Integrand size = 40, antiderivative size = 664 \[ \int \frac {1}{x^4 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {2 e (a e+c d x)}{3 d \left (c d^2-a e^2\right ) x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {2 \left (3 c^3 d^6+a c^2 d^4 e^2+13 a^2 c d^2 e^4-9 a^3 e^6+c d e \left (3 c^2 d^4+14 a c d^2 e^2-9 a^2 e^4\right ) x\right )}{3 a d^2 e \left (c d^2-a e^2\right )^3 x^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {\left (7 c^3 d^6-3 a c^2 d^4 e^2+33 a^2 c d^2 e^4-21 a^3 e^6\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 a^2 d^3 e^2 \left (c d^2-a e^2\right )^3 x^3}+\frac {\left (35 c^4 d^8-16 a c^3 d^6 e^2-18 a^2 c^2 d^4 e^4+168 a^3 c d^2 e^6-105 a^4 e^8\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 a^3 d^4 e^3 \left (c d^2-a e^2\right )^3 x^2}-\frac {\left (105 c^5 d^{10}-55 a c^4 d^8 e^2-54 a^2 c^3 d^6 e^4-78 a^3 c^2 d^4 e^6+525 a^4 c d^2 e^8-315 a^5 e^{10}\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 a^4 d^5 e^4 \left (c d^2-a e^2\right )^3 x}+\frac {5 \left (7 c^3 d^6+15 a c^2 d^4 e^2+21 a^2 c d^2 e^4+21 a^3 e^6\right ) \text {arctanh}\left (\frac {2 a d e+\left (c d^2+a e^2\right ) x}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{16 a^{9/2} d^{11/2} e^{9/2}} \]
-2/3*e*(c*d*x+a*e)/d/(-a*e^2+c*d^2)/x^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^ (3/2)+5/16*(21*a^3*e^6+21*a^2*c*d^2*e^4+15*a*c^2*d^4*e^2+7*c^3*d^6)*arctan h(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^(9/2)/d^(11/2)/e^(9/2)+2/3*(3*c^3*d^6+a*c^2*d^4*e ^2+13*a^2*c*d^2*e^4-9*a^3*e^6+c*d*e*(-9*a^2*e^4+14*a*c*d^2*e^2+3*c^2*d^4)* x)/a/d^2/e/(-a*e^2+c*d^2)^3/x^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-1/ 3*(-21*a^3*e^6+33*a^2*c*d^2*e^4-3*a*c^2*d^4*e^2+7*c^3*d^6)*(a*d*e+(a*e^2+c *d^2)*x+c*d*e*x^2)^(1/2)/a^2/d^3/e^2/(-a*e^2+c*d^2)^3/x^3+1/12*(-105*a^4*e ^8+168*a^3*c*d^2*e^6-18*a^2*c^2*d^4*e^4-16*a*c^3*d^6*e^2+35*c^4*d^8)*(a*d* e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/a^3/d^4/e^3/(-a*e^2+c*d^2)^3/x^2-1/24*( -315*a^5*e^10+525*a^4*c*d^2*e^8-78*a^3*c^2*d^4*e^6-54*a^2*c^3*d^6*e^4-55*a *c^4*d^8*e^2+105*c^5*d^10)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/a^4/d^5 /e^4/(-a*e^2+c*d^2)^3/x
