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

Optimal. Leaf size=101 \[ -\frac{x \left (4 c d^2-e (2 a e+b d)\right )}{3 d^2 e^2 \sqrt{d+e x^2}}+\frac{x \left (a+\frac{d (c d-b e)}{e^2}\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac{c \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{e^{5/2}} \]

[Out]

((a + (d*(c*d - b*e))/e^2)*x)/(3*d*(d + e*x^2)^(3/2)) - ((4*c*d^2 - e*(b*d + 2*a
*e))*x)/(3*d^2*e^2*Sqrt[d + e*x^2]) + (c*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/e
^(5/2)

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Rubi [A]  time = 0.148578, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{x \left (4 c d^2-e (2 a e+b d)\right )}{3 d^2 e^2 \sqrt{d+e x^2}}+\frac{x \left (a+\frac{d (c d-b e)}{e^2}\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac{c \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{e^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

((a + (d*(c*d - b*e))/e^2)*x)/(3*d*(d + e*x^2)^(3/2)) - ((4*c*d^2 - e*(b*d + 2*a
*e))*x)/(3*d^2*e^2*Sqrt[d + e*x^2]) + (c*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/e
^(5/2)

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Rubi in Sympy [A]  time = 28.7816, size = 97, normalized size = 0.96 \[ \frac{c \operatorname{atanh}{\left (\frac{\sqrt{e} x}{\sqrt{d + e x^{2}}} \right )}}{e^{\frac{5}{2}}} + \frac{x \left (a e^{2} - b d e + c d^{2}\right )}{3 d e^{2} \left (d + e x^{2}\right )^{\frac{3}{2}}} + \frac{x \left (2 a e^{2} + b d e - 4 c d^{2}\right )}{3 d^{2} e^{2} \sqrt{d + e x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c*atanh(sqrt(e)*x/sqrt(d + e*x**2))/e**(5/2) + x*(a*e**2 - b*d*e + c*d**2)/(3*d*
e**2*(d + e*x**2)**(3/2)) + x*(2*a*e**2 + b*d*e - 4*c*d**2)/(3*d**2*e**2*sqrt(d
+ e*x**2))

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Mathematica [A]  time = 0.243741, size = 91, normalized size = 0.9 \[ \frac{x \left (e^2 \left (3 a d+2 a e x^2+b d x^2\right )-c d^2 \left (3 d+4 e x^2\right )\right )}{3 d^2 e^2 \left (d+e x^2\right )^{3/2}}+\frac{c \log \left (\sqrt{e} \sqrt{d+e x^2}+e x\right )}{e^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(-(c*d^2*(3*d + 4*e*x^2)) + e^2*(3*a*d + b*d*x^2 + 2*a*e*x^2)))/(3*d^2*e^2*(d
+ e*x^2)^(3/2)) + (c*Log[e*x + Sqrt[e]*Sqrt[d + e*x^2]])/e^(5/2)

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Maple [A]  time = 0.011, size = 124, normalized size = 1.2 \[{\frac{ax}{3\,d} \left ( e{x}^{2}+d \right ) ^{-{\frac{3}{2}}}}+{\frac{2\,ax}{3\,{d}^{2}}{\frac{1}{\sqrt{e{x}^{2}+d}}}}-{\frac{bx}{3\,e} \left ( e{x}^{2}+d \right ) ^{-{\frac{3}{2}}}}+{\frac{bx}{3\,de}{\frac{1}{\sqrt{e{x}^{2}+d}}}}-{\frac{c{x}^{3}}{3\,e} \left ( e{x}^{2}+d \right ) ^{-{\frac{3}{2}}}}-{\frac{cx}{{e}^{2}}{\frac{1}{\sqrt{e{x}^{2}+d}}}}+{c\ln \left ( x\sqrt{e}+\sqrt{e{x}^{2}+d} \right ){e}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*a*x/d/(e*x^2+d)^(3/2)+2/3*a/d^2*x/(e*x^2+d)^(1/2)-1/3*b/e*x/(e*x^2+d)^(3/2)+
1/3*b/d/e*x/(e*x^2+d)^(1/2)-1/3*c*x^3/e/(e*x^2+d)^(3/2)-c/e^2*x/(e*x^2+d)^(1/2)+
c/e^(5/2)*ln(x*e^(1/2)+(e*x^2+d)^(1/2))

