Integrand size = 40, antiderivative size = 512 \[ \int \frac {1}{x^3 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {c \left (15 c^2 d^4-7 a^2 e^4\right )}{4 a^3 d e^3 \left (c d^2-a e^2\right ) (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {1}{2 a d e x^2 (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\frac {7 a}{d^2}+\frac {5 c}{e^2}}{4 a^2 x (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\left (45 c^3 d^6-15 a c^2 d^4 e^2-33 a^2 c d^2 e^4+35 a^3 e^6\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 a^3 d^3 e^2 \left (c d^2-a e^2\right )^2 (d+e x)^2}+\frac {\left (45 c^4 d^8-30 a c^3 d^6 e^2-36 a^2 c^2 d^4 e^4+190 a^3 c d^2 e^6-105 a^4 e^8\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 a^3 d^4 e^2 \left (c d^2-a e^2\right )^3 (d+e x)}-\frac {5 \left (3 c^2 d^4+6 a c d^2 e^2+7 a^2 e^4\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e} (d+e x)}{\sqrt {d} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{4 a^{7/2} d^{9/2} e^{7/2}} \] Output:
1/4*c*(-7*a^2*e^4+15*c^2*d^4)/a^3/d/e^3/(-a*e^2+c*d^2)/(e*x+d)/(a*d*e+(a*e ^2+c*d^2)*x+c*d*e*x^2)^(1/2)-1/2/a/d/e/x^2/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+ c*d*e*x^2)^(1/2)+1/4*(7*a/d^2+5*c/e^2)/a^2/x/(e*x+d)/(a*d*e+(a*e^2+c*d^2)* x+c*d*e*x^2)^(1/2)+1/12*(35*a^3*e^6-33*a^2*c*d^2*e^4-15*a*c^2*d^4*e^2+45*c ^3*d^6)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/a^3/d^3/e^2/(-a*e^2+c*d^2) ^2/(e*x+d)^2+1/12*(-105*a^4*e^8+190*a^3*c*d^2*e^6-36*a^2*c^2*d^4*e^4-30*a* c^3*d^6*e^2+45*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^ 2/(-a*e^2+c*d^2)^3/(e*x+d)-5/4*(7*a^2*e^4+6*a*c*d^2*e^2+3*c^2*d^4)*arctanh (a^(1/2)*e^(1/2)*(e*x+d)/d^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/ a^(7/2)/d^(9/2)/e^(7/2)
Time = 1.24 (sec) , antiderivative size = 390, normalized size of antiderivative = 0.76 \[ \int \frac {1}{x^3 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {\frac {\sqrt {a} \sqrt {d} \sqrt {e} (a e+c d x) \left (-45 c^5 d^9 x^2 (d+e x)^2-15 a c^4 d^7 e x (d-2 e x) (d+e x)^2+6 a^2 c^3 d^5 e^2 (d+e x)^2 \left (d^2+2 d e x+6 e^2 x^2\right )+a^5 e^8 \left (-6 d^3+21 d^2 e x+140 d e^2 x^2+105 e^3 x^3\right )-2 a^3 c^2 d^3 e^4 \left (9 d^4-9 d^3 e x-6 d^2 e^2 x^2+111 d e^3 x^3+95 e^4 x^4\right )+a^4 c d e^6 \left (18 d^4-48 d^3 e x-237 d^2 e^2 x^2-50 d e^3 x^3+105 e^4 x^4\right )\right )}{\left (-c d^2+a e^2\right )^3 x^2}-15 \left (3 c^2 d^4+6 a c d^2 e^2+7 a^2 e^4\right ) (a e+c d x)^{3/2} (d+e x)^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e} \sqrt {d+e x}}{\sqrt {d} \sqrt {a e+c d x}}\right )}{12 a^{7/2} d^{9/2} e^{7/2} ((a e+c d x) (d+e x))^{3/2}} \] Input:
Integrate[1/(x^3*(d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)), x]
Output:
((Sqrt[a]*Sqrt[d]*Sqrt[e]*(a*e + c*d*x)*(-45*c^5*d^9*x^2*(d + e*x)^2 - 15* a*c^4*d^7*e*x*(d - 2*e*x)*(d + e*x)^2 + 6*a^2*c^3*d^5*e^2*(d + e*x)^2*(d^2 + 2*d*e*x + 6*e^2*x^2) + a^5*e^8*(-6*d^3 + 21*d^2*e*x + 140*d*e^2*x^2 + 1 05*e^3*x^3) - 2*a^3*c^2*d^3*e^4*(9*d^4 - 9*d^3*e*x - 6*d^2*e^2*x^2 + 111*d *e^3*x^3 + 95*e^4*x^4) + a^4*c*d*e^6*(18*d^4 - 48*d^3*e*x - 237*d^2*e^2*x^ 2 - 50*d*e^3*x^3 + 105*e^4*x^4)))/((-(c*d^2) + a*e^2)^3*x^2) - 15*(3*c^2*d ^4 + 6*a*c*d^2*e^2 + 7*a^2*e^4)*(a*e + c*d*x)^(3/2)*(d + e*x)^(3/2)*ArcTan h[(Sqrt[a]*Sqrt[e]*Sqrt[d + e*x])/(Sqrt[d]*Sqrt[a*e + c*d*x])])/(12*a^(7/2 )*d^(9/2)*e^(7/2)*((a*e + c*d*x)*(d + e*x))^(3/2))
Time = 1.68 (sec) , antiderivative size = 565, normalized size of antiderivative = 1.10, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.225, Rules used = {1246, 27, 1235, 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.
