\(\int \frac {x}{(a+b x^8)^2 \sqrt {c+d x^8}} \, dx\) [126]

Optimal result

Integrand size = 22, antiderivative size = 735 \[ \int \frac {x}{\left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx=\frac {b x^2 \sqrt {c+d x^8}}{8 a (b c-a d) \left (a+b x^8\right )}-\frac {\sqrt [4]{b} (3 b c-5 a d) \arctan \left (\frac {\sqrt {-b c+a d} x^2}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {c+d x^8}}\right )}{32 (-a)^{7/4} (-b c+a d)^{3/2}}-\frac {\sqrt [4]{b} (3 b c-5 a d) \text {arctanh}\left (\frac {\sqrt {-b c+a d} x^2}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {c+d x^8}}\right )}{32 (-a)^{7/4} (-b c+a d)^{3/2}}+\frac {d^{3/4} \left (\sqrt {c}+\sqrt {d} x^4\right ) \sqrt {\frac {c+d x^8}{\left (\sqrt {c}+\sqrt {d} x^4\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{4 a \sqrt [4]{c} (b c+a d) \sqrt {c+d x^8}}+\frac {\left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right ) (3 b c-5 a d) \left (\sqrt {c}+\sqrt {d} x^4\right ) \sqrt {\frac {c+d x^8}{\left (\sqrt {c}+\sqrt {d} x^4\right )^2}} \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right )^2}{4 \sqrt {-a} \sqrt {b} \sqrt {c} \sqrt {d}},2 \arctan \left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{64 a^2 \sqrt [4]{c} \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right ) \sqrt [4]{d} (b c-a d) \sqrt {c+d x^8}}+\frac {\left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right ) (3 b c-5 a d) \left (\sqrt {c}+\sqrt {d} x^4\right ) \sqrt {\frac {c+d x^8}{\left (\sqrt {c}+\sqrt {d} x^4\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right )^2}{4 \sqrt {-a} \sqrt {b} \sqrt {c} \sqrt {d}},2 \arctan \left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{64 a^2 \sqrt [4]{c} \left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right ) \sqrt [4]{d} (b c-a d) \sqrt {c+d x^8}} \] Output:

1/8*b*x^2*(d*x^8+c)^(1/2)/a/(-a*d+b*c)/(b*x^8+a)-1/32*b^(1/4)*(-5*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)^(7/ 
4)/(a*d-b*c)^(3/2)-1/32*b^(1/4)*(-5*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)^(7/4)/(a*d-b*c)^(3/2)+1/4*d^(3/4 
)*(c^(1/2)+d^(1/2)*x^4)*((d*x^8+c)/(c^(1/2)+d^(1/2)*x^4)^2)^(1/2)*InverseJ 
acobiAM(2*arctan(d^(1/4)*x^2/c^(1/4)),1/2*2^(1/2))/a/c^(1/4)/(a*d+b*c)/(d* 
x^8+c)^(1/2)+1/64*(b^(1/2)*c^(1/2)+(-a)^(1/2)*d^(1/2))*(-5*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))/a^2/c^(1/4)/(b^(1/2)*c^( 
1/2)-(-a)^(1/2)*d^(1/2))/d^(1/4)/(-a*d+b*c)/(d*x^8+c)^(1/2)+1/64*(b^(1/2)* 
c^(1/2)-(-a)^(1/2)*d^(1/2))*(-5*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))/a^2/c^(1/4)/(b^(1/2)*c^(1/2)+(-a)^(1/2)*d^(1/2))/d^( 
1/4)/(-a*d+b*c)/(d*x^8+c)^(1/2)
 

Mathematica [C] (warning: unable to verify)

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

Time = 10.14 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.23 \[ \int \frac {x}{\left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx=\frac {x^2 \left (5 a b \left (c+d x^8\right )+5 (3 b c-4 a d) \left (a+b x^8\right ) \sqrt {1+\frac {d x^8}{c}} \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},-\frac {d x^8}{c},-\frac {b x^8}{a}\right )+b d x^8 \left (a+b x^8\right ) \sqrt {1+\frac {d x^8}{c}} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},-\frac {d x^8}{c},-\frac {b x^8}{a}\right )\right )}{40 a^2 (b c-a d) \left (a+b x^8\right ) \sqrt {c+d x^8}} \] Input:

