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

Optimal. Leaf size=105 \[ -\frac {\sqrt {\sqrt {a^2 x^4+b}+a x^2}}{x}+\frac {\sqrt {a} \log \left (i \sqrt {a^2 x^4+b}+i \sqrt {2} \sqrt {a} x \sqrt {\sqrt {a^2 x^4+b}+a x^2}+i a x^2\right )}{\sqrt {2}} \]

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Rubi [F]  time = 0.20, 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 \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [F]  time = 0.09, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.54, size = 105, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{x}+\frac {\sqrt {a} \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 )}{\sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\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((a*x^2+(a^2*x^4+b)^(1/2))^(1/2)/x^2,x, algorithm="giac")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\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((a*x^2+(a^2*x^4+b)^(1/2))^(1/2)/x^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 \frac {\sqrt {\sqrt {a^2\,x^4+b}+a\,x^2}}{x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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