Integrand size = 29, antiderivative size = 87 \[ \int \left (b^2+a x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}} \, dx=\frac {2 x \sqrt {b^2+a x^2} \left (13 b^2+5 a x^2\right )}{35 \sqrt {b+\sqrt {b^2+a x^2}}}+\frac {4 x \left (11 b^3+3 a b x^2\right )}{35 \sqrt {b+\sqrt {b^2+a x^2}}} \]
2/35*x*(a*x^2+b^2)^(1/2)*(5*a*x^2+13*b^2)/(b+(a*x^2+b^2)^(1/2))^(1/2)+4/35 *x*(3*a*b*x^2+11*b^3)/(b+(a*x^2+b^2)^(1/2))^(1/2)
Time = 0.13 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.85 \[ \int \left (b^2+a x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}} \, dx=\frac {2 x \left (22 b^3+6 a b x^2+13 b^2 \sqrt {b^2+a x^2}+5 a x^2 \sqrt {b^2+a x^2}\right )}{35 \sqrt {b+\sqrt {b^2+a x^2}}} \]
(2*x*(22*b^3 + 6*a*b*x^2 + 13*b^2*Sqrt[b^2 + a*x^2] + 5*a*x^2*Sqrt[b^2 + a *x^2]))/(35*Sqrt[b + Sqrt[b^2 + a*x^2]])
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \left (a x^2+b^2\right ) \sqrt {\sqrt {a x^2+b^2}+b} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (b^2 \sqrt {\sqrt {a x^2+b^2}+b}+a x^2 \sqrt {\sqrt {a x^2+b^2}+b}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle a \int x^2 \sqrt {b+\sqrt {b^2+a x^2}}dx+\frac {2 a b^2 x^3}{3 \left (\sqrt {a x^2+b^2}+b\right )^{3/2}}+\frac {2 b^3 x}{\sqrt {\sqrt {a x^2+b^2}+b}}\) |
3.12.96.3.1 Defintions of rubi rules used
Result contains higher order function than in optimal. Order 5 vs. order 2.
Time = 0.12 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.76
method | result | size |
meijerg | \(\frac {\left (b^{2}\right )^{\frac {1}{4}} a \sqrt {2}\, x^{3} \operatorname {hypergeom}\left (\left [-\frac {1}{4}, \frac {1}{4}, \frac {3}{2}\right ], \left [\frac {1}{2}, \frac {5}{2}\right ], -\frac {x^{2} a}{b^{2}}\right )}{3}-\frac {b^{2} \left (b^{2}\right )^{\frac {1}{4}} \left (-\frac {32 \sqrt {\pi }\, \sqrt {2}\, x^{3} \sqrt {\frac {a}{b^{2}}}\, a \cosh \left (\frac {3 \,\operatorname {arcsinh}\left (\frac {x \sqrt {a}}{b}\right )}{2}\right )}{3 b^{2}}-\frac {8 \sqrt {\pi }\, \sqrt {2}\, \sqrt {\frac {a}{b^{2}}}\, \left (-\frac {4 x^{4} a^{2}}{3 b^{4}}-\frac {2 x^{2} a}{3 b^{2}}+\frac {2}{3}\right ) \sinh \left (\frac {3 \,\operatorname {arcsinh}\left (\frac {x \sqrt {a}}{b}\right )}{2}\right ) b}{\sqrt {a}\, \sqrt {\frac {x^{2} a}{b^{2}}+1}}\right )}{8 \sqrt {\pi }\, \sqrt {\frac {a}{b^{2}}}}\) | \(153\) |
1/3*(b^2)^(1/4)*a*2^(1/2)*x^3*hypergeom([-1/4,1/4,3/2],[1/2,5/2],-1/b^2*x^ 2*a)-1/8*b^2*(b^2)^(1/4)/Pi^(1/2)/(a/b^2)^(1/2)*(-32/3*Pi^(1/2)*2^(1/2)*x^ 3*(a/b^2)^(1/2)/b^2*a*cosh(3/2*arcsinh(1/b*x*a^(1/2)))-8*Pi^(1/2)*2^(1/2)* (a/b^2)^(1/2)*(-4/3/b^4*x^4*a^2-2/3/b^2*x^2*a+2/3)*sinh(3/2*arcsinh(1/b*x* a^(1/2)))*b/a^(1/2)/(1/b^2*x^2*a+1)^(1/2))
Time = 0.29 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.80 \[ \int \left (b^2+a x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}} \, dx=\frac {2 \, {\left (5 \, a^{2} x^{4} + 12 \, a b^{2} x^{2} - 9 \, b^{4} + {\left (a b x^{2} + 9 \, b^{3}\right )} \sqrt {a x^{2} + b^{2}}\right )} \sqrt {b + \sqrt {a x^{2} + b^{2}}}}{35 \, a x} \]
2/35*(5*a^2*x^4 + 12*a*b^2*x^2 - 9*b^4 + (a*b*x^2 + 9*b^3)*sqrt(a*x^2 + b^ 2))*sqrt(b + sqrt(a*x^2 + b^2))/(a*x)
Leaf count of result is larger than twice the leaf count of optimal. 581 vs. \(2 (80) = 160\).
