3.195 \(\int \frac{1}{(d+e x)^3 \sqrt{a+c x^4}} \, dx\)

Optimal. Leaf size=1969 \[ \text{result too large to display} \]

[Out]

(-3*Sqrt[c]*e^2*x*Sqrt[a + c*x^4])/(2*d*(c*d^4 + a*e^4)*(Sqrt[a] + Sqrt[c]*x^2))
+ (3*Sqrt[c]*e^2*(3*c*d^4 + a*e^4)*x*Sqrt[a + c*x^4])/(2*d*(c*d^4 + a*e^4)^2*(S
qrt[a] + Sqrt[c]*x^2)) - (d^2*e^3*Sqrt[a + c*x^4])/((c*d^4 + a*e^4)*(d^2 - e^2*x
^2)^2) + (d*e^4*x*Sqrt[a + c*x^4])/((c*d^4 + a*e^4)*(d^2 - e^2*x^2)^2) - (9*c*d^
4*e^3*Sqrt[a + c*x^4])/(4*(c*d^4 + a*e^4)^2*(d^2 - e^2*x^2)) - (e^3*(c*d^4 - 2*a
*e^4)*Sqrt[a + c*x^4])/(4*(c*d^4 + a*e^4)^2*(d^2 - e^2*x^2)) - (3*e^4*x*Sqrt[a +
c*x^4])/(2*d*(c*d^4 + a*e^4)*(d^2 - e^2*x^2)) + (3*e^4*(3*c*d^4 + a*e^4)*x*Sqrt
[a + c*x^4])/(2*d*(c*d^4 + a*e^4)^2*(d^2 - e^2*x^2)) + (3*(3*c*d^4 + a*e^4)*ArcT
an[(Sqrt[-((c*d^4 + a*e^4)/(d^2*e^2))]*x)/Sqrt[a + c*x^4]])/(4*d^5*e^2*(-((c*d^4
+ a*e^4)/(d^2*e^2)))^(3/2)) + (3*(5*c^2*d^8 + 2*a*c*d^4*e^4 + a^2*e^8)*ArcTan[(
Sqrt[-((c*d^4 + a*e^4)/(d^2*e^2))]*x)/Sqrt[a + c*x^4]])/(4*d^3*(c*d^4 + a*e^4)^2
*Sqrt[-((c*d^4 + a*e^4)/(d^2*e^2))]) + (3*a*c*d^2*e^5*ArcTanh[(a*e^2 + c*d^2*x^2
)/(Sqrt[c*d^4 + a*e^4]*Sqrt[a + c*x^4])])/(4*(c*d^4 + a*e^4)^(5/2)) - (3*c*d^2*e
*(2*c*d^4 - a*e^4)*ArcTanh[(a*e^2 + c*d^2*x^2)/(Sqrt[c*d^4 + a*e^4]*Sqrt[a + c*x
^4])])/(4*(c*d^4 + a*e^4)^(5/2)) + (3*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])/(2*d*(c*d^4 + a*e^4)*Sqrt[a + c*x^4]) - (3*a^(1/4)*c^(1/4)*e^2*(3*c
*d^4 + a*e^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])/(2*d*(c*d^4 + a*e^4)^2*Sqrt[a +
c*x^4]) + (3*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])/(4*d*(c*d^4 + a*e^4)*Sqrt[a + c*x^4]) - (3*c^(1/4)*(3*c*d^4 + a*e
^4)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*Elliptic
F[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(4*a^(1/4)*d*(Sqrt[c]*d^2 + Sqrt[a]*e^2)*
(c*d^4 + a*e^4)*Sqrt[a + c*x^4]) - (c^(1/4)*(7*c^(3/2)*d^6 - 9*Sqrt[a]*c*d^4*e^2
+ a*Sqrt[c]*d^2*e^4 - 3*a^(3/2)*e^6)*(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^(1
/4)*d*(c*d^4 + a*e^4)^2*Sqrt[a + c*x^4]) + (3*c^(1/4)*(5*c^2*d^8 + 2*a*c*d^4*e^4
+ a^2*e^8)*(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^(1/4)*d*(Sqrt[c]*d^2 + Sqrt[
a]*e^2)*(c*d^4 + a*e^4)^2*Sqrt[a + c*x^4]) + (3*(Sqrt[c]*d^2 - Sqrt[a]*e^2)*(3*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])/(8*a^(1/4)*c^(1/4)*d^3*(Sqrt[c]*d^2 + Sqrt[a]*e^2)*
(c*d^4 + a*e^4)*Sqrt[a + c*x^4]) - (3*(Sqrt[c]*d^2 - Sqrt[a]*e^2)*(5*c^2*d^8 + 2
*a*c*d^4*e^4 + a^2*e^8)*(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])/(8*a^(1/4)*c^(1/4)*d^3*(Sqrt[c]*d^2 + Sqr
t[a]*e^2)*(c*d^4 + a*e^4)^2*Sqrt[a + c*x^4])

