Integrand size = 40, antiderivative size = 628 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^9 (d+e x)} \, dx=-\frac {3 \left (c d^2-a e^2\right )^3 \left (15 c^3 d^6+35 a c^2 d^4 e^2+45 a^2 c d^2 e^4+33 a^3 e^6\right ) \left (2 a d e+\left (c d^2+a e^2\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16384 a^5 d^6 e^5 x^2}+\frac {\left (c d^2-a e^2\right ) \left (15 c^3 d^6+35 a c^2 d^4 e^2+45 a^2 c d^2 e^4+33 a^3 e^6\right ) \left (2 a d e+\left (c d^2+a e^2\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{2048 a^4 d^5 e^4 x^4}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{8 d x^8}-\frac {\left (\frac {5 c}{a e}-\frac {11 e}{d^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{112 x^7}+\frac {\left (15 c^2 d^4+10 a c d^2 e^2-33 a^2 e^4\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{448 a^2 d^3 e^2 x^6}-\frac {\left (105 c^3 d^6+95 a c^2 d^4 e^2+15 a^2 c d^2 e^4-231 a^3 e^6\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{4480 a^3 d^4 e^3 x^5}+\frac {3 \left (c d^2-a e^2\right )^5 \left (15 c^3 d^6+35 a c^2 d^4 e^2+45 a^2 c d^2 e^4+33 a^3 e^6\right ) \text {arctanh}\left (\frac {2 a d e+\left (c d^2+a e^2\right ) x}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{32768 a^{11/2} d^{13/2} e^{11/2}} \]
1/2048*(-a*e^2+c*d^2)*(33*a^3*e^6+45*a^2*c*d^2*e^4+35*a*c^2*d^4*e^2+15*c^3 *d^6)*(2*a*d*e+(a*e^2+c*d^2)*x)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/a^ 4/d^5/e^4/x^4-1/8*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/d/x^8-1/112*(5*c /a/e-11*e/d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/x^7+1/448*(-33*a^2* e^4+10*a*c*d^2*e^2+15*c^2*d^4)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/a^2 /d^3/e^2/x^6-1/4480*(-231*a^3*e^6+15*a^2*c*d^2*e^4+95*a*c^2*d^4*e^2+105*c^ 3*d^6)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/a^3/d^4/e^3/x^5+3/32768*(-a *e^2+c*d^2)^5*(33*a^3*e^6+45*a^2*c*d^2*e^4+35*a*c^2*d^4*e^2+15*c^3*d^6)*ar ctanh(1/2*(2*a*d*e+(a*e^2+c*d^2)*x)/a^(1/2)/d^(1/2)/e^(1/2)/(a*d*e+(a*e^2+ c*d^2)*x+c*d*e*x^2)^(1/2))/a^(11/2)/d^(13/2)/e^(11/2)-3/16384*(-a*e^2+c*d^ 2)^3*(33*a^3*e^6+45*a^2*c*d^2*e^4+35*a*c^2*d^4*e^2+15*c^3*d^6)*(2*a*d*e+(a *e^2+c*d^2)*x)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/a^5/d^6/e^5/x^2
