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

Optimal. Leaf size=390 \[ -\frac{\left (-\frac{3 a b c e-2 a c^2 d+b^3 (-e)+b^2 c d}{\sqrt{b^2-4 a c}}+a c e+b^2 (-e)+b c d\right ) \tan ^{-1}\left (\frac{x \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right )}{c^2 \sqrt{b-\sqrt{b^2-4 a c}} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}-\frac{\left (\frac{3 a b c e-2 a c^2 d+b^3 (-e)+b^2 c d}{\sqrt{b^2-4 a c}}+a c e+b^2 (-e)+b c d\right ) \tan ^{-1}\left (\frac{x \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{d+e x^2}}\right )}{c^2 \sqrt{\sqrt{b^2-4 a c}+b} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}+\frac{(c d-2 b e) \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{2 c^2 \sqrt{e}}+\frac{x \sqrt{d+e x^2}}{2 c} \]

[Out]

(x*Sqrt[d + e*x^2])/(2*c) - ((b*c*d - b^2*e + a*c*e - (b^2*c*d - 2*a*c^2*d - b^3
*e + 3*a*b*c*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*
e]*x)/(Sqrt[b - Sqrt[b^2 - 4*a*c]]*Sqrt[d + e*x^2])])/(c^2*Sqrt[b - Sqrt[b^2 - 4
*a*c]]*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]) - ((b*c*d - b^2*e + a*c*e + (b^2
*c*d - 2*a*c^2*d - b^3*e + 3*a*b*c*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2*c*d - (b
+ Sqrt[b^2 - 4*a*c])*e]*x)/(Sqrt[b + Sqrt[b^2 - 4*a*c]]*Sqrt[d + e*x^2])])/(c^2
*Sqrt[b + Sqrt[b^2 - 4*a*c]]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]) + ((c*d -
2*b*e)*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(2*c^2*Sqrt[e])

_______________________________________________________________________________________

Rubi [A]  time = 6.52865, antiderivative size = 390, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.241 \[ -\frac{\left (-\frac{3 a b c e-2 a c^2 d+b^3 (-e)+b^2 c d}{\sqrt{b^2-4 a c}}+a c e+b^2 (-e)+b c d\right ) \tan ^{-1}\left (\frac{x \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right )}{c^2 \sqrt{b-\sqrt{b^2-4 a c}} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}-\frac{\left (\frac{3 a b c e-2 a c^2 d+b^3 (-e)+b^2 c d}{\sqrt{b^2-4 a c}}+a c e+b^2 (-e)+b c d\right ) \tan ^{-1}\left (\frac{x \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{d+e x^2}}\right )}{c^2 \sqrt{\sqrt{b^2-4 a c}+b} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}+\frac{(c d-2 b e) \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{2 c^2 \sqrt{e}}+\frac{x \sqrt{d+e x^2}}{2 c} \]

Antiderivative was successfully verified.

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

[Out]

(x*Sqrt[d + e*x^2])/(2*c) - ((b*c*d - b^2*e + a*c*e - (b^2*c*d - 2*a*c^2*d - b^3
*e + 3*a*b*c*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*
e]*x)/(Sqrt[b - Sqrt[b^2 - 4*a*c]]*Sqrt[d + e*x^2])])/(c^2*Sqrt[b - Sqrt[b^2 - 4
*a*c]]*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]) - ((b*c*d - b^2*e + a*c*e + (b^2
*c*d - 2*a*c^2*d - b^3*e + 3*a*b*c*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2*c*d - (b
+ Sqrt[b^2 - 4*a*c])*e]*x)/(Sqrt[b + Sqrt[b^2 - 4*a*c]]*Sqrt[d + e*x^2])])/(c^2
*Sqrt[b + Sqrt[b^2 - 4*a*c]]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]) + ((c*d -
2*b*e)*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(2*c^2*Sqrt[e])

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [A]  time = 0.808581, size = 0, normalized size = 0. \[ \int \frac{x^4 \sqrt{d+e x^2}}{a+b x^2+c x^4} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

