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

Optimal. Leaf size=866 \[ -\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{e} \sqrt{f x}}{\sqrt [4]{d} \sqrt{f}}\right ) e^{7/4}}{\sqrt{2} d^{3/4} \left (c d^2-b e d+a e^2\right ) \sqrt{f}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{e} \sqrt{f x}}{\sqrt [4]{d} \sqrt{f}}+1\right ) e^{7/4}}{\sqrt{2} d^{3/4} \left (c d^2-b e d+a e^2\right ) \sqrt{f}}-\frac{\log \left (\sqrt{e} \sqrt{f} x+\sqrt{d} \sqrt{f}-\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{f x}\right ) e^{7/4}}{2 \sqrt{2} d^{3/4} \left (c d^2-b e d+a e^2\right ) \sqrt{f}}+\frac{\log \left (\sqrt{e} \sqrt{f} x+\sqrt{d} \sqrt{f}+\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{f x}\right ) e^{7/4}}{2 \sqrt{2} d^{3/4} \left (c d^2-b e d+a e^2\right ) \sqrt{f}}+\frac{c^{3/4} \left (2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{f x}}{\sqrt [4]{-b-\sqrt{b^2-4 a c}} \sqrt{f}}\right )}{\sqrt [4]{2} \sqrt{b^2-4 a c} \left (-b-\sqrt{b^2-4 a c}\right )^{3/4} \left (c d^2-b e d+a e^2\right ) \sqrt{f}}-\frac{c^{3/4} \left (2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{f x}}{\sqrt [4]{\sqrt{b^2-4 a c}-b} \sqrt{f}}\right )}{\sqrt [4]{2} \sqrt{b^2-4 a c} \left (\sqrt{b^2-4 a c}-b\right )^{3/4} \left (c d^2-b e d+a e^2\right ) \sqrt{f}}+\frac{c^{3/4} \left (2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{f x}}{\sqrt [4]{-b-\sqrt{b^2-4 a c}} \sqrt{f}}\right )}{\sqrt [4]{2} \sqrt{b^2-4 a c} \left (-b-\sqrt{b^2-4 a c}\right )^{3/4} \left (c d^2-b e d+a e^2\right ) \sqrt{f}}-\frac{c^{3/4} \left (2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{f x}}{\sqrt [4]{\sqrt{b^2-4 a c}-b} \sqrt{f}}\right )}{\sqrt [4]{2} \sqrt{b^2-4 a c} \left (\sqrt{b^2-4 a c}-b\right )^{3/4} \left (c d^2-b e d+a e^2\right ) \sqrt{f}} \]

[Out]

(c^(3/4)*(2*c*d - (b - Sqrt[b^2 - 4*a*c])*e)*ArcTan[(2^(1/4)*c^(1/4)*Sqrt[f*x])/
((-b - Sqrt[b^2 - 4*a*c])^(1/4)*Sqrt[f])])/(2^(1/4)*Sqrt[b^2 - 4*a*c]*(-b - Sqrt
[b^2 - 4*a*c])^(3/4)*(c*d^2 - b*d*e + a*e^2)*Sqrt[f]) - (c^(3/4)*(2*c*d - (b + S
qrt[b^2 - 4*a*c])*e)*ArcTan[(2^(1/4)*c^(1/4)*Sqrt[f*x])/((-b + Sqrt[b^2 - 4*a*c]
)^(1/4)*Sqrt[f])])/(2^(1/4)*Sqrt[b^2 - 4*a*c]*(-b + Sqrt[b^2 - 4*a*c])^(3/4)*(c*
d^2 - b*d*e + a*e^2)*Sqrt[f]) - (e^(7/4)*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[f*x])/
(d^(1/4)*Sqrt[f])])/(Sqrt[2]*d^(3/4)*(c*d^2 - b*d*e + a*e^2)*Sqrt[f]) + (e^(7/4)
*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[f*x])/(d^(1/4)*Sqrt[f])])/(Sqrt[2]*d^(3/4)*(c*
d^2 - b*d*e + a*e^2)*Sqrt[f]) + (c^(3/4)*(2*c*d - (b - Sqrt[b^2 - 4*a*c])*e)*Arc
Tanh[(2^(1/4)*c^(1/4)*Sqrt[f*x])/((-b - Sqrt[b^2 - 4*a*c])^(1/4)*Sqrt[f])])/(2^(
1/4)*Sqrt[b^2 - 4*a*c]*(-b - Sqrt[b^2 - 4*a*c])^(3/4)*(c*d^2 - b*d*e + a*e^2)*Sq
rt[f]) - (c^(3/4)*(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*ArcTanh[(2^(1/4)*c^(1/4)*S
qrt[f*x])/((-b + Sqrt[b^2 - 4*a*c])^(1/4)*Sqrt[f])])/(2^(1/4)*Sqrt[b^2 - 4*a*c]*
(-b + Sqrt[b^2 - 4*a*c])^(3/4)*(c*d^2 - b*d*e + a*e^2)*Sqrt[f]) - (e^(7/4)*Log[S
qrt[d]*Sqrt[f] + Sqrt[e]*Sqrt[f]*x - Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[f*x]])/(2*Sqrt
[2]*d^(3/4)*(c*d^2 - b*d*e + a*e^2)*Sqrt[f]) + (e^(7/4)*Log[Sqrt[d]*Sqrt[f] + Sq
rt[e]*Sqrt[f]*x + Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[f*x]])/(2*Sqrt[2]*d^(3/4)*(c*d^2
- b*d*e + a*e^2)*Sqrt[f])

