Integrand size = 20, antiderivative size = 480 \[ \int \frac {x (c+d x)^{4/3}}{a+b x^2} \, dx=\frac {3 \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \sqrt [3]{c+d x}}{2 b}+\frac {3 \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \sqrt [3]{c+d x}}{2 b}+\frac {3 (c+d x)^{4/3}}{4 b}-\frac {\sqrt {3} \left (\sqrt {b} c-\sqrt {-a} d\right )^{4/3} \arctan \left (\frac {1+\frac {2 \sqrt [6]{b} \sqrt [3]{c+d x}}{\sqrt [3]{\sqrt {b} c-\sqrt {-a} d}}}{\sqrt {3}}\right )}{2 b^{5/3}}-\frac {\sqrt {3} \left (\sqrt {b} c+\sqrt {-a} d\right )^{4/3} \arctan \left (\frac {1+\frac {2 \sqrt [6]{b} \sqrt [3]{c+d x}}{\sqrt [3]{\sqrt {b} c+\sqrt {-a} d}}}{\sqrt {3}}\right )}{2 b^{5/3}}-\frac {\left (\sqrt {b} c+\sqrt {-a} d\right )^{4/3} \log \left (\sqrt {-a}-\sqrt {b} x\right )}{4 b^{5/3}}-\frac {\left (\sqrt {b} c-\sqrt {-a} d\right )^{4/3} \log \left (\sqrt {-a}+\sqrt {b} x\right )}{4 b^{5/3}}+\frac {3 \left (\sqrt {b} c-\sqrt {-a} d\right )^{4/3} \log \left (\sqrt [3]{\sqrt {b} c-\sqrt {-a} d}-\sqrt [6]{b} \sqrt [3]{c+d x}\right )}{4 b^{5/3}}+\frac {3 \left (\sqrt {b} c+\sqrt {-a} d\right )^{4/3} \log \left (\sqrt [3]{\sqrt {b} c+\sqrt {-a} d}-\sqrt [6]{b} \sqrt [3]{c+d x}\right )}{4 b^{5/3}} \] Output:
3/2*(c-(-a)^(1/2)*d/b^(1/2))*(d*x+c)^(1/3)/b+3/2*(c+(-a)^(1/2)*d/b^(1/2))* (d*x+c)^(1/3)/b+3/4*(d*x+c)^(4/3)/b-1/2*3^(1/2)*(b^(1/2)*c-(-a)^(1/2)*d)^( 4/3)*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^(5/3)-1/2*3^(1/2)*(b^(1/2)*c+(-a)^(1/2)*d)^(4/3)*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^(5/3)- 1/4*(b^(1/2)*c+(-a)^(1/2)*d)^(4/3)*ln((-a)^(1/2)-b^(1/2)*x)/b^(5/3)-1/4*(b ^(1/2)*c-(-a)^(1/2)*d)^(4/3)*ln((-a)^(1/2)+b^(1/2)*x)/b^(5/3)+3/4*(b^(1/2) *c-(-a)^(1/2)*d)^(4/3)*ln((b^(1/2)*c-(-a)^(1/2)*d)^(1/3)-b^(1/6)*(d*x+c)^( 1/3))/b^(5/3)+3/4*(b^(1/2)*c+(-a)^(1/2)*d)^(4/3)*ln((b^(1/2)*c+(-a)^(1/2)* d)^(1/3)-b^(1/6)*(d*x+c)^(1/3))/b^(5/3)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.12 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.34 \[ \int \frac {x (c+d x)^{4/3}}{a+b x^2} \, dx=\frac {3 b \sqrt [3]{c+d x} (5 c+d x)+2 \text {RootSum}\left [b c^2+a d^2-2 b c \text {$\#$1}^3+b \text {$\#$1}^6\&,\frac {b c^3 \log \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )+a c d^2 \log \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )-b c^2 \log \left (\sqrt [3]{c+d x}-\text {$\#$1}\right ) \text {$\#$1}^3+a d^2 \log \left (\sqrt [3]{c+d x}-\text {$\#$1}\right ) \text {$\#$1}^3}{c \text {$\#$1}^2-\text {$\#$1}^5}\&\right ]}{4 b^2} \] Input:
Integrate[(x*(c + d*x)^(4/3))/(a + b*x^2),x]
Output:
(3*b*(c + d*x)^(1/3)*(5*c + d*x) + 2*RootSum[b*c^2 + a*d^2 - 2*b*c*#1^3 + b*#1^6 & , (b*c^3*Log[(c + d*x)^(1/3) - #1] + a*c*d^2*Log[(c + d*x)^(1/3) - #1] - b*c^2*Log[(c + d*x)^(1/3) - #1]*#1^3 + a*d^2*Log[(c + d*x)^(1/3) - #1]*#1^3)/(c*#1^2 - #1^5) & ])/(4*b^2)
Time = 2.16 (sec) , antiderivative size = 633, normalized size of antiderivative = 1.32, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.850, Rules used = {561, 25, 27, 1826, 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.