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

Output:

(x^2*(5*a*b*(c + d*x^8) + 5*(3*b*c - 4*a*d)*(a + b*x^8)*Sqrt[1 + (d*x^8)/c 
]*AppellF1[1/4, 1/2, 1, 5/4, -((d*x^8)/c), -((b*x^8)/a)] + b*d*x^8*(a + b* 
x^8)*Sqrt[1 + (d*x^8)/c]*AppellF1[5/4, 1/2, 1, 9/4, -((d*x^8)/c), -((b*x^8 
)/a)]))/(40*a^2*(b*c - a*d)*(a + b*x^8)*Sqrt[c + d*x^8])
 

Rubi [A] (warning: unable to verify)

Time = 2.44 (sec) , antiderivative size = 1036, normalized size of antiderivative = 1.41, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {965, 931, 25, 1021, 761, 925, 1541, 27, 761, 2221, 2223}

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.

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

\(\Big \downarrow \) 965

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

\(\Big \downarrow \) 931

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1021

\(\displaystyle \frac {1}{2} \left (\frac {(3 b c-5 a d) \int \frac {1}{\left (b x^8+a\right ) \sqrt {d x^8+c}}dx^2+d \int \frac {1}{\sqrt {d x^8+c}}dx^2}{4 a (b c-a d)}+\frac {b x^2 \sqrt {c+d x^8}}{4 a \left (a+b x^8\right ) (b c-a d)}\right )\)

\(\Big \downarrow \) 761

\(\displaystyle \frac {1}{2} \left (\frac {(3 b c-5 a d) \int \frac {1}{\left (b x^8+a\right ) \sqrt {d x^8+c}}dx^2+\frac {d^{3/4} \left (\sqrt {c}+\sqrt {d} x^4\right ) \sqrt {\frac {c+d x^8}{\left (\sqrt {c}+\sqrt {d} x^4\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{c} \sqrt {c+d x^8}}}{4 a (b c-a d)}+\frac {b x^2 \sqrt {c+d x^8}}{4 a \left (a+b x^8\right ) (b c-a d)}\right )\)

\(\Big \downarrow \) 925

\(\displaystyle \frac {1}{2} \left (\frac {(3 b c-5 a d) \left (\frac {\int \frac {1}{\left (1-\frac {\sqrt {b} x^4}{\sqrt {-a}}\right ) \sqrt {d x^8+c}}dx^2}{2 a}+\frac {\int \frac {1}{\left (\frac {\sqrt {b} x^4}{\sqrt {-a}}+1\right ) \sqrt {d x^8+c}}dx^2}{2 a}\right )+\frac {d^{3/4} \left (\sqrt {c}+\sqrt {d} x^4\right ) \sqrt {\frac {c+d x^8}{\left (\sqrt {c}+\sqrt {d} x^4\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{c} \sqrt {c+d x^8}}}{4 a (b c-a d)}+\frac {b x^2 \sqrt {c+d x^8}}{4 a \left (a+b x^8\right ) (b c-a d)}\right )\)

