3.1.2 \(\int \frac {1}{(d+e x)^2 \sqrt {a+c x^4}} \, dx\) [2]

3.1.2.1 Optimal result

Integrand size = 19, antiderivative size = 610 \[ \int \frac {1}{(d+e x)^2 \sqrt {a+c x^4}} \, dx=-\frac {e^3 \sqrt {a+c x^4}}{\left (c d^4+a e^4\right ) (d+e x)}+\frac {\sqrt {c} e^2 x \sqrt {a+c x^4}}{\left (c d^4+a e^4\right ) \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {c d^3 e \arctan \left (\frac {\sqrt {-c d^4-a e^4} x}{d e \sqrt {a+c x^4}}\right )}{\left (-c d^4-a e^4\right )^{3/2}}-\frac {c d^3 e \text {arctanh}\left (\frac {a e^2+c d^2 x^2}{\sqrt {c d^4+a e^4} \sqrt {a+c x^4}}\right )}{\left (c d^4+a e^4\right )^{3/2}}-\frac {\sqrt [4]{a} \sqrt [4]{c} e^2 \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\left (c d^4+a e^4\right ) \sqrt {a+c x^4}}+\frac {\sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \sqrt {a+c x^4}}-\frac {c^{3/4} d^2 \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {c} d^2+\sqrt {a} e^2\right )^2}{4 \sqrt {a} \sqrt {c} d^2 e^2},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \left (c d^4+a e^4\right ) \sqrt {a+c x^4}} \]

output

-c*d^3*e*arctan(x*(-a*e^4-c*d^4)^(1/2)/d/e/(c*x^4+a)^(1/2))/(-a*e^4-c*d^4) 
^(3/2)-c*d^3*e*arctanh((c*d^2*x^2+a*e^2)/(a*e^4+c*d^4)^(1/2)/(c*x^4+a)^(1/ 
2))/(a*e^4+c*d^4)^(3/2)-e^3*(c*x^4+a)^(1/2)/(a*e^4+c*d^4)/(e*x+d)+e^2*x*c^ 
(1/2)*(c*x^4+a)^(1/2)/(a*e^4+c*d^4)/(a^(1/2)+x^2*c^(1/2))-a^(1/4)*c^(1/4)* 
e^2*(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)^(1/2)/cos(2*arctan(c^(1/4)*x/a^(1 
/4)))*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),1/2*2^(1/2))*(a^(1/2)+x^2 
*c^(1/2))*((c*x^4+a)/(a^(1/2)+x^2*c^(1/2))^2)^(1/2)/(a*e^4+c*d^4)/(c*x^4+a 
)^(1/2)+1/2*c^(1/4)*(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)^(1/2)/cos(2*arcta 
n(c^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x/a^(1/4))),1/2*2^(1/ 
2))*(a^(1/2)+x^2*c^(1/2))*((c*x^4+a)/(a^(1/2)+x^2*c^(1/2))^2)^(1/2)/a^(1/4 
)/(e^2*a^(1/2)+d^2*c^(1/2))/(c*x^4+a)^(1/2)-1/2*c^(3/4)*d^2*(cos(2*arctan( 
c^(1/4)*x/a^(1/4)))^2)^(1/2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticPi(s 
in(2*arctan(c^(1/4)*x/a^(1/4))),1/4*(e^2*a^(1/2)+d^2*c^(1/2))^2/d^2/e^2/a^ 
(1/2)/c^(1/2),1/2*2^(1/2))*(-e^2*a^(1/2)+d^2*c^(1/2))*(a^(1/2)+x^2*c^(1/2) 
)*((c*x^4+a)/(a^(1/2)+x^2*c^(1/2))^2)^(1/2)/a^(1/4)/(a*e^4+c*d^4)/(e^2*a^( 
1/2)+d^2*c^(1/2))/(c*x^4+a)^(1/2)
 

