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

Optimal. Leaf size=570 \[ \frac{e^2 x \sqrt{a+c x^4}}{2 d \left (d+e x^2\right ) \left (a e^2+c d^2\right )}-\frac{\sqrt{c} e x \sqrt{a+c x^4}}{2 d \left (\sqrt{a}+\sqrt{c} x^2\right ) \left (a e^2+c d^2\right )}+\frac{\sqrt [4]{a} \sqrt [4]{c} e \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 d \sqrt{a+c x^4} \left (a e^2+c d^2\right )}-\frac{\sqrt [4]{a} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (\frac{\sqrt{c} d}{\sqrt{a}}+e\right ) \left (a e^2+3 c d^2\right ) \Pi \left (-\frac{\left (\sqrt{c} d-\sqrt{a} e\right )^2}{4 \sqrt{a} \sqrt{c} d e};2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{8 \sqrt [4]{c} d^2 \sqrt{a+c x^4} \left (\sqrt{c} d-\sqrt{a} e\right ) \left (a e^2+c d^2\right )}+\frac{\left (a e^2+3 c d^2\right ) \tan ^{-1}\left (\frac{x \sqrt{\frac{a e}{d}+\frac{c d}{e}}}{\sqrt{a+c x^4}}\right )}{4 d^3 e \left (\frac{a e}{d}+\frac{c d}{e}\right )^{3/2}}-\frac{\sqrt [4]{c} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 \sqrt [4]{a} d \sqrt{a+c x^4} \left (\sqrt{a} e-\sqrt{c} d\right )} \]

[Out]

-(Sqrt[c]*e*x*Sqrt[a + c*x^4])/(2*d*(c*d^2 + a*e^2)*(Sqrt[a] + Sqrt[c]*x^2)) + (
e^2*x*Sqrt[a + c*x^4])/(2*d*(c*d^2 + a*e^2)*(d + e*x^2)) + ((3*c*d^2 + a*e^2)*Ar
cTan[(Sqrt[(c*d)/e + (a*e)/d]*x)/Sqrt[a + c*x^4]])/(4*d^3*e*((c*d)/e + (a*e)/d)^
(3/2)) + (a^(1/4)*c^(1/4)*e*(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])/(2*d*(c*d^2 + a*e
^2)*Sqrt[a + c*x^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*a^(1/4)*d*
(-(Sqrt[c]*d) + Sqrt[a]*e)*Sqrt[a + c*x^4]) - (a^(1/4)*((Sqrt[c]*d)/Sqrt[a] + e)
*(3*c*d^2 + a*e^2)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x
^2)^2]*EllipticPi[-(Sqrt[c]*d - Sqrt[a]*e)^2/(4*Sqrt[a]*Sqrt[c]*d*e), 2*ArcTan[(
c^(1/4)*x)/a^(1/4)], 1/2])/(8*c^(1/4)*d^2*(Sqrt[c]*d - Sqrt[a]*e)*(c*d^2 + a*e^2
)*Sqrt[a + c*x^4])

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Rubi [A]  time = 0.918835, antiderivative size = 713, normalized size of antiderivative = 1.25, number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ \frac{e^2 x \sqrt{a+c x^4}}{2 d \left (d+e x^2\right ) \left (a e^2+c d^2\right )}-\frac{\sqrt{c} e x \sqrt{a+c x^4}}{2 d \left (\sqrt{a}+\sqrt{c} x^2\right ) \left (a e^2+c d^2\right )}+\frac{\sqrt [4]{c} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (a e^2+3 c d^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 \sqrt [4]{a} d \sqrt{a+c x^4} \left (\sqrt{c} d-\sqrt{a} e\right ) \left (a e^2+c d^2\right )}-\frac{\sqrt [4]{a} \sqrt [4]{c} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (\frac{\sqrt{c} d}{\sqrt{a}}+e\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 d \sqrt{a+c x^4} \left (a e^2+c d^2\right )}+\frac{\sqrt [4]{a} \sqrt [4]{c} e \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 d \sqrt{a+c x^4} \left (a e^2+c d^2\right )}-\frac{\sqrt [4]{a} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (\frac{\sqrt{c} d}{\sqrt{a}}+e\right ) \left (a e^2+3 c d^2\right ) \Pi \left (-\frac{\left (\sqrt{c} d-\sqrt{a} e\right )^2}{4 \sqrt{a} \sqrt{c} d e};2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{8 \sqrt [4]{c} d^2 \sqrt{a+c x^4} \left (\sqrt{c} d-\sqrt{a} e\right ) \left (a e^2+c d^2\right )}+\frac{\left (a e^2+3 c d^2\right ) \tan ^{-1}\left (\frac{x \sqrt{\frac{a e}{d}+\frac{c d}{e}}}{\sqrt{a+c x^4}}\right )}{4 d^3 e \left (\frac{a e}{d}+\frac{c d}{e}\right )^{3/2}} \]

