Integrand size = 37, antiderivative size = 285 \[ \int \frac {(d+e x)^4}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/4}} \, dx=-\frac {4 \left (c d^2-a e^2\right )^2 (d+e x)}{c^3 d^3 \sqrt [4]{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {17 e \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/4}}{15 c^3 d^3}+\frac {2 e (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/4}}{5 c^2 d^2}+\frac {77 \left (c d^2-a e^2\right )^3 \sqrt [4]{-\frac {c d e (a e+c d x) (d+e x)}{\left (c d^2-a e^2\right )^2}} E\left (\left .\frac {1}{2} \arcsin \left (\frac {a e^2+c d (d+2 e x)}{c d^2-a e^2}\right )\right |2\right )}{10 \sqrt {2} c^4 d^4 \sqrt [4]{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \] Output:
-4*(-a*e^2+c*d^2)^2*(e*x+d)/c^3/d^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/4 )+17/15*e*(-a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/4)/c^3/d^3+2 /5*e*(e*x+d)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/4)/c^2/d^2+77/20*(-a*e^2 +c*d^2)^3*(-c*d*e*(c*d*x+a*e)*(e*x+d)/(-a*e^2+c*d^2)^2)^(1/4)*EllipticE(si n(1/2*arcsin((a*e^2+c*d*(2*e*x+d))/(-a*e^2+c*d^2))),2^(1/2))*2^(1/2)/c^4/d ^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/4)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.05 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.35 \[ \int \frac {(d+e x)^4}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/4}} \, dx=-\frac {4 \left (c d^2-a e^2\right )^3 \sqrt [4]{\frac {c d (d+e x)}{c d^2-a e^2}} \operatorname {Hypergeometric2F1}\left (-\frac {11}{4},-\frac {1}{4},\frac {3}{4},\frac {e (a e+c d x)}{-c d^2+a e^2}\right )}{c^4 d^4 \sqrt [4]{(a e+c d x) (d+e x)}} \] Input:
Integrate[(d + e*x)^4/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/4),x]
Output:
(-4*(c*d^2 - a*e^2)^3*((c*d*(d + e*x))/(c*d^2 - a*e^2))^(1/4)*Hypergeometr ic2F1[-11/4, -1/4, 3/4, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/(c^4*d^4*(( a*e + c*d*x)*(d + e*x))^(1/4))
Leaf count is larger than twice the leaf count of optimal. \(1198\) vs. \(2(285)=570\).
Time = 1.10 (sec) , antiderivative size = 1198, normalized size of antiderivative = 4.20, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.324, Rules used = {1164, 27, 27, 1166, 27, 1166, 27, 1160, 1094, 834, 761, 1510}
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.
| \(\displaystyle \int \frac {(d+e x)^4}{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/4}} \, dx\) |
| \(\Big \downarrow \) 1164 |
| \(\displaystyle -\frac {4 \int -\frac {7 e (d+e x)^2 \left (d \left (c d^2-a e^2\right )+e x \left (c d^2-a e^2\right )\right )}{2 \sqrt [4]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{\left (c d^2-a e^2\right )^2}-\frac {4 (d+e x)^4}{\left (c d^2-a e^2\right ) \sqrt [4]{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {14 e \int \frac {\left (c d^2-a e^2\right ) (d+e x)^3}{\sqrt [4]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{\left (c d^2-a e^2\right )^2}-\frac {4 (d+e x)^4}{\left (c d^2-a e^2\right ) \sqrt [4]{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {14 e \int \frac {(d+e x)^3}{\sqrt [4]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{c d^2-a e^2}-\frac {4 (d+e x)^4}{\left (c d^2-a e^2\right ) \sqrt [4]{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
| \(\Big \downarrow \) 1166 |
| \(\displaystyle \frac {14 e \left (\frac {2 \int \frac {11 e \left (c d^2-a e^2\right ) (d+e x)^2}{4 \sqrt [4]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{7 c d e}+\frac {2 (d+e x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/4}}{7 c d}\right )}{c d^2-a e^2}-\frac {4 (d+e x)^4}{\left (c d^2-a e^2\right ) \sqrt [4]{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {14 e \left (\frac {11 \left (c d^2-a e^2\right ) \int \frac {(d+e x)^2}{\sqrt [4]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{14 c d}+\frac {2 (d+e x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/4}}{7 c d}\right )}{c d^2-a e^2}-\frac {4 (d+e x)^4}{\left (c d^2-a e^2\right ) \sqrt [4]{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