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.305514, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \,{\left ({\left (4 \, c d^{2} e - b d e^{2} - 2 \, a e^{3}\right )} x^{3} + 3 \,{\left (c d^{3} - a d e^{2}\right )} x\right )} \sqrt{e x^{2} + d} \sqrt{e} - 3 \,{\left (c d^{2} e^{2} x^{4} + 2 \, c d^{3} e x^{2} + c d^{4}\right )} \log \left (-2 \, \sqrt{e x^{2} + d} e x -{\left (2 \, e x^{2} + d\right )} \sqrt{e}\right )}{6 \,{\left (d^{2} e^{4} x^{4} + 2 \, d^{3} e^{3} x^{2} + d^{4} e^{2}\right )} \sqrt{e}}, -\frac{{\left ({\left (4 \, c d^{2} e - b d e^{2} - 2 \, a e^{3}\right )} x^{3} + 3 \,{\left (c d^{3} - a d e^{2}\right )} x\right )} \sqrt{e x^{2} + d} \sqrt{-e} - 3 \,{\left (c d^{2} e^{2} x^{4} + 2 \, c d^{3} e x^{2} + c d^{4}\right )} \arctan \left (\frac{\sqrt{-e} x}{\sqrt{e x^{2} + d}}\right )}{3 \,{\left (d^{2} e^{4} x^{4} + 2 \, d^{3} e^{3} x^{2} + d^{4} e^{2}\right )} \sqrt{-e}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/6*(2*((4*c*d^2*e - b*d*e^2 - 2*a*e^3)*x^3 + 3*(c*d^3 - a*d*e^2)*x)*sqrt(e*x^
2 + d)*sqrt(e) - 3*(c*d^2*e^2*x^4 + 2*c*d^3*e*x^2 + c*d^4)*log(-2*sqrt(e*x^2 + d
)*e*x - (2*e*x^2 + d)*sqrt(e)))/((d^2*e^4*x^4 + 2*d^3*e^3*x^2 + d^4*e^2)*sqrt(e)
), -1/3*(((4*c*d^2*e - b*d*e^2 - 2*a*e^3)*x^3 + 3*(c*d^3 - a*d*e^2)*x)*sqrt(e*x^
2 + d)*sqrt(-e) - 3*(c*d^2*e^2*x^4 + 2*c*d^3*e*x^2 + c*d^4)*arctan(sqrt(-e)*x/sq
rt(e*x^2 + d)))/((d^2*e^4*x^4 + 2*d^3*e^3*x^2 + d^4*e^2)*sqrt(-e))]

