\(\int \frac {1}{x^3 (a+b x^8) \sqrt {c+d x^8}} \, dx\) [107]

Optimal result

Integrand size = 24, antiderivative size = 1053 \[ \int \frac {1}{x^3 \left (a+b x^8\right ) \sqrt {c+d x^8}} \, dx =\text {Too large to display} \] Output:

-1/2*(d*x^8+c)^(1/2)/a/c/x^2+1/2*d^(1/2)*x^2*(d*x^8+c)^(1/2)/a/c/(c^(1/2)+ 
d^(1/2)*x^4)+1/8*b^(3/4)*arctan((-a*d+b*c)^(1/2)*x^2/(-a)^(1/4)/b^(1/4)/(d 
*x^8+c)^(1/2))/(-a)^(5/4)/(-a*d+b*c)^(1/2)-1/8*b^(3/4)*arctanh((-a*d+b*c)^ 
(1/2)*x^2/(-a)^(1/4)/b^(1/4)/(d*x^8+c)^(1/2))/(-a)^(5/4)/(-a*d+b*c)^(1/2)- 
1/2*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(sin(2*arctan(d^(1/4)*x^2/c^(1/4))),1/2*2^(1/2))/a/c^(3/4)/(d*x 
^8+c)^(1/2)+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 
/c^(3/4)/(d*x^8+c)^(1/2)+1/8*b^(1/2)*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/c^(1/4)/(b^(1/2)*c^(1/2)-(-a)^(1/2)*d^(1/2))/(d*x^8+ 
c)^(1/2)+1/8*b^(1/2)*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/c^(1/4)/(b^(1/2)*c^(1/2)+(-a)^(1/2)*d^(1/2))/(d*x^8+c)^(1/2)-1/16*b^ 
(1/2)*(b^(1/2)*c^(1/2)+(-a)^(1/2)*d^(1/2))*(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/c^(1/4)/((-a)^(1/2)*b^(1/2)*c^(1/2)+a*d^(1/2))/d^ 
(1/4)/(d*x^8+c)^(1/2)-1/16*b^(1/2)*((-a)^(1/2)*b^(1/2)*c^(1/2)+a*d^(1/2))* 
(c^(1/2)+d^(1/2)*x^4)*((d*x^8+c)/(c^(1/2)+d^(1/2)*x^4)^2)^(1/2)*Ellipti...
 

Mathematica [C] (verified)

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

Time = 10.10 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.13 \[ \int \frac {1}{x^3 \left (a+b x^8\right ) \sqrt {c+d x^8}} \, dx=\frac {-21 a \left (c+d x^8\right )+7 (-b c+a d) x^8 \sqrt {1+\frac {d x^8}{c}} \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},-\frac {d x^8}{c},-\frac {b x^8}{a}\right )+3 b d x^{16} \sqrt {1+\frac {d x^8}{c}} \operatorname {AppellF1}\left (\frac {7}{4},\frac {1}{2},1,\frac {11}{4},-\frac {d x^8}{c},-\frac {b x^8}{a}\right )}{42 a^2 c x^2 \sqrt {c+d x^8}} \] Input:

Integrate[1/(x^3*(a + b*x^8)*Sqrt[c + d*x^8]),x]
 

Output:

(-21*a*(c + d*x^8) + 7*(-(b*c) + a*d)*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^16*Sqrt[1 + (d*x^8)/c 
]*AppellF1[7/4, 1/2, 1, 11/4, -((d*x^8)/c), -((b*x^8)/a)])/(42*a^2*c*x^2*S 
qrt[c + d*x^8])
 

Rubi [A] (verified)

Time = 2.28 (sec) , antiderivative size = 1021, normalized size of antiderivative = 0.97, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {965, 980, 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.

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

\(\Big \downarrow \) 965

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

\(\Big \downarrow \) 980

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1054

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

\(\Big \downarrow \) 2009

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

Input:

Int[1/(x^3*(a + b*x^8)*Sqrt[c + d*x^8]),x]
 

Output:

(-(Sqrt[c + d*x^8]/(a*c*x^2)) - (-((Sqrt[d]*x^2*Sqrt[c + d*x^8])/(Sqrt[c] 
+ Sqrt[d]*x^4)) + (b^(3/4)*c*ArcTan[(Sqrt[b*c - a*d]*x^2)/((-a)^(1/4)*b^(1 
/4)*Sqrt[c + d*x^8])])/(4*(-a)^(1/4)*Sqrt[b*c - a*d]) - (b^(3/4)*c*ArcTanh 
[(Sqrt[b*c - a*d]*x^2)/((-a)^(1/4)*b^(1/4)*Sqrt[c + d*x^8])])/(4*(-a)^(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]) - (b*c^(3/4)*(Sqrt[c] - (Sqrt[-a]*Sqrt[d])/ 
Sqrt[b])*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])/(4*(b*c + a*d) 
*Sqrt[c + d*x^8]) - (b*c^(3/4)*(Sqrt[c] + (Sqrt[-a]*Sqrt[d])/Sqrt[b])*d^(1 
/4)*(Sqrt[c] + Sqrt[d]*x^4)*Sqrt[(c + d*x^8)/(Sqrt[c] + Sqrt[d]*x^4)^2]*El 
lipticF[2*ArcTan[(d^(1/4)*x^2)/c^(1/4)], 1/2])/(4*(b*c + a*d)*Sqrt[c + d*x 
^8]) - (Sqrt[b]*c^(3/4)*(Sqrt[b]*Sqrt[c] - Sqrt[-a]*Sqrt[d])^2*(Sqrt[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*A 
rcTan[(d^(1/4)*x^2)/c^(1/4)], 1/2])/(8*Sqrt[-a]*d^(1/4)*(b*c + a*d)*Sqrt[c 
 + d*x^8]) + (Sqrt[b]*c^(3/4)*(Sqrt[b]*Sqrt[c] + Sqrt[-a]*Sqrt[d])^2*(Sqrt 
[c] + Sqrt[d]*x^4)*Sqrt[(c + d*x^8)/(Sqrt[c] + Sqrt[d]*x^4)^2]*Elliptic...
 

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 980

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

\[\int \frac {1}{x^{3} \left (b \,x^{8}+a \right ) \sqrt {d \,x^{8}+c}}d x\]

Input:

int(1/x^3/(b*x^8+a)/(d*x^8+c)^(1/2),x)
 

Output:

int(1/x^3/(b*x^8+a)/(d*x^8+c)^(1/2),x)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{x^3 \left (a+b x^8\right ) \sqrt {c+d x^8}} \, dx=\text {Timed out} \] Input:

integrate(1/x^3/(b*x^8+a)/(d*x^8+c)^(1/2),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {1}{x^3 \left (a+b x^8\right ) \sqrt {c+d x^8}} \, dx=\int \frac {1}{x^{3} \left (a + b x^{8}\right ) \sqrt {c + d x^{8}}}\, dx \] Input:

integrate(1/x**3/(b*x**8+a)/(d*x**8+c)**(1/2),x)
 

Output:

Integral(1/(x**3*(a + b*x**8)*sqrt(c + d*x**8)), x)
 

Maxima [F]

\[ \int \frac {1}{x^3 \left (a+b x^8\right ) \sqrt {c+d x^8}} \, dx=\int { \frac {1}{{\left (b x^{8} + a\right )} \sqrt {d x^{8} + c} x^{3}} \,d x } \] Input:

integrate(1/x^3/(b*x^8+a)/(d*x^8+c)^(1/2),x, algorithm="maxima")
 

Output:

integrate(1/((b*x^8 + a)*sqrt(d*x^8 + c)*x^3), x)
 

Giac [F]

\[ \int \frac {1}{x^3 \left (a+b x^8\right ) \sqrt {c+d x^8}} \, dx=\int { \frac {1}{{\left (b x^{8} + a\right )} \sqrt {d x^{8} + c} x^{3}} \,d x } \] Input:

integrate(1/x^3/(b*x^8+a)/(d*x^8+c)^(1/2),x, algorithm="giac")
 

Output:

integrate(1/((b*x^8 + a)*sqrt(d*x^8 + c)*x^3), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^3 \left (a+b x^8\right ) \sqrt {c+d x^8}} \, dx=\int \frac {1}{x^3\,\left (b\,x^8+a\right )\,\sqrt {d\,x^8+c}} \,d x \] Input:

int(1/(x^3*(a + b*x^8)*(c + d*x^8)^(1/2)),x)
 

Output:

int(1/(x^3*(a + b*x^8)*(c + d*x^8)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {1}{x^3 \left (a+b x^8\right ) \sqrt {c+d x^8}} \, dx=\int \frac {\sqrt {d \,x^{8}+c}}{b d \,x^{19}+a d \,x^{11}+b c \,x^{11}+a c \,x^{3}}d x \] Input:

int(1/x^3/(b*x^8+a)/(d*x^8+c)^(1/2),x)
 

Output:

int(sqrt(c + d*x**8)/(a*c*x**3 + a*d*x**11 + b*c*x**11 + b*d*x**19),x)