Time = 0.98 (sec) , antiderivative size = 493, normalized size of antiderivative = 0.74 \[ \int \frac {1}{x^4 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {\frac {\sqrt {a} \sqrt {d} \sqrt {e} \left (-105 c^6 d^{11} x^3 (d+e x)^2-5 a c^5 d^9 e x^2 (7 d-11 e x) (d+e x)^2+a^2 c^4 d^7 e^2 x (d+e x)^2 \left (14 d^2+23 d e x+54 e^2 x^2\right )-2 a^3 c^3 d^5 e^3 (d+e x)^2 \left (4 d^3+4 d^2 e x-9 d e^2 x^2-39 e^3 x^3\right )+a^6 e^9 \left (8 d^4-18 d^3 e x+63 d^2 e^2 x^2+420 d e^3 x^3+315 e^4 x^4\right )+a^4 c^2 d^3 e^5 \left (24 d^5-12 d^4 e x+62 d^3 e^2 x^2+3 d^2 e^3 x^3-636 d e^4 x^4-525 e^5 x^5\right )-a^5 c d e^7 \left (24 d^5-40 d^4 e x+135 d^3 e^2 x^2+651 d^2 e^3 x^3+105 d e^4 x^4-315 e^5 x^5\right )\right )}{\left (c d^2-a e^2\right )^3 x^3 (d+e x)}+15 \left (7 c^3 d^6+15 a c^2 d^4 e^2+21 a^2 c d^2 e^4+21 a^3 e^6\right ) \sqrt {a e+c d x} \sqrt {d+e x} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e} \sqrt {d+e x}}{\sqrt {d} \sqrt {a e+c d x}}\right )}{24 a^{9/2} d^{11/2} e^{9/2} \sqrt {(a e+c d x) (d+e x)}} \]
((Sqrt[a]*Sqrt[d]*Sqrt[e]*(-105*c^6*d^11*x^3*(d + e*x)^2 - 5*a*c^5*d^9*e*x ^2*(7*d - 11*e*x)*(d + e*x)^2 + a^2*c^4*d^7*e^2*x*(d + e*x)^2*(14*d^2 + 23 *d*e*x + 54*e^2*x^2) - 2*a^3*c^3*d^5*e^3*(d + e*x)^2*(4*d^3 + 4*d^2*e*x - 9*d*e^2*x^2 - 39*e^3*x^3) + a^6*e^9*(8*d^4 - 18*d^3*e*x + 63*d^2*e^2*x^2 + 420*d*e^3*x^3 + 315*e^4*x^4) + a^4*c^2*d^3*e^5*(24*d^5 - 12*d^4*e*x + 62* d^3*e^2*x^2 + 3*d^2*e^3*x^3 - 636*d*e^4*x^4 - 525*e^5*x^5) - a^5*c*d*e^7*( 24*d^5 - 40*d^4*e*x + 135*d^3*e^2*x^2 + 651*d^2*e^3*x^3 + 105*d*e^4*x^4 - 315*e^5*x^5)))/((c*d^2 - a*e^2)^3*x^3*(d + e*x)) + 15*(7*c^3*d^6 + 15*a*c^ 2*d^4*e^2 + 21*a^2*c*d^2*e^4 + 21*a^3*e^6)*Sqrt[a*e + c*d*x]*Sqrt[d + e*x] *ArcTanh[(Sqrt[a]*Sqrt[e]*Sqrt[d + e*x])/(Sqrt[d]*Sqrt[a*e + c*d*x])])/(24 *a^(9/2)*d^(11/2)*e^(9/2)*Sqrt[(a*e + c*d*x)*(d + e*x)])
Time = 1.21 (sec) , antiderivative size = 708, normalized size of antiderivative = 1.07, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.275, Rules used = {1246, 27, 1235, 27, 1237, 27, 1237, 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.