Time = 1.54 (sec) , antiderivative size = 572, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^9 (d+e x)} \, dx=\frac {\sqrt {(a e+c d x) (d+e x)} \left (-\frac {\sqrt {a} \sqrt {d} \sqrt {e} \left (1575 c^7 d^{14} x^7-525 a c^6 d^{12} e x^6 (2 d+7 e x)+35 a^2 c^5 d^{10} e^2 x^5 \left (24 d^2+68 d e x+29 e^2 x^2\right )-5 a^3 c^4 d^8 e^3 x^4 \left (144 d^3+376 d^2 e x+110 d e^2 x^2-185 e^3 x^3\right )+5 a^4 c^3 d^6 e^4 x^3 \left (128 d^4+320 d^3 e x+80 d^2 e^2 x^2-120 d e^3 x^3+265 e^4 x^4\right )+a^5 c^2 d^4 e^5 x^2 \left (103680 d^5+137600 d^4 e x+4640 d^3 e^2 x^2-5488 d^2 e^3 x^3+7034 d e^4 x^4-11193 e^5 x^5\right )+a^6 c d^2 e^6 x \left (168960 d^6+212480 d^5 e x+4480 d^4 e^2 x^2-5056 d^3 e^3 x^3+5928 d^2 e^4 x^4-7476 d e^5 x^5+11445 e^6 x^6\right )+a^7 e^7 \left (71680 d^7+87040 d^6 e x+1280 d^5 e^2 x^2-1408 d^4 e^3 x^3+1584 d^3 e^4 x^4-1848 d^2 e^5 x^5+2310 d e^6 x^6-3465 e^7 x^7\right )\right )}{x^8}+\frac {105 \left (c d^2-a e^2\right )^5 \left (15 c^3 d^6+35 a c^2 d^4 e^2+45 a^2 c d^2 e^4+33 a^3 e^6\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a e+c d x}}{\sqrt {a} \sqrt {e} \sqrt {d+e x}}\right )}{\sqrt {a e+c d x} \sqrt {d+e x}}\right )}{573440 a^{11/2} d^{13/2} e^{11/2}} \]
(Sqrt[(a*e + c*d*x)*(d + e*x)]*(-((Sqrt[a]*Sqrt[d]*Sqrt[e]*(1575*c^7*d^14* x^7 - 525*a*c^6*d^12*e*x^6*(2*d + 7*e*x) + 35*a^2*c^5*d^10*e^2*x^5*(24*d^2 + 68*d*e*x + 29*e^2*x^2) - 5*a^3*c^4*d^8*e^3*x^4*(144*d^3 + 376*d^2*e*x + 110*d*e^2*x^2 - 185*e^3*x^3) + 5*a^4*c^3*d^6*e^4*x^3*(128*d^4 + 320*d^3*e *x + 80*d^2*e^2*x^2 - 120*d*e^3*x^3 + 265*e^4*x^4) + a^5*c^2*d^4*e^5*x^2*( 103680*d^5 + 137600*d^4*e*x + 4640*d^3*e^2*x^2 - 5488*d^2*e^3*x^3 + 7034*d *e^4*x^4 - 11193*e^5*x^5) + a^6*c*d^2*e^6*x*(168960*d^6 + 212480*d^5*e*x + 4480*d^4*e^2*x^2 - 5056*d^3*e^3*x^3 + 5928*d^2*e^4*x^4 - 7476*d*e^5*x^5 + 11445*e^6*x^6) + a^7*e^7*(71680*d^7 + 87040*d^6*e*x + 1280*d^5*e^2*x^2 - 1408*d^4*e^3*x^3 + 1584*d^3*e^4*x^4 - 1848*d^2*e^5*x^5 + 2310*d*e^6*x^6 - 3465*e^7*x^7)))/x^8) + (105*(c*d^2 - a*e^2)^5*(15*c^3*d^6 + 35*a*c^2*d^4*e ^2 + 45*a^2*c*d^2*e^4 + 33*a^3*e^6)*ArcTanh[(Sqrt[d]*Sqrt[a*e + c*d*x])/(S qrt[a]*Sqrt[e]*Sqrt[d + e*x])])/(Sqrt[a*e + c*d*x]*Sqrt[d + e*x])))/(57344 0*a^(11/2)*d^(13/2)*e^(11/2))
Time = 1.04 (sec) , antiderivative size = 608, normalized size of antiderivative = 0.97, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1215, 1237, 27, 1237, 27, 1237, 27, 1228, 1152, 1152, 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.