| \(\displaystyle \int \frac {x (c+d x)^{4/3}}{a+b x^2} \, dx\) |
| \(\Big \downarrow \) 561 |
| \(\displaystyle \frac {3 \int \frac {x (c+d x)^2}{\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a}d\sqrt [3]{c+d x}}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {3 \int -\frac {x (c+d x)^2}{\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a}d\sqrt [3]{c+d x}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {3 \int -\frac {d x (c+d x)^2}{\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a}d\sqrt [3]{c+d x}}{d^2}\) |
| \(\Big \downarrow \) 1826 |
| \(\displaystyle -\frac {3 \left (-\frac {d^2 \int -\frac {4 (c+d x) \left (\left (\frac {b c^2}{d^2}+a\right ) d^2-b c (c+d x)\right )}{d^2 \left (\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a\right )}d\sqrt [3]{c+d x}}{4 b}-\frac {d^2 (c+d x)^{4/3}}{4 b}\right )}{d^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {3 \left (\frac {\int \frac {(c+d x) \left (b c^2-b (c+d x) c+a d^2\right )}{\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a}d\sqrt [3]{c+d x}}{b}-\frac {d^2 (c+d x)^{4/3}}{4 b}\right )}{d^2}\) |
| \(\Big \downarrow \) 1826 |
| \(\displaystyle -\frac {3 \left (\frac {-\frac {d^2 \int -\frac {b \left (c \left (\frac {b c^2}{d^2}+a\right ) d^2-\left (b c^2-a d^2\right ) (c+d x)\right )}{d^2 \left (\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a\right )}d\sqrt [3]{c+d x}}{b}-c d^2 \sqrt [3]{c+d x}}{b}-\frac {d^2 (c+d x)^{4/3}}{4 b}\right )}{d^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {3 \left (\frac {\frac {d^2 \int \frac {b \left (c \left (b c^2+a d^2\right )-\left (b c^2-a d^2\right ) (c+d x)\right )}{d^2 \left (\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a\right )}d\sqrt [3]{c+d x}}{b}-c d^2 \sqrt [3]{c+d x}}{b}-\frac {d^2 (c+d x)^{4/3}}{4 b}\right )}{d^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {3 \left (\frac {\int \frac {c \left (b c^2+a d^2\right )-\left (b c^2-a d^2\right ) (c+d x)}{\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a}d\sqrt [3]{c+d x}-c d^2 \sqrt [3]{c+d x}}{b}-\frac {d^2 (c+d x)^{4/3}}{4 b}\right )}{d^2}\) |
| \(\Big \downarrow \) 1752 |
| \(\displaystyle -\frac {3 \left (\frac {-\frac {1}{2} \left (-2 \sqrt {-a} \sqrt {b} c d-a d^2+b c^2\right ) \int \frac {1}{\frac {b (c+d x)}{d^2}-\frac {\sqrt {b} \left (\sqrt {b} c-\sqrt {-a} d\right )}{d^2}}d\sqrt [3]{c+d x}-\frac {1}{2} \left (2 \sqrt {-a} \sqrt {b} c d-a d^2+b c^2\right ) \int \frac {1}{\frac {b (c+d x)}{d^2}-\frac {\sqrt {b} \left (\sqrt {b} c+\sqrt {-a} d\right )}{d^2}}d\sqrt [3]{c+d x}-c d^2 \sqrt [3]{c+d x}}{b}-\frac {d^2 (c+d x)^{4/3}}{4 b}\right )}{d^2}\) |
| \(\Big \downarrow \) 750 |
| \(\displaystyle -\frac {3 \left (\frac {-\frac {1}{2} \left (-2 \sqrt {-a} \sqrt {b} c d-a d^2+b c^2\right ) \left (\frac {d^{4/3} \int -\frac {\sqrt [6]{b} \left (2 \sqrt [3]{\sqrt {b} c-\sqrt {-a} d}+\sqrt [6]{b} \sqrt [3]{c+d x}\right )}{d^{2/3} \left (\frac {\sqrt [3]{b} \left (\sqrt {b} c-\sqrt {-a} d\right )^{2/3}}{d^{4/3}}+\frac {\sqrt {b} \sqrt [3]{c+d x} \sqrt [3]{\sqrt {b} c-\sqrt {-a} d}}{d^{4/3}}+\frac {b^{2/3} (c+d x)^{2/3}}{d^{4/3}}\right )}d\sqrt [3]{c+d x}}{3 \sqrt [3]{b} \left (\sqrt {b} c-\sqrt {-a} d\right )^{2/3}}+\frac {d^{4/3} \int \frac {1}{\frac {\sqrt [3]{b} \sqrt [3]{c+d x}}{d^{2/3}}-\frac {\sqrt [6]{b} \sqrt [3]{\sqrt {b} c-\sqrt {-a} d}}{d^{2/3}}}d\sqrt [3]{c+d x}}{3 \sqrt [3]{b} \left (\sqrt {b} c-\sqrt {-a} d\right )^{2/3}}\right )-\frac {1}{2} \left (2 \sqrt {-a} \sqrt {b} c d-a d^2+b c^2\right ) \left (\frac {d^{4/3} \int -\frac {\sqrt [6]{b} \left (2 \sqrt [3]{\sqrt {b} c+\sqrt {-a} d}+\sqrt [6]{b} \sqrt [3]{c+d x}\right )}{d^{2/3} \left (\frac {\sqrt [3]{b} \left (\sqrt {b} c+\sqrt {-a} d\right )^{2/3}}{d^{4/3}}+\frac {\sqrt {b} \sqrt [3]{c+d x} \sqrt [3]{\sqrt {b} c+\sqrt {-a} d}}{d^{4/3}}+\frac {b^{2/3} (c+d x)^{2/3}}{d^{4/3}}\right )}d\sqrt [3]{c+d x}}{3 \sqrt [3]{b} \left (\sqrt {-a} d+\sqrt {b} c\right )^{2/3}}+\frac {d^{4/3} \int \frac {1}{\frac {\sqrt [3]{b} \sqrt [3]{c+d x}}{d^{2/3}}-\frac {\sqrt [6]{b} \sqrt [3]{\sqrt {b} c+\sqrt {-a} d}}{d^{2/3}}}d\sqrt [3]{c+d x}}{3 \sqrt [3]{b} \left (\sqrt {-a} d+\sqrt {b} c\right )^{2/3}}\right )-c d^2 \sqrt [3]{c+d x}}{b}-\frac {d^2 (c+d x)^{4/3}}{4 b}\right )}{d^2}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle -\frac {3 \left (\frac {-\frac {1}{2} \left (-2 \sqrt {-a} \sqrt {b} c d-a d^2+b c^2\right ) \left (\frac {d^{4/3} \int -\frac {\sqrt [6]{b} \left (2 \sqrt [3]{\sqrt {b} c-\sqrt {-a} d}+\sqrt [6]{b} \sqrt [3]{c+d x}\right )}{d^{2/3} \left (\frac {\sqrt [3]{b} \left (\sqrt {b} c-\sqrt {-a} d\right )^{2/3}}{d^{4/3}}+\frac {\sqrt {b} \sqrt [3]{c+d x} \sqrt [3]{\sqrt {b} c-\sqrt {-a} d}}{d^{4/3}}+\frac {b^{2/3} (c+d x)^{2/3}}{d^{4/3}}\right )}d\sqrt [3]{c+d x}}{3 \sqrt [3]{b} \left (\sqrt {b} c-\sqrt {-a} d\right )^{2/3}}+\frac {d^2 \log \left (\sqrt [3]{\sqrt {b} c-\sqrt {-a} d}-\sqrt [6]{b} \sqrt [3]{c+d x}\right )}{3 b^{2/3} \left (\sqrt {b} c-\sqrt {-a} d\right )^{2/3}}\right )-\frac {1}{2} \left (2 \sqrt {-a} \sqrt {b} c d-a d^2+b c^2\right ) \left (\frac {d^{4/3} \int -\frac {\sqrt [6]{b} \left (2 \sqrt [3]{\sqrt {b} c+\sqrt {-a} d}+\sqrt [6]{b} \sqrt [3]{c+d x}\right )}{d^{2/3} \left (\frac {\sqrt [3]{b} \left (\sqrt {b} c+\sqrt {-a} d\right )^{2/3}}{d^{4/3}}+\frac {\sqrt {b} \sqrt [3]{c+d x} \sqrt [3]{\sqrt {b} c+\sqrt {-a} d}}{d^{4/3}}+\frac {b^{2/3} (c+d x)^{2/3}}{d^{4/3}}\right )}d\sqrt [3]{c+d x}}{3 \sqrt [3]{b} \left (\sqrt {-a} d+\sqrt {b} c\right )^{2/3}}+\frac {d^2 \log \left (\sqrt [3]{\sqrt {-a} d+\sqrt {b} c}-\sqrt [6]{b} \sqrt [3]{c+d x}\right )}{3 b^{2/3} \left (\sqrt {-a} d+\sqrt {b} c\right )^{2/3}}\right )-c d^2 \sqrt [3]{c+d x}}{b}-\frac {d^2 (c+d x)^{4/3}}{4 b}\right )}{d^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {3 \left (\frac {-\frac {1}{2} \left (-2 \sqrt {-a} \sqrt {b} c d-a d^2+b c^2\right ) \left (\frac {d^2 \log \left (\sqrt [3]{\sqrt {b} c-\sqrt {-a} d}-\sqrt [6]{b} \sqrt [3]{c+d x}\right )}{3 b^{2/3} \left (\sqrt {b} c-\sqrt {-a} d\right )^{2/3}}-\frac {d^{4/3} \int \frac {2 \sqrt [3]{\sqrt {b} c-\sqrt {-a} d}+\sqrt [6]{b} \sqrt [3]{c+d x}}{\frac {\sqrt [6]{b} \left (\sqrt {b} c-\sqrt {-a} d\right )^{2/3}}{d^{2/3}}+\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{\sqrt {b} c-\sqrt {-a} d}}{d^{2/3}}+\frac {\sqrt {b} (c+d x)^{2/3}}{d^{2/3}}}d\sqrt [3]{c+d x}}{3 \sqrt [3]{b} \left (\sqrt {b} c-\sqrt {-a} d\right )^{2/3}}\right )-\frac {1}{2} \left (2 \sqrt {-a} \sqrt {b} c d-a d^2+b c^2\right ) \left (\frac {d^2 \log \left (\sqrt [3]{\sqrt {-a} d+\sqrt {b} c}-\sqrt [6]{b} \sqrt [3]{c+d x}\right )}{3 b^{2/3} \left (\sqrt {-a} d+\sqrt {b} c\right )^{2/3}}-\frac {d^{4/3} \int \frac {2 \sqrt [3]{\sqrt {b} c+\sqrt {-a} d}+\sqrt [6]{b} \sqrt [3]{c+d x}}{\frac {\sqrt [6]{b} \left (\sqrt {b} c+\sqrt {-a} d\right )^{2/3}}{d^{2/3}}+\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{\sqrt {b} c+\sqrt {-a} d}}{d^{2/3}}+\frac {\sqrt {b} (c+d x)^{2/3}}{d^{2/3}}}d\sqrt [3]{c+d x}}{3 \sqrt [3]{b} \left (\sqrt {-a} d+\sqrt {b} c\right )^{2/3}}\right )-c d^2 \sqrt [3]{c+d x}}{b}-\frac {d^2 (c+d x)^{4/3}}{4 b}\right )}{d^2}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle -\frac {3 \left (\frac {-c \sqrt [3]{c+d x} d^2-\frac {1}{2} \left (b c^2-2 \sqrt {-a} \sqrt {b} d c-a d^2\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} \left (b c^2+2 \sqrt {-a} \sqrt {b} d c-a d^2\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 (c+d x)^{4/3}}{4 b}\right )}{d^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {3 \left (\frac {-c \sqrt [3]{c+d x} d^2-\frac {1}{2} \left (b c^2-2 \sqrt {-a} \sqrt {b} d c-a d^2\right ) \left (\frac {d^2 \log \left (\sqrt [3]{\sqrt {b} c-\sqrt {-a} d}-\sqrt [6]{b} \sqrt [3]{c+d x}\right )}{3 b^{2/3} \left (\sqrt {b} c-\sqrt {-a} d\right )^{2/3}}-\frac {d^{4/3} \left (\frac {3}{2} \sqrt [3]{\sqrt {b} c-\sqrt {-a} d} \int \frac {1}{\frac {\sqrt [6]{b} \left (\sqrt {b} c-\sqrt {-a} d\right )^{2/3}}{d^{2/3}}+\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{\sqrt {b} c-\sqrt {-a} d}}{d^{2/3}}+\frac {\sqrt {b} (c+d x)^{2/3}}{d^{2/3}}}d\sqrt [3]{c+d x}+\frac {1}{2} \int \frac {\sqrt [3]{\sqrt {b} c-\sqrt {-a} d}+2 \sqrt [6]{b} \sqrt [3]{c+d x}}{\frac {\sqrt [6]{b} \left (\sqrt {b} c-\sqrt {-a} d\right )^{2/3}}{d^{2/3}}+\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{\sqrt {b} c-\sqrt {-a} d}}{d^{2/3}}+\frac {\sqrt {b} (c+d x)^{2/3}}{d^{2/3}}}d\sqrt [3]{c+d x}\right )}{3 \sqrt [3]{b} \left (\sqrt {b} c-\sqrt {-a} d\right )^{2/3}}\right )-\frac {1}{2} \left (b c^2+2 \sqrt {-a} \sqrt {b} d c-a d^2\right ) \left (\frac {d^2 \log \left (\sqrt [3]{\sqrt {b} c+\sqrt {-a} d}-\sqrt [6]{b} \sqrt [3]{c+d x}\right )}{3 b^{2/3} \left (\sqrt {b} c+\sqrt {-a} d\right )^{2/3}}-\frac {d^{4/3} \left (\frac {3}{2} \sqrt [3]{\sqrt {b} c+\sqrt {-a} d} \int \frac {1}{\frac {\sqrt [6]{b} \left (\sqrt {b} c+\sqrt {-a} d\right )^{2/3}}{d^{2/3}}+\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{\sqrt {b} c+\sqrt {-a} d}}{d^{2/3}}+\frac {\sqrt {b} (c+d x)^{2/3}}{d^{2/3}}}d\sqrt [3]{c+d x}+\frac {1}{2} \int \frac {\sqrt [3]{\sqrt {b} c+\sqrt {-a} d}+2 \sqrt [6]{b} \sqrt [3]{c+d x}}{\frac {\sqrt [6]{b} \left (\sqrt {b} c+\sqrt {-a} d\right )^{2/3}}{d^{2/3}}+\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{\sqrt {b} c+\sqrt {-a} d}}{d^{2/3}}+\frac {\sqrt {b} (c+d x)^{2/3}}{d^{2/3}}}d\sqrt [3]{c+d x}\right )}{3 \sqrt [3]{b} \left (\sqrt {b} c+\sqrt {-a} d\right )^{2/3}}\right )}{b}-\frac {d^2 (c+d x)^{4/3}}{4 b}\right )}{d^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle -\frac {3 \left (\frac {-\frac {1}{2} \left (-2 \sqrt {-a} \sqrt {b} c d-a d^2+b c^2\right ) \left (\frac {d^2 \log \left (\sqrt [3]{\sqrt {b} c-\sqrt {-a} d}-\sqrt [6]{b} \sqrt [3]{c+d x}\right )}{3 b^{2/3} \left (\sqrt {b} c-\sqrt {-a} d\right )^{2/3}}-\frac {d^{4/3} \left (\frac {1}{2} \int \frac {\sqrt [3]{\sqrt {b} c-\sqrt {-a} d}+2 \sqrt [6]{b} \sqrt [3]{c+d x}}{\frac {\sqrt [6]{b} \left (\sqrt {b} c-\sqrt {-a} d\right )^{2/3}}{d^{2/3}}+\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{\sqrt {b} c-\sqrt {-a} d}}{d^{2/3}}+\frac {\sqrt {b} (c+d x)^{2/3}}{d^{2/3}}}d\sqrt [3]{c+d x}-\frac {3 d^{2/3} \int \frac {1}{-(c+d x)^{2/3}-3}d\left (\frac {2 \sqrt [6]{b} \sqrt [3]{c+d x}}{\sqrt [3]{\sqrt {b} c-\sqrt {-a} d}}+1\right )}{\sqrt [3]{b}}\right )}{3 \sqrt [3]{b} \left (\sqrt {b} c-\sqrt {-a} d\right )^{2/3}}\right )-\frac {1}{2} \left (2 \sqrt {-a} \sqrt {b} c d-a d^2+b c^2\right ) \left (\frac {d^2 \log \left (\sqrt [3]{\sqrt {-a} d+\sqrt {b} c}-\sqrt [6]{b} \sqrt [3]{c+d x}\right )}{3 b^{2/3} \left (\sqrt {-a} d+\sqrt {b} c\right )^{2/3}}-\frac {d^{4/3} \left (\frac {1}{2} \int \frac {\sqrt [3]{\sqrt {b} c+\sqrt {-a} d}+2 \sqrt [6]{b} \sqrt [3]{c+d x}}{\frac {\sqrt [6]{b} \left (\sqrt {b} c+\sqrt {-a} d\right )^{2/3}}{d^{2/3}}+\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{\sqrt {b} c+\sqrt {-a} d}}{d^{2/3}}+\frac {\sqrt {b} (c+d x)^{2/3}}{d^{2/3}}}d\sqrt [3]{c+d x}-\frac {3 d^{2/3} \int \frac {1}{-(c+d x)^{2/3}-3}d\left (\frac {2 \sqrt [6]{b} \sqrt [3]{c+d x}}{\sqrt [3]{\sqrt {b} c+\sqrt {-a} d}}+1\right )}{\sqrt [3]{b}}\right )}{3 \sqrt [3]{b} \left (\sqrt {-a} d+\sqrt {b} c\right )^{2/3}}\right )-c d^2 \sqrt [3]{c+d x}}{b}-\frac {d^2 (c+d x)^{4/3}}{4 b}\right )}{d^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {3 \left (\frac {-\frac {1}{2} \left (-2 \sqrt {-a} \sqrt {b} c d-a d^2+b c^2\right ) \left (\frac {d^2 \log \left (\sqrt [3]{\sqrt {b} c-\sqrt {-a} d}-\sqrt [6]{b} \sqrt [3]{c+d x}\right )}{3 b^{2/3} \left (\sqrt {b} c-\sqrt {-a} d\right )^{2/3}}-\frac {d^{4/3} \left (\frac {1}{2} \int \frac {\sqrt [3]{\sqrt {b} c-\sqrt {-a} d}+2 \sqrt [6]{b} \sqrt [3]{c+d x}}{\frac {\sqrt [6]{b} \left (\sqrt {b} c-\sqrt {-a} d\right )^{2/3}}{d^{2/3}}+\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{\sqrt {b} c-\sqrt {-a} d}}{d^{2/3}}+\frac {\sqrt {b} (c+d x)^{2/3}}{d^{2/3}}}d\sqrt [3]{c+d x}+\frac {\sqrt {3} d^{2/3} \arctan \left (\frac {\frac {2 \sqrt [6]{b} \sqrt [3]{c+d x}}{\sqrt [3]{\sqrt {b} c-\sqrt {-a} d}}+1}{\sqrt {3}}\right )}{\sqrt [3]{b}}\right )}{3 \sqrt [3]{b} \left (\sqrt {b} c-\sqrt {-a} d\right )^{2/3}}\right )-\frac {1}{2} \left (2 \sqrt {-a} \sqrt {b} c d-a d^2+b c^2\right ) \left (\frac {d^2 \log \left (\sqrt [3]{\sqrt {-a} d+\sqrt {b} c}-\sqrt [6]{b} \sqrt [3]{c+d x}\right )}{3 b^{2/3} \left (\sqrt {-a} d+\sqrt {b} c\right )^{2/3}}-\frac {d^{4/3} \left (\frac {1}{2} \int \frac {\sqrt [3]{\sqrt {b} c+\sqrt {-a} d}+2 \sqrt [6]{b} \sqrt [3]{c+d x}}{\frac {\sqrt [6]{b} \left (\sqrt {b} c+\sqrt {-a} d\right )^{2/3}}{d^{2/3}}+\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{\sqrt {b} c+\sqrt {-a} d}}{d^{2/3}}+\frac {\sqrt {b} (c+d x)^{2/3}}{d^{2/3}}}d\sqrt [3]{c+d x}+\frac {\sqrt {3} d^{2/3} \arctan \left (\frac {\frac {2 \sqrt [6]{b} \sqrt [3]{c+d x}}{\sqrt [3]{\sqrt {-a} d+\sqrt {b} c}}+1}{\sqrt {3}}\right )}{\sqrt [3]{b}}\right )}{3 \sqrt [3]{b} \left (\sqrt {-a} d+\sqrt {b} c\right )^{2/3}}\right )-c d^2 \sqrt [3]{c+d x}}{b}-\frac {d^2 (c+d x)^{4/3}}{4 b}\right )}{d^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -\frac {3 \left (\frac {-\frac {1}{2} \left (-2 \sqrt {-a} \sqrt {b} c d-a d^2+b c^2\right ) \left (\frac {d^2 \log \left (\sqrt [3]{\sqrt {b} c-\sqrt {-a} d}-\sqrt [6]{b} \sqrt [3]{c+d x}\right )}{3 b^{2/3} \left (\sqrt {b} c-\sqrt {-a} d\right )^{2/3}}-\frac {d^{4/3} \left (\frac {\sqrt {3} d^{2/3} \arctan \left (\frac {\frac {2 \sqrt [6]{b} \sqrt [3]{c+d x}}{\sqrt [3]{\sqrt {b} c-\sqrt {-a} d}}+1}{\sqrt {3}}\right )}{\sqrt [3]{b}}+\frac {d^{2/3} \log \left (\sqrt [6]{b} \sqrt [3]{c+d x} \sqrt [3]{\sqrt {b} c-\sqrt {-a} d}+\left (\sqrt {b} c-\sqrt {-a} d\right )^{2/3}+\sqrt [3]{b} (c+d x)^{2/3}\right )}{2 \sqrt [3]{b}}\right )}{3 \sqrt [3]{b} \left (\sqrt {b} c-\sqrt {-a} d\right )^{2/3}}\right )-\frac {1}{2} \left (2 \sqrt {-a} \sqrt {b} c d-a d^2+b c^2\right ) \left (\frac {d^2 \log \left (\sqrt [3]{\sqrt {-a} d+\sqrt {b} c}-\sqrt [6]{b} \sqrt [3]{c+d x}\right )}{3 b^{2/3} \left (\sqrt {-a} d+\sqrt {b} c\right )^{2/3}}-\frac {d^{4/3} \left (\frac {\sqrt {3} d^{2/3} \arctan \left (\frac {\frac {2 \sqrt [6]{b} \sqrt [3]{c+d x}}{\sqrt [3]{\sqrt {-a} d+\sqrt {b} c}}+1}{\sqrt {3}}\right )}{\sqrt [3]{b}}+\frac {d^{2/3} \log \left (\sqrt [6]{b} \sqrt [3]{c+d x} \sqrt [3]{\sqrt {-a} d+\sqrt {b} c}+\left (\sqrt {-a} d+\sqrt {b} c\right )^{2/3}+\sqrt [3]{b} (c+d x)^{2/3}\right )}{2 \sqrt [3]{b}}\right )}{3 \sqrt [3]{b} \left (\sqrt {-a} d+\sqrt {b} c\right )^{2/3}}\right )-c d^2 \sqrt [3]{c+d x}}{b}-\frac {d^2 (c+d x)^{4/3}}{4 b}\right )}{d^2}\) |
Input:
Int[(x*(c + d*x)^(4/3))/(a + b*x^2),x]
Output:
(-3*(-1/4*(d^2*(c + d*x)^(4/3))/b + (-(c*d^2*(c + d*x)^(1/3)) - ((b*c^2 - 2*Sqrt[-a]*Sqrt[b]*c*d - a*d^2)*((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*c^2 + 2*Sqrt[-a]*Sqrt[b]*c*d - a*d^2)*((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)*(S qrt[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 = 1.60 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.22
| method | result | size |
| default | \(\frac {-2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6} b -2 b c \,\textit {\_Z}^{3}+a \,d^{2}+b \,c^{2}\right )}{\sum }\frac {\left (\textit {\_R}^{3} a \,d^{2}-\textit {\_R}^{3} b \,c^{2}+a \,d^{2} c +b \,c^{3}\right ) \ln \left (\left (d x +c \right )^{\frac {1}{3}}-\textit {\_R} \right )}{\textit {\_R}^{2} \left (\textit {\_R}^{3}-c \right )}\right )+3 \left (d x +5 c \right ) b \left (d x +c \right )^{\frac {1}{3}}}{4 b^{2}}\) | \(107\) |
| pseudoelliptic | \(\frac {-2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6} b -2 b c \,\textit {\_Z}^{3}+a \,d^{2}+b \,c^{2}\right )}{\sum }\frac {\left (\textit {\_R}^{3} a \,d^{2}-\textit {\_R}^{3} b \,c^{2}+a \,d^{2} c +b \,c^{3}\right ) \ln \left (\left (d x +c \right )^{\frac {1}{3}}-\textit {\_R} \right )}{\textit {\_R}^{2} \left (\textit {\_R}^{3}-c \right )}\right )+3 \left (d x +5 c \right ) b \left (d x +c \right )^{\frac {1}{3}}}{4 b^{2}}\) | \(107\) |
| derivativedivides | \(\frac {\frac {3 \left (d x +c \right )^{\frac {4}{3}}}{4}+3 c \left (d x +c \right )^{\frac {1}{3}}}{b}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6} b -2 b c \,\textit {\_Z}^{3}+a \,d^{2}+b \,c^{2}\right )}{\sum }\frac {\left (\left (-a \,d^{2}+b \,c^{2}\right ) \textit {\_R}^{3}-a \,d^{2} c -b \,c^{3}\right ) \ln \left (\left (d x +c \right )^{\frac {1}{3}}-\textit {\_R} \right )}{\textit {\_R}^{5}-\textit {\_R}^{2} c}}{2 b^{2}}\) | \(113\) |
Input:
int(x*(d*x+c)^(4/3)/(b*x^2+a),x,method=_RETURNVERBOSE)
Output:
1/4*(-2*sum((_R^3*a*d^2-_R^3*b*c^2+a*c*d^2+b*c^3)*ln((d*x+c)^(1/3)-_R)/_R^ 2/(_R^3-c),_R=RootOf(_Z^6*b-2*_Z^3*b*c+a*d^2+b*c^2))+3*(d*x+5*c)*b*(d*x+c) ^(1/3))/b^2
Leaf count of result is larger than twice the leaf count of optimal. 1695 vs. \(2 (346) = 692\).