| \(\displaystyle \int \frac {1}{x^3 (d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1246 |
| \(\displaystyle \frac {2 \int \frac {e \left (3 c d^2-8 c e x d-7 a e^2\right )}{2 x^3 \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}dx}{3 d e \left (c d^2-a e^2\right )}-\frac {2 e (a e+c d x)}{3 d x^2 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {3 c d^2-8 c e x d-7 a e^2}{x^3 \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}dx}{3 d \left (c d^2-a e^2\right )}-\frac {2 e (a e+c d x)}{3 d x^2 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle \frac {\frac {2 \left (-7 a^3 e^6+c d e x \left (-7 a^2 e^4+12 a c d^2 e^2+3 c^2 d^4\right )+11 a^2 c d^2 e^4+a c^2 d^4 e^2+3 c^3 d^6\right )}{a d e x^2 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {2 \int -\frac {15 c^3 d^6-9 a c^2 e^2 d^4+61 a^2 c e^4 d^2+4 c e \left (3 c^2 d^4+12 a c e^2 d^2-7 a^2 e^4\right ) x d-35 a^3 e^6}{2 x^3 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{a d e \left (c d^2-a e^2\right )^2}}{3 d \left (c d^2-a e^2\right )}-\frac {2 e (a e+c d x)}{3 d x^2 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {15 c^3 d^6-9 a c^2 e^2 d^4+61 a^2 c e^4 d^2+4 c e \left (3 c^2 d^4+12 a c e^2 d^2-7 a^2 e^4\right ) x d-35 a^3 e^6}{x^3 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{a d e \left (c d^2-a e^2\right )^2}+\frac {2 \left (-7 a^3 e^6+c d e x \left (-7 a^2 e^4+12 a c d^2 e^2+3 c^2 d^4\right )+11 a^2 c d^2 e^4+a c^2 d^4 e^2+3 c^3 d^6\right )}{a d e x^2 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}}{3 d \left (c d^2-a e^2\right )}-\frac {2 e (a e+c d x)}{3 d x^2 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle \frac {\frac {-\frac {\int \frac {45 c^4 d^8-30 a c^3 e^2 d^6-36 a^2 c^2 e^4 d^4+190 a^3 c e^6 d^2+2 c e \left (15 c^3 d^6-9 a c^2 e^2 d^4+61 a^2 c e^4 d^2-35 a^3 e^6\right ) x d-105 a^4 e^8}{2 x^2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 a d e}-\frac {\left (-35 a^3 e^6+61 a^2 c d^2 e^4-9 a c^2 d^4 e^2+15 c^3 d^6\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 a d e x^2}}{a d e \left (c d^2-a e^2\right )^2}+\frac {2 \left (-7 a^3 e^6+c d e x \left (-7 a^2 e^4+12 a c d^2 e^2+3 c^2 d^4\right )+11 a^2 c d^2 e^4+a c^2 d^4 e^2+3 c^3 d^6\right )}{a d e x^2 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}}{3 d \left (c d^2-a e^2\right )}-\frac {2 e (a e+c d x)}{3 d x^2 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {-\frac {\int \frac {45 c^4 d^8-30 a c^3 e^2 d^6-36 a^2 c^2 e^4 d^4+190 a^3 c e^6 d^2+2 c e \left (15 c^3 d^6-9 a c^2 e^2 d^4+61 a^2 c e^4 d^2-35 a^3 e^6\right ) x d-105 a^4 e^8}{x^2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{4 a d e}-\frac {\left (-35 a^3 e^6+61 a^2 c d^2 e^4-9 a c^2 d^4 e^2+15 c^3 d^6\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 a d e x^2}}{a d e \left (c d^2-a e^2\right )^2}+\frac {2 \left (-7 a^3 e^6+c d e x \left (-7 a^2 e^4+12 a c d^2 e^2+3 c^2 d^4\right )+11 a^2 c d^2 e^4+a c^2 d^4 e^2+3 c^3 d^6\right )}{a d e x^2 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}}{3 d \left (c d^2-a e^2\right )}-\frac {2 e (a e+c d x)}{3 d x^2 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1228 |