| \(\Big \downarrow \) 1166 |
| \(\displaystyle \frac {14 e \left (\frac {11 \left (c d^2-a e^2\right ) \left (\frac {2 \int \frac {7 e \left (c d^2-a e^2\right ) (d+e x)}{4 \sqrt [4]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{5 c d e}+\frac {2 (d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/4}}{5 c d}\right )}{14 c d}+\frac {2 (d+e x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/4}}{7 c d}\right )}{c d^2-a e^2}-\frac {4 (d+e x)^4}{\left (c d^2-a e^2\right ) \sqrt [4]{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {14 e \left (\frac {11 \left (c d^2-a e^2\right ) \left (\frac {7 \left (c d^2-a e^2\right ) \int \frac {d+e x}{\sqrt [4]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{10 c d}+\frac {2 (d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/4}}{5 c d}\right )}{14 c d}+\frac {2 (d+e x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/4}}{7 c d}\right )}{c d^2-a e^2}-\frac {4 (d+e x)^4}{\left (c d^2-a e^2\right ) \sqrt [4]{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
| \(\Big \downarrow \) 1160 |
| \(\displaystyle \frac {14 e \left (\frac {11 \left (c d^2-a e^2\right ) \left (\frac {7 \left (c d^2-a e^2\right ) \left (\frac {\left (d^2-\frac {a e^2}{c}\right ) \int \frac {1}{\sqrt [4]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 d}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/4}}{3 c d}\right )}{10 c d}+\frac {2 (d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/4}}{5 c d}\right )}{14 c d}+\frac {2 (d+e x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/4}}{7 c d}\right )}{c d^2-a e^2}-\frac {4 (d+e x)^4}{\left (c d^2-a e^2\right ) \sqrt [4]{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
| \(\Big \downarrow \) 1094 |
| \(\displaystyle \frac {14 e \left (\frac {11 \left (c d^2-a e^2\right ) \left (\frac {7 \left (c d^2-a e^2\right ) \left (\frac {2 \left (d^2-\frac {a e^2}{c}\right ) \sqrt {\left (a e^2+c d^2+2 c d e x\right )^2} \int \frac {\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{\sqrt {\left (c d^2-a e^2\right )^2+4 c d e \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )}}d\sqrt [4]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{d \left (a e^2+c d^2+2 c d e x\right )}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/4}}{3 c d}\right )}{10 c d}+\frac {2 (d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/4}}{5 c d}\right )}{14 c d}+\frac {2 (d+e x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/4}}{7 c d}\right )}{c d^2-a e^2}-\frac {4 (d+e x)^4}{\left (c d^2-a e^2\right ) \sqrt [4]{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
| \(\Big \downarrow \) 834 |
| \(\displaystyle \frac {14 e \left (\frac {11 \left (c d^2-a e^2\right ) \left (\frac {7 \left (c d^2-a e^2\right ) \left (\frac {2 \left (d^2-\frac {a e^2}{c}\right ) \sqrt {\left (a e^2+c d^2+2 c d e x\right )^2} \left (\frac {\left (c d^2-a e^2\right ) \int \frac {1}{\sqrt {\left (c d^2-a e^2\right )^2+4 c d e \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )}}d\sqrt [4]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{2 \sqrt {c} \sqrt {d} \sqrt {e}}-\frac {\left (c d^2-a e^2\right ) \int \frac {1-\frac {2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{c d^2-a e^2}}{\sqrt {\left (c d^2-a e^2\right )^2+4 c d e \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )}}d\sqrt [4]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{2 \sqrt {c} \sqrt {d} \sqrt {e}}\right )}{d \left (a e^2+c d^2+2 c d e x\right )}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/4}}{3 c d}\right )}{10 c d}+\frac {2 (d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/4}}{5 c d}\right )}{14 c d}+\frac {2 (d+e x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/4}}{7 c d}\right )}{c d^2-a e^2}-\frac {4 (d+e x)^4}{\left (c d^2-a e^2\right ) \sqrt [4]{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {14 e \left (\frac {2 \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/4} (d+e x)^2}{7 c d}+\frac {11 \left (c d^2-a e^2\right ) \left (\frac {2 \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/4} (d+e x)}{5 c d}+\frac {7 \left (c d^2-a e^2\right ) \left (\frac {2 \left (d^2-\frac {a e^2}{c}\right ) \sqrt {\left (c d^2+2 c e x d+a e^2\right )^2} \left (\frac {\left (c d^2-a e^2\right )^{3/2} \left (\frac {2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{c d^2-a e^2}+1\right ) \sqrt {\frac {\left (c d^2-a e^2\right )^2+4 c d e \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )}{\left (c d^2-a e^2\right )^2 \left (\frac {2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{c d^2-a e^2}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt [4]{e} \sqrt [4]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{\sqrt {c d^2-a e^2}}\right ),\frac {1}{2}\right )}{4 \sqrt {2} c^{3/4} d^{3/4} e^{3/4} \sqrt {\left (c d^2-a e^2\right )^2+4 c d e \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )}}-\frac {\left (c d^2-a e^2\right ) \int \frac {1-\frac {2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{c d^2-a e^2}}{\sqrt {\left (c d^2-a e^2\right )^2+4 c d e \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )}}d\sqrt [4]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{2 \sqrt {c} \sqrt {d} \sqrt {e}}\right )}{d \left (c d^2+2 c e x d+a e^2\right )}+\frac {2 \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/4}}{3 c d}\right )}{10 c d}\right )}{14 c d}\right )}{c d^2-a e^2}-\frac {4 (d+e x)^4}{\left (c d^2-a e^2\right ) \sqrt [4]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle \frac {14 e \left (\frac {2 \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/4} (d+e x)^2}{7 c d}+\frac {11 \left (c d^2-a e^2\right ) \left (\frac {2 \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/4} (d+e x)}{5 c d}+\frac {7 \left (c d^2-a e^2\right ) \left (\frac {2 \left (d^2-\frac {a e^2}{c}\right ) \sqrt {\left (c d^2+2 c e x d+a e^2\right )^2} \left (\frac {\left (c d^2-a e^2\right )^{3/2} \left (\frac {2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{c d^2-a e^2}+1\right ) \sqrt {\frac {\left (c d^2-a e^2\right )^2+4 c d e \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )}{\left (c d^2-a e^2\right )^2 \left (\frac {2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{c d^2-a e^2}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt [4]{e} \sqrt [4]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{\sqrt {c d^2-a e^2}}\right ),\frac {1}{2}\right )}{4 \sqrt {2} c^{3/4} d^{3/4} e^{3/4} \sqrt {\left (c d^2-a e^2\right )^2+4 c d e \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )}}-\frac {\left (c d^2-a e^2\right ) \left (\frac {\sqrt {c d^2-a e^2} \left (\frac {2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{c d^2-a e^2}+1\right ) \sqrt {\frac {\left (c d^2-a e^2\right )^2+4 c d e \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )}{\left (c d^2-a e^2\right )^2 \left (\frac {2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{c d^2-a e^2}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt [4]{e} \sqrt [4]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{\sqrt {c d^2-a e^2}}\right )|\frac {1}{2}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt [4]{e} \sqrt {\left (c d^2-a e^2\right )^2+4 c d e \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )}}-\frac {\sqrt [4]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e} \sqrt {\left (c d^2-a e^2\right )^2+4 c d e \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )}}{\left (c d^2-a e^2\right )^2 \left (\frac {2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{c d^2-a e^2}+1\right )}\right )}{2 \sqrt {c} \sqrt {d} \sqrt {e}}\right )}{d \left (c d^2+2 c e x d+a e^2\right )}+\frac {2 \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/4}}{3 c d}\right )}{10 c d}\right )}{14 c d}\right )}{c d^2-a e^2}-\frac {4 (d+e x)^4}{\left (c d^2-a e^2\right ) \sqrt [4]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}\) |