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Rubi [A]  time = 4.77455, antiderivative size = 866, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 12, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.387 \[ -\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{e} \sqrt{f x}}{\sqrt [4]{d} \sqrt{f}}\right ) e^{7/4}}{\sqrt{2} d^{3/4} \left (c d^2-b e d+a e^2\right ) \sqrt{f}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{e} \sqrt{f x}}{\sqrt [4]{d} \sqrt{f}}+1\right ) e^{7/4}}{\sqrt{2} d^{3/4} \left (c d^2-b e d+a e^2\right ) \sqrt{f}}-\frac{\log \left (\sqrt{e} \sqrt{f} x+\sqrt{d} \sqrt{f}-\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{f x}\right ) e^{7/4}}{2 \sqrt{2} d^{3/4} \left (c d^2-b e d+a e^2\right ) \sqrt{f}}+\frac{\log \left (\sqrt{e} \sqrt{f} x+\sqrt{d} \sqrt{f}+\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{f x}\right ) e^{7/4}}{2 \sqrt{2} d^{3/4} \left (c d^2-b e d+a e^2\right ) \sqrt{f}}+\frac{c^{3/4} \left (2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{f x}}{\sqrt [4]{-b-\sqrt{b^2-4 a c}} \sqrt{f}}\right )}{\sqrt [4]{2} \sqrt{b^2-4 a c} \left (-b-\sqrt{b^2-4 a c}\right )^{3/4} \left (c d^2-b e d+a e^2\right ) \sqrt{f}}-\frac{c^{3/4} \left (2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{f x}}{\sqrt [4]{\sqrt{b^2-4 a c}-b} \sqrt{f}}\right )}{\sqrt [4]{2} \sqrt{b^2-4 a c} \left (\sqrt{b^2-4 a c}-b\right )^{3/4} \left (c d^2-b e d+a e^2\right ) \sqrt{f}}+\frac{c^{3/4} \left (2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{f x}}{\sqrt [4]{-b-\sqrt{b^2-4 a c}} \sqrt{f}}\right )}{\sqrt [4]{2} \sqrt{b^2-4 a c} \left (-b-\sqrt{b^2-4 a c}\right )^{3/4} \left (c d^2-b e d+a e^2\right ) \sqrt{f}}-\frac{c^{3/4} \left (2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{f x}}{\sqrt [4]{\sqrt{b^2-4 a c}-b} \sqrt{f}}\right )}{\sqrt [4]{2} \sqrt{b^2-4 a c} \left (\sqrt{b^2-4 a c}-b\right )^{3/4} \left (c d^2-b e d+a e^2\right ) \sqrt{f}} \]

Antiderivative was successfully verified.

