrt[b])*d^(1/4)*(b*c - 3*a*d)*(Sqr 
t[c] + Sqrt[d]*x^4)*Sqrt[(c + d*x^8)/(Sqrt[c] + Sqrt[d]*x^4)^2]*EllipticF[ 
2*ArcTan[(d^(1/4)*x^2)/c^(1/4)], 1/2])/(4*c^(1/4)*(b*c + a*d)*Sqrt[c + d*x 
^8]) - ((Sqrt[c] + (Sqrt[-a]*Sqrt[d])/Sqrt[b])*d^(1/4)*(b*c - 3*a*d)*(Sqrt 
[c] + Sqrt[d]*x^4)*Sqrt[(c + d*x^8)/(Sqrt[c] + Sqrt[d]*x^4)^2]*EllipticF[2 
*ArcTan[(d^(1/4)*x^2)/c^(1/4)], 1/2])/(4*c^(1/4)*(b*c + a*d)*Sqrt[c + d*x^ 
8]) - ((Sqrt[b]*Sqrt[c] - Sqrt[-a]*Sqrt[d])^2*(b*c - 3*a*d)*(Sqrt[c] + Sqr 
t[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*ArcT 
an[(d^(1/4)*x^2)/c^(1/4)], 1/2])/(8*Sqrt[-a]*Sqrt[b]*c^(1/4)*d^(1/4)*(b*c 
+ a*d)*Sqrt[c + d*x^8]) + ((Sqrt[b]*Sqrt[c] + Sqrt[-a]*Sqrt[d])^2*(b*c ...
 

Defintions of rubi rules used

rule 25

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

rule 972

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

\[\int \frac {x^{5}}{\left (b \,x^{8}+a \right )^{2} \sqrt {d \,x^{8}+c}}d x\]

Input:

int(x^5/(b*x^8+a)^2/(d*x^8+c)^(1/2),x)
 

Output:

int(x^5/(b*x^8+a)^2/(d*x^8+c)^(1/2),x)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {x^5}{\left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx=\text {Timed out} \] Input:

integrate(x^5/(b*x^8+a)^2/(d*x^8+c)^(1/2),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {x^5}{\left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx=\int \frac {x^{5}}{\left (a + b x^{8}\right )^{2} \sqrt {c + d x^{8}}}\, dx \] Input:

integrate(x**5/(b*x**8+a)**2/(d*x**8+c)**(1/2),x)
 

Output:

Integral(x**5/((a + b*x**8)**2*sqrt(c + d*x**8)), x)
 

Maxima [F]

\[ \int \frac {x^5}{\left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx=\int { \frac {x^{5}}{{\left (b x^{8} + a\right )}^{2} \sqrt {d x^{8} + c}} \,d x } \] Input:

integrate(x^5/(b*x^8+a)^2/(d*x^8+c)^(1/2),x, algorithm="maxima")
 

Output:

integrate(x^5/((b*x^8 + a)^2*sqrt(d*x^8 + c)), x)
 

Giac [F]

\[ \int \frac {x^5}{\left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx=\int { \frac {x^{5}}{{\left (b x^{8} + a\right )}^{2} \sqrt {d x^{8} + c}} \,d x } \] Input:

integrate(x^5/(b*x^8+a)^2/(d*x^8+c)^(1/2),x, algorithm="giac")
 

Output:

integrate(x^5/((b*x^8 + a)^2*sqrt(d*x^8 + c)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^5}{\left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx=\int \frac {x^5}{{\left (b\,x^8+a\right )}^2\,\sqrt {d\,x^8+c}} \,d x \] Input:

int(x^5/((a + b*x^8)^2*(c + d*x^8)^(1/2)),x)
 

Output:

int(x^5/((a + b*x^8)^2*(c + d*x^8)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {x^5}{\left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx=\int \frac {\sqrt {d \,x^{8}+c}\, x^{5}}{b^{2} d \,x^{24}+2 a b d \,x^{16}+b^{2} c \,x^{16}+a^{2} d \,x^{8}+2 a b c \,x^{8}+a^{2} c}d x \] Input:

int(x^5/(b*x^8+a)^2/(d*x^8+c)^(1/2),x)
 

Output:

int((sqrt(c + d*x**8)*x**5)/(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)