3.328 \(\int \frac{1}{x^3 \left (d+e x^2\right ) \sqrt{a+b x^2+c x^4}} \, dx\)

Optimal. Leaf size=218 \[ \frac{b \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{4 a^{3/2} d}+\frac{e^2 \tanh ^{-1}\left (\frac{-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}}\right )}{2 d^2 \sqrt{a e^2-b d e+c d^2}}+\frac{e \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{2 \sqrt{a} d^2}-\frac{\sqrt{a+b x^2+c x^4}}{2 a d x^2} \]

[Out]

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

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Rubi [A]  time = 0.701368, antiderivative size = 218, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172 \[ \frac{b \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{4 a^{3/2} d}+\frac{e^2 \tanh ^{-1}\left (\frac{-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}}\right )}{2 d^2 \sqrt{a e^2-b d e+c d^2}}+\frac{e \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{2 \sqrt{a} d^2}-\frac{\sqrt{a+b x^2+c x^4}}{2 a d x^2} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*(d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]),x]

[Out]

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

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Rubi in Sympy [A]  time = 64.3852, size = 194, normalized size = 0.89 \[ - \frac{e^{2} \operatorname{atanh}{\left (\frac{2 a e - b d + x^{2} \left (b e - 2 c d\right )}{2 \sqrt{a + b x^{2} + c x^{4}} \sqrt{a e^{2} - b d e + c d^{2}}} \right )}}{2 d^{2} \sqrt{a e^{2} - b d e + c d^{2}}} - \frac{\sqrt{a + b x^{2} + c x^{4}}}{2 a d x^{2}} + \frac{e \operatorname{atanh}{\left (\frac{2 a + b x^{2}}{2 \sqrt{a} \sqrt{a + b x^{2} + c x^{4}}} \right )}}{2 \sqrt{a} d^{2}} + \frac{b \operatorname{atanh}{\left (\frac{2 a + b x^{2}}{2 \sqrt{a} \sqrt{a + b x^{2} + c x^{4}}} \right )}}{4 a^{\frac{3}{2}} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**3/(e*x**2+d)/(c*x**4+b*x**2+a)**(1/2),x)

[Out]

-e**2*atanh((2*a*e - b*d + x**2*(b*e - 2*c*d))/(2*sqrt(a + b*x**2 + c*x**4)*sqrt
(a*e**2 - b*d*e + c*d**2)))/(2*d**2*sqrt(a*e**2 - b*d*e + c*d**2)) - sqrt(a + b*
x**2 + c*x**4)/(2*a*d*x**2) + e*atanh((2*a + b*x**2)/(2*sqrt(a)*sqrt(a + b*x**2
+ c*x**4)))/(2*sqrt(a)*d**2) + b*atanh((2*a + b*x**2)/(2*sqrt(a)*sqrt(a + b*x**2
+ c*x**4)))/(4*a**(3/2)*d)

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Mathematica [A]  time = 1.85177, size = 256, normalized size = 1.17 \[ \frac{\frac{2 a^{3/2} e^2 \log \left (d+e x^2\right )}{\sqrt{a e^2-b d e+c d^2}}-\frac{2 a^{3/2} e^2 \log \left (2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}+2 a e-b d+b e x^2-2 c d x^2\right )}{\sqrt{a e^2-b d e+c d^2}}-\frac{2 \sqrt{a} d \sqrt{a+b x^2+c x^4}}{x^2}+b d \log \left (2 \sqrt{a} \sqrt{a+b x^2+c x^4}+2 a+b x^2\right )+2 a e \log \left (2 \sqrt{a} \sqrt{a+b x^2+c x^4}+2 a+b x^2\right )-\log \left (x^2\right ) (2 a e+b d)}{4 a^{3/2} d^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^3*(d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]),x]

[Out]

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

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Maple [A]  time = 0.019, size = 276, normalized size = 1.3 \[ -{\frac{1}{2\,ad{x}^{2}}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{b}{4\,d}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+b{x}^{2}+2\,\sqrt{a}\sqrt{c{x}^{4}+b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}-{\frac{e}{2\,{d}^{2}}\ln \left ({1 \left ( 2\,{\frac{a{e}^{2}-bde+c{d}^{2}}{{e}^{2}}}+{\frac{be-2\,cd}{e} \left ({x}^{2}+{\frac{d}{e}} \right ) }+2\,\sqrt{{\frac{a{e}^{2}-bde+c{d}^{2}}{{e}^{2}}}}\sqrt{ \left ({x}^{2}+{\frac{d}{e}} \right ) ^{2}c+{\frac{be-2\,cd}{e} \left ({x}^{2}+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}-bde+c{d}^{2}}{{e}^{2}}}} \right ) \left ({x}^{2}+{\frac{d}{e}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{e}^{2}-bde+c{d}^{2}}{{e}^{2}}}}}}}+{\frac{e}{2\,{d}^{2}}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+b{x}^{2}+2\,\sqrt{a}\sqrt{c{x}^{4}+b{x}^{2}+a} \right ) } \right ){\frac{1}{\sqrt{a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(e*x^2+d)/(c*x^4+b*x^2+a)^(1/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{c x^{4} + b x^{2} + a}{\left (e x^{2} + d\right )} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^4 + b*x^2 + a)*(e*x^2 + d)*x^3),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(c*x^4 + b*x^2 + a)*(e*x^2 + d)*x^3), x)