Time = 0.16 (sec) , antiderivative size = 1695, normalized size of antiderivative = 3.53 \[ \int \frac {x (c+d x)^{4/3}}{a+b x^2} \, dx=\text {Too large to display} \] Input:
integrate(x*(d*x+c)^(4/3)/(b*x^2+a),x, algorithm="fricas")
Output:
1/4*(2*b*((b^2*c^4 - 6*a*b*c^2*d^2 + a^2*d^4 + 4*b^5*sqrt(-(a*b^2*c^6*d^2 - 2*a^2*b*c^4*d^4 + a^3*c^2*d^6)/b^9))/b^5)^(1/3)*log(-(b^2*c^5 - a^2*c*d^ 4)*(d*x + c)^(1/3) + (b^3*c^4 - a*b^2*c^2*d^2 - b^6*sqrt(-(a*b^2*c^6*d^2 - 2*a^2*b*c^4*d^4 + a^3*c^2*d^6)/b^9))*((b^2*c^4 - 6*a*b*c^2*d^2 + a^2*d^4 + 4*b^5*sqrt(-(a*b^2*c^6*d^2 - 2*a^2*b*c^4*d^4 + a^3*c^2*d^6)/b^9))/b^5)^( 1/3)) - (sqrt(-3)*b + b)*((b^2*c^4 - 6*a*b*c^2*d^2 + a^2*d^4 + 4*b^5*sqrt( -(a*b^2*c^6*d^2 - 2*a^2*b*c^4*d^4 + a^3*c^2*d^6)/b^9))/b^5)^(1/3)*log(-(b^ 2*c^5 - a^2*c*d^4)*(d*x + c)^(1/3) - 1/2*(b^3*c^4 - a*b^2*c^2*d^2 + sqrt(- 3)*(b^3*c^4 - a*b^2*c^2*d^2) - (sqrt(-3)*b^6 + b^6)*sqrt(-(a*b^2*c^6*d^2 - 2*a^2*b*c^4*d^4 + a^3*c^2*d^6)/b^9))*((b^2*c^4 - 6*a*b*c^2*d^2 + a^2*d^4 + 4*b^5*sqrt(-(a*b^2*c^6*d^2 - 2*a^2*b*c^4*d^4 + a^3*c^2*d^6)/b^9))/b^5)^( 1/3)) + (sqrt(-3)*b - b)*((b^2*c^4 - 6*a*b*c^2*d^2 + a^2*d^4 + 4*b^5*sqrt( -(a*b^2*c^6*d^2 - 2*a^2*b*c^4*d^4 + a^3*c^2*d^6)/b^9))/b^5)^(1/3)*log(-(b^ 2*c^5 - a^2*c*d^4)*(d*x + c)^(1/3) - 1/2*(b^3*c^4 - a*b^2*c^2*d^2 - sqrt(- 3)*(b^3*c^4 - a*b^2*c^2*d^2) + (sqrt(-3)*b^6 - b^6)*sqrt(-(a*b^2*c^6*d^2 - 2*a^2*b*c^4*d^4 + a^3*c^2*d^6)/b^9))*((b^2*c^4 - 6*a*b*c^2*d^2 + a^2*d^4 + 4*b^5*sqrt(-(a*b^2*c^6*d^2 - 2*a^2*b*c^4*d^4 + a^3*c^2*d^6)/b^9))/b^5)^( 1/3)) + 2*b*((b^2*c^4 - 6*a*b*c^2*d^2 + a^2*d^4 - 4*b^5*sqrt(-(a*b^2*c^6*d ^2 - 2*a^2*b*c^4*d^4 + a^3*c^2*d^6)/b^9))/b^5)^(1/3)*log(-(b^2*c^5 - a^2*c *d^4)*(d*x + c)^(1/3) + (b^3*c^4 - a*b^2*c^2*d^2 + b^6*sqrt(-(a*b^2*c^6...