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Sympy [A]  time = 55.0696, size = 450, normalized size = 4.46 \[ a \left (\frac{3 d x}{3 d^{\frac{7}{2}} \sqrt{1 + \frac{e x^{2}}{d}} + 3 d^{\frac{5}{2}} e x^{2} \sqrt{1 + \frac{e x^{2}}{d}}} + \frac{2 e x^{3}}{3 d^{\frac{7}{2}} \sqrt{1 + \frac{e x^{2}}{d}} + 3 d^{\frac{5}{2}} e x^{2} \sqrt{1 + \frac{e x^{2}}{d}}}\right ) + \frac{b x^{3}}{3 d^{\frac{5}{2}} \sqrt{1 + \frac{e x^{2}}{d}} + 3 d^{\frac{3}{2}} e x^{2} \sqrt{1 + \frac{e x^{2}}{d}}} + c \left (\frac{3 d^{\frac{39}{2}} e^{11} \sqrt{1 + \frac{e x^{2}}{d}} \operatorname{asinh}{\left (\frac{\sqrt{e} x}{\sqrt{d}} \right )}}{3 d^{\frac{39}{2}} e^{\frac{27}{2}} \sqrt{1 + \frac{e x^{2}}{d}} + 3 d^{\frac{37}{2}} e^{\frac{29}{2}} x^{2} \sqrt{1 + \frac{e x^{2}}{d}}} + \frac{3 d^{\frac{37}{2}} e^{12} x^{2} \sqrt{1 + \frac{e x^{2}}{d}} \operatorname{asinh}{\left (\frac{\sqrt{e} x}{\sqrt{d}} \right )}}{3 d^{\frac{39}{2}} e^{\frac{27}{2}} \sqrt{1 + \frac{e x^{2}}{d}} + 3 d^{\frac{37}{2}} e^{\frac{29}{2}} x^{2} \sqrt{1 + \frac{e x^{2}}{d}}} - \frac{3 d^{19} e^{\frac{23}{2}} x}{3 d^{\frac{39}{2}} e^{\frac{27}{2}} \sqrt{1 + \frac{e x^{2}}{d}} + 3 d^{\frac{37}{2}} e^{\frac{29}{2}} x^{2} \sqrt{1 + \frac{e x^{2}}{d}}} - \frac{4 d^{18} e^{\frac{25}{2}} x^{3}}{3 d^{\frac{39}{2}} e^{\frac{27}{2}} \sqrt{1 + \frac{e x^{2}}{d}} + 3 d^{\frac{37}{2}} e^{\frac{29}{2}} x^{2} \sqrt{1 + \frac{e x^{2}}{d}}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*(3*d*x/(3*d**(7/2)*sqrt(1 + e*x**2/d) + 3*d**(5/2)*e*x**2*sqrt(1 + e*x**2/d))
+ 2*e*x**3/(3*d**(7/2)*sqrt(1 + e*x**2/d) + 3*d**(5/2)*e*x**2*sqrt(1 + e*x**2/d)
)) + b*x**3/(3*d**(5/2)*sqrt(1 + e*x**2/d) + 3*d**(3/2)*e*x**2*sqrt(1 + e*x**2/d
)) + c*(3*d**(39/2)*e**11*sqrt(1 + e*x**2/d)*asinh(sqrt(e)*x/sqrt(d))/(3*d**(39/
2)*e**(27/2)*sqrt(1 + e*x**2/d) + 3*d**(37/2)*e**(29/2)*x**2*sqrt(1 + e*x**2/d))
+ 3*d**(37/2)*e**12*x**2*sqrt(1 + e*x**2/d)*asinh(sqrt(e)*x/sqrt(d))/(3*d**(39/
2)*e**(27/2)*sqrt(1 + e*x**2/d) + 3*d**(37/2)*e**(29/2)*x**2*sqrt(1 + e*x**2/d))
- 3*d**19*e**(23/2)*x/(3*d**(39/2)*e**(27/2)*sqrt(1 + e*x**2/d) + 3*d**(37/2)*e
**(29/2)*x**2*sqrt(1 + e*x**2/d)) - 4*d**18*e**(25/2)*x**3/(3*d**(39/2)*e**(27/2
)*sqrt(1 + e*x**2/d) + 3*d**(37/2)*e**(29/2)*x**2*sqrt(1 + e*x**2/d)))

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GIAC/XCAS [A]  time = 0.271994, size = 119, normalized size = 1.18 \[ -c e^{\left (-\frac{5}{2}\right )}{\rm ln}\left ({\left | -x e^{\frac{1}{2}} + \sqrt{x^{2} e + d} \right |}\right ) - \frac{{\left (\frac{{\left (4 \, c d^{2} e^{2} - b d e^{3} - 2 \, a e^{4}\right )} x^{2} e^{\left (-3\right )}}{d^{2}} + \frac{3 \,{\left (c d^{3} e - a d e^{3}\right )} e^{\left (-3\right )}}{d^{2}}\right )} x}{3 \,{\left (x^{2} e + d\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-c*e^(-5/2)*ln(abs(-x*e^(1/2) + sqrt(x^2*e + d))) - 1/3*((4*c*d^2*e^2 - b*d*e^3
- 2*a*e^4)*x^2*e^(-3)/d^2 + 3*(c*d^3*e - a*d*e^3)*e^(-3)/d^2)*x/(x^2*e + d)^(3/2
)