Integrand size = 40, antiderivative size = 607 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^6 (d+e x)^4} \, dx=\frac {\left (15 c^4 d^8+80 a c^3 d^6 e^2-3318 a^2 c^2 d^4 e^4+6720 a^3 c d^2 e^6-3465 a^4 e^8\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{640 a^2 d^6 e (d+e x)}-\frac {a e \left (13 c d^2-11 a e^2\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{40 d^2 x^4 (d+e x)}-\frac {\left (31 c d^2-33 a e^2\right ) \left (c d^2-a e^2\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{80 d^3 x^3 (d+e x)}-\frac {\left (c d^2-a e^2\right ) \left (5 c^2 d^4-228 a c d^2 e^2+231 a^2 e^4\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{320 a d^4 e x^2 (d+e x)}+\frac {\left (c d^2-a e^2\right ) \left (15 c^3 d^6+85 a c^2 d^4 e^2-1239 a^2 c d^2 e^4+1155 a^3 e^6\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{640 a^2 d^5 e^2 x (d+e x)}-\frac {a e \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{5 d x^5 (d+e x)^2}-\frac {3 \left (c d^2-a e^2\right )^2 \left (c^3 d^6+7 a c^2 d^4 e^2+63 a^2 c d^2 e^4-231 a^3 e^6\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {a} \sqrt {e} (d+e x)}\right )}{128 a^{5/2} d^{13/2} e^{5/2}} \] Output:
1/640*(-3465*a^4*e^8+6720*a^3*c*d^2*e^6-3318*a^2*c^2*d^4*e^4+80*a*c^3*d^6* e^2+15*c^4*d^8)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/a^2/d^6/e/(e*x+d)- 1/40*a*e*(-11*a*e^2+13*c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/d^2/ x^4/(e*x+d)-1/80*(-33*a*e^2+31*c*d^2)*(-a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)* x+c*d*e*x^2)^(1/2)/d^3/x^3/(e*x+d)-1/320*(-a*e^2+c*d^2)*(231*a^2*e^4-228*a *c*d^2*e^2+5*c^2*d^4)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/a/d^4/e/x^2/ (e*x+d)+1/640*(-a*e^2+c*d^2)*(1155*a^3*e^6-1239*a^2*c*d^2*e^4+85*a*c^2*d^4 *e^2+15*c^3*d^6)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/a^2/d^5/e^2/x/(e* x+d)-1/5*a*e*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/d/x^5/(e*x+d)^2-3/128 *(-a*e^2+c*d^2)^2*(-231*a^3*e^6+63*a^2*c*d^2*e^4+7*a*c^2*d^4*e^2+c^3*d^6)* arctanh(d^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/a^(1/2)/e^(1/2)/(e *x+d))/a^(5/2)/d^(13/2)/e^(5/2)
Time = 1.81 (sec) , antiderivative size = 387, normalized size of antiderivative = 0.64 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^6 (d+e x)^4} \, dx=\frac {\sqrt {(a e+c d x) (d+e x)} \left (-\frac {\sqrt {a} \sqrt {d} \sqrt {e} \left (-15 c^4 d^8 x^4 (d+e x)+10 a c^3 d^6 e x^3 \left (d^2-7 d e x-8 e^2 x^2\right )+2 a^2 c^2 d^4 e^2 x^2 \left (124 d^3-233 d^2 e x+662 d e^2 x^2+1659 e^3 x^3\right )+2 a^3 c d^2 e^3 x \left (168 d^4-256 d^3 e x+459 d^2 e^2 x^2-1197 d e^3 x^3-3360 e^4 x^4\right )+a^4 e^4 \left (128 d^5-176 d^4 e x+264 d^3 e^2 x^2-462 d^2 e^3 x^3+1155 d e^4 x^4+3465 e^5 x^5\right )\right )}{x^5 (d+e x)}-\frac {15 \left (c d^2-a e^2\right )^2 \left (c^3 d^6+7 a c^2 d^4 e^2+63 a^2 c d^2 e^4-231 a^3 e^6\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a e+c d x}}{\sqrt {a} \sqrt {e} \sqrt {d+e x}}\right )}{\sqrt {a e+c d x} \sqrt {d+e x}}\right )}{640 a^{5/2} d^{13/2} e^{5/2}} \] Input:
Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(x^6*(d + e*x)^4), x]
Output:
(Sqrt[(a*e + c*d*x)*(d + e*x)]*(-((Sqrt[a]*Sqrt[d]*Sqrt[e]*(-15*c^4*d^8*x^ 4*(d + e*x) + 10*a*c^3*d^6*e*x^3*(d^2 - 7*d*e*x - 8*e^2*x^2) + 2*a^2*c^2*d ^4*e^2*x^2*(124*d^3 - 233*d^2*e*x + 662*d*e^2*x^2 + 1659*e^3*x^3) + 2*a^3* c*d^2*e^3*x*(168*d^4 - 256*d^3*e*x + 459*d^2*e^2*x^2 - 1197*d*e^3*x^3 - 33 60*e^4*x^4) + a^4*e^4*(128*d^5 - 176*d^4*e*x + 264*d^3*e^2*x^2 - 462*d^2*e ^3*x^3 + 1155*d*e^4*x^4 + 3465*e^5*x^5)))/(x^5*(d + e*x))) - (15*(c*d^2 - a*e^2)^2*(c^3*d^6 + 7*a*c^2*d^4*e^2 + 63*a^2*c*d^2*e^4 - 231*a^3*e^6)*ArcT anh[(Sqrt[d]*Sqrt[a*e + c*d*x])/(Sqrt[a]*Sqrt[e]*Sqrt[d + e*x])])/(Sqrt[a* e + c*d*x]*Sqrt[d + e*x])))/(640*a^(5/2)*d^(13/2)*e^(5/2))
Time = 4.15 (sec) , antiderivative size = 615, normalized size of antiderivative = 1.01, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.325, Rules used = {1214, 25, 2181, 27, 2181, 27, 2181, 27, 2181, 27, 1228, 1154, 219}
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.
| \(\displaystyle \int \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{x^6 (d+e x)^4} \, dx\) |
| \(\Big \downarrow \) 1214 |
| \(\displaystyle -\frac {\int -\frac {\frac {a^3 e^9}{d}+\frac {\left (c d^2-a e^2\right )^3 x^5 e^8}{d^6}+\frac {a^2 \left (3 c d^2-a e^2\right ) x e^8}{d^2}-\frac {\left (c d^2-a e^2\right )^3 x^4 e^7}{d^5}+\frac {a \left (3 c^2 d^4-3 a c e^2 d^2+a^2 e^4\right ) x^2 e^7}{d^3}+\frac {\left (c d^2-a e^2\right )^3 x^3 e^6}{d^4}}{x^6 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{e^6}-\frac {2 e^3 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^6 (d+e x)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\frac {a^3 e^9}{d}+\frac {\left (c d^2-a e^2\right )^3 x^5 e^8}{d^6}+\frac {a^2 \left (3 c d^2-a e^2\right ) x e^8}{d^2}-\frac {\left (c d^2-a e^2\right )^3 x^4 e^7}{d^5}+\frac {a \left (3 c^2 d^4-3 a c e^2 d^2+a^2 e^4\right ) x^2 e^7}{d^3}+\frac {\left (c d^2-a e^2\right )^3 x^3 e^6}{d^4}}{x^6 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{e^6}-\frac {2 e^3 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^6 (d+e x)}\) |
| \(\Big \downarrow \) 2181 |
| \(\displaystyle \frac {-\frac {\int -\frac {\frac {10 a \left (c d^2-a e^2\right )^3 x^4 e^9}{d^5}+\frac {a^3 \left (21 c d^2-19 a e^2\right ) e^9}{d}-\frac {10 a \left (c d^2-a e^2\right )^3 x^3 e^8}{d^4}+2 a^2 \left (\frac {5 a^2 e^4}{d^2}-19 a c e^2+15 c^2 d^2\right ) x e^8+\frac {10 a \left (c d^2-a e^2\right )^3 x^2 e^7}{d^3}}{2 x^5 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{5 a d e}-\frac {a^2 e^8 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 d^2 x^5}}{e^6}-\frac {2 e^3 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^6 (d+e x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {\frac {10 a \left (c d^2-a e^2\right )^3 x^4 e^9}{d^5}+\frac {a^3 \left (21 c d^2-19 a e^2\right ) e^9}{d}-\frac {10 a \left (c d^2-a e^2\right )^3 x^3 e^8}{d^4}+2 a^2 \left (\frac {5 a^2 e^4}{d^2}-19 a c e^2+15 c^2 d^2\right ) x e^8+\frac {10 a \left (c d^2-a e^2\right )^3 x^2 e^7}{d^3}}{x^5 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{10 a d e}-\frac {a^2 e^8 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 d^2 x^5}}{e^6}-\frac {2 e^3 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^6 (d+e x)}\) |
