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

Optimal. Leaf size=97 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right ) \left (8 a e^2-4 b d e+3 c d^2\right )}{8 e^{5/2}}-\frac{x \sqrt{d+e x^2} (3 c d-4 b e)}{8 e^2}+\frac{c x^3 \sqrt{d+e x^2}}{4 e} \]

[Out]

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

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Rubi [A]  time = 0.13211, antiderivative size = 97, 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 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right ) \left (8 a e^2-4 b d e+3 c d^2\right )}{8 e^{5/2}}-\frac{x \sqrt{d+e x^2} (3 c d-4 b e)}{8 e^2}+\frac{c x^3 \sqrt{d+e x^2}}{4 e} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 16.893, size = 90, normalized size = 0.93 \[ \frac{c x^{3} \sqrt{d + e x^{2}}}{4 e} + \frac{x \sqrt{d + e x^{2}} \left (4 b e - 3 c d\right )}{8 e^{2}} + \frac{\left (8 a e^{2} - 4 b d e + 3 c d^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{e} x}{\sqrt{d + e x^{2}}} \right )}}{8 e^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c*x**3*sqrt(d + e*x**2)/(4*e) + x*sqrt(d + e*x**2)*(4*b*e - 3*c*d)/(8*e**2) + (8
*a*e**2 - 4*b*d*e + 3*c*d**2)*atanh(sqrt(e)*x/sqrt(d + e*x**2))/(8*e**(5/2))

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Mathematica [A]  time = 0.0928843, size = 85, normalized size = 0.88 \[ \frac{\log \left (\sqrt{e} \sqrt{d+e x^2}+e x\right ) \left (8 a e^2-4 b d e+3 c d^2\right )+\sqrt{e} x \sqrt{d+e x^2} \left (4 b e-3 c d+2 c e x^2\right )}{8 e^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.011, size = 122, normalized size = 1.3 \[{a\ln \left ( x\sqrt{e}+\sqrt{e{x}^{2}+d} \right ){\frac{1}{\sqrt{e}}}}+{\frac{bx}{2\,e}\sqrt{e{x}^{2}+d}}-{\frac{bd}{2}\ln \left ( x\sqrt{e}+\sqrt{e{x}^{2}+d} \right ){e}^{-{\frac{3}{2}}}}+{\frac{c{x}^{3}}{4\,e}\sqrt{e{x}^{2}+d}}-{\frac{3\,cdx}{8\,{e}^{2}}\sqrt{e{x}^{2}+d}}+{\frac{3\,c{d}^{2}}{8}\ln \left ( x\sqrt{e}+\sqrt{e{x}^{2}+d} \right ){e}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*ln(x*e^(1/2)+(e*x^2+d)^(1/2))/e^(1/2)+1/2*b*x/e*(e*x^2+d)^(1/2)-1/2*b*d/e^(3/2
)*ln(x*e^(1/2)+(e*x^2+d)^(1/2))+1/4*c*x^3*(e*x^2+d)^(1/2)/e-3/8*c*d/e^2*x*(e*x^2
+d)^(1/2)+3/8*c*d^2/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)/sqrt(e*x^2 + d),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.2953, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (2 \, c e x^{3} -{\left (3 \, c d - 4 \, b e\right )} x\right )} \sqrt{e x^{2} + d} \sqrt{e} +{\left (3 \, c d^{2} - 4 \, b d e + 8 \, a e^{2}\right )} \log \left (-2 \, \sqrt{e x^{2} + d} e x -{\left (2 \, e x^{2} + d\right )} \sqrt{e}\right )}{16 \, e^{\frac{5}{2}}}, \frac{{\left (2 \, c e x^{3} -{\left (3 \, c d - 4 \, b e\right )} x\right )} \sqrt{e x^{2} + d} \sqrt{-e} +{\left (3 \, c d^{2} - 4 \, b d e + 8 \, a e^{2}\right )} \arctan \left (\frac{\sqrt{-e} x}{\sqrt{e x^{2} + d}}\right )}{8 \, \sqrt{-e} e^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/16*(2*(2*c*e*x^3 - (3*c*d - 4*b*e)*x)*sqrt(e*x^2 + d)*sqrt(e) + (3*c*d^2 - 4*
b*d*e + 8*a*e^2)*log(-2*sqrt(e*x^2 + d)*e*x - (2*e*x^2 + d)*sqrt(e)))/e^(5/2), 1
/8*((2*c*e*x^3 - (3*c*d - 4*b*e)*x)*sqrt(e*x^2 + d)*sqrt(-e) + (3*c*d^2 - 4*b*d*
e + 8*a*e^2)*arctan(sqrt(-e)*x/sqrt(e*x^2 + d)))/(sqrt(-e)*e^2)]