| \(\displaystyle \int \frac {1}{x^4 (d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1246 |
| \(\displaystyle \frac {2 \int \frac {e \left (3 \left (c d^2-3 a e^2\right )-10 c d e x\right )}{2 x^4 \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}dx}{3 d e \left (c d^2-a e^2\right )}-\frac {2 e (a e+c d x)}{3 d x^3 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {3 \left (c d^2-3 a e^2\right )-10 c d e x}{x^4 \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}dx}{3 d \left (c d^2-a e^2\right )}-\frac {2 e (a e+c d x)}{3 d x^3 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle \frac {\frac {2 \left (-9 a^3 e^6+c d e x \left (-9 a^2 e^4+14 a c d^2 e^2+3 c^2 d^4\right )+13 a^2 c d^2 e^4+a c^2 d^4 e^2+3 c^3 d^6\right )}{a d e x^3 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {2 \int -\frac {3 \left (7 c^3 d^6-3 a c^2 e^2 d^4+33 a^2 c e^4 d^2+2 c e \left (3 c^2 d^4+14 a c e^2 d^2-9 a^2 e^4\right ) x d-21 a^3 e^6\right )}{2 x^4 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{a d e \left (c d^2-a e^2\right )^2}}{3 d \left (c d^2-a e^2\right )}-\frac {2 e (a e+c d x)}{3 d x^3 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {3 \int \frac {7 c^3 d^6-3 a c^2 e^2 d^4+33 a^2 c e^4 d^2+2 c e \left (3 c^2 d^4+14 a c e^2 d^2-9 a^2 e^4\right ) x d-21 a^3 e^6}{x^4 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{a d e \left (c d^2-a e^2\right )^2}+\frac {2 \left (-9 a^3 e^6+c d e x \left (-9 a^2 e^4+14 a c d^2 e^2+3 c^2 d^4\right )+13 a^2 c d^2 e^4+a c^2 d^4 e^2+3 c^3 d^6\right )}{a d e x^3 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}}{3 d \left (c d^2-a e^2\right )}-\frac {2 e (a e+c d x)}{3 d x^3 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle \frac {\frac {3 \left (-\frac {\int \frac {35 c^4 d^8-16 a c^3 e^2 d^6-18 a^2 c^2 e^4 d^4+168 a^3 c e^6 d^2+4 c e \left (7 c^3 d^6-3 a c^2 e^2 d^4+33 a^2 c e^4 d^2-21 a^3 e^6\right ) x d-105 a^4 e^8}{2 x^3 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{3 a d e}-\frac {\left (-21 a^3 e^6+33 a^2 c d^2 e^4-3 a c^2 d^4 e^2+7 c^3 d^6\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 a d e x^3}\right )}{a d e \left (c d^2-a e^2\right )^2}+\frac {2 \left (-9 a^3 e^6+c d e x \left (-9 a^2 e^4+14 a c d^2 e^2+3 c^2 d^4\right )+13 a^2 c d^2 e^4+a c^2 d^4 e^2+3 c^3 d^6\right )}{a d e x^3 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}}{3 d \left (c d^2-a e^2\right )}-\frac {2 e (a e+c d x)}{3 d x^3 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {3 \left (-\frac {\int \frac {35 c^4 d^8-16 a c^3 e^2 d^6-18 a^2 c^2 e^4 d^4+168 a^3 c e^6 d^2+4 c e \left (7 c^3 d^6-3 a c^2 e^2 d^4+33 a^2 c e^4 d^2-21 a^3 e^6\right ) x d-105 a^4 e^8}{x^3 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{6 a d e}-\frac {\left (-21 a^3 e^6+33 a^2 c d^2 e^4-3 a c^2 d^4 e^2+7 c^3 d^6\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 a d e x^3}\right )}{a d e \left (c d^2-a e^2\right )^2}+\frac {2 \left (-9 a^3 e^6+c d e x \left (-9 a^2 e^4+14 a c d^2 e^2+3 c^2 d^4\right )+13 a^2 c d^2 e^4+a c^2 d^4 e^2+3 c^3 d^6\right )}{a d e x^3 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}}{3 d \left (c d^2-a e^2\right )}-\frac {2 e (a e+c d x)}{3 d x^3 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle \frac {\frac {3 \left (-\frac {-\frac {\int \frac {105 c^5 d^{10}-55 a c^4 e^2 d^8-54 a^2 c^3 e^4 d^6-78 a^3 c^2 e^6 d^4+525 a^4 c e^8 d^2+2 c e \left (35 c^4 d^8-16 a c^3 e^2 d^6-18 a^2 c^2 e^4 d^4+168 a^3 c e^6 d^2-105 a^4 e^8\right ) x d-315 a^5 e^{10}}{2 x^2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 a d e}-\frac {\left (-105 a^4 e^8+168 a^3 c d^2 e^6-18 a^2 c^2 d^4 e^4-16 a c^3 d^6 e^2+35 c^4 d^8\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 a d e x^2}}{6 a d e}-\frac {\left (-21 a^3 e^6+33 a^2 c d^2 e^4-3 a c^2 d^4 e^2+7 c^3 d^6\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 a d e x^3}\right )}{a d e \left (c d^2-a e^2\right )^2}+\frac {2 \left (-9 a^3 e^6+c d e x \left (-9 a^2 e^4+14 a c d^2 e^2+3 c^2 d^4\right )+13 a^2 c d^2 e^4+a c^2 d^4 e^2+3 c^3 d^6\right )}{a d e x^3 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}}{3 d \left (c d^2-a e^2\right )}-\frac {2 e (a e+c d x)}{3 d x^3 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {3 \left (-\frac {-\frac {\int \frac {105 c^5 d^{10}-55 a c^4 e^2 d^8-54 a^2 c^3 e^4 d^6-78 a^3 c^2 e^6 d^4+525 a^4 c e^8 d^2+2 c e \left (35 c^4 d^8-16 a c^3 e^2 d^6-18 a^2 c^2 e^4 d^4+168 a^3 c e^6 d^2-105 a^4 e^8\right ) x d-315 a^5 e^{10}}{x^2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{4 a d e}-\frac {\left (-105 a^4 e^8+168 a^3 c d^2 e^6-18 a^2 c^2 d^4 e^4-16 a c^3 d^6 e^2+35 c^4 d^8\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 a d e x^2}}{6 a d e}-\frac {\left (-21 a^3 e^6+33 a^2 c d^2 e^4-3 a c^2 d^4 e^2+7 c^3 d^6\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 a d e x^3}\right )}{a d e \left (c d^2-a e^2\right )^2}+\frac {2 \left (-9 a^3 e^6+c d e x \left (-9 a^2 e^4+14 a c d^2 e^2+3 c^2 d^4\right )+13 a^2 c d^2 e^4+a c^2 d^4 e^2+3 c^3 d^6\right )}{a d e x^3 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}}{3 d \left (c d^2-a e^2\right )}-\frac {2 e (a e+c d x)}{3 d x^3 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1228 |
| \(\displaystyle \frac {\frac {3 \left (-\frac {-\frac {-\frac {15 \left (21 a^3 e^6+21 a^2 c d^2 e^4+15 a c^2 d^4 e^2+7 c^3 d^6\right ) \left (c d^2-a e^2\right )^3 \int \frac {1}{x \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 a d e}-\frac {\left (-315 a^5 e^{10}+525 a^4 c d^2 e^8-78 a^3 c^2 d^4 e^6-54 a^2 c^3 d^6 e^4-55 a c^4 d^8 e^2+105 c^5 d^{10}\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{a d e x}}{4 a d e}-\frac {\left (-105 a^4 e^8+168 a^3 c d^2 e^6-18 a^2 c^2 d^4 e^4-16 a c^3 d^6 e^2+35 c^4 d^8\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 a d e x^2}}{6 a d e}-\frac {\left (-21 a^3 e^6+33 a^2 c d^2 e^4-3 a c^2 d^4 e^2+7 c^3 d^6\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 a d e x^3}\right )}{a d e \left (c d^2-a e^2\right )^2}+\frac {2 \left (-9 a^3 e^6+c d e x \left (-9 a^2 e^4+14 a c d^2 e^2+3 c^2 d^4\right )+13 a^2 c d^2 e^4+a c^2 d^4 e^2+3 c^3 d^6\right )}{a d e x^3 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}}{3 d \left (c d^2-a e^2\right )}-\frac {2 e (a e+c d x)}{3 d x^3 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {\frac {3 \left (-\frac {-\frac {\frac {15 \left (c d^2-a e^2\right )^3 \left (21 a^3 e^6+21 a^2 c d^2 e^4+15 a c^2 d^4 e^2+7 c^3 d^6\right ) \int \frac {1}{4 a d e-\frac {\left (2 a d e+\left (c d^2+a e^2\right ) x\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {2 a d e+\left (c d^2+a e^2\right ) x}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}}{a d e}-\frac {\left (-315 a^5 e^{10}+525 a^4 c d^2 e^8-78 a^3 c^2 d^4 e^6-54 a^2 c^3 d^6 e^4-55 a c^4 d^8 e^2+105 c^5 d^{10}\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{a d e x}}{4 a d e}-\frac {\left (-105 a^4 e^8+168 a^3 c d^2 e^6-18 a^2 c^2 d^4 e^4-16 a c^3 d^6 e^2+35 c^4 d^8\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 a d e x^2}}{6 a d e}-\frac {\left (-21 a^3 e^6+33 a^2 c d^2 e^4-3 a c^2 d^4 e^2+7 c^3 d^6\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 a d e x^3}\right )}{a d e \left (c d^2-a e^2\right )^2}+\frac {2 \left (-9 a^3 e^6+c d e x \left (-9 a^2 e^4+14 a c d^2 e^2+3 c^2 d^4\right )+13 a^2 c d^2 e^4+a c^2 d^4 e^2+3 c^3 d^6\right )}{a d e x^3 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}}{3 d \left (c d^2-a e^2\right )}-\frac {2 e (a e+c d x)}{3 d x^3 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {2 \left (-9 a^3 e^6+c d e x \left (-9 a^2 e^4+14 a c d^2 e^2+3 c^2 d^4\right )+13 a^2 c d^2 e^4+a c^2 d^4 e^2+3 c^3 d^6\right )}{a d e x^3 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {3 \left (-\frac {\left (-21 a^3 e^6+33 a^2 c d^2 e^4-3 a c^2 d^4 e^2+7 c^3 d^6\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 a d e x^3}-\frac {-\frac {\left (-105 a^4 e^8+168 a^3 c d^2 e^6-18 a^2 c^2 d^4 e^4-16 a c^3 d^6 e^2+35 c^4 d^8\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 a d e x^2}-\frac {\frac {15 \left (c d^2-a e^2\right )^3 \left (21 a^3 e^6+21 a^2 c d^2 e^4+15 a c^2 d^4 e^2+7 c^3 d^6\right ) \text {arctanh}\left (\frac {x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 a^{3/2} d^{3/2} e^{3/2}}-\frac {\left (-315 a^5 e^{10}+525 a^4 c d^2 e^8-78 a^3 c^2 d^4 e^6-54 a^2 c^3 d^6 e^4-55 a c^4 d^8 e^2+105 c^5 d^{10}\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{a d e x}}{4 a d e}}{6 a d e}\right )}{a d e \left (c d^2-a e^2\right )^2}}{3 d \left (c d^2-a e^2\right )}-\frac {2 e (a e+c d x)}{3 d x^3 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\) |
(-2*e*(a*e + c*d*x))/(3*d*(c*d^2 - a*e^2)*x^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) + ((2*(3*c^3*d^6 + a*c^2*d^4*e^2 + 13*a^2*c*d^2*e^4 - 9 *a^3*e^6 + c*d*e*(3*c^2*d^4 + 14*a*c*d^2*e^2 - 9*a^2*e^4)*x))/(a*d*e*(c*d^ 2 - a*e^2)^2*x^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (3*(-1/3*( (7*c^3*d^6 - 3*a*c^2*d^4*e^2 + 33*a^2*c*d^2*e^4 - 21*a^3*e^6)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(a*d*e*x^3) - (-1/2*((35*c^4*d^8 - 16*a*c ^3*d^6*e^2 - 18*a^2*c^2*d^4*e^4 + 168*a^3*c*d^2*e^6 - 105*a^4*e^8)*Sqrt[a* d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(a*d*e*x^2) - (-(((105*c^5*d^10 - 55 *a*c^4*d^8*e^2 - 54*a^2*c^3*d^6*e^4 - 78*a^3*c^2*d^4*e^6 + 525*a^4*c*d^2*e ^8 - 315*a^5*e^10)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(a*d*e*x)) + (15*(c*d^2 - a*e^2)^3*(7*c^3*d^6 + 15*a*c^2*d^4*e^2 + 21*a^2*c*d^2*e^4 + 21*a^3*e^6)*ArcTanh[(2*a*d*e + (c*d^2 + a*e^2)*x)/(2*Sqrt[a]*Sqrt[d]*Sqr t[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])])/(2*a^(3/2)*d^(3/2)*e^( 3/2)))/(4*a*d*e))/(6*a*d*e)))/(a*d*e*(c*d^2 - a*e^2)^2))/(3*d*(c*d^2 - a*e ^2))
