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

Optimal. Leaf size=132 \[ \frac{x \sqrt{d+e x^2} \left (8 a e^2-2 b d e+c d^2\right )}{16 e^2}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right ) \left (8 a e^2-2 b d e+c d^2\right )}{16 e^{5/2}}-\frac{x \left (d+e x^2\right )^{3/2} (c d-2 b e)}{8 e^2}+\frac{c x^3 \left (d+e x^2\right )^{3/2}}{6 e} \]

[Out]

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

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Rubi [A]  time = 0.211031, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{x \sqrt{d+e x^2} \left (8 a e^2-2 b d e+c d^2\right )}{16 e^2}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right ) \left (8 a e^2-2 b d e+c d^2\right )}{16 e^{5/2}}-\frac{x \left (d+e x^2\right )^{3/2} (c d-2 b e)}{8 e^2}+\frac{c x^3 \left (d+e x^2\right )^{3/2}}{6 e} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 19.8546, size = 124, normalized size = 0.94 \[ \frac{c x^{3} \left (d + e x^{2}\right )^{\frac{3}{2}}}{6 e} + \frac{d \left (8 a e^{2} - 2 b d e + c d^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{e} x}{\sqrt{d + e x^{2}}} \right )}}{16 e^{\frac{5}{2}}} + \frac{x \left (d + e x^{2}\right )^{\frac{3}{2}} \left (2 b e - c d\right )}{8 e^{2}} + \frac{x \sqrt{d + e x^{2}} \left (8 a e^{2} - 2 b d e + c d^{2}\right )}{16 e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c*x**3*(d + e*x**2)**(3/2)/(6*e) + d*(8*a*e**2 - 2*b*d*e + c*d**2)*atanh(sqrt(e)
*x/sqrt(d + e*x**2))/(16*e**(5/2)) + x*(d + e*x**2)**(3/2)*(2*b*e - c*d)/(8*e**2
) + x*sqrt(d + e*x**2)*(8*a*e**2 - 2*b*d*e + c*d**2)/(16*e**2)

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Mathematica [A]  time = 0.126588, size = 112, normalized size = 0.85 \[ \frac{3 d \log \left (\sqrt{e} \sqrt{d+e x^2}+e x\right ) \left (8 a e^2-2 b d e+c d^2\right )+\sqrt{e} x \sqrt{d+e x^2} \left (6 e \left (4 a e+b \left (d+2 e x^2\right )\right )+c \left (-3 d^2+2 d e x^2+8 e^2 x^4\right )\right )}{48 e^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.012, size = 175, normalized size = 1.3 \[{\frac{ax}{2}\sqrt{e{x}^{2}+d}}+{\frac{ad}{2}\ln \left ( x\sqrt{e}+\sqrt{e{x}^{2}+d} \right ){\frac{1}{\sqrt{e}}}}+{\frac{bx}{4\,e} \left ( e{x}^{2}+d \right ) ^{{\frac{3}{2}}}}-{\frac{bdx}{8\,e}\sqrt{e{x}^{2}+d}}-{\frac{b{d}^{2}}{8}\ln \left ( x\sqrt{e}+\sqrt{e{x}^{2}+d} \right ){e}^{-{\frac{3}{2}}}}+{\frac{c{x}^{3}}{6\,e} \left ( e{x}^{2}+d \right ) ^{{\frac{3}{2}}}}-{\frac{cdx}{8\,{e}^{2}} \left ( e{x}^{2}+d \right ) ^{{\frac{3}{2}}}}+{\frac{c{d}^{2}x}{16\,{e}^{2}}\sqrt{e{x}^{2}+d}}+{\frac{c{d}^{3}}{16}\ln \left ( x\sqrt{e}+\sqrt{e{x}^{2}+d} \right ){e}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*a*x*(e*x^2+d)^(1/2)+1/2*a*d/e^(1/2)*ln(x*e^(1/2)+(e*x^2+d)^(1/2))+1/4*b*x*(e
*x^2+d)^(3/2)/e-1/8*b*d/e*x*(e*x^2+d)^(1/2)-1/8*b*d^2/e^(3/2)*ln(x*e^(1/2)+(e*x^
2+d)^(1/2))+1/6*c*x^3*(e*x^2+d)^(3/2)/e-1/8*c*d/e^2*x*(e*x^2+d)^(3/2)+1/16*c*d^2
/e^2*x*(e*x^2+d)^(1/2)+1/16*c*d^3/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 = 0.328305, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (8 \, c e^{2} x^{5} + 2 \,{\left (c d e + 6 \, b e^{2}\right )} x^{3} - 3 \,{\left (c d^{2} - 2 \, b d e - 8 \, a e^{2}\right )} x\right )} \sqrt{e x^{2} + d} \sqrt{e} + 3 \,{\left (c d^{3} - 2 \, b d^{2} e + 8 \, a d e^{2}\right )} \log \left (-2 \, \sqrt{e x^{2} + d} e x -{\left (2 \, e x^{2} + d\right )} \sqrt{e}\right )}{96 \, e^{\frac{5}{2}}}, \frac{{\left (8 \, c e^{2} x^{5} + 2 \,{\left (c d e + 6 \, b e^{2}\right )} x^{3} - 3 \,{\left (c d^{2} - 2 \, b d e - 8 \, a e^{2}\right )} x\right )} \sqrt{e x^{2} + d} \sqrt{-e} + 3 \,{\left (c d^{3} - 2 \, b d^{2} e + 8 \, a d e^{2}\right )} \arctan \left (\frac{\sqrt{-e} x}{\sqrt{e x^{2} + d}}\right )}{48 \, \sqrt{-e} e^{2}}\right ] \]