3.1.2.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.89 (sec) , antiderivative size = 448, normalized size of antiderivative = 0.73 \[ \int \frac {1}{(d+e x)^2 \sqrt {a+c x^4}} \, dx=-\frac {\sqrt {a} \sqrt {c} e^2 \sqrt {-c d^4-a e^4} (d+e x) \sqrt {1+\frac {c x^4}{a}} E\left (\left .i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )+i \sqrt {c} \left (\sqrt {c} d^2+i \sqrt {a} e^2\right ) \sqrt {-c d^4-a e^4} (d+e x) \sqrt {1+\frac {c x^4}{a}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right ),-1\right )-\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} \left (e^3 \sqrt {-c d^4-a e^4} \left (a+c x^4\right )-2 c d^3 e (d+e x) \sqrt {a+c x^4} \arctan \left (\frac {\sqrt {c} \left (d^2-e^2 x^2\right )+e^2 \sqrt {a+c x^4}}{\sqrt {-c d^4-a e^4}}\right )+2 \sqrt [4]{-1} \sqrt [4]{a} c^{3/4} d^2 \sqrt {-c d^4-a e^4} (d+e x) \sqrt {1+\frac {c x^4}{a}} \operatorname {EllipticPi}\left (\frac {i \sqrt {a} e^2}{\sqrt {c} d^2},\arcsin \left (\frac {(-1)^{3/4} \sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} \left (-c d^4-a e^4\right )^{3/2} (d+e x) \sqrt {a+c x^4}} \]

input

Integrate[1/((d + e*x)^2*Sqrt[a + c*x^4]),x]
 

output

-((Sqrt[a]*Sqrt[c]*e^2*Sqrt[-(c*d^4) - a*e^4]*(d + e*x)*Sqrt[1 + (c*x^4)/a 
]*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1] + I*Sqrt[c]*(Sqrt[ 
c]*d^2 + I*Sqrt[a]*e^2)*Sqrt[-(c*d^4) - a*e^4]*(d + e*x)*Sqrt[1 + (c*x^4)/ 
a]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1] - Sqrt[(I*Sqrt[c] 
)/Sqrt[a]]*(e^3*Sqrt[-(c*d^4) - a*e^4]*(a + c*x^4) - 2*c*d^3*e*(d + e*x)*S 
qrt[a + c*x^4]*ArcTan[(Sqrt[c]*(d^2 - e^2*x^2) + e^2*Sqrt[a + c*x^4])/Sqrt 
[-(c*d^4) - a*e^4]] + 2*(-1)^(1/4)*a^(1/4)*c^(3/4)*d^2*Sqrt[-(c*d^4) - a*e 
^4]*(d + e*x)*Sqrt[1 + (c*x^4)/a]*EllipticPi[(I*Sqrt[a]*e^2)/(Sqrt[c]*d^2) 
, ArcSin[((-1)^(3/4)*c^(1/4)*x)/a^(1/4)], -1]))/(Sqrt[(I*Sqrt[c])/Sqrt[a]] 
*(-(c*d^4) - a*e^4)^(3/2)*(d + e*x)*Sqrt[a + c*x^4]))
 

3.1.2.3 Rubi [A] (verified)

Time = 1.41 (sec) , antiderivative size = 634, normalized size of antiderivative = 1.04, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.789, Rules used = {2265, 25, 2280, 27, 1577, 488, 219, 2233, 25, 27, 1510, 2227, 27, 761, 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.

\(\displaystyle \int \frac {1}{\sqrt {a+c x^4} (d+e x)^2} \, dx\)

\(\Big \downarrow \) 2265

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2280

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1577

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

\(\Big \downarrow \) 488

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

\(\Big \downarrow \) 219

\(\displaystyle \frac {c \left (\int \frac {d^4+2 e^2 x^2 d^2-e^4 x^4}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx+\frac {d^3 e \text {arctanh}\left (\frac {-a e^2-c d^2 x^2}{\sqrt {a+c x^4} \sqrt {a e^4+c d^4}}\right )}{\sqrt {a e^4+c d^4}}\right )}{a e^4+c d^4}-\frac {e^3 \sqrt {a+c x^4}}{(d+e x) \left (a e^4+c d^4\right )}\)