_______________________________________________________________________________________

Maple [C]  time = 0.048, size = 290, normalized size = 0.7 \[{\frac{x}{2\,c}\sqrt{e{x}^{2}+d}}+{\frac{d}{2\,c}\ln \left ( x\sqrt{e}+\sqrt{e{x}^{2}+d} \right ){\frac{1}{\sqrt{e}}}}+{\frac{1}{2\,{c}^{2}}\sqrt{e}\sum _{{\it \_R}={\it RootOf} \left ( c{{\it \_Z}}^{4}+ \left ( 4\,be-4\,cd \right ){{\it \_Z}}^{3}+ \left ( 16\,a{e}^{2}-8\,bde+6\,c{d}^{2} \right ){{\it \_Z}}^{2}+ \left ( 4\,b{d}^{2}e-4\,c{d}^{3} \right ){\it \_Z}+c{d}^{4} \right ) }{\frac{ \left ( ace-{b}^{2}e+bcd \right ){{\it \_R}}^{2}+2\, \left ( -2\,ab{e}^{2}+acde+{b}^{2}de-bc{d}^{2} \right ){\it \_R}+ac{d}^{2}e-{b}^{2}{d}^{2}e+bc{d}^{3}}{{{\it \_R}}^{3}c+3\,{{\it \_R}}^{2}be-3\,{{\it \_R}}^{2}cd+8\,{\it \_R}\,a{e}^{2}-4\,{\it \_R}\,bde+3\,{\it \_R}\,c{d}^{2}+b{d}^{2}e-c{d}^{3}}\ln \left ( \left ( \sqrt{e{x}^{2}+d}-x\sqrt{e} \right ) ^{2}-{\it \_R} \right ) }}+{\frac{b}{{c}^{2}}\sqrt{e}\ln \left ( \sqrt{e{x}^{2}+d}-x\sqrt{e} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*x*(e*x^2+d)^(1/2)/c+1/2/c*d/e^(1/2)*ln(x*e^(1/2)+(e*x^2+d)^(1/2))+1/2/c^2*e^
(1/2)*sum(((a*c*e-b^2*e+b*c*d)*_R^2+2*(-2*a*b*e^2+a*c*d*e+b^2*d*e-b*c*d^2)*_R+a*
c*d^2*e-b^2*d^2*e+b*c*d^3)/(_R^3*c+3*_R^2*b*e-3*_R^2*c*d+8*_R*a*e^2-4*_R*b*d*e+3
*_R*c*d^2+b*d^2*e-c*d^3)*ln(((e*x^2+d)^(1/2)-x*e^(1/2))^2-_R),_R=RootOf(c*_Z^4+(
4*b*e-4*c*d)*_Z^3+(16*a*e^2-8*b*d*e+6*c*d^2)*_Z^2+(4*b*d^2*e-4*c*d^3)*_Z+c*d^4))
+1/c^2*e^(1/2)*b*ln((e*x^2+d)^(1/2)-x*e^(1/2))