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

[Out]

(c^(3/4)*(2*c*d - (b - Sqrt[b^2 - 4*a*c])*e)*ArcTan[(2^(1/4)*c^(1/4)*Sqrt[f*x])/
((-b - Sqrt[b^2 - 4*a*c])^(1/4)*Sqrt[f])])/(2^(1/4)*Sqrt[b^2 - 4*a*c]*(-b - Sqrt
[b^2 - 4*a*c])^(3/4)*(c*d^2 - b*d*e + a*e^2)*Sqrt[f]) - (c^(3/4)*(2*c*d - (b + S
qrt[b^2 - 4*a*c])*e)*ArcTan[(2^(1/4)*c^(1/4)*Sqrt[f*x])/((-b + Sqrt[b^2 - 4*a*c]
)^(1/4)*Sqrt[f])])/(2^(1/4)*Sqrt[b^2 - 4*a*c]*(-b + Sqrt[b^2 - 4*a*c])^(3/4)*(c*
d^2 - b*d*e + a*e^2)*Sqrt[f]) - (e^(7/4)*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[f*x])/
(d^(1/4)*Sqrt[f])])/(Sqrt[2]*d^(3/4)*(c*d^2 - b*d*e + a*e^2)*Sqrt[f]) + (e^(7/4)
*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[f*x])/(d^(1/4)*Sqrt[f])])/(Sqrt[2]*d^(3/4)*(c*
d^2 - b*d*e + a*e^2)*Sqrt[f]) + (c^(3/4)*(2*c*d - (b - Sqrt[b^2 - 4*a*c])*e)*Arc
Tanh[(2^(1/4)*c^(1/4)*Sqrt[f*x])/((-b - Sqrt[b^2 - 4*a*c])^(1/4)*Sqrt[f])])/(2^(
1/4)*Sqrt[b^2 - 4*a*c]*(-b - Sqrt[b^2 - 4*a*c])^(3/4)*(c*d^2 - b*d*e + a*e^2)*Sq
rt[f]) - (c^(3/4)*(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*ArcTanh[(2^(1/4)*c^(1/4)*S
qrt[f*x])/((-b + Sqrt[b^2 - 4*a*c])^(1/4)*Sqrt[f])])/(2^(1/4)*Sqrt[b^2 - 4*a*c]*
(-b + Sqrt[b^2 - 4*a*c])^(3/4)*(c*d^2 - b*d*e + a*e^2)*Sqrt[f]) - (e^(7/4)*Log[S
qrt[d]*Sqrt[f] + Sqrt[e]*Sqrt[f]*x - Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[f*x]])/(2*Sqrt
[2]*d^(3/4)*(c*d^2 - b*d*e + a*e^2)*Sqrt[f]) + (e^(7/4)*Log[Sqrt[d]*Sqrt[f] + Sq
rt[e]*Sqrt[f]*x + Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[f*x]])/(2*Sqrt[2]*d^(3/4)*(c*d^2
- b*d*e + a*e^2)*Sqrt[f])

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

[Out]

Timed out

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Mathematica [C]  time = 0.514124, size = 267, normalized size = 0.31 \[ \frac{\sqrt{x} \left (\sqrt{2} e^{7/4} \left (-\log \left (-\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{x}+\sqrt{d}+\sqrt{e} x\right )+\log \left (\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{x}+\sqrt{d}+\sqrt{e} x\right )-2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{e} \sqrt{x}}{\sqrt [4]{d}}\right )+2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{e} \sqrt{x}}{\sqrt [4]{d}}+1\right )\right )-2 d^{3/4} \text{RootSum}\left [\text{$\#$1}^8 c+\text{$\#$1}^4 b+a\&,\frac{\text{$\#$1}^4 c e \log \left (\sqrt{x}-\text{$\#$1}\right )+b e \log \left (\sqrt{x}-\text{$\#$1}\right )-c d \log \left (\sqrt{x}-\text{$\#$1}\right )}{2 \text{$\#$1}^7 c+\text{$\#$1}^3 b}\&\right ]\right )}{4 d^{3/4} \sqrt{f x} \left (e (a e-b d)+c d^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x]*(Sqrt[2]*e^(7/4)*(-2*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[x])/d^(1/4)] + 2*
ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[x])/d^(1/4)] - Log[Sqrt[d] - Sqrt[2]*d^(1/4)*e^
(1/4)*Sqrt[x] + Sqrt[e]*x] + Log[Sqrt[d] + Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[x] + Sqr
t[e]*x]) - 2*d^(3/4)*RootSum[a + b*#1^4 + c*#1^8 & , (-(c*d*Log[Sqrt[x] - #1]) +
b*e*Log[Sqrt[x] - #1] + c*e*Log[Sqrt[x] - #1]*#1^4)/(b*#1^3 + 2*c*#1^7) & ]))/(
4*d^(3/4)*(c*d^2 + e*(-(b*d) + a*e))*Sqrt[f*x])

