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

Optimal. Leaf size=749 \[ \frac{c^{5/4} d^4 \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{\sqrt [4]{a} \sqrt{a+c x^4} \left (\sqrt{a} e^2+\sqrt{c} d^2\right ) \left (a e^4+c d^4\right )}-\frac{c^{3/4} d^2 \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (\sqrt{c} d^2-\sqrt{a} e^2\right ) \Pi \left (\frac{\left (\sqrt{c} d^2+\sqrt{a} e^2\right )^2}{4 \sqrt{a} \sqrt{c} d^2 e^2};2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 \sqrt [4]{a} \sqrt{a+c x^4} \left (\sqrt{a} e^2+\sqrt{c} d^2\right ) \left (a e^4+c d^4\right )}-\frac{e^3 \sqrt{a+c x^4}}{(d+e x) \left (a e^4+c d^4\right )}+\frac{\sqrt{c} e^2 x \sqrt{a+c x^4}}{\left (\sqrt{a}+\sqrt{c} x^2\right ) \left (a e^4+c d^4\right )}-\frac{\sqrt [4]{a} \sqrt [4]{c} e^2 \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{\sqrt{a+c x^4} \left (a e^4+c d^4\right )}-\frac{c \tan ^{-1}\left (\frac{x \sqrt{-\frac{a e^4+c d^4}{d^2 e^2}}}{\sqrt{a+c x^4}}\right )}{e^2 \left (-\frac{a e^4+c d^4}{d^2 e^2}\right )^{3/2}}-\frac{\sqrt [4]{a} \sqrt [4]{c} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (\frac{\sqrt{c} d^2}{\sqrt{a}}-e^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 \sqrt{a+c x^4} \left (a e^4+c d^4\right )}-\frac{c d^3 e \tanh ^{-1}\left (\frac{a e^2+c d^2 x^2}{\sqrt{a+c x^4} \sqrt{a e^4+c d^4}}\right )}{\left (a e^4+c d^4\right )^{3/2}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 1.16176, antiderivative size = 749, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.474 \[ \frac{c^{5/4} d^4 \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{\sqrt [4]{a} \sqrt{a+c x^4} \left (\sqrt{a} e^2+\sqrt{c} d^2\right ) \left (a e^4+c d^4\right )}-\frac{c^{3/4} d^2 \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (\sqrt{c} d^2-\sqrt{a} e^2\right ) \Pi \left (\frac{\left (\sqrt{c} d^2+\sqrt{a} e^2\right )^2}{4 \sqrt{a} \sqrt{c} d^2 e^2};2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 \sqrt [4]{a} \sqrt{a+c x^4} \left (\sqrt{a} e^2+\sqrt{c} d^2\right ) \left (a e^4+c d^4\right )}-\frac{e^3 \sqrt{a+c x^4}}{(d+e x) \left (a e^4+c d^4\right )}+\frac{\sqrt{c} e^2 x \sqrt{a+c x^4}}{\left (\sqrt{a}+\sqrt{c} x^2\right ) \left (a e^4+c d^4\right )}-\frac{\sqrt [4]{a} \sqrt [4]{c} e^2 \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{\sqrt{a+c x^4} \left (a e^4+c d^4\right )}-\frac{c \tan ^{-1}\left (\frac{x \sqrt{-\frac{a e^4+c d^4}{d^2 e^2}}}{\sqrt{a+c x^4}}\right )}{e^2 \left (-\frac{a e^4+c d^4}{d^2 e^2}\right )^{3/2}}-\frac{\sqrt [4]{a} \sqrt [4]{c} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (\frac{\sqrt{c} d^2}{\sqrt{a}}-e^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 \sqrt{a+c x^4} \left (a e^4+c d^4\right )}-\frac{c d^3 e \tanh ^{-1}\left (\frac{a e^2+c d^2 x^2}{\sqrt{a+c x^4} \sqrt{a e^4+c d^4}}\right )}{\left (a e^4+c d^4\right )^{3/2}} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 119.817, size = 666, normalized size = 0.89 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**(1/4)*c**(1/4)*e**2*sqrt((a + c*x**4)/(sqrt(a) + sqrt(c)*x**2)**2)*(sqrt(a)
+ sqrt(c)*x**2)*elliptic_e(2*atan(c**(1/4)*x/a**(1/4)), 1/2)/(sqrt(a + c*x**4)*(
a*e**4 + c*d**4)) + sqrt(c)*e**2*x*sqrt(a + c*x**4)/((sqrt(a) + sqrt(c)*x**2)*(a
*e**4 + c*d**4)) + c*d**3*e*atanh((-a*e**2 - c*d**2*x**2)/(sqrt(a + c*x**4)*sqrt
(a*e**4 + c*d**4)))/(a*e**4 + c*d**4)**(3/2) - c*atan(x*sqrt(-a*e**2/d**2 - c*d*
*2/e**2)/sqrt(a + c*x**4))/(e**2*(-a*e**2/d**2 - c*d**2/e**2)**(3/2)) - e**3*sqr
t(a + c*x**4)/((d + e*x)*(a*e**4 + c*d**4)) + c**(5/4)*d**4*sqrt((a + c*x**4)/(s
qrt(a) + sqrt(c)*x**2)**2)*(sqrt(a) + sqrt(c)*x**2)*elliptic_f(2*atan(c**(1/4)*x
/a**(1/4)), 1/2)/(a**(1/4)*sqrt(a + c*x**4)*(sqrt(a)*e**2 + sqrt(c)*d**2)*(a*e**
4 + c*d**4)) + c**(3/4)*d**2*sqrt((a + c*x**4)/(sqrt(a) + sqrt(c)*x**2)**2)*(sqr
t(a) + sqrt(c)*x**2)*(sqrt(a)*e**2 - sqrt(c)*d**2)*elliptic_pi((sqrt(a)*e**2 + s
qrt(c)*d**2)**2/(4*sqrt(a)*sqrt(c)*d**2*e**2), 2*atan(c**(1/4)*x/a**(1/4)), 1/2)
/(2*a**(1/4)*sqrt(a + c*x**4)*(sqrt(a)*e**2 + sqrt(c)*d**2)*(a*e**4 + c*d**4)) +
c**(1/4)*sqrt((a + c*x**4)/(sqrt(a) + sqrt(c)*x**2)**2)*(sqrt(a) + sqrt(c)*x**2
)*(sqrt(a)*e**2 - sqrt(c)*d**2)*elliptic_f(2*atan(c**(1/4)*x/a**(1/4)), 1/2)/(2*
a**(1/4)*sqrt(a + c*x**4)*(a*e**4 + c*d**4))

