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

Optimal. Leaf size=1385 \[ \text{result too large to display} \]

[Out]

(2*Sqrt[c]*x*Sqrt[a + c*x^4])/(e^2*(Sqrt[a] + Sqrt[c]*x^2)) - (d*Sqrt[a + c*x^4]
)/(e*(d^2 - e^2*x^2)) + (x*Sqrt[a + c*x^4])/(d^2 - e^2*x^2) - ((c*d^4 - a*e^4)*A
rcTan[(Sqrt[-((c*d^4 + a*e^4)/(d^2*e^2))]*x)/Sqrt[a + c*x^4]])/(2*d^2*e^4*Sqrt[-
((c*d^4 + a*e^4)/(d^2*e^2))]) + (Sqrt[-((c*d^4 + a*e^4)/(d^2*e^2))]*ArcTan[(Sqrt
[-((c*d^4 + a*e^4)/(d^2*e^2))]*x)/Sqrt[a + c*x^4]])/(2*e^2) - (Sqrt[c]*d*ArcTanh
[(Sqrt[c]*x^2)/Sqrt[a + c*x^4]])/e^3 + (c*d^3*ArcTanh[(a*e^2 + c*d^2*x^2)/(Sqrt[
c*d^4 + a*e^4]*Sqrt[a + c*x^4])])/(e^3*Sqrt[c*d^4 + a*e^4]) - (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*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*e^4*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]
)/(2*a^(1/4)*e^4*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^(1/4)*e^4*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]*Ellip
ticF[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]) + ((Sqrt[c]*d^2 - Sqrt[a]*e^2)^2*(Sqrt[a] + Sqrt[c]*x^2)*S
qrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticPi[(Sqrt[a]*((Sqrt[c]*d^2)/Sq
rt[a] + e^2)^2)/(4*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^2*e^4*Sqrt[a + c*x^4]) + ((Sqrt[c]*d^2 - Sqrt[a]*e^2)*(c*d^4 + a*
e^4)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*Ellipti
cPi[(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*e^4*(Sqrt[c]*d^2 + Sqrt[a]*e^2)*S
qrt[a + c*x^4]) + ((Sqrt[c]*d^2 - Sqrt[a]*e^2)^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])/(8*a^(1/4)*c^(1/
4)*d^2*e^4*Sqrt[a + c*x^4])

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Rubi [A]  time = 4.06253, antiderivative size = 1385, normalized size of antiderivative = 1., number of steps used = 29, number of rules used = 14, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.737 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

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

[Out]

(2*Sqrt[c]*x*Sqrt[a + c*x^4])/(e^2*(Sqrt[a] + Sqrt[c]*x^2)) - (d*Sqrt[a + c*x^4]
)/(e*(d^2 - e^2*x^2)) + (x*Sqrt[a + c*x^4])/(d^2 - e^2*x^2) - ((c*d^4 - a*e^4)*A
rcTan[(Sqrt[-((c*d^4 + a*e^4)/(d^2*e^2))]*x)/Sqrt[a + c*x^4]])/(2*d^2*e^4*Sqrt[-
((c*d^4 + a*e^4)/(d^2*e^2))]) + (Sqrt[-((c*d^4 + a*e^4)/(d^2*e^2))]*ArcTan[(Sqrt
[-((c*d^4 + a*e^4)/(d^2*e^2))]*x)/Sqrt[a + c*x^4]])/(2*e^2) - (Sqrt[c]*d*ArcTanh
[(Sqrt[c]*x^2)/Sqrt[a + c*x^4]])/e^3 + (c*d^3*ArcTanh[(a*e^2 + c*d^2*x^2)/(Sqrt[
c*d^4 + a*e^4]*Sqrt[a + c*x^4])])/(e^3*Sqrt[c*d^4 + a*e^4]) - (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*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*e^4*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]
)/(2*a^(1/4)*e^4*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^(1/4)*e^4*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]*Ellip
ticF[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]) + ((Sqrt[c]*d^2 - Sqrt[a]*e^2)^2*(Sqrt[a] + Sqrt[c]*x^2)*S
qrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticPi[(Sqrt[a]*((Sqrt[c]*d^2)/Sq
rt[a] + e^2)^2)/(4*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^2*e^4*Sqrt[a + c*x^4]) + ((Sqrt[c]*d^2 - Sqrt[a]*e^2)*(c*d^4 + a*
e^4)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*Ellipti
cPi[(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*e^4*(Sqrt[c]*d^2 + Sqrt[a]*e^2)*S
qrt[a + c*x^4]) + ((Sqrt[c]*d^2 - Sqrt[a]*e^2)^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])/(8*a^(1/4)*c^(1/
4)*d^2*e^4*Sqrt[a + c*x^4])