| \(\Big \downarrow \) 2181 |
| \(\displaystyle \frac {\frac {-\frac {\int -\frac {\frac {80 a^2 \left (c d^2-a e^2\right )^3 x^3 e^{10}}{d^4}-\frac {80 a^2 \left (c d^2-a e^2\right )^3 x^2 e^9}{d^3}+\frac {3 a^3 \left (31 c^2 d^4-106 a c e^2 d^2+71 a^2 e^4\right ) e^9}{d}+2 a^2 \left (-\frac {40 a^3 e^6}{d^2}+177 a^2 c e^4-183 a c^2 d^2 e^2+40 c^3 d^4\right ) x e^8}{2 x^4 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{4 a d e}-\frac {a^2 e^8 \left (21 c d^2-19 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 d^2 x^4}}{10 a d e}-\frac {a^2 e^8 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 d^2 x^5}}{e^6}-\frac {2 e^3 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^6 (d+e x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {\frac {80 a^2 \left (c d^2-a e^2\right )^3 x^3 e^{10}}{d^4}-\frac {80 a^2 \left (c d^2-a e^2\right )^3 x^2 e^9}{d^3}+\frac {3 a^3 \left (31 c^2 d^4-106 a c e^2 d^2+71 a^2 e^4\right ) e^9}{d}+2 a^2 \left (-\frac {40 a^3 e^6}{d^2}+177 a^2 c e^4-183 a c^2 d^2 e^2+40 c^3 d^4\right ) x e^8}{x^4 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{8 a d e}-\frac {a^2 e^8 \left (21 c d^2-19 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 d^2 x^4}}{10 a d e}-\frac {a^2 e^8 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 d^2 x^5}}{e^6}-\frac {2 e^3 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^6 (d+e x)}\) |
| \(\Big \downarrow \) 2181 |
| \(\displaystyle \frac {\frac {\frac {-\frac {\int -\frac {3 \left (\frac {160 a^3 \left (c d^2-a e^2\right )^3 x^2 e^{11}}{d^3}-4 a^3 \left (-\frac {40 a^3 e^6}{d^2}+191 a^2 c e^4-226 a c^2 d^2 e^2+71 c^3 d^4\right ) x e^{10}+\frac {a^3 \left (5 c^3 d^6-357 a c^2 e^2 d^4+883 a^2 c e^4 d^2-515 a^3 e^6\right ) e^9}{d}\right )}{2 x^3 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{3 a d e}-\frac {a^2 e^8 \left (71 a^2 e^4-106 a c d^2 e^2+31 c^2 d^4\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^2 x^3}}{8 a d e}-\frac {a^2 e^8 \left (21 c d^2-19 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 d^2 x^4}}{10 a d e}-\frac {a^2 e^8 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 d^2 x^5}}{e^6}-\frac {2 e^3 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^6 (d+e x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\frac {\int \frac {\frac {160 a^3 \left (c d^2-a e^2\right )^3 x^2 e^{11}}{d^3}-4 a^3 \left (-\frac {40 a^3 e^6}{d^2}+191 a^2 c e^4-226 a c^2 d^2 e^2+71 c^3 d^4\right ) x e^{10}+\frac {a^3 \left (5 c^3 d^6-357 a c^2 e^2 d^4+883 a^2 c e^4 d^2-515 a^3 e^6\right ) e^9}{d}}{x^3 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 a d e}-\frac {a^2 e^8 \left (71 a^2 e^4-106 a c d^2 e^2+31 c^2 d^4\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^2 x^3}}{8 a d e}-\frac {a^2 e^8 \left (21 c d^2-19 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 d^2 x^4}}{10 a d e}-\frac {a^2 e^8 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 d^2 x^5}}{e^6}-\frac {2 e^3 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^6 (d+e x)}\) |
| \(\Big \downarrow \) 2181 |
