Integrand size = 22, antiderivative size = 399 \[ \int \frac {c+d x}{\sqrt [3]{e x} \left (a+b x^2\right )} \, dx=\frac {d \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{\sqrt [6]{a} b^{5/6} \sqrt [3]{e}}-\frac {d \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{2 \sqrt [6]{a} b^{5/6} \sqrt [3]{e}}+\frac {d \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{2 \sqrt [6]{a} b^{5/6} \sqrt [3]{e}}-\frac {\sqrt {3} c \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} (e x)^{2/3}}{\sqrt [3]{a} e^{2/3}}}{\sqrt {3}}\right )}{2 a^{2/3} \sqrt [3]{b} \sqrt [3]{e}}-\frac {\sqrt {3} d \text {arctanh}\left (\frac {\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}}\right )}{2 \sqrt [6]{a} b^{5/6} \sqrt [3]{e}}+\frac {c \log \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{2 a^{2/3} \sqrt [3]{b} \sqrt [3]{e}}-\frac {c \log \left (a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} e^{2/3} (e x)^{2/3}+b^{2/3} (e x)^{4/3}\right )}{4 a^{2/3} \sqrt [3]{b} \sqrt [3]{e}} \] Output:
d*arctan(b^(1/6)*(e*x)^(1/3)/a^(1/6)/e^(1/3))/a^(1/6)/b^(5/6)/e^(1/3)+1/2* d*arctan(-3^(1/2)+2*b^(1/6)*(e*x)^(1/3)/a^(1/6)/e^(1/3))/a^(1/6)/b^(5/6)/e ^(1/3)+1/2*d*arctan(3^(1/2)+2*b^(1/6)*(e*x)^(1/3)/a^(1/6)/e^(1/3))/a^(1/6) /b^(5/6)/e^(1/3)-1/2*3^(1/2)*c*arctan(1/3*(1-2*b^(1/3)*(e*x)^(2/3)/a^(1/3) /e^(2/3))*3^(1/2))/a^(2/3)/b^(1/3)/e^(1/3)-1/2*3^(1/2)*d*arctanh(3^(1/2)*a ^(1/6)*b^(1/6)*e^(1/3)*(e*x)^(1/3)/(a^(1/3)*e^(2/3)+b^(1/3)*(e*x)^(2/3)))/ a^(1/6)/b^(5/6)/e^(1/3)+1/2*c*ln(a^(1/3)*e^(2/3)+b^(1/3)*(e*x)^(2/3))/a^(2 /3)/b^(1/3)/e^(1/3)-1/4*c*ln(a^(2/3)*e^(4/3)-a^(1/3)*b^(1/3)*e^(2/3)*(e*x) ^(2/3)+b^(2/3)*(e*x)^(4/3))/a^(2/3)/b^(1/3)/e^(1/3)
Time = 0.28 (sec) , antiderivative size = 368, normalized size of antiderivative = 0.92 \[ \int \frac {c+d x}{\sqrt [3]{e x} \left (a+b x^2\right )} \, dx=-\frac {\sqrt [3]{x} \left (2 \left (\sqrt {3} \sqrt {b} c+\sqrt {a} d\right ) \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{b} \sqrt [3]{x}}{\sqrt [6]{a}}\right )+2 \left (\sqrt {3} \sqrt {b} c-\sqrt {a} d\right ) \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{b} \sqrt [3]{x}}{\sqrt [6]{a}}\right )-4 \sqrt {a} d \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{x}}{\sqrt [6]{a}}\right )-2 \sqrt {b} c \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^{2/3}\right )+\sqrt {b} c \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{x}+\sqrt [3]{b} x^{2/3}\right )-\sqrt {3} \sqrt {a} d \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{x}+\sqrt [3]{b} x^{2/3}\right )+\sqrt {b} c \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{x}+\sqrt [3]{b} x^{2/3}\right )+\sqrt {3} \sqrt {a} d \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{x}+\sqrt [3]{b} x^{2/3}\right )\right )}{4 a^{2/3} b^{5/6} \sqrt [3]{e x}} \] Input:
Integrate[(c + d*x)/((e*x)^(1/3)*(a + b*x^2)),x]
Output:
-1/4*(x^(1/3)*(2*(Sqrt[3]*Sqrt[b]*c + Sqrt[a]*d)*ArcTan[Sqrt[3] - (2*b^(1/ 6)*x^(1/3))/a^(1/6)] + 2*(Sqrt[3]*Sqrt[b]*c - Sqrt[a]*d)*ArcTan[Sqrt[3] + (2*b^(1/6)*x^(1/3))/a^(1/6)] - 4*Sqrt[a]*d*ArcTan[(b^(1/6)*x^(1/3))/a^(1/6 )] - 2*Sqrt[b]*c*Log[a^(1/3) + b^(1/3)*x^(2/3)] + Sqrt[b]*c*Log[a^(1/3) - Sqrt[3]*a^(1/6)*b^(1/6)*x^(1/3) + b^(1/3)*x^(2/3)] - Sqrt[3]*Sqrt[a]*d*Log [a^(1/3) - Sqrt[3]*a^(1/6)*b^(1/6)*x^(1/3) + b^(1/3)*x^(2/3)] + Sqrt[b]*c* Log[a^(1/3) + Sqrt[3]*a^(1/6)*b^(1/6)*x^(1/3) + b^(1/3)*x^(2/3)] + Sqrt[3] *Sqrt[a]*d*Log[a^(1/3) + Sqrt[3]*a^(1/6)*b^(1/6)*x^(1/3) + b^(1/3)*x^(2/3) ]))/(a^(2/3)*b^(5/6)*(e*x)^(1/3))
Time = 1.32 (sec) , antiderivative size = 485, normalized size of antiderivative = 1.22, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.682, Rules used = {557, 266, 27, 807, 750, 16, 824, 27, 218, 1142, 25, 27, 1082, 217, 1103}
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.