_______________________________________________________________________________________

Mathematica [C]  time = 2.41905, size = 462, normalized size = 0.62 \[ \frac{-\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} \left (2 \sqrt [4]{-1} \sqrt [4]{a} c^{3/4} d^2 \sqrt{\frac{c x^4}{a}+1} (d+e x) \sqrt{a e^4+c d^4} \Pi \left (\frac{i \sqrt{a} e^2}{\sqrt{c} d^2};\left .\sin ^{-1}\left (\frac{(-1)^{3/4} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )+e^3 \left (a+c x^4\right ) \sqrt{a e^4+c d^4}-c d^3 e \sqrt{a+c x^4} (d+e x) \log \left (e^2 x^2-d^2\right )+c d^3 e \sqrt{a+c x^4} (d+e x) \log \left (\sqrt{a+c x^4} \sqrt{a e^4+c d^4}+a e^2+c d^2 x^2\right )\right )+\sqrt{a} \sqrt{c} e^2 \sqrt{\frac{c x^4}{a}+1} (d+e x) \sqrt{a e^4+c d^4} E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )+i \sqrt{c} \sqrt{\frac{c x^4}{a}+1} (d+e x) \left (\sqrt{c} d^2+i \sqrt{a} e^2\right ) \sqrt{a e^4+c d^4} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )}{\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} \sqrt{a+c x^4} (d+e x) \left (a e^4+c d^4\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(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[Sqr
t[(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*(-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] - c*d^3*e*(d + e*x)*Sqrt[a + c*x^4]*Log[-d^2 + e^2*x^
2] + c*d^3*e*(d + e*x)*Sqrt[a + c*x^4]*Log[a*e^2 + c*d^2*x^2 + Sqrt[c*d^4 + a*e^
4]*Sqrt[a + c*x^4]]))/(Sqrt[(I*Sqrt[c])/Sqrt[a]]*(c*d^4 + a*e^4)^(3/2)*(d + e*x)
*Sqrt[a + c*x^4])

