Integrand size = 30, antiderivative size = 81 \[ \int \frac {x^2 \left (x^2+\sqrt {1+x^4}\right )^{3/2}}{\sqrt {1+x^4}} \, dx=\frac {x}{4 \sqrt {x^2+\sqrt {1+x^4}}}+\frac {1}{2} x^3 \sqrt {x^2+\sqrt {1+x^4}}-\frac {\text {arctanh}\left (\frac {\sqrt {2} x}{\sqrt {x^2+\sqrt {1+x^4}}}\right )}{4 \sqrt {2}} \] Output:
1/4*x/(x^2+(x^4+1)^(1/2))^(1/2)+1/2*x^3*(x^2+(x^4+1)^(1/2))^(1/2)-1/8*arct anh(2^(1/2)*x/(x^2+(x^4+1)^(1/2))^(1/2))*2^(1/2)
Time = 0.54 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.89 \[ \int \frac {x^2 \left (x^2+\sqrt {1+x^4}\right )^{3/2}}{\sqrt {1+x^4}} \, dx=\frac {1}{4} \left (x \left (x^2+\sqrt {1+x^4}\right )^{3/2}-\sqrt {2} \text {arctanh}\left (\frac {-1+x^2+\sqrt {1+x^4}}{\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}\right )\right ) \] Input:
Integrate[(x^2*(x^2 + Sqrt[1 + x^4])^(3/2))/Sqrt[1 + x^4],x]
Output:
(x*(x^2 + Sqrt[1 + x^4])^(3/2) - Sqrt[2]*ArcTanh[(-1 + x^2 + Sqrt[1 + x^4] )/(Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]])])/4
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.
\(\displaystyle \int \frac {x^2 \left (\sqrt {x^4+1}+x^2\right )^{3/2}}{\sqrt {x^4+1}} \, dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \frac {x^2 \left (\sqrt {x^4+1}+x^2\right )^{3/2}}{\sqrt {x^4+1}}dx\) |
Input:
Int[(x^2*(x^2 + Sqrt[1 + x^4])^(3/2))/Sqrt[1 + x^4],x]
Output:
$Aborted
\[\int \frac {x^{2} \left (x^{2}+\sqrt {x^{4}+1}\right )^{\frac {3}{2}}}{\sqrt {x^{4}+1}}d x\]
Input:
int(x^2*(x^2+(x^4+1)^(1/2))^(3/2)/(x^4+1)^(1/2),x)
Output:
int(x^2*(x^2+(x^4+1)^(1/2))^(3/2)/(x^4+1)^(1/2),x)
Time = 0.22 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.10 \[ \int \frac {x^2 \left (x^2+\sqrt {1+x^4}\right )^{3/2}}{\sqrt {1+x^4}} \, dx=\frac {1}{4} \, {\left (x^{3} + \sqrt {x^{4} + 1} x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} + \frac {1}{16} \, \sqrt {2} \log \left (4 \, x^{4} + 4 \, \sqrt {x^{4} + 1} x^{2} - 2 \, {\left (\sqrt {2} x^{3} + \sqrt {2} \sqrt {x^{4} + 1} x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} + 1\right ) \] Input:
integrate(x^2*(x^2+(x^4+1)^(1/2))^(3/2)/(x^4+1)^(1/2),x, algorithm="fricas ")
Output:
1/4*(x^3 + sqrt(x^4 + 1)*x)*sqrt(x^2 + sqrt(x^4 + 1)) + 1/16*sqrt(2)*log(4 *x^4 + 4*sqrt(x^4 + 1)*x^2 - 2*(sqrt(2)*x^3 + sqrt(2)*sqrt(x^4 + 1)*x)*sqr t(x^2 + sqrt(x^4 + 1)) + 1)