| \(\displaystyle \int \frac {c+d x}{\sqrt [3]{e x} \left (a+b x^2\right )} \, dx\) |
| \(\Big \downarrow \) 557 |
| \(\displaystyle c \int \frac {1}{\sqrt [3]{e x} \left (b x^2+a\right )}dx+\frac {d \int \frac {(e x)^{2/3}}{b x^2+a}dx}{e}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {3 c \int \frac {e^2 \sqrt [3]{e x}}{b x^2 e^2+a e^2}d\sqrt [3]{e x}}{e}+\frac {3 d \int \frac {e^2 (e x)^{4/3}}{b x^2 e^2+a e^2}d\sqrt [3]{e x}}{e^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 3 c e \int \frac {\sqrt [3]{e x}}{b x^2 e^2+a e^2}d\sqrt [3]{e x}+3 d \int \frac {(e x)^{4/3}}{b x^2 e^2+a e^2}d\sqrt [3]{e x}\) |
| \(\Big \downarrow \) 807 |
| \(\displaystyle \frac {3}{2} c e \int \frac {1}{a e^2+b x e}d(e x)^{2/3}+3 d \int \frac {(e x)^{4/3}}{b x^2 e^2+a e^2}d\sqrt [3]{e x}\) |
| \(\Big \downarrow \) 750 |
| \(\displaystyle \frac {3}{2} c e \left (\frac {\int \frac {2 \sqrt [3]{a} e^{2/3}-\sqrt [3]{b} (e x)^{2/3}}{a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} (e x)^{2/3} e^{2/3}+b^{2/3} (e x)^{2/3}}d(e x)^{2/3}}{3 a^{2/3} e^{4/3}}+\frac {\int \frac {1}{\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}}d(e x)^{2/3}}{3 a^{2/3} e^{4/3}}\right )+3 d \int \frac {(e x)^{4/3}}{b x^2 e^2+a e^2}d\sqrt [3]{e x}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {3}{2} c e \left (\frac {\int \frac {2 \sqrt [3]{a} e^{2/3}-\sqrt [3]{b} (e x)^{2/3}}{a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} (e x)^{2/3} e^{2/3}+b^{2/3} (e x)^{2/3}}d(e x)^{2/3}}{3 a^{2/3} e^{4/3}}+\frac {\log \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{3 a^{2/3} \sqrt [3]{b} e^{4/3}}\right )+3 d \int \frac {(e x)^{4/3}}{b x^2 e^2+a e^2}d\sqrt [3]{e x}\) |
| \(\Big \downarrow \) 824 |
| \(\displaystyle \frac {3}{2} c e \left (\frac {\int \frac {2 \sqrt [3]{a} e^{2/3}-\sqrt [3]{b} (e x)^{2/3}}{a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} (e x)^{2/3} e^{2/3}+b^{2/3} (e x)^{2/3}}d(e x)^{2/3}}{3 a^{2/3} e^{4/3}}+\frac {\log \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{3 a^{2/3} \sqrt [3]{b} e^{4/3}}\right )+3 d \left (\frac {\int \frac {1}{\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{3 b^{2/3}}+\frac {\int -\frac {\sqrt [6]{a} \sqrt [3]{e}-\sqrt {3} \sqrt [6]{b} \sqrt [3]{e x}}{2 \left (\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}\right )}d\sqrt [3]{e x}}{3 \sqrt [6]{a} b^{2/3} \sqrt [3]{e}}+\frac {\int -\frac {\sqrt [6]{a} \sqrt [3]{e}+\sqrt {3} \sqrt [6]{b} \sqrt [3]{e x}}{2 \left (\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}\right )}d\sqrt [3]{e x}}{3 \sqrt [6]{a} b^{2/3} \sqrt [3]{e}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{2} c e \left (\frac {\int \frac {2 \sqrt [3]{a} e^{2/3}-\sqrt [3]{b} (e x)^{2/3}}{a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} (e x)^{2/3} e^{2/3}+b^{2/3} (e x)^{2/3}}d(e x)^{2/3}}{3 a^{2/3} e^{4/3}}+\frac {\log \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{3 a^{2/3} \sqrt [3]{b} e^{4/3}}\right )+3 d \left (\frac {\int \frac {1}{\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{3 b^{2/3}}-\frac {\int \frac {\sqrt [6]{a} \sqrt [3]{e}-\sqrt {3} \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{6 \sqrt [6]{a} b^{2/3} \sqrt [3]{e}}-\frac {\int \frac {\sqrt [6]{a} \sqrt [3]{e}+\sqrt {3} \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{6 \sqrt [6]{a} b^{2/3} \sqrt [3]{e}}\right )\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {3}{2} c e \left (\frac {\int \frac {2 \sqrt [3]{a} e^{2/3}-\sqrt [3]{b} (e x)^{2/3}}{a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} (e x)^{2/3} e^{2/3}+b^{2/3} (e x)^{2/3}}d(e x)^{2/3}}{3 a^{2/3} e^{4/3}}+\frac {\log \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{3 a^{2/3} \sqrt [3]{b} e^{4/3}}\right )+3 d \left (-\frac {\int \frac {\sqrt [6]{a} \sqrt [3]{e}-\sqrt {3} \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{6 \sqrt [6]{a} b^{2/3} \sqrt [3]{e}}-\frac {\int \frac {\sqrt [6]{a} \sqrt [3]{e}+\sqrt {3} \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{6 \sqrt [6]{a} b^{2/3} \sqrt [3]{e}}+\frac {\arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{3 \sqrt [6]{a} b^{5/6} \sqrt [3]{e}}\right )\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {3}{2} c e \left (\frac {\frac {3}{2} \sqrt [3]{a} e^{2/3} \int \frac {1}{a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} (e x)^{2/3} e^{2/3}+b^{2/3} (e x)^{2/3}}d(e x)^{2/3}-\frac {\int -\frac {\sqrt [3]{b} \left (\sqrt [3]{a} e^{2/3}-2 \sqrt [3]{b} (e x)^{2/3}\right )}{a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} (e x)^{2/3} e^{2/3}+b^{2/3} (e x)^{2/3}}d(e x)^{2/3}}{2 \sqrt [3]{b}}}{3 a^{2/3} e^{4/3}}+\frac {\log \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{3 a^{2/3} \sqrt [3]{b} e^{4/3}}\right )+3 d \left (-\frac {-\frac {1}{2} \sqrt [6]{a} \sqrt [3]{e} \int \frac {1}{\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}-\frac {\sqrt {3} \int -\frac {\sqrt [6]{b} \left (\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}-2 \sqrt [6]{b} \sqrt [3]{e x}\right )}{\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{2 \sqrt [6]{b}}}{6 \sqrt [6]{a} b^{2/3} \sqrt [3]{e}}-\frac {\frac {\sqrt {3} \int \frac {\sqrt [6]{b} \left (\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}+2 \sqrt [6]{b} \sqrt [3]{e x}\right )}{\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{2 \sqrt [6]{b}}-\frac {1}{2} \sqrt [6]{a} \sqrt [3]{e} \int \frac {1}{\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{6 \sqrt [6]{a} b^{2/3} \sqrt [3]{e}}+\frac {\arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{3 \sqrt [6]{a} b^{5/6} \sqrt [3]{e}}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {3}{2} c e \left (\frac {\frac {3}{2} \sqrt [3]{a} e^{2/3} \int \frac {1}{a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} (e x)^{2/3} e^{2/3}+b^{2/3} (e x)^{2/3}}d(e x)^{2/3}+\frac {\int \frac {\sqrt [3]{b} \left (\sqrt [3]{a} e^{2/3}-2 \sqrt [3]{b} (e x)^{2/3}\right )}{a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} (e x)^{2/3} e^{2/3}+b^{2/3} (e x)^{2/3}}d(e x)^{2/3}}{2 \sqrt [3]{b}}}{3 a^{2/3} e^{4/3}}+\frac {\log \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{3 a^{2/3} \sqrt [3]{b} e^{4/3}}\right )+3 d \left (-\frac {\frac {\sqrt {3} \int \frac {\sqrt [6]{b} \left (\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}-2 \sqrt [6]{b} \sqrt [3]{e x}\right )}{\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{2 \sqrt [6]{b}}-\frac {1}{2} \sqrt [6]{a} \sqrt [3]{e} \int \frac {1}{\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{6 \sqrt [6]{a} b^{2/3} \sqrt [3]{e}}-\frac {\frac {\sqrt {3} \int \frac {\sqrt [6]{b} \left (\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}+2 \sqrt [6]{b} \sqrt [3]{e x}\right )}{\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{2 \sqrt [6]{b}}-\frac {1}{2} \sqrt [6]{a} \sqrt [3]{e} \int \frac {1}{\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{6 \sqrt [6]{a} b^{2/3} \sqrt [3]{e}}+\frac {\arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{3 \sqrt [6]{a} b^{5/6} \sqrt [3]{e}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{2} c e \left (\frac {\frac {3}{2} \sqrt [3]{a} e^{2/3} \int \frac {1}{a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} (e x)^{2/3} e^{2/3}+b^{2/3} (e x)^{2/3}}d(e x)^{2/3}+\frac {1}{2} \int \frac {\sqrt [3]{a} e^{2/3}-2 \sqrt [3]{b} (e x)^{2/3}}{a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} (e x)^{2/3} e^{2/3}+b^{2/3} (e x)^{2/3}}d(e x)^{2/3}}{3 a^{2/3} e^{4/3}}+\frac {\log \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{3 a^{2/3} \sqrt [3]{b} e^{4/3}}\right )+3 d \left (-\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}-2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}-\frac {1}{2} \sqrt [6]{a} \sqrt [3]{e} \int \frac {1}{\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{6 \sqrt [6]{a} b^{2/3} \sqrt [3]{e}}-\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}+2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}-\frac {1}{2} \sqrt [6]{a} \sqrt [3]{e} \int \frac {1}{\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{6 \sqrt [6]{a} b^{2/3} \sqrt [3]{e}}+\frac {\arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{3 \sqrt [6]{a} b^{5/6} \sqrt [3]{e}}\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {3}{2} c e \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{a} e^{2/3}-2 \sqrt [3]{b} (e x)^{2/3}}{a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} (e x)^{2/3} e^{2/3}+b^{2/3} (e x)^{2/3}}d(e x)^{2/3}+\frac {3 \int \frac {1}{\frac {2 \sqrt [3]{b} (e x)^{2/3}}{\sqrt [3]{a} e^{2/3}}-4}d\left (1-\frac {2 \sqrt [3]{b} (e x)^{2/3}}{\sqrt [3]{a} e^{2/3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3} e^{4/3}}+\frac {\log \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{3 a^{2/3} \sqrt [3]{b} e^{4/3}}\right )+3 d \left (-\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}-2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}-\frac {\int \frac {1}{-(e x)^{2/3}-\frac {1}{3}}d\left (1-\frac {2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}}\right )}{\sqrt {3} \sqrt [6]{b}}}{6 \sqrt [6]{a} b^{2/3} \sqrt [3]{e}}-\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}+2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}+\frac {\int \frac {1}{-(e x)^{2/3}-\frac {1}{3}}d\left (\frac {2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}}+1\right )}{\sqrt {3} \sqrt [6]{b}}}{6 \sqrt [6]{a} b^{2/3} \sqrt [3]{e}}+\frac {\arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{3 \sqrt [6]{a} b^{5/6} \sqrt [3]{e}}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {3}{2} c e \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{a} e^{2/3}-2 \sqrt [3]{b} (e x)^{2/3}}{a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} (e x)^{2/3} e^{2/3}+b^{2/3} (e x)^{2/3}}d(e x)^{2/3}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} (e x)^{2/3}}{\sqrt [3]{a} e^{2/3}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3} e^{4/3}}+\frac {\log \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{3 a^{2/3} \sqrt [3]{b} e^{4/3}}\right )+3 d \left (-\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}-2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}+\frac {\arctan \left (\sqrt {3} \left (1-\frac {2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}}\right )\right )}{\sqrt [6]{b}}}{6 \sqrt [6]{a} b^{2/3} \sqrt [3]{e}}-\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}+2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}-\frac {\arctan \left (\sqrt {3} \left (\frac {2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}}+1\right )\right )}{\sqrt [6]{b}}}{6 \sqrt [6]{a} b^{2/3} \sqrt [3]{e}}+\frac {\arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{3 \sqrt [6]{a} b^{5/6} \sqrt [3]{e}}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {3}{2} c e \left (\frac {-\frac {\log \left (a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} e^{2/3} (e x)^{2/3}+b^{2/3} (e x)^{2/3}\right )}{2 \sqrt [3]{b}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} (e x)^{2/3}}{\sqrt [3]{a} e^{2/3}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3} e^{4/3}}+\frac {\log \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{3 a^{2/3} \sqrt [3]{b} e^{4/3}}\right )+3 d \left (-\frac {\frac {\arctan \left (\sqrt {3} \left (1-\frac {2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}}\right )\right )}{\sqrt [6]{b}}-\frac {\sqrt {3} \log \left (-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e} \sqrt [3]{e x}+\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{2 \sqrt [6]{b}}}{6 \sqrt [6]{a} b^{2/3} \sqrt [3]{e}}-\frac {\frac {\sqrt {3} \log \left (\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e} \sqrt [3]{e x}+\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{2 \sqrt [6]{b}}-\frac {\arctan \left (\sqrt {3} \left (\frac {2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}}+1\right )\right )}{\sqrt [6]{b}}}{6 \sqrt [6]{a} b^{2/3} \sqrt [3]{e}}+\frac {\arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{3 \sqrt [6]{a} b^{5/6} \sqrt [3]{e}}\right )\) |
Input:
Int[(c + d*x)/((e*x)^(1/3)*(a + b*x^2)),x]
Output:
3*d*(ArcTan[(b^(1/6)*(e*x)^(1/3))/(a^(1/6)*e^(1/3))]/(3*a^(1/6)*b^(5/6)*e^ (1/3)) - (ArcTan[Sqrt[3]*(1 - (2*b^(1/6)*(e*x)^(1/3))/(Sqrt[3]*a^(1/6)*e^( 1/3)))]/b^(1/6) - (Sqrt[3]*Log[a^(1/3)*e^(2/3) - Sqrt[3]*a^(1/6)*b^(1/6)*e ^(1/3)*(e*x)^(1/3) + b^(1/3)*(e*x)^(2/3)])/(2*b^(1/6)))/(6*a^(1/6)*b^(2/3) *e^(1/3)) - (-(ArcTan[Sqrt[3]*(1 + (2*b^(1/6)*(e*x)^(1/3))/(Sqrt[3]*a^(1/6 )*e^(1/3)))]/b^(1/6)) + (Sqrt[3]*Log[a^(1/3)*e^(2/3) + Sqrt[3]*a^(1/6)*b^( 1/6)*e^(1/3)*(e*x)^(1/3) + b^(1/3)*(e*x)^(2/3)])/(2*b^(1/6)))/(6*a^(1/6)*b ^(2/3)*e^(1/3))) + (3*c*e*(Log[a^(1/3)*e^(2/3) + b^(1/3)*(e*x)^(2/3)]/(3*a ^(2/3)*b^(1/3)*e^(4/3)) + (-((Sqrt[3]*ArcTan[(1 - (2*b^(1/3)*(e*x)^(2/3))/ (a^(1/3)*e^(2/3)))/Sqrt[3]])/b^(1/3)) - Log[a^(2/3)*e^(4/3) + b^(2/3)*(e*x )^(2/3) - a^(1/3)*b^(1/3)*e^(2/3)*(e*x)^(2/3)]/(2*b^(1/3)))/(3*a^(2/3)*e^( 4/3))))/2
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Sym bol] :> Simp[c Int[(e*x)^m*(a + b*x^2)^p, x], x] + Simp[d/e Int[(e*x)^( m + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x]
Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Simp[1/(3*Rt[a, 3]^2) Int[1/ (Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]^2) Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x] /; FreeQ[{a, b}, x]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]
Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Module[{r = Numerator [Rt[a/b, n]], s = Denominator[Rt[a/b, n]], k, u}, Simp[u = Int[(r*Cos[(2*k - 1)*m*(Pi/n)] - s*Cos[(2*k - 1)*(m + 1)*(Pi/n)]*x)/(r^2 - 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2*x^2), x] + Int[(r*Cos[(2*k - 1)*m*(Pi/n)] + s*Cos[(2*k - 1)*(m + 1)*(Pi/n)]*x)/(r^2 + 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2*x^2), x] ; 2*(-1)^(m/2)*(r^(m + 2)/(a*n*s^m)) Int[1/(r^2 + s^2*x^2), x] + 2*(r^(m + 1)/(a*n*s^m)) Sum[u, {k, 1, (n - 2)/4}], x]] /; FreeQ[{a, b}, x] && IGt Q[(n - 2)/4, 0] && IGtQ[m, 0] && LtQ[m, n - 1] && PosQ[a/b]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Time = 0.58 (sec) , antiderivative size = 451, normalized size of antiderivative = 1.13
| method | result | size |
| pseudoelliptic | \(\frac {-2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}+2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}\right ) b c \sqrt {\frac {a \,e^{2}}{b}}-2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}-2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}\right ) b c \sqrt {\frac {a \,e^{2}}{b}}-\ln \left (\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \left (e x \right )^{\frac {1}{3}}-\left (e x \right )^{\frac {2}{3}}-\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right ) b c \sqrt {\frac {a \,e^{2}}{b}}-\ln \left (\left (e x \right )^{\frac {2}{3}}+\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \left (e x \right )^{\frac {1}{3}}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right ) b c \sqrt {\frac {a \,e^{2}}{b}}+2 \ln \left (\left (e x \right )^{\frac {2}{3}}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right ) b c \sqrt {\frac {a \,e^{2}}{b}}+\sqrt {3}\, \ln \left (\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \left (e x \right )^{\frac {1}{3}}-\left (e x \right )^{\frac {2}{3}}-\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right ) a d e -\sqrt {3}\, \ln \left (\left (e x \right )^{\frac {2}{3}}+\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \left (e x \right )^{\frac {1}{3}}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right ) a d e +2 \arctan \left (\frac {\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}+2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}\right ) a d e -2 \arctan \left (\frac {\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}-2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}\right ) a d e +4 \arctan \left (\frac {\left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}\right ) a d e}{4 a e b \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}\) | \(451\) |