\(\Big \downarrow \) 2233

\(\displaystyle \frac {c \left (-\frac {\int -\frac {\sqrt {c} e^2 \left (\left (\sqrt {c} d^2+\sqrt {a} e^2\right ) d^2+e^2 \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) x^2\right )}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx}{c e^2}-\frac {\sqrt {a} e^2 \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {a} \sqrt {c x^4+a}}dx}{\sqrt {c}}+\frac {d^3 e \text {arctanh}\left (\frac {-a e^2-c d^2 x^2}{\sqrt {a+c x^4} \sqrt {a e^4+c d^4}}\right )}{\sqrt {a e^4+c d^4}}\right )}{a e^4+c d^4}-\frac {e^3 \sqrt {a+c x^4}}{(d+e x) \left (a e^4+c d^4\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {c \left (\frac {\int \frac {\sqrt {c} e^2 \left (\left (\sqrt {c} d^2+\sqrt {a} e^2\right ) d^2+e^2 \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) x^2\right )}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx}{c e^2}-\frac {\sqrt {a} e^2 \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {a} \sqrt {c x^4+a}}dx}{\sqrt {c}}+\frac {d^3 e \text {arctanh}\left (\frac {-a e^2-c d^2 x^2}{\sqrt {a+c x^4} \sqrt {a e^4+c d^4}}\right )}{\sqrt {a e^4+c d^4}}\right )}{a e^4+c d^4}-\frac {e^3 \sqrt {a+c x^4}}{(d+e x) \left (a e^4+c d^4\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {c \left (\frac {\int \frac {\left (\sqrt {c} d^2+\sqrt {a} e^2\right ) d^2+e^2 \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) x^2}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx}{\sqrt {c}}-\frac {e^2 \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+a}}dx}{\sqrt {c}}+\frac {d^3 e \text {arctanh}\left (\frac {-a e^2-c d^2 x^2}{\sqrt {a+c x^4} \sqrt {a e^4+c d^4}}\right )}{\sqrt {a e^4+c d^4}}\right )}{a e^4+c d^4}-\frac {e^3 \sqrt {a+c x^4}}{(d+e x) \left (a e^4+c d^4\right )}\)

\(\Big \downarrow \) 1510

\(\displaystyle \frac {c \left (\frac {\int \frac {\left (\sqrt {c} d^2+\sqrt {a} e^2\right ) d^2+e^2 \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) x^2}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx}{\sqrt {c}}-\frac {e^2 \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^4}}-\frac {x \sqrt {a+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{\sqrt {c}}+\frac {d^3 e \text {arctanh}\left (\frac {-a e^2-c d^2 x^2}{\sqrt {a+c x^4} \sqrt {a e^4+c d^4}}\right )}{\sqrt {a e^4+c d^4}}\right )}{a e^4+c d^4}-\frac {e^3 \sqrt {a+c x^4}}{(d+e x) \left (a e^4+c d^4\right )}\)

