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

Optimal. Leaf size=157 \[ \frac {b \log \left (i \sqrt {a^2 x^4+b}+i \sqrt {2} \sqrt {a} x \sqrt {\sqrt {a^2 x^4+b}+a x^2}+i a x^2\right )}{8 \sqrt {2} a^{3/2}}-\frac {i \left (2 i a x^4 \sqrt {a^2 x^4+b}+i x^2 \left (2 a^2 x^4-b\right )\right )}{8 a x \sqrt {\sqrt {a^2 x^4+b}+a x^2}} \]

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Rubi [F]  time = 0.19, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int x^2 \sqrt {a x^2+\sqrt {b+a^2 x^4}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[x^2*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]],x]

[Out]

Defer[Int][x^2*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]], x]

Rubi steps

\begin {align*} \int x^2 \sqrt {a x^2+\sqrt {b+a^2 x^4}} \, dx &=\int x^2 \sqrt {a x^2+\sqrt {b+a^2 x^4}} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.10, size = 0, normalized size = 0.00 \begin {gather*} \int x^2 \sqrt {a x^2+\sqrt {b+a^2 x^4}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[x^2*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]],x]

[Out]

Integrate[x^2*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]], x]

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IntegrateAlgebraic [A]  time = 0.56, size = 157, normalized size = 1.00 \begin {gather*} -\frac {i \left (2 i a x^4 \sqrt {b+a^2 x^4}+i x^2 \left (-b+2 a^2 x^4\right )\right )}{8 a x \sqrt {a x^2+\sqrt {b+a^2 x^4}}}+\frac {b \log \left (i a x^2+i \sqrt {b+a^2 x^4}+i \sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}\right )}{8 \sqrt {2} a^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^2*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]],x]

[Out]

((-1/8*I)*((2*I)*a*x^4*Sqrt[b + a^2*x^4] + I*x^2*(-b + 2*a^2*x^4)))/(a*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]) + (b
*Log[I*a*x^2 + I*Sqrt[b + a^2*x^4] + I*Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]])/(8*Sqrt[2]*a^(3/2))

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fricas [A]  time = 1.82, size = 236, normalized size = 1.50 \begin {gather*} \left [\frac {\sqrt {2} \sqrt {a} b \log \left (4 \, a^{2} x^{4} + 4 \, \sqrt {a^{2} x^{4} + b} a x^{2} + 2 \, {\left (\sqrt {2} a^{\frac {3}{2}} x^{3} + \sqrt {2} \sqrt {a^{2} x^{4} + b} \sqrt {a} x\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}} + b\right ) + 4 \, {\left (3 \, a^{2} x^{3} - \sqrt {a^{2} x^{4} + b} a x\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{32 \, a^{2}}, -\frac {\sqrt {2} \sqrt {-a} b \arctan \left (\frac {\sqrt {2} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}} \sqrt {-a}}{2 \, a x}\right ) - 2 \, {\left (3 \, a^{2} x^{3} - \sqrt {a^{2} x^{4} + b} a x\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{16 \, a^{2}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/32*(sqrt(2)*sqrt(a)*b*log(4*a^2*x^4 + 4*sqrt(a^2*x^4 + b)*a*x^2 + 2*(sqrt(2)*a^(3/2)*x^3 + sqrt(2)*sqrt(a^2
*x^4 + b)*sqrt(a)*x)*sqrt(a*x^2 + sqrt(a^2*x^4 + b)) + b) + 4*(3*a^2*x^3 - sqrt(a^2*x^4 + b)*a*x)*sqrt(a*x^2 +
sqrt(a^2*x^4 + b)))/a^2, -1/16*(sqrt(2)*sqrt(-a)*b*arctan(1/2*sqrt(2)*sqrt(a*x^2 + sqrt(a^2*x^4 + b))*sqrt(-a
)/(a*x)) - 2*(3*a^2*x^3 - sqrt(a^2*x^4 + b)*a*x)*sqrt(a*x^2 + sqrt(a^2*x^4 + b)))/a^2]

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> An error occurred running a Giac command:INPUT:sage2OUTPUT:Unable to divide,
perhaps due to rounding error%%%{%%%{%%{[-2,%%%{4,[1]%%%}]:[1,0,%%%{1,[1]%%%}]%%},[0]%%%}/%%%{%%{[2,1]:[1,0,%
%%{1,[1]%%%}]%%},[0]%%%},[1]%%%}+%%%{%%%{1,[1]%%%},[0]%%%} / %%%{%%%{1/%%%{%%{[2,1]:[1,0,%%%{1,[1]%%%}]%%},[0]
%%%},[0]%%%},[0]%%%} Error: Bad Argument ValueUnable to divide, perhaps due to rounding error%%%{%%%{%%{[2,%%%
{4,[1]%%%}]:[1,0,%%%{1,[1]%%%}]%%},[0]%%%}/%%%{%%{[-2,1]:[1,0,%%%{1,[1]%%%}]%%},[0]%%%},[1]%%%}+%%%{%%%{1,[1]%
%%},[0]%%%} / %%%{%%%{1/%%%{%%{[-2,1]:[1,0,%%%{1,[1]%%%}]%%},[0]%%%},[0]%%%},[0]%%%} Error: Bad Argument Value
Evaluation time: 1integrate((4*a^2*x^6*sqrt(sqrt(a^2*x^4+b)+a*x^2)-4*a^2*x^6-4*a*x^4*sqrt(sqrt(a^2*x^4+b)+a*x^
2)+4*b*x^2*sqrt(sqrt(a^2*x^4+b)+a*x^2)-4*b*x^2+x^2*sqrt(sqrt(a^2*x^4+b)+a*x^2)+4*x^2*(a^2*x^4+b))/(4*a^2*x^4-4
*a*x^2+4*b+1),x)

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maple [F]  time = 180.00, size = 0, normalized size = 0.00 \[\int x^{2} \sqrt {a \,x^{2}+\sqrt {a^{2} x^{4}+b}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}} x^{2}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^2\,\sqrt {\sqrt {a^2\,x^4+b}+a\,x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{2} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**2*sqrt(a*x**2 + sqrt(a**2*x**4 + b)), x)

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