\[ \int \frac {x (c+d x)^{4/3}}{a+b x^2} \, dx=\int \frac {x \left (c + d x\right )^{\frac {4}{3}}}{a + b x^{2}}\, dx \] Input:
integrate(x*(d*x+c)**(4/3)/(b*x**2+a),x)
Output:
Integral(x*(c + d*x)**(4/3)/(a + b*x**2), x)
\[ \int \frac {x (c+d x)^{4/3}}{a+b x^2} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {4}{3}} x}{b x^{2} + a} \,d x } \] Input:
integrate(x*(d*x+c)^(4/3)/(b*x^2+a),x, algorithm="maxima")
Output:
integrate((d*x + c)^(4/3)*x/(b*x^2 + a), x)
\[ \int \frac {x (c+d x)^{4/3}}{a+b x^2} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {4}{3}} x}{b x^{2} + a} \,d x } \] Input:
integrate(x*(d*x+c)^(4/3)/(b*x^2+a),x, algorithm="giac")
Output:
integrate((d*x + c)^(4/3)*x/(b*x^2 + a), x)
Time = 3.71 (sec) , antiderivative size = 2788, normalized size of antiderivative = 5.81 \[ \int \frac {x (c+d x)^{4/3}}{a+b x^2} \, dx=\text {Too large to display} \] Input:
int((x*(c + d*x)^(4/3))/(a + b*x^2),x)
Output:
log((486*a^2*d^4*(a*d^2 + b*c^2)^2*(c + d*x)^(1/3)*(a^2*d^4 + b^2*c^4 - 6* a*b*c^2*d^2))/b - (((b^7*c^4 + a^2*b^5*d^4 - 6*a*b^6*c^2*d^2 - 4*a*c*d^3*( -a*b^11)^(1/2) + 4*b*c^3*d*(-a*b^11)^(1/2))/b^10)^(1/3)*(((3888*a^2*b^4*d^ 4*(a*d^2 + b*c^2)^2*(c + d*x)^(1/3) - 3888*a^2*b^6*c*d^4*(a*d^2 + b*c^2)*( (b^7*c^4 + a^2*b^5*d^4 - 6*a*b^6*c^2*d^2 - 4*a*c*d^3*(-a*b^11)^(1/2) + 4*b *c^3*d*(-a*b^11)^(1/2))/b^10)^(1/3))*((b^7*c^4 + a^2*b^5*d^4 - 6*a*b^6*c^2 *d^2 - 4*a*c*d^3*(-a*b^11)^(1/2) + 4*b*c^3*d*(-a*b^11)^(1/2))/b^10)^(2/3)) /4 + 972*a^2*b^4*c^7*d^4 - 8748*a^3*b^3*c^5*d^6 - 4860*a^4*b^2*c^3*d^8 + 4 860*a^5*b*c*d^10))/2)*((b^7*c^4 + a^2*b^5*d^4 - 6*a*b^6*c^2*d^2 - 4*a*c*d^ 3*(-a*b^11)^(1/2) + 4*b*c^3*d*(-a*b^11)^(1/2))/(8*b^10))^(1/3) + log((486* a^2*d^4*(a*d^2 + b*c^2)^2*(c + d*x)^(1/3)*(a^2*d^4 + b^2*c^4 - 6*a*b*c^2*d ^2))/b - (((b^7*c^4 + a^2*b^5*d^4 - 6*a*b^6*c^2*d^2 + 4*a*c*d^3*(-a*b^11)^ (1/2) - 4*b*c^3*d*(-a*b^11)^(1/2))/b^10)^(1/3)*(((3888*a^2*b^4*d^4*(a*d^2 + b*c^2)^2*(c + d*x)^(1/3) - 3888*a^2*b^6*c*d^4*(a*d^2 + b*c^2)*((b^7*c^4 + a^2*b^5*d^4 - 6*a*b^6*c^2*d^2 + 4*a*c*d^3*(-a*b^11)^(1/2) - 4*b*c^3*d*(- a*b^11)^(1/2))/b^10)^(1/3))*((b^7*c^4 + a^2*b^5*d^4 - 6*a*b^6*c^2*d^2 + 4* a*c*d^3*(-a*b^11)^(1/2) - 4*b*c^3*d*(-a*b^11)^(1/2))/b^10)^(2/3))/4 + 972* a^2*b^4*c^7*d^4 - 8748*a^3*b^3*c^5*d^6 - 4860*a^4*b^2*c^3*d^8 + 4860*a^5*b *c*d^10))/2)*((b^7*c^4 + a^2*b^5*d^4 - 6*a*b^6*c^2*d^2 + 4*a*c*d^3*(-a*b^1 1)^(1/2) - 4*b*c^3*d*(-a*b^11)^(1/2))/(8*b^10))^(1/3) + (3*(c + d*x)^(4...
\[ \int \frac {x (c+d x)^{4/3}}{a+b x^2} \, dx=\frac {15 \left (d x +c \right )^{\frac {1}{3}} c +3 \left (d x +c \right )^{\frac {1}{3}} d x -8 \left (\int \frac {\left (d x +c \right )^{\frac {1}{3}}}{b d \,x^{3}+b c \,x^{2}+a d x +a c}d x \right ) a c d -4 \left (\int \frac {\left (d x +c \right )^{\frac {1}{3}} x}{b d \,x^{3}+b c \,x^{2}+a d x +a c}d x \right ) a \,d^{2}+4 \left (\int \frac {\left (d x +c \right )^{\frac {1}{3}} x}{b d \,x^{3}+b c \,x^{2}+a d x +a c}d x \right ) b \,c^{2}}{4 b} \] Input:
int(x*(d*x+c)^(4/3)/(b*x^2+a),x)
Output:
(15*(c + d*x)**(1/3)*c + 3*(c + d*x)**(1/3)*d*x - 8*int((c + d*x)**(1/3)/( a*c + a*d*x + b*c*x**2 + b*d*x**3),x)*a*c*d - 4*int(((c + d*x)**(1/3)*x)/( a*c + a*d*x + b*c*x**2 + b*d*x**3),x)*a*d**2 + 4*int(((c + d*x)**(1/3)*x)/ (a*c + a*d*x + b*c*x**2 + b*d*x**3),x)*b*c**2)/(4*b)