\(\Big \downarrow \) 2227

\(\displaystyle \frac {c \left (\frac {\frac {2 \sqrt {a} \sqrt {c} d^4 e^2 \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a} \left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx}{\sqrt {a} e^2+\sqrt {c} d^2}+\frac {\left (a e^4+c d^4\right ) \int \frac {1}{\sqrt {c x^4+a}}dx}{\sqrt {a} e^2+\sqrt {c} d^2}}{\sqrt {c}}-\frac {e^2 \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^4}}-\frac {x \sqrt {a+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{\sqrt {c}}+\frac {d^3 e \text {arctanh}\left (\frac {-a e^2-c d^2 x^2}{\sqrt {a+c x^4} \sqrt {a e^4+c d^4}}\right )}{\sqrt {a e^4+c d^4}}\right )}{a e^4+c d^4}-\frac {e^3 \sqrt {a+c x^4}}{(d+e x) \left (a e^4+c d^4\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {c \left (\frac {\frac {2 \sqrt {c} d^4 e^2 \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx}{\sqrt {a} e^2+\sqrt {c} d^2}+\frac {\left (a e^4+c d^4\right ) \int \frac {1}{\sqrt {c x^4+a}}dx}{\sqrt {a} e^2+\sqrt {c} d^2}}{\sqrt {c}}-\frac {e^2 \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^4}}-\frac {x \sqrt {a+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{\sqrt {c}}+\frac {d^3 e \text {arctanh}\left (\frac {-a e^2-c d^2 x^2}{\sqrt {a+c x^4} \sqrt {a e^4+c d^4}}\right )}{\sqrt {a e^4+c d^4}}\right )}{a e^4+c d^4}-\frac {e^3 \sqrt {a+c x^4}}{(d+e x) \left (a e^4+c d^4\right )}\)

\(\Big \downarrow \) 761

\(\displaystyle \frac {c \left (\frac {\frac {2 \sqrt {c} d^4 e^2 \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx}{\sqrt {a} e^2+\sqrt {c} d^2}+\frac {\left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (a e^4+c d^4\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+c x^4} \left (\sqrt {a} e^2+\sqrt {c} d^2\right )}}{\sqrt {c}}-\frac {e^2 \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^4}}-\frac {x \sqrt {a+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{\sqrt {c}}+\frac {d^3 e \text {arctanh}\left (\frac {-a e^2-c d^2 x^2}{\sqrt {a+c x^4} \sqrt {a e^4+c d^4}}\right )}{\sqrt {a e^4+c d^4}}\right )}{a e^4+c d^4}-\frac {e^3 \sqrt {a+c x^4}}{(d+e x) \left (a e^4+c d^4\right )}\)

\(\Big \downarrow \) 2223

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

input

Int[1/((d + e*x)^2*Sqrt[a + c*x^4]),x]
 

output

-((e^3*Sqrt[a + c*x^4])/((c*d^4 + a*e^4)*(d + e*x))) + (c*((d^3*e*ArcTanh[ 
(-(a*e^2) - c*d^2*x^2)/(Sqrt[c*d^4 + a*e^4]*Sqrt[a + c*x^4])])/Sqrt[c*d^4 
+ a*e^4] - (e^2*(-((x*Sqrt[a + c*x^4])/(Sqrt[a] + Sqrt[c]*x^2)) + (a^(1/4) 
*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*Ellip 
ticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(c^(1/4)*Sqrt[a + c*x^4])))/Sqrt 
[c] + (((c*d^4 + a*e^4)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] 
+ Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(2*a^(1/4 
)*c^(1/4)*(Sqrt[c]*d^2 + Sqrt[a]*e^2)*Sqrt[a + c*x^4]) + (2*Sqrt[c]*d^4*e^ 
2*(((Sqrt[c]*d^2 + Sqrt[a]*e^2)*ArcTanh[(Sqrt[c*d^4 + a*e^4]*x)/(d*e*Sqrt[ 
a + c*x^4])])/(2*d*e*Sqrt[c*d^4 + a*e^4]) + ((Sqrt[a]/d^2 - Sqrt[c]/e^2)*( 
Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*Ellipti 
cPi[(Sqrt[c]*d^2 + Sqrt[a]*e^2)^2/(4*Sqrt[a]*Sqrt[c]*d^2*e^2), 2*ArcTan[(c 
^(1/4)*x)/a^(1/4)], 1/2])/(4*a^(1/4)*c^(1/4)*Sqrt[a + c*x^4])))/(Sqrt[c]*d 
^2 + Sqrt[a]*e^2))/Sqrt[c]))/(c*d^4 + a*e^4)
 

3.1.2.3.1 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 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 488

Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> -Subst[ 
Int[1/(b*c^2 + a*d^2 - x^2), x], x, (a*d - b*c*x)/Sqrt[a + b*x^2]] /; FreeQ 
[{a, b, c, d}, 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 1510