| \(\displaystyle \frac {\frac {-\frac {-\frac {15 \left (7 a^2 e^4+6 a c d^2 e^2+3 c^2 d^4\right ) \left (c d^2-a e^2\right )^3 \int \frac {1}{x \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 a d e}-\frac {\left (-105 a^4 e^8+190 a^3 c d^2 e^6-36 a^2 c^2 d^4 e^4-30 a c^3 d^6 e^2+45 c^4 d^8\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{a d e x}}{4 a d e}-\frac {\left (-35 a^3 e^6+61 a^2 c d^2 e^4-9 a c^2 d^4 e^2+15 c^3 d^6\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 a d e x^2}}{a d e \left (c d^2-a e^2\right )^2}+\frac {2 \left (-7 a^3 e^6+c d e x \left (-7 a^2 e^4+12 a c d^2 e^2+3 c^2 d^4\right )+11 a^2 c d^2 e^4+a c^2 d^4 e^2+3 c^3 d^6\right )}{a d e x^2 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}}{3 d \left (c d^2-a e^2\right )}-\frac {2 e (a e+c d x)}{3 d x^2 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {\frac {-\frac {\frac {15 \left (c d^2-a e^2\right )^3 \left (7 a^2 e^4+6 a c d^2 e^2+3 c^2 d^4\right ) \int \frac {1}{4 a d e-\frac {\left (2 a d e+\left (c d^2+a e^2\right ) x\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {2 a d e+\left (c d^2+a e^2\right ) x}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}}{a d e}-\frac {\left (-105 a^4 e^8+190 a^3 c d^2 e^6-36 a^2 c^2 d^4 e^4-30 a c^3 d^6 e^2+45 c^4 d^8\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{a d e x}}{4 a d e}-\frac {\left (-35 a^3 e^6+61 a^2 c d^2 e^4-9 a c^2 d^4 e^2+15 c^3 d^6\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 a d e x^2}}{a d e \left (c d^2-a e^2\right )^2}+\frac {2 \left (-7 a^3 e^6+c d e x \left (-7 a^2 e^4+12 a c d^2 e^2+3 c^2 d^4\right )+11 a^2 c d^2 e^4+a c^2 d^4 e^2+3 c^3 d^6\right )}{a d e x^2 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}}{3 d \left (c d^2-a e^2\right )}-\frac {2 e (a e+c d x)}{3 d x^2 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {2 \left (-7 a^3 e^6+c d e x \left (-7 a^2 e^4+12 a c d^2 e^2+3 c^2 d^4\right )+11 a^2 c d^2 e^4+a c^2 d^4 e^2+3 c^3 d^6\right )}{a d e x^2 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {-\frac {\left (-35 a^3 e^6+61 a^2 c d^2 e^4-9 a c^2 d^4 e^2+15 c^3 d^6\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 a d e x^2}-\frac {\frac {15 \left (c d^2-a e^2\right )^3 \left (7 a^2 e^4+6 a c d^2 e^2+3 c^2 d^4\right ) \text {arctanh}\left (\frac {x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 a^{3/2} d^{3/2} e^{3/2}}-\frac {\left (-105 a^4 e^8+190 a^3 c d^2 e^6-36 a^2 c^2 d^4 e^4-30 a c^3 d^6 e^2+45 c^4 d^8\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{a d e x}}{4 a d e}}{a d e \left (c d^2-a e^2\right )^2}}{3 d \left (c d^2-a e^2\right )}-\frac {2 e (a e+c d x)}{3 d x^2 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\) |
Input:
Int[1/(x^3*(d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]
Output:
(-2*e*(a*e + c*d*x))/(3*d*(c*d^2 - a*e^2)*x^2*(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 + 11*a^2*c*d^2*e^4 - 7 *a^3*e^6 + c*d*e*(3*c^2*d^4 + 12*a*c*d^2*e^2 - 7*a^2*e^4)*x))/(a*d*e*(c*d^ 2 - a*e^2)^2*x^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (-1/2*((15 *c^3*d^6 - 9*a*c^2*d^4*e^2 + 61*a^2*c*d^2*e^4 - 35*a^3*e^6)*Sqrt[a*d*e + ( c*d^2 + a*e^2)*x + c*d*e*x^2])/(a*d*e*x^2) - (-(((45*c^4*d^8 - 30*a*c^3*d^ 6*e^2 - 36*a^2*c^2*d^4*e^4 + 190*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)) + (15*(c*d^2 - a*e^2)^3*(3*c^2 *d^4 + 6*a*c*d^2*e^2 + 7*a^2*e^4)*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)*d^(3/2)*e^(3/2)))/(4*a*d*e))/(a*d*e*(c*d^2 - a*e^2)^2))/(3*d*(c*d ^2 - a*e^2))
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. \(1353\) vs. \(2(476)=952\).