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Rubi [A]  time = 9.25775, antiderivative size = 1969, normalized size of antiderivative = 1., number of steps used = 36, number of rules used = 17, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.895 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-3*Sqrt[c]*e^2*x*Sqrt[a + c*x^4])/(2*d*(c*d^4 + a*e^4)*(Sqrt[a] + Sqrt[c]*x^2))
+ (3*Sqrt[c]*e^2*(3*c*d^4 + a*e^4)*x*Sqrt[a + c*x^4])/(2*d*(c*d^4 + a*e^4)^2*(S
qrt[a] + Sqrt[c]*x^2)) - (d^2*e^3*Sqrt[a + c*x^4])/((c*d^4 + a*e^4)*(d^2 - e^2*x
^2)^2) + (d*e^4*x*Sqrt[a + c*x^4])/((c*d^4 + a*e^4)*(d^2 - e^2*x^2)^2) - (9*c*d^
4*e^3*Sqrt[a + c*x^4])/(4*(c*d^4 + a*e^4)^2*(d^2 - e^2*x^2)) - (e^3*(c*d^4 - 2*a
*e^4)*Sqrt[a + c*x^4])/(4*(c*d^4 + a*e^4)^2*(d^2 - e^2*x^2)) - (3*e^4*x*Sqrt[a +
c*x^4])/(2*d*(c*d^4 + a*e^4)*(d^2 - e^2*x^2)) + (3*e^4*(3*c*d^4 + a*e^4)*x*Sqrt
[a + c*x^4])/(2*d*(c*d^4 + a*e^4)^2*(d^2 - e^2*x^2)) + (3*(3*c*d^4 + a*e^4)*ArcT
an[(Sqrt[-((c*d^4 + a*e^4)/(d^2*e^2))]*x)/Sqrt[a + c*x^4]])/(4*d^5*e^2*(-((c*d^4
+ a*e^4)/(d^2*e^2)))^(3/2)) + (3*(5*c^2*d^8 + 2*a*c*d^4*e^4 + a^2*e^8)*ArcTan[(
Sqrt[-((c*d^4 + a*e^4)/(d^2*e^2))]*x)/Sqrt[a + c*x^4]])/(4*d^3*(c*d^4 + a*e^4)^2
*Sqrt[-((c*d^4 + a*e^4)/(d^2*e^2))]) + (3*a*c*d^2*e^5*ArcTanh[(a*e^2 + c*d^2*x^2
)/(Sqrt[c*d^4 + a*e^4]*Sqrt[a + c*x^4])])/(4*(c*d^4 + a*e^4)^(5/2)) - (3*c*d^2*e
*(2*c*d^4 - a*e^4)*ArcTanh[(a*e^2 + c*d^2*x^2)/(Sqrt[c*d^4 + a*e^4]*Sqrt[a + c*x
^4])])/(4*(c*d^4 + a*e^4)^(5/2)) + (3*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])/(2*d*(c*d^4 + a*e^4)*Sqrt[a + c*x^4]) - (3*a^(1/4)*c^(1/4)*e^2*(3*c
*d^4 + a*e^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])/(2*d*(c*d^4 + a*e^4)^2*Sqrt[a +
c*x^4]) + (3*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])/(4*d*(c*d^4 + a*e^4)*Sqrt[a + c*x^4]) - (3*c^(1/4)*(3*c*d^4 + a*e
^4)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*Elliptic
F[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(4*a^(1/4)*d*(Sqrt[c]*d^2 + Sqrt[a]*e^2)*
(c*d^4 + a*e^4)*Sqrt[a + c*x^4]) - (c^(1/4)*(7*c^(3/2)*d^6 - 9*Sqrt[a]*c*d^4*e^2
+ a*Sqrt[c]*d^2*e^4 - 3*a^(3/2)*e^6)*(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^(1
/4)*d*(c*d^4 + a*e^4)^2*Sqrt[a + c*x^4]) + (3*c^(1/4)*(5*c^2*d^8 + 2*a*c*d^4*e^4
+ a^2*e^8)*(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^(1/4)*d*(Sqrt[c]*d^2 + Sqrt[