| \(\displaystyle \frac {\frac {\frac {\frac {-\frac {\int \frac {a^3 e^9 \left (d \left (15 c^4 d^8+80 a c^3 e^2 d^6-2038 a^2 c^2 e^4 d^4+4160 a^3 c e^6 d^2-2185 a^4 e^8\right )+2 e \left (5 c^4 d^8-677 a c^3 e^2 d^6+1843 a^2 c^2 e^4 d^4-1475 a^3 c e^6 d^2+320 a^4 e^8\right ) x\right )}{2 d^2 x^2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 a d e}-\frac {a^2 e^8 \left (-515 a^3 e^6+883 a^2 c d^2 e^4-357 a c^2 d^4 e^2+5 c^3 d^6\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 d^2 x^2}}{2 a d e}-\frac {a^2 e^8 \left (71 a^2 e^4-106 a c d^2 e^2+31 c^2 d^4\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^2 x^3}}{8 a d e}-\frac {a^2 e^8 \left (21 c d^2-19 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 d^2 x^4}}{10 a d e}-\frac {a^2 e^8 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 d^2 x^5}}{e^6}-\frac {2 e^3 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^6 (d+e x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\frac {-\frac {a^2 e^8 \int \frac {15 c^4 d^9+80 a c^3 e^2 d^7-2038 a^2 c^2 e^4 d^5+4160 a^3 c e^6 d^3-2185 a^4 e^8 d+2 e \left (5 c^4 d^8-677 a c^3 e^2 d^6+1843 a^2 c^2 e^4 d^4-1475 a^3 c e^6 d^2+320 a^4 e^8\right ) x}{x^2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{4 d^3}-\frac {a^2 e^8 \left (-515 a^3 e^6+883 a^2 c d^2 e^4-357 a c^2 d^4 e^2+5 c^3 d^6\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 d^2 x^2}}{2 a d e}-\frac {a^2 e^8 \left (71 a^2 e^4-106 a c d^2 e^2+31 c^2 d^4\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^2 x^3}}{8 a d e}-\frac {a^2 e^8 \left (21 c d^2-19 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 d^2 x^4}}{10 a d e}-\frac {a^2 e^8 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 d^2 x^5}}{e^6}-\frac {2 e^3 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^6 (d+e x)}\) |
| \(\Big \downarrow \) 1228 |
| \(\displaystyle \frac {\frac {\frac {\frac {-\frac {a^2 e^8 \left (-\frac {15 \left (-231 a^3 e^6+63 a^2 c d^2 e^4+7 a c^2 d^4 e^2+c^3 d^6\right ) \left (c d^2-a e^2\right )^2 \int \frac {1}{x \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 a e}-\frac {\left (-2185 a^4 e^8+4160 a^3 c d^2 e^6-2038 a^2 c^2 d^4 e^4+80 a c^3 d^6 e^2+15 c^4 d^8\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{a e x}\right )}{4 d^3}-\frac {a^2 e^8 \left (-515 a^3 e^6+883 a^2 c d^2 e^4-357 a c^2 d^4 e^2+5 c^3 d^6\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 d^2 x^2}}{2 a d e}-\frac {a^2 e^8 \left (71 a^2 e^4-106 a c d^2 e^2+31 c^2 d^4\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^2 x^3}}{8 a d e}-\frac {a^2 e^8 \left (21 c d^2-19 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 d^2 x^4}}{10 a d e}-\frac {a^2 e^8 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 d^2 x^5}}{e^6}-\frac {2 e^3 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^6 (d+e x)}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {\frac {\frac {\frac {-\frac {a^2 e^8 \left (\frac {15 \left (c d^2-a e^2\right )^2 \left (-231 a^3 e^6+63 a^2 c d^2 e^4+7 a c^2 d^4 e^2+c^3 d^6\right ) \int \frac {1}{4 a d e-\frac {\left (2 a d e+\left (c d^2+a e^2\right ) x\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {2 a d e+\left (c d^2+a e^2\right ) x}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}}{a e}-\frac {\left (-2185 a^4 e^8+4160 a^3 c d^2 