3.352 \(\int \frac{\sqrt{d+e x^2}}{x^5 \left (a+b x^2+c x^4\right )} \, dx\)

Optimal. Leaf size=552 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right ) \left (-a b e-a c d+b^2 d\right )}{a^3 \sqrt{d}}+\frac{\sqrt{c} \left (b^2 \left (d \sqrt{b^2-4 a c}-a e\right )-a b \left (e \sqrt{b^2-4 a c}+3 c d\right )-a c \left (d \sqrt{b^2-4 a c}-2 a e\right )+b^3 d\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{\sqrt{2} a^3 \sqrt{b^2-4 a c} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}-\frac{\sqrt{c} \left (-b^2 \left (d \sqrt{b^2-4 a c}+a e\right )-a b \left (3 c d-e \sqrt{b^2-4 a c}\right )+a c \left (d \sqrt{b^2-4 a c}+2 a e\right )+b^3 d\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{2} a^3 \sqrt{b^2-4 a c} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}-\frac{e (b d-a e) \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{2 a^2 d^{3/2}}+\frac{\sqrt{d+e x^2} (b d-a e)}{2 a^2 d x^2}-\frac{3 e^2 \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{8 a d^{3/2}}+\frac{3 e \sqrt{d+e x^2}}{8 a d x^2}-\frac{\sqrt{d+e x^2}}{4 a x^4} \]

[Out]

-Sqrt[d + e*x^2]/(4*a*x^4) + (3*e*Sqrt[d + e*x^2])/(8*a*d*x^2) + ((b*d - a*e)*Sq
rt[d + e*x^2])/(2*a^2*d*x^2) - (3*e^2*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(8*a*d^(
3/2)) - (e*(b*d - a*e)*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(2*a^2*d^(3/2)) - ((b^2
*d - a*c*d - a*b*e)*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(a^3*Sqrt[d]) + (Sqrt[c]*(
b^3*d - a*c*(Sqrt[b^2 - 4*a*c]*d - 2*a*e) + b^2*(Sqrt[b^2 - 4*a*c]*d - a*e) - a*
b*(3*c*d + Sqrt[b^2 - 4*a*c]*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x^2])/Sqrt[
2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[2]*a^3*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d -
(b - Sqrt[b^2 - 4*a*c])*e]) - (Sqrt[c]*(b^3*d - b^2*(Sqrt[b^2 - 4*a*c]*d + a*e)
+ a*c*(Sqrt[b^2 - 4*a*c]*d + 2*a*e) - a*b*(3*c*d - Sqrt[b^2 - 4*a*c]*e))*ArcTan
h[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x^2])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(S
qrt[2]*a^3*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e])

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Rubi [A]  time = 9.29025, antiderivative size = 552, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 7, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.241 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right ) \left (-a b e-a c d+b^2 d\right )}{a^3 \sqrt{d}}+\frac{\sqrt{c} \left (b^2 \left (d \sqrt{b^2-4 a c}-a e\right )-a b \left (e \sqrt{b^2-4 a c}+3 c d\right )-a c \left (d \sqrt{b^2-4 a c}-2 a e\right )+b^3 d\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{\sqrt{2} a^3 \sqrt{b^2-4 a c} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}-\frac{\sqrt{c} \left (-b^2 \left (d \sqrt{b^2-4 a c}+a e\right )-a b \left (3 c d-e \sqrt{b^2-4 a c}\right )+a c \left (d \sqrt{b^2-4 a c}+2 a e\right )+b^3 d\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{2} a^3 \sqrt{b^2-4 a c} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}-\frac{e (b d-a e) \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{2 a^2 d^{3/2}}+\frac{\sqrt{d+e x^2} (b d-a e)}{2 a^2 d x^2}-\frac{3 e^2 \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{8 a d^{3/2}}+\frac{3 e \sqrt{d+e x^2}}{8 a d x^2}-\frac{\sqrt{d+e x^2}}{4 a x^4} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[d + e*x^2]/(x^5*(a + b*x^2 + c*x^4)),x]

[Out]

-Sqrt[d + e*x^2]/(4*a*x^4) + (3*e*Sqrt[d + e*x^2])/(8*a*d*x^2) + ((b*d - a*e)*Sq
rt[d + e*x^2])/(2*a^2*d*x^2) - (3*e^2*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(8*a*d^(
3/2)) - (e*(b*d - a*e)*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(2*a^2*d^(3/2)) - ((b^2
*d - a*c*d - a*b*e)*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(a^3*Sqrt[d]) + (Sqrt[c]*(
b^3*d - a*c*(Sqrt[b^2 - 4*a*c]*d - 2*a*e) + b^2*(Sqrt[b^2 - 4*a*c]*d - a*e) - a*
b*(3*c*d + Sqrt[b^2 - 4*a*c]*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x^2])/Sqrt[
2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[2]*a^3*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d -
(b - Sqrt[b^2 - 4*a*c])*e]) - (Sqrt[c]*(b^3*d - b^2*(Sqrt[b^2 - 4*a*c]*d + a*e)
+ a*c*(Sqrt[b^2 - 4*a*c]*d + 2*a*e) - a*b*(3*c*d - Sqrt[b^2 - 4*a*c]*e))*ArcTan
h[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x^2])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(S
qrt[2]*a^3*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e])

