3.201 \(\int \frac{1}{(d+e x)^2 \left (a+c x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=2413 \[ \text{result too large to display} \]

[Out]

(d*e*(a*e^2 - c*d^2*x^2))/(a*(c*d^4 + a*e^4)*(d^2 - e^2*x^2)*Sqrt[a + c*x^4]) -
(c*x*(d^2 + e^2*x^2))/(2*a*(c*d^4 + a*e^4)*Sqrt[a + c*x^4]) + (c*d^2*x*(c*d^4 -
a*e^4 + 2*c*d^2*e^2*x^2))/(a*(c*d^4 + a*e^4)^2*Sqrt[a + c*x^4]) + (e^7*Sqrt[a +
c*x^4])/(2*(c*d^4 + a*e^4)^2*(d - e*x)) - (e^7*Sqrt[a + c*x^4])/(2*(c*d^4 + a*e^
4)^2*(d + e*x)) - (2*c^(3/2)*d^4*e^2*x*Sqrt[a + c*x^4])/(a*(c*d^4 + a*e^4)^2*(Sq
rt[a] + Sqrt[c]*x^2)) + (Sqrt[c]*e^6*x*Sqrt[a + c*x^4])/((c*d^4 + a*e^4)^2*(Sqrt
[a] + Sqrt[c]*x^2)) + (Sqrt[c]*e^2*x*Sqrt[a + c*x^4])/(2*a*(c*d^4 + a*e^4)*(Sqrt
[a] + Sqrt[c]*x^2)) + (d*e^3*(c*d^4 - 2*a*e^4)*Sqrt[a + c*x^4])/(a*(c*d^4 + a*e^
4)^2*(d^2 - e^2*x^2)) + (c*ArcTan[(Sqrt[-((c*d^4 + a*e^4)/(d^2*e^2))]*x)/Sqrt[a
+ c*x^4]])/(d^2*(-((c*d^4 + a*e^4)/(d^2*e^2)))^(5/2)) + (e^2*ArcTan[(Sqrt[-((c*d
^4 + a*e^4)/(d^2*e^2))]*x)/Sqrt[a + c*x^4]])/(2*d^4*(-((c*d^4 + a*e^4)/(d^2*e^2)
))^(3/2)) + (e^4*(5*c*d^4 + a*e^4)*ArcTan[(Sqrt[-((c*d^4 + a*e^4)/(d^2*e^2))]*x)
/Sqrt[a + c*x^4]])/(2*d^2*(c*d^4 + a*e^4)^2*Sqrt[-((c*d^4 + a*e^4)/(d^2*e^2))])
- (3*c*d^3*e^5*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)^(5/2) + (2*c^(5/4)*d^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])
/(a^(3/4)*(c*d^4 + a*e^4)^2*Sqrt[a + c*x^4]) - (a^(1/4)*c^(1/4)*e^6*(Sqrt[a] + S
qrt[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)^2*Sqrt[a + c*x^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])/(2*a^(3/4)*(c*d^4 + a*e^4)*Sqrt[a + c*x^4]) - (c^(1
/4)*e^4*(Sqrt[c]*d^2 - Sqrt[a]*e^2)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sq
rt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(2*a^(1/4
)*(c*d^4 + a*e^4)^2*Sqrt[a + c*x^4]) + (c^(5/4)*d^4*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])/(a^(1/4)*(Sqrt[c]*d^2 + Sqrt[a]*e^2)*(c*d^4 + a*e^4)^2*Sqrt[a + c*x^4
]) + (c^(3/4)*d^2*(c*d^4 - 2*Sqrt[a]*Sqrt[c]*d^2*e^2 - 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^(5/4)*(c*d^4 + a*e^4)^2*Sqrt[a + c*x^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])/(4*a^(5/4)*(c*d^4 + a*e
^4)*Sqrt[a + c*x^4]) - (c^(1/4)*e^4*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sq
rt[a] + Sqrt[c]*x^2)^2]*EllipticF[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]) + (c^(1/4)*e^4*(5
*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)*(Sqrt[c]*d^2 + Sqr
t[a]*e^2)*(c*d^4 + a*e^4)^2*Sqrt[a + c*x^4]) + (e^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]*EllipticPi[(
Sqrt[a]*((Sqrt[c]*d^2)/Sqrt[a] + e^2)^2)/(4*Sqrt[c]*d^2*e^2), 2*ArcTan[(c^(1/4)*
x)/a^(1/4)], 1/2])/(4*a^(1/4)*c^(1/4)*d^2*(Sqrt[c]*d^2 + Sqrt[a]*e^2)*(c*d^4 + a
*e^4)*Sqrt[a + c*x^4]) - (c^(3/4)*d^2*e^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]*EllipticPi[(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])/(2*a^(1/4)*(Sqrt[c]*d^2 + Sqrt[a]*e^2)*(c*d^4 + a*e^4)^2*Sqrt[a + c*x^4]) -
(e^4*(Sqrt[c]*d^2 - Sqrt[a]*e^2)*(5*c*d^4 + a*e^4)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt
[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticPi[(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)*d^2*(Sqrt[c]*d^2 + Sqrt[a]*e^2)*(c*d^4 + a*e^4)^2*Sqrt[a + c*x^4])

