\(\int \frac {\sqrt {a+c x^4}}{(d+e x)^2} \, dx\) [196]

Optimal result

Integrand size = 19, antiderivative size = 648 \[ \int \frac {\sqrt {a+c x^4}}{(d+e x)^2} \, dx=\frac {2 \sqrt {c} x \sqrt {a+c x^4}}{e^2 \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {d \sqrt {a+c x^4}}{e \left (d^2-e^2 x^2\right )}+\frac {x \sqrt {a+c x^4}}{d^2-e^2 x^2}-\frac {c d^3 \text {arctanh}\left (\frac {\sqrt {c d^4+a e^4} x}{d e \sqrt {a+c x^4}}\right )}{e^3 \sqrt {c d^4+a e^4}}-\frac {\sqrt {c} d \text {arctanh}\left (\frac {\sqrt {c} x^2}{\sqrt {a+c x^4}}\right )}{e^3}+\frac {c d^3 \text {arctanh}\left (\frac {a e^2+c d^2 x^2}{\sqrt {c d^4+a e^4} \sqrt {a+c x^4}}\right )}{e^3 \sqrt {c d^4+a e^4}}-\frac {2 \sqrt [4]{a} \sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{e^2 \sqrt {a+c x^4}}+\frac {\sqrt [4]{a} \sqrt [4]{c} \left (2 \sqrt {c} d^2+\sqrt {a} e^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{e^2 \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \sqrt {a+c x^4}}+\frac {c^{3/4} d^2 \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {c} d^2+\sqrt {a} e^2\right )^2}{4 \sqrt {a} \sqrt {c} d^2 e^2},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} e^4 \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \sqrt {a+c x^4}} \] Output:

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

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 12.73 (sec) , antiderivative size = 394, normalized size of antiderivative = 0.61 \[ \int \frac {\sqrt {a+c x^4}}{(d+e x)^2} \, dx=\frac {-\frac {e^3 \left (a+c x^4\right )}{d+e x}-\frac {2 c d^3 e \sqrt {a+c x^4} \arctan \left (\frac {\sqrt {c} \left (d^2-e^2 x^2\right )+e^2 \sqrt {a+c x^4}}{\sqrt {-c d^4-a e^4}}\right )}{\sqrt {-c d^4-a e^4}}-2 i a \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} e^2 \sqrt {1+\frac {c x^4}{a}} E\left (\left .i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )-\frac {2 \sqrt {c} \left (i \sqrt {c} d^2+\sqrt {a} e^2\right ) \sqrt {1+\frac {c x^4}{a}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right ),-1\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}}+2 \sqrt [4]{-1} \sqrt [4]{a} c^{3/4} d^2 \sqrt {1+\frac {c x^4}{a}} \operatorname {EllipticPi}\left (\frac {i \sqrt {a} e^2}{\sqrt {c} d^2},\arcsin \left (\frac {(-1)^{3/4} \sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )+\sqrt {c} d e \sqrt {a+c x^4} \log \left (-\sqrt {c} x^2+\sqrt {a+c x^4}\right )}{e^4 \sqrt {a+c x^4}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 4.37 (sec) , antiderivative size = 975, normalized size of antiderivative = 1.50, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.632, Rules used = {2584, 2255, 27, 1577, 492, 605, 224, 219, 488, 219, 2259, 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.

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

\(\Big \downarrow \) 2584

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

\(\Big \downarrow \) 2255

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1577

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

\(\Big \downarrow \) 492

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

\(\Big \downarrow \) 605

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 488

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 2259

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

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 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], 
x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] &&  !GtQ[a, 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 492

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

rule 605

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

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 2009

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

rule 2255

Int[(Pr_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] 
 :> Module[{r = Expon[Pr, x], k}, Int[Sum[Coeff[Pr, x, 2*k]*x^(2*k), {k, 0, 
 r/2}]*(d + e*x^2)^q*(a + c*x^4)^p, x] + Int[x*Sum[Coeff[Pr, x, 2*k + 1]*x^ 
(2*k), {k, 0, (r - 1)/2}]*(d + e*x^2)^q*(a + c*x^4)^p, x]] /; FreeQ[{a, c, 
d, e, p, q}, x] && PolyQ[Pr, x] &&  !PolyQ[Pr, x^2]
 

rule 2259

Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] 
 :> Int[ExpandIntegrand[1/Sqrt[a + c*x^4], Px*(d + e*x^2)^q*(a + c*x^4)^(p 
+ 1/2), x], x] /; FreeQ[{a, c, d, e}, x] && PolyQ[Px, x] && IntegerQ[p + 1/ 
2] && IntegerQ[q]
 