Warning: Unable to verify antiderivative.

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

[Out]

-(Sqrt[c]*e*x*Sqrt[a + c*x^4])/(2*d*(c*d^2 + a*e^2)*(Sqrt[a] + Sqrt[c]*x^2)) + (
e^2*x*Sqrt[a + c*x^4])/(2*d*(c*d^2 + a*e^2)*(d + e*x^2)) + ((3*c*d^2 + a*e^2)*Ar
cTan[(Sqrt[(c*d)/e + (a*e)/d]*x)/Sqrt[a + c*x^4]])/(4*d^3*e*((c*d)/e + (a*e)/d)^
(3/2)) + (a^(1/4)*c^(1/4)*e*(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])/(2*d*(c*d^2 + a*e
^2)*Sqrt[a + c*x^4]) - (a^(1/4)*c^(1/4)*((Sqrt[c]*d)/Sqrt[a] + e)*(Sqrt[a] + Sqr
t[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])/(4*d*(c*d^2 + a*e^2)*Sqrt[a + c*x^4]) + (c^(1/4)*(3*c*d^2 +
a*e^2)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*Elli
pticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(4*a^(1/4)*d*(Sqrt[c]*d - Sqrt[a]*e)*
(c*d^2 + a*e^2)*Sqrt[a + c*x^4]) - (a^(1/4)*((Sqrt[c]*d)/Sqrt[a] + e)*(3*c*d^2 +
a*e^2)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*Elli
pticPi[-(Sqrt[c]*d - Sqrt[a]*e)^2/(4*Sqrt[a]*Sqrt[c]*d*e), 2*ArcTan[(c^(1/4)*x)/
a^(1/4)], 1/2])/(8*c^(1/4)*d^2*(Sqrt[c]*d - Sqrt[a]*e)*(c*d^2 + a*e^2)*Sqrt[a +
c*x^4])

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Rubi in Sympy [A]  time = 82.3448, size = 619, normalized size = 1.09 \[ \frac{\sqrt [4]{a} \sqrt [4]{c} e \sqrt{\frac{a + c x^{4}}{\left (\sqrt{a} + \sqrt{c} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x^{2}\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{2 d \sqrt{a + c x^{4}} \left (a e^{2} + c d^{2}\right )} - \frac{\sqrt{c} e x \sqrt{a + c x^{4}}}{2 d \left (\sqrt{a} + \sqrt{c} x^{2}\right ) \left (a e^{2} + c d^{2}\right )} + \frac{e^{2} x \sqrt{a + c x^{4}}}{2 d \left (d + e x^{2}\right ) \left (a e^{2} + c d^{2}\right )} + \frac{\left (a e^{2} + 3 c d^{2}\right ) \operatorname{atan}{\left (\frac{x \sqrt{\frac{a e}{d} + \frac{c d}{e}}}{\sqrt{a + c x^{4}}} \right )}}{4 d^{3} e \left (\frac{a e}{d} + \frac{c d}{e}\right )^{\frac{3}{2}}} - \frac{\sqrt [4]{c} \sqrt{\frac{a + c x^{4}}{\left (\sqrt{a} + \sqrt{c} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x^{2}\right ) \left (\sqrt{a} e + \sqrt{c} d\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{4 \sqrt [4]{a} d \sqrt{a + c x^{4}} \left (a e^{2} + c d^{2}\right )} - \frac{\sqrt [4]{c} \sqrt{\frac{a + c x^{4}}{\left (\sqrt{a} + \sqrt{c} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x^{2}\right ) \left (a e^{2} + 3 c d^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{4 \sqrt [4]{a} d \sqrt{a + c x^{4}} \left (\sqrt{a} e - \sqrt{c} d\right ) \left (a e^{2} + c d^{2}\right )} + \frac{\sqrt{\frac{a + c x^{4}}{\left (\sqrt{a} + \sqrt{c} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x^{2}\right ) \left (\sqrt{a} e + \sqrt{c} d\right ) \left (a e^{2} + 3 c d^{2}\right ) \Pi \left (- \frac{\sqrt{a} \left (e - \frac{\sqrt{c} d}{\sqrt{a}}\right )^{2}}{4 \sqrt{c} d e}; 2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{8 \sqrt [4]{a} \sqrt [4]{c} d^{2} \sqrt{a + c x^{4}} \left (\sqrt{a} e - \sqrt{c} d\right ) \left (a e^{2} + c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x**2+d)**2/(c*x**4+a)**(1/2),x)