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

rule 1577

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] 
 :> Simp[1/2   Subst[Int[(d + e*x)^q*(a + c*x^2)^p, x], x, x^2], x] /; Free 
Q[{a, c, d, e, p, q}, x]
 

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)]
 

rule 2227

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

rule 2233

Int[(P4x_)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> 
With[{q = Rt[c/a, 2], A = Coeff[P4x, x, 0], B = Coeff[P4x, x, 2], C = Coeff 
[P4x, x, 4]}, Simp[-C/(e*q)   Int[(1 - q*x^2)/Sqrt[a + c*x^4], x], x] + Sim 
p[1/(c*e)   Int[(A*c*e + a*C*d*q + (B*c*e - C*(c*d - a*e*q))*x^2)/((d + e*x 
^2)*Sqrt[a + c*x^4]), x], x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[P4x, x^2, 
2] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a]
 

rule 2265

Int[((d_) + (e_.)*(x_))^(q_)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[e 
^3*(d + e*x)^(q + 1)*(Sqrt[a + c*x^4]/((q + 1)*(c*d^4 + a*e^4))), x] + Simp 
[c/((q + 1)*(c*d^4 + a*e^4))   Int[((d + e*x)^(q + 1)/Sqrt[a + c*x^4])*Simp 
[d^3*(q + 1) - d^2*e*(q + 1)*x + d*e^2*(q + 1)*x^2 - e^3*(q + 3)*x^3, x], x 
], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^4 + a*e^4, 0] && ILtQ[q, -1]
 

rule 2280

Int[(Px_)/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Wit 
h[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1], C = Coeff[Px, x, 2], D = Coeff 
[Px, x, 3]}, Int[(x*(B*d - A*e + (d*D - C*e)*x^2))/((d^2 - e^2*x^2)*Sqrt[a 
+ c*x^4]), x] + Int[(A*d + (C*d - B*e)*x^2 - D*e*x^4)/((d^2 - e^2*x^2)*Sqrt 
[a + c*x^4]), x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[Px, x] && LeQ[Expon[Px 
, x], 3] && NeQ[c*d^4 + a*e^4, 0]
 

3.1.2.4 Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.55 (sec) , antiderivative size = 421, normalized size of antiderivative = 0.69

method	result	size
default	\(-\frac {e^{3} \sqrt {c \,x^{4}+a}}{\left (e^{4} a +d^{4} c \right ) \left (e x +d \right )}-\frac {c \,d^{2} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{\left (e^{4} a +d^{4} c \right ) \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {i e^{2} \sqrt {c}\, \sqrt {a}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\left (e^{4} a +d^{4} c \right ) \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {2 d^{3} c \left (-\frac {\operatorname {arctanh}\left (\frac {\frac {2 c \,x^{2} d^{2}}{e^{2}}+2 a}{2 \sqrt {\frac {c \,d^{4}}{e^{4}}+a}\, \sqrt {c \,x^{4}+a}}\right )}{2 \sqrt {\frac {c \,d^{4}}{e^{4}}+a}}+\frac {e \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \Pi \left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, -\frac {i \sqrt {a}\, e^{2}}{\sqrt {c}\, d^{2}}, \frac {\sqrt {-\frac {i \sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}}\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, d \sqrt {c \,x^{4}+a}}\right )}{\left (e^{4} a +d^{4} c \right ) e}\)	\(421\)
elliptic	\(-\frac {e^{3} \sqrt {c \,x^{4}+a}}{\left (e^{4} a +d^{4} c \right ) \left (e x +d \right )}-\frac {c \,d^{2} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{\left (e^{4} a +d^{4} c \right ) \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {i e^{2} \sqrt {c}\, \sqrt {a}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\left (e^{4} a +d^{4} c \right ) \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {2 d^{3} c \left (-\frac {\operatorname {arctanh}\left (\frac {\frac {2 c \,x^{2} d^{2}}{e^{2}}+2 a}{2 \sqrt {\frac {c \,d^{4}}{e^{4}}+a}\, \sqrt {c \,x^{4}+a}}\right )}{2 \sqrt {\frac {c \,d^{4}}{e^{4}}+a}}+\frac {e \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \Pi \left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, -\frac {i \sqrt {a}\, e^{2}}{\sqrt {c}\, d^{2}}, \frac {\sqrt {-\frac {i \sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}}\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, d \sqrt {c \,x^{4}+a}}\right )}{\left (e^{4} a +d^{4} c \right ) e}\)	\(421\)