rule 2584

Int[((c_) + (d_.)*(x_)^(n_.))^(q_)*((a_) + (b_.)*(x_)^(nn_.))^(p_), x_Symbo 
l] :> Int[ExpandToSum[(c - d*x^n)^(-q), x]*((a + b*x^nn)^p/(c^2 - d^2*x^(2* 
n))^(-q)), x] /; FreeQ[{a, b, c, d, n, nn, p}, x] &&  !IntegerQ[p] && ILtQ[ 
q, 0] && IGtQ[Log[2, nn/n], 0]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.51 (sec) , antiderivative size = 402, normalized size of antiderivative = 0.62

method	result	size
default	\(-\frac {\sqrt {c \,x^{4}+a}}{e \left (e x +d \right )}+\frac {2 c \,d^{2} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{e^{4} \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}-\frac {\sqrt {c}\, d \ln \left (2 \sqrt {c}\, x^{2}+2 \sqrt {c \,x^{4}+a}\right )}{e^{3}}+\frac {2 i \sqrt {c}\, \sqrt {a}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{e^{2} \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}-\frac {2 c \,d^{3} \left (-\frac {\operatorname {arctanh}\left (\frac {\frac {2 c \,x^{2} d^{2}}{e^{2}}+2 a}{2 \sqrt {a +\frac {c \,d^{4}}{e^{4}}}\, \sqrt {c \,x^{4}+a}}\right )}{2 \sqrt {a +\frac {c \,d^{4}}{e^{4}}}}+\frac {e \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, -\frac {i \sqrt {a}\, e^{2}}{\sqrt {c}\, d^{2}}, \frac {\sqrt {-\frac {i \sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}}\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, d \sqrt {c \,x^{4}+a}}\right )}{e^{5}}\)	\(402\)
elliptic	\(-\frac {\sqrt {c \,x^{4}+a}}{e \left (e x +d \right )}+\frac {2 c \,d^{2} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{e^{4} \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}-\frac {\sqrt {c}\, d \ln \left (2 \sqrt {c}\, x^{2}+2 \sqrt {c \,x^{4}+a}\right )}{e^{3}}+\frac {2 i \sqrt {c}\, \sqrt {a}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{e^{2} \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}-\frac {2 c \,d^{3} \left (-\frac {\operatorname {arctanh}\left (\frac {\frac {2 c \,x^{2} d^{2}}{e^{2}}+2 a}{2 \sqrt {a +\frac {c \,d^{4}}{e^{4}}}\, \sqrt {c \,x^{4}+a}}\right )}{2 \sqrt {a +\frac {c \,d^{4}}{e^{4}}}}+\frac {e \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, -\frac {i \sqrt {a}\, e^{2}}{\sqrt {c}\, d^{2}}, \frac {\sqrt {-\frac {i \sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}}\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, d \sqrt {c \,x^{4}+a}}\right )}{e^{5}}\)	\(402\)

Input:

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

Output:

-1/e*(c*x^4+a)^(1/2)/(e*x+d)+2*c*d^2/e^4/(I*c^(1/2)/a^(1/2))^(1/2)*(1-I*c^ 
(1/2)*x^2/a^(1/2))^(1/2)*(1+I*c^(1/2)*x^2/a^(1/2))^(1/2)/(c*x^4+a)^(1/2)*E 
llipticF(x*(I*c^(1/2)/a^(1/2))^(1/2),I)-c^(1/2)*d/e^3*ln(2*c^(1/2)*x^2+2*( 
c*x^4+a)^(1/2))+2*I*c^(1/2)/e^2*a^(1/2)/(I*c^(1/2)/a^(1/2))^(1/2)*(1-I*c^( 
1/2)*x^2/a^(1/2))^(1/2)*(1+I*c^(1/2)*x^2/a^(1/2))^(1/2)/(c*x^4+a)^(1/2)*(E 
llipticF(x*(I*c^(1/2)/a^(1/2))^(1/2),I)-EllipticE(x*(I*c^(1/2)/a^(1/2))^(1 
/2),I))-2*c*d^3/e^5*(-1/2/(a+c*d^4/e^4)^(1/2)*arctanh(1/2*(2*c*x^2*d^2/e^2 
+2*a)/(a+c*d^4/e^4)^(1/2)/(c*x^4+a)^(1/2))+1/(I*c^(1/2)/a^(1/2))^(1/2)/d*e 
*(1-I*c^(1/2)*x^2/a^(1/2))^(1/2)*(1+I*c^(1/2)*x^2/a^(1/2))^(1/2)/(c*x^4+a) 
^(1/2)*EllipticPi(x*(I*c^(1/2)/a^(1/2))^(1/2),-I/c^(1/2)*a^(1/2)/d^2*e^2,( 
-I/a^(1/2)*c^(1/2))^(1/2)/(I*c^(1/2)/a^(1/2))^(1/2)))
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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