Integrand size = 20, antiderivative size = 413 \[ \int \frac {x \sqrt [3]{c+d x}}{a+b x^2} \, dx=\frac {3 \sqrt [3]{c+d x}}{b}-\frac {\sqrt {3} \sqrt [3]{\sqrt {b} c-\sqrt {-a} d} \arctan \left (\frac {1+\frac {2 \sqrt [6]{b} \sqrt [3]{c+d x}}{\sqrt [3]{\sqrt {b} c-\sqrt {-a} d}}}{\sqrt {3}}\right )}{2 b^{7/6}}-\frac {\sqrt {3} \sqrt [3]{\sqrt {b} c+\sqrt {-a} d} \arctan \left (\frac {1+\frac {2 \sqrt [6]{b} \sqrt [3]{c+d x}}{\sqrt [3]{\sqrt {b} c+\sqrt {-a} d}}}{\sqrt {3}}\right )}{2 b^{7/6}}-\frac {\sqrt [3]{\sqrt {b} c+\sqrt {-a} d} \log \left (\sqrt {-a}-\sqrt {b} x\right )}{4 b^{7/6}}-\frac {\sqrt [3]{\sqrt {b} c-\sqrt {-a} d} \log \left (\sqrt {-a}+\sqrt {b} x\right )}{4 b^{7/6}}+\frac {3 \sqrt [3]{\sqrt {b} c-\sqrt {-a} d} \log \left (\sqrt [3]{\sqrt {b} c-\sqrt {-a} d}-\sqrt [6]{b} \sqrt [3]{c+d x}\right )}{4 b^{7/6}}+\frac {3 \sqrt [3]{\sqrt {b} c+\sqrt {-a} d} \log \left (\sqrt [3]{\sqrt {b} c+\sqrt {-a} d}-\sqrt [6]{b} \sqrt [3]{c+d x}\right )}{4 b^{7/6}} \] Output:
3*(d*x+c)^(1/3)/b-1/2*3^(1/2)*(b^(1/2)*c-(-a)^(1/2)*d)^(1/3)*arctan(1/3*(1 +2*b^(1/6)*(d*x+c)^(1/3)/(b^(1/2)*c-(-a)^(1/2)*d)^(1/3))*3^(1/2))/b^(7/6)- 1/2*3^(1/2)*(b^(1/2)*c+(-a)^(1/2)*d)^(1/3)*arctan(1/3*(1+2*b^(1/6)*(d*x+c) ^(1/3)/(b^(1/2)*c+(-a)^(1/2)*d)^(1/3))*3^(1/2))/b^(7/6)-1/4*(b^(1/2)*c+(-a )^(1/2)*d)^(1/3)*ln((-a)^(1/2)-b^(1/2)*x)/b^(7/6)-1/4*(b^(1/2)*c-(-a)^(1/2 )*d)^(1/3)*ln((-a)^(1/2)+b^(1/2)*x)/b^(7/6)+3/4*(b^(1/2)*c-(-a)^(1/2)*d)^( 1/3)*ln((b^(1/2)*c-(-a)^(1/2)*d)^(1/3)-b^(1/6)*(d*x+c)^(1/3))/b^(7/6)+3/4* (b^(1/2)*c+(-a)^(1/2)*d)^(1/3)*ln((b^(1/2)*c+(-a)^(1/2)*d)^(1/3)-b^(1/6)*( d*x+c)^(1/3))/b^(7/6)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.08 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.32 \[ \int \frac {x \sqrt [3]{c+d x}}{a+b x^2} \, dx=\frac {3 \sqrt [3]{c+d x}}{b}-\frac {\text {RootSum}\left [b c^2+a d^2-2 b c \text {$\#$1}^3+b \text {$\#$1}^6\&,\frac {-b c^2 \log \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )-a d^2 \log \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )+b c \log \left (\sqrt [3]{c+d x}-\text {$\#$1}\right ) \text {$\#$1}^3}{c \text {$\#$1}^2-\text {$\#$1}^5}\&\right ]}{2 b^2} \] Input:
Integrate[(x*(c + d*x)^(1/3))/(a + b*x^2),x]
Output:
(3*(c + d*x)^(1/3))/b - RootSum[b*c^2 + a*d^2 - 2*b*c*#1^3 + b*#1^6 & , (- (b*c^2*Log[(c + d*x)^(1/3) - #1]) - a*d^2*Log[(c + d*x)^(1/3) - #1] + b*c* Log[(c + d*x)^(1/3) - #1]*#1^3)/(c*#1^2 - #1^5) & ]/(2*b^2)
Time = 1.70 (sec) , antiderivative size = 605, normalized size of antiderivative = 1.46, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {561, 25, 27, 1826, 25, 27, 1752, 750, 16, 25, 1142, 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.