3.5.86.3.1 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[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[((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[(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2 *a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^m *(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d* m + b*e*m) - b*d*(3*c*d - b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p] )
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])
Int[(((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_))/(( d_) + (e_.)*(x_)), x_Symbol] :> Simp[(f + g*x)^(n + 1)*(a + b*x + c*x^2)^p* ((c*d - b*e - c*e*x)/(p*(2*c*d - b*e)*(e*f - d*g))), x] + Simp[1/(p*(2*c*d - b*e)*(e*f - d*g)) Int[(f + g*x)^n*(a + b*x + c*x^2)^p*(b*e*g*(n + p + 1 ) + c*e*f*(2*p + 1) - c*d*g*(n + 2*p + 1) + c*e*g*(n + 2*p + 2)*x), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && ILtQ [n, 0] && ILtQ[n + 2*p, 0] && !IGtQ[n, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(2408\) vs. \(2(626)=1252\).
Time = 1.14 (sec) , antiderivative size = 2409, normalized size of antiderivative = 3.63
1/d*(-1/3/a/d/e/x^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-7/6*(a*e^2+c*d ^2)/a/d/e*(-1/2/a/d/e/x^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-5/4*(a*e ^2+c*d^2)/a/d/e*(-1/a/d/e/x/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-3/2*(a *e^2+c*d^2)/a/d/e*(1/a/d/e/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-(a*e^2+ c*d^2)/a/d/e*(2*c*d*e*x+a*e^2+c*d^2)/(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/(a*d* e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-1/a/d/e/(a*d*e)^(1/2)*ln((2*a*d*e+(a*e^ 2+c*d^2)*x+2*(a*d*e)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/x))-4* c/a*(2*c*d*e*x+a*e^2+c*d^2)/(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/(a*d*e+(a*e^2+ c*d^2)*x+c*d*e*x^2)^(1/2))-3/2*c/a*(1/a/d/e/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x ^2)^(1/2)-(a*e^2+c*d^2)/a/d/e*(2*c*d*e*x+a*e^2+c*d^2)/(4*a*c*d^2*e^2-(a*e^ 2+c*d^2)^2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-1/a/d/e/(a*d*e)^(1/2)* ln((2*a*d*e+(a*e^2+c*d^2)*x+2*(a*d*e)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x ^2)^(1/2))/x)))-4/3*c/a*(-1/a/d/e/x/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2 )-3/2*(a*e^2+c*d^2)/a/d/e*(1/a/d/e/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2) -(a*e^2+c*d^2)/a/d/e*(2*c*d*e*x+a*e^2+c*d^2)/(4*a*c*d^2*e^2-(a*e^2+c*d^2)^ 2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-1/a/d/e/(a*d*e)^(1/2)*ln((2*a*d *e+(a*e^2+c*d^2)*x+2*(a*d*e)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2) )/x))-4*c/a*(2*c*d*e*x+a*e^2+c*d^2)/(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/(a*d*e +(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)))+e^2/d^3*(-1/a/d/e/x/(a*d*e+(a*e^2+c*d^ 2)*x+c*d*e*x^2)^(1/2)-3/2*(a*e^2+c*d^2)/a/d/e*(1/a/d/e/(a*d*e+(a*e^2+c*...
