Integrand size = 45, antiderivative size = 193 \[ \int \frac {1}{\sqrt {-b+a^2 x^2} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=\frac {4 \sqrt {3} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {3} \sqrt [3]{c}}\right )}{a \sqrt [3]{c}}+\frac {4 \log \left (-\sqrt [3]{c}+\sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )}{a \sqrt [3]{c}}-\frac {2 \log \left (c^{2/3}+\sqrt [3]{c} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}+\left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}\right )}{a \sqrt [3]{c}} \]
4*3^(1/2)*arctan(1/3*3^(1/2)+2/3*(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(1/3)*3 ^(1/2)/c^(1/3))/a/c^(1/3)+4*ln(-c^(1/3)+(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^ (1/3))/a/c^(1/3)-2*ln(c^(2/3)+c^(1/3)*(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(1 /3)+(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(2/3))/a/c^(1/3)
Time = 0.51 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\sqrt {-b+a^2 x^2} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=\frac {4 \sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [3]{c}}}{\sqrt {3}}\right )+4 \log \left (-\sqrt [3]{c}+\sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )-2 \log \left (c^{2/3}+\sqrt [3]{c} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}+\left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}\right )}{a \sqrt [3]{c}} \]
(4*Sqrt[3]*ArcTan[(1 + (2*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3))/c^ (1/3))/Sqrt[3]] + 4*Log[-c^(1/3) + (c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^ (1/3)] - 2*Log[c^(2/3) + c^(1/3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1 /3) + (c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(2/3)])/(a*c^(1/3))
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.
\(\displaystyle \int \frac {1}{\sqrt {a^2 x^2-b} \sqrt [3]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}} \, dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \frac {1}{\sqrt {a^2 x^2-b} \sqrt [3]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}}dx\) |
3.25.12.3.1 Defintions of rubi rules used
Time = 0.15 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.75
method | result | size |
derivativedivides | \(\frac {\frac {4 \ln \left (-c^{\frac {1}{3}}+{\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )}^{\frac {1}{3}}\right )}{c^{\frac {1}{3}}}-\frac {2 \ln \left (c^{\frac {2}{3}}+c^{\frac {1}{3}} {\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )}^{\frac {1}{3}}+{\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )}^{\frac {2}{3}}\right )}{c^{\frac {1}{3}}}+\frac {4 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 {\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )}^{\frac {1}{3}}}{c^{\frac {1}{3}}}+1\right )}{3}\right )}{c^{\frac {1}{3}}}}{a}\) | \(144\) |
default | \(\frac {\frac {4 \ln \left (-c^{\frac {1}{3}}+{\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )}^{\frac {1}{3}}\right )}{c^{\frac {1}{3}}}-\frac {2 \ln \left (c^{\frac {2}{3}}+c^{\frac {1}{3}} {\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )}^{\frac {1}{3}}+{\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )}^{\frac {2}{3}}\right )}{c^{\frac {1}{3}}}+\frac {4 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 {\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )}^{\frac {1}{3}}}{c^{\frac {1}{3}}}+1\right )}{3}\right )}{c^{\frac {1}{3}}}}{a}\) | \(144\) |
4/a*(1/c^(1/3)*ln(-c^(1/3)+(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(1/3))-1/2/c^ (1/3)*ln(c^(2/3)+c^(1/3)*(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(1/3)+(c+(a*x+( a^2*x^2-b)^(1/2))^(1/4))^(2/3))+3^(1/2)/c^(1/3)*arctan(1/3*3^(1/2)*(2*(c+( a*x+(a^2*x^2-b)^(1/2))^(1/4))^(1/3)/c^(1/3)+1)))