a]*e^2)*(c*d^4 + a*e^4)^2*Sqrt[a + c*x^4]) + (3*(Sqrt[c]*d^2 - Sqrt[a]*e^2)*(3*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])/(8*a^(1/4)*c^(1/4)*d^3*(Sqrt[c]*d^2 + Sqrt[a]*e^2)*
(c*d^4 + a*e^4)*Sqrt[a + c*x^4]) - (3*(Sqrt[c]*d^2 - Sqrt[a]*e^2)*(5*c^2*d^8 + 2
*a*c*d^4*e^4 + a^2*e^8)*(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])/(8*a^(1/4)*c^(1/4)*d^3*(Sqrt[c]*d^2 + Sqr
t[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. \[ \int \frac{1}{\sqrt{a + c x^{4}} \left (d + e x\right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 2.04995, size = 884, normalized size = 0.45 \[ \frac{3 c^2 e (d+e x)^2 \sqrt{c x^4+a} \log \left (e^2 x^2-d^2\right ) d^6-3 c^2 e (d+e x)^2 \sqrt{c x^4+a} \log \left (a e^2+c d^2 x^2+\sqrt{c d^4+a e^4} \sqrt{c x^4+a}\right ) d^6+\frac{4 i c^2 \sqrt{c d^4+a e^4} (d+e x)^2 \sqrt{\frac{c x^4}{a}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right ) d^5}{\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}}}-6 \sqrt [4]{-1} \sqrt [4]{a} c^{7/4} \sqrt{c d^4+a e^4} (d+e x)^2 \sqrt{\frac{c x^4}{a}+1} \Pi \left (\frac{i \sqrt{a} e^2}{\sqrt{c} d^2};\left .\sin ^{-1}\left (\frac{(-1)^{3/4} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right ) d^5-6 c e^3 \sqrt{c d^4+a e^4} (d+e x) \left (c x^4+a\right ) d^3-6 i a \sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} c e^2 \sqrt{c d^4+a e^4} (d+e x)^2 \sqrt{\frac{c x^4}{a}+1} E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right ) d^3+6 i a \sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} c e^2 \sqrt{c d^4+a e^4} (d+e x)^2 \sqrt{\frac{c x^4}{a}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right ) d^3-3 a c e^5 (d+e x)^2 \sqrt{c x^4+a} \log \left (e^2 x^2-d^2\right ) d^2+3 a c e^5 (d+e x)^2 \sqrt{c x^4+a} \log \left (a e^2+c d^2 x^2+\sqrt{c d^4+a e^4} \sqrt{c x^4+a}\right ) d^2-\frac{2 i a c e^4 \sqrt{c d^4+a e^4} (d+e x)^2 \sqrt{\frac{c x^4}{a}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right ) d}{\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}}}+6 \sqrt [4]{-1} a^{5/4} c^{3/4} e^4 \sqrt{c d^4+a e^4} (d+e x)^2 \sqrt{\frac{c x^4}{a}+1} \Pi \left (\frac{i \sqrt{a} e^2}{\sqrt{c} d^2};\left .\sin ^{-1}\left (\frac{(-1)^{3/4} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right ) d-e^3 \left (c d^4+a e^4\right )^{3/2} \left (c x^4+a\right )}{2 \left (c d^4+a e^4\right )^{5/2} (d+e x)^2 \sqrt{c x^4+a}} \]

