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

Optimal. Leaf size=595 \[ -\frac{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )} \left (-\frac{3 a b c e-2 a c^2 d+b^3 (-e)+b^2 c d}{\sqrt{b^2-4 a c}}+a c e+b^2 (-e)+b c d\right ) \tan ^{-1}\left (\frac{x \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right )}{2 c^3 \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )} \left (\frac{3 a b c e-2 a c^2 d+b^3 (-e)+b^2 c d}{\sqrt{b^2-4 a c}}+a c e+b^2 (-e)+b c d\right ) \tan ^{-1}\left (\frac{x \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{d+e x^2}}\right )}{2 c^3 \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{\sqrt{e} \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right ) \left (-\frac{3 a b c e-2 a c^2 d+b^3 (-e)+b^2 c d}{\sqrt{b^2-4 a c}}+a c e+b^2 (-e)+b c d\right )}{2 c^3}-\frac{\sqrt{e} \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right ) \left (\frac{3 a b c e-2 a c^2 d+b^3 (-e)+b^2 c d}{\sqrt{b^2-4 a c}}+a c e+b^2 (-e)+b c d\right )}{2 c^3}+\frac{x \sqrt{d+e x^2} (3 c d-4 b e)}{8 c^2}+\frac{d (3 c d-4 b e) \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{8 c^2 \sqrt{e}}+\frac{x \left (d+e x^2\right )^{3/2}}{4 c} \]

[Out]

((3*c*d - 4*b*e)*x*Sqrt[d + e*x^2])/(8*c^2) + (x*(d + e*x^2)^(3/2))/(4*c) - (Sqr
t[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])])/(2*c^3*Sqrt[b - Sq
rt[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])])/(2*c^3*Sqrt[b + Sqrt[b^2 - 4*a*c]]) + (d*(3*c*d - 4*b*e)*ArcTanh[(Sqrt[e]*
x)/Sqrt[d + e*x^2]])/(8*c^2*Sqrt[e]) - (Sqrt[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])*ArcTanh[(Sqrt[e]*x)/Sqrt[d
+ e*x^2]])/(2*c^3) - (Sqrt[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])*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(2*c^
3)

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Rubi [A]  time = 7.25336, antiderivative size = 595, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 9, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.31 \[ -\frac{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )} \left (-\frac{3 a b c e-2 a c^2 d+b^3 (-e)+b^2 c d}{\sqrt{b^2-4 a c}}+a c e+b^2 (-e)+b c d\right ) \tan ^{-1}\left (\frac{x \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right )}{2 c^3 \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )} \left (\frac{3 a b c e-2 a c^2 d+b^3 (-e)+b^2 c d}{\sqrt{b^2-4 a c}}+a c e+b^2 (-e)+b c d\right ) \tan ^{-1}\left (\frac{x \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{d+e x^2}}\right )}{2 c^3 \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{\sqrt{e} \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right ) \left (-\frac{3 a b c e-2 a c^2 d+b^3 (-e)+b^2 c d}{\sqrt{b^2-4 a c}}+a c e+b^2 (-e)+b c d\right )}{2 c^3}-\frac{\sqrt{e} \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right ) \left (\frac{3 a b c e-2 a c^2 d+b^3 (-e)+b^2 c d}{\sqrt{b^2-4 a c}}+a c e+b^2 (-e)+b c d\right )}{2 c^3}+\frac{x \sqrt{d+e x^2} (3 c d-4 b e)}{8 c^2}+\frac{d (3 c d-4 b e) \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{8 c^2 \sqrt{e}}+\frac{x \left (d+e x^2\right )^{3/2}}{4 c} \]

Antiderivative was successfully verified.

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

[Out]

((3*c*d - 4*b*e)*x*Sqrt[d + e*x^2])/(8*c^2) + (x*(d + e*x^2)^(3/2))/(4*c) - (Sqr
t[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])])/(2*c^3*Sqrt[b - Sq
rt[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])])/(2*c^3*Sqrt[b + Sqrt[b^2 - 4*a*c]]) + (d*(3*c*d - 4*b*e)*ArcTanh[(Sqrt[e]*
x)/Sqrt[d + e*x^2]])/(8*c^2*Sqrt[e]) - (Sqrt[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])*ArcTanh[(Sqrt[e]*x)/Sqrt[d
+ e*x^2]])/(2*c^3) - (Sqrt[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])*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(2*c^
3)

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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)**(3/2)/(c*x**4+b*x**2+a),x)

[Out]