Time = 0.38 (sec) , antiderivative size = 532, normalized size of antiderivative = 2.76 \[ \int \frac {1}{\sqrt {-b+a^2 x^2} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=\left [\frac {2 \, {\left (\sqrt {3} c \sqrt {-\frac {1}{c^{\frac {2}{3}}}} \log \left (2 \, \sqrt {3} {\left (a c^{\frac {2}{3}} x - \sqrt {a^{2} x^{2} - b} c^{\frac {2}{3}}\right )} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {3}{4}} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {2}{3}} \sqrt {-\frac {1}{c^{\frac {2}{3}}}} - {\left (3 \, a c^{\frac {2}{3}} x + \sqrt {3} {\left (a c x - \sqrt {a^{2} x^{2} - b} c\right )} \sqrt {-\frac {1}{c^{\frac {2}{3}}}} - 3 \, \sqrt {a^{2} x^{2} - b} c^{\frac {2}{3}}\right )} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {3}{4}} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{3}} + {\left (3 \, a c x - \sqrt {3} {\left (a c^{\frac {4}{3}} x - \sqrt {a^{2} x^{2} - b} c^{\frac {4}{3}}\right )} \sqrt {-\frac {1}{c^{\frac {2}{3}}}} - 3 \, \sqrt {a^{2} x^{2} - b} c\right )} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {3}{4}} + 2 \, b\right ) - c^{\frac {2}{3}} \log \left ({\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {2}{3}} + {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{3}} c^{\frac {1}{3}} + c^{\frac {2}{3}}\right ) + 2 \, c^{\frac {2}{3}} \log \left ({\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{3}} - c^{\frac {1}{3}}\right )\right )}}{a c}, \frac {2 \, {\left (2 \, \sqrt {3} c^{\frac {2}{3}} \arctan \left (\frac {1}{3} \, \sqrt {3} + \frac {2 \, \sqrt {3} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{3}}}{3 \, c^{\frac {1}{3}}}\right ) - c^{\frac {2}{3}} \log \left ({\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {2}{3}} + {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{3}} c^{\frac {1}{3}} + c^{\frac {2}{3}}\right ) + 2 \, c^{\frac {2}{3}} \log \left ({\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{3}} - c^{\frac {1}{3}}\right )\right )}}{a c}\right ] \]
[2*(sqrt(3)*c*sqrt(-1/c^(2/3))*log(2*sqrt(3)*(a*c^(2/3)*x - sqrt(a^2*x^2 - b)*c^(2/3))*(a*x + sqrt(a^2*x^2 - b))^(3/4)*(c + (a*x + sqrt(a^2*x^2 - b) )^(1/4))^(2/3)*sqrt(-1/c^(2/3)) - (3*a*c^(2/3)*x + sqrt(3)*(a*c*x - sqrt(a ^2*x^2 - b)*c)*sqrt(-1/c^(2/3)) - 3*sqrt(a^2*x^2 - b)*c^(2/3))*(a*x + sqrt (a^2*x^2 - b))^(3/4)*(c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/3) + (3*a*c* x - sqrt(3)*(a*c^(4/3)*x - sqrt(a^2*x^2 - b)*c^(4/3))*sqrt(-1/c^(2/3)) - 3 *sqrt(a^2*x^2 - b)*c)*(a*x + sqrt(a^2*x^2 - b))^(3/4) + 2*b) - c^(2/3)*log ((c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(2/3) + (c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/3)*c^(1/3) + c^(2/3)) + 2*c^(2/3)*log((c + (a*x + sqrt(a^2*x ^2 - b))^(1/4))^(1/3) - c^(1/3)))/(a*c), 2*(2*sqrt(3)*c^(2/3)*arctan(1/3*s qrt(3) + 2/3*sqrt(3)*(c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/3)/c^(1/3)) - c^(2/3)*log((c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(2/3) + (c + (a*x + sq rt(a^2*x^2 - b))^(1/4))^(1/3)*c^(1/3) + c^(2/3)) + 2*c^(2/3)*log((c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/3) - c^(1/3)))/(a*c)]
\[ \int \frac {1}{\sqrt {-b+a^2 x^2} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=\int \frac {1}{\sqrt [3]{c + \sqrt [4]{a x + \sqrt {a^{2} x^{2} - b}}} \sqrt {a^{2} x^{2} - b}}\, dx \]
\[ \int \frac {1}{\sqrt {-b+a^2 x^2} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=\int { \frac {1}{\sqrt {a^{2} x^{2} - b} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{3}}} \,d x } \]
Timed out. \[ \int \frac {1}{\sqrt {-b+a^2 x^2} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {1}{\sqrt {-b+a^2 x^2} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=\int \frac {1}{{\left (c+{\left (a\,x+\sqrt {a^2\,x^2-b}\right )}^{1/4}\right )}^{1/3}\,\sqrt {a^2\,x^2-b}} \,d x \]