input

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

output

-e^3*(c*x^4+a)^(1/2)/(a*e^4+c*d^4)/(e*x+d)-c*d^2/(a*e^4+c*d^4)/(I/a^(1/2)* 
c^(1/2))^(1/2)*(1-I/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+I/a^(1/2)*c^(1/2)*x^2)^( 
1/2)/(c*x^4+a)^(1/2)*EllipticF(x*(I/a^(1/2)*c^(1/2))^(1/2),I)+I*e^2*c^(1/2 
)/(a*e^4+c*d^4)*a^(1/2)/(I/a^(1/2)*c^(1/2))^(1/2)*(1-I/a^(1/2)*c^(1/2)*x^2 
)^(1/2)*(1+I/a^(1/2)*c^(1/2)*x^2)^(1/2)/(c*x^4+a)^(1/2)*(EllipticF(x*(I/a^ 
(1/2)*c^(1/2))^(1/2),I)-EllipticE(x*(I/a^(1/2)*c^(1/2))^(1/2),I))+2*d^3*c/ 
(a*e^4+c*d^4)/e*(-1/2/(c/e^4*d^4+a)^(1/2)*arctanh(1/2*(2*c*x^2/e^2*d^2+2*a 
)/(c/e^4*d^4+a)^(1/2)/(c*x^4+a)^(1/2))+1/(I/a^(1/2)*c^(1/2))^(1/2)*e/d*(1- 
I/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+I/a^(1/2)*c^(1/2)*x^2)^(1/2)/(c*x^4+a)^(1/ 
2)*EllipticPi(x*(I/a^(1/2)*c^(1/2))^(1/2),-I*a^(1/2)/c^(1/2)*e^2/d^2,(-I/a 
^(1/2)*c^(1/2))^(1/2)/(I/a^(1/2)*c^(1/2))^(1/2)))
 

3.1.2.5 Fricas [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x)^2 \sqrt {a+c x^4}} \, dx=\text {Timed out} \]

input

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

output

Timed out
 

3.1.2.6 Sympy [F]

\[ \int \frac {1}{(d+e x)^2 \sqrt {a+c x^4}} \, dx=\int \frac {1}{\sqrt {a + c x^{4}} \left (d + e x\right )^{2}}\, dx \]

input

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

output

Integral(1/(sqrt(a + c*x**4)*(d + e*x)**2), x)
 

3.1.2.7 Maxima [F]

\[ \int \frac {1}{(d+e x)^2 \sqrt {a+c x^4}} \, dx=\int { \frac {1}{\sqrt {c x^{4} + a} {\left (e x + d\right )}^{2}} \,d x } \]

input

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

output

integrate(1/(sqrt(c*x^4 + a)*(e*x + d)^2), x)
 

3.1.2.8 Giac [F]

\[ \int \frac {1}{(d+e x)^2 \sqrt {a+c x^4}} \, dx=\int { \frac {1}{\sqrt {c x^{4} + a} {\left (e x + d\right )}^{2}} \,d x } \]

input

integrate(1/(e*x+d)^2/(c*x^4+a)^(1/2),x, algorithm="giac")
 

output

integrate(1/(sqrt(c*x^4 + a)*(e*x + d)^2), x)
 

3.1.2.9 Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x)^2 \sqrt {a+c x^4}} \, dx=\int \frac {1}{\sqrt {c\,x^4+a}\,{\left (d+e\,x\right )}^2} \,d x \]

input

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

output

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