Timed out

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Mathematica [A]  time = 1.24665, size = 0, normalized size = 0. \[ \int \frac{x^4 \left (d+e x^2\right )^{3/2}}{a+b x^2+c x^4} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [C]  time = 0.048, size = 516, normalized size = 0.9 \[{\frac{x}{4\,c} \left ( e{x}^{2}+d \right ) ^{{\frac{3}{2}}}}+{\frac{3\,dx}{8\,c}\sqrt{e{x}^{2}+d}}+{\frac{3\,{d}^{2}}{8\,c}\ln \left ( x\sqrt{e}+\sqrt{e{x}^{2}+d} \right ){\frac{1}{\sqrt{e}}}}+{\frac{b{x}^{2}}{4\,{c}^{2}}{e}^{{\frac{3}{2}}}}-{\frac{bex}{4\,{c}^{2}}\sqrt{e{x}^{2}+d}}+{\frac{bd}{8\,{c}^{2}}\sqrt{e}}-{\frac{1}{2\,{c}^{3}}\sqrt{e}\sum _{{\it \_R}={\it RootOf} \left ( c{{\it \_Z}}^{4}+ \left ( 4\,be-4\,cd \right ){{\it \_Z}}^{3}+ \left ( 16\,a{e}^{2}-8\,bde+6\,c{d}^{2} \right ){{\it \_Z}}^{2}+ \left ( 4\,b{d}^{2}e-4\,c{d}^{3} \right ){\it \_Z}+c{d}^{4} \right ) }{\frac{ \left ( 2\,ab{e}^{2}c-2\,a{c}^{2}de-{b}^{3}{e}^{2}+2\,{b}^{2}dec-b{c}^{2}{d}^{2} \right ){{\it \_R}}^{2}+2\, \left ( 2\,{a}^{2}c{e}^{3}-2\,a{b}^{2}{e}^{3}+2\,abcd{e}^{2}+{b}^{3}d{e}^{2}-2\,{b}^{2}{d}^{2}ec+b{c}^{2}{d}^{3} \right ){\it \_R}+2\,abc{d}^{2}{e}^{2}-2\,a{c}^{2}{d}^{3}e-{b}^{3}{d}^{2}{e}^{2}+2\,{b}^{2}c{d}^{3}e-{c}^{2}{d}^{4}b}{{{\it \_R}}^{3}c+3\,{{\it \_R}}^{2}be-3\,{{\it \_R}}^{2}cd+8\,{\it \_R}\,a{e}^{2}-4\,{\it \_R}\,bde+3\,{\it \_R}\,c{d}^{2}+b{d}^{2}e-c{d}^{3}}\ln \left ( \left ( \sqrt{e{x}^{2}+d}-x\sqrt{e} \right ) ^{2}-{\it \_R} \right ) }}+{\frac{a}{{c}^{2}}{e}^{{\frac{3}{2}}}\ln \left ( \sqrt{e{x}^{2}+d}-x\sqrt{e} \right ) }-{\frac{{b}^{2}}{{c}^{3}}{e}^{{\frac{3}{2}}}\ln \left ( \sqrt{e{x}^{2}+d}-x\sqrt{e} \right ) }+{\frac{3\,bd}{2\,{c}^{2}}\sqrt{e}\ln \left ( \sqrt{e{x}^{2}+d}-x\sqrt{e} \right ) }-{\frac{b{d}^{2}}{8\,{c}^{2}}\sqrt{e} \left ( \sqrt{e{x}^{2}+d}-x\sqrt{e} \right ) ^{-2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*x*(e*x^2+d)^(3/2)/c+3/8/c*d*x*(e*x^2+d)^(1/2)+3/8/c*d^2/e^(1/2)*ln(x*e^(1/2)
+(e*x^2+d)^(1/2))+1/4/c^2*e^(3/2)*b*x^2-1/4/c^2*e*b*(e*x^2+d)^(1/2)*x+1/8/c^2*e^
(1/2)*b*d-1/2/c^3*e^(1/2)*sum(((2*a*b*c*e^2-2*a*c^2*d*e-b^3*e^2+2*b^2*c*d*e-b*c^
2*d^2)*_R^2+2*(2*a^2*c*e^3-2*a*b^2*e^3+2*a*b*c*d*e^2+b^3*d*e^2-2*b^2*c*d^2*e+b*c
^2*d^3)*_R+2*a*b*c*d^2*e^2-2*a*c^2*d^3*e-b^3*d^2*e^2+2*b^2*c*d^3*e-c^2*d^4*b)/(_
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^(3/2)*ln((e*x^2+d)^(
1/2)-x*e^(1/2))*a-1/c^3*e^(3/2)*ln((e*x^2+d)^(1/2)-x*e^(1/2))*b^2+3/2/c^2*e^(1/2
)*ln((e*x^2+d)^(1/2)-x*e^(1/2))*b*d-1/8/c^2*e^(1/2)*b*d^2/((e*x^2+d)^(1/2)-x*e^(
1/2))^2

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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((e*x^2 + d)^(3/2)*x^4/(c*x^4 + b*x^2 + a),x, algorithm="giac")

[Out]

Timed out