| derivativedivides | \(\frac {\ln \left (\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \left (e x \right )^{\frac {1}{3}}-\left (e x \right )^{\frac {2}{3}}-\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right ) \sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {5}{6}} d}{4 a \,e^{2}}-\frac {\ln \left (\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \left (e x \right )^{\frac {1}{3}}-\left (e x \right )^{\frac {2}{3}}-\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}} c}{4 a e}+\frac {d \arctan \left (\frac {2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{2 b \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}+\frac {\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}} \sqrt {3}\, c \arctan \left (\frac {2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{2 a e}+\frac {\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}} c \ln \left (\left (e x \right )^{\frac {2}{3}}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right )}{2 a e}+\frac {d \arctan \left (\frac {\left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}\right )}{b \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}-\frac {\ln \left (\left (e x \right )^{\frac {2}{3}}+\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \left (e x \right )^{\frac {1}{3}}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right ) \sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {5}{6}} d}{4 a \,e^{2}}-\frac {\ln \left (\left (e x \right )^{\frac {2}{3}}+\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \left (e x \right )^{\frac {1}{3}}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}} c}{4 a e}+\frac {\arctan \left (\frac {2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) d}{2 b \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}-\frac {\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) \sqrt {3}\, c}{2 a e}\) | \(472\) |
| default | \(\frac {\ln \left (\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \left (e x \right )^{\frac {1}{3}}-\left (e x \right )^{\frac {2}{3}}-\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right ) \sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {5}{6}} d}{4 a \,e^{2}}-\frac {\ln \left (\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \left (e x \right )^{\frac {1}{3}}-\left (e x \right )^{\frac {2}{3}}-\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}} c}{4 a e}+\frac {d \arctan \left (\frac {2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{2 b \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}+\frac {\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}} \sqrt {3}\, c \arctan \left (\frac {2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{2 a e}+\frac {\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}} c \ln \left (\left (e x \right )^{\frac {2}{3}}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right )}{2 a e}+\frac {d \arctan \left (\frac {\left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}\right )}{b \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}-\frac {\ln \left (\left (e x \right )^{\frac {2}{3}}+\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \left (e x \right )^{\frac {1}{3}}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right ) \sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {5}{6}} d}{4 a \,e^{2}}-\frac {\ln \left (\left (e x \right )^{\frac {2}{3}}+\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \left (e x \right )^{\frac {1}{3}}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}} c}{4 a e}+\frac {\arctan \left (\frac {2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) d}{2 b \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}-\frac {\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) \sqrt {3}\, c}{2 a e}\) | \(472\) |
Input:
int((d*x+c)/(e*x)^(1/3)/(b*x^2+a),x,method=_RETURNVERBOSE)
Output:
1/4/a/e*(-2*3^(1/2)*arctan((3^(1/2)*(a*e^2/b)^(1/6)+2*(e*x)^(1/3))/(a*e^2/ b)^(1/6))*b*c*(a*e^2/b)^(1/2)-2*3^(1/2)*arctan((3^(1/2)*(a*e^2/b)^(1/6)-2* (e*x)^(1/3))/(a*e^2/b)^(1/6))*b*c*(a*e^2/b)^(1/2)-ln(3^(1/2)*(a*e^2/b)^(1/ 6)*(e*x)^(1/3)-(e*x)^(2/3)-(a*e^2/b)^(1/3))*b*c*(a*e^2/b)^(1/2)-ln((e*x)^( 2/3)+3^(1/2)*(a*e^2/b)^(1/6)*(e*x)^(1/3)+(a*e^2/b)^(1/3))*b*c*(a*e^2/b)^(1 /2)+2*ln((e*x)^(2/3)+(a*e^2/b)^(1/3))*b*c*(a*e^2/b)^(1/2)+3^(1/2)*ln(3^(1/ 2)*(a*e^2/b)^(1/6)*(e*x)^(1/3)-(e*x)^(2/3)-(a*e^2/b)^(1/3))*a*d*e-3^(1/2)* ln((e*x)^(2/3)+3^(1/2)*(a*e^2/b)^(1/6)*(e*x)^(1/3)+(a*e^2/b)^(1/3))*a*d*e+ 2*arctan((3^(1/2)*(a*e^2/b)^(1/6)+2*(e*x)^(1/3))/(a*e^2/b)^(1/6))*a*d*e-2* arctan((3^(1/2)*(a*e^2/b)^(1/6)-2*(e*x)^(1/3))/(a*e^2/b)^(1/6))*a*d*e+4*ar ctan((e*x)^(1/3)/(a*e^2/b)^(1/6))*a*d*e)/b/(a*e^2/b)^(1/6)
Leaf count of result is larger than twice the leaf count of optimal. 2110 vs. \(2 (264) = 528\).