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Sympy [A]  time = 19.5, size = 230, normalized size = 2.37 \[ a \left (\begin{cases} \frac{\sqrt{- \frac{d}{e}} \operatorname{asin}{\left (x \sqrt{- \frac{e}{d}} \right )}}{\sqrt{d}} & \text{for}\: d > 0 \wedge e < 0 \\\frac{\sqrt{\frac{d}{e}} \operatorname{asinh}{\left (x \sqrt{\frac{e}{d}} \right )}}{\sqrt{d}} & \text{for}\: d > 0 \wedge e > 0 \\\frac{\sqrt{- \frac{d}{e}} \operatorname{acosh}{\left (x \sqrt{- \frac{e}{d}} \right )}}{\sqrt{- d}} & \text{for}\: e > 0 \wedge d < 0 \end{cases}\right ) + \frac{b \sqrt{d} x \sqrt{1 + \frac{e x^{2}}{d}}}{2 e} - \frac{b d \operatorname{asinh}{\left (\frac{\sqrt{e} x}{\sqrt{d}} \right )}}{2 e^{\frac{3}{2}}} - \frac{3 c d^{\frac{3}{2}} x}{8 e^{2} \sqrt{1 + \frac{e x^{2}}{d}}} - \frac{c \sqrt{d} x^{3}}{8 e \sqrt{1 + \frac{e x^{2}}{d}}} + \frac{3 c d^{2} \operatorname{asinh}{\left (\frac{\sqrt{e} x}{\sqrt{d}} \right )}}{8 e^{\frac{5}{2}}} + \frac{c x^{5}}{4 \sqrt{d} \sqrt{1 + \frac{e x^{2}}{d}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*Piecewise((sqrt(-d/e)*asin(x*sqrt(-e/d))/sqrt(d), (d > 0) & (e < 0)), (sqrt(d/
e)*asinh(x*sqrt(e/d))/sqrt(d), (d > 0) & (e > 0)), (sqrt(-d/e)*acosh(x*sqrt(-e/d
))/sqrt(-d), (e > 0) & (d < 0))) + b*sqrt(d)*x*sqrt(1 + e*x**2/d)/(2*e) - b*d*as
inh(sqrt(e)*x/sqrt(d))/(2*e**(3/2)) - 3*c*d**(3/2)*x/(8*e**2*sqrt(1 + e*x**2/d))
 - c*sqrt(d)*x**3/(8*e*sqrt(1 + e*x**2/d)) + 3*c*d**2*asinh(sqrt(e)*x/sqrt(d))/(
8*e**(5/2)) + c*x**5/(4*sqrt(d)*sqrt(1 + e*x**2/d))

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GIAC/XCAS [A]  time = 0.268649, size = 107, normalized size = 1.1 \[ -\frac{1}{8} \,{\left (3 \, c d^{2} - 4 \, b d e + 8 \, a e^{2}\right )} e^{\left (-\frac{5}{2}\right )}{\rm ln}\left ({\left | -x e^{\frac{1}{2}} + \sqrt{x^{2} e + d} \right |}\right ) + \frac{1}{8} \,{\left (2 \, c x^{2} e^{\left (-1\right )} -{\left (3 \, c d e - 4 \, b e^{2}\right )} e^{\left (-3\right )}\right )} \sqrt{x^{2} e + d} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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