| \(\displaystyle \int \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{x^9 (d+e x)} \, dx\) |
| \(\Big \downarrow \) 1215 |
| \(\displaystyle \int \frac {(a e+c d x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{x^9}dx\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle -\frac {\int -\frac {a e \left (5 c d^2-6 c e x d-11 a e^2\right ) \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}{2 x^8}dx}{8 a d e}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{8 d x^8}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\left (5 c d^2-6 c e x d-11 a e^2\right ) \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}{x^8}dx}{16 d}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{8 d x^8}\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle \frac {-\frac {\int \frac {\left (3 \left (15 c^2 d^4+10 a c e^2 d^2-33 a^2 e^4\right )+4 c d e \left (5 c d^2-11 a e^2\right ) x\right ) \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}{2 x^7}dx}{7 a d e}-\frac {\left (\frac {5 c d}{a e}-\frac {11 e}{d}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{7 x^7}}{16 d}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{8 d x^8}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\int \frac {\left (3 \left (15 c^2 d^4+10 a c e^2 d^2-33 a^2 e^4\right )+4 c d e \left (5 c d^2-11 a e^2\right ) x\right ) \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}{x^7}dx}{14 a d e}-\frac {\left (\frac {5 c d}{a e}-\frac {11 e}{d}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{7 x^7}}{16 d}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{8 d x^8}\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle \frac {-\frac {-\frac {\int \frac {3 \left (105 c^3 d^6+95 a c^2 e^2 d^4+15 a^2 c e^4 d^2+2 c e \left (15 c^2 d^4+10 a c e^2 d^2-33 a^2 e^4\right ) x d-231 a^3 e^6\right ) \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}{2 x^6}dx}{6 a d e}-\frac {\left (\frac {15 c^2 d^4}{a}-33 a e^4+10 c d^2 e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{2 d e x^6}}{14 a d e}-\frac {\left (\frac {5 c d}{a e}-\frac {11 e}{d}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{7 x^7}}{16 d}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{8 d x^8}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {-\frac {\int \frac {\left (105 c^3 d^6+95 a c^2 e^2 d^4+15 a^2 c e^4 d^2+2 c e \left (15 c^2 d^4+10 a c e^2 d^2-33 a^2 e^4\right ) x d-231 a^3 e^6\right ) \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}{x^6}dx}{4 a d e}-\frac {\left (\frac {15 c^2 d^4}{a}-33 a e^4+10 c d^2 e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{2 d e x^6}}{14 a d e}-\frac {\left (\frac {5 c d}{a e}-\frac {11 e}{d}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{7 x^7}}{16 d}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{8 d x^8}\) |
| \(\Big \downarrow \) 1228 |
| \(\displaystyle \frac {-\frac {-\frac {-\frac {7 \left (c d^2-a e^2\right ) \left (33 a^3 e^6+45 a^2 c d^2 e^4+35 a c^2 d^4 e^2+15 c^3 d^6\right ) \int \frac {\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}{x^5}dx}{2 a d e}-\frac {\left (-231 a^3 e^6+15 a^2 c d^2 e^4+95 a c^2 d^4 e^2+105 c^3 d^6\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 a d e x^5}}{4 a d e}-\frac {\left (\frac {15 c^2 d^4}{a}-33 a e^4+10 c d^2 e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{2 d e x^6}}{14 a d e}-\frac {\left (\frac {5 c d}{a e}-\frac {11 e}{d}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{7 x^7}}{16 d}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{8 d x^8}\) |
