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

Optimal. Leaf size=211 \[ \frac {2 x \left (a x^2+2 b^2\right )}{\left (a x^2+b^2\right ) \sqrt {\sqrt {a x^2+b^2}+b}}-\frac {b x}{\sqrt {a x^2+b^2} \sqrt {\sqrt {a x^2+b^2}+b}}-\frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b} \sqrt {\sqrt {a x^2+b^2}+b}}\right )}{\sqrt {a}}-\frac {2 \sqrt {2} \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt {b} \sqrt {\sqrt {a x^2+b^2}+b}}-\frac {\sqrt {\sqrt {a x^2+b^2}+b}}{\sqrt {2} \sqrt {b}}\right )}{\sqrt {a}} \]

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Rubi [F]  time = 2.48, 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 {\left (-b^2+a x^2\right )^2}{\left (b^2+a x^2\right )^2 \sqrt {b+\sqrt {b^2+a x^2}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

Defer[Int][1/Sqrt[b + Sqrt[b^2 + a*x^2]], x] - b*Defer[Int][1/((b - Sqrt[-a]*x)*Sqrt[b + Sqrt[b^2 + a*x^2]]),
x] - b*Defer[Int][1/((b + Sqrt[-a]*x)*Sqrt[b + Sqrt[b^2 + a*x^2]]), x] - a*b^2*Defer[Int][1/((Sqrt[-a]*b - a*x
)^2*Sqrt[b + Sqrt[b^2 + a*x^2]]), x] - a*b^2*Defer[Int][1/((Sqrt[-a]*b + a*x)^2*Sqrt[b + Sqrt[b^2 + a*x^2]]),
x]

