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

Optimal. Leaf size=142 \[ \frac {x \left (2 a x^2+11 b^2\right )}{3 \sqrt {a x^2+b^2} \sqrt {\sqrt {a x^2+b^2}+b}}-\frac {5 b^{3/2} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b} \sqrt {\sqrt {a x^2+b^2}+b}}\right )}{\sqrt {a}}+\frac {2 x \left (2 a b x^2+5 b^3\right )}{3 \left (a x^2+b^2\right ) \sqrt {\sqrt {a x^2+b^2}+b}} \]

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Rubi [F]  time = 2.28, 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*De
fer[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]]/(
Sqrt[-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 = 0.59, 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 = 0.33, size = 142, normalized size = 1.00 \begin {gather*} \frac {x \left (11 b^2+2 a x^2\right )}{3 \sqrt {b^2+a x^2} \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 {5 b^{3/2} \tan ^{-1}\left (\frac {\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)^2*Sqrt[b + Sqrt[b^2 + a*x^2]])/(b^2 + a*x^2)^2,x]

[Out]

(x*(11*b^2 + 2*a*x^2))/(3*Sqrt[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]]) - (5*b^(3/2)*ArcTan[(Sqrt[a]*x)/(Sqrt[b]*Sqrt[b + Sqrt[b^2 + a*x^2]])])/S
qrt[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 = 0.05, 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 = -0.01 \begin {gather*} \int \frac {{\left (a\,x^2-b^2\right )}^2\,\sqrt {b+\sqrt {b^2+a\,x^2}}}{{\left (b^2+a\,x^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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