| \(\Big \downarrow \) 1152 |
| \(\displaystyle \frac {-\frac {-\frac {-\frac {7 \left (c d^2-a e^2\right ) \left (33 a^3 e^6+45 a^2 c d^2 e^4+35 a c^2 d^4 e^2+15 c^3 d^6\right ) \left (-\frac {3 \left (c d^2-a e^2\right )^2 \int \frac {\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{x^3}dx}{16 a d e}-\frac {\left (x \left (a e^2+c d^2\right )+2 a d e\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{8 a d e x^4}\right )}{2 a d e}-\frac {\left (-231 a^3 e^6+15 a^2 c d^2 e^4+95 a c^2 d^4 e^2+105 c^3 d^6\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 a d e x^5}}{4 a d e}-\frac {\left (\frac {15 c^2 d^4}{a}-33 a e^4+10 c d^2 e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{2 d e x^6}}{14 a d e}-\frac {\left (\frac {5 c d}{a e}-\frac {11 e}{d}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{7 x^7}}{16 d}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{8 d x^8}\) |
| \(\Big \downarrow \) 1152 |
| \(\displaystyle \frac {-\frac {-\frac {-\frac {7 \left (c d^2-a e^2\right ) \left (33 a^3 e^6+45 a^2 c d^2 e^4+35 a c^2 d^4 e^2+15 c^3 d^6\right ) \left (-\frac {3 \left (c d^2-a e^2\right )^2 \left (-\frac {\left (c d^2-a e^2\right )^2 \int \frac {1}{x \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{8 a d e}-\frac {\left (x \left (a e^2+c d^2\right )+2 a d e\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 a d e x^2}\right )}{16 a d e}-\frac {\left (x \left (a e^2+c d^2\right )+2 a d e\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{8 a d e x^4}\right )}{2 a d e}-\frac {\left (-231 a^3 e^6+15 a^2 c d^2 e^4+95 a c^2 d^4 e^2+105 c^3 d^6\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 a d e x^5}}{4 a d e}-\frac {\left (\frac {15 c^2 d^4}{a}-33 a e^4+10 c d^2 e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{2 d e x^6}}{14 a d e}-\frac {\left (\frac {5 c d}{a e}-\frac {11 e}{d}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{7 x^7}}{16 d}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{8 d x^8}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {-\frac {-\frac {-\frac {7 \left (c d^2-a e^2\right ) \left (33 a^3 e^6+45 a^2 c d^2 e^4+35 a c^2 d^4 e^2+15 c^3 d^6\right ) \left (-\frac {3 \left (c d^2-a e^2\right )^2 \left (\frac {\left (c d^2-a e^2\right )^2 \int \frac {1}{4 a d e-\frac {\left (2 a d e+\left (c d^2+a e^2\right ) x\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {2 a d e+\left (c d^2+a e^2\right ) x}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}}{4 a d e}-\frac {\left (x \left (a e^2+c d^2\right )+2 a d e\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 a d e x^2}\right )}{16 a d e}-\frac {\left (x \left (a e^2+c d^2\right )+2 a d e\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{8 a d e x^4}\right )}{2 a d e}-\frac {\left (-231 a^3 e^6+15 a^2 c d^2 e^4+95 a c^2 d^4 e^2+105 c^3 d^6\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 a d e x^5}}{4 a d e}-\frac {\left (\frac {15 c^2 d^4}{a}-33 a e^4+10 c d^2 e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{2 d e x^6}}{14 a d e}-\frac {\left (\frac {5 c d}{a e}-\frac {11 e}{d}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{7 x^7}}{16 d}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{8 d x^8}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {-\frac {-\frac {-\frac {\left (-231 a^3 e^6+15 