\(\Big \downarrow \) 1541

\(\displaystyle \frac {1}{2} \left (\frac {(3 b c-5 a d) \left (\frac {\frac {\sqrt {d} \left (\sqrt {-a} \sqrt {b} \sqrt {c}+a \sqrt {d}\right ) \int \frac {1}{\sqrt {d x^8+c}}dx^2}{a d+b c}+\frac {\sqrt {b} \sqrt {c} \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right ) \int \frac {\sqrt {d} x^4+\sqrt {c}}{\sqrt {c} \left (1-\frac {\sqrt {b} x^4}{\sqrt {-a}}\right ) \sqrt {d x^8+c}}dx^2}{a d+b c}}{2 a}+\frac {\frac {a \sqrt {d} \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {-a}}+\sqrt {d}\right ) \int \frac {1}{\sqrt {d x^8+c}}dx^2}{a d+b c}+\frac {\sqrt {b} \sqrt {c} \left (\sqrt {-a} \sqrt {d}+\sqrt {b} \sqrt {c}\right ) \int \frac {\sqrt {d} x^4+\sqrt {c}}{\sqrt {c} \left (\frac {\sqrt {b} x^4}{\sqrt {-a}}+1\right ) \sqrt {d x^8+c}}dx^2}{a d+b c}}{2 a}\right )+\frac {d^{3/4} \left (\sqrt {c}+\sqrt {d} x^4\right ) \sqrt {\frac {c+d x^8}{\left (\sqrt {c}+\sqrt {d} x^4\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{c} \sqrt {c+d x^8}}}{4 a (b c-a d)}+\frac {b x^2 \sqrt {c+d x^8}}{4 a \left (a+b x^8\right ) (b c-a d)}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} \left (\frac {(3 b c-5 a d) \left (\frac {\frac {\sqrt {d} \left (\sqrt {-a} \sqrt {b} \sqrt {c}+a \sqrt {d}\right ) \int \frac {1}{\sqrt {d x^8+c}}dx^2}{a d+b c}+\frac {\sqrt {b} \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right ) \int \frac {\sqrt {d} x^4+\sqrt {c}}{\left (1-\frac {\sqrt {b} x^4}{\sqrt {-a}}\right ) \sqrt {d x^8+c}}dx^2}{a d+b c}}{2 a}+\frac {\frac {a \sqrt {d} \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {-a}}+\sqrt {d}\right ) \int \frac {1}{\sqrt {d x^8+c}}dx^2}{a d+b c}+\frac {\sqrt {b} \left (\sqrt {-a} \sqrt {d}+\sqrt {b} \sqrt {c}\right ) \int \frac {\sqrt {d} x^4+\sqrt {c}}{\left (\frac {\sqrt {b} x^4}{\sqrt {-a}}+1\right ) \sqrt {d x^8+c}}dx^2}{a d+b c}}{2 a}\right )+\frac {d^{3/4} \left (\sqrt {c}+\sqrt {d} x^4\right ) \sqrt {\frac {c+d x^8}{\left (\sqrt {c}+\sqrt {d} x^4\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{c} \sqrt {c+d x^8}}}{4 a (b c-a d)}+\frac {b x^2 \sqrt {c+d x^8}}{4 a \left (a+b x^8\right ) (b c-a d)}\right )\)

\(\Big \downarrow \) 761

\(\displaystyle \frac {1}{2} \left (\frac {(3 b c-5 a d) \left (\frac {\frac {\sqrt {b} \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right ) \int \frac {\sqrt {d} x^4+\sqrt {c}}{\left (1-\frac {\sqrt {b} x^4}{\sqrt {-a}}\right ) \sqrt {d x^8+c}}dx^2}{a d+b c}+\frac {\sqrt [4]{d} \left (\sqrt {c}+\sqrt {d} x^4\right ) \sqrt {\frac {c+d x^8}{\left (\sqrt {c}+\sqrt {d} x^4\right )^2}} \left (\sqrt {-a} \sqrt {b} \sqrt {c}+a \sqrt {d}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{c} \sqrt {c+d x^8} (a d+b c)}}{2 a}+\frac {\frac {\sqrt {b} \left (\sqrt {-a} \sqrt {d}+\sqrt {b} \sqrt {c}\right ) \int \frac {\sqrt {d} x^4+\sqrt {c}}{\left (\frac {\sqrt {b} x^4}{\sqrt {-a}}+1\right ) \sqrt {d x^8+c}}dx^2}{a d+b c}+\frac {a \sqrt [4]{d} \left (\sqrt {c}+\sqrt {d} x^4\right ) \sqrt {\frac {c+d x^8}{\left (\sqrt {c}+\sqrt {d} x^4\right )^2}} \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {-a}}+\sqrt {d}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{c} \sqrt {c+d x^8} (a d+b c)}}{2 a}\right )+\frac {d^{3/4} \left (\sqrt {c}+\sqrt {d} x^4\right ) \sqrt {\frac {c+d x^8}{\left (\sqrt {c}+\sqrt {d} x^4\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{c} \sqrt {c+d x^8}}}{4 a (b c-a d)}+\frac {b x^2 \sqrt {c+d x^8}}{4 a \left (a+b x^8\right ) (b c-a d)}\right )\)

