Integrand size = 25, antiderivative size = 601 \[ \int \frac {x \sqrt {c+d x^3}}{8 c-d x^3} \, dx=-\frac {2 \sqrt {c+d x^3}}{d^{2/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}-\frac {\sqrt {3} \sqrt [6]{c} \arctan \left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt {c+d x^3}}\right )}{2 d^{2/3}}+\frac {\sqrt [6]{c} \text {arctanh}\left (\frac {\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt {c+d x^3}}\right )}{2 d^{2/3}}-\frac {\sqrt [6]{c} \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{2 d^{2/3}}+\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt {3}\right )}{d^{2/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}-\frac {2 \sqrt {2} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} d^{2/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}} \] Output:
-2*(d*x^3+c)^(1/2)/d^(2/3)/((1+3^(1/2))*c^(1/3)+d^(1/3)*x)-1/2*3^(1/2)*c^( 1/6)*arctan(3^(1/2)*c^(1/6)*(c^(1/3)+d^(1/3)*x)/(d*x^3+c)^(1/2))/d^(2/3)+1 /2*c^(1/6)*arctanh(1/3*(c^(1/3)+d^(1/3)*x)^2/c^(1/6)/(d*x^3+c)^(1/2))/d^(2 /3)-1/2*c^(1/6)*arctanh(1/3*(d*x^3+c)^(1/2)/c^(1/2))/d^(2/3)+3^(1/4)*(1/2* 6^(1/2)-1/2*2^(1/2))*c^(1/3)*(c^(1/3)+d^(1/3)*x)*((c^(2/3)-c^(1/3)*d^(1/3) *x+d^(2/3)*x^2)/((1+3^(1/2))*c^(1/3)+d^(1/3)*x)^2)^(1/2)*EllipticE(((1-3^( 1/2))*c^(1/3)+d^(1/3)*x)/((1+3^(1/2))*c^(1/3)+d^(1/3)*x),I*3^(1/2)+2*I)/d^ (2/3)/(c^(1/3)*(c^(1/3)+d^(1/3)*x)/((1+3^(1/2))*c^(1/3)+d^(1/3)*x)^2)^(1/2 )/(d*x^3+c)^(1/2)-2/3*2^(1/2)*c^(1/3)*(c^(1/3)+d^(1/3)*x)*((c^(2/3)-c^(1/3 )*d^(1/3)*x+d^(2/3)*x^2)/((1+3^(1/2))*c^(1/3)+d^(1/3)*x)^2)^(1/2)*Elliptic F(((1-3^(1/2))*c^(1/3)+d^(1/3)*x)/((1+3^(1/2))*c^(1/3)+d^(1/3)*x),I*3^(1/2 )+2*I)*3^(3/4)/d^(2/3)/(c^(1/3)*(c^(1/3)+d^(1/3)*x)/((1+3^(1/2))*c^(1/3)+d ^(1/3)*x)^2)^(1/2)/(d*x^3+c)^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 8.90 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.10 \[ \int \frac {x \sqrt {c+d x^3}}{8 c-d x^3} \, dx=\frac {x^2 \sqrt {1+\frac {d x^3}{c}} \operatorname {AppellF1}\left (\frac {2}{3},-\frac {1}{2},1,\frac {5}{3},-\frac {d x^3}{c},\frac {d x^3}{8 c}\right )}{16 \sqrt {c+d x^3}} \] Input:
Integrate[(x*Sqrt[c + d*x^3])/(8*c - d*x^3),x]
Output:
(x^2*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, -1/2, 1, 5/3, -((d*x^3)/c), (d*x^3) /(8*c)])/(16*Sqrt[c + d*x^3])
Time = 2.95 (sec) , antiderivative size = 629, normalized size of antiderivative = 1.05, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {984, 832, 759, 988, 946, 73, 219, 2416, 2563, 219, 2570, 218}
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.