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Rubi [A]  time = 13.5254, antiderivative size = 2413, normalized size of antiderivative = 1., number of steps used = 72, number of rules used = 16, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.842 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]  Int[1/((d + e*x)^2*(a + c*x^4)^(3/2)),x]

[Out]

(d*e*(a*e^2 - c*d^2*x^2))/(a*(c*d^4 + a*e^4)*(d^2 - e^2*x^2)*Sqrt[a + c*x^4]) -
(c*x*(d^2 + e^2*x^2))/(2*a*(c*d^4 + a*e^4)*Sqrt[a + c*x^4]) + (c*d^2*x*(c*d^4 -
a*e^4 + 2*c*d^2*e^2*x^2))/(a*(c*d^4 + a*e^4)^2*Sqrt[a + c*x^4]) + (e^7*Sqrt[a +
c*x^4])/(2*(c*d^4 + a*e^4)^2*(d - e*x)) - (e^7*Sqrt[a + c*x^4])/(2*(c*d^4 + a*e^
4)^2*(d + e*x)) - (2*c^(3/2)*d^4*e^2*x*Sqrt[a + c*x^4])/(a*(c*d^4 + a*e^4)^2*(Sq
rt[a] + Sqrt[c]*x^2)) + (Sqrt[c]*e^6*x*Sqrt[a + c*x^4])/((c*d^4 + a*e^4)^2*(Sqrt
[a] + Sqrt[c]*x^2)) + (Sqrt[c]*e^2*x*Sqrt[a + c*x^4])/(2*a*(c*d^4 + a*e^4)*(Sqrt
[a] + Sqrt[c]*x^2)) + (d*e^3*(c*d^4 - 2*a*e^4)*Sqrt[a + c*x^4])/(a*(c*d^4 + a*e^
4)^2*(d^2 - e^2*x^2)) + (c*ArcTan[(Sqrt[-((c*d^4 + a*e^4)/(d^2*e^2))]*x)/Sqrt[a
+ c*x^4]])/(d^2*(-((c*d^4 + a*e^4)/(d^2*e^2)))^(5/2)) + (e^2*ArcTan[(Sqrt[-((c*d
^4 + a*e^4)/(d^2*e^2))]*x)/Sqrt[a + c*x^4]])/(2*d^4*(-((c*d^4 + a*e^4)/(d^2*e^2)
))^(3/2)) + (e^4*(5*c*d^4 + a*e^4)*ArcTan[(Sqrt[-((c*d^4 + a*e^4)/(d^2*e^2))]*x)
/Sqrt[a + c*x^4]])/(2*d^2*(c*d^4 + a*e^4)^2*Sqrt[-((c*d^4 + a*e^4)/(d^2*e^2))])
- (3*c*d^3*e^5*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)^(5/2) + (2*c^(5/4)*d^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])
/(a^(3/4)*(c*d^4 + a*e^4)^2*Sqrt[a + c*x^4]) - (a^(1/4)*c^(1/4)*e^6*(Sqrt[a] + S
qrt[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)^2*Sqrt[a + c*x^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])/(2*a^(3/4)*(c*d^4 + a*e^4)*Sqrt[a + c*x^4]) - (c^(1
/4)*e^4*(Sqrt[c]*d^2 - Sqrt[a]*e^2)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sq
rt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(2*a^(1/4
)*(c*d^4 + a*e^4)^2*Sqrt[a + c*x^4]) + (c^(5/4)*d^4*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])/(a^(1/4)*(Sqrt[c]*d^2 + Sqrt[a]*e^2)*(c*d^4 + a*e^4)^2*Sqrt[a + c*x^4
]) + (c^(3/4)*d^2*(c*d^4 - 2*Sqrt[a]*Sqrt[c]*d^2*e^2 - 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^(5/4)*(c*d^4 + a*e^4)^2*Sqrt[a + c*x^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])/(4*a^(5/4)*(c*d^4 + a*e
^4)*Sqrt[a + c*x^4]) - (c^(1/4)*e^4*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sq
rt[a] + Sqrt[c]*x^2)^2]*EllipticF[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]) + (c^(1/4)*e^4*(5
*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)*(Sqrt[c]*d^2 + Sqr
t[a]*e^2)*(c*d^4 + a*e^4)^2*Sqrt[a + c*x^4]) + (e^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]*EllipticPi[(
Sqrt[a]*((Sqrt[c]*d^2)/Sqrt[a] + e^2)^2)/(4*Sqrt[c]*d^2*e^2), 2*ArcTan[(c^(1/4)*
x)/a^(1/4)], 1/2])/(4*a^(1/4)*c^(1/4)*d^2*(Sqrt[c]*d^2 + Sqrt[a]*e^2)*(c*d^4 + a
*e^4)*Sqrt[a + c*x^4]) - (c^(3/4)*d^2*e^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]*EllipticPi[(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])/(2*a^(1/4)*(Sqrt[c]*d^2 + Sqrt[a]*e^2)*(c*d^4 + a*e^4)^2*Sqrt[a + c*x^4]) -
(e^4*(Sqrt[c]*d^2 - Sqrt[a]*e^2)*(5*c*d^4 + a*e^4)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt
[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticPi[(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)*d^2*(Sqrt[c]*d^2 + Sqrt[a]*e^2)*(c*d^4 + a*e^4)^2*Sqrt[a + c*x^4])