\(\Big \downarrow \) 2221

\(\displaystyle \frac {1}{2} \left (\frac {b \sqrt {d x^8+c} x^2}{4 a (b c-a d) \left (b x^8+a\right )}+\frac {\frac {d^{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 {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{c} \sqrt {d x^8+c}}+(3 b c-5 a d) \left (\frac {\frac {a \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {-a}}+\sqrt {d}\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 )}{2 \sqrt [4]{c} (b c+a d) \sqrt {d x^8+c}}+\frac {\sqrt {b} \left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right ) \left (\frac {(-a)^{3/4} \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {-a}}-\sqrt {d}\right ) \arctan \left (\frac {\sqrt {b c-a d} x^2}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {d x^8+c}}\right )}{2 \sqrt [4]{b} \sqrt {b c-a d}}+\frac {\left (\sqrt {c}+\frac {\sqrt {-a} \sqrt {d}}{\sqrt {b}}\right ) \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 )}{4 \sqrt [4]{c} \sqrt [4]{d} \sqrt {d x^8+c}}\right )}{b c+a d}}{2 a}+\frac {\frac {\left (\sqrt {d} a+\sqrt {-a} \sqrt {b} \sqrt {c}\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 )}{2 \sqrt [4]{c} (b c+a d) \sqrt {d x^8+c}}+\frac {\sqrt {b} \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right ) \int \frac {\sqrt {d} x^4+\sqrt {c}}{\left (1-\frac {\sqrt {b} x^4}{\sqrt {-a}}\right ) \sqrt {d x^8+c}}dx^2}{b c+a d}}{2 a}\right )}{4 a (b c-a d)}\right )\)

\(\Big \downarrow \) 2223

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

Input:

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

Output:

((b*x^2*Sqrt[c + d*x^8])/(4*a*(b*c - a*d)*(a + b*x^8)) + ((d^(3/4)*(Sqrt[c 
] + Sqrt[d]*x^4)*Sqrt[(c + d*x^8)/(Sqrt[c] + Sqrt[d]*x^4)^2]*EllipticF[2*A 
rcTan[(d^(1/4)*x^2)/c^(1/4)], 1/2])/(2*c^(1/4)*Sqrt[c + d*x^8]) + (3*b*c - 
 5*a*d)*(((a*((Sqrt[b]*Sqrt[c])/Sqrt[-a] + Sqrt[d])*d^(1/4)*(Sqrt[c] + Sqr 
t[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*c^(1/4)*(b*c + a*d)*Sqrt[c + d*x^8]) + (Sq 
rt[b]*(Sqrt[b]*Sqrt[c] + Sqrt[-a]*Sqrt[d])*(((-a)^(3/4)*((Sqrt[b]*Sqrt[c]) 
/Sqrt[-a] - Sqrt[d])*ArcTan[(Sqrt[b*c - a*d]*x^2)/((-a)^(1/4)*b^(1/4)*Sqrt 
[c + d*x^8])])/(2*b^(1/4)*Sqrt[b*c - a*d]) + ((Sqrt[c] + (Sqrt[-a]*Sqrt[d] 
)/Sqrt[b])*(Sqrt[c] + Sqrt[d]*x^4)*Sqrt[(c + d*x^8)/(Sqrt[c] + Sqrt[d]*x^4 
)^2]*EllipticPi[-1/4*(Sqrt[b]*Sqrt[c] - Sqrt[-a]*Sqrt[d])^2/(Sqrt[-a]*Sqrt 
[b]*Sqrt[c]*Sqrt[d]), 2*ArcTan[(d^(1/4)*x^2)/c^(1/4)], 1/2])/(4*c^(1/4)*d^ 
(1/4)*Sqrt[c + d*x^8])))/(b*c + a*d))/(2*a) + (((Sqrt[-a]*Sqrt[b]*Sqrt[c] 
+ a*Sqrt[d])*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*c^(1/4) 
*(b*c + a*d)*Sqrt[c + d*x^8]) + (Sqrt[b]*(Sqrt[b]*Sqrt[c] - Sqrt[-a]*Sqrt[ 
d])*(((-a)^(1/4)*(Sqrt[b]*Sqrt[c] + Sqrt[-a]*Sqrt[d])*ArcTanh[(Sqrt[b*c - 
a*d]*x^2)/((-a)^(1/4)*b^(1/4)*Sqrt[c + d*x^8])])/(2*b^(1/4)*Sqrt[b*c - a*d 
]) + ((Sqrt[c] - (Sqrt[-a]*Sqrt[d])/Sqrt[b])*(Sqrt[c] + Sqrt[d]*x^4)*Sqrt[ 
(c + d*x^8)/(Sqrt[c] + Sqrt[d]*x^4)^2]*EllipticPi[(Sqrt[b]*Sqrt[c] + Sq...
 