Time = 0.23 (sec) , antiderivative size = 2110, normalized size of antiderivative = 5.29 \[ \int \frac {c+d x}{\sqrt [3]{e x} \left (a+b x^2\right )} \, dx=\text {Too large to display} \] Input:
integrate((d*x+c)/(e*x)^(1/3)/(b*x^2+a),x, algorithm="fricas")
Output:
1/4*(sqrt(-3) - 1)*((a^2*b^2*e*sqrt(-(9*b^2*c^4*d^2 - 6*a*b*c^2*d^4 + a^2* d^6)/(a^3*b^5*e^2)) + b*c^3 - 3*a*c*d^2)/(a^2*b^2*e))^(1/3)*log(1/2*(2*sqr t(-3)*(3*a^2*b^3*c^3*d^2 - a^3*b^2*c*d^4)*e + 2*(3*a^2*b^3*c^3*d^2 - a^3*b ^2*c*d^4)*e + (sqrt(-3)*(a^3*b^5*c^2 - a^4*b^4*d^2)*e^2 + (a^3*b^5*c^2 - a ^4*b^4*d^2)*e^2)*sqrt(-(9*b^2*c^4*d^2 - 6*a*b*c^2*d^4 + a^2*d^6)/(a^3*b^5* e^2)))*((a^2*b^2*e*sqrt(-(9*b^2*c^4*d^2 - 6*a*b*c^2*d^4 + a^2*d^6)/(a^3*b^ 5*e^2)) + b*c^3 - 3*a*c*d^2)/(a^2*b^2*e))^(2/3) - (3*b^3*c^6*d + 5*a*b^2*c ^4*d^3 + a^2*b*c^2*d^5 - a^3*d^7)*(e*x)^(1/3)) - 1/4*(sqrt(-3) + 1)*((a^2* b^2*e*sqrt(-(9*b^2*c^4*d^2 - 6*a*b*c^2*d^4 + a^2*d^6)/(a^3*b^5*e^2)) + b*c ^3 - 3*a*c*d^2)/(a^2*b^2*e))^(1/3)*log(-1/2*(2*sqrt(-3)*(3*a^2*b^3*c^3*d^2 - a^3*b^2*c*d^4)*e - 2*(3*a^2*b^3*c^3*d^2 - a^3*b^2*c*d^4)*e + (sqrt(-3)* (a^3*b^5*c^2 - a^4*b^4*d^2)*e^2 - (a^3*b^5*c^2 - a^4*b^4*d^2)*e^2)*sqrt(-( 9*b^2*c^4*d^2 - 6*a*b*c^2*d^4 + a^2*d^6)/(a^3*b^5*e^2)))*((a^2*b^2*e*sqrt( -(9*b^2*c^4*d^2 - 6*a*b*c^2*d^4 + a^2*d^6)/(a^3*b^5*e^2)) + b*c^3 - 3*a*c* d^2)/(a^2*b^2*e))^(2/3) - (3*b^3*c^6*d + 5*a*b^2*c^4*d^3 + a^2*b*c^2*d^5 - a^3*d^7)*(e*x)^(1/3)) + 1/4*(sqrt(-3) - 1)*(-(a^2*b^2*e*sqrt(-(9*b^2*c^4* d^2 - 6*a*b*c^2*d^4 + a^2*d^6)/(a^3*b^5*e^2)) - b*c^3 + 3*a*c*d^2)/(a^2*b^ 2*e))^(1/3)*log(1/2*(2*sqrt(-3)*(3*a^2*b^3*c^3*d^2 - a^3*b^2*c*d^4)*e + 2* (3*a^2*b^3*c^3*d^2 - a^3*b^2*c*d^4)*e - (sqrt(-3)*(a^3*b^5*c^2 - a^4*b^4*d ^2)*e^2 + (a^3*b^5*c^2 - a^4*b^4*d^2)*e^2)*sqrt(-(9*b^2*c^4*d^2 - 6*a*b...
\[ \int \frac {c+d x}{\sqrt [3]{e x} \left (a+b x^2\right )} \, dx=\int \frac {c + d x}{\sqrt [3]{e x} \left (a + b x^{2}\right )}\, dx \] Input:
integrate((d*x+c)/(e*x)**(1/3)/(b*x**2+a),x)
Output:
Integral((c + d*x)/((e*x)**(1/3)*(a + b*x**2)), x)
Exception generated. \[ \int \frac {c+d x}{\sqrt [3]{e x} \left (a+b x^2\right )} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((d*x+c)/(e*x)^(1/3)/(b*x^2+a),x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
Time = 0.48 (sec) , antiderivative size = 351, normalized size of antiderivative = 0.88 \[ \int \frac {c+d x}{\sqrt [3]{e x} \left (a+b x^2\right )} \, dx=-\frac {{\left (\sqrt {3} b^{3} c e - \sqrt {a b} b^{2} d e\right )} \arctan \left (\frac {\sqrt {3} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{6}} + 2 \, \left (e x\right )^{\frac {1}{3}}}{\left (\frac {a e^{2}}{b}\right )^{\frac {1}{6}}}\right )}{2 \, \left (a b^{5} e^{2}\right )^{\frac {2}{3}}} + \frac {{\left (\sqrt {3} b^{3} c e - \sqrt {a b} b^{2} d e\right )} \arctan \left (-\frac {\sqrt {3} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{6}} - 2 \, \left (e x\right )^{\frac {1}{3}}}{\left (\frac {a e^{2}}{b}\right )^{\frac {1}{6}}}\right )}{2 \, \left (a b^{5} e^{2}\right )^{\frac {2}{3}}} - \frac {{\left (b^{3} c e + \sqrt {3} \sqrt {a b} b^{2} d e\right )} \log \left (\sqrt {3} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{6}} \left (e x\right )^{\frac {1}{3}} + \left (e x\right )^{\frac {2}{3}} + \left (\frac {a e^{2}}{b}\right )^{\frac {1}{3}}\right )}{4 \, \left (a b^{5} e^{2}\right )^{\frac {2}{3}}} - \frac {{\left (b^{3} c e + \sqrt {3} \sqrt {a b} b^{2} d e\right )} \log \left (-\sqrt {3} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{6}} \left (e x\right )^{\frac {1}{3}} + \left (e x\right )^{\frac {2}{3}} + \left (\frac {a e^{2}}{b}\right )^{\frac {1}{3}}\right )}{4 \, \left (a b^{5} e^{2}\right )^{\frac {2}{3}}} + \frac {\left (\frac {a e^{2}}{b}\right )^{\frac {5}{6}} d \arctan \left (\frac {\left (e x\right )^{\frac {1}{3}}}{\left (\frac {a e^{2}}{b}\right )^{\frac {1}{6}}}\right )}{a e^{2}} + \frac {\left (a b^{5} e^{2}\right )^{\frac {1}{3}} c \log \left (\left (e x\right )^{\frac {2}{3}} + \left (\frac {a e^{2}}{b}\right )^{\frac {1}{3}}\right )}{2 \, a b^{2} e} \] Input:
integrate((d*x+c)/(e*x)^(1/3)/(b*x^2+a),x, algorithm="giac")
Output:
-1/2*(sqrt(3)*b^3*c*e - sqrt(a*b)*b^2*d*e)*arctan((sqrt(3)*(a*e^2/b)^(1/6) + 2*(e*x)^(1/3))/(a*e^2/b)^(1/6))/(a*b^5*e^2)^(2/3) + 1/2*(sqrt(3)*b^3*c* e - sqrt(a*b)*b^2*d*e)*arctan(-(sqrt(3)*(a*e^2/b)^(1/6) - 2*(e*x)^(1/3))/( a*e^2/b)^(1/6))/(a*b^5*e^2)^(2/3) - 1/4*(b^3*c*e + sqrt(3)*sqrt(a*b)*b^2*d *e)*log(sqrt(3)*(a*e^2/b)^(1/6)*(e*x)^(1/3) + (e*x)^(2/3) + (a*e^2/b)^(1/3 ))/(a*b^5*e^2)^(2/3) - 1/4*(b^3*c*e + sqrt(3)*sqrt(a*b)*b^2*d*e)*log(-sqrt (3)*(a*e^2/b)^(1/6)*(e*x)^(1/3) + (e*x)^(2/3) + (a*e^2/b)^(1/3))/(a*b^5*e^ 2)^(2/3) + (a*e^2/b)^(5/6)*d*arctan((e*x)^(1/3)/(a*e^2/b)^(1/6))/(a*e^2) + 1/2*(a*b^5*e^2)^(1/3)*c*log((e*x)^(2/3) + (a*e^2/b)^(1/3))/(a*b^2*e)
Time = 0.38 (sec) , antiderivative size = 2094, normalized size of antiderivative = 5.25 \[ \int \frac {c+d x}{\sqrt [3]{e x} \left (a+b x^2\right )} \, dx=\text {Too large to display} \] Input:
int((c + d*x)/((e*x)^(1/3)*(a + b*x^2)),x)
Output:
log((e*x)^(1/3)*(243*b^4*c^5*e^5 + 486*a*b^3*c^3*d^2*e^5 + 243*a^2*b^2*c*d ^4*e^5) - ((1944*a^2*b^5*c^2*e^6 - 1944*a^3*b^4*d^2*e^6)*(e*x)^(1/3)*((a^2 *b^4*c^3 + a*d^3*(-a^5*b^5)^(1/2) - 3*a^3*b^3*c*d^2 - 3*b*c^2*d*(-a^5*b^5) ^(1/2))/(8*a^4*b^5*e))^(1/3) + 972*a^3*b^3*d^3*e^6 - 2916*a^2*b^4*c^2*d*e^ 6)*((a^2*b^4*c^3 + a*d^3*(-a^5*b^5)^(1/2) - 3*a^3*b^3*c*d^2 - 3*b*c^2*d*(- a^5*b^5)^(1/2))/(8*a^4*b^5*e))^(2/3))*((a^2*b^4*c^3 + a*d^3*(-a^5*b^5)^(1/ 2) - 3*a^3*b^3*c*d^2 - 3*b*c^2*d*(-a^5*b^5)^(1/2))/(8*a^4*b^5*e))^(1/3) + log((e*x)^(1/3)*(243*b^4*c^5*e^5 + 486*a*b^3*c^3*d^2*e^5 + 243*a^2*b^2*c*d ^4*e^5) - ((1944*a^2*b^5*c^2*e^6 - 1944*a^3*b^4*d^2*e^6)*(e*x)^(1/3)*((a^2 *b^4*c^3 - a*d^3*(-a^5*b^5)^(1/2) - 3*a^3*b^3*c*d^2 + 3*b*c^2*d*(-a^5*b^5) ^(1/2))/(8*a^4*b^5*e))^(1/3) + 972*a^3*b^3*d^3*e^6 - 2916*a^2*b^4*c^2*d*e^ 6)*((a^2*b^4*c^3 - a*d^3*(-a^5*b^5)^(1/2) - 3*a^3*b^3*c*d^2 + 3*b*c^2*d*(- a^5*b^5)^(1/2))/(8*a^4*b^5*e))^(2/3))*((a^2*b^4*c^3 - a*d^3*(-a^5*b^5)^(1/ 2) - 3*a^3*b^3*c*d^2 + 3*b*c^2*d*(-a^5*b^5)^(1/2))/(8*a^4*b^5*e))^(1/3) - log((e*x)^(1/3)*(243*b^4*c^5*e^5 + 486*a*b^3*c^3*d^2*e^5 + 243*a^2*b^2*c*d ^4*e^5) + ((3^(1/2)*1i)/2 + 1/2)^2*(((3^(1/2)*1i)/2 + 1/2)*(1944*a^2*b^5*c ^2*e^6 - 1944*a^3*b^4*d^2*e^6)*(e*x)^(1/3)*((a^2*b^4*c^3 + a*d^3*(-a^5*b^5 )^(1/2) - 3*a^3*b^3*c*d^2 - 3*b*c^2*d*(-a^5*b^5)^(1/2))/(8*a^4*b^5*e))^(1/ 3) - 972*a^3*b^3*d^3*e^6 + 2916*a^2*b^4*c^2*d*e^6)*((a^2*b^4*c^3 + a*d^3*( -a^5*b^5)^(1/2) - 3*a^3*b^3*c*d^2 - 3*b*c^2*d*(-a^5*b^5)^(1/2))/(8*a^4*...
