Optimal. Leaf size=87 \[ \frac {2 x \sqrt {a x^2+b^2} \left (5 a x^2+13 b^2\right )}{35 \sqrt {\sqrt {a x^2+b^2}+b}}+\frac {4 x \left (3 a b x^2+11 b^3\right )}{35 \sqrt {\sqrt {a x^2+b^2}+b}} \]
________________________________________________________________________________________
Rubi [F] time = 0.28, 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 \left (b^2+a x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \left (b^2+a x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}} \, dx &=\int \left (b^2 \sqrt {b+\sqrt {b^2+a x^2}}+a x^2 \sqrt {b+\sqrt {b^2+a x^2}}\right ) \, dx\\ &=a \int x^2 \sqrt {b+\sqrt {b^2+a x^2}} \, dx+b^2 \int \sqrt {b+\sqrt {b^2+a x^2}} \, dx\\ &=\frac {2 a b^2 x^3}{3 \left (b+\sqrt {b^2+a x^2}\right )^{3/2}}+\frac {2 b^3 x}{\sqrt {b+\sqrt {b^2+a x^2}}}+a \int x^2 \sqrt {b+\sqrt {b^2+a x^2}} \, dx\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.21, size = 85, normalized size = 0.98 \begin {gather*} \frac {2 x \left (5 a^2 x^4+24 a b^2 x^2+11 a b x^2 \sqrt {a x^2+b^2}+35 b^3 \sqrt {a x^2+b^2}+35 b^4\right )}{35 \left (\sqrt {a x^2+b^2}+b\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.16, size = 87, normalized size = 1.00 \begin {gather*} \frac {2 x \sqrt {a x^2+b^2} \left (5 a x^2+13 b^2\right )}{35 \sqrt {\sqrt {a x^2+b^2}+b}}+\frac {4 x \left (3 a b x^2+11 b^3\right )}{35 \sqrt {\sqrt {a x^2+b^2}+b}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.47, size = 70, normalized size = 0.80 \begin {gather*} \frac {2 \, {\left (5 \, a^{2} x^{4} + 12 \, a b^{2} x^{2} - 9 \, b^{4} + {\left (a b x^{2} + 9 \, b^{3}\right )} \sqrt {a x^{2} + b^{2}}\right )} \sqrt {b + \sqrt {a x^{2} + b^{2}}}}{35 \, a x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int {\left (a x^{2} + b^{2}\right )} \sqrt {b + \sqrt {a x^{2} + b^{2}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.04, size = 153, normalized size = 1.76 \begin {gather*} \frac {\left (b^{2}\right )^{\frac {1}{4}} a \sqrt {2}\, x^{3} \hypergeom \left (\left [-\frac {1}{4}, \frac {1}{4}, \frac {3}{2}\right ], \left [\frac {1}{2}, \frac {5}{2}\right ], -\frac {x^{2} a}{b^{2}}\right )}{3}-\frac {b^{2} \left (b^{2}\right )^{\frac {1}{4}} \left (-\frac {32 \sqrt {\pi }\, \sqrt {2}\, x^{3} \sqrt {\frac {a}{b^{2}}}\, a \cosh \left (\frac {3 \arcsinh \left (\frac {x \sqrt {a}}{b}\right )}{2}\right )}{3 b^{2}}-\frac {8 \sqrt {\pi }\, \sqrt {2}\, \sqrt {\frac {a}{b^{2}}}\, \left (-\frac {4 x^{4} a^{2}}{3 b^{4}}-\frac {2 x^{2} a}{3 b^{2}}+\frac {2}{3}\right ) \sinh \left (\frac {3 \arcsinh \left (\frac {x \sqrt {a}}{b}\right )}{2}\right ) b}{\sqrt {a}\, \sqrt {\frac {x^{2} a}{b^{2}}+1}}\right )}{8 \sqrt {\pi }\, \sqrt {\frac {a}{b^{2}}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int {\left (a x^{2} + b^{2}\right )} \sqrt {b + \sqrt {a x^{2} + b^{2}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \left (b^2+a\,x^2\right )\,\sqrt {b+\sqrt {b^2+a\,x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 3.71, size = 581, normalized size = 6.68 \begin {gather*} - \frac {15 \sqrt {2} a^{2} \sqrt {b} x^{5} \Gamma \left (- \frac {1}{4}\right ) \Gamma \left (\frac {1}{4}\right )}{420 \pi b^{2} \sqrt {\frac {a x^{2}}{b^{2}} + 1} \sqrt {\sqrt {\frac {a x^{2}}{b^{2}} + 1} + 1} + 420 \pi b^{2} \sqrt {\sqrt {\frac {a x^{2}}{b^{2}} + 1} + 1}} - \frac {33 \sqrt {2} a b^{\frac {5}{2}} x^{3} \sqrt {\frac {a x^{2}}{b^{2}} + 1} \Gamma \left (- \frac {1}{4}\right ) \Gamma \left (\frac {1}{4}\right )}{420 \pi b^{2} \sqrt {\frac {a x^{2}}{b^{2}} + 1} \sqrt {\sqrt {\frac {a x^{2}}{b^{2}} + 1} + 1} + 420 \pi b^{2} \sqrt {\sqrt {\frac {a x^{2}}{b^{2}} + 1} + 1}} - \frac {37 \sqrt {2} a b^{\frac {5}{2}} x^{3} \Gamma \left (- \frac {1}{4}\right ) \Gamma \left (\frac {1}{4}\right )}{420 \pi b^{2} \sqrt {\frac {a x^{2}}{b^{2}} + 1} \sqrt {\sqrt {\frac {a x^{2}}{b^{2}} + 1} + 1} + 420 \pi b^{2} \sqrt {\sqrt {\frac {a x^{2}}{b^{2}} + 1} + 1}} - \frac {\sqrt {2} a b^{\frac {5}{2}} x^{3} \Gamma \left (- \frac {1}{4}\right ) \Gamma \left (\frac {1}{4}\right )}{12 \pi b^{2} \sqrt {\frac {a x^{2}}{b^{2}} + 1} \sqrt {\sqrt {\frac {a x^{2}}{b^{2}} + 1} + 1} + 12 \pi b^{2} \sqrt {\sqrt {\frac {a x^{2}}{b^{2}} + 1} + 1}} - \frac {3 \sqrt {2} b^{\frac {9}{2}} x \sqrt {\frac {a x^{2}}{b^{2}} + 1} \Gamma \left (- \frac {1}{4}\right ) \Gamma \left (\frac {1}{4}\right )}{12 \pi b^{2} \sqrt {\frac {a x^{2}}{b^{2}} + 1} \sqrt {\sqrt {\frac {a x^{2}}{b^{2}} + 1} + 1} + 12 \pi b^{2} \sqrt {\sqrt {\frac {a x^{2}}{b^{2}} + 1} + 1}} - \frac {3 \sqrt {2} b^{\frac {9}{2}} x \Gamma \left (- \frac {1}{4}\right ) \Gamma \left (\frac {1}{4}\right )}{12 \pi b^{2} \sqrt {\frac {a x^{2}}{b^{2}} + 1} \sqrt {\sqrt {\frac {a x^{2}}{b^{2}} + 1} + 1} + 12 \pi b^{2} \sqrt {\sqrt {\frac {a x^{2}}{b^{2}} + 1} + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________