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 25.8277, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/4*(sqrt(1/2)*c^2*sqrt(e)*sqrt(-((b^3*c - 3*a*b*c^2)*d - (b^4 - 4*a*b^2*c + 2*
a^2*c^2)*e + (b^2*c^4 - 4*a*c^5)*sqrt(((b^4*c^2 - 2*a*b^2*c^3 + a^2*c^4)*d^2 - 2
*(b^5*c - 3*a*b^3*c^2 + 2*a^2*b*c^3)*d*e + (b^6 - 4*a*b^4*c + 4*a^2*b^2*c^2)*e^2
)/(b^2*c^8 - 4*a*c^9)))/(b^2*c^4 - 4*a*c^5))*log(((a*b^2*c^4 - 4*a^2*c^5)*d*x^2*
sqrt(((b^4*c^2 - 2*a*b^2*c^3 + a^2*c^4)*d^2 - 2*(b^5*c - 3*a*b^3*c^2 + 2*a^2*b*c
^3)*d*e + (b^6 - 4*a*b^4*c + 4*a^2*b^2*c^2)*e^2)/(b^2*c^8 - 4*a*c^9)) + 2*(a^2*b
^2*c - a^3*c^2)*d^2 - 2*(a^2*b^3 - 2*a^3*b*c)*d*e - ((a*b^3*c - a^2*b*c^2)*d^2 -
(a*b^4 + 2*a^2*b^2*c - 4*a^3*c^2)*d*e + 4*(a^2*b^3 - 2*a^3*b*c)*e^2)*x^2 + 2*sq
rt(1/2)*sqrt(e*x^2 + d)*((b^4*c^4 - 6*a*b^2*c^5 + 8*a^2*c^6)*x*sqrt(((b^4*c^2 -
2*a*b^2*c^3 + a^2*c^4)*d^2 - 2*(b^5*c - 3*a*b^3*c^2 + 2*a^2*b*c^3)*d*e + (b^6 -
4*a*b^4*c + 4*a^2*b^2*c^2)*e^2)/(b^2*c^8 - 4*a*c^9)) - ((b^5*c - 5*a*b^3*c^2 + 4
*a^2*b*c^3)*d - (b^6 - 6*a*b^4*c + 8*a^2*b^2*c^2)*e)*x)*sqrt(-((b^3*c - 3*a*b*c^
2)*d - (b^4 - 4*a*b^2*c + 2*a^2*c^2)*e + (b^2*c^4 - 4*a*c^5)*sqrt(((b^4*c^2 - 2*
a*b^2*c^3 + a^2*c^4)*d^2 - 2*(b^5*c - 3*a*b^3*c^2 + 2*a^2*b*c^3)*d*e + (b^6 - 4*
a*b^4*c + 4*a^2*b^2*c^2)*e^2)/(b^2*c^8 - 4*a*c^9)))/(b^2*c^4 - 4*a*c^5)))/x^2) -
sqrt(1/2)*c^2*sqrt(e)*sqrt(-((b^3*c - 3*a*b*c^2)*d - (b^4 - 4*a*b^2*c + 2*a^2*c
^2)*e + (b^2*c^4 - 4*a*c^5)*sqrt(((b^4*c^2 - 2*a*b^2*c^3 + a^2*c^4)*d^2 - 2*(b^5
*c - 3*a*b^3*c^2 + 2*a^2*b*c^3)*d*e + (b^6 - 4*a*b^4*c + 4*a^2*b^2*c^2)*e^2)/(b^
2*c^8 - 4*a*c^9)))/(b^2*c^4 - 4*a*c^5))*log(((a*b^2*c^4 - 4*a^2*c^5)*d*x^2*sqrt(
((b^4*c^2 - 2*a*b^2*c^3 + a^2*c^4)*d^2 - 2*(b^5*c - 3*a*b^3*c^2 + 2*a^2*b*c^3)*d
*e + (b^6 - 4*a*b^4*c + 4*a^2*b^2*c^2)*e^2)/(b^2*c^8 - 4*a*c^9)) + 2*(a^2*b^2*c
- a^3*c^2)*d^2 - 2*(a^2*b^3 - 2*a^3*b*c)*d*e - ((a*b^3*c - a^2*b*c^2)*d^2 - (a*b
^4 + 2*a^2*b^2*c - 4*a^3*c^2)*d*e + 4*(a^2*b^3 - 2*a^3*b*c)*e^2)*x^2 - 2*sqrt(1/
