3.2 \(\int \sqrt [4]{a^2+2 a b x^2+b^2 x^4} \, dx\)

Optimal. Leaf size=91 \[ \frac{1}{2} x \sqrt [4]{a^2+2 a b x^2+b^2 x^4}+\frac{\sqrt{a} \sqrt [4]{a^2+2 a b x^2+b^2 x^4} \sinh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{b} \sqrt{\frac{b x^2}{a}+1}} \]

[Out]

(x*(a^2 + 2*a*b*x^2 + b^2*x^4)^(1/4))/2 + (Sqrt[a]*(a^2 + 2*a*b*x^2 + b^2*x^4)^(
1/4)*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/(2*Sqrt[b]*Sqrt[1 + (b*x^2)/a])

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Rubi [A]  time = 0.0578536, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{1}{2} x \sqrt [4]{a^2+2 a b x^2+b^2 x^4}+\frac{\sqrt{a} \sqrt [4]{a^2+2 a b x^2+b^2 x^4} \sinh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{b} \sqrt{\frac{b x^2}{a}+1}} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^(1/4),x]

[Out]

(x*(a^2 + 2*a*b*x^2 + b^2*x^4)^(1/4))/2 + (Sqrt[a]*(a^2 + 2*a*b*x^2 + b^2*x^4)^(
1/4)*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/(2*Sqrt[b]*Sqrt[1 + (b*x^2)/a])

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Rubi in Sympy [A]  time = 21.9298, size = 99, normalized size = 1.09 \[ \frac{\sqrt{2} a \sqrt [4]{a^{2} + 2 a b x^{2} + b^{2} x^{4}} \operatorname{atanh}{\left (\frac{\sqrt{2} b x}{\sqrt{2 a b + 2 b^{2} x^{2}}} \right )}}{2 \sqrt{2 a b + 2 b^{2} x^{2}}} + \frac{x \sqrt [4]{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(2)*a*(a**2 + 2*a*b*x**2 + b**2*x**4)**(1/4)*atanh(sqrt(2)*b*x/sqrt(2*a*b +
2*b**2*x**2))/(2*sqrt(2*a*b + 2*b**2*x**2)) + x*(a**2 + 2*a*b*x**2 + b**2*x**4)*
*(1/4)/2

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Mathematica [A]  time = 0.0577432, size = 59, normalized size = 0.65 \[ \frac{1}{2} \sqrt [4]{\left (a+b x^2\right )^2} \left (\frac{a \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{\sqrt{b} \sqrt{a+b x^2}}+x\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(1/4),x]

[Out]

(((a + b*x^2)^2)^(1/4)*(x + (a*Log[b*x + Sqrt[b]*Sqrt[a + b*x^2]])/(Sqrt[b]*Sqrt
[a + b*x^2])))/2

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Maple [A]  time = 0.02, size = 58, normalized size = 0.6 \[{\frac{x}{2}\sqrt [4]{ \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{a}{2}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ) \sqrt [4]{ \left ( b{x}^{2}+a \right ) ^{2}}{\frac{1}{\sqrt{b}}}{\frac{1}{\sqrt{b{x}^{2}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*x*((b*x^2+a)^2)^(1/4)+1/2*a*ln(x*b^(1/2)+(b*x^2+a)^(1/2))/b^(1/2)*((b*x^2+a)
^2)^(1/4)/(b*x^2+a)^(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((b^2*x^4 + 2*a*b*x^2 + a^2)^(1/4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.287826, size = 1, normalized size = 0.01 \[ \left [\frac{a \sqrt{b} \log \left (-2 \,{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac{1}{4}} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right ) + 2 \,{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac{1}{4}} b x}{4 \, b}, -\frac{a \sqrt{-b} \arctan \left (\frac{\sqrt{-b} x}{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac{1}{4}}}\right ) -{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac{1}{4}} b x}{2 \, b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(1/4),x, algorithm="fricas")

[Out]

[1/4*(a*sqrt(b)*log(-2*(b^2*x^4 + 2*a*b*x^2 + a^2)^(1/4)*b*x - (2*b*x^2 + a)*sqr
t(b)) + 2*(b^2*x^4 + 2*a*b*x^2 + a^2)^(1/4)*b*x)/b, -1/2*(a*sqrt(-b)*arctan(sqrt
(-b)*x/(b^2*x^4 + 2*a*b*x^2 + a^2)^(1/4)) - (b^2*x^4 + 2*a*b*x^2 + a^2)^(1/4)*b*
x)/b]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((a**2 + 2*a*b*x**2 + b**2*x**4)**(1/4), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac{1}{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(1/4),x, algorithm="giac")

[Out]

integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(1/4), x)