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

Optimal. Leaf size=1310 \[ \frac {5 \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {2 \left (-1+\sqrt {2}\right )} \sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}-\frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\sqrt {2 \left (-1+\sqrt {2}\right )} \sqrt {b}}\right ) b^{3/2}}{\sqrt {2 \left (-1+\sqrt {2}\right )} \sqrt {a}}-\frac {2 \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {2 \left (-1+\sqrt {2}\right )} \sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}-\frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\sqrt {2 \left (-1+\sqrt {2}\right )} \sqrt {b}}\right ) b^{3/2}}{\sqrt {-1+\sqrt {2}} \sqrt {a}}+\frac {\tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}-\frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {b}}\right ) b^{3/2}}{\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {a}}+\frac {3 \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}-\frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {b}}\right ) b^{3/2}}{\sqrt {1+\sqrt {2}} \sqrt {a}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {2 \left (-1+\sqrt {2}\right )} \sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}-\frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\sqrt {2 \left (-1+\sqrt {2}\right )} \sqrt {b}}\right ) b^{3/2}}{\sqrt {2 \left (-1+\sqrt {2}\right )} \sqrt {a}}+\frac {3 \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {2 \left (-1+\sqrt {2}\right )} \sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}-\frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\sqrt {2 \left (-1+\sqrt {2}\right )} \sqrt {b}}\right ) b^{3/2}}{\sqrt {-1+\sqrt {2}} \sqrt {a}}-\frac {5 \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}-\frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {b}}\right ) b^{3/2}}{\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {a}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}-\frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {b}}\right ) b^{3/2}}{\sqrt {1+\sqrt {2}} \sqrt {a}}+\frac {-2 i x \left (-9 b^7-46 a x^2 b^5-20 a^2 x^4 b^3+24 a^3 x^6 b\right ) a^{3/2}+2 i \left (a b^8-13 a^2 x^2 b^6-75 a^3 x^4 b^4-52 a^4 x^6 b^2+16 a^5 x^8\right )+\sqrt {b^2+a x^2} \left (2 i \left (-a b^7-27 a^2 x^2 b^5-32 a^3 x^4 b^3+24 a^4 x^6 b\right )-2 i a^{3/2} x \left (16 a^3 x^6-60 a^2 b^2 x^4-43 a b^4 x^2\right )\right )}{\frac {3 i x \left (a x^2-b^2\right ) \left (5 b^4+20 a x^2 b^2+16 a^2 x^4\right ) a^2}{\sqrt {b+\sqrt {b^2+a x^2}}}+3 i \left (a x^2-b^2\right ) \left (-5 b^4-20 a x^2 b^2-16 a^2 x^4\right ) \sqrt {b+\sqrt {b^2+a x^2}} a^{3/2}+\sqrt {b^2+a x^2} \left (\frac {3 i x \left (a x^2-b^2\right ) \left (4 b^3+8 a x^2 b\right ) a^2}{\sqrt {b+\sqrt {b^2+a x^2}}}+3 i \left (a x^2-b^2\right ) \left (4 b^3+8 a x^2 b\right ) \sqrt {b+\sqrt {b^2+a x^2}} a^{3/2}\right )} \]

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Rubi [F]  time = 2.07, 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 \sqrt {b+\sqrt {b^2+a x^2}}}{\left (-b^2+a x^2\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(2*a*x^3)/(3*(b + Sqrt[b^2 + a*x^2])^(3/2)) + (2*b*x)/Sqrt[b + Sqrt[b^2 + a*x^2]] - b*Defer[Int][Sqrt[b + Sqrt
[b^2 + a*x^2]]/(b - Sqrt[a]*x), x] - b*Defer[Int][Sqrt[b + Sqrt[b^2 + a*x^2]]/(b + Sqrt[a]*x), x] + a*b^2*Defe
r[Int][Sqrt[b + Sqrt[b^2 + a*x^2]]/(Sqrt[a]*b - a*x)^2, x] + a*b^2*Defer[Int][Sqrt[b + Sqrt[b^2 + a*x^2]]/(Sqr
t[a]*b + a*x)^2, x]