2)*sqrt(e*x^2 + d)*((b^4*c^4 - 6*a*b^2*c^5 + 8*a^2*c^6)*x*sqrt(((b^4*c^2 - 2*a*b
^2*c^3 + a^2*c^4)*d^2 - 2*(b^5*c - 3*a*b^3*c^2 + 2*a^2*b*c^3)*d*e + (b^6 - 4*a*b
^4*c + 4*a^2*b^2*c^2)*e^2)/(b^2*c^8 - 4*a*c^9)) - ((b^5*c - 5*a*b^3*c^2 + 4*a^2*
b*c^3)*d - (b^6 - 6*a*b^4*c + 8*a^2*b^2*c^2)*e)*x)*sqrt(-((b^3*c - 3*a*b*c^2)*d
- (b^4 - 4*a*b^2*c + 2*a^2*c^2)*e + (b^2*c^4 - 4*a*c^5)*sqrt(((b^4*c^2 - 2*a*b^2
*c^3 + a^2*c^4)*d^2 - 2*(b^5*c - 3*a*b^3*c^2 + 2*a^2*b*c^3)*d*e + (b^6 - 4*a*b^4
*c + 4*a^2*b^2*c^2)*e^2)/(b^2*c^8 - 4*a*c^9)))/(b^2*c^4 - 4*a*c^5)))/x^2) + sqrt
(1/2)*c^2*sqrt(e)*sqrt(-((b^3*c - 3*a*b*c^2)*d - (b^4 - 4*a*b^2*c + 2*a^2*c^2)*e
- (b^2*c^4 - 4*a*c^5)*sqrt(((b^4*c^2 - 2*a*b^2*c^3 + a^2*c^4)*d^2 - 2*(b^5*c -
3*a*b^3*c^2 + 2*a^2*b*c^3)*d*e + (b^6 - 4*a*b^4*c + 4*a^2*b^2*c^2)*e^2)/(b^2*c^8
- 4*a*c^9)))/(b^2*c^4 - 4*a*c^5))*log(-((a*b^2*c^4 - 4*a^2*c^5)*d*x^2*sqrt(((b^
4*c^2 - 2*a*b^2*c^3 + a^2*c^4)*d^2 - 2*(b^5*c - 3*a*b^3*c^2 + 2*a^2*b*c^3)*d*e +
(b^6 - 4*a*b^4*c + 4*a^2*b^2*c^2)*e^2)/(b^2*c^8 - 4*a*c^9)) - 2*(a^2*b^2*c - a^
3*c^2)*d^2 + 2*(a^2*b^3 - 2*a^3*b*c)*d*e + ((a*b^3*c - a^2*b*c^2)*d^2 - (a*b^4 +
2*a^2*b^2*c - 4*a^3*c^2)*d*e + 4*(a^2*b^3 - 2*a^3*b*c)*e^2)*x^2 + 2*sqrt(1/2)*s
qrt(e*x^2 + d)*((b^4*c^4 - 6*a*b^2*c^5 + 8*a^2*c^6)*x*sqrt(((b^4*c^2 - 2*a*b^2*c
^3 + a^2*c^4)*d^2 - 2*(b^5*c - 3*a*b^3*c^2 + 2*a^2*b*c^3)*d*e + (b^6 - 4*a*b^4*c
+ 4*a^2*b^2*c^2)*e^2)/(b^2*c^8 - 4*a*c^9)) + ((b^5*c - 5*a*b^3*c^2 + 4*a^2*b*c^
3)*d - (b^6 - 6*a*b^4*c + 8*a^2*b^2*c^2)*e)*x)*sqrt(-((b^3*c - 3*a*b*c^2)*d - (b
^4 - 4*a*b^2*c + 2*a^2*c^2)*e - (b^2*c^4 - 4*a*c^5)*sqrt(((b^4*c^2 - 2*a*b^2*c^3
+ a^2*c^4)*d^2 - 2*(b^5*c - 3*a*b^3*c^2 + 2*a^2*b*c^3)*d*e + (b^6 - 4*a*b^4*c +
4*a^2*b^2*c^2)*e^2)/(b^2*c^8 - 4*a*c^9)))/(b^2*c^4 - 4*a*c^5)))/x^2) - sqrt(1/2
)*c^2*sqrt(e)*sqrt(-((b^3*c - 3*a*b*c^2)*d - (b^4 - 4*a*b^2*c + 2*a^2*c^2)*e - (
b^2*c^4 - 4*a*c^5)*sqrt(((b^4*c^2 - 2*a*b^2*c^3 + a^2*c^4)*d^2 - 2*(b^5*c - 3*a*
b^3*c^2 + 2*a^2*b*c^3)*d*e + (b^6 - 4*a*b^4*c + 4*a^2*b^2*c^2)*e^2)/(b^2*c^8 - 4