Antiderivative was successfully verified.

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

[Out]

(-(e^3*(c*d^4 + a*e^4)^(3/2)*(a + c*x^4)) - 6*c*d^3*e^3*Sqrt[c*d^4 + a*e^4]*(d +
e*x)*(a + c*x^4) - (6*I)*a*Sqrt[(I*Sqrt[c])/Sqrt[a]]*c*d^3*e^2*Sqrt[c*d^4 + a*e
^4]*(d + e*x)^2*Sqrt[1 + (c*x^4)/a]*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]
]*x], -1] + ((4*I)*c^2*d^5*Sqrt[c*d^4 + a*e^4]*(d + e*x)^2*Sqrt[1 + (c*x^4)/a]*E
llipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1])/Sqrt[(I*Sqrt[c])/Sqrt[a]]
+ (6*I)*a*Sqrt[(I*Sqrt[c])/Sqrt[a]]*c*d^3*e^2*Sqrt[c*d^4 + a*e^4]*(d + e*x)^2*Sq
rt[1 + (c*x^4)/a]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1] - ((2*I)
*a*c*d*e^4*Sqrt[c*d^4 + a*e^4]*(d + e*x)^2*Sqrt[1 + (c*x^4)/a]*EllipticF[I*ArcSi
nh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1])/Sqrt[(I*Sqrt[c])/Sqrt[a]] - 6*(-1)^(1/4)*a
^(1/4)*c^(7/4)*d^5*Sqrt[c*d^4 + a*e^4]*(d + e*x)^2*Sqrt[1 + (c*x^4)/a]*EllipticP
i[(I*Sqrt[a]*e^2)/(Sqrt[c]*d^2), ArcSin[((-1)^(3/4)*c^(1/4)*x)/a^(1/4)], -1] + 6
*(-1)^(1/4)*a^(5/4)*c^(3/4)*d*e^4*Sqrt[c*d^4 + a*e^4]*(d + e*x)^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] + 3*c^2*d^6*e*(d + e*x)^2*Sqrt[a + c*x^4]*Log[-d^2 + e^2*x^2] - 3*a*
c*d^2*e^5*(d + e*x)^2*Sqrt[a + c*x^4]*Log[-d^2 + e^2*x^2] - 3*c^2*d^6*e*(d + e*x
)^2*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]]
+ 3*a*c*d^2*e^5*(d + e*x)^2*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]])/(2*(c*d^4 + a*e^4)^(5/2)*(d + e*x)^2*Sqrt[a + c*x^4])

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Maple [C]  time = 0.025, size = 483, normalized size = 0.3 \[ -{\frac{{e}^{3}}{ \left ( 2\,a{e}^{4}+2\,c{d}^{4} \right ) \left ( ex+d \right ) ^{2}}\sqrt{c{x}^{4}+a}}-3\,{\frac{c{e}^{3}{d}^{3}\sqrt{c{x}^{4}+a}}{ \left ( a{e}^{4}+c{d}^{4} \right ) ^{2} \left ( ex+d \right ) }}+{\frac{cd \left ( a{e}^{4}-2\,c{d}^{4} \right ) }{ \left ( a{e}^{4}+c{d}^{4} \right ) ^{2}}\sqrt{1-{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{c{x}^{4}+a}}}}+{\frac{3\,i{d}^{3}{e}^{2}}{ \left ( a{e}^{4}+c{d}^{4} \right ) ^{2}}{c}^{{\frac{3}{2}}}\sqrt{a}\sqrt{1-{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}} \left ({\it EllipticF} \left ( x\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ) -{\it EllipticE} \left ( x\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ) \right ){\frac{1}{\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{c{x}^{4}+a}}}}-3\,{\frac{c{d}^{2} \left ( a{e}^{4}-c{d}^{4} \right ) }{ \left ( a{e}^{4}+c{d}^{4} \right ) ^{2}e} \left ( -1/2\,{1{\it Artanh} \left ( 1/2\,{\frac{1}{\sqrt{c{x}^{4}+a}} \left ( 2\,{\frac{c{d}^{2}{x}^{2}}{{e}^{2}}}+2\,a \right ){\frac{1}{\sqrt{{\frac{c{d}^{4}}{{e}^{4}}}+a}}}} \right ){\frac{1}{\sqrt{{\frac{c{d}^{4}}{{e}^{4}}}+a}}}}+{\frac{e}{d\sqrt{c{x}^{4}+a}}\sqrt{1-{\frac{i\sqrt{c}{x}^{2}}{\sqrt{a}}}}\sqrt{1+{\frac{i\sqrt{c}{x}^{2}}{\sqrt{a}}}}{\it EllipticPi} \left ( x\sqrt{{\frac{i\sqrt{c}}{\sqrt{a}}}},{\frac{-i\sqrt{a}{e}^{2}}{{d}^{2}\sqrt{c}}},{1\sqrt{{\frac{-i\sqrt{c}}{\sqrt{a}}}}{\frac{1}{\sqrt{{\frac{i\sqrt{c}}{\sqrt{a}}}}}}} \right ){\frac{1}{\sqrt{{\frac{i\sqrt{c}}{\sqrt{a}}}}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*e^3/(a*e^4+c*d^4)*(c*x^4+a)^(1/2)/(e*x+d)^2-3*c*e^3*d^3/(a*e^4+c*d^4)^2*(c*
x^4+a)^(1/2)/(e*x+d)+c*d*(a*e^4-2*c*d^4)/(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)+3*I*c^(3/2)*d^3*e^2/(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)*(EllipticF(x*(I/a^(1/2)*c^(1/2))^(1/2),I)-Elli
pticE(x*(I/a^(1/2)*c^(1/2))^(1/2),I))-3*c*d^2*(a*e^4-c*d^4)/(a*e^4+c*d^4)^2/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)))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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