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 6.29073, size = 924, normalized size = 0.67 \[ \frac{2 c \left (\frac{\left (-\frac{e \left (-2 \sqrt [4]{-1} \sqrt [4]{a} \sqrt{\frac{c d^4}{a e^4}+1} e \sqrt{1-\frac{i \sqrt{c} x^2}{\sqrt{a}}} \sqrt{\frac{i \sqrt{c} x^2}{\sqrt{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 )-\sqrt [4]{c} d \sqrt{\frac{c x^4}{a}+1} \log \left (\frac{e^2 x^2-d^2}{a e^2+a \sqrt{\frac{c d^4}{a e^4}+1} \sqrt{\frac{c x^4}{a}+1} e^2+c d^2 x^2}\right )\right )}{4 \sqrt [4]{c} d^2 \sqrt{\frac{c d^4}{a e^4}+1} \sqrt{c x^4+a}}-\frac{e \left (\sqrt [4]{c} d \sqrt{\frac{c x^4}{a}+1} \log \left (\frac{e^2 x^2-d^2}{a e^2+a \sqrt{\frac{c d^4}{a e^4}+1} \sqrt{\frac{c x^4}{a}+1} e^2+c d^2 x^2}\right )-2 \sqrt [4]{-1} \sqrt [4]{a} \sqrt{\frac{c d^4}{a e^4}+1} e \sqrt{1-\frac{i \sqrt{c} x^2}{\sqrt{a}}} \sqrt{\frac{i \sqrt{c} x^2}{\sqrt{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 )\right )}{4 \sqrt [4]{c} d^2 \sqrt{\frac{c d^4}{a e^4}+1} \sqrt{c x^4+a}}\right ) d^4}{e^5}-\frac{i \sqrt{1-\frac{i \sqrt{c} x^2}{\sqrt{a}}} \sqrt{\frac{i \sqrt{c} x^2}{\sqrt{a}}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right ) d^2}{\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} e^3 \sqrt{c x^4+a}}-\frac{\left (\frac{\log \left (e^2 x^2-d^2\right ) d^2}{\sqrt{c d^4+a e^4}}-\frac{\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}{\sqrt{c d^4+a e^4}}+\frac{\log \left (c x^2+\sqrt{c} \sqrt{c x^4+a}\right )}{\sqrt{c}}\right ) d}{2 e^2}+\frac{\sqrt{a} \sqrt{1-\frac{i \sqrt{c} x^2}{\sqrt{a}}} \sqrt{\frac{i \sqrt{c} x^2}{\sqrt{a}}+1} \left (E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )\right )}{\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} \sqrt{c} e \sqrt{c x^4+a}}\right )}{e}-\frac{\sqrt{c x^4+a}}{e (d+e x)} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[a + c*x^4]/(e*(d + e*x))) + (2*c*((Sqrt[a]*Sqrt[1 - (I*Sqrt[c]*x^2)/Sqrt[
a]]*Sqrt[1 + (I*Sqrt[c]*x^2)/Sqrt[a]]*(EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt
[a]]*x], -1] - EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1]))/(Sqrt[(I*
Sqrt[c])/Sqrt[a]]*Sqrt[c]*e*Sqrt[a + c*x^4]) - (I*d^2*Sqrt[1 - (I*Sqrt[c]*x^2)/S
qrt[a]]*Sqrt[1 + (I*Sqrt[c]*x^2)/Sqrt[a]]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/S
qrt[a]]*x], -1])/(Sqrt[(I*Sqrt[c])/Sqrt[a]]*e^3*Sqrt[a + c*x^4]) - (d*((d^2*Log[
-d^2 + e^2*x^2])/Sqrt[c*d^4 + a*e^4] + Log[c*x^2 + Sqrt[c]*Sqrt[a + c*x^4]]/Sqrt
[c] - (d^2*Log[a*e^2 + c*d^2*x^2 + Sqrt[c*d^4 + a*e^4]*Sqrt[a + c*x^4]])/Sqrt[c*
d^4 + a*e^4]))/(2*e^2) + (d^4*(-(e*(-2*(-1)^(1/4)*a^(1/4)*Sqrt[1 + (c*d^4)/(a*e^
4)]*e*Sqrt[1 - (I*Sqrt[c]*x^2)/Sqrt[a]]*Sqrt[1 + (I*Sqrt[c]*x^2)/Sqrt[a]]*Ellipt
icPi[(I*Sqrt[a]*e^2)/(Sqrt[c]*d^2), ArcSin[((-1)^(3/4)*c^(1/4)*x)/a^(1/4)], -1]
- c^(1/4)*d*Sqrt[1 + (c*x^4)/a]*Log[(-d^2 + e^2*x^2)/(a*e^2 + c*d^2*x^2 + a*Sqrt
[1 + (c*d^4)/(a*e^4)]*e^2*Sqrt[1 + (c*x^4)/a])]))/(4*c^(1/4)*d^2*Sqrt[1 + (c*d^4
)/(a*e^4)]*Sqrt[a + c*x^4]) - (e*(-2*(-1)^(1/4)*a^(1/4)*Sqrt[1 + (c*d^4)/(a*e^4)
]*e*Sqrt[1 - (I*Sqrt[c]*x^2)/Sqrt[a]]*Sqrt[1 + (I*Sqrt[c]*x^2)/Sqrt[a]]*Elliptic
Pi[(I*Sqrt[a]*e^2)/(Sqrt[c]*d^2), ArcSin[((-1)^(3/4)*c^(1/4)*x)/a^(1/4)], -1] +
c^(1/4)*d*Sqrt[1 + (c*x^4)/a]*Log[(-d^2 + e^2*x^2)/(a*e^2 + c*d^2*x^2 + a*Sqrt[1
+ (c*d^4)/(a*e^4)]*e^2*Sqrt[1 + (c*x^4)/a])]))/(4*c^(1/4)*d^2*Sqrt[1 + (c*d^4)/
(a*e^4)]*Sqrt[a + c*x^4])))/e^5))/e

