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

Optimal. Leaf size=530 \[ \frac {4 b x}{3 \sqrt {\sqrt {a x^2+b^2}+b}}+\frac {2 x \sqrt {a x^2+b^2}}{3 \sqrt {\sqrt {a x^2+b^2}+b}}+\frac {2 \sqrt {\sqrt {2}-1} b^{3/2} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {b} \sqrt {\sqrt {a x^2+b^2}+b}}-\frac {\sqrt {\sqrt {a x^2+b^2}+b}}{\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {b}}\right )}{\sqrt {a}}+\frac {2 i \left (\sqrt {2 \left (\sqrt {2}-1\right )} b^{3/2}+\sqrt {\sqrt {2}-1} b^{3/2}\right ) \tan ^{-1}\left (\frac {\frac {i a x}{\sqrt {\sqrt {a x^2+b^2}+b}}-i \sqrt {a} \sqrt {\sqrt {a x^2+b^2}+b}}{\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {a} \sqrt {b}}\right )}{\sqrt {a}}+\frac {2 \sqrt {1+\sqrt {2}} b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {2 \left (\sqrt {2}-1\right )} \sqrt {b} \sqrt {\sqrt {a x^2+b^2}+b}}-\frac {\sqrt {\sqrt {a x^2+b^2}+b}}{\sqrt {2 \left (\sqrt {2}-1\right )} \sqrt {b}}\right )}{\sqrt {a}}-\frac {2 i \left (\sqrt {2 \left (1+\sqrt {2}\right )} b^{3/2}-\sqrt {1+\sqrt {2}} b^{3/2}\right ) \tanh ^{-1}\left (\frac {\frac {i a x}{\sqrt {\sqrt {a x^2+b^2}+b}}-i \sqrt {a} \sqrt {\sqrt {a x^2+b^2}+b}}{\sqrt {2 \left (\sqrt {2}-1\right )} \sqrt {a} \sqrt {b}}\right )}{\sqrt {a}} \]

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Rubi [F]  time = 0.92, 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 ) \sqrt {b+\sqrt {b^2+a x^2}}}{-b^2+a x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

Rubi steps

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

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Mathematica [F]  time = 0.30, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (b^2+a x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}}}{-b^2+a x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.51, size = 195, normalized size = 0.37 \begin {gather*} \frac {4 b x}{3 \sqrt {b+\sqrt {b^2+a x^2}}}+\frac {2 x \sqrt {b^2+a x^2}}{3 \sqrt {b+\sqrt {b^2+a x^2}}}+\frac {2 \sqrt {-1+\sqrt {2}} 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 )}{\sqrt {a}}-\frac {2 \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 )}{\sqrt {a}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*b*x)/(3*Sqrt[b + Sqrt[b^2 + a*x^2]]) + (2*x*Sqrt[b^2 + a*x^2])/(3*Sqrt[b + Sqrt[b^2 + a*x^2]]) + (2*Sqrt[-1
+ Sqrt[2]]*b^(3/2)*ArcTan[(Sqrt[-1 + Sqrt[2]]*Sqrt[a]*x)/(Sqrt[b]*Sqrt[b + Sqrt[b^2 + a*x^2]])])/Sqrt[a] - (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]])])/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)*(b+(a*x^2+b^2)^(1/2))^(1/2)/(a*x^2-b^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 )} \sqrt {b + \sqrt {a x^{2} + b^{2}}}}{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)*(b+(a*x^2+b^2)^(1/2))^(1/2)/(a*x^2-b^2),x, algorithm="giac")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((a*x^2+b^2)*(b+(a*x^2+b^2)^(1/2))^(1/2)/(a*x^2-b^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 )} \sqrt {b + \sqrt {a x^{2} + b^{2}}}}{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)*(b+(a*x^2+b^2)^(1/2))^(1/2)/(a*x^2-b^2),x, algorithm="maxima")

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(((a*x^2 + b^2)*(b + (a*x^2 + b^2)^(1/2))^(1/2))/(a*x^2 - b^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 )}{a x^{2} - b^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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