*a*c^9)))/(b^2*c^4 - 4*a*c^5))*log(-((a*b^2*c^4 - 4*a^2*c^5)*d*x^2*sqrt(((b^4*c^
2 - 2*a*b^2*c^3 + a^2*c^4)*d^2 - 2*(b^5*c - 3*a*b^3*c^2 + 2*a^2*b*c^3)*d*e + (b^
6 - 4*a*b^4*c + 4*a^2*b^2*c^2)*e^2)/(b^2*c^8 - 4*a*c^9)) - 2*(a^2*b^2*c - a^3*c^
2)*d^2 + 2*(a^2*b^3 - 2*a^3*b*c)*d*e + ((a*b^3*c - a^2*b*c^2)*d^2 - (a*b^4 + 2*a
^2*b^2*c - 4*a^3*c^2)*d*e + 4*(a^2*b^3 - 2*a^3*b*c)*e^2)*x^2 - 2*sqrt(1/2)*sqrt(
e*x^2 + d)*((b^4*c^4 - 6*a*b^2*c^5 + 8*a^2*c^6)*x*sqrt(((b^4*c^2 - 2*a*b^2*c^3 +
a^2*c^4)*d^2 - 2*(b^5*c - 3*a*b^3*c^2 + 2*a^2*b*c^3)*d*e + (b^6 - 4*a*b^4*c + 4
*a^2*b^2*c^2)*e^2)/(b^2*c^8 - 4*a*c^9)) + ((b^5*c - 5*a*b^3*c^2 + 4*a^2*b*c^3)*d
- (b^6 - 6*a*b^4*c + 8*a^2*b^2*c^2)*e)*x)*sqrt(-((b^3*c - 3*a*b*c^2)*d - (b^4 -
4*a*b^2*c + 2*a^2*c^2)*e - (b^2*c^4 - 4*a*c^5)*sqrt(((b^4*c^2 - 2*a*b^2*c^3 + a
^2*c^4)*d^2 - 2*(b^5*c - 3*a*b^3*c^2 + 2*a^2*b*c^3)*d*e + (b^6 - 4*a*b^4*c + 4*a
^2*b^2*c^2)*e^2)/(b^2*c^8 - 4*a*c^9)))/(b^2*c^4 - 4*a*c^5)))/x^2) + 2*sqrt(e*x^2
+ d)*c*sqrt(e)*x - (c*d - 2*b*e)*log(2*sqrt(e*x^2 + d)*e*x - (2*e*x^2 + d)*sqrt
(e)))/(c^2*sqrt(e)), 1/4*(sqrt(1/2)*c^2*sqrt(-e)*sqrt(-((b^3*c - 3*a*b*c^2)*d -
(b^4 - 4*a*b^2*c + 2*a^2*c^2)*e + (b^2*c^4 - 4*a*c^5)*sqrt(((b^4*c^2 - 2*a*b^2*c
^3 + a^2*c^4)*d^2 - 2*(b^5*c - 3*a*b^3*c^2 + 2*a^2*b*c^3)*d*e + (b^6 - 4*a*b^4*c
+ 4*a^2*b^2*c^2)*e^2)/(b^2*c^8 - 4*a*c^9)))/(b^2*c^4 - 4*a*c^5))*log(((a*b^2*c^
4 - 4*a^2*c^5)*d*x^2*sqrt(((b^4*c^2 - 2*a*b^2*c^3 + a^2*c^4)*d^2 - 2*(b^5*c - 3*
a*b^3*c^2 + 2*a^2*b*c^3)*d*e + (b^6 - 4*a*b^4*c + 4*a^2*b^2*c^2)*e^2)/(b^2*c^8 -
4*a*c^9)) + 2*(a^2*b^2*c - a^3*c^2)*d^2 - 2*(a^2*b^3 - 2*a^3*b*c)*d*e - ((a*b^3
*c - a^2*b*c^2)*d^2 - (a*b^4 + 2*a^2*b^2*c - 4*a^3*c^2)*d*e + 4*(a^2*b^3 - 2*a^3
*b*c)*e^2)*x^2 + 2*sqrt(1/2)*sqrt(e*x^2 + d)*((b^4*c^4 - 6*a*b^2*c^5 + 8*a^2*c^6
)*x*sqrt(((b^4*c^2 - 2*a*b^2*c^3 + a^2*c^4)*d^2 - 2*(b^5*c - 3*a*b^3*c^2 + 2*a^2
*b*c^3)*d*e + (b^6 - 4*a*b^4*c + 4*a^2*b^2*c^2)*e^2)/(b^2*c^8 - 4*a*c^9)) - ((b^
5*c - 5*a*b^3*c^2 + 4*a^2*b*c^3)*d - (b^6 - 6*a*b^4*c + 8*a^2*b^2*c^2)*e)*x)*sqr