_______________________________________________________________________________________

Maple [C]  time = 0.009, size = 421, normalized size = 0.6 \[ -{\frac{{e}^{3}}{ \left ( a{e}^{4}+c{d}^{4} \right ) \left ( ex+d \right ) }\sqrt{c{x}^{4}+a}}-{\frac{c{d}^{2}}{a{e}^{4}+c{d}^{4}}\sqrt{1-{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{c{x}^{4}+a}}}}+{\frac{i{e}^{2}}{a{e}^{4}+c{d}^{4}}\sqrt{a}\sqrt{c}\sqrt{1-{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}} \left ({\it EllipticF} \left ( x\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ) -{\it EllipticE} \left ( x\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ) \right ){\frac{1}{\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{c{x}^{4}+a}}}}+2\,{\frac{c{d}^{3}}{ \left ( a{e}^{4}+c{d}^{4} \right ) e} \left ( -1/2\,{1{\it Artanh} \left ( 1/2\,{\frac{1}{\sqrt{c{x}^{4}+a}} \left ( 2\,{\frac{c{d}^{2}{x}^{2}}{{e}^{2}}}+2\,a \right ){\frac{1}{\sqrt{{\frac{c{d}^{4}}{{e}^{4}}}+a}}}} \right ){\frac{1}{\sqrt{{\frac{c{d}^{4}}{{e}^{4}}}+a}}}}+{\frac{e}{d\sqrt{c{x}^{4}+a}}\sqrt{1-{\frac{i\sqrt{c}{x}^{2}}{\sqrt{a}}}}\sqrt{1+{\frac{i\sqrt{c}{x}^{2}}{\sqrt{a}}}}{\it EllipticPi} \left ( x\sqrt{{\frac{i\sqrt{c}}{\sqrt{a}}}},{\frac{-i\sqrt{a}{e}^{2}}{{d}^{2}\sqrt{c}}},{1\sqrt{{\frac{-i\sqrt{c}}{\sqrt{a}}}}{\frac{1}{\sqrt{{\frac{i\sqrt{c}}{\sqrt{a}}}}}}} \right ){\frac{1}{\sqrt{{\frac{i\sqrt{c}}{\sqrt{a}}}}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-e^3*(c*x^4+a)^(1/2)/(a*e^4+c*d^4)/(e*x+d)-d^2*c/(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*c^(1/2)*e^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)-Ellipt
icE(x*(I/a^(1/2)*c^(1/2))^(1/2),I))+2*c*d^3/(a*e^4+c*d^4)/e*(-1/2/(c*d^4/e^4+a)^
(1/2)*arctanh(1/2*(2*c*x^2*d^2/e^2+2*a)/(c*d^4/e^4+a)^(1/2)/(c*x^4+a)^(1/2))+1/(
I/a^(1/2)*c^(1/2))^(1/2)/d*e*(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)/d^2*e^2,(-I/a^(1/2)*c^(1/2))^(1/2)/(I/a^(1/2)*c^(1/2))^(1/2)))

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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 + d)^2),x, algorithm="fricas")

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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