Rubi steps

\begin {align*} \int \frac {\left (-b^2+a x^2\right )^2}{\left (b^2+a x^2\right )^2 \sqrt {b+\sqrt {b^2+a x^2}}} \, dx &=\int \left (\frac {1}{\sqrt {b+\sqrt {b^2+a x^2}}}+\frac {4 b^4}{\left (b^2+a x^2\right )^2 \sqrt {b+\sqrt {b^2+a x^2}}}-\frac {4 b^2}{\left (b^2+a x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}}}\right ) \, dx\\ &=-\left (\left (4 b^2\right ) \int \frac {1}{\left (b^2+a x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}}} \, dx\right )+\left (4 b^4\right ) \int \frac {1}{\left (b^2+a x^2\right )^2 \sqrt {b+\sqrt {b^2+a x^2}}} \, dx+\int \frac {1}{\sqrt {b+\sqrt {b^2+a x^2}}} \, dx\\ &=-\left (\left (4 b^2\right ) \int \left (\frac {1}{2 b \left (b-\sqrt {-a} x\right ) \sqrt {b+\sqrt {b^2+a x^2}}}+\frac {1}{2 b \left (b+\sqrt {-a} x\right ) \sqrt {b+\sqrt {b^2+a x^2}}}\right ) \, dx\right )+\left (4 b^4\right ) \int \left (-\frac {a}{4 b^2 \left (\sqrt {-a} b-a x\right )^2 \sqrt {b+\sqrt {b^2+a x^2}}}-\frac {a}{4 b^2 \left (\sqrt {-a} b+a x\right )^2 \sqrt {b+\sqrt {b^2+a x^2}}}-\frac {a}{2 b^2 \left (-a b^2-a^2 x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}}}\right ) \, dx+\int \frac {1}{\sqrt {b+\sqrt {b^2+a x^2}}} \, dx\\ &=-\left ((2 b) \int \frac {1}{\left (b-\sqrt {-a} x\right ) \sqrt {b+\sqrt {b^2+a x^2}}} \, dx\right )-(2 b) \int \frac {1}{\left (b+\sqrt {-a} x\right ) \sqrt {b+\sqrt {b^2+a x^2}}} \, dx-\left (a b^2\right ) \int \frac {1}{\left (\sqrt {-a} b-a x\right )^2 \sqrt {b+\sqrt {b^2+a x^2}}} \, dx-\left (a b^2\right ) \int \frac {1}{\left (\sqrt {-a} b+a x\right )^2 \sqrt {b+\sqrt {b^2+a x^2}}} \, dx-\left (2 a b^2\right ) \int \frac {1}{\left (-a b^2-a^2 x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}}} \, dx+\int \frac {1}{\sqrt {b+\sqrt {b^2+a x^2}}} \, dx\\ &=-\left ((2 b) \int \frac {1}{\left (b-\sqrt {-a} x\right ) \sqrt {b+\sqrt {b^2+a x^2}}} \, dx\right )-(2 b) \int \frac {1}{\left (b+\sqrt {-a} x\right ) \sqrt {b+\sqrt {b^2+a x^2}}} \, dx-\left (a b^2\right ) \int \frac {1}{\left (\sqrt {-a} b-a x\right )^2 \sqrt {b+\sqrt {b^2+a x^2}}} \, dx-\left (a b^2\right ) \int \frac {1}{\left (\sqrt {-a} b+a x\right )^2 \sqrt {b+\sqrt {b^2+a x^2}}} \, dx-\left (2 a b^2\right ) \int \left (-\frac {1}{2 a b \left (b-\sqrt {-a} x\right ) \sqrt {b+\sqrt {b^2+a x^2}}}-\frac {1}{2 a b \left (b+\sqrt {-a} x\right ) \sqrt {b+\sqrt {b^2+a x^2}}}\right ) \, dx+\int \frac {1}{\sqrt {b+\sqrt {b^2+a x^2}}} \, dx\\ &=b \int \frac {1}{\left (b-\sqrt {-a} x\right ) \sqrt {b+\sqrt {b^2+a x^2}}} \, dx+b \int \frac {1}{\left (b+\sqrt {-a} x\right ) \sqrt {b+\sqrt {b^2+a x^2}}} \, dx-(2 b) \int \frac {1}{\left (b-\sqrt {-a} x\right ) \sqrt {b+\sqrt {b^2+a x^2}}} \, dx-(2 b) \int \frac {1}{\left (b+\sqrt {-a} x\right ) \sqrt {b+\sqrt {b^2+a x^2}}} \, dx-\left (a b^2\right ) \int \frac {1}{\left (\sqrt {-a} b-a x\right )^2 \sqrt {b+\sqrt {b^2+a x^2}}} \, dx-\left (a b^2\right ) \int \frac {1}{\left (\sqrt {-a} b+a x\right )^2 \sqrt {b+\sqrt {b^2+a x^2}}} \, dx+\int \frac {1}{\sqrt {b+\sqrt {b^2+a x^2}}} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.74, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-b^2+a x^2\right )^2}{\left (b^2+a x^2\right )^2 \sqrt {b+\sqrt {b^2+a x^2}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.44, size = 179, normalized size = 0.85 \begin {gather*} -\frac {b x}{\sqrt {b^2+a x^2} \sqrt {b+\sqrt {b^2+a x^2}}}+\frac {2 x \left (2 b^2+a x^2\right )}{\left (b^2+a x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}}}-\frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}\right )}{\sqrt {a}}-\frac {\sqrt {2} \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}\right )}{\sqrt {a}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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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-b^2)^2/(a*x^2+b^2)^2/(b+(a*x^2+b^2)^(1/2))^(1/2),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 {{\left (a x^{2} - b^{2}\right )}^{2}}{{\left (a x^{2} + b^{2}\right )}^{2} \sqrt {b + \sqrt {a x^{2} + b^{2}}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.07, size = 0, normalized size = 0.00 \[\int \frac {\left (a \,x^{2}-b^{2}\right )^{2}}{\left (a \,x^{2}+b^{2}\right )^{2} \sqrt {b +\sqrt {a \,x^{2}+b^{2}}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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