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

Optimal. Leaf size=272 \[ -\frac{\left (-8 c^2 d e (b d-a e)-2 b c e^2 (b d-2 a e)-b^3 e^3+16 c^3 d^3\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{32 c^{5/2} e^4}+\frac{\sqrt{a+b x^2+c x^4} \left ((2 c d-b e) (b e+4 c d)-2 c e x^2 (b e+2 c d)\right )}{16 c^2 e^3}+\frac{d^2 \sqrt{a e^2-b d e+c d^2} \tanh ^{-1}\left (\frac{-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}}\right )}{2 e^4}+\frac{\left (a+b x^2+c x^4\right )^{3/2}}{6 c e} \]

[Out]

(((2*c*d - b*e)*(4*c*d + b*e) - 2*c*e*(2*c*d + b*e)*x^2)*Sqrt[a + b*x^2 + c*x^4]
)/(16*c^2*e^3) + (a + b*x^2 + c*x^4)^(3/2)/(6*c*e) - ((16*c^3*d^3 - b^3*e^3 - 2*
b*c*e^2*(b*d - 2*a*e) - 8*c^2*d*e*(b*d - a*e))*ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*
Sqrt[a + b*x^2 + c*x^4])])/(32*c^(5/2)*e^4) + (d^2*Sqrt[c*d^2 - b*d*e + a*e^2]*A
rcTanh[(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^4)

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Rubi [A]  time = 1.16697, antiderivative size = 272, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.241 \[ -\frac{\left (-8 c^2 d e (b d-a e)-2 b c e^2 (b d-2 a e)-b^3 e^3+16 c^3 d^3\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{32 c^{5/2} e^4}+\frac{\sqrt{a+b x^2+c x^4} \left ((2 c d-b e) (b e+4 c d)-2 c e x^2 (b e+2 c d)\right )}{16 c^2 e^3}+\frac{d^2 \sqrt{a e^2-b d e+c d^2} \tanh ^{-1}\left (\frac{-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}}\right )}{2 e^4}+\frac{\left (a+b x^2+c x^4\right )^{3/2}}{6 c e} \]

Antiderivative was successfully verified.

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

[Out]

(((2*c*d - b*e)*(4*c*d + b*e) - 2*c*e*(2*c*d + b*e)*x^2)*Sqrt[a + b*x^2 + c*x^4]
)/(16*c^2*e^3) + (a + b*x^2 + c*x^4)^(3/2)/(6*c*e) - ((16*c^3*d^3 - b^3*e^3 - 2*
b*c*e^2*(b*d - 2*a*e) - 8*c^2*d*e*(b*d - a*e))*ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*
Sqrt[a + b*x^2 + c*x^4])])/(32*c^(5/2)*e^4) + (d^2*Sqrt[c*d^2 - b*d*e + a*e^2]*A
rcTanh[(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^4)

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Rubi in Sympy [A]  time = 104.923, size = 350, normalized size = 1.29 \[ - \frac{b \left (b + 2 c x^{2}\right ) \sqrt{a + b x^{2} + c x^{4}}}{16 c^{2} e} + \frac{b \left (- 4 a c + b^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{2 \sqrt{c} \sqrt{a + b x^{2} + c x^{4}}} \right )}}{32 c^{\frac{5}{2}} e} + \frac{d^{2} \sqrt{a + b x^{2} + c x^{4}}}{2 e^{3}} - \frac{d^{2} \sqrt{a e^{2} - b d e + c d^{2}} \operatorname{atanh}{\left (\frac{2 a e - b d + x^{2} \left (b e - 2 c d\right )}{2 \sqrt{a + b x^{2} + c x^{4}} \sqrt{a e^{2} - b d e + c d^{2}}} \right )}}{2 e^{4}} - \frac{d \left (b + 2 c x^{2}\right ) \sqrt{a + b x^{2} + c x^{4}}}{8 c e^{2}} + \frac{\left (a + b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{6 c e} + \frac{d^{2} \left (b e - 2 c d\right ) \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{2 \sqrt{c} \sqrt{a + b x^{2} + c x^{4}}} \right )}}{4 \sqrt{c} e^{4}} + \frac{d \left (- 4 a c + b^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{2 \sqrt{c} \sqrt{a + b x^{2} + c x^{4}}} \right )}}{16 c^{\frac{3}{2}} e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b*(b + 2*c*x**2)*sqrt(a + b*x**2 + c*x**4)/(16*c**2*e) + b*(-4*a*c + b**2)*atan
h((b + 2*c*x**2)/(2*sqrt(c)*sqrt(a + b*x**2 + c*x**4)))/(32*c**(5/2)*e) + d**2*s
qrt(a + b*x**2 + c*x**4)/(2*e**3) - d**2*sqrt(a*e**2 - b*d*e + c*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**4) - d*(b + 2*c*x**2)*sqrt(a + b*x**2 + c*x**4)/(8*c*e**2) +
(a + b*x**2 + c*x**4)**(3/2)/(6*c*e) + d**2*(b*e - 2*c*d)*atanh((b + 2*c*x**2)/(
2*sqrt(c)*sqrt(a + b*x**2 + c*x**4)))/(4*sqrt(c)*e**4) + d*(-4*a*c + b**2)*atanh
((b + 2*c*x**2)/(2*sqrt(c)*sqrt(a + b*x**2 + c*x**4)))/(16*c**(3/2)*e**2)