| \(\displaystyle \int \frac {x \sqrt {c+d x^3}}{8 c-d x^3} \, dx\) |
| \(\Big \downarrow \) 984 |
| \(\displaystyle 9 c \int \frac {x}{\left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx-\int \frac {x}{\sqrt {d x^3+c}}dx\) |
| \(\Big \downarrow \) 832 |
| \(\displaystyle \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c} \int \frac {1}{\sqrt {d x^3+c}}dx}{\sqrt [3]{d}}-\frac {\int \frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt {d x^3+c}}dx}{\sqrt [3]{d}}+9 c \int \frac {x}{\left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx\) |
| \(\Big \downarrow \) 759 |
| \(\displaystyle -\frac {\int \frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt {d x^3+c}}dx}{\sqrt [3]{d}}+9 c \int \frac {x}{\left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx+\frac {2 \left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} d^{2/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}\) |
| \(\Big \downarrow \) 988 |
| \(\displaystyle 9 c \left (-\frac {\int \frac {-\frac {d^{4/3} x^2}{\sqrt [3]{c}}-2 d x+2 \sqrt [3]{c} d^{2/3}}{\left (\frac {d^{2/3} x^2}{c^{2/3}}+\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}+4\right ) \sqrt {d x^3+c}}dx}{12 c d}+\frac {\int \frac {\sqrt [3]{d} x+\sqrt [3]{c}}{\left (2 \sqrt [3]{c}-\sqrt [3]{d} x\right ) \sqrt {d x^3+c}}dx}{12 c^{2/3} \sqrt [3]{d}}-\frac {\sqrt [3]{d} \int \frac {x^2}{\left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{4 \sqrt [3]{c}}\right )-\frac {\int \frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt {d x^3+c}}dx}{\sqrt [3]{d}}+\frac {2 \left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} d^{2/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}\) |
| \(\Big \downarrow \) 946 |
| \(\displaystyle 9 c \left (-\frac {\int \frac {-\frac {d^{4/3} x^2}{\sqrt [3]{c}}-2 d x+2 \sqrt [3]{c} d^{2/3}}{\left (\frac {d^{2/3} x^2}{c^{2/3}}+\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}+4\right ) \sqrt {d x^3+c}}dx}{12 c d}+\frac {\int \frac {\sqrt [3]{d} x+\sqrt [3]{c}}{\left (2 \sqrt [3]{c}-\sqrt [3]{d} x\right ) \sqrt {d x^3+c}}dx}{12 c^{2/3} \sqrt [3]{d}}-\frac {\sqrt [3]{d} \int \frac {1}{\left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx^3}{12 \sqrt [3]{c}}\right )-\frac {\int \frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt {d x^3+c}}dx}{\sqrt [3]{d}}+\frac {2 \left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} d^{2/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle 9 c \left (-\frac {\int \frac {-\frac {d^{4/3} x^2}{\sqrt [3]{c}}-2 d x+2 \sqrt [3]{c} d^{2/3}}{\left (\frac {d^{2/3} x^2}{c^{2/3}}+\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}+4\right ) \sqrt {d x^3+c}}dx}{12 c d}+\frac {\int \frac {\sqrt [3]{d} x+\sqrt [3]{c}}{\left (2 \sqrt [3]{c}-\sqrt [3]{d} x\right ) \sqrt {d x^3+c}}dx}{12 