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + c x^{4}\right )^{\frac{3}{2}} \left (d + e x\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)**2/(c*x**4+a)**(3/2),x)

[Out]

Integral(1/((a + c*x**4)**(3/2)*(d + e*x)**2), x)

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Mathematica [C]  time = 4.06581, size = 809, normalized size = 0.34 \[ \frac{3 \sqrt{a} \sqrt{c} \left (a e^4-c d^4\right ) \sqrt{c d^4+a e^4} (d+e x) \sqrt{\frac{c x^4}{a}+1} E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right ) e^2+\sqrt{c} \sqrt{c d^4+a e^4} \left (-i c^{3/2} d^6+3 \sqrt{a} c e^2 d^4+5 i a \sqrt{c} e^4 d^2-3 a^{3/2} e^6\right ) (d+e x) \sqrt{\frac{c x^4}{a}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )+\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} \left (c^2 \sqrt{c d^4+a e^4} x d^7-c^2 e \sqrt{c d^4+a e^4} x^2 d^6+c^2 e^2 \sqrt{c d^4+a e^4} x^3 d^5+3 c^2 e^3 \sqrt{c d^4+a e^4} x^4 d^4-6 a c e^5 \sqrt{c x^4+a} \log \left (a e^2+c d^2 x^2+\sqrt{c d^4+a e^4} \sqrt{c x^4+a}\right ) d^4+4 a c e^3 \sqrt{c d^4+a e^4} d^4+a c e^4 \sqrt{c d^4+a e^4} x d^3+6 a c e^5 (d+e x) \sqrt{c x^4+a} \log \left (e^2 x^2-d^2\right ) d^3-6 a c e^6 x \sqrt{c x^4+a} \log \left (a e^2+c d^2 x^2+\sqrt{c d^4+a e^4} \sqrt{c x^4+a}\right ) d^3-a c e^5 \sqrt{c d^4+a e^4} x^2 d^2-12 \sqrt [4]{-1} a^{5/4} c^{3/4} e^4 \sqrt{c d^4+a e^4} (d+e x) \sqrt{\frac{c x^4}{a}+1} \Pi \left (\frac{i \sqrt{a} e^2}{\sqrt{c} d^2};\left .\sin ^{-1}\left (\frac{(-1)^{3/4} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right ) d^2+a c e^6 \sqrt{c d^4+a e^4} x^3 d-3 a c e^7 \sqrt{c d^4+a e^4} x^4-2 a^2 e^7 \sqrt{c d^4+a e^4}\right )}{2 a \sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} \left (c d^4+a e^4\right )^{5/2} (d+e x) \sqrt{c x^4+a}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^2*(a + c*x^4)^(3/2)),x]

[Out]