Time = 2.68 (sec) , antiderivative size = 1354, normalized size of antiderivative = 2.64
Input:
int(1/x^3/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2),x,method=_RETURN VERBOSE)
Output:
1/d*(-1/2/a/d/e/x^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(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*x^2*e)^(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*x^2*e)^(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*x^2*e)^(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*x^2*e)^(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*x^2*e)^(1/2))-3/2*c/a*(1/a/d/e/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(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*x^2*e)^(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*x^2*e)^(1 /2))/x)))+e^2/d^3*(1/a/d/e/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(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*x^2*e)^(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*x^2*e)^(1/2))/x))-e/ d^2*(-1/a/d/e/x/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(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*x^2*e)^(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*x^2*e)^(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*x^2*e)^(1/2))/x))-4*c/a*(2*c*...
Leaf count of result is larger than twice the leaf count of optimal. 1071 vs. \(2 (476) = 952\).
Time = 23.67 (sec) , antiderivative size = 2162, normalized size of antiderivative = 4.22 \[ \int \frac {1}{x^3 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:
integrate(1/x^3/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorit hm="fricas")
Output:
[1/48*(15*((3*c^6*d^11*e^2 - 3*a*c^5*d^9*e^4 - 2*a^2*c^4*d^7*e^6 - 6*a^3*c ^3*d^5*e^8 + 15*a^4*c^2*d^3*e^10 - 7*a^5*c*d*e^12)*x^5 + (6*c^6*d^12*e - 3 *a*c^5*d^10*e^3 - 7*a^2*c^4*d^8*e^5 - 14*a^3*c^3*d^6*e^7 + 24*a^4*c^2*d^4* e^9 + a^5*c*d^2*e^11 - 7*a^6*e^13)*x^4 + (3*c^6*d^13 + 3*a*c^5*d^11*e^2 - 8*a^2*c^4*d^9*e^4 - 10*a^3*c^3*d^7*e^6 + 3*a^4*c^2*d^5*e^8 + 23*a^5*c*d^3* e^10 - 14*a^6*d*e^12)*x^3 + (3*a*c^5*d^12*e - 3*a^2*c^4*d^10*e^3 - 2*a^3*c ^3*d^8*e^5 - 6*a^4*c^2*d^6*e^7 + 15*a^5*c*d^4*e^9 - 7*a^6*d^2*e^11)*x^2)*s qrt(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*(6*a^3*c^3*d^10*e^3 - 18*a^4*c^2*d^8*e^5 + 18*a^5*c*d^6*e^7 - 6*a^6*d^4*e^9 - (45*a*c^5*d^10*e ^3 - 30*a^2*c^4*d^8*e^5 - 36*a^3*c^3*d^6*e^7 + 190*a^4*c^2*d^4*e^9 - 105*a ^5*c*d^2*e^11)*x^4 - (90*a*c^5*d^11*e^2 - 45*a^2*c^4*d^9*e^4 - 84*a^3*c^3* d^7*e^6 + 222*a^4*c^2*d^5*e^8 + 50*a^5*c*d^3*e^10 - 105*a^6*d*e^12)*x^3 - (45*a*c^5*d^12*e - 66*a^3*c^3*d^8*e^5 - 12*a^4*c^2*d^6*e^7 + 237*a^5*c*d^4 *e^9 - 140*a^6*d^2*e^11)*x^2 - 3*(5*a^2*c^4*d^11*e^2 - 8*a^3*c^3*d^9*e^4 - 6*a^4*c^2*d^7*e^6 + 16*a^5*c*d^5*e^8 - 7*a^6*d^3*e^10)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/((a^4*c^4*d^12*e^6 - 3*a^5*c^3*d^10*e^8 + 3* a^6*c^2*d^8*e^10 - a^7*c*d^6*e^12)*x^5 + (2*a^4*c^4*d^13*e^5 - 5*a^5*c^3*d ^11*e^7 + 3*a^6*c^2*d^9*e^9 + a^7*c*d^7*e^11 - a^8*d^5*e^13)*x^4 + (a^4...
