3.23.80 \(\int \frac {(b^2+a x^2)^{3/2}}{\sqrt {b+\sqrt {b^2+a x^2}}} \, dx\)

Optimal. Leaf size=173 \[ \frac {2 x \sqrt {a x^2+b^2} \left (15 a x^2+46 b^2\right )}{105 \sqrt {\sqrt {a x^2+b^2}+b}}+\frac {2 \sqrt {2} b^{7/2} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt {b} \sqrt {\sqrt {a x^2+b^2}+b}}-\frac {\sqrt {\sqrt {a x^2+b^2}+b}}{\sqrt {2} \sqrt {b}}\right )}{\sqrt {a}}-\frac {2 x \left (3 a b x^2+46 b^3\right )}{105 \sqrt {\sqrt {a x^2+b^2}+b}} \]

________________________________________________________________________________________

Rubi [F]  time = 0.24, 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 )^{3/2}}{\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)^(3/2)/Sqrt[b + Sqrt[b^2 + a*x^2]],x]

[Out]

Defer[Int][(b^2 + a*x^2)^(3/2)/Sqrt[b + Sqrt[b^2 + a*x^2]], x]

Rubi steps

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

________________________________________________________________________________________

Mathematica [C]  time = 0.64, size = 260, normalized size = 1.50 \begin {gather*} \frac {60 a^3 x^6+232 a^2 b^2 x^4+48 a^2 b x^4 \sqrt {a x^2+b^2}-210 a b^4 x^2+105 \sqrt {2} b^{7/2} \sqrt {\sqrt {a x^2+b^2}-b} \left (2 b \sqrt {a x^2+b^2}+a x^2+2 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {\sqrt {a x^2+b^2}-b}}{\sqrt {2} \sqrt {b}}\right )-420 b^5 \sqrt {a x^2+b^2}+210 b^4 \left (2 b \sqrt {a x^2+b^2}+a x^2+2 b^2\right ) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {b-\sqrt {b^2+a x^2}}{2 b}\right )-420 b^6}{210 a x \left (\sqrt {a x^2+b^2}+b\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b^2 + a*x^2)^(3/2)/Sqrt[b + Sqrt[b^2 + a*x^2]],x]

[Out]

(-420*b^6 - 210*a*b^4*x^2 + 232*a^2*b^2*x^4 + 60*a^3*x^6 - 420*b^5*Sqrt[b^2 + a*x^2] + 48*a^2*b*x^4*Sqrt[b^2 +
a*x^2] + 105*Sqrt[2]*b^(7/2)*Sqrt[-b + Sqrt[b^2 + a*x^2]]*(2*b^2 + a*x^2 + 2*b*Sqrt[b^2 + a*x^2])*ArcTan[Sqrt
[-b + Sqrt[b^2 + a*x^2]]/(Sqrt[2]*Sqrt[b])] + 210*b^4*(2*b^2 + a*x^2 + 2*b*Sqrt[b^2 + a*x^2])*Hypergeometric2F
1[-1/2, 1, 1/2, (b - Sqrt[b^2 + a*x^2])/(2*b)])/(210*a*x*(b + Sqrt[b^2 + a*x^2])^(3/2))

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.26, size = 140, normalized size = 0.81 \begin {gather*} \frac {2 x \sqrt {b^2+a x^2} \left (46 b^2+15 a x^2\right )}{105 \sqrt {b+\sqrt {b^2+a x^2}}}-\frac {2 x \left (46 b^3+3 a b x^2\right )}{105 \sqrt {b+\sqrt {b^2+a x^2}}}+\frac {\sqrt {2} b^{7/2} \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)^(3/2)/Sqrt[b + Sqrt[b^2 + a*x^2]],x]

[Out]

(2*x*Sqrt[b^2 + a*x^2]*(46*b^2 + 15*a*x^2))/(105*Sqrt[b + Sqrt[b^2 + a*x^2]]) - (2*x*(46*b^3 + 3*a*b*x^2))/(10
5*Sqrt[b + Sqrt[b^2 + a*x^2]]) + (Sqrt[2]*b^(7/2)*ArcTan[(Sqrt[a]*x)/(Sqrt[2]*Sqrt[b]*Sqrt[b + Sqrt[b^2 + a*x^
2]])])/Sqrt[a]

________________________________________________________________________________________

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)^(3/2)/(b+(a*x^2+b^2)^(1/2))^(1/2),x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (a x^{2} + b^{2}\right )}^{\frac {3}{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)^(3/2)/(b+(a*x^2+b^2)^(1/2))^(1/2),x, algorithm="giac")

[Out]

integrate((a*x^2 + b^2)^(3/2)/sqrt(b + sqrt(a*x^2 + b^2)), x)

________________________________________________________________________________________

maple [F]  time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\left (a \,x^{2}+b^{2}\right )^{\frac {3}{2}}}{\sqrt {b +\sqrt {a \,x^{2}+b^{2}}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x^2+b^2)^(3/2)/(b+(a*x^2+b^2)^(1/2))^(1/2),x)

[Out]

int((a*x^2+b^2)^(3/2)/(b+(a*x^2+b^2)^(1/2))^(1/2),x)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (a x^{2} + b^{2}\right )}^{\frac {3}{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)^(3/2)/(b+(a*x^2+b^2)^(1/2))^(1/2),x, algorithm="maxima")

[Out]

integrate((a*x^2 + b^2)^(3/2)/sqrt(b + sqrt(a*x^2 + b^2)), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (b^2+a\,x^2\right )}^{3/2}}{\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)^(3/2)/(b + (a*x^2 + b^2)^(1/2))^(1/2),x)

[Out]

int((a*x^2 + b^2)^(3/2)/(b + (a*x^2 + b^2)^(1/2))^(1/2), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a x^{2} + b^{2}\right )^{\frac {3}{2}}}{\sqrt {b + \sqrt {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)**(3/2)/(b+(a*x**2+b**2)**(1/2))**(1/2),x)

[Out]

Integral((a*x**2 + b**2)**(3/2)/sqrt(b + sqrt(a*x**2 + b**2)), x)

________________________________________________________________________________________