| \(\displaystyle \int \frac {x \sqrt [3]{c+d x}}{a+b x^2} \, dx\) |
| \(\Big \downarrow \) 561 |
| \(\displaystyle \frac {3 \int \frac {x (c+d x)}{\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a}d\sqrt [3]{c+d x}}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {3 \int -\frac {x (c+d x)}{\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a}d\sqrt [3]{c+d x}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {3 \int -\frac {d x (c+d x)}{\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a}d\sqrt [3]{c+d x}}{d^2}\) |
| \(\Big \downarrow \) 1826 |
| \(\displaystyle -\frac {3 \left (-\frac {d^2 \int -\frac {\left (\frac {b c^2}{d^2}+a\right ) d^2-b c (c+d x)}{d^2 \left (\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a\right )}d\sqrt [3]{c+d x}}{b}-\frac {d^2 \sqrt [3]{c+d x}}{b}\right )}{d^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {3 \left (\frac {d^2 \int \frac {b c^2-b (c+d x) c+a d^2}{d^2 \left (\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a\right )}d\sqrt [3]{c+d x}}{b}-\frac {d^2 \sqrt [3]{c+d x}}{b}\right )}{d^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {3 \left (\frac {\int \frac {b c^2-b (c+d x) c+a d^2}{\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a}d\sqrt [3]{c+d x}}{b}-\frac {d^2 \sqrt [3]{c+d x}}{b}\right )}{d^2}\) |
| \(\Big \downarrow \) 1752 |
| \(\displaystyle -\frac {3 \left (\frac {-\frac {1}{2} \sqrt {b} \left (\sqrt {b} c-\sqrt {-a} d\right ) \int \frac {1}{\frac {b (c+d x)}{d^2}-\frac {\sqrt {b} \left (\sqrt {b} c-\sqrt {-a} d\right )}{d^2}}d\sqrt [3]{c+d x}-\frac {1}{2} \sqrt {b} \left (\sqrt {-a} d+\sqrt {b} c\right ) \int \frac {1}{\frac {b (c+d x)}{d^2}-\frac {\sqrt {b} \left (\sqrt {b} c+\sqrt {-a} d\right )}{d^2}}d\sqrt [3]{c+d x}}{b}-\frac {d^2 \sqrt [3]{c+d x}}{b}\right )}{d^2}\) |
| \(\Big \downarrow \) 750 |
| \(\displaystyle -\frac {3 \left (\frac {-\frac {1}{2} \sqrt {b} \left (\sqrt {b} c-\sqrt {-a} d\right ) \left (\frac {d^{4/3} \int -\frac {\sqrt [6]{b} \left (2 \sqrt [3]{\sqrt {b} c-\sqrt {-a} d}+\sqrt [6]{b} \sqrt [3]{c+d x}\right )}{d^{2/3} \left (\frac {\sqrt [3]{b} \left (\sqrt {b} c-\sqrt {-a} d\right )^{2/3}}{d^{4/3}}+\frac {\sqrt {b} \sqrt [3]{c+d x} \sqrt [3]{\sqrt {b} c-\sqrt {-a} d}}{d^{4/3}}+\frac {b^{2/3} (c+d x)^{2/3}}{d^{4/3}}\right )}d\sqrt [3]{c+d x}}{3 \sqrt [3]{b} \left (\sqrt {b} c-\sqrt {-a} d\right )^{2/3}}+\frac {d^{4/3} \int \frac {1}{\frac {\sqrt [3]{b} \sqrt [3]{c+d x}}{d^{2/3}}-\frac {\sqrt [6]{b} \sqrt [3]{\sqrt {b} c-\sqrt {-a} d}}{d^{2/3}}}d\sqrt [3]{c+d x}}{3 \sqrt [3]{b} \left (\sqrt {b} c-\sqrt {-a} d\right )^{2/3}}\right )-\frac {1}{2} \sqrt {b} \left (\sqrt {-a} d+\sqrt {b} c\right ) \left (\frac {d^{4/3} \int -\frac {\sqrt [6]{b} \left (2 \sqrt [3]{\sqrt {b} c+\sqrt {-a} d}+\sqrt [6]{b} \sqrt [3]{c+d x}\right )}{d^{2/3} \left (\frac {\sqrt [3]{b} \left (\sqrt {b} c+\sqrt {-a} d\right )^{2/3}}{d^{4/3}}+\frac {\sqrt {b} \sqrt [3]{c+d x} \sqrt [3]{\sqrt {b} c+\sqrt {-a} d}}{d^{4/3}}+\frac {b^{2/3} (c+d x)^{2/3}}{d^{4/3}}\right )}d\sqrt [3]{c+d x}}{3 \sqrt [3]{b} \left (\sqrt {-a} d+\sqrt {b} c\right )^{2/3}}+\frac {d^{4/3} \int \frac {1}{\frac {\sqrt [3]{b} \sqrt [3]{c+d x}}{d^{2/3}}-\frac {\sqrt [6]{b} \sqrt [3]{\sqrt {b} c+\sqrt {-a} d}}{d^{2/3}}}d\sqrt [3]{c+d x}}{3 \sqrt [3]{b} \left (\sqrt {-a} d+\sqrt {b} c\right )^{2/3}}\right )}{b}-\frac {d^2 \sqrt [3]{c+d x}}{b}\right )}{d^2}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle -\frac {3 \left (\frac {-\frac {1}{2} \sqrt {b} \left (\sqrt {b} c-\sqrt {-a} d\right ) \left (\frac {d^{4/3} \int -\frac {\sqrt [6]{b} \left (2 \sqrt [3]{\sqrt {b} c-\sqrt {-a} d}+\sqrt [6]{b} \sqrt [3]{c+d x}\right )}{d^{2/3} \left (\frac {\sqrt [3]{b} \left (\sqrt {b} c-\sqrt {-a} d\right )^{2/3}}{d^{4/3}}+\frac {\sqrt {b} \sqrt [3]{c+d x} \sqrt [3]{\sqrt {b} c-\sqrt {-a} d}}{d^{4/3}}+\frac {b^{2/3} (c+d x)^{2/3}}{d^{4/3}}\right )}d\sqrt [3]{c+d x}}{3 \sqrt [3]{b} \left (\sqrt {b} c-\sqrt {-a} d\right )^{2/3}}+\frac {d^2 \log \left (\sqrt [3]{\sqrt {b} c-\sqrt {-a} d}-\sqrt [6]{b} \sqrt [3]{c+d x}\right )}{3 b^{2/3} \left (\sqrt {b} c-\sqrt {-a} d\right )^{2/3}}\right )-\frac {1}{2} \sqrt {b} \left (\sqrt {-a} d+\sqrt {b} c\right ) \left (\frac {d^{4/3} \int -\frac {\sqrt [6]{b} \left (2 \sqrt [3]{\sqrt {b} c+\sqrt {-a} d}+\sqrt [6]{b} \sqrt [3]{c+d x}\right )}{d^{2/3} \left (\frac {\sqrt [3]{b} \left (\sqrt {b} c+\sqrt {-a} d\right )^{2/3}}{d^{4/3}}+\frac {\sqrt {b} \sqrt [3]{c+d x} \sqrt [3]{\sqrt {b} c+\sqrt {-a} d}}{d^{4/3}}+\frac {b^{2/3} (c+d x)^{2/3}}{d^{4/3}}\right )}d\sqrt [3]{c+d x}}{3 \sqrt [3]{b} \left (\sqrt {-a} d+\sqrt {b} c\right )^{2/3}}+\frac {d^2 \log \left (\sqrt [3]{\sqrt {-a} d+\sqrt {b} c}-\sqrt [6]{b} \sqrt [3]{c+d x}\right )}{3 b^{2/3} \left (\sqrt {-a} d+\sqrt {b} c\right )^{2/3}}\right )}{b}-\frac {d^2 \sqrt [3]{c+d x}}{b}\right )}{d^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {3 \left (\frac {-\frac {1}{2} \sqrt {b} \left (\sqrt {b} c-\sqrt {-a} d\right ) \left (\frac {d^2 \log \left (\sqrt [3]{\sqrt {b} c-\sqrt {-a} d}-\sqrt [6]{b} \sqrt [3]{c+d x}\right )}{3 b^{2/3} \left (\sqrt {b} c-\sqrt {-a} d\right )^{2/3}}-\frac {d^{4/3} \int \frac {2 \sqrt [3]{\sqrt {b} c-\sqrt {-a} d}+\sqrt [6]{b} \sqrt [3]{c+d x}}{\frac {\sqrt [6]{b} \left (\sqrt {b} c-\sqrt {-a} d\right )^{2/3}}{d^{2/3}}+\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{\sqrt {b} c-\sqrt {-a} d}}{d^{2/3}}+\frac {\sqrt {b} (c+d x)^{2/3}}{d^{2/3}}}d\sqrt [3]{c+d x}}{3 \sqrt [3]{b} \left (\sqrt {b} c-\sqrt {-a} d\right )^{2/3}}\right )-\frac {1}{2} \sqrt {b} \left (\sqrt {-a} d+\sqrt {b} c\right ) \left (\frac {d^2 \log \left (\sqrt [3]{\sqrt {-a} d+\sqrt {b} c}-\sqrt [6]{b} \sqrt [3]{c+d x}\right )}{3 b^{2/3} \left (\sqrt {-a} d+\sqrt {b} c\right )^{2/3}}-\frac {d^{4/3} \int \frac {2 \sqrt [3]{\sqrt {b} c+\sqrt {-a} d}+\sqrt [6]{b} \sqrt [3]{c+d x}}{\frac {\sqrt [6]{b} \left (\sqrt {b} c+\sqrt {-a} d\right )^{2/3}}{d^{2/3}}+\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{\sqrt {b} c+\sqrt {-a} d}}{d^{2/3}}+\frac {\sqrt {b} (c+d x)^{2/3}}{d^{2/3}}}d\sqrt [3]{c+d x}}{3 \sqrt [3]{b} \left (\sqrt {-a} d+\sqrt {b} c\right )^{2/3}}\right )}{b}-\frac {d^2 \sqrt [3]{c+d x}}{b}\right )}{d^2}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle -\frac {3 \left (\frac {-\frac {1}{2} \sqrt {b} \left (\sqrt {b} c-\sqrt {-a} d\right ) \left (\frac {d^2 \log \left (\sqrt [3]{\sqrt {b} c-\sqrt {-a} d}-\sqrt [6]{b} \sqrt [3]{c+d x}\right )}{3 b^{2/3} \left (\sqrt {b} c-\sqrt {-a} d\right )^{2/3}}-\frac {d^{4/3} \left (\frac {3}{2} \sqrt [3]{\sqrt {b} c-\sqrt {-a} d} \int \frac {1}{\frac {\sqrt [6]{b} \left (\sqrt {b} c-\sqrt {-a} d\right )^{2/3}}{d^{2/3}}+\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{\sqrt {b} c-\sqrt {-a} d}}{d^{2/3}}+\frac {\sqrt {b} (c+d x)^{2/3}}{d^{2/3}}}d\sqrt [3]{c+d x}+\frac {d^{2/3} \int \frac {\sqrt [3]{b} \left (\sqrt [3]{\sqrt {b} c-\sqrt {-a} d}+2 \sqrt [6]{b} \sqrt [3]{c+d x}\right )}{d^{2/3} \left (\frac {\sqrt [6]{b} \left (\sqrt {b} c-\sqrt {-a} d\right )^{2/3}}{d^{2/3}}+\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{\sqrt {b} c-\sqrt {-a} d}}{d^{2/3}}+\frac {\sqrt {b} (c+d x)^{2/3}}{d^{2/3}}\right )}d\sqrt [3]{c+d