Time = 0.21 (sec) , antiderivative size = 334, normalized size of antiderivative = 0.84 \[ \int \frac {c+d x}{\sqrt [3]{e x} \left (a+b x^2\right )} \, dx=\frac {-2 b^{\frac {1}{6}} a^{\frac {7}{6}} \mathit {atan} \left (\frac {b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}-2 x^{\frac {1}{3}} b^{\frac {1}{3}}}{b^{\frac {1}{6}} a^{\frac {1}{6}}}\right ) d -2 b^{\frac {2}{3}} a^{\frac {2}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}-2 x^{\frac {1}{3}} b^{\frac {1}{3}}}{b^{\frac {1}{6}} a^{\frac {1}{6}}}\right ) c +2 b^{\frac {1}{6}} a^{\frac {7}{6}} \mathit {atan} \left (\frac {b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+2 x^{\frac {1}{3}} b^{\frac {1}{3}}}{b^{\frac {1}{6}} a^{\frac {1}{6}}}\right ) d -2 b^{\frac {2}{3}} a^{\frac {2}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+2 x^{\frac {1}{3}} b^{\frac {1}{3}}}{b^{\frac {1}{6}} a^{\frac {1}{6}}}\right ) c +4 b^{\frac {1}{6}} a^{\frac {7}{6}} \mathit {atan} \left (\frac {x^{\frac {1}{3}} b^{\frac {1}{6}}}{a^{\frac {1}{6}}}\right ) d +b^{\frac {1}{6}} a^{\frac {7}{6}} \sqrt {3}\, \mathrm {log}\left (-x^{\frac {1}{3}} b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+a^{\frac {1}{3}}+x^{\frac {2}{3}} b^{\frac {1}{3}}\right ) d -b^{\frac {1}{6}} a^{\frac {7}{6}} \sqrt {3}\, \mathrm {log}\left (x^{\frac {1}{3}} b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+a^{\frac {1}{3}}+x^{\frac {2}{3}} b^{\frac {1}{3}}\right ) d +2 b^{\frac {2}{3}} a^{\frac {2}{3}} \mathrm {log}\left (a^{\frac {1}{3}}+x^{\frac {2}{3}} b^{\frac {1}{3}}\right ) c -b^{\frac {2}{3}} a^{\frac {2}{3}} \mathrm {log}\left (-x^{\frac {1}{3}} b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+a^{\frac {1}{3}}+x^{\frac {2}{3}} b^{\frac {1}{3}}\right ) c -b^{\frac {2}{3}} a^{\frac {2}{3}} \mathrm {log}\left (x^{\frac {1}{3}} b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+a^{\frac {1}{3}}+x^{\frac {2}{3}} b^{\frac {1}{3}}\right ) c}{4 e^{\frac {1}{3}} a^{\frac {4}{3}} b} \] Input:
int((d*x+c)/(e*x)^(1/3)/(b*x^2+a),x)
Output:
( - 2*b**(1/6)*a**(1/6)*atan((b**(1/6)*a**(1/6)*sqrt(3) - 2*x**(1/3)*b**(1 /3))/(b**(1/6)*a**(1/6)))*a*d - 2*b**(2/3)*a**(2/3)*sqrt(3)*atan((b**(1/6) *a**(1/6)*sqrt(3) - 2*x**(1/3)*b**(1/3))/(b**(1/6)*a**(1/6)))*c + 2*b**(1/ 6)*a**(1/6)*atan((b**(1/6)*a**(1/6)*sqrt(3) + 2*x**(1/3)*b**(1/3))/(b**(1/ 6)*a**(1/6)))*a*d - 2*b**(2/3)*a**(2/3)*sqrt(3)*atan((b**(1/6)*a**(1/6)*sq rt(3) + 2*x**(1/3)*b**(1/3))/(b**(1/6)*a**(1/6)))*c + 4*b**(1/6)*a**(1/6)* atan((x**(1/3)*b**(1/3))/(b**(1/6)*a**(1/6)))*a*d + b**(1/6)*a**(1/6)*sqrt (3)*log( - x**(1/3)*b**(1/6)*a**(1/6)*sqrt(3) + a**(1/3) + x**(2/3)*b**(1/ 3))*a*d - b**(1/6)*a**(1/6)*sqrt(3)*log(x**(1/3)*b**(1/6)*a**(1/6)*sqrt(3) + a**(1/3) + x**(2/3)*b**(1/3))*a*d + 2*b**(2/3)*a**(2/3)*log(a**(1/3) + x**(2/3)*b**(1/3))*c - b**(2/3)*a**(2/3)*log( - x**(1/3)*b**(1/6)*a**(1/6) *sqrt(3) + a**(1/3) + x**(2/3)*b**(1/3))*c - b**(2/3)*a**(2/3)*log(x**(1/3 )*b**(1/6)*a**(1/6)*sqrt(3) + a**(1/3) + x**(2/3)*b**(1/3))*c)/(4*e**(1/3) *a**(1/3)*a*b)