a^2 c d^2 e^4+95 a c^2 d^4 e^2+105 c^3 d^6\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 a d e x^5}-\frac {7 \left (c d^2-a e^2\right ) \left (33 a^3 e^6+45 a^2 c d^2 e^4+35 a c^2 d^4 e^2+15 c^3 d^6\right ) \left (-\frac {3 \left (c d^2-a e^2\right )^2 \left (\frac {\left (c d^2-a e^2\right )^2 \text {arctanh}\left (\frac {x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 a^{3/2} d^{3/2} e^{3/2}}-\frac {\left (x \left (a e^2+c d^2\right )+2 a d e\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 a d e x^2}\right )}{16 a d e}-\frac {\left (x \left (a e^2+c d^2\right )+2 a d e\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{8 a d e x^4}\right )}{2 a d e}}{4 a d e}-\frac {\left (\frac {15 c^2 d^4}{a}-33 a e^4+10 c d^2 e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{2 d e x^6}}{14 a d e}-\frac {\left (\frac {5 c d}{a e}-\frac {11 e}{d}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{7 x^7}}{16 d}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{8 d x^8}\) |
-1/8*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d*x^8) + (-1/7*(((5*c* d)/(a*e) - (11*e)/d)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/x^7 - (-1/2*(((15*c^2*d^4)/a + 10*c*d^2*e^2 - 33*a*e^4)*(a*d*e + (c*d^2 + a*e^2) *x + c*d*e*x^2)^(5/2))/(d*e*x^6) - (-1/5*((105*c^3*d^6 + 95*a*c^2*d^4*e^2 + 15*a^2*c*d^2*e^4 - 231*a^3*e^6)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^ (5/2))/(a*d*e*x^5) - (7*(c*d^2 - a*e^2)*(15*c^3*d^6 + 35*a*c^2*d^4*e^2 + 4 5*a^2*c*d^2*e^4 + 33*a^3*e^6)*(-1/8*((2*a*d*e + (c*d^2 + a*e^2)*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(a*d*e*x^4) - (3*(c*d^2 - a*e^2)^2 *(-1/4*((2*a*d*e + (c*d^2 + a*e^2)*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d *e*x^2])/(a*d*e*x^2) + ((c*d^2 - a*e^2)^2*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])])/(8*a^(3/2)*d^(3/2)*e^(3/2))))/(16*a*d*e)))/(2*a*d*e))/(4*a*d*e))/(14 *a*d*e))/(16*d)
3.5.69.3.1 Defintions of rubi rules used
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[((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/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[p*((b^2 - 4*a *c)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))) Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[m + 2*p + 2, 0] && GtQ[p, 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[(((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_))/( (d_) + (e_.)*(x_)), x_Symbol] :> Int[(a/d + c*(x/e))*(f + g*x)^n*(a + b*x + c*x^2)^(p - 1), x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0]
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[((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)/((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*Simp[ (c*d*f - f*b*e + a*e*g)*(m + 1) + b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && LtQ[m, -1 ] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Leaf count of result is larger than twice the leaf count of optimal. \(70735\) vs. \(2(586)=1172\).
Time = 3.29 (sec) , antiderivative size = 70736, normalized size of antiderivative = 112.64
Time = 176.45 (sec) , antiderivative size = 1550, normalized size of antiderivative = 2.47 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^9 (d+e x)} \, dx=\text {Too large to display} \]
