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

Optimal. Leaf size=89 \[ \frac{x \left (a+\frac{d (c d-b e)}{e^2}\right )}{d \sqrt{d+e x^2}}-\frac{(3 c d-2 b e) \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{2 e^{5/2}}+\frac{c x \sqrt{d+e x^2}}{2 e^2} \]

[Out]

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

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Rubi [A]  time = 0.152003, antiderivative size = 89, 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 \[ \frac{x \left (a+\frac{d (c d-b e)}{e^2}\right )}{d \sqrt{d+e x^2}}-\frac{(3 c d-2 b e) \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{2 e^{5/2}}+\frac{c x \sqrt{d+e x^2}}{2 e^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 19.5912, size = 85, normalized size = 0.96 \[ \frac{c x \sqrt{d + e x^{2}}}{2 e^{2}} + \frac{\left (2 b e - 3 c d\right ) \operatorname{atanh}{\left (\frac{\sqrt{e} x}{\sqrt{d + e x^{2}}} \right )}}{2 e^{\frac{5}{2}}} + \frac{x \left (a e^{2} - b d e + c d^{2}\right )}{d e^{2} \sqrt{d + e x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.152854, size = 87, normalized size = 0.98 \[ \frac{\frac{\sqrt{e} x \left (2 e (a e-b d)+c d \left (3 d+e x^2\right )\right )}{d \sqrt{d+e x^2}}+(2 b e-3 c d) \log \left (\sqrt{e} \sqrt{d+e x^2}+e x\right )}{2 e^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.011, size = 112, normalized size = 1.3 \[{\frac{ax}{d}{\frac{1}{\sqrt{e{x}^{2}+d}}}}-{\frac{bx}{e}{\frac{1}{\sqrt{e{x}^{2}+d}}}}+{b\ln \left ( x\sqrt{e}+\sqrt{e{x}^{2}+d} \right ){e}^{-{\frac{3}{2}}}}+{\frac{c{x}^{3}}{2\,e}{\frac{1}{\sqrt{e{x}^{2}+d}}}}+{\frac{3\,cdx}{2\,{e}^{2}}{\frac{1}{\sqrt{e{x}^{2}+d}}}}-{\frac{3\,cd}{2}\ln \left ( x\sqrt{e}+\sqrt{e{x}^{2}+d} \right ){e}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*x/d/(e*x^2+d)^(1/2)-b*x/e/(e*x^2+d)^(1/2)+b/e^(3/2)*ln(x*e^(1/2)+(e*x^2+d)^(1/
2))+1/2*c*x^3/e/(e*x^2+d)^(1/2)+3/2*c*d/e^2*x/(e*x^2+d)^(1/2)-3/2*c*d/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)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.298928, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (c d e x^{3} +{\left (3 \, c d^{2} - 2 \, b d e + 2 \, a e^{2}\right )} x\right )} \sqrt{e x^{2} + d} \sqrt{e} -{\left (3 \, c d^{3} - 2 \, b d^{2} e +{\left (3 \, c d^{2} e - 2 \, b d e^{2}\right )} x^{2}\right )} \log \left (-2 \, \sqrt{e x^{2} + d} e x -{\left (2 \, e x^{2} + d\right )} \sqrt{e}\right )}{4 \,{\left (d e^{3} x^{2} + d^{2} e^{2}\right )} \sqrt{e}}, \frac{{\left (c d e x^{3} +{\left (3 \, c d^{2} - 2 \, b d e + 2 \, a e^{2}\right )} x\right )} \sqrt{e x^{2} + d} \sqrt{-e} -{\left (3 \, c d^{3} - 2 \, b d^{2} e +{\left (3 \, c d^{2} e - 2 \, b d e^{2}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{-e} x}{\sqrt{e x^{2} + d}}\right )}{2 \,{\left (d e^{3} x^{2} + d^{2} e^{2}\right )} \sqrt{-e}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 22.1744, size = 134, normalized size = 1.51 \[ \frac{a x}{d^{\frac{3}{2}} \sqrt{1 + \frac{e x^{2}}{d}}} + b \left (\frac{\operatorname{asinh}{\left (\frac{\sqrt{e} x}{\sqrt{d}} \right )}}{e^{\frac{3}{2}}} - \frac{x}{\sqrt{d} e \sqrt{1 + \frac{e x^{2}}{d}}}\right ) + c \left (\frac{3 \sqrt{d} x}{2 e^{2} \sqrt{1 + \frac{e x^{2}}{d}}} - \frac{3 d \operatorname{asinh}{\left (\frac{\sqrt{e} x}{\sqrt{d}} \right )}}{2 e^{\frac{5}{2}}} + \frac{x^{3}}{2 \sqrt{d} e \sqrt{1 + \frac{e x^{2}}{d}}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*x/(d**(3/2)*sqrt(1 + e*x**2/d)) + b*(asinh(sqrt(e)*x/sqrt(d))/e**(3/2) - x/(sq
rt(d)*e*sqrt(1 + e*x**2/d))) + c*(3*sqrt(d)*x/(2*e**2*sqrt(1 + e*x**2/d)) - 3*d*
asinh(sqrt(e)*x/sqrt(d))/(2*e**(5/2)) + x**3/(2*sqrt(d)*e*sqrt(1 + e*x**2/d)))

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GIAC/XCAS [A]  time = 0.26993, size = 108, normalized size = 1.21 \[ \frac{1}{2} \,{\left (3 \, c d - 2 \, b e\right )} e^{\left (-\frac{5}{2}\right )}{\rm ln}\left ({\left | -x e^{\frac{1}{2}} + \sqrt{x^{2} e + d} \right |}\right ) + \frac{{\left (c x^{2} e^{\left (-1\right )} + \frac{{\left (3 \, c d^{2} e - 2 \, b d e^{2} + 2 \, a e^{3}\right )} e^{\left (-3\right )}}{d}\right )} x}{2 \, \sqrt{x^{2} e + d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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