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Mathematica [A]  time = 1.02148, size = 301, normalized size = 1.11 \[ \frac{-3 \left (8 c^2 d e (a e-b d)-2 b c e^2 (b d-2 a e)-b^3 e^3+16 c^3 d^3\right ) \log \left (2 \sqrt{c} \sqrt{a+b x^2+c x^4}+b+2 c x^2\right )+2 \sqrt{c} \left (e \sqrt{a+b x^2+c x^4} \left (2 c e \left (4 a e-3 b d+b e x^2\right )-3 b^2 e^2+4 c^2 \left (6 d^2-3 d e x^2+2 e^2 x^4\right )\right )-24 c^2 d^2 \sqrt{e (a e-b d)+c d^2} \log \left (2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}+2 a e-b d+b e x^2-2 c d x^2\right )\right )+48 c^{5/2} d^2 \log \left (d+e x^2\right ) \sqrt{e (a e-b d)+c d^2}}{96 c^{5/2} e^4} \]

Antiderivative was successfully verified.

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

[Out]

(48*c^(5/2)*d^2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Log[d + e*x^2] - 3*(16*c^3*d^3 -
b^3*e^3 - 2*b*c*e^2*(b*d - 2*a*e) + 8*c^2*d*e*(-(b*d) + a*e))*Log[b + 2*c*x^2 +
2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4]] + 2*Sqrt[c]*(e*Sqrt[a + b*x^2 + c*x^4]*(-3*b^
2*e^2 + 2*c*e*(-3*b*d + 4*a*e + b*e*x^2) + 4*c^2*(6*d^2 - 3*d*e*x^2 + 2*e^2*x^4)
) - 24*c^2*d^2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*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]]))/(96*c^(5/2)*e^
4)

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Maple [B]  time = 0.03, size = 1049, normalized size = 3.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(c*x^4+b*x^2+a)^(3/2)/c/e-1/8/e*b/c*(c*x^4+b*x^2+a)^(1/2)*x^2-1/16/e*b^2/c^2
*(c*x^4+b*x^2+a)^(1/2)-1/8/e*b/c^(3/2)*ln((1/2*b+c*x^2)/c^(1/2)+(c*x^4+b*x^2+a)^
(1/2))*a+1/32/e*b^3/c^(5/2)*ln((1/2*b+c*x^2)/c^(1/2)+(c*x^4+b*x^2+a)^(1/2))-1/4/
e^2*d*(c*x^4+b*x^2+a)^(1/2)*x^2-1/8/e^2*d/c*(c*x^4+b*x^2+a)^(1/2)*b-1/4/e^2*d/c^
(1/2)*ln((1/2*b+c*x^2)/c^(1/2)+(c*x^4+b*x^2+a)^(1/2))*a+1/16/e^2*d/c^(3/2)*ln((1
/2*b+c*x^2)/c^(1/2)+(c*x^4+b*x^2+a)^(1/2))*b^2+1/2*d^2/e^3*((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)+1/4*d^2/e^3*ln((1/2*(b*e-2*c*d)
/e+c*(x^2+d/e))/c^(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))/c^(1/2)*b-1/2*d^3/e^4*ln((1/2*(b*e-2*c*d)/e+c*(x^2+d/e))/c^(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))*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))*a+1/2*d^3/e^4/((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))*b-1/2*d^4/e^5/((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))*c

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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