x}}{2 \sqrt [3]{b}}\right )}{3 \sqrt [3]{b} \left (\sqrt {b} c-\sqrt {-a} d\right )^{2/3}}\right )-\frac {1}{2} \sqrt {b} \left (\sqrt {b} c+\sqrt {-a} d\right ) \left (\frac {d^2 \log \left (\sqrt [3]{\sqrt {b} c+\sqrt {-a} d}-\sqrt [6]{b} \sqrt [3]{c+d x}\right )}{3 b^{2/3} \left (\sqrt {b} c+\sqrt {-a} d\right )^{2/3}}-\frac {d^{4/3} \left (\frac {3}{2} \sqrt [3]{\sqrt {b} c+\sqrt {-a} d} \int \frac {1}{\frac {\sqrt [6]{b} \left (\sqrt {b} c+\sqrt {-a} d\right )^{2/3}}{d^{2/3}}+\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{\sqrt {b} c+\sqrt {-a} d}}{d^{2/3}}+\frac {\sqrt {b} (c+d x)^{2/3}}{d^{2/3}}}d\sqrt [3]{c+d x}+\frac {d^{2/3} \int \frac {\sqrt [3]{b} \left (\sqrt [3]{\sqrt {b} c+\sqrt {-a} d}+2 \sqrt [6]{b} \sqrt [3]{c+d x}\right )}{d^{2/3} \left (\frac {\sqrt [6]{b} \left (\sqrt {b} c+\sqrt {-a} d\right )^{2/3}}{d^{2/3}}+\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{\sqrt {b} c+\sqrt {-a} d}}{d^{2/3}}+\frac {\sqrt {b} (c+d x)^{2/3}}{d^{2/3}}\right )}d\sqrt [3]{c+d x}}{2 \sqrt [3]{b}}\right )}{3 \sqrt [3]{b} \left (\sqrt {b} c+\sqrt {-a} d\right )^{2/3}}\right )}{b}-\frac {d^2 \sqrt [3]{c+d x}}{b}\right )}{d^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {3 \left (\frac {-\frac {1}{2} \sqrt {b} \left (\sqrt {b} c-\sqrt {-a} d\right ) \left (\frac {d^2 \log \left (\sqrt [3]{\sqrt {b} c-\sqrt {-a} d}-\sqrt [6]{b} \sqrt [3]{c+d x}\right )}{3 b^{2/3} \left (\sqrt {b} c-\sqrt {-a} d\right )^{2/3}}-\frac {d^{4/3} \left (\frac {3}{2} \sqrt [3]{\sqrt {b} c-\sqrt {-a} d} \int \frac {1}{\frac {\sqrt [6]{b} \left (\sqrt {b} c-\sqrt {-a} d\right )^{2/3}}{d^{2/3}}+\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{\sqrt {b} c-\sqrt {-a} d}}{d^{2/3}}+\frac {\sqrt {b} (c+d x)^{2/3}}{d^{2/3}}}d\sqrt [3]{c+d x}+\frac {1}{2} \int \frac {\sqrt [3]{\sqrt {b} c-\sqrt {-a} d}+2 \sqrt [6]{b} \sqrt [3]{c+d x}}{\frac {\sqrt [6]{b} \left (\sqrt {b} c-\sqrt {-a} d\right )^{2/3}}{d^{2/3}}+\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{\sqrt {b} c-\sqrt {-a} d}}{d^{2/3}}+\frac {\sqrt {b} (c+d x)^{2/3}}{d^{2/3}}}d\sqrt [3]{c+d x}\right )}{3 \sqrt [3]{b} \left (\sqrt {b} c-\sqrt {-a} d\right )^{2/3}}\right )-\frac {1}{2} \sqrt {b} \left (\sqrt {b} c+\sqrt {-a} d\right ) \left (\frac {d^2 \log \left (\sqrt [3]{\sqrt {b} c+\sqrt {-a} d}-\sqrt [6]{b} \sqrt [3]{c+d x}\right )}{3 b^{2/3} \left (\sqrt {b} c+\sqrt {-a} d\right )^{2/3}}-\frac {d^{4/3} \left (\frac {3}{2} \sqrt [3]{\sqrt {b} c+\sqrt {-a} d} \int \frac {1}{\frac {\sqrt [6]{b} \left (\sqrt {b} c+\sqrt {-a} d\right )^{2/3}}{d^{2/3}}+\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{\sqrt {b} c+\sqrt {-a} d}}{d^{2/3}}+\frac {\sqrt {b} (c+d x)^{2/3}}{d^{2/3}}}d\sqrt [3]{c+d x}+\frac {1}{2} \int \frac {\sqrt [3]{\sqrt {b} c+\sqrt {-a} d}+2 \sqrt [6]{b} \sqrt [3]{c+d x}}{\frac {\sqrt [6]{b} \left (\sqrt {b} c+\sqrt {-a} d\right )^{2/3}}{d^{2/3}}+\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{\sqrt {b} c+\sqrt {-a} d}}{d^{2/3}}+\frac {\sqrt {b} (c+d x)^{2/3}}{d^{2/3}}}d\sqrt [3]{c+d x}\right )}{3 \sqrt [3]{b} \left (\sqrt {b} c+\sqrt {-a} d\right )^{2/3}}\right )}{b}-\frac {d^2 \sqrt [3]{c+d x}}{b}\right )}{d^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle -\frac {3 \left (\frac {-\frac {1}{2} \sqrt {b} \left (\sqrt {b} c-\sqrt {-a} d\right ) \left (\frac {d^2 \log \left (\sqrt [3]{\sqrt {b} c-\sqrt {-a} d}-\sqrt [6]{b} \sqrt [3]{c+d x}\right )}{3 b^{2/3} \left (\sqrt {b} c-\sqrt {-a} d\right )^{2/3}}-\frac {d^{4/3} \left (\frac {1}{2} \int \frac {\sqrt [3]{\sqrt {b} c-\sqrt {-a} d}+2 \sqrt [6]{b} \sqrt [3]{c+d x}}{\frac {\sqrt [6]{b} \left (\sqrt {b} c-\sqrt {-a} d\right )^{2/3}}{d^{2/3}}+\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{\sqrt {b} c-\sqrt {-a} d}}{d^{2/3}}+\frac {\sqrt {b} (c+d x)^{2/3}}{d^{2/3}}}d\sqrt [3]{c+d x}-\frac {3 d^{2/3} \int \frac {1}{-(c+d x)^{2/3}-3}d\left (\frac {2 \sqrt [6]{b} \sqrt [3]{c+d x}}{\sqrt [3]{\sqrt {b} c-\sqrt {-a} d}}+1\right )}{\sqrt [3]{b}}\right )}{3 \sqrt [3]{b} \left (\sqrt {b} c-\sqrt {-a} d\right )^{2/3}}\right )-\frac {1}{2} \sqrt {b} \left (\sqrt {-a} d+\sqrt {b} c\right ) \left (\frac {d^2 \log \left (\sqrt [3]{\sqrt {-a} d+\sqrt {b} c}-\sqrt [6]{b} \sqrt [3]{c+d x}\right )}{3 b^{2/3} \left (\sqrt {-a} d+\sqrt {b} c\right )^{2/3}}-\frac {d^{4/3} \left (\frac {1}{2} \int \frac {\sqrt [3]{\sqrt {b} c+\sqrt {-a} d}+2 \sqrt [6]{b} \sqrt [3]{c+d x}}{\frac {\sqrt [6]{b} \left (\sqrt {b} c+\sqrt {-a} d\right )^{2/3}}{d^{2/3}}+\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{\sqrt {b} c+\sqrt {-a} d}}{d^{2/3}}+\frac {\sqrt {b} (c+d x)^{2/3}}{d^{2/3}}}d\sqrt [3]{c+d x}-\frac {3 d^{2/3} \int \frac {1}{-(c+d x)^{2/3}-3}d\left (\frac {2 \sqrt [6]{b} \sqrt [3]{c+d x}}{\sqrt [3]{\sqrt {b} c+\sqrt {-a} d}}+1\right )}{\sqrt [3]{b}}\right )}{3 \sqrt [3]{b} \left (\sqrt {-a} d+\sqrt {b} c\right )^{2/3}}\right )}{b}-\frac {d^2 \sqrt [3]{c+d x}}{b}\right )}{d^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {3 \left (\frac {-\frac {1}{2} \sqrt {b} \left (\sqrt {b} c-\sqrt {-a} d\right ) \left (\frac {d^2 \log \left (\sqrt [3]{\sqrt {b} c-\sqrt {-a} d}-\sqrt [6]{b} \sqrt [3]{c+d x}\right )}{3 b^{2/3} \left (\sqrt {b} c-\sqrt {-a} d\right )^{2/3}}-\frac {d^{4/3} \left (\frac {1}{2} \int \frac {\sqrt [3]{\sqrt {b} c-\sqrt {-a} d}+2 \sqrt [6]{b} \sqrt [3]{c+d x}}{\frac {\sqrt [6]{b} \left (\sqrt {b} c-\sqrt {-a} d\right )^{2/3}}{d^{2/3}}+\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{\sqrt {b} c-\sqrt {-a} d}}{d^{2/3}}+\frac {\sqrt {b} (c+d x)^{2/3}}{d^{2/3}}}d\sqrt [3]{c+d x}+\frac {\sqrt {3} d^{2/3} \arctan \left (\frac {\frac {2 \sqrt [6]{b} \sqrt [3]{c+d x}}{\sqrt [3]{\sqrt {b} c-\sqrt {-a} d}}+1}{\sqrt {3}}\right )}{\sqrt [3]{b}}\right )}{3 \sqrt [3]{b} \left (\sqrt {b} c-\sqrt {-a} d\right )^{2/3}}\right )-\frac {1}{2} \sqrt {b} \left (\sqrt {-a} d+\sqrt {b} c\right ) \left (\frac {d^2 \log \left (\sqrt [3]{\sqrt {-a} d+\sqrt {b} c}-\sqrt [6]{b} \sqrt [3]{c+d x}\right )}{3 b^{2/3} \left (\sqrt {-a} d+\sqrt {b} c\right )^{2/3}}-\frac {d^{4/3} \left (\frac {1}{2} \int \frac {\sqrt [3]{\sqrt {b} c+\sqrt {-a} d}+2 \sqrt [6]{b} \sqrt [3]{c+d x}}{\frac {\sqrt [6]{b} \left (\sqrt {b} c+\sqrt {-a} d\right )^{2/3}}{d^{2/3}}+\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{\sqrt {b} c+\sqrt {-a} d}}{d^{2/3}}+\frac {\sqrt {b} (c+d x)^{2/3}}{d^{2/3}}}d\sqrt [3]{c+d x}+\frac {\sqrt {3} d^{2/3} \arctan \left (\frac {\frac {2 \sqrt [6]{b} \sqrt [3]{c+d x}}{\sqrt [3]{\sqrt {-a} d+\sqrt {b} c}}+1}{\sqrt {3}}\right )}{\sqrt [3]{b}}\right )}{3 \sqrt [3]{b} \left (\sqrt {-a} d+\sqrt {b} c\right )^{2/3}}\right )}{b}-\frac {d^2 \sqrt [3]{c+d x}}{b}\right )}{d^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -\frac {3 \left (\frac {-\frac {1}{2} \sqrt {b} \left (\sqrt {b} c-\sqrt {-a} d\right ) \left (\frac {d^2 \log \left (\sqrt [3]{\sqrt {b} c-\sqrt {-a} d}-\sqrt [6]{b} \sqrt [3]{c+d x}\right )}{3 b^{2/3} \left (\sqrt {b} c-\sqrt {-a} d\right )^{2/3}}-\frac {d^{4/3} \left (\frac {\sqrt {3} d^{2/3} \arctan \left (\frac {\frac {2 \sqrt [6]{b} \sqrt [3]{c+d x}}{\sqrt [3]{\sqrt {b} c-\sqrt {-a} d}}+1}{\sqrt {3}}\right )}{\sqrt [3]{b}}+\frac {d^{2/3} \log \left (\sqrt [6]{b} \sqrt [3]{c+d x} \sqrt [3]{\sqrt {b} c-\sqrt {-a} d}+\left (\sqrt {b} c-\sqrt {-a} d\right )^{2/3}+\sqrt [3]{b} (c+d x)^{2/3}\right )}{2 \sqrt [3]{b}}\right )}{3 \sqrt [3]{b} \left (\sqrt {b} c-\sqrt {-a} d\right )^{2/3}}\right )-\frac {1}{2} \sqrt {b} \left (\sqrt {-a} d+\sqrt {b} c\right ) \left (\frac {d^2 \log \left (\sqrt [3]{\sqrt {-a} d+\sqrt {b} c}-\sqrt [6]{b} \sqrt [3]{c+d x}\right )}{3 b^{2/3} \left (\sqrt {-a} d+\sqrt {b} c\right )^{2/3}}-\frac {d^{4/3} \left (\frac {\sqrt {3} d^{2/3} \arctan \left (\frac {\frac {2 \sqrt [6]{b} \sqrt [3]{c+d x}}{\sqrt [3]{\sqrt {-a} d+\sqrt {b} c}}+1}{\sqrt {3}}\right )}{\sqrt [3]{b}}+\frac {d^{2/3} \log \left (\sqrt [6]{b} \sqrt [3]{c+d x} \sqrt [3]{\sqrt {-a} d+\sqrt {b} c}+\left (\sqrt {-a} d+\sqrt {b} c\right )^{2/3}+\sqrt [3]{b} (c+d x)^{2/3}\right )}{2 \sqrt [3]{b}}\right )}{3 \sqrt [3]{b} \left (\sqrt {-a} d+\sqrt {b} c\right )^{2/3}}\right )}{b}-\frac {d^2 \sqrt [3]{c+d x}}{b}\right )}{d^2}\) |
Input:
Int[(x*(c + d*x)^(1/3))/(a + b*x^2),x]
Output:
(-3*(-((d^2*(c + d*x)^(1/3))/b) + (-1/2*(Sqrt[b]*(Sqrt[b]*c - Sqrt[-a]*d)* ((d^2*Log[(Sqrt[b]*c - Sqrt[-a]*d)^(1/3) - b^(1/6)*(c + d*x)^(1/3)])/(3*b^ (2/3)*(Sqrt[b]*c - Sqrt[-a]*d)^(2/3)) - (d^(4/3)*((Sqrt[3]*d^(2/3)*ArcTan[ (1 + (2*b^(1/6)*(c + d*x)^(1/3))/(Sqrt[b]*c - Sqrt[-a]*d)^(1/3))/Sqrt[3]]) /b^(1/3) + (d^(2/3)*Log[(Sqrt[b]*c - Sqrt[-a]*d)^(2/3) + b^(1/6)*(Sqrt[b]* c - Sqrt[-a]*d)^(1/3)*(c + d*x)^(1/3) + b^(1/3)*(c + d*x)^(2/3)])/(2*b^(1/ 3))))/(3*b^(1/3)*(Sqrt[b]*c - Sqrt[-a]*d)^(2/3)))) - (Sqrt[b]*(Sqrt[b]*c + Sqrt[-a]*d)*((d^2*Log[(Sqrt[b]*c + Sqrt[-a]*d)^(1/3) - b^(1/6)*(c + d*x)^ (1/3)])/(3*b^(2/3)*(Sqrt[b]*c + Sqrt[-a]*d)^(2/3)) - (d^(4/3)*((Sqrt[3]*d^ (2/3)*ArcTan[(1 + (2*b^(1/6)*(c + d*x)^(1/3))/(Sqrt[b]*c + Sqrt[-a]*d)^(1/ 3))/Sqrt[3]])/b^(1/3) + (d^(2/3)*Log[(Sqrt[b]*c + Sqrt[-a]*d)^(2/3) + b^(1 /6)*(Sqrt[b]*c + Sqrt[-a]*d)^(1/3)*(c + d*x)^(1/3) + b^(1/3)*(c + d*x)^(2/ 3)])/(2*b^(1/3))))/(3*b^(1/3)*(Sqrt[b]*c + Sqrt[-a]*d)^(2/3))))/2)/b))/d^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[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> With[{k = Denominator[n]}, Simp[k/d Subst[Int[x^(k*(n + 1) - 1)*(-c /d + x^k/d)^m*Simp[(b*c^2 + a*d^2)/d^2 - 2*b*c*(x^k/d^2) + b*(x^(2*k)/d^2), x]^p, x], x, (c + d*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, m, p}, x] && Frac tionQ[n] && IntegerQ[p] && IntegerQ[m]
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[((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]
Int[((d_) + (e_.)*(x_)^(n_))/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x _Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q)) Int[1/(b/2 - q/2 + c*x^n), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q)) I nt[1/(b/2 + q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2 , 2*n] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && (PosQ[b^2 - 4*a*c] || !IGtQ[n/2, 0])
Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))*((a_) + (b_.)*(x_)^(n_) + ( c_.)*(x_)^(n2_))^(p_), x_Symbol] :> Simp[e*f^(n - 1)*(f*x)^(m - n + 1)*((a + b*x^n + c*x^(2*n))^(p + 1)/(c*(m + n*(2*p + 1) + 1))), x] - Simp[f^n/(c*( m + n*(2*p + 1) + 1)) Int[(f*x)^(m - n)*(a + b*x^n + c*x^(2*n))^p*Simp[a* e*(m - n + 1) + (b*e*(m + n*p + 1) - c*d*(m + n*(2*p + 1) + 1))*x^n, x], x] , x] /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*(2*p + 1) + 1, 0] && Intege rQ[p]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.03 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.21
| method | result | size |