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Maple [C]  time = 0.103, size = 336, normalized size = 0.4 \[{\frac{{e}^{2}\sqrt{2}}{4\,f \left ( a{e}^{2}-bde+c{d}^{2} \right ) d}\sqrt [4]{{\frac{d{f}^{2}}{e}}}\ln \left ({1 \left ( fx+\sqrt [4]{{\frac{d{f}^{2}}{e}}}\sqrt{fx}\sqrt{2}+\sqrt{{\frac{d{f}^{2}}{e}}} \right ) \left ( fx-\sqrt [4]{{\frac{d{f}^{2}}{e}}}\sqrt{fx}\sqrt{2}+\sqrt{{\frac{d{f}^{2}}{e}}} \right ) ^{-1}} \right ) }+{\frac{{e}^{2}\sqrt{2}}{2\,f \left ( a{e}^{2}-bde+c{d}^{2} \right ) d}\sqrt [4]{{\frac{d{f}^{2}}{e}}}\arctan \left ({\sqrt{2}\sqrt{fx}{\frac{1}{\sqrt [4]{{\frac{d{f}^{2}}{e}}}}}}+1 \right ) }+{\frac{{e}^{2}\sqrt{2}}{2\,f \left ( a{e}^{2}-bde+c{d}^{2} \right ) d}\sqrt [4]{{\frac{d{f}^{2}}{e}}}\arctan \left ({\sqrt{2}\sqrt{fx}{\frac{1}{\sqrt [4]{{\frac{d{f}^{2}}{e}}}}}}-1 \right ) }+{\frac{f}{2\,a{e}^{2}-2\,bde+2\,c{d}^{2}}\sum _{{\it \_R}={\it RootOf} \left ( c{{\it \_Z}}^{8}+b{f}^{2}{{\it \_Z}}^{4}+a{f}^{4} \right ) }{\frac{-{{\it \_R}}^{4}ce-be{f}^{2}+cd{f}^{2}}{2\,{{\it \_R}}^{7}c+{{\it \_R}}^{3}b{f}^{2}}\ln \left ( \sqrt{fx}-{\it \_R} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4/f*e^2/(a*e^2-b*d*e+c*d^2)*(d*f^2/e)^(1/4)/d*2^(1/2)*ln((f*x+(d*f^2/e)^(1/4)*
(f*x)^(1/2)*2^(1/2)+(d*f^2/e)^(1/2))/(f*x-(d*f^2/e)^(1/4)*(f*x)^(1/2)*2^(1/2)+(d
*f^2/e)^(1/2)))+1/2/f*e^2/(a*e^2-b*d*e+c*d^2)*(d*f^2/e)^(1/4)/d*2^(1/2)*arctan(2
^(1/2)/(d*f^2/e)^(1/4)*(f*x)^(1/2)+1)+1/2/f*e^2/(a*e^2-b*d*e+c*d^2)*(d*f^2/e)^(1
/4)/d*2^(1/2)*arctan(2^(1/2)/(d*f^2/e)^(1/4)*(f*x)^(1/2)-1)+1/2*f/(a*e^2-b*d*e+c
*d^2)*sum((-_R^4*c*e-b*e*f^2+c*d*f^2)/(2*_R^7*c+_R^3*b*f^2)*ln((f*x)^(1/2)-_R),_
R=RootOf(_Z^8*c+_Z^4*b*f^2+a*f^4))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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