Input:
Int[(d + e*x)^4/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/4),x]
Output:
(-4*(d + e*x)^4)/((c*d^2 - a*e^2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^ (1/4)) + (14*e*((2*(d + e*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/ 4))/(7*c*d) + (11*(c*d^2 - a*e^2)*((2*(d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/4))/(5*c*d) + (7*(c*d^2 - a*e^2)*((2*(a*d*e + (c*d^2 + a* e^2)*x + c*d*e*x^2)^(3/4))/(3*c*d) + (2*(d^2 - (a*e^2)/c)*Sqrt[(c*d^2 + a* e^2 + 2*c*d*e*x)^2]*(-1/2*((c*d^2 - a*e^2)*(-(((a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(1/4)*Sqrt[(c*d^2 - a*e^2)^2 + 4*c*d*e*(a*d*e + (c*d^2 + a*e^ 2)*x + c*d*e*x^2)])/((c*d^2 - a*e^2)^2*(1 + (2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqr t[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(c*d^2 - a*e^2)))) + (Sqrt[c*d^2 - a*e^2]*(1 + (2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(c*d^2 - a*e^2))*Sqrt[((c*d^2 - a*e^2)^2 + 4*c*d*e*(a*d*e + ( c*d^2 + a*e^2)*x + c*d*e*x^2))/((c*d^2 - a*e^2)^2*(1 + (2*Sqrt[c]*Sqrt[d]* Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(c*d^2 - a*e^2))^2)]* EllipticE[2*ArcTan[(Sqrt[2]*c^(1/4)*d^(1/4)*e^(1/4)*(a*d*e + (c*d^2 + a*e^ 2)*x + c*d*e*x^2)^(1/4))/Sqrt[c*d^2 - a*e^2]], 1/2])/(Sqrt[2]*c^(1/4)*d^(1 /4)*e^(1/4)*Sqrt[(c*d^2 - a*e^2)^2 + 4*c*d*e*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)])))/(Sqrt[c]*Sqrt[d]*Sqrt[e]) + ((c*d^2 - a*e^2)^(3/2)*(1 + (2* Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(c*d^ 2 - a*e^2))*Sqrt[((c*d^2 - a*e^2)^2 + 4*c*d*e*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/((c*d^2 - a*e^2)^2*(1 + (2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*...
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[( 1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))* EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 2]}, S imp[1/q Int[1/Sqrt[a + b*x^4], x], x] - Simp[1/q Int[(1 - q*x^2)/Sqrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[4*(Sqrt[(b + 2*c*x)^2]/(b + 2*c*x)) Subst[Int[x^(4*(p + 1) - 1)/Sqrt[b^2 - 4*a*c + 4 *c*x^4], x], x, (a + b*x + c*x^2)^(1/4)], x] /; FreeQ[{a, b, c}, x] && Inte gerQ[4*p]
Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol ] :> Simp[e*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + Simp[(2*c*d - b *e)/(2*c) Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[p, -1]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(d + e*x)^(m - 1)*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Simp[1/((p + 1)*(b^2 - 4*a* c)) Int[(d + e*x)^(m - 2)*Simp[e*(2*a*e*(m - 1) + b*d*(2*p - m + 4)) - 2* c*d^2*(2*p + 3) + e*(b*e - 2*d*c)*(m + 2*p + 2)*x, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[p, -1] && GtQ[m, 1] && Int QuadraticQ[a, b, c, d, e, m, p, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[e*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 1))), x] + Simp[1/(c*(m + 2*p + 1)) Int[(d + e*x)^(m - 2)*Simp[c*d^2*(m + 2*p + 1) - e*(a*e*(m - 1) + b*d*(p + 1)) + e*(2*c*d - b*e)*(m + p)*x, x]* (a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && If[Ration alQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadrat icQ[a, b, c, d, e, m, p, x]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + c*x^4]/(a*(1 + q^2*x^2))), x] + Simp[d* (1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(q*Sqrt[a + c*x^4]))*E llipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e }, x] && PosQ[c/a]