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Fricas [A]  time = 0.950815, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^4 + b*x^2 + a)*(e*x^2 + d)*x^3),x, algorithm="fricas")

[Out]

[1/8*(2*a^(3/2)*e^2*x^2*log(-(4*(b*c*d^3 + 3*a*b*d*e^2 - 2*a^2*e^3 - (b^2 + 2*a*
c)*d^2*e + (2*c^2*d^3 - 3*b*c*d^2*e - a*b*e^3 + (b^2 + 2*a*c)*d*e^2)*x^2)*sqrt(c
*x^4 + b*x^2 + a) + ((8*c^2*d^2 - 8*b*c*d*e + (b^2 + 4*a*c)*e^2)*x^4 - 8*a*b*d*e
+ 8*a^2*e^2 + (b^2 + 4*a*c)*d^2 + 2*(4*b*c*d^2 + 4*a*b*e^2 - (3*b^2 + 4*a*c)*d*
e)*x^2)*sqrt(c*d^2 - b*d*e + a*e^2))/(e^2*x^4 + 2*d*e*x^2 + d^2)) + sqrt(c*d^2 -
b*d*e + a*e^2)*(b*d + 2*a*e)*x^2*log(-(4*sqrt(c*x^4 + b*x^2 + a)*(a*b*x^2 + 2*a
^2) + ((b^2 + 4*a*c)*x^4 + 8*a*b*x^2 + 8*a^2)*sqrt(a))/x^4) - 4*sqrt(c*x^4 + b*x
^2 + a)*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(a)*d)/(sqrt(c*d^2 - b*d*e + a*e^2)*a^(3
/2)*d^2*x^2), -1/8*(4*a^(3/2)*e^2*x^2*arctan(-1/2*sqrt(-c*d^2 + b*d*e - a*e^2)*(
(2*c*d - b*e)*x^2 + b*d - 2*a*e)/(sqrt(c*x^4 + b*x^2 + a)*(c*d^2 - b*d*e + a*e^2
))) - sqrt(-c*d^2 + b*d*e - a*e^2)*(b*d + 2*a*e)*x^2*log(-(4*sqrt(c*x^4 + b*x^2
+ a)*(a*b*x^2 + 2*a^2) + ((b^2 + 4*a*c)*x^4 + 8*a*b*x^2 + 8*a^2)*sqrt(a))/x^4) +
4*sqrt(c*x^4 + b*x^2 + a)*sqrt(-c*d^2 + b*d*e - a*e^2)*sqrt(a)*d)/(sqrt(-c*d^2
+ b*d*e - a*e^2)*a^(3/2)*d^2*x^2), 1/4*(sqrt(-a)*a*e^2*x^2*log(-(4*(b*c*d^3 + 3*
a*b*d*e^2 - 2*a^2*e^3 - (b^2 + 2*a*c)*d^2*e + (2*c^2*d^3 - 3*b*c*d^2*e - a*b*e^3
+ (b^2 + 2*a*c)*d*e^2)*x^2)*sqrt(c*x^4 + b*x^2 + a) + ((8*c^2*d^2 - 8*b*c*d*e +
(b^2 + 4*a*c)*e^2)*x^4 - 8*a*b*d*e + 8*a^2*e^2 + (b^2 + 4*a*c)*d^2 + 2*(4*b*c*d
^2 + 4*a*b*e^2 - (3*b^2 + 4*a*c)*d*e)*x^2)*sqrt(c*d^2 - b*d*e + a*e^2))/(e^2*x^4
+ 2*d*e*x^2 + d^2)) + sqrt(c*d^2 - b*d*e + a*e^2)*(b*d + 2*a*e)*x^2*arctan(1/2*
(b*x^2 + 2*a)*sqrt(-a)/(sqrt(c*x^4 + b*x^2 + a)*a)) - 2*sqrt(c*x^4 + b*x^2 + a)*
sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(-a)*d)/(sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(-a)*a*
d^2*x^2), -1/4*(2*sqrt(-a)*a*e^2*x^2*arctan(-1/2*sqrt(-c*d^2 + b*d*e - a*e^2)*((
2*c*d - b*e)*x^2 + b*d - 2*a*e)/(sqrt(c*x^4 + b*x^2 + a)*(c*d^2 - b*d*e + a*e^2)
)) - sqrt(-c*d^2 + b*d*e - a*e^2)*(b*d + 2*a*e)*x^2*arctan(1/2*(b*x^2 + 2*a)*sqr
t(-a)/(sqrt(c*x^4 + b*x^2 + a)*a)) + 2*sqrt(c*x^4 + b*x^2 + a)*sqrt(-c*d^2 + b*d
*e - a*e^2)*sqrt(-a)*d)/(sqrt(-c*d^2 + b*d*e - a*e^2)*sqrt(-a)*a*d^2*x^2)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{3} \left (d + e x^{2}\right ) \sqrt{a + b x^{2} + c x^{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**3/(e*x**2+d)/(c*x**4+b*x**2+a)**(1/2),x)

[Out]

Integral(1/(x**3*(d + e*x**2)*sqrt(a + b*x**2 + c*x**4)), x)

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^4 + b*x^2 + a)*(e*x^2 + d)*x^3),x, algorithm="giac")

[Out]

Exception raised: RuntimeError