c^{2/3} \sqrt [3]{d}}-\frac {\int \frac {1}{9 c-x^6}d\sqrt {d x^3+c}}{6 \sqrt [3]{c} d^{2/3}}\right )-\frac {\int \frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt {d x^3+c}}dx}{\sqrt [3]{d}}+\frac {2 \left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} d^{2/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle 9 c \left (-\frac {\int \frac {-\frac {d^{4/3} x^2}{\sqrt [3]{c}}-2 d x+2 \sqrt [3]{c} d^{2/3}}{\left (\frac {d^{2/3} x^2}{c^{2/3}}+\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}+4\right ) \sqrt {d x^3+c}}dx}{12 c d}+\frac {\int \frac {\sqrt [3]{d} x+\sqrt [3]{c}}{\left (2 \sqrt [3]{c}-\sqrt [3]{d} x\right ) \sqrt {d x^3+c}}dx}{12 c^{2/3} \sqrt [3]{d}}-\frac {\text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{18 c^{5/6} d^{2/3}}\right )-\frac {\int \frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt {d x^3+c}}dx}{\sqrt [3]{d}}+\frac {2 \left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} d^{2/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}\) |
| \(\Big \downarrow \) 2416 |
| \(\displaystyle 9 c \left (-\frac {\int \frac {-\frac {d^{4/3} x^2}{\sqrt [3]{c}}-2 d x+2 \sqrt [3]{c} d^{2/3}}{\left (\frac {d^{2/3} x^2}{c^{2/3}}+\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}+4\right ) \sqrt {d x^3+c}}dx}{12 c d}+\frac {\int \frac {\sqrt [3]{d} x+\sqrt [3]{c}}{\left (2 \sqrt [3]{c}-\sqrt [3]{d} x\right ) \sqrt {d x^3+c}}dx}{12 c^{2/3} \sqrt [3]{d}}-\frac {\text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{18 c^{5/6} d^{2/3}}\right )+\frac {2 \left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} d^{2/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}-\frac {\frac {2 \sqrt {c+d x^3}}{\sqrt [3]{d} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt {3}\right )}{\sqrt [3]{d} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}}{\sqrt [3]{d}}\) |
| \(\Big \downarrow \) 2563 |
| \(\displaystyle 9 c \left (\frac {\int \frac {1}{9-\frac {\left (\sqrt [3]{d} x+\sqrt [3]{c}\right )^4}{\sqrt [3]{c} \left (d x^3+c\right )}}d\frac {\left (\sqrt [3]{d} x+\sqrt [3]{c}\right )^2}{c^{2/3} \sqrt {d x^3+c}}}{6 \sqrt [3]{c} d^{2/3}}-\frac {\int \frac {-\frac {d^{4/3} x^2}{\sqrt [3]{c}}-2 d x+2 \sqrt [3]{c} d^{2/3}}{\left (\frac {d^{2/3} x^2}{c^{2/3}}+\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}+4\right ) \sqrt {d x^3+c}}dx}{12 c d}-\frac {\text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{18 c^{5/6} d^{2/3}}\right )+\frac {2 \left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} d^{2/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}-\frac {\frac {2 \sqrt {c+d x^3}}{\sqrt [3]{d} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt {3}\right )}{\sqrt [3]{d} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}}{\sqrt [3]{d}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle 9 c \left (-\frac {\int \frac {-\frac {d^{4/3} x^2}{\sqrt [3]{c}}-2 d x+2 \sqrt [3]{c} d^{2/3}}{\left (\frac {d^{2/3} x^2}{c^{2/3}}+\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}+4\right ) \sqrt {d x^3+c}}dx}{12 c d}+\frac {\text {arctanh}\left (\frac {\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt {c+d x^3}}\right )}{18 c^{5/6} d^{2/3}}-\frac {\text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{18 c^{5/6} d^{2/3}}\right )+\frac {2 \left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} d^{2/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}-\frac {\frac {2 \sqrt {c+d x^3}}{\sqrt [3]{d} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt {3}\right )}{\sqrt [3]{d} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}}{\sqrt [3]{d}}\) |
| \(\Big \downarrow \) 2570 |
| \(\displaystyle 9 c \left (\frac {d^{4/3} \int \frac {1}{-\frac {2 d^2}{c}-\frac {6 \left (\sqrt [3]{d} x+\sqrt [3]{c}\right )^2 d^2}{c^{2/3} \left (d x^3+c\right )}}d\frac {\sqrt [3]{d} x+\sqrt [3]{c}}{\sqrt [3]{c} \sqrt {d x^3+c}}}{3 c^{4/3}}+\frac {\text {arctanh}\left (\frac {\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt {c+d x^3}}\right )}{18 c^{5/6} d^{2/3}}-\frac {\text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{18 c^{5/6} d^{2/3}}\right )+\frac {2 \left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} d^{2/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}-\frac {\frac {2 \sqrt {c+d x^3}}{\sqrt [3]{d} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt {3}\right )}{\sqrt [3]{d} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}}{\sqrt [3]{d}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {2 \left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} d^{2/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}-\frac {\frac {2 \sqrt {c+d x^3}}{\sqrt [3]{d} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt {3}\right )}{\sqrt [3]{d} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}}{\sqrt [3]{d}}+9 c \left (-\frac {\arctan \left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt {c+d x^3}}\right )}{6 \sqrt {3} c^{5/6} d^{2/3}}+\frac {\text {arctanh}\left (\frac {\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt {c+d x^3}}\right )}{18 c^{5/6} d^{2/3}}-\frac {\text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{18 c^{5/6} d^{2/3}}\right )\) |
Input:
Int[(x*Sqrt[c + d*x^3])/(8*c - d*x^3),x]
Output:
9*c*(-1/6*ArcTan[(Sqrt[3]*c^(1/6)*(c^(1/3) + d^(1/3)*x))/Sqrt[c + d*x^3]]/ (Sqrt[3]*c^(5/6)*d^(2/3)) + ArcTanh[(c^(1/3) + d^(1/3)*x)^2/(3*c^(1/6)*Sqr t[c + d*x^3])]/(18*c^(5/6)*d^(2/3)) - ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])] /(18*c^(5/6)*d^(2/3))) - ((2*Sqrt[c + d*x^3])/(d^(1/3)*((1 + Sqrt[3])*c^(1 /3) + d^(1/3)*x)) - (3^(1/4)*Sqrt[2 - Sqrt[3]]*c^(1/3)*(c^(1/3) + d^(1/3)* x)*Sqrt[(c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2)/((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)^2]*EllipticE[ArcSin[((1 - Sqrt[3])*c^(1/3) + d^(1/3)*x)/((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)], -7 - 4*Sqrt[3]])/(d^(1/3)*Sqrt[(c^(1/3)* (c^(1/3) + d^(1/3)*x))/((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)^2]*Sqrt[c + d*x ^3]))/d^(1/3) + (2*(1 - Sqrt[3])*Sqrt[2 + Sqrt[3]]*c^(1/3)*(c^(1/3) + d^(1 /3)*x)*Sqrt[(c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2)/((1 + Sqrt[3])*c^( 1/3) + d^(1/3)*x)^2]*EllipticF[ArcSin[((1 - Sqrt[3])*c^(1/3) + d^(1/3)*x)/ ((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)], -7 - 4*Sqrt[3]])/(3^(1/4)*d^(2/3)*Sq rt[(c^(1/3)*(c^(1/3) + d^(1/3)*x))/((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)^2]* Sqrt[c + d*x^3])
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt[2 + Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s *x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[s* ((s + r*x)/((1 + Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 - Sqrt[3])*s + r*x)/((1 + Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x] & & PosQ[a]
Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3] ], s = Denom[Rt[b/a, 3]]}, Simp[(-(1 - Sqrt[3]))*(s/r) Int[1/Sqrt[a + b*x ^3], x], x] + Simp[1/r Int[((1 - Sqrt[3])*s + r*x)/Sqrt[a + b*x^3], x], x ]] /; FreeQ[{a, b}, x] && PosQ[a]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. ), x_Symbol] :> Simp[1/n Subst[Int[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m - n + 1, 0]
Int[((x_)*((a_) + (b_.)*(x_)^(n_))^(p_))/((c_) + (d_.)*(x_)^(n_)), x_Symbol ] :> Simp[b/d Int[x*(a + b*x^n)^(p - 1), x], x] - Simp[(b*c - a*d)/d In t[x*((a + b*x^n)^(p - 1)/(c + d*x^n)), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && GtQ[p, 0] && IntBinomialQ[a, b, c, d, 1, 1, n, p, -1, x]
Int[(x_)/(((a_) + (b_.)*(x_)^3)*Sqrt[(c_) + (d_.)*(x_)^3]), x_Symbol] :> Wi th[{q = Rt[d/c, 3]}, Simp[d*(q/(4*b)) Int[x^2/((8*c - d*x^3)*Sqrt[c + d*x ^3]), x], x] + (-Simp[q^2/(12*b) Int[(1 + q*x)/((2 - q*x)*Sqrt[c + d*x^3] ), x], x] + Simp[1/(12*b*c) Int[(2*c*q^2 - 2*d*x - d*q*x^2)/((4 + 2*q*x + q^2*x^2)*Sqrt[c + d*x^3]), x], x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[8*b*c + a*d, 0]
Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = N umer[Simplify[(1 - Sqrt[3])*(d/c)]], s = Denom[Simplify[(1 - Sqrt[3])*(d/c) ]]}, Simp[2*d*s^3*(Sqrt[a + b*x^3]/(a*r^2*((1 + Sqrt[3])*s + r*x))), x] - S imp[3^(1/4)*Sqrt[2 - Sqrt[3]]*d*s*(s + r*x)*(Sqrt[(s^2 - r*s*x + r^2*x^2)/( (1 + Sqrt[3])*s + r*x)^2]/(r^2*Sqrt[a + b*x^3]*Sqrt[s*((s + r*x)/((1 + Sqrt [3])*s + r*x)^2)]))*EllipticE[ArcSin[((1 - Sqrt[3])*s + r*x)/((1 + Sqrt[3]) *s + r*x)], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b, c, d}, x] && PosQ[a] && Eq Q[b*c^3 - 2*(5 - 3*Sqrt[3])*a*d^3, 0]