\[\int \frac {\left (e x +d \right )^{4}}{{\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{\frac {5}{4}}}d x\]
Input:
int((e*x+d)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(5/4),x)
Output:
int((e*x+d)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(5/4),x)
\[ \int \frac {(d+e x)^4}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/4}} \, dx=\int { \frac {{\left (e x + d\right )}^{4}}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {5}{4}}} \,d x } \] Input:
integrate((e*x+d)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/4),x, algorithm=" fricas")
Output:
integral((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/4)*(e^2*x^2 + 2*d*e*x + d^2)/(c^2*d^2*x^2 + 2*a*c*d*e*x + a^2*e^2), x)
\[ \int \frac {(d+e x)^4}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/4}} \, dx=\int \frac {\left (d + e x\right )^{4}}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {5}{4}}}\, dx \] Input:
integrate((e*x+d)**4/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/4),x)
Output:
Integral((d + e*x)**4/((d + e*x)*(a*e + c*d*x))**(5/4), x)
\[ \int \frac {(d+e x)^4}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/4}} \, dx=\int { \frac {{\left (e x + d\right )}^{4}}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {5}{4}}} \,d x } \] Input:
integrate((e*x+d)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/4),x, algorithm=" maxima")
Output:
integrate((e*x + d)^4/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/4), x)
\[ \int \frac {(d+e x)^4}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/4}} \, dx=\int { \frac {{\left (e x + d\right )}^{4}}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {5}{4}}} \,d x } \] Input:
integrate((e*x+d)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/4),x, algorithm=" giac")
Output:
integrate((e*x + d)^4/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/4), x)
Timed out. \[ \int \frac {(d+e x)^4}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/4}} \, dx=\int \frac {{\left (d+e\,x\right )}^4}{{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/4}} \,d x \] Input:
int((d + e*x)^4/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/4),x)
Output:
int((d + e*x)^4/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/4), x)
\[ \int \frac {(d+e x)^4}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/4}} \, dx=\left (\int \frac {x^{3}}{\left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {1}{4}} a e +\left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {1}{4}} c d x}d x \right ) e^{3}+3 \left (\int \frac {x^{2}}{\left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {1}{4}} a e +\left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {1}{4}} c d x}d x \right ) d \,e^{2}+3 \left (\int \frac {x}{\left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {1}{4}} a e +\left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {1}{4}} c d x}d x \right ) d^{2} e +\left (\int \frac {1}{\left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {1}{4}} a e +\left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {1}{4}} c d x}d x \right ) d^{3} \] Input:
int((e*x+d)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/4),x)
Output:
int(x**3/((a*d*e + a*e**2*x + c*d**2*x + c*d*e*x**2)**(1/4)*a*e + (a*d*e + a*e**2*x + c*d**2*x + c*d*e*x**2)**(1/4)*c*d*x),x)*e**3 + 3*int(x**2/((a* d*e + a*e**2*x + c*d**2*x + c*d*e*x**2)**(1/4)*a*e + (a*d*e + a*e**2*x + c *d**2*x + c*d*e*x**2)**(1/4)*c*d*x),x)*d*e**2 + 3*int(x/((a*d*e + a*e**2*x + c*d**2*x + c*d*e*x**2)**(1/4)*a*e + (a*d*e + a*e**2*x + c*d**2*x + c*d* e*x**2)**(1/4)*c*d*x),x)*d**2*e + int(1/((a*d*e + a*e**2*x + c*d**2*x + c* d*e*x**2)**(1/4)*a*e + (a*d*e + a*e**2*x + c*d**2*x + c*d*e*x**2)**(1/4)*c *d*x),x)*d**3