Leaf count of result is larger than twice the leaf count of optimal. 1253 vs. \(2 (626) = 1252\).
Time = 64.20 (sec) , antiderivative size = 2526, normalized size of antiderivative = 3.80 \[ \int \frac {1}{x^4 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\text {Too large to display} \]
[1/96*(15*((7*c^7*d^13*e^2 - 6*a*c^6*d^11*e^4 - 3*a^2*c^5*d^9*e^6 - 4*a^3* c^4*d^7*e^8 - 15*a^4*c^3*d^5*e^10 + 42*a^5*c^2*d^3*e^12 - 21*a^6*c*d*e^14) *x^6 + (14*c^7*d^14*e - 5*a*c^6*d^12*e^3 - 12*a^2*c^5*d^10*e^5 - 11*a^3*c^ 4*d^8*e^7 - 34*a^4*c^3*d^6*e^9 + 69*a^5*c^2*d^4*e^11 - 21*a^7*e^15)*x^5 + (7*c^7*d^15 + 8*a*c^6*d^13*e^2 - 15*a^2*c^5*d^11*e^4 - 10*a^3*c^4*d^9*e^6 - 23*a^4*c^3*d^7*e^8 + 12*a^5*c^2*d^5*e^10 + 63*a^6*c*d^3*e^12 - 42*a^7*d* e^14)*x^4 + (7*a*c^6*d^14*e - 6*a^2*c^5*d^12*e^3 - 3*a^3*c^4*d^10*e^5 - 4* a^4*c^3*d^8*e^7 - 15*a^5*c^2*d^6*e^9 + 42*a^6*c*d^4*e^11 - 21*a^7*d^2*e^13 )*x^3)*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*(8*a^4*c^3*d ^11*e^4 - 24*a^5*c^2*d^9*e^6 + 24*a^6*c*d^7*e^8 - 8*a^7*d^5*e^10 + (105*a* c^6*d^12*e^3 - 55*a^2*c^5*d^10*e^5 - 54*a^3*c^4*d^8*e^7 - 78*a^4*c^3*d^6*e ^9 + 525*a^5*c^2*d^4*e^11 - 315*a^6*c*d^2*e^13)*x^5 + (210*a*c^6*d^13*e^2 - 75*a^2*c^5*d^11*e^4 - 131*a^3*c^4*d^9*e^6 - 174*a^4*c^3*d^7*e^8 + 636*a^ 5*c^2*d^5*e^10 + 105*a^6*c*d^3*e^12 - 315*a^7*d*e^14)*x^4 + (105*a*c^6*d^1 4*e + 15*a^2*c^5*d^12*e^3 - 114*a^3*c^4*d^10*e^5 - 106*a^4*c^3*d^8*e^7 - 3 *a^5*c^2*d^6*e^9 + 651*a^6*c*d^4*e^11 - 420*a^7*d^2*e^13)*x^3 + (35*a^2*c^ 5*d^13*e^2 - 51*a^3*c^4*d^11*e^4 + 6*a^4*c^3*d^9*e^6 - 62*a^5*c^2*d^7*e^8 + 135*a^6*c*d^5*e^10 - 63*a^7*d^3*e^12)*x^2 - 2*(7*a^3*c^4*d^12*e^3 - 1...
\[ \int \frac {1}{x^4 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int \frac {1}{x^{4} \left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}} \left (d + e x\right )}\, dx \]
\[ \int \frac {1}{x^4 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{2}} {\left (e x + d\right )} x^{4}} \,d x } \]
\[ \int \frac {1}{x^4 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{2}} {\left (e x + d\right )} x^{4}} \,d x } \]
Timed out. \[ \int \frac {1}{x^4 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int \frac {1}{x^4\,\left (d+e\,x\right )\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}} \,d x \]