Int[((e_) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_ Symbol] :> Simp[-2*(e/d) Subst[Int[1/(9 - a*x^2), x], x, (1 + f*(x/e))^2/ Sqrt[a + b*x^3]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] & & EqQ[b*c^3 + 8*a*d^3, 0] && EqQ[2*d*e + c*f, 0]
Int[((f_) + (g_.)*(x_) + (h_.)*(x_)^2)/(((c_) + (d_.)*(x_) + (e_.)*(x_)^2)* Sqrt[(a_) + (b_.)*(x_)^3]), x_Symbol] :> Simp[-2*g*h Subst[Int[1/(2*e*h - (b*d*f - 2*a*e*h)*x^2), x], x, (1 + 2*h*(x/g))/Sqrt[a + b*x^3]], x] /; Fre eQ[{a, b, c, d, e, f, g, h}, x] && NeQ[b*d*f - 2*a*e*h, 0] && EqQ[b*g^3 - 8 *a*h^3, 0] && EqQ[g^2 + 2*f*h, 0] && EqQ[b*d*f + b*c*g - 4*a*e*h, 0]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 1.04 (sec) , antiderivative size = 848, normalized size of antiderivative = 1.41
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(848\) |
| elliptic | \(\text {Expression too large to display}\) | \(848\) |
Input:
int(x*(d*x^3+c)^(1/2)/(-d*x^3+8*c),x,method=_RETURNVERBOSE)
Output:
2/3*I*3^(1/2)/d*(-c*d^2)^(1/3)*(I*(x+1/2/d*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/d* (-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2)*((x-1/d*(-c*d^2)^(1/3))/(- 3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3)))^(1/2)*(-I*(x+1/2/d*( -c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1 /2)/(d*x^3+c)^(1/2)*((-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3) )*EllipticE(1/3*3^(1/2)*(I*(x+1/2/d*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d^2 )^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2),(I*3^(1/2)/d*(-c*d^2)^(1/3)/(-3/2 /d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3)))^(1/2))+1/d*(-c*d^2)^(1/ 3)*EllipticF(1/3*3^(1/2)*(I*(x+1/2/d*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d^ 2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2),(I*3^(1/2)/d*(-c*d^2)^(1/3)/(-3/ 2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3)))^(1/2)))-1/3*I/d^3*2^(1 /2)*sum(1/_alpha*(-c*d^2)^(1/3)*(1/2*I*d*(2*x+1/d*(-I*3^(1/2)*(-c*d^2)^(1/ 3)+(-c*d^2)^(1/3)))/(-c*d^2)^(1/3))^(1/2)*(d*(x-1/d*(-c*d^2)^(1/3))/(-3*(- c*d^2)^(1/3)+I*3^(1/2)*(-c*d^2)^(1/3)))^(1/2)*(-1/2*I*d*(2*x+1/d*(I*3^(1/2 )*(-c*d^2)^(1/3)+(-c*d^2)^(1/3)))/(-c*d^2)^(1/3))^(1/2)/(d*x^3+c)^(1/2)*(I *(-c*d^2)^(1/3)*_alpha*3^(1/2)*d-I*3^(1/2)*(-c*d^2)^(2/3)+2*_alpha^2*d^2-( -c*d^2)^(1/3)*_alpha*d-(-c*d^2)^(2/3))*EllipticPi(1/3*3^(1/2)*(I*(x+1/2/d* (-c*d^2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^( 1/2),-1/18/d*(2*I*(-c*d^2)^(1/3)*_alpha^2*3^(1/2)*d-I*(-c*d^2)^(2/3)*_alph a*3^(1/2)+I*3^(1/2)*c*d-3*(-c*d^2)^(2/3)*_alpha-3*c*d)/c,(I*3^(1/2)/d*(...
Leaf count of result is larger than twice the leaf count of optimal. 2194 vs. \(2 (424) = 848\).
