Integrand size = 42, antiderivative size = 152 \[ \int \frac {\left (-b+a x^5\right )^{3/4} \left (4 b+a x^5\right )}{x^4 \left (-b+c x^4+a x^5\right )} \, dx=\frac {4 \left (-b+a x^5\right )^{3/4}}{3 x^3}+\sqrt {2} c^{3/4} \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x \sqrt [4]{-b+a x^5}}{-\sqrt {c} x^2+\sqrt {-b+a x^5}}\right )+\sqrt {2} c^{3/4} \text {arctanh}\left (\frac {\frac {\sqrt [4]{c} x^2}{\sqrt {2}}+\frac {\sqrt {-b+a x^5}}{\sqrt {2} \sqrt [4]{c}}}{x \sqrt [4]{-b+a x^5}}\right ) \]
4/3*(a*x^5-b)^(3/4)/x^3+2^(1/2)*c^(3/4)*arctan(2^(1/2)*c^(1/4)*x*(a*x^5-b) ^(1/4)/(-c^(1/2)*x^2+(a*x^5-b)^(1/2)))+2^(1/2)*c^(3/4)*arctanh((1/2*c^(1/4 )*x^2*2^(1/2)+1/2*(a*x^5-b)^(1/2)*2^(1/2)/c^(1/4))/x/(a*x^5-b)^(1/4))
Time = 1.13 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.96 \[ \int \frac {\left (-b+a x^5\right )^{3/4} \left (4 b+a x^5\right )}{x^4 \left (-b+c x^4+a x^5\right )} \, dx=\frac {4 \left (-b+a x^5\right )^{3/4}}{3 x^3}+\sqrt {2} c^{3/4} \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x \sqrt [4]{-b+a x^5}}{-\sqrt {c} x^2+\sqrt {-b+a x^5}}\right )+\sqrt {2} c^{3/4} \text {arctanh}\left (\frac {\sqrt {c} x^2+\sqrt {-b+a x^5}}{\sqrt {2} \sqrt [4]{c} x \sqrt [4]{-b+a x^5}}\right ) \]
(4*(-b + a*x^5)^(3/4))/(3*x^3) + Sqrt[2]*c^(3/4)*ArcTan[(Sqrt[2]*c^(1/4)*x *(-b + a*x^5)^(1/4))/(-(Sqrt[c]*x^2) + Sqrt[-b + a*x^5])] + Sqrt[2]*c^(3/4 )*ArcTanh[(Sqrt[c]*x^2 + Sqrt[-b + a*x^5])/(Sqrt[2]*c^(1/4)*x*(-b + a*x^5) ^(1/4))]
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.
\(\displaystyle \int \frac {\left (a x^5-b\right )^{3/4} \left (a x^5+4 b\right )}{x^4 \left (a x^5-b+c x^4\right )} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {\left (a x^5-b\right )^{3/4} (5 a x+4 c)}{a x^5-b+c x^4}-\frac {4 \left (a x^5-b\right )^{3/4}}{x^4}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 4 c \int \frac {\left (a x^5-b\right )^{3/4}}{a x^5+c x^4-b}dx+5 a \int \frac {x \left (a x^5-b\right )^{3/4}}{a x^5+c x^4-b}dx+\frac {4 \left (a x^5-b\right )^{3/4} \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},-\frac {3}{5},\frac {2}{5},\frac {a x^5}{b}\right )}{3 x^3 \left (1-\frac {a x^5}{b}\right )^{3/4}}\) |
3.21.98.3.1 Defintions of rubi rules used
Time = 0.47 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.27
method | result | size |
pseudoelliptic | \(\frac {-3 \ln \left (\frac {\sqrt {a \,x^{5}-b}-\left (a \,x^{5}-b \right )^{\frac {1}{4}} x \,c^{\frac {1}{4}} \sqrt {2}+\sqrt {c}\, x^{2}}{\left (a \,x^{5}-b \right )^{\frac {1}{4}} x \,c^{\frac {1}{4}} \sqrt {2}+\sqrt {c}\, x^{2}+\sqrt {a \,x^{5}-b}}\right ) c^{\frac {3}{4}} \sqrt {2}\, x^{3}-6 \arctan \left (\frac {\sqrt {2}\, \left (a \,x^{5}-b \right )^{\frac {1}{4}}+c^{\frac {1}{4}} x}{c^{\frac {1}{4}} x}\right ) c^{\frac {3}{4}} \sqrt {2}\, x^{3}-6 \arctan \left (\frac {\sqrt {2}\, \left (a \,x^{5}-b \right )^{\frac {1}{4}}-c^{\frac {1}{4}} x}{c^{\frac {1}{4}} x}\right ) c^{\frac {3}{4}} \sqrt {2}\, x^{3}+8 \left (a \,x^{5}-b \right )^{\frac {3}{4}}}{6 x^{3}}\) | \(193\) |
1/6*(-3*ln(((a*x^5-b)^(1/2)-(a*x^5-b)^(1/4)*x*c^(1/4)*2^(1/2)+c^(1/2)*x^2) /((a*x^5-b)^(1/4)*x*c^(1/4)*2^(1/2)+c^(1/2)*x^2+(a*x^5-b)^(1/2)))*c^(3/4)* 2^(1/2)*x^3-6*arctan((2^(1/2)*(a*x^5-b)^(1/4)+c^(1/4)*x)/c^(1/4)/x)*c^(3/4 )*2^(1/2)*x^3-6*arctan((2^(1/2)*(a*x^5-b)^(1/4)-c^(1/4)*x)/c^(1/4)/x)*c^(3 /4)*2^(1/2)*x^3+8*(a*x^5-b)^(3/4))/x^3
Timed out. \[ \int \frac {\left (-b+a x^5\right )^{3/4} \left (4 b+a x^5\right )}{x^4 \left (-b+c x^4+a x^5\right )} \, dx=\text {Timed out} \]
\[ \int \frac {\left (-b+a x^5\right )^{3/4} \left (4 b+a x^5\right )}{x^4 \left (-b+c x^4+a x^5\right )} \, dx=\int \frac {\left (a x^{5} - b\right )^{\frac {3}{4}} \left (a x^{5} + 4 b\right )}{x^{4} \left (a x^{5} - b + c x^{4}\right )}\, dx \]
\[ \int \frac {\left (-b+a x^5\right )^{3/4} \left (4 b+a x^5\right )}{x^4 \left (-b+c x^4+a x^5\right )} \, dx=\int { \frac {{\left (a x^{5} + 4 \, b\right )} {\left (a x^{5} - b\right )}^{\frac {3}{4}}}{{\left (a x^{5} + c x^{4} - b\right )} x^{4}} \,d x } \]
\[ \int \frac {\left (-b+a x^5\right )^{3/4} \left (4 b+a x^5\right )}{x^4 \left (-b+c x^4+a x^5\right )} \, dx=\int { \frac {{\left (a x^{5} + 4 \, b\right )} {\left (a x^{5} - b\right )}^{\frac {3}{4}}}{{\left (a x^{5} + c x^{4} - b\right )} x^{4}} \,d x } \]
Timed out. \[ \int \frac {\left (-b+a x^5\right )^{3/4} \left (4 b+a x^5\right )}{x^4 \left (-b+c x^4+a x^5\right )} \, dx=\int \frac {{\left (a\,x^5-b\right )}^{3/4}\,\left (a\,x^5+4\,b\right )}{x^4\,\left (a\,x^5+c\,x^4-b\right )} \,d x \]