e^6-2038 a^2 c^2 d^4 e^4+80 a c^3 d^6 e^2+15 c^4 d^8\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{a e x}\right )}{4 d^3}-\frac {a^2 e^8 \left (-515 a^3 e^6+883 a^2 c d^2 e^4-357 a c^2 d^4 e^2+5 c^3 d^6\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 d^2 x^2}}{2 a d e}-\frac {a^2 e^8 \left (71 a^2 e^4-106 a c d^2 e^2+31 c^2 d^4\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^2 x^3}}{8 a d e}-\frac {a^2 e^8 \left (21 c d^2-19 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 d^2 x^4}}{10 a d e}-\frac {a^2 e^8 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 d^2 x^5}}{e^6}-\frac {2 e^3 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^6 (d+e x)}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\frac {\frac {-\frac {a^2 e^8 \left (-515 a^3 e^6+883 a^2 c d^2 e^4-357 a c^2 d^4 e^2+5 c^3 d^6\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 d^2 x^2}-\frac {a^2 e^8 \left (\frac {15 \left (c d^2-a e^2\right )^2 \left (-231 a^3 e^6+63 a^2 c d^2 e^4+7 a c^2 d^4 e^2+c^3 d^6\right ) \text {arctanh}\left (\frac {x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 a^{3/2} \sqrt {d} e^{3/2}}-\frac {\left (-2185 a^4 e^8+4160 a^3 c d^2 e^6-2038 a^2 c^2 d^4 e^4+80 a c^3 d^6 e^2+15 c^4 d^8\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{a e x}\right )}{4 d^3}}{2 a d e}-\frac {a^2 e^8 \left (71 a^2 e^4-106 a c d^2 e^2+31 c^2 d^4\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^2 x^3}}{8 a d e}-\frac {a^2 e^8 \left (21 c d^2-19 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 d^2 x^4}}{10 a d e}-\frac {a^2 e^8 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 d^2 x^5}}{e^6}-\frac {2 e^3 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^6 (d+e x)}\) |
Input:
Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(x^6*(d + e*x)^4),x]
Output:
(-2*e^3*(c*d^2 - a*e^2)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(d^ 6*(d + e*x)) + (-1/5*(a^2*e^8*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) /(d^2*x^5) + (-1/4*(a^2*e^8*(21*c*d^2 - 19*a*e^2)*Sqrt[a*d*e + (c*d^2 + a* e^2)*x + c*d*e*x^2])/(d^2*x^4) + (-((a^2*e^8*(31*c^2*d^4 - 106*a*c*d^2*e^2 + 71*a^2*e^4)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(d^2*x^3)) + ( -1/2*(a^2*e^8*(5*c^3*d^6 - 357*a*c^2*d^4*e^2 + 883*a^2*c*d^2*e^4 - 515*a^3 *e^6)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(d^2*x^2) - (a^2*e^8*(- (((15*c^4*d^8 + 80*a*c^3*d^6*e^2 - 2038*a^2*c^2*d^4*e^4 + 4160*a^3*c*d^2*e ^6 - 2185*a^4*e^8)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(a*e*x)) + (15*(c*d^2 - a*e^2)^2*(c^3*d^6 + 7*a*c^2*d^4*e^2 + 63*a^2*c*d^2*e^4 - 231 *a^3*e^6)*ArcTanh[(2*a*d*e + (c*d^2 + a*e^2)*x)/(2*Sqrt[a]*Sqrt[d]*Sqrt[e] *Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])])/(2*a^(3/2)*Sqrt[d]*e^(3/2) )))/(4*d^3))/(2*a*d*e))/(8*a*d*e))/(10*a*d*e))/e^6
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[(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/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym bol] :> Simp[-2 Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c , d, e}, x]
