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

Optimal. Leaf size=569 \[ \frac {2 x \left (a x^2-2 b^2\right )}{\left (a x^2-b^2\right ) \sqrt {\sqrt {a x^2+b^2}+b}}-\frac {2 \sqrt {2} \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt {b} \sqrt {\sqrt {a x^2+b^2}+b}}-\frac {\sqrt {\sqrt {a x^2+b^2}+b}}{\sqrt {2} \sqrt {b}}\right )}{\sqrt {a}}-\frac {\sqrt {\frac {1}{2} \left (\sqrt {2}-1\right )} \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {2 \left (\sqrt {2}-1\right )} \sqrt {b} \sqrt {\sqrt {a x^2+b^2}+b}}-\frac {\sqrt {\sqrt {a x^2+b^2}+b}}{\sqrt {2 \left (\sqrt {2}-1\right )} \sqrt {b}}\right )}{\sqrt {a}}-\frac {\sqrt {\frac {1}{2} \left (\sqrt {2}-1\right )} \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {b} \sqrt {\sqrt {a x^2+b^2}+b}}-\frac {\sqrt {\sqrt {a x^2+b^2}+b}}{\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {b}}\right )}{\sqrt {a}}-\frac {\sqrt {\sqrt {2}-1} \left (2+\sqrt {2}\right ) \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {\sqrt {a x^2+b^2}+b}-\frac {a x}{\sqrt {\sqrt {a x^2+b^2}+b}}}{\sqrt {2 \left (\sqrt {2}-1\right )} \sqrt {a} \sqrt {b}}\right )}{2 \sqrt {a}}+\frac {\sqrt {\sqrt {2}-1} \left (2+\sqrt {2}\right ) \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {\sqrt {a x^2+b^2}+b}-\frac {a x}{\sqrt {\sqrt {a x^2+b^2}+b}}}{\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {a} \sqrt {b}}\right )}{2 \sqrt {a}} \]

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Rubi [F]  time = 2.33, 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 = 0.72, 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 = 1.53, size = 241, normalized size = 0.42 \begin {gather*} \frac {2 \left (2 b^2 x-a x^3\right )}{\left (b^2-a x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}}}+\frac {\sqrt {2} \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}{\sqrt {a} x}\right )}{\sqrt {a}}+\frac {\sqrt {-\frac {1}{2}+\frac {1}{\sqrt {2}}} \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {1+\sqrt {2}} \sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}{\sqrt {a} x}\right )}{\sqrt {a}}-\frac {\sqrt {\frac {1}{2}+\frac {1}{\sqrt {2}}} \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {-1+\sqrt {2}} \sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}{\sqrt {a} x}\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]

(2*(2*b^2*x - a*x^3))/((b^2 - a*x^2)*Sqrt[b + Sqrt[b^2 + a*x^2]]) + (Sqrt[2]*Sqrt[b]*ArcTan[(Sqrt[2]*Sqrt[b]*S
qrt[b + Sqrt[b^2 + a*x^2]])/(Sqrt[a]*x)])/Sqrt[a] + (Sqrt[-1/2 + 1/Sqrt[2]]*Sqrt[b]*ArcTan[(Sqrt[1 + Sqrt[2]]*
Sqrt[b]*Sqrt[b + Sqrt[b^2 + a*x^2]])/(Sqrt[a]*x)])/Sqrt[a] - (Sqrt[1/2 + 1/Sqrt[2]]*Sqrt[b]*ArcTanh[(Sqrt[-1 +
Sqrt[2]]*Sqrt[b]*Sqrt[b + Sqrt[b^2 + a*x^2]])/(Sqrt[a]*x)])/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 = 0.02, 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 (b^2+a\,x^2\right )}^2}{{\left (a\,x^2-b^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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