| default | \(\frac {6 \left (d x +c \right )^{\frac {1}{3}} b +\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6} b -2 b c \,\textit {\_Z}^{3}+a \,d^{2}+b \,c^{2}\right )}{\sum }\frac {\ln \left (\left (d x +c \right )^{\frac {1}{3}}-\textit {\_R} \right ) \left (c \left (-\textit {\_R}^{3}+c \right ) b +a \,d^{2}\right )}{\textit {\_R}^{2} \left (-\textit {\_R}^{3}+c \right )}\right )}{2 b^{2}}\) | \(85\) |
| pseudoelliptic | \(\frac {6 \left (d x +c \right )^{\frac {1}{3}} b +\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6} b -2 b c \,\textit {\_Z}^{3}+a \,d^{2}+b \,c^{2}\right )}{\sum }\frac {\ln \left (\left (d x +c \right )^{\frac {1}{3}}-\textit {\_R} \right ) \left (c \left (-\textit {\_R}^{3}+c \right ) b +a \,d^{2}\right )}{\textit {\_R}^{2} \left (-\textit {\_R}^{3}+c \right )}\right )}{2 b^{2}}\) | \(85\) |
| derivativedivides | \(\frac {3 \left (d x +c \right )^{\frac {1}{3}}}{b}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6} b -2 b c \,\textit {\_Z}^{3}+a \,d^{2}+b \,c^{2}\right )}{\sum }\frac {\left (\textit {\_R}^{3} b c -a \,d^{2}-b \,c^{2}\right ) \ln \left (\left (d x +c \right )^{\frac {1}{3}}-\textit {\_R} \right )}{\textit {\_R}^{5}-\textit {\_R}^{2} c}}{2 b^{2}}\) | \(90\) |
Input:
int(x*(d*x+c)^(1/3)/(b*x^2+a),x,method=_RETURNVERBOSE)
Output:
1/2*(6*(d*x+c)^(1/3)*b+sum(ln((d*x+c)^(1/3)-_R)*(c*(-_R^3+c)*b+a*d^2)/_R^2 /(-_R^3+c),_R=RootOf(_Z^6*b-2*_Z^3*b*c+a*d^2+b*c^2)))/b^2
Time = 0.13 (sec) , antiderivative size = 453, normalized size of antiderivative = 1.10 \[ \int \frac {x \sqrt [3]{c+d x}}{a+b x^2} \, dx=-\frac {{\left (\sqrt {-3} b + b\right )} \left (\frac {b^{3} \sqrt {-\frac {a d^{2}}{b^{7}}} + c}{b^{3}}\right )^{\frac {1}{3}} \log \left (\frac {1}{2} \, {\left (\sqrt {-3} b + b\right )} \left (\frac {b^{3} \sqrt {-\frac {a d^{2}}{b^{7}}} + c}{b^{3}}\right )^{\frac {1}{3}} + {\left (d x + c\right )}^{\frac {1}{3}}\right ) - {\left (\sqrt {-3} b - b\right )} \left (\frac {b^{3} \sqrt {-\frac {a d^{2}}{b^{7}}} + c}{b^{3}}\right )^{\frac {1}{3}} \log \left (-\frac {1}{2} \, {\left (\sqrt {-3} b - b\right )} \left (\frac {b^{3} \sqrt {-\frac {a d^{2}}{b^{7}}} + c}{b^{3}}\right )^{\frac {1}{3}} + {\left (d x + c\right )}^{\frac {1}{3}}\right ) - 2 \, b \left (\frac {b^{3} \sqrt {-\frac {a d^{2}}{b^{7}}} + c}{b^{3}}\right )^{\frac {1}{3}} \log \left (-b \left (\frac {b^{3} \sqrt {-\frac {a d^{2}}{b^{7}}} + c}{b^{3}}\right )^{\frac {1}{3}} + {\left (d x + c\right )}^{\frac {1}{3}}\right ) + {\left (\sqrt {-3} b + b\right )} \left (-\frac {b^{3} \sqrt {-\frac {a d^{2}}{b^{7}}} - c}{b^{3}}\right )^{\frac {1}{3}} \log \left (\frac {1}{2} \, {\left (\sqrt {-3} b + b\right )} \left (-\frac {b^{3} \sqrt {-\frac {a d^{2}}{b^{7}}} - c}{b^{3}}\right )^{\frac {1}{3}} + {\left (d x + c\right )}^{\frac {1}{3}}\right ) - {\left (\sqrt {-3} b - b\right )} \left (-\frac {b^{3} \sqrt {-\frac {a d^{2}}{b^{7}}} - c}{b^{3}}\right )^{\frac {1}{3}} \log \left (-\frac {1}{2} \, {\left (\sqrt {-3} b - b\right )} \left (-\frac {b^{3} \sqrt {-\frac {a d^{2}}{b^{7}}} - c}{b^{3}}\right )^{\frac {1}{3}} + {\left (d x + c\right )}^{\frac {1}{3}}\right ) - 2 \, b \left (-\frac {b^{3} \sqrt {-\frac {a d^{2}}{b^{7}}} - c}{b^{3}}\right )^{\frac {1}{3}} \log \left (-b \left (-\frac {b^{3} \sqrt {-\frac {a d^{2}}{b^{7}}} - c}{b^{3}}\right )^{\frac {1}{3}} + {\left (d x + c\right )}^{\frac {1}{3}}\right ) - 12 \, {\left (d x + c\right )}^{\frac {1}{3}}}{4 \, b} \] Input:
integrate(x*(d*x+c)^(1/3)/(b*x^2+a),x, algorithm="fricas")
Output:
-1/4*((sqrt(-3)*b + b)*((b^3*sqrt(-a*d^2/b^7) + c)/b^3)^(1/3)*log(1/2*(sqr t(-3)*b + b)*((b^3*sqrt(-a*d^2/b^7) + c)/b^3)^(1/3) + (d*x + c)^(1/3)) - ( sqrt(-3)*b - b)*((b^3*sqrt(-a*d^2/b^7) + c)/b^3)^(1/3)*log(-1/2*(sqrt(-3)* b - b)*((b^3*sqrt(-a*d^2/b^7) + c)/b^3)^(1/3) + (d*x + c)^(1/3)) - 2*b*((b ^3*sqrt(-a*d^2/b^7) + c)/b^3)^(1/3)*log(-b*((b^3*sqrt(-a*d^2/b^7) + c)/b^3 )^(1/3) + (d*x + c)^(1/3)) + (sqrt(-3)*b + b)*(-(b^3*sqrt(-a*d^2/b^7) - c) /b^3)^(1/3)*log(1/2*(sqrt(-3)*b + b)*(-(b^3*sqrt(-a*d^2/b^7) - c)/b^3)^(1/ 3) + (d*x + c)^(1/3)) - (sqrt(-3)*b - b)*(-(b^3*sqrt(-a*d^2/b^7) - c)/b^3) ^(1/3)*log(-1/2*(sqrt(-3)*b - b)*(-(b^3*sqrt(-a*d^2/b^7) - c)/b^3)^(1/3) + (d*x + c)^(1/3)) - 2*b*(-(b^3*sqrt(-a*d^2/b^7) - c)/b^3)^(1/3)*log(-b*(-( b^3*sqrt(-a*d^2/b^7) - c)/b^3)^(1/3) + (d*x + c)^(1/3)) - 12*(d*x + c)^(1/ 3))/b
\[ \int \frac {x \sqrt [3]{c+d x}}{a+b x^2} \, dx=\int \frac {x \sqrt [3]{c + d x}}{a + b x^{2}}\, dx \] Input:
integrate(x*(d*x+c)**(1/3)/(b*x**2+a),x)
Output:
Integral(x*(c + d*x)**(1/3)/(a + b*x**2), x)
\[ \int \frac {x \sqrt [3]{c+d x}}{a+b x^2} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {1}{3}} x}{b x^{2} + a} \,d x } \] Input:
integrate(x*(d*x+c)^(1/3)/(b*x^2+a),x, algorithm="maxima")
Output:
integrate((d*x + c)^(1/3)*x/(b*x^2 + a), x)
\[ \int \frac {x \sqrt [3]{c+d x}}{a+b x^2} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {1}{3}} x}{b x^{2} + a} \,d x } \] Input:
integrate(x*(d*x+c)^(1/3)/(b*x^2+a),x, algorithm="giac")
Output:
undef
Time = 0.30 (sec) , antiderivative size = 1551, normalized size of antiderivative = 3.76 \[ \int \frac {x \sqrt [3]{c+d x}}{a+b x^2} \, dx=\text {Too large to display} \] Input:
int((x*(c + d*x)^(1/3))/(a + b*x^2),x)
Output:
log(((b^4*c + d*(-a*b^7)^(1/2))/(8*b^7))^(1/3)*(((7776*a^3*b^6*c*d^6 + 777 6*a^2*b^7*c^3*d^4)*((b^4*c + d*(-a*b^7)^(1/2))/(8*b^7))^(1/3) - (3888*a^3* b^5*c*d^6 + 3888*a^2*b^6*c^3*d^4)*(c + d*x)^(1/3))*((b^4*c + d*(-a*b^7)^(1 /2))/(8*b^7))^(2/3) + 972*a^4*b^2*d^8 - 972*a^2*b^4*c^4*d^4) - (486*a^4*b* d^8 - 486*a^2*b^3*c^4*d^4)*(c + d*x)^(1/3))*((b^4*c + d*(-a*b^7)^(1/2))/(8 *b^7))^(1/3) + log(((b^4*c - d*(-a*b^7)^(1/2))/(8*b^7))^(1/3)*(((7776*a^3* b^6*c*d^6 + 7776*a^2*b^7*c^3*d^4)*((b^4*c - d*(-a*b^7)^(1/2))/(8*b^7))^(1/ 3) - (3888*a^3*b^5*c*d^6 + 3888*a^2*b^6*c^3*d^4)*(c + d*x)^(1/3))*((b^4*c - d*(-a*b^7)^(1/2))/(8*b^7))^(2/3) + 972*a^4*b^2*d^8 - 972*a^2*b^4*c^4*d^4 ) - (486*a^4*b*d^8 - 486*a^2*b^3*c^4*d^4)*(c + d*x)^(1/3))*((b^4*c - d*(-a *b^7)^(1/2))/(8*b^7))^(1/3) + (3*(c + d*x)^(1/3))/b + log((486*a^4*b*d^8 - 486*a^2*b^3*c^4*d^4)*(c + d*x)^(1/3) + ((3^(1/2)*1i)/2 - 1/2)*(((3^(1/2)* 1i)/2 - 1/2)^2*((3888*a^3*b^5*c*d^6 + 3888*a^2*b^6*c^3*d^4)*(c + d*x)^(1/3 ) - ((3^(1/2)*1i)/2 - 1/2)*(7776*a^3*b^6*c*d^6 + 7776*a^2*b^7*c^3*d^4)*((b ^4*c + d*(-a*b^7)^(1/2))/(8*b^7))^(1/3))*((b^4*c + d*(-a*b^7)^(1/2))/(8*b^ 7))^(2/3) - 972*a^4*b^2*d^8 + 972*a^2*b^4*c^4*d^4)*((b^4*c + d*(-a*b^7)^(1 /2))/(8*b^7))^(1/3))*((3^(1/2)*1i)/2 - 1/2)*((b^4*c + d*(-a*b^7)^(1/2))/(8 *b^7))^(1/3) - log((486*a^4*b*d^8 - 486*a^2*b^3*c^4*d^4)*(c + d*x)^(1/3) - ((3^(1/2)*1i)/2 + 1/2)*(((3^(1/2)*1i)/2 + 1/2)^2*((3888*a^3*b^5*c*d^6 + 3 888*a^2*b^6*c^3*d^4)*(c + d*x)^(1/3) + ((3^(1/2)*1i)/2 + 1/2)*(7776*a^3...
\[ \int \frac {x \sqrt [3]{c+d x}}{a+b x^2} \, dx=\frac {3 \left (d x +c \right )^{\frac {1}{3}}-\left (\int \frac {\left (d x +c \right )^{\frac {1}{3}}}{b d \,x^{3}+b c \,x^{2}+a d x +a c}d x \right ) a d +\left (\int \frac {\left (d x +c \right )^{\frac {1}{3}} x}{b d \,x^{3}+b c \,x^{2}+a d x +a c}d x \right ) b c}{b} \] Input:
int(x*(d*x+c)^(1/3)/(b*x^2+a),x)
Output:
(3*(c + d*x)**(1/3) - int((c + d*x)**(1/3)/(a*c + a*d*x + b*c*x**2 + b*d*x **3),x)*a*d + int(((c + d*x)**(1/3)*x)/(a*c + a*d*x + b*c*x**2 + b*d*x**3) ,x)*b*c)/b