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 761

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[( 
1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))* 
EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
 

rule 925

Int[1/(Sqrt[(a_) + (b_.)*(x_)^4]*((c_) + (d_.)*(x_)^4)), x_Symbol] :> Simp[ 
1/(2*c)   Int[1/(Sqrt[a + b*x^4]*(1 - Rt[-d/c, 2]*x^2)), x], x] + Simp[1/(2 
*c)   Int[1/(Sqrt[a + b*x^4]*(1 + Rt[-d/c, 2]*x^2)), x], x] /; FreeQ[{a, b, 
 c, d}, x] && NeQ[b*c - a*d, 0]
 

rule 931

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

rule 1021

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*Sqrt[(c_) + (d_.)*(x 
_)^(n_)]), x_Symbol] :> Simp[f/b   Int[1/Sqrt[c + d*x^n], x], x] + Simp[(b* 
e - a*f)/b   Int[1/((a + b*x^n)*Sqrt[c + d*x^n]), x], x] /; FreeQ[{a, b, c, 
 d, e, f, n}, x]
 

rule 1541

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[ 
{q = Rt[c/a, 2]}, Simp[(c*d + a*e*q)/(c*d^2 - a*e^2)   Int[1/Sqrt[a + c*x^4 
], x], x] - Simp[(a*e*(e + d*q))/(c*d^2 - a*e^2)   Int[(1 + q*x^2)/((d + e* 
x^2)*Sqrt[a + c*x^4]), x], x]] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e 
^2, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a]
 

rule 2221

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]) 
, x_Symbol] :> With[{q = Rt[B/A, 2]}, Simp[(-(B*d - A*e))*(ArcTan[Rt[c*(d/e 
) + a*(e/d), 2]*(x/Sqrt[a + c*x^4])]/(2*d*e*Rt[c*(d/e) + a*(e/d), 2])), x] 
+ Simp[(B*d + A*e)*(1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(4* 
d*e*q*Sqrt[a + c*x^4]))*EllipticPi[-(e - d*q^2)^2/(4*d*e*q^2), 2*ArcTan[q*x 
], 1/2], x]] /; FreeQ[{a, c, d, e, A, B}, x] && NeQ[c*d^2 - a*e^2, 0] && Po 
sQ[c/a] && EqQ[c*A^2 - a*B^2, 0] && PosQ[B/A] && PosQ[c*(d/e) + a*(e/d)]
 

rule 2223

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]) 
, x_Symbol] :> With[{q = Rt[B/A, 2]}, Simp[(-(B*d - A*e))*(ArcTanh[Rt[(-c)* 
(d/e) - a*(e/d), 2]*(x/Sqrt[a + c*x^4])]/(2*d*e*Rt[(-c)*(d/e) - a*(e/d), 2] 
)), x] + Simp[(B*d + A*e)*(1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^ 
2)]/(4*d*e*q*Sqrt[a + c*x^4]))*EllipticPi[-(e - d*q^2)^2/(4*d*e*q^2), 2*Arc 
Tan[q*x], 1/2], x]] /; FreeQ[{a, c, d, e, A, B}, x] && NeQ[c*d^2 - a*e^2, 0 
] && PosQ[c/a] && EqQ[c*A^2 - a*B^2, 0] && PosQ[B/A] && NegQ[c*(d/e) + a*(e 
/d)]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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