Time = 1.17 (sec) , antiderivative size = 2194, normalized size of antiderivative = 3.65 \[ \int \frac {x \sqrt {c+d x^3}}{8 c-d x^3} \, dx=\text {Too large to display} \] Input:
integrate(x*(d*x^3+c)^(1/2)/(-d*x^3+8*c),x, algorithm="fricas")
Output:
-1/24*((sqrt(-3)*d - d)*(c/d^4)^(1/6)*log(1/4*((d^6*x^9 + 318*c*d^5*x^6 + 1200*c^2*d^4*x^3 + 640*c^3*d^3 + sqrt(-3)*(d^6*x^9 + 318*c*d^5*x^6 + 1200* c^2*d^4*x^3 + 640*c^3*d^3))*(c/d^4)^(5/6) + 6*(2*c*d^2*x^7 + 160*c^2*d*x^4 + 320*c^3*x - 6*(5*c*d^4*x^5 + 32*c^2*d^3*x^2 - sqrt(-3)*(5*c*d^4*x^5 + 3 2*c^2*d^3*x^2))*(c/d^4)^(2/3) - (7*c*d^3*x^6 + 152*c^2*d^2*x^3 + 64*c^3*d + sqrt(-3)*(7*c*d^3*x^6 + 152*c^2*d^2*x^3 + 64*c^3*d))*(c/d^4)^(1/3))*sqrt (d*x^3 + c) - 36*(5*c*d^4*x^7 + 64*c^2*d^3*x^4 + 32*c^3*d^2*x)*sqrt(c/d^4) + 18*(c*d^3*x^8 + 38*c^2*d^2*x^5 + 64*c^3*d*x^2 - sqrt(-3)*(c*d^3*x^8 + 3 8*c^2*d^2*x^5 + 64*c^3*d*x^2))*(c/d^4)^(1/6))/(d^3*x^9 - 24*c*d^2*x^6 + 19 2*c^2*d*x^3 - 512*c^3)) - (sqrt(-3)*d - d)*(c/d^4)^(1/6)*log(-1/4*((d^6*x^ 9 + 318*c*d^5*x^6 + 1200*c^2*d^4*x^3 + 640*c^3*d^3 + sqrt(-3)*(d^6*x^9 + 3 18*c*d^5*x^6 + 1200*c^2*d^4*x^3 + 640*c^3*d^3))*(c/d^4)^(5/6) - 6*(2*c*d^2 *x^7 + 160*c^2*d*x^4 + 320*c^3*x - 6*(5*c*d^4*x^5 + 32*c^2*d^3*x^2 - sqrt( -3)*(5*c*d^4*x^5 + 32*c^2*d^3*x^2))*(c/d^4)^(2/3) - (7*c*d^3*x^6 + 152*c^2 *d^2*x^3 + 64*c^3*d + sqrt(-3)*(7*c*d^3*x^6 + 152*c^2*d^2*x^3 + 64*c^3*d)) *(c/d^4)^(1/3))*sqrt(d*x^3 + c) - 36*(5*c*d^4*x^7 + 64*c^2*d^3*x^4 + 32*c^ 3*d^2*x)*sqrt(c/d^4) + 18*(c*d^3*x^8 + 38*c^2*d^2*x^5 + 64*c^3*d*x^2 - sqr t(-3)*(c*d^3*x^8 + 38*c^2*d^2*x^5 + 64*c^3*d*x^2))*(c/d^4)^(1/6))/(d^3*x^9 - 24*c*d^2*x^6 + 192*c^2*d*x^3 - 512*c^3)) - (sqrt(-3)*d + d)*(c/d^4)^(1/ 6)*log(1/4*((d^6*x^9 + 318*c*d^5*x^6 + 1200*c^2*d^4*x^3 + 640*c^3*d^3 -...
\[ \int \frac {x \sqrt {c+d x^3}}{8 c-d x^3} \, dx=- \int \frac {x \sqrt {c + d x^{3}}}{- 8 c + d x^{3}}\, dx \] Input:
integrate(x*(d*x**3+c)**(1/2)/(-d*x**3+8*c),x)
Output:
-Integral(x*sqrt(c + d*x**3)/(-8*c + d*x**3), x)
\[ \int \frac {x \sqrt {c+d x^3}}{8 c-d x^3} \, dx=\int { -\frac {\sqrt {d x^{3} + c} x}{d x^{3} - 8 \, c} \,d x } \] Input:
integrate(x*(d*x^3+c)^(1/2)/(-d*x^3+8*c),x, algorithm="maxima")
Output:
-integrate(sqrt(d*x^3 + c)*x/(d*x^3 - 8*c), x)
\[ \int \frac {x \sqrt {c+d x^3}}{8 c-d x^3} \, dx=\int { -\frac {\sqrt {d x^{3} + c} x}{d x^{3} - 8 \, c} \,d x } \] Input:
integrate(x*(d*x^3+c)^(1/2)/(-d*x^3+8*c),x, algorithm="giac")
Output:
integrate(-sqrt(d*x^3 + c)*x/(d*x^3 - 8*c), x)
Timed out. \[ \int \frac {x \sqrt {c+d x^3}}{8 c-d x^3} \, dx=\int \frac {x\,\sqrt {d\,x^3+c}}{8\,c-d\,x^3} \,d x \] Input:
int((x*(c + d*x^3)^(1/2))/(8*c - d*x^3),x)
Output:
int((x*(c + d*x^3)^(1/2))/(8*c - d*x^3), x)
\[ \int \frac {x \sqrt {c+d x^3}}{8 c-d x^3} \, dx=\int \frac {\sqrt {d \,x^{3}+c}\, x}{-d \,x^{3}+8 c}d x \] Input:
int(x*(d*x^3+c)^(1/2)/(-d*x^3+8*c),x)
Output:
int((sqrt(c + d*x**3)*x)/(8*c - d*x**3),x)