t(-((b^3*c - 3*a*b*c^2)*d - (b^4 - 4*a*b^2*c + 2*a^2*c^2)*e + (b^2*c^4 - 4*a*c^5
)*sqrt(((b^4*c^2 - 2*a*b^2*c^3 + a^2*c^4)*d^2 - 2*(b^5*c - 3*a*b^3*c^2 + 2*a^2*b
*c^3)*d*e + (b^6 - 4*a*b^4*c + 4*a^2*b^2*c^2)*e^2)/(b^2*c^8 - 4*a*c^9)))/(b^2*c^
4 - 4*a*c^5)))/x^2) - sqrt(1/2)*c^2*sqrt(-e)*sqrt(-((b^3*c - 3*a*b*c^2)*d - (b^4
- 4*a*b^2*c + 2*a^2*c^2)*e + (b^2*c^4 - 4*a*c^5)*sqrt(((b^4*c^2 - 2*a*b^2*c^3 +
a^2*c^4)*d^2 - 2*(b^5*c - 3*a*b^3*c^2 + 2*a^2*b*c^3)*d*e + (b^6 - 4*a*b^4*c + 4
*a^2*b^2*c^2)*e^2)/(b^2*c^8 - 4*a*c^9)))/(b^2*c^4 - 4*a*c^5))*log(((a*b^2*c^4 -
4*a^2*c^5)*d*x^2*sqrt(((b^4*c^2 - 2*a*b^2*c^3 + a^2*c^4)*d^2 - 2*(b^5*c - 3*a*b^
3*c^2 + 2*a^2*b*c^3)*d*e + (b^6 - 4*a*b^4*c + 4*a^2*b^2*c^2)*e^2)/(b^2*c^8 - 4*a
*c^9)) + 2*(a^2*b^2*c - a^3*c^2)*d^2 - 2*(a^2*b^3 - 2*a^3*b*c)*d*e - ((a*b^3*c -
a^2*b*c^2)*d^2 - (a*b^4 + 2*a^2*b^2*c - 4*a^3*c^2)*d*e + 4*(a^2*b^3 - 2*a^3*b*c
)*e^2)*x^2 - 2*sqrt(1/2)*sqrt(e*x^2 + d)*((b^4*c^4 - 6*a*b^2*c^5 + 8*a^2*c^6)*x*
sqrt(((b^4*c^2 - 2*a*b^2*c^3 + a^2*c^4)*d^2 - 2*(b^5*c - 3*a*b^3*c^2 + 2*a^2*b*c
^3)*d*e + (b^6 - 4*a*b^4*c + 4*a^2*b^2*c^2)*e^2)/(b^2*c^8 - 4*a*c^9)) - ((b^5*c
- 5*a*b^3*c^2 + 4*a^2*b*c^3)*d - (b^6 - 6*a*b^4*c + 8*a^2*b^2*c^2)*e)*x)*sqrt(-(
(b^3*c - 3*a*b*c^2)*d - (b^4 - 4*a*b^2*c + 2*a^2*c^2)*e + (b^2*c^4 - 4*a*c^5)*sq
rt(((b^4*c^2 - 2*a*b^2*c^3 + a^2*c^4)*d^2 - 2*(b^5*c - 3*a*b^3*c^2 + 2*a^2*b*c^3
)*d*e + (b^6 - 4*a*b^4*c + 4*a^2*b^2*c^2)*e^2)/(b^2*c^8 - 4*a*c^9)))/(b^2*c^4 -
4*a*c^5)))/x^2) + sqrt(1/2)*c^2*sqrt(-e)*sqrt(-((b^3*c - 3*a*b*c^2)*d - (b^4 - 4
*a*b^2*c + 2*a^2*c^2)*e - (b^2*c^4 - 4*a*c^5)*sqrt(((b^4*c^2 - 2*a*b^2*c^3 + a^2
*c^4)*d^2 - 2*(b^5*c - 3*a*b^3*c^2 + 2*a^2*b*c^3)*d*e + (b^6 - 4*a*b^4*c + 4*a^2
*b^2*c^2)*e^2)/(b^2*c^8 - 4*a*c^9)))/(b^2*c^4 - 4*a*c^5))*log(-((a*b^2*c^4 - 4*a
^2*c^5)*d*x^2*sqrt(((b^4*c^2 - 2*a*b^2*c^3 + a^2*c^4)*d^2 - 2*(b^5*c - 3*a*b^3*c
^2 + 2*a^2*b*c^3)*d*e + (b^6 - 4*a*b^4*c + 4*a^2*b^2*c^2)*e^2)/(b^2*c^8 - 4*a*c^
9)) - 2*(a^2*b^2*c - a^3*c^2)*d^2 + 2*(a^2*b^3 - 2*a^3*b*c)*d*e + ((a*b^3*c - a^