Rubi steps

\begin {align*} \int \frac {\left (b^2+a x^2\right )^2 \sqrt {b+\sqrt {b^2+a x^2}}}{\left (-b^2+a x^2\right )^2} \, dx &=\int \left (\sqrt {b+\sqrt {b^2+a x^2}}+\frac {4 b^4 \sqrt {b+\sqrt {b^2+a x^2}}}{\left (b^2-a x^2\right )^2}-\frac {4 b^2 \sqrt {b+\sqrt {b^2+a x^2}}}{b^2-a x^2}\right ) \, dx\\ &=-\left (\left (4 b^2\right ) \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{b^2-a x^2} \, dx\right )+\left (4 b^4\right ) \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (b^2-a x^2\right )^2} \, dx+\int \sqrt {b+\sqrt {b^2+a x^2}} \, dx\\ &=\frac {2 a x^3}{3 \left (b+\sqrt {b^2+a x^2}\right )^{3/2}}+\frac {2 b x}{\sqrt {b+\sqrt {b^2+a x^2}}}-\left (4 b^2\right ) \int \left (\frac {\sqrt {b+\sqrt {b^2+a x^2}}}{2 b \left (b-\sqrt {a} x\right )}+\frac {\sqrt {b+\sqrt {b^2+a x^2}}}{2 b \left (b+\sqrt {a} x\right )}\right ) \, dx+\left (4 b^4\right ) \int \left (\frac {a \sqrt {b+\sqrt {b^2+a x^2}}}{4 b^2 \left (\sqrt {a} b-a x\right )^2}+\frac {a \sqrt {b+\sqrt {b^2+a x^2}}}{4 b^2 \left (\sqrt {a} b+a x\right )^2}+\frac {a \sqrt {b+\sqrt {b^2+a x^2}}}{2 b^2 \left (a b^2-a^2 x^2\right )}\right ) \, dx\\ &=\frac {2 a x^3}{3 \left (b+\sqrt {b^2+a x^2}\right )^{3/2}}+\frac {2 b x}{\sqrt {b+\sqrt {b^2+a x^2}}}-(2 b) \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{b-\sqrt {a} x} \, dx-(2 b) \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{b+\sqrt {a} x} \, dx+\left (a b^2\right ) \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (\sqrt {a} b-a x\right )^2} \, dx+\left (a b^2\right ) \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (\sqrt {a} b+a x\right )^2} \, dx+\left (2 a b^2\right ) \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{a b^2-a^2 x^2} \, dx\\ &=\frac {2 a x^3}{3 \left (b+\sqrt {b^2+a x^2}\right )^{3/2}}+\frac {2 b x}{\sqrt {b+\sqrt {b^2+a x^2}}}-(2 b) \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{b-\sqrt {a} x} \, dx-(2 b) \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{b+\sqrt {a} x} \, dx+\left (a b^2\right ) \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (\sqrt {a} b-a x\right )^2} \, dx+\left (a b^2\right ) \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (\sqrt {a} b+a x\right )^2} \, dx+\left (2 a b^2\right ) \int \left (\frac {\sqrt {b+\sqrt {b^2+a x^2}}}{2 a b \left (b-\sqrt {a} x\right )}+\frac {\sqrt {b+\sqrt {b^2+a x^2}}}{2 a b \left (b+\sqrt {a} x\right )}\right ) \, dx\\ &=\frac {2 a x^3}{3 \left (b+\sqrt {b^2+a x^2}\right )^{3/2}}+\frac {2 b x}{\sqrt {b+\sqrt {b^2+a x^2}}}+b \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{b-\sqrt {a} x} \, dx+b \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{b+\sqrt {a} x} \, dx-(2 b) \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{b-\sqrt {a} x} \, dx-(2 b) \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{b+\sqrt {a} x} \, dx+\left (a b^2\right ) \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (\sqrt {a} b-a x\right )^2} \, dx+\left (a b^2\right ) \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (\sqrt {a} b+a x\right )^2} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.62, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (b^2+a x^2\right )^2 \sqrt {b+\sqrt {b^2+a x^2}}}{\left (-b^2+a x^2\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 1.02, size = 262, normalized size = 0.20 \begin {gather*} \frac {2 x \left (-4 b^2+a x^2\right ) \sqrt {b^2+a x^2}}{3 \left (-b^2+a x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}}}+\frac {2 x \left (-5 b^3+2 a b x^2\right )}{3 \left (-b^2+a x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}}}+\frac {\sqrt {-1+\sqrt {2}} \left (6+\sqrt {2}\right ) b^{3/2} \tan ^{-1}\left (\frac {\sqrt {-1+\sqrt {2}} \sqrt {a} x}{\sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}\right )}{2 \sqrt {a}}+\frac {\left (-6+\sqrt {2}\right ) \sqrt {1+\sqrt {2}} b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {1+\sqrt {2}} \sqrt {a} x}{\sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}\right )}{2 \sqrt {a}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x*(-4*b^2 + a*x^2)*Sqrt[b^2 + a*x^2])/(3*(-b^2 + a*x^2)*Sqrt[b + Sqrt[b^2 + a*x^2]]) + (2*x*(-5*b^3 + 2*a*b
*x^2))/(3*(-b^2 + a*x^2)*Sqrt[b + Sqrt[b^2 + a*x^2]]) + (Sqrt[-1 + Sqrt[2]]*(6 + Sqrt[2])*b^(3/2)*ArcTan[(Sqrt
[-1 + Sqrt[2]]*Sqrt[a]*x)/(Sqrt[b]*Sqrt[b + Sqrt[b^2 + a*x^2]])])/(2*Sqrt[a]) + ((-6 + Sqrt[2])*Sqrt[1 + Sqrt[
2]]*b^(3/2)*ArcTanh[(Sqrt[1 + Sqrt[2]]*Sqrt[a]*x)/(Sqrt[b]*Sqrt[b + Sqrt[b^2 + a*x^2]])])/(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*(b+(a*x^2+b^2)^(1/2))^(1/2)/(a*x^2-b^2)^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} \sqrt {b + \sqrt {a x^{2} + b^{2}}}}{{\left (a x^{2} - b^{2}\right )}^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((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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maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (a \,x^{2}+b^{2}\right )^{2} \sqrt {b +\sqrt {a \,x^{2}+b^{2}}}}{\left (a \,x^{2}-b^{2}\right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {b + \sqrt {a x^{2} + b^{2}}} \left (a x^{2} + b^{2}\right )^{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*(b+(a*x**2+b**2)**(1/2))**(1/2)/(a*x**2-b**2)**2,x)

[Out]

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

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