[Out]

a**(1/4)*c**(1/4)*e*sqrt((a + c*x**4)/(sqrt(a) + sqrt(c)*x**2)**2)*(sqrt(a) + sq
rt(c)*x**2)*elliptic_e(2*atan(c**(1/4)*x/a**(1/4)), 1/2)/(2*d*sqrt(a + c*x**4)*(
a*e**2 + c*d**2)) - sqrt(c)*e*x*sqrt(a + c*x**4)/(2*d*(sqrt(a) + sqrt(c)*x**2)*(
a*e**2 + c*d**2)) + e**2*x*sqrt(a + c*x**4)/(2*d*(d + e*x**2)*(a*e**2 + c*d**2))
+ (a*e**2 + 3*c*d**2)*atan(x*sqrt(a*e/d + c*d/e)/sqrt(a + c*x**4))/(4*d**3*e*(a
*e/d + c*d/e)**(3/2)) - c**(1/4)*sqrt((a + c*x**4)/(sqrt(a) + sqrt(c)*x**2)**2)*
(sqrt(a) + sqrt(c)*x**2)*(sqrt(a)*e + sqrt(c)*d)*elliptic_f(2*atan(c**(1/4)*x/a*
*(1/4)), 1/2)/(4*a**(1/4)*d*sqrt(a + c*x**4)*(a*e**2 + c*d**2)) - c**(1/4)*sqrt(
(a + c*x**4)/(sqrt(a) + sqrt(c)*x**2)**2)*(sqrt(a) + sqrt(c)*x**2)*(a*e**2 + 3*c
*d**2)*elliptic_f(2*atan(c**(1/4)*x/a**(1/4)), 1/2)/(4*a**(1/4)*d*sqrt(a + c*x**
4)*(sqrt(a)*e - sqrt(c)*d)*(a*e**2 + c*d**2)) + sqrt((a + c*x**4)/(sqrt(a) + sqr
t(c)*x**2)**2)*(sqrt(a) + sqrt(c)*x**2)*(sqrt(a)*e + sqrt(c)*d)*(a*e**2 + 3*c*d*
*2)*elliptic_pi(-sqrt(a)*(e - sqrt(c)*d/sqrt(a))**2/(4*sqrt(c)*d*e), 2*atan(c**(
1/4)*x/a**(1/4)), 1/2)/(8*a**(1/4)*c**(1/4)*d**2*sqrt(a + c*x**4)*(sqrt(a)*e - s
qrt(c)*d)*(a*e**2 + c*d**2))