2*b*c^2)*d^2 - (a*b^4 + 2*a^2*b^2*c - 4*a^3*c^2)*d*e + 4*(a^2*b^3 - 2*a^3*b*c)*e
^2)*x^2 + 2*sqrt(1/2)*sqrt(e*x^2 + d)*((b^4*c^4 - 6*a*b^2*c^5 + 8*a^2*c^6)*x*sqr
t(((b^4*c^2 - 2*a*b^2*c^3 + a^2*c^4)*d^2 - 2*(b^5*c - 3*a*b^3*c^2 + 2*a^2*b*c^3)
*d*e + (b^6 - 4*a*b^4*c + 4*a^2*b^2*c^2)*e^2)/(b^2*c^8 - 4*a*c^9)) + ((b^5*c - 5
*a*b^3*c^2 + 4*a^2*b*c^3)*d - (b^6 - 6*a*b^4*c + 8*a^2*b^2*c^2)*e)*x)*sqrt(-((b^
3*c - 3*a*b*c^2)*d - (b^4 - 4*a*b^2*c + 2*a^2*c^2)*e - (b^2*c^4 - 4*a*c^5)*sqrt(
((b^4*c^2 - 2*a*b^2*c^3 + a^2*c^4)*d^2 - 2*(b^5*c - 3*a*b^3*c^2 + 2*a^2*b*c^3)*d
*e + (b^6 - 4*a*b^4*c + 4*a^2*b^2*c^2)*e^2)/(b^2*c^8 - 4*a*c^9)))/(b^2*c^4 - 4*a
*c^5)))/x^2) - sqrt(1/2)*c^2*sqrt(-e)*sqrt(-((b^3*c - 3*a*b*c^2)*d - (b^4 - 4*a*
b^2*c + 2*a^2*c^2)*e - (b^2*c^4 - 4*a*c^5)*sqrt(((b^4*c^2 - 2*a*b^2*c^3 + a^2*c^
4)*d^2 - 2*(b^5*c - 3*a*b^3*c^2 + 2*a^2*b*c^3)*d*e + (b^6 - 4*a*b^4*c + 4*a^2*b^
2*c^2)*e^2)/(b^2*c^8 - 4*a*c^9)))/(b^2*c^4 - 4*a*c^5))*log(-((a*b^2*c^4 - 4*a^2*
c^5)*d*x^2*sqrt(((b^4*c^2 - 2*a*b^2*c^3 + a^2*c^4)*d^2 - 2*(b^5*c - 3*a*b^3*c^2
+ 2*a^2*b*c^3)*d*e + (b^6 - 4*a*b^4*c + 4*a^2*b^2*c^2)*e^2)/(b^2*c^8 - 4*a*c^9))
- 2*(a^2*b^2*c - a^3*c^2)*d^2 + 2*(a^2*b^3 - 2*a^3*b*c)*d*e + ((a*b^3*c - a^2*b
*c^2)*d^2 - (a*b^4 + 2*a^2*b^2*c - 4*a^3*c^2)*d*e + 4*(a^2*b^3 - 2*a^3*b*c)*e^2)
*x^2 - 2*sqrt(1/2)*sqrt(e*x^2 + d)*((b^4*c^4 - 6*a*b^2*c^5 + 8*a^2*c^6)*x*sqrt((
(b^4*c^2 - 2*a*b^2*c^3 + a^2*c^4)*d^2 - 2*(b^5*c - 3*a*b^3*c^2 + 2*a^2*b*c^3)*d*
e + (b^6 - 4*a*b^4*c + 4*a^2*b^2*c^2)*e^2)/(b^2*c^8 - 4*a*c^9)) + ((b^5*c - 5*a*
b^3*c^2 + 4*a^2*b*c^3)*d - (b^6 - 6*a*b^4*c + 8*a^2*b^2*c^2)*e)*x)*sqrt(-((b^3*c
- 3*a*b*c^2)*d - (b^4 - 4*a*b^2*c + 2*a^2*c^2)*e - (b^2*c^4 - 4*a*c^5)*sqrt(((b
^4*c^2 - 2*a*b^2*c^3 + a^2*c^4)*d^2 - 2*(b^5*c - 3*a*b^3*c^2 + 2*a^2*b*c^3)*d*e
+ (b^6 - 4*a*b^4*c + 4*a^2*b^2*c^2)*e^2)/(b^2*c^8 - 4*a*c^9)))/(b^2*c^4 - 4*a*c^
5)))/x^2) + 2*sqrt(e*x^2 + d)*c*sqrt(-e)*x + 2*(c*d - 2*b*e)*arctan(sqrt(-e)*x/s
qrt(e*x^2 + d)))/(c^2*sqrt(-e))]

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out