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

Optimal. Leaf size=173 \[ -\frac{(b e+2 c d) \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{4 c^{3/2} e^2}+\frac{d^2 \tanh ^{-1}\left (\frac{-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}}\right )}{2 e^2 \sqrt{a e^2-b d e+c d^2}}+\frac{\sqrt{a+b x^2+c x^4}}{2 c e} \]

[Out]

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

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Rubi [A]  time = 0.756937, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207 \[ -\frac{(b e+2 c d) \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{4 c^{3/2} e^2}+\frac{d^2 \tanh ^{-1}\left (\frac{-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}}\right )}{2 e^2 \sqrt{a e^2-b d e+c d^2}}+\frac{\sqrt{a+b x^2+c x^4}}{2 c e} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 62.1153, size = 190, normalized size = 1.1 \[ - \frac{b \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{2 \sqrt{c} \sqrt{a + b x^{2} + c x^{4}}} \right )}}{4 c^{\frac{3}{2}} e} - \frac{d^{2} \operatorname{atanh}{\left (\frac{2 a e - b d + x^{2} \left (b e - 2 c d\right )}{2 \sqrt{a + b x^{2} + c x^{4}} \sqrt{a e^{2} - b d e + c d^{2}}} \right )}}{2 e^{2} \sqrt{a e^{2} - b d e + c d^{2}}} + \frac{\sqrt{a + b x^{2} + c x^{4}}}{2 c e} - \frac{d \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{2 \sqrt{c} \sqrt{a + b x^{2} + c x^{4}}} \right )}}{2 \sqrt{c} e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.75862, size = 196, normalized size = 1.13 \[ \frac{-\frac{(b e+2 c d) \log \left (2 \sqrt{c} \sqrt{a+b x^2+c x^4}+b+2 c x^2\right )}{c^{3/2}}-\frac{2 d^2 \log \left (2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}+2 a e-b d+b e x^2-2 c d x^2\right )}{\sqrt{a e^2-b d e+c d^2}}+\frac{2 d^2 \log \left (d+e x^2\right )}{\sqrt{e (a e-b d)+c d^2}}+\frac{2 e \sqrt{a+b x^2+c x^4}}{c}}{4 e^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.026, size = 267, normalized size = 1.5 \[{\frac{1}{2\,ce}\sqrt{c{x}^{4}+b{x}^{2}+a}}-{\frac{b}{4\,e}\ln \left ({1 \left ({\frac{b}{2}}+c{x}^{2} \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{4}+b{x}^{2}+a} \right ){c}^{-{\frac{3}{2}}}}-{\frac{d}{2\,{e}^{2}}\ln \left ({1 \left ({\frac{b}{2}}+c{x}^{2} \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{4}+b{x}^{2}+a} \right ){\frac{1}{\sqrt{c}}}}-{\frac{{d}^{2}}{2\,{e}^{3}}\ln \left ({1 \left ( 2\,{\frac{a{e}^{2}-bde+c{d}^{2}}{{e}^{2}}}+{\frac{be-2\,cd}{e} \left ({x}^{2}+{\frac{d}{e}} \right ) }+2\,\sqrt{{\frac{a{e}^{2}-bde+c{d}^{2}}{{e}^{2}}}}\sqrt{ \left ({x}^{2}+{\frac{d}{e}} \right ) ^{2}c+{\frac{be-2\,cd}{e} \left ({x}^{2}+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}-bde+c{d}^{2}}{{e}^{2}}}} \right ) \left ({x}^{2}+{\frac{d}{e}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{e}^{2}-bde+c{d}^{2}}{{e}^{2}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(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)/c/e-1/4/e*b/c^(3/2)*ln((1/2*b+c*x^2)/c^(1/2)+(c*x^4+b*
x^2+a)^(1/2))-1/2/e^2*d*ln((1/2*b+c*x^2)/c^(1/2)+(c*x^4+b*x^2+a)^(1/2))/c^(1/2)-
1/2*d^2/e^3/((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))

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 21.3666, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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