\[ \int \frac {1}{x^3 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int \frac {1}{x^{3} \left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}} \left (d + e x\right )}\, dx \] Input:
integrate(1/x**3/(e*x+d)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2),x)
Output:
Integral(1/(x**3*((d + e*x)*(a*e + c*d*x))**(3/2)*(d + e*x)), x)
\[ \int \frac {1}{x^3 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{2}} {\left (e x + d\right )} x^{3}} \,d x } \] Input:
integrate(1/x^3/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorit hm="maxima")
Output:
integrate(1/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)*(e*x + d)*x^3), x)
\[ \int \frac {1}{x^3 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{2}} {\left (e x + d\right )} x^{3}} \,d x } \] Input:
integrate(1/x^3/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorit hm="giac")
Output:
integrate(1/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)*(e*x + d)*x^3), x)
Timed out. \[ \int \frac {1}{x^3 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int \frac {1}{x^3\,\left (d+e\,x\right )\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}} \,d x \] Input:
int(1/(x^3*(d + e*x)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2)),x)
Output:
int(1/(x^3*(d + e*x)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2)), x)
Time = 3.42 (sec) , antiderivative size = 5923, normalized size of antiderivative = 11.57 \[ \int \frac {1}{x^3 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:
int(1/x^3/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x)
Output:
(315*sqrt(e)*sqrt(d)*sqrt(a)*sqrt(a*e + c*d*x)*log(sqrt(e)*sqrt(a*e + c*d* x) - sqrt(2*sqrt(c)*sqrt(a)*d*e + a*e**2 + c*d**2) + sqrt(d)*sqrt(c)*sqrt( d + e*x))*a**6*d**2*e**12*x**2 + 630*sqrt(e)*sqrt(d)*sqrt(a)*sqrt(a*e + c* d*x)*log(sqrt(e)*sqrt(a*e + c*d*x) - sqrt(2*sqrt(c)*sqrt(a)*d*e + a*e**2 + c*d**2) + sqrt(d)*sqrt(c)*sqrt(d + e*x))*a**6*d*e**13*x**3 + 315*sqrt(e)* sqrt(d)*sqrt(a)*sqrt(a*e + c*d*x)*log(sqrt(e)*sqrt(a*e + c*d*x) - sqrt(2*s qrt(c)*sqrt(a)*d*e + a*e**2 + c*d**2) + sqrt(d)*sqrt(c)*sqrt(d + e*x))*a** 6*e**14*x**4 - 570*sqrt(e)*sqrt(d)*sqrt(a)*sqrt(a*e + c*d*x)*log(sqrt(e)*s qrt(a*e + c*d*x) - sqrt(2*sqrt(c)*sqrt(a)*d*e + a*e**2 + c*d**2) + sqrt(d) *sqrt(c)*sqrt(d + e*x))*a**5*c*d**4*e**10*x**2 - 1140*sqrt(e)*sqrt(d)*sqrt (a)*sqrt(a*e + c*d*x)*log(sqrt(e)*sqrt(a*e + c*d*x) - sqrt(2*sqrt(c)*sqrt( a)*d*e + a*e**2 + c*d**2) + sqrt(d)*sqrt(c)*sqrt(d + e*x))*a**5*c*d**3*e** 11*x**3 - 570*sqrt(e)*sqrt(d)*sqrt(a)*sqrt(a*e + c*d*x)*log(sqrt(e)*sqrt(a *e + c*d*x) - sqrt(2*sqrt(c)*sqrt(a)*d*e + a*e**2 + c*d**2) + sqrt(d)*sqrt (c)*sqrt(d + e*x))*a**5*c*d**2*e**12*x**4 + 45*sqrt(e)*sqrt(d)*sqrt(a)*sqr t(a*e + c*d*x)*log(sqrt(e)*sqrt(a*e + c*d*x) - sqrt(2*sqrt(c)*sqrt(a)*d*e + a*e**2 + c*d**2) + sqrt(d)*sqrt(c)*sqrt(d + e*x))*a**4*c**2*d**6*e**8*x* *2 + 90*sqrt(e)*sqrt(d)*sqrt(a)*sqrt(a*e + c*d*x)*log(sqrt(e)*sqrt(a*e + c *d*x) - sqrt(2*sqrt(c)*sqrt(a)*d*e + a*e**2 + c*d**2) + sqrt(d)*sqrt(c)*sq rt(d + e*x))*a**4*c**2*d**5*e**9*x**3 + 45*sqrt(e)*sqrt(d)*sqrt(a)*sqrt...