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Maple [C]  time = 0.025, size = 402, normalized size = 0.3 \[ -{\frac{1}{e \left ( ex+d \right ) }\sqrt{c{x}^{4}+a}}+2\,{\frac{c{d}^{2}}{{e}^{4}\sqrt{c{x}^{4}+a}}\sqrt{1-{\frac{i\sqrt{c}{x}^{2}}{\sqrt{a}}}}\sqrt{1+{\frac{i\sqrt{c}{x}^{2}}{\sqrt{a}}}}{\it EllipticF} \left ( x\sqrt{{\frac{i\sqrt{c}}{\sqrt{a}}}},i \right ){\frac{1}{\sqrt{{\frac{i\sqrt{c}}{\sqrt{a}}}}}}}-{\frac{d}{{e}^{3}}\sqrt{c}\ln \left ( 2\,{x}^{2}\sqrt{c}+2\,\sqrt{c{x}^{4}+a} \right ) }+{\frac{2\,i}{{e}^{2}}\sqrt{a}\sqrt{c}\sqrt{1-{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}} \left ({\it EllipticF} \left ( x\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ) -{\it EllipticE} \left ( x\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ) \right ){\frac{1}{\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{c{x}^{4}+a}}}}+{\frac{c{d}^{3}}{{e}^{5}}{\it Artanh} \left ({\frac{1}{2} \left ( 2\,{\frac{c{d}^{2}{x}^{2}}{{e}^{2}}}+2\,a \right ){\frac{1}{\sqrt{{\frac{c{d}^{4}}{{e}^{4}}}+a}}}{\frac{1}{\sqrt{c{x}^{4}+a}}}} \right ){\frac{1}{\sqrt{{\frac{c{d}^{4}}{{e}^{4}}}+a}}}}-2\,{\frac{c{d}^{2}}{{e}^{4}\sqrt{c{x}^{4}+a}}\sqrt{1-{\frac{i\sqrt{c}{x}^{2}}{\sqrt{a}}}}\sqrt{1+{\frac{i\sqrt{c}{x}^{2}}{\sqrt{a}}}}{\it EllipticPi} \left ( x\sqrt{{\frac{i\sqrt{c}}{\sqrt{a}}}},{\frac{-i\sqrt{a}{e}^{2}}{{d}^{2}\sqrt{c}}},{1\sqrt{{\frac{-i\sqrt{c}}{\sqrt{a}}}}{\frac{1}{\sqrt{{\frac{i\sqrt{c}}{\sqrt{a}}}}}}} \right ){\frac{1}{\sqrt{{\frac{i\sqrt{c}}{\sqrt{a}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/e*(c*x^4+a)^(1/2)/(e*x+d)+2*c*d^2/e^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)-c^(1/2)*d/e^3*ln(2*x^2*c^(1/2)+2*(c*x^4+a)^(1/2))+2*
I*c^(1/2)/e^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)-EllipticE(x*(I/a^(1/2)*c^(1/2))^(1/2),I))+c*d^3/e^5/(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))-2*c*d^2
/e^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)*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. \[ \int \frac{\sqrt{c x^{4} + a}}{{\left (e x + d\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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