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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((e*x**2+d)**(1/2)/x**5/(c*x**4+b*x**2+a),x)

[Out]

Timed out

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Mathematica [A]  time = 2.95803, size = 468, normalized size = 0.85 \[ \frac{\frac{\log \left (\sqrt{d} \sqrt{d+e x^2}+d\right ) \left (4 a b d e+a \left (a e^2+8 c d^2\right )-8 b^2 d^2\right )}{d^{3/2}}-\frac{\log (x) \left (4 a b d e+a \left (a e^2+8 c d^2\right )-8 b^2 d^2\right )}{d^{3/2}}-\frac{4 \sqrt{2} \left (\frac{c \left (b^2 \left (a e-d \sqrt{b^2-4 a c}\right )+a b \left (e \sqrt{b^2-4 a c}+3 c d\right )+a c \left (d \sqrt{b^2-4 a c}-2 a e\right )+b^3 (-d)\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right )}{\sqrt{e \left (\sqrt{b^2-4 a c}-b\right )+2 c d}}+\frac{c \left (-b^2 \left (d \sqrt{b^2-4 a c}+a e\right )+a b \left (e \sqrt{b^2-4 a c}-3 c d\right )+a c \left (d \sqrt{b^2-4 a c}+2 a e\right )+b^3 d\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{c} \sqrt{b^2-4 a c}}+\frac{a \sqrt{d+e x^2} \left (4 b d x^2-a \left (2 d+e x^2\right )\right )}{d x^4}}{8 a^3} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[d + e*x^2]/(x^5*(a + b*x^2 + c*x^4)),x]

[Out]

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

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Maple [C]  time = 0.047, size = 655, normalized size = 1.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x^2+d)^(1/2)/x^5/(c*x^4+b*x^2+a),x)

[Out]

-1/4/a/d/x^4*(e*x^2+d)^(3/2)+1/8/a/d^2*e/x^2*(e*x^2+d)^(3/2)+1/8/a/d^(3/2)*e^2*l
n((2*d+2*d^(1/2)*(e*x^2+d)^(1/2))/x)-1/8/a/d^2*e^2*(e*x^2+d)^(1/2)+1/a^2*d^(1/2)
*ln((2*d+2*d^(1/2)*(e*x^2+d)^(1/2))/x)*c-1/a^3*d^(1/2)*ln((2*d+2*d^(1/2)*(e*x^2+
d)^(1/2))/x)*b^2-1/2/a^2*(e*x^2+d)^(1/2)*c+1/2/a^3*(e*x^2+d)^(1/2)*b^2-1/2/a^2*e
^(1/2)*x*c+1/2/a^3*e^(1/2)*x*b^2+1/4/a^3*sum((c*(a*b*e+a*c*d-b^2*d)*_R^6+(-4*a^2
*c*e^2+4*a*b^2*e^2+5*a*b*c*d*e-3*a*c^2*d^2-4*b^3*d*e+3*b^2*c*d^2)*_R^4+d*(4*a^2*
c*e^2-4*a*b^2*e^2-5*a*b*c*d*e+3*a*c^2*d^2+4*b^3*d*e-3*b^2*c*d^2)*_R^2-a*b*c*d^3*
e-a*c^2*d^4+b^2*c*d^4)/(_R^7*c+3*_R^5*b*e-3*_R^5*c*d+8*_R^3*a*e^2-4*_R^3*b*d*e+3
*_R^3*c*d^2+_R*b*d^2*e-_R*c*d^3)*ln((e*x^2+d)^(1/2)-x*e^(1/2)-_R),_R=RootOf(c*_Z
^8+(4*b*e-4*c*d)*_Z^6+(16*a*e^2-8*b*d*e+6*c*d^2)*_Z^4+(4*b*d^2*e-4*c*d^3)*_Z^2+c
*d^4))+1/2/a^2*d/((e*x^2+d)^(1/2)-x*e^(1/2))*c-1/2/a^3*d/((e*x^2+d)^(1/2)-x*e^(1
/2))*b^2+1/2*b/a^2/d/x^2*(e*x^2+d)^(3/2)+1/2*b/a^2/d^(1/2)*e*ln((2*d+2*d^(1/2)*(
e*x^2+d)^(1/2))/x)-1/2*b/a^2/d*e*(e*x^2+d)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{e x^{2} + d}}{{\left (c x^{4} + b x^{2} + a\right )} x^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(e*x^2 + d)/((c*x^4 + b*x^2 + a)*x^5),x, algorithm="maxima")

[Out]

integrate(sqrt(e*x^2 + d)/((c*x^4 + b*x^2 + a)*x^5), 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(sqrt(e*x^2 + d)/((c*x^4 + b*x^2 + a)*x^5),x, algorithm="fricas")

[Out]

Timed out

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Sympy [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)**(1/2)/x**5/(c*x**4+b*x**2+a),x)

[Out]

Timed out

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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(sqrt(e*x^2 + d)/((c*x^4 + b*x^2 + a)*x^5),x, algorithm="giac")

[Out]

Timed out