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]

[1/96*(2*(8*c*e^2*x^5 + 2*(c*d*e + 6*b*e^2)*x^3 - 3*(c*d^2 - 2*b*d*e - 8*a*e^2)*
x)*sqrt(e*x^2 + d)*sqrt(e) + 3*(c*d^3 - 2*b*d^2*e + 8*a*d*e^2)*log(-2*sqrt(e*x^2
+ d)*e*x - (2*e*x^2 + d)*sqrt(e)))/e^(5/2), 1/48*((8*c*e^2*x^5 + 2*(c*d*e + 6*b
*e^2)*x^3 - 3*(c*d^2 - 2*b*d*e - 8*a*e^2)*x)*sqrt(e*x^2 + d)*sqrt(-e) + 3*(c*d^3
- 2*b*d^2*e + 8*a*d*e^2)*arctan(sqrt(-e)*x/sqrt(e*x^2 + d)))/(sqrt(-e)*e^2)]

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Sympy [A]  time = 33.4551, size = 272, normalized size = 2.06 \[ \frac{a \sqrt{d} x \sqrt{1 + \frac{e x^{2}}{d}}}{2} + \frac{a d \operatorname{asinh}{\left (\frac{\sqrt{e} x}{\sqrt{d}} \right )}}{2 \sqrt{e}} + \frac{b d^{\frac{3}{2}} x}{8 e \sqrt{1 + \frac{e x^{2}}{d}}} + \frac{3 b \sqrt{d} x^{3}}{8 \sqrt{1 + \frac{e x^{2}}{d}}} - \frac{b d^{2} \operatorname{asinh}{\left (\frac{\sqrt{e} x}{\sqrt{d}} \right )}}{8 e^{\frac{3}{2}}} + \frac{b e x^{5}}{4 \sqrt{d} \sqrt{1 + \frac{e x^{2}}{d}}} - \frac{c d^{\frac{5}{2}} x}{16 e^{2} \sqrt{1 + \frac{e x^{2}}{d}}} - \frac{c d^{\frac{3}{2}} x^{3}}{48 e \sqrt{1 + \frac{e x^{2}}{d}}} + \frac{5 c \sqrt{d} x^{5}}{24 \sqrt{1 + \frac{e x^{2}}{d}}} + \frac{c d^{3} \operatorname{asinh}{\left (\frac{\sqrt{e} x}{\sqrt{d}} \right )}}{16 e^{\frac{5}{2}}} + \frac{c e x^{7}}{6 \sqrt{d} \sqrt{1 + \frac{e x^{2}}{d}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*sqrt(d)*x*sqrt(1 + e*x**2/d)/2 + a*d*asinh(sqrt(e)*x/sqrt(d))/(2*sqrt(e)) + b*
d**(3/2)*x/(8*e*sqrt(1 + e*x**2/d)) + 3*b*sqrt(d)*x**3/(8*sqrt(1 + e*x**2/d)) -
b*d**2*asinh(sqrt(e)*x/sqrt(d))/(8*e**(3/2)) + b*e*x**5/(4*sqrt(d)*sqrt(1 + e*x*
*2/d)) - c*d**(5/2)*x/(16*e**2*sqrt(1 + e*x**2/d)) - c*d**(3/2)*x**3/(48*e*sqrt(
1 + e*x**2/d)) + 5*c*sqrt(d)*x**5/(24*sqrt(1 + e*x**2/d)) + c*d**3*asinh(sqrt(e)
*x/sqrt(d))/(16*e**(5/2)) + c*e*x**7/(6*sqrt(d)*sqrt(1 + e*x**2/d))

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GIAC/XCAS [A]  time = 0.266153, size = 143, normalized size = 1.08 \[ -\frac{1}{16} \,{\left (c d^{3} - 2 \, b d^{2} e + 8 \, a d e^{2}\right )} e^{\left (-\frac{5}{2}\right )}{\rm ln}\left ({\left | -x e^{\frac{1}{2}} + \sqrt{x^{2} e + d} \right |}\right ) + \frac{1}{48} \,{\left (2 \,{\left (4 \, c x^{2} +{\left (c d e^{3} + 6 \, b e^{4}\right )} e^{\left (-4\right )}\right )} x^{2} - 3 \,{\left (c d^{2} e^{2} - 2 \, b d e^{3} - 8 \, a e^{4}\right )} e^{\left (-4\right )}\right )} \sqrt{x^{2} e + d} x \]

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]

-1/16*(c*d^3 - 2*b*d^2*e + 8*a*d*e^2)*e^(-5/2)*ln(abs(-x*e^(1/2) + sqrt(x^2*e +
d))) + 1/48*(2*(4*c*x^2 + (c*d*e^3 + 6*b*e^4)*e^(-4))*x^2 - 3*(c*d^2*e^2 - 2*b*d
*e^3 - 8*a*e^4)*e^(-4))*sqrt(x^2*e + d)*x