Int[(x_)^(n_.)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^ 2)^(p_), x_Symbol] :> Simp[-2*(-d)^n*e^(2*m - n + 3)*(Sqrt[a + b*x + c*x^2] /((-2*c*d + b*e)^(m + 2)*(d + e*x))), x] - Simp[e^(2*m + 2) Int[ExpandToS um[(((-d)^n*(-2*c*d + b*e)^(-m - 1))/(e^n*x^n) - ((-c)*d + b*e + c*e*x)^(-m - 1))/(d + e*x), x]/(Sqrt[a + b*x + c*x^2]/x^n), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && ILtQ[m, 0] && ILtQ[n, 0] && EqQ[m + p, -3/2]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Simp[(b*(e *f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^ (m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x ] && EqQ[Simplify[m + 2*p + 3], 0]
Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_ ), x_Symbol] :> With[{Qx = PolynomialQuotient[Pq, d + e*x, x], R = Polynomi alRemainder[Pq, d + e*x, x]}, Simp[(e*R*(d + e*x)^(m + 1)*(a + b*x + c*x^2) ^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Simp[1/((m + 1)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*ExpandToSum[(m + 1)*(c*d^2 - b*d*e + a*e^2)*Qx + c*d*R*(m + 1) - b*e*R*(m + p + 2) - c*e*R *(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, e, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(21016\) vs. \(2(567)=1134\).
Time = 9.05 (sec) , antiderivative size = 21017, normalized size of antiderivative = 34.62
Input:
int((a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(5/2)/x^6/(e*x+d)^4,x,method=_RETURN VERBOSE)
Output:
result too large to display
Time = 35.08 (sec) , antiderivative size = 1198, normalized size of antiderivative = 1.97 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^6 (d+e x)^4} \, dx=\text {Too large to display} \] Input:
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/x^6/(e*x+d)^4,x, algorit hm="fricas")
Output:
[-1/2560*(15*((c^5*d^10*e + 5*a*c^4*d^8*e^3 + 50*a^2*c^3*d^6*e^5 - 350*a^3 *c^2*d^4*e^7 + 525*a^4*c*d^2*e^9 - 231*a^5*e^11)*x^6 + (c^5*d^11 + 5*a*c^4 *d^9*e^2 + 50*a^2*c^3*d^7*e^4 - 350*a^3*c^2*d^5*e^6 + 525*a^4*c*d^3*e^8 - 231*a^5*d*e^10)*x^5)*sqrt(a*d*e)*log((8*a^2*d^2*e^2 + (c^2*d^4 + 6*a*c*d^2 *e^2 + a^2*e^4)*x^2 + 4*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*a*d *e + (c*d^2 + a*e^2)*x)*sqrt(a*d*e) + 8*(a*c*d^3*e + a^2*d*e^3)*x)/x^2) + 4*(128*a^5*d^6*e^5 - (15*a*c^4*d^9*e^2 + 80*a^2*c^3*d^7*e^4 - 3318*a^3*c^2 *d^5*e^6 + 6720*a^4*c*d^3*e^8 - 3465*a^5*d*e^10)*x^5 - (15*a*c^4*d^10*e + 70*a^2*c^3*d^8*e^3 - 1324*a^3*c^2*d^6*e^5 + 2394*a^4*c*d^4*e^7 - 1155*a^5* d^2*e^9)*x^4 + 2*(5*a^2*c^3*d^9*e^2 - 233*a^3*c^2*d^7*e^4 + 459*a^4*c*d^5* e^6 - 231*a^5*d^3*e^8)*x^3 + 8*(31*a^3*c^2*d^8*e^3 - 64*a^4*c*d^6*e^5 + 33 *a^5*d^4*e^7)*x^2 + 16*(21*a^4*c*d^7*e^4 - 11*a^5*d^5*e^6)*x)*sqrt(c*d*e*x ^2 + a*d*e + (c*d^2 + a*e^2)*x))/(a^3*d^7*e^4*x^6 + a^3*d^8*e^3*x^5), 1/12 80*(15*((c^5*d^10*e + 5*a*c^4*d^8*e^3 + 50*a^2*c^3*d^6*e^5 - 350*a^3*c^2*d ^4*e^7 + 525*a^4*c*d^2*e^9 - 231*a^5*e^11)*x^6 + (c^5*d^11 + 5*a*c^4*d^9*e ^2 + 50*a^2*c^3*d^7*e^4 - 350*a^3*c^2*d^5*e^6 + 525*a^4*c*d^3*e^8 - 231*a^ 5*d*e^10)*x^5)*sqrt(-a*d*e)*arctan(1/2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a *e^2)*x)*(2*a*d*e + (c*d^2 + a*e^2)*x)*sqrt(-a*d*e)/(a*c*d^2*e^2*x^2 + a^2 *d^2*e^2 + (a*c*d^3*e + a^2*d*e^3)*x)) - 2*(128*a^5*d^6*e^5 - (15*a*c^4*d^ 9*e^2 + 80*a^2*c^3*d^7*e^4 - 3318*a^3*c^2*d^5*e^6 + 6720*a^4*c*d^3*e^8 ...
Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^6 (d+e x)^4} \, dx=\text {Timed out} \] Input:
integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2)/x**6/(e*x+d)**4,x)
Output:
Timed out
\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^6 (d+e x)^4} \, dx=\int { \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{4} x^{6}} \,d x } \] Input:
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/x^6/(e*x+d)^4,x, algorit hm="maxima")
Output:
integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)/((e*x + d)^4*x^6), x)
Exception generated. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^6 (d+e x)^4} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/x^6/(e*x+d)^4,x, algorit hm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro unding error%%%{%%%{1,[0,6,15]%%%},[2,7]%%%}+%%%{%%%{-7,[1,8,13]%%%},[2,6] %%%}+%%%{
Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^6 (d+e x)^4} \, dx=\int \frac {{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}}{x^6\,{\left (d+e\,x\right )}^4} \,d x \] Input:
int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)/(x^6*(d + e*x)^4),x)
Output:
int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)/(x^6*(d + e*x)^4), x)
Time = 199.31 (sec) , antiderivative size = 3796, normalized size of antiderivative = 6.25 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^6 (d+e x)^4} \, dx =\text {Too large to display} \] Input:
int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/x^6/(e*x+d)^4,x)
Output:
( - 2816*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**6*d**6*e**7 + 3872*sqrt(d + e* x)*sqrt(a*e + c*d*x)*a**6*d**5*e**8*x - 5808*sqrt(d + e*x)*sqrt(a*e + c*d* x)*a**6*d**4*e**9*x**2 + 10164*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**6*d**3*e **10*x**3 - 25410*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**6*d**2*e**11*x**4 - 7 6230*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**6*d*e**12*x**5 - 2304*sqrt(d + e*x )*sqrt(a*e + c*d*x)*a**5*c*d**8*e**5 - 4224*sqrt(d + e*x)*sqrt(a*e + c*d*x )*a**5*c*d**7*e**6*x + 6512*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**5*c*d**6*e* *7*x**2 - 11880*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**5*c*d**5*e**8*x**3 + 31 878*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**5*c*d**4*e**9*x**4 + 85470*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**5*c*d**3*e**10*x**5 - 6048*sqrt(d + e*x)*sqrt(a *e + c*d*x)*a**4*c**2*d**9*e**4*x + 3760*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a **4*c**2*d**8*e**5*x**2 - 6272*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**4*c**2*d **7*e**6*x**3 + 13964*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**4*c**2*d**6*e**7* x**4 + 47964*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**4*c**2*d**5*e**8*x**5 - 44 64*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**3*c**3*d**10*e**3*x**2 + 8168*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**3*c**3*d**9*e**4*x**3 - 22292*sqrt(d + e*x)*s qrt(a*e + c*d*x)*a**3*c**3*d**8*e**5*x**4 - 57964*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**3*c**3*d**7*e**6*x**5 - 180*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a** 2*c**4*d**11*e**2*x**3 + 1590*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**2*c**4*d* *10*e**3*x**4 + 1770*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**2*c**4*d**9*e**...