3.386 \(\int \frac{x^6}{\left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx\)

Optimal. Leaf size=350 \[ \frac{2 \left (-\frac{b \left (b^2-3 a c\right )}{\sqrt{b^2-4 a c}}-a c+b^2\right ) \tan ^{-1}\left (\frac{x \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right )}{c \sqrt{b-\sqrt{b^2-4 a c}} \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right )^{3/2}}+\frac{2 \left (\frac{b \left (b^2-3 a c\right )}{\sqrt{b^2-4 a c}}-a c+b^2\right ) \tan ^{-1}\left (\frac{x \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{d+e x^2}}\right )}{c \sqrt{\sqrt{b^2-4 a c}+b} \left (2 c d-e \left (\sqrt{b^2-4 a c}+b\right )\right )^{3/2}}-\frac{d^2 x}{e \sqrt{d+e x^2} \left (a e^2-b d e+c d^2\right )}+\frac{\tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{c e^{3/2}} \]

[Out]

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

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Rubi [A]  time = 10.5529, antiderivative size = 507, normalized size of antiderivative = 1.45, number of steps used = 14, number of rules used = 7, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.241 \[ \frac{\left (-\frac{2 a^2 c e-a b^2 e-3 a b c d+b^3 d}{\sqrt{b^2-4 a c}}-a b e-a c d+b^2 d\right ) \tan ^{-1}\left (\frac{x \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right )}{c \sqrt{b-\sqrt{b^2-4 a c}} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )} \left (a e^2-b d e+c d^2\right )}+\frac{\left (\frac{2 a^2 c e-a b^2 e-3 a b c d+b^3 d}{\sqrt{b^2-4 a c}}-a b e-a c d+b^2 d\right ) \tan ^{-1}\left (\frac{x \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{d+e x^2}}\right )}{c \sqrt{\sqrt{b^2-4 a c}+b} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )} \left (a e^2-b d e+c d^2\right )}-\frac{d^2 x}{e \sqrt{d+e x^2} \left (a e^2-b d e+c d^2\right )}-\frac{(b d-a e) \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{c \sqrt{e} \left (a e^2-b d e+c d^2\right )}+\frac{d^2 \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{e^{3/2} \left (a e^2-b d e+c d^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[x^6/((d + e*x^2)^(3/2)*(a + b*x^2 + c*x^4)),x]

[Out]

-((d^2*x)/(e*(c*d^2 - b*d*e + a*e^2)*Sqrt[d + e*x^2])) + ((b^2*d - a*c*d - a*b*e
- (b^3*d - 3*a*b*c*d - a*b^2*e + 2*a^2*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*Sqrt[b - Sqrt[b^2 - 4*a*c]]*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]*(c*d^2
- b*d*e + a*e^2)) + ((b^2*d - a*c*d - a*b*e + (b^3*d - 3*a*b*c*d - a*b^2*e + 2*
a^2*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*Sqrt[b + Sqrt[b^2 - 4*a*c]]*Sq
rt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]*(c*d^2 - b*d*e + a*e^2)) + (d^2*ArcTanh[(S
qrt[e]*x)/Sqrt[d + e*x^2]])/(e^(3/2)*(c*d^2 - b*d*e + a*e^2)) - ((b*d - a*e)*Arc
Tanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(c*Sqrt[e]*(c*d^2 - b*d*e + a*e^2))

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

[Out]

Timed out

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Mathematica [A]  time = 1.08464, size = 0, normalized size = 0. \[ \int \frac{x^6}{\left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[x^6/((d + e*x^2)^(3/2)*(a + b*x^2 + c*x^4)),x]

[Out]

Integrate[x^6/((d + e*x^2)^(3/2)*(a + b*x^2 + c*x^4)), x]

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Maple [C]  time = 0.051, size = 480, normalized size = 1.4 \[ -{\frac{x}{ce}{\frac{1}{\sqrt{e{x}^{2}+d}}}}+{\frac{1}{c}\ln \left ( x\sqrt{e}+\sqrt{e{x}^{2}+d} \right ){e}^{-{\frac{3}{2}}}}-{\frac{bx}{{c}^{2}d}{\frac{1}{\sqrt{e{x}^{2}+d}}}}+8\,{\frac{{e}^{3/2}ab}{{c}^{2} \left ( 4\,a{e}^{2}-4\,bde+4\,c{d}^{2} \right ) \left ( 2\,e{x}^{2}-2\,\sqrt{e}\sqrt{e{x}^{2}+d}x+2\,d \right ) }}+8\,{\frac{\sqrt{e}ad}{c \left ( 4\,a{e}^{2}-4\,bde+4\,c{d}^{2} \right ) \left ( 2\,e{x}^{2}-2\,\sqrt{e}\sqrt{e{x}^{2}+d}x+2\,d \right ) }}-8\,{\frac{\sqrt{e}{b}^{2}d}{{c}^{2} \left ( 4\,a{e}^{2}-4\,bde+4\,c{d}^{2} \right ) \left ( 2\,e{x}^{2}-2\,\sqrt{e}\sqrt{e{x}^{2}+d}x+2\,d \right ) }}+2\,{\frac{\sqrt{e}}{c \left ( 4\,a{e}^{2}-4\,bde+4\,c{d}^{2} \right ) }\sum _{{\it \_R}={\it RootOf} \left ( c{{\it \_Z}}^{4}+ \left ( 4\,be-4\,cd \right ){{\it \_Z}}^{3}+ \left ( 16\,a{e}^{2}-8\,bde+6\,c{d}^{2} \right ){{\it \_Z}}^{2}+ \left ( 4\,b{d}^{2}e-4\,c{d}^{3} \right ){\it \_Z}+c{d}^{4} \right ) }{\frac{ \left ( \left ( abe+acd-{b}^{2}d \right ){{\it \_R}}^{2}+2\, \left ( 2\,{a}^{2}{e}^{2}-3\,abde-ac{d}^{2}+{b}^{2}{d}^{2} \right ){\it \_R}+ab{d}^{2}e+ac{d}^{3}-{b}^{2}{d}^{3} \right ) \ln \left ( \left ( \sqrt{e{x}^{2}+d}-x\sqrt{e} \right ) ^{2}-{\it \_R} \right ) }{{{\it \_R}}^{3}c+3\,{{\it \_R}}^{2}be-3\,{{\it \_R}}^{2}cd+8\,{\it \_R}\,a{e}^{2}-4\,{\it \_R}\,bde+3\,{\it \_R}\,c{d}^{2}+b{d}^{2}e-c{d}^{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^6/(e*x^2+d)^(3/2)/(c*x^4+b*x^2+a),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{6}}{{\left (c x^{4} + b x^{2} + a\right )}{\left (e x^{2} + d\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/((c*x^4 + b*x^2 + a)*(e*x^2 + d)^(3/2)),x, algorithm="maxima")

[Out]

integrate(x^6/((c*x^4 + b*x^2 + a)*(e*x^2 + d)^(3/2)), 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(x^6/((c*x^4 + b*x^2 + a)*(e*x^2 + d)^(3/2)),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{6}}{\left (d + e x^{2}\right )^{\frac{3}{2}} \left (a + b x^{2} + c x^{4}\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**6/(e*x**2+d)**(3/2)/(c*x**4+b*x**2+a),x)

[Out]

Integral(x**6/((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(x^6/((c*x^4 + b*x^2 + a)*(e*x^2 + d)^(3/2)),x, algorithm="giac")

[Out]

Timed out