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Mathematica [C]  time = 1.00079, size = 522, normalized size = 0.92 \[ \frac{-3 i c d^3 \sqrt{\frac{c x^4}{a}+1} \Pi \left (-\frac{i \sqrt{a} e}{\sqrt{c} d};\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )-3 i c d^2 e x^2 \sqrt{\frac{c x^4}{a}+1} \Pi \left (-\frac{i \sqrt{a} e}{\sqrt{c} d};\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )-i a e^3 x^2 \sqrt{\frac{c x^4}{a}+1} \Pi \left (-\frac{i \sqrt{a} e}{\sqrt{c} d};\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )+c d e^2 x^5 \sqrt{\frac{i \sqrt{c}}{\sqrt{a}}}-i a d e^2 \sqrt{\frac{c x^4}{a}+1} \Pi \left (-\frac{i \sqrt{a} e}{\sqrt{c} d};\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )+a d e^2 x \sqrt{\frac{i \sqrt{c}}{\sqrt{a}}}+\sqrt{c} d \sqrt{\frac{c x^4}{a}+1} \left (d+e x^2\right ) \left (\sqrt{a} e+i \sqrt{c} d\right ) F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )-\sqrt{a} \sqrt{c} d e \sqrt{\frac{c x^4}{a}+1} \left (d+e x^2\right ) E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )}{2 d^2 \sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} \sqrt{a+c x^4} \left (d+e x^2\right ) \left (a e^2+c d^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

(a*Sqrt[(I*Sqrt[c])/Sqrt[a]]*d*e^2*x + Sqrt[(I*Sqrt[c])/Sqrt[a]]*c*d*e^2*x^5 - S
qrt[a]*Sqrt[c]*d*e*(d + e*x^2)*Sqrt[1 + (c*x^4)/a]*EllipticE[I*ArcSinh[Sqrt[(I*S
qrt[c])/Sqrt[a]]*x], -1] + Sqrt[c]*d*(I*Sqrt[c]*d + Sqrt[a]*e)*(d + e*x^2)*Sqrt[
1 + (c*x^4)/a]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1] - (3*I)*c*d
^3*Sqrt[1 + (c*x^4)/a]*EllipticPi[((-I)*Sqrt[a]*e)/(Sqrt[c]*d), I*ArcSinh[Sqrt[(
I*Sqrt[c])/Sqrt[a]]*x], -1] - I*a*d*e^2*Sqrt[1 + (c*x^4)/a]*EllipticPi[((-I)*Sqr
t[a]*e)/(Sqrt[c]*d), I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1] - (3*I)*c*d^2*e
*x^2*Sqrt[1 + (c*x^4)/a]*EllipticPi[((-I)*Sqrt[a]*e)/(Sqrt[c]*d), I*ArcSinh[Sqrt
[(I*Sqrt[c])/Sqrt[a]]*x], -1] - I*a*e^3*x^2*Sqrt[1 + (c*x^4)/a]*EllipticPi[((-I)
*Sqrt[a]*e)/(Sqrt[c]*d), I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1])/(2*Sqrt[(I
*Sqrt[c])/Sqrt[a]]*d^2*(c*d^2 + a*e^2)*(d + e*x^2)*Sqrt[a + c*x^4])

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Maple [C]  time = 0.032, size = 556, normalized size = 1. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*e^2*x*(c*x^4+a)^(1/2)/d/(a*e^2+c*d^2)/(e*x^2+d)-1/2*c/(a*e^2+c*d^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)-1/2*I*e*c^(1/2)/(a*e^
2+c*d^2)/d*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)+1/2*I*e*c^(1/2)/(a*e^2+c*d^2)/d*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)*Elliptic
E(x*(I/a^(1/2)*c^(1/2))^(1/2),I)+1/2/(a*e^2+c*d^2)/d^2*e^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)*EllipticPi(x*(I/a^(1/2)*c^(1/2))^(1/2),I*a^(1/2)/c^(1/2)*e/d,(-I/a^(1/2)*c^
(1/2))^(1/2)/(I/a^(1/2)*c^(1/2))^(1/2))*a+3/2/(a*e^2+c*d^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)*EllipticPi(x*(I/a^(1/2)*c^(1/2))^(1/2),I*a^(1/2)/c^(1/2)*e/d,(-I/a^(1/2)*c
^(1/2))^(1/2)/(I/a^(1/2)*c^(1/2))^(1/2))*c

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{c x^{4} + a}{\left (e x^{2} + d\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a + c x^{4}} \left (d + e x^{2}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{c x^{4} + a}{\left (e x^{2} + d\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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