Time = 2.01 (sec) , antiderivative size = 581, normalized size of antiderivative = 6.68 \[ \int \left (b^2+a x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}} \, dx=- \frac {15 \sqrt {2} a^{2} \sqrt {b} x^{5} \Gamma \left (- \frac {1}{4}\right ) \Gamma \left (\frac {1}{4}\right )}{420 \pi b^{2} \sqrt {\frac {a x^{2}}{b^{2}} + 1} \sqrt {\sqrt {\frac {a x^{2}}{b^{2}} + 1} + 1} + 420 \pi b^{2} \sqrt {\sqrt {\frac {a x^{2}}{b^{2}} + 1} + 1}} - \frac {33 \sqrt {2} a b^{\frac {5}{2}} x^{3} \sqrt {\frac {a x^{2}}{b^{2}} + 1} \Gamma \left (- \frac {1}{4}\right ) \Gamma \left (\frac {1}{4}\right )}{420 \pi b^{2} \sqrt {\frac {a x^{2}}{b^{2}} + 1} \sqrt {\sqrt {\frac {a x^{2}}{b^{2}} + 1} + 1} + 420 \pi b^{2} \sqrt {\sqrt {\frac {a x^{2}}{b^{2}} + 1} + 1}} - \frac {37 \sqrt {2} a b^{\frac {5}{2}} x^{3} \Gamma \left (- \frac {1}{4}\right ) \Gamma \left (\frac {1}{4}\right )}{420 \pi b^{2} \sqrt {\frac {a x^{2}}{b^{2}} + 1} \sqrt {\sqrt {\frac {a x^{2}}{b^{2}} + 1} + 1} + 420 \pi b^{2} \sqrt {\sqrt {\frac {a x^{2}}{b^{2}} + 1} + 1}} - \frac {\sqrt {2} a b^{\frac {5}{2}} x^{3} \Gamma \left (- \frac {1}{4}\right ) \Gamma \left (\frac {1}{4}\right )}{12 \pi b^{2} \sqrt {\frac {a x^{2}}{b^{2}} + 1} \sqrt {\sqrt {\frac {a x^{2}}{b^{2}} + 1} + 1} + 12 \pi b^{2} \sqrt {\sqrt {\frac {a x^{2}}{b^{2}} + 1} + 1}} - \frac {3 \sqrt {2} b^{\frac {9}{2}} x \sqrt {\frac {a x^{2}}{b^{2}} + 1} \Gamma \left (- \frac {1}{4}\right ) \Gamma \left (\frac {1}{4}\right )}{12 \pi b^{2} \sqrt {\frac {a x^{2}}{b^{2}} + 1} \sqrt {\sqrt {\frac {a x^{2}}{b^{2}} + 1} + 1} + 12 \pi b^{2} \sqrt {\sqrt {\frac {a x^{2}}{b^{2}} + 1} + 1}} - \frac {3 \sqrt {2} b^{\frac {9}{2}} x \Gamma \left (- \frac {1}{4}\right ) \Gamma \left (\frac {1}{4}\right )}{12 \pi b^{2} \sqrt {\frac {a x^{2}}{b^{2}} + 1} \sqrt {\sqrt {\frac {a x^{2}}{b^{2}} + 1} + 1} + 12 \pi b^{2} \sqrt {\sqrt {\frac {a x^{2}}{b^{2}} + 1} + 1}} \]
-15*sqrt(2)*a**2*sqrt(b)*x**5*gamma(-1/4)*gamma(1/4)/(420*pi*b**2*sqrt(a*x **2/b**2 + 1)*sqrt(sqrt(a*x**2/b**2 + 1) + 1) + 420*pi*b**2*sqrt(sqrt(a*x* *2/b**2 + 1) + 1)) - 33*sqrt(2)*a*b**(5/2)*x**3*sqrt(a*x**2/b**2 + 1)*gamm a(-1/4)*gamma(1/4)/(420*pi*b**2*sqrt(a*x**2/b**2 + 1)*sqrt(sqrt(a*x**2/b** 2 + 1) + 1) + 420*pi*b**2*sqrt(sqrt(a*x**2/b**2 + 1) + 1)) - 37*sqrt(2)*a* b**(5/2)*x**3*gamma(-1/4)*gamma(1/4)/(420*pi*b**2*sqrt(a*x**2/b**2 + 1)*sq rt(sqrt(a*x**2/b**2 + 1) + 1) + 420*pi*b**2*sqrt(sqrt(a*x**2/b**2 + 1) + 1 )) - sqrt(2)*a*b**(5/2)*x**3*gamma(-1/4)*gamma(1/4)/(12*pi*b**2*sqrt(a*x** 2/b**2 + 1)*sqrt(sqrt(a*x**2/b**2 + 1) + 1) + 12*pi*b**2*sqrt(sqrt(a*x**2/ b**2 + 1) + 1)) - 3*sqrt(2)*b**(9/2)*x*sqrt(a*x**2/b**2 + 1)*gamma(-1/4)*g amma(1/4)/(12*pi*b**2*sqrt(a*x**2/b**2 + 1)*sqrt(sqrt(a*x**2/b**2 + 1) + 1 ) + 12*pi*b**2*sqrt(sqrt(a*x**2/b**2 + 1) + 1)) - 3*sqrt(2)*b**(9/2)*x*gam ma(-1/4)*gamma(1/4)/(12*pi*b**2*sqrt(a*x**2/b**2 + 1)*sqrt(sqrt(a*x**2/b** 2 + 1) + 1) + 12*pi*b**2*sqrt(sqrt(a*x**2/b**2 + 1) + 1))
\[ \int \left (b^2+a x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}} \, dx=\int { {\left (a x^{2} + b^{2}\right )} \sqrt {b + \sqrt {a x^{2} + b^{2}}} \,d x } \]
\[ \int \left (b^2+a x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}} \, dx=\int { {\left (a x^{2} + b^{2}\right )} \sqrt {b + \sqrt {a x^{2} + b^{2}}} \,d x } \]
Timed out. \[ \int \left (b^2+a x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}} \, dx=\int \left (b^2+a\,x^2\right )\,\sqrt {b+\sqrt {b^2+a\,x^2}} \,d x \]