[1/2293760*(105*(15*c^8*d^16 - 40*a*c^7*d^14*e^2 + 20*a^2*c^6*d^12*e^4 + 8 *a^3*c^5*d^10*e^6 + 10*a^4*c^4*d^8*e^8 + 40*a^5*c^3*d^6*e^10 - 140*a^6*c^2 *d^4*e^12 + 120*a^7*c*d^2*e^14 - 33*a^8*e^16)*sqrt(a*d*e)*x^8*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*(71680*a^8*d^8*e^8 + (1575*a*c^7*d^15*e - 3 675*a^2*c^6*d^13*e^3 + 1015*a^3*c^5*d^11*e^5 + 925*a^4*c^4*d^9*e^7 + 1325* a^5*c^3*d^7*e^9 - 11193*a^6*c^2*d^5*e^11 + 11445*a^7*c*d^3*e^13 - 3465*a^8 *d*e^15)*x^7 - 2*(525*a^2*c^6*d^14*e^2 - 1190*a^3*c^5*d^12*e^4 + 275*a^4*c ^4*d^10*e^6 + 300*a^5*c^3*d^8*e^8 - 3517*a^6*c^2*d^6*e^10 + 3738*a^7*c*d^4 *e^12 - 1155*a^8*d^2*e^14)*x^6 + 8*(105*a^3*c^5*d^13*e^3 - 235*a^4*c^4*d^1 1*e^5 + 50*a^5*c^3*d^9*e^7 - 686*a^6*c^2*d^7*e^9 + 741*a^7*c*d^5*e^11 - 23 1*a^8*d^3*e^13)*x^5 - 16*(45*a^4*c^4*d^12*e^4 - 100*a^5*c^3*d^10*e^6 - 290 *a^6*c^2*d^8*e^8 + 316*a^7*c*d^6*e^10 - 99*a^8*d^4*e^12)*x^4 + 128*(5*a^5* c^3*d^11*e^5 + 1075*a^6*c^2*d^9*e^7 + 35*a^7*c*d^7*e^9 - 11*a^8*d^5*e^11)* x^3 + 1280*(81*a^6*c^2*d^10*e^6 + 166*a^7*c*d^8*e^8 + a^8*d^6*e^10)*x^2 + 5120*(33*a^7*c*d^9*e^7 + 17*a^8*d^7*e^9)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^ 2 + a*e^2)*x))/(a^6*d^7*e^6*x^8), -1/1146880*(105*(15*c^8*d^16 - 40*a*c^7* d^14*e^2 + 20*a^2*c^6*d^12*e^4 + 8*a^3*c^5*d^10*e^6 + 10*a^4*c^4*d^8*e^8 + 40*a^5*c^3*d^6*e^10 - 140*a^6*c^2*d^4*e^12 + 120*a^7*c*d^2*e^14 - 33*a...
Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^9 (d+e x)} \, dx=\text {Timed out} \]
\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^9 (d+e x)} \, dx=\int { \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {5}{2}}}{{\left (e x + d\right )} x^{9}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 5681 vs. \(2 (586) = 1172\).
Time = 1.30 (sec) , antiderivative size = 5681, normalized size of antiderivative = 9.05 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^9 (d+e x)} \, dx=\text {Too large to display} \]
-3/16384*(15*c^8*d^16 - 40*a*c^7*d^14*e^2 + 20*a^2*c^6*d^12*e^4 + 8*a^3*c^ 5*d^10*e^6 + 10*a^4*c^4*d^8*e^8 + 40*a^5*c^3*d^6*e^10 - 140*a^6*c^2*d^4*e^ 12 + 120*a^7*c*d^2*e^14 - 33*a^8*e^16)*arctan(-(sqrt(c*d*e)*x - sqrt(c*d*e *x^2 + c*d^2*x + a*e^2*x + a*d*e))/sqrt(-a*d*e))/(sqrt(-a*d*e)*a^5*d^6*e^5 ) + 1/573440*(1575*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a *d*e))*a^7*c^8*d^23*e^7 - 4200*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))*a^8*c^7*d^21*e^9 + 2100*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))*a^9*c^6*d^19*e^11 + 1147720*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))*a^10*c^5*d^17*e^13 + 344169 0*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))*a^11*c^4*d ^15*e^15 + 3444840*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a *d*e))*a^12*c^3*d^13*e^17 + 1132180*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^ 2*x + a*e^2*x + a*d*e))*a^13*c^2*d^11*e^19 + 12600*(sqrt(c*d*e)*x - sqrt(c *d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))*a^14*c*d^9*e^21 - 3465*(sqrt(c*d*e) *x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))*a^15*d^7*e^23 + 163840*s qrt(c*d*e)*a^11*c^4*d^16*e^14 + 327680*sqrt(c*d*e)*a^12*c^3*d^14*e^16 + 22 9376*sqrt(c*d*e)*a^13*c^2*d^12*e^18 - 12075*(sqrt(c*d*e)*x - sqrt(c*d*e*x^ 2 + c*d^2*x + a*e^2*x + a*d*e))^3*a^6*c^8*d^22*e^6 + 32200*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^3*a^7*c^7*d^20*e^8 + 571830 0*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^3*a^8*c...
Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^9 (d+e x)} \, dx=\int \frac {{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}}{x^9\,\left (d+e\,x\right )} \,d x \]