Time = 3.87 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.19 \[ \int \frac {x^2 \left (x^2+\sqrt {1+x^4}\right )^{3/2}}{\sqrt {1+x^4}} \, dx=\frac {{G_{3, 3}^{2, 2}\left (\begin {matrix} 2, 1 & \frac {1}{2} \\\frac {3}{4}, \frac {5}{4} & 0 \end {matrix} \middle | {x^{4}} \right )}}{4 \sqrt {\pi }} \] Input:
integrate(x**2*(x**2+(x**4+1)**(1/2))**(3/2)/(x**4+1)**(1/2),x)
Output:
meijerg(((2, 1), (1/2,)), ((3/4, 5/4), (0,)), x**4)/(4*sqrt(pi))
\[ \int \frac {x^2 \left (x^2+\sqrt {1+x^4}\right )^{3/2}}{\sqrt {1+x^4}} \, dx=\int { \frac {{\left (x^{2} + \sqrt {x^{4} + 1}\right )}^{\frac {3}{2}} x^{2}}{\sqrt {x^{4} + 1}} \,d x } \] Input:
integrate(x^2*(x^2+(x^4+1)^(1/2))^(3/2)/(x^4+1)^(1/2),x, algorithm="maxima ")
Output:
integrate((x^2 + sqrt(x^4 + 1))^(3/2)*x^2/sqrt(x^4 + 1), x)
\[ \int \frac {x^2 \left (x^2+\sqrt {1+x^4}\right )^{3/2}}{\sqrt {1+x^4}} \, dx=\int { \frac {{\left (x^{2} + \sqrt {x^{4} + 1}\right )}^{\frac {3}{2}} x^{2}}{\sqrt {x^{4} + 1}} \,d x } \] Input:
integrate(x^2*(x^2+(x^4+1)^(1/2))^(3/2)/(x^4+1)^(1/2),x, algorithm="giac")
Output:
integrate((x^2 + sqrt(x^4 + 1))^(3/2)*x^2/sqrt(x^4 + 1), x)
Timed out. \[ \int \frac {x^2 \left (x^2+\sqrt {1+x^4}\right )^{3/2}}{\sqrt {1+x^4}} \, dx=\int \frac {x^2\,{\left (\sqrt {x^4+1}+x^2\right )}^{3/2}}{\sqrt {x^4+1}} \,d x \] Input:
int((x^2*((x^4 + 1)^(1/2) + x^2)^(3/2))/(x^4 + 1)^(1/2),x)
Output:
int((x^2*((x^4 + 1)^(1/2) + x^2)^(3/2))/(x^4 + 1)^(1/2), x)
\[ \int \frac {x^2 \left (x^2+\sqrt {1+x^4}\right )^{3/2}}{\sqrt {1+x^4}} \, dx=\frac {\sqrt {\sqrt {x^{4}+1}+x^{2}}\, \sqrt {x^{4}+1}\, x}{3}+\frac {\sqrt {2}\, \mathrm {log}\left (\sqrt {\sqrt {x^{4}+1}+x^{2}}-\sqrt {2}\, x \right )}{16}-\frac {\sqrt {2}\, \mathrm {log}\left (\sqrt {\sqrt {x^{4}+1}+x^{2}}+\sqrt {2}\, x \right )}{16}+\frac {2 \left (\int \frac {\sqrt {\sqrt {x^{4}+1}+x^{2}}\, x^{6}}{x^{4}+1}d x \right )}{3}+\frac {2 \left (\int \frac {\sqrt {\sqrt {x^{4}+1}+x^{2}}\, x^{2}}{x^{4}+1}d x \right )}{3}-\frac {\left (\int \frac {\sqrt {\sqrt {x^{4}+1}+x^{2}}\, \sqrt {x^{4}+1}}{x^{4}+1}d x \right )}{12} \] Input:
int(x^2*(x^2+(x^4+1)^(1/2))^(3/2)/(x^4+1)^(1/2),x)
Output:
(16*sqrt(sqrt(x**4 + 1) + x**2)*sqrt(x**4 + 1)*x + 3*sqrt(2)*log(sqrt(sqrt (x**4 + 1) + x**2) - sqrt(2)*x) - 3*sqrt(2)*log(sqrt(sqrt(x**4 + 1) + x**2 ) + sqrt(2)*x) + 32*int((sqrt(sqrt(x**4 + 1) + x**2)*x**6)/(x**4 + 1),x) + 32*int((sqrt(sqrt(x**4 + 1) + x**2)*x**2)/(x**4 + 1),x) - 4*int((sqrt(sqr t(x**4 + 1) + x**2)*sqrt(x**4 + 1))/(x**4 + 1),x))/48