(3*Sqrt[a]*Sqrt[c]*e^2*(-(c*d^4) + a*e^4)*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] + Sqrt[c]*Sqrt
[c*d^4 + a*e^4]*((-I)*c^(3/2)*d^6 + 3*Sqrt[a]*c*d^4*e^2 + (5*I)*a*Sqrt[c]*d^2*e^
4 - 3*a^(3/2)*e^6)*(d + e*x)*Sqrt[1 + (c*x^4)/a]*EllipticF[I*ArcSinh[Sqrt[(I*Sqr
t[c])/Sqrt[a]]*x], -1] + Sqrt[(I*Sqrt[c])/Sqrt[a]]*(4*a*c*d^4*e^3*Sqrt[c*d^4 + a
*e^4] - 2*a^2*e^7*Sqrt[c*d^4 + a*e^4] + c^2*d^7*Sqrt[c*d^4 + a*e^4]*x + a*c*d^3*
e^4*Sqrt[c*d^4 + a*e^4]*x - c^2*d^6*e*Sqrt[c*d^4 + a*e^4]*x^2 - a*c*d^2*e^5*Sqrt
[c*d^4 + a*e^4]*x^2 + c^2*d^5*e^2*Sqrt[c*d^4 + a*e^4]*x^3 + a*c*d*e^6*Sqrt[c*d^4
+ a*e^4]*x^3 + 3*c^2*d^4*e^3*Sqrt[c*d^4 + a*e^4]*x^4 - 3*a*c*e^7*Sqrt[c*d^4 + a
*e^4]*x^4 - 12*(-1)^(1/4)*a^(5/4)*c^(3/4)*d^2*e^4*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] + 6*a*c*d^3*e^5*(d + e*x)*Sqrt[a + c*x^4]*Log[-d^2 + e
^2*x^2] - 6*a*c*d^4*e^5*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]] - 6*a*c*d^3*e^6*x*Sqrt[a + c*x^4]*Log[a*e^2 + c*d^2*x^2 + S
qrt[c*d^4 + a*e^4]*Sqrt[a + c*x^4]]))/(2*a*Sqrt[(I*Sqrt[c])/Sqrt[a]]*(c*d^4 + a*
e^4)^(5/2)*(d + e*x)*Sqrt[a + c*x^4])

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Maple [C]  time = 0.038, size = 642, normalized size = 0.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^2/(c*x^4+a)^(3/2),x)

[Out]

-e^7*(c*x^4+a)^(1/2)/(a*e^4+c*d^4)^2/(e*x+d)-2*c*(1/4*e^2*(a*e^4-3*c*d^4)/a/(a*e
^4+c*d^4)^2*x^3-1/2*d*e*(a*e^4-c*d^4)/a/(a*e^4+c*d^4)^2*x^2+1/4*d^2*(3*a*e^4-c*d
^4)/a/(a*e^4+c*d^4)^2*x-d^3*e^3/(a*e^4+c*d^4)^2)/((x^4+1/c*a)*c)^(1/2)+(-d^2*e^4
*c/(a*e^4+c*d^4)^2-1/2*c*d^2*(3*a*e^4-c*d^4)/a/(a*e^4+c*d^4)^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)+I*(c*e^6/(a*e^4+c*d^4)^2+1/2*
c*e^2*(a*e^4-3*c*d^4)/a/(a*e^4+c*d^4)^2)*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)/c^(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))+6*c*d^3*e^3/(a*e^4+c*d^4)^2*(-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)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{4} + a\right )}^{\frac{3}{2}}{\left (e x + d\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + a)^(3/2)*(e*x + d)^2),x, algorithm="maxima")

[Out]

integrate(1/((c*x^4 + a)^(3/2)*(e*x + d)^2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (c e^{2} x^{6} + 2 \, c d e x^{5} + c d^{2} x^{4} + a e^{2} x^{2} + 2 \, a d e x + a d^{2}\right )} \sqrt{c x^{4} + a}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + a)^(3/2)*(e*x + d)^2),x, algorithm="fricas")

[Out]

integral(1/((c*e^2*x^6 + 2*c*d*e*x^5 + c*d^2*x^4 + a*e^2*x^2 + 2*a*d*e*x + a*d^2
)*sqrt(c*x^4 + a)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + c x^{4}\right )^{\frac{3}{2}} \left (d + e x\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x+d)**2/(c*x**4+a)**(3/2),x)

[Out]

Integral(1/((a + c*x**4)**(3/2)*(d + e*x)**2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{4} + a\right )}^{\frac{3}{2}}{\left (e x + d\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + a)^(3/2)*(e*x + d)^2),x, algorithm="giac")

[Out]

integrate(1/((c*x^4 + a)^(3/2)*(e*x + d)^2), x)