Integrand size = 40, antiderivative size = 471 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^5 (d+e x)^4} \, dx=-\frac {\left (15 c^3 d^6-839 a c^2 d^4 e^2+1785 a^2 c d^2 e^4-945 a^3 e^6\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{192 a d^5 (d+e x)}-\frac {a e \left (11 c d^2-9 a e^2\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 d^2 x^3 (d+e x)}-\frac {\left (59 c d^2-63 a e^2\right ) \left (c d^2-a e^2\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{96 d^3 x^2 (d+e x)}-\frac {\left (c d^2-a e^2\right ) \left (15 c^2 d^4-322 a c d^2 e^2+315 a^2 e^4\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{192 a d^4 e x (d+e x)}-\frac {a e \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{4 d x^4 (d+e x)^2}+\frac {5 \left (c d^2-a e^2\right )^2 \left (c^2 d^4+14 a c d^2 e^2-63 a^2 e^4\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {a} \sqrt {e} (d+e x)}\right )}{64 a^{3/2} d^{11/2} e^{3/2}} \] Output:
-1/192*(-945*a^3*e^6+1785*a^2*c*d^2*e^4-839*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/d^5/(e*x+d)-1/24*a*e*(-9*a*e^2+11*c* d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/d^2/x^3/(e*x+d)-1/96*(-63*a*e ^2+59*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^ 2/(e*x+d)-1/192*(-a*e^2+c*d^2)*(315*a^2*e^4-322*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)^(1/2)/a/d^4/e/x/(e*x+d)-1/4*a*e*(a*d*e+(a* e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/d/x^4/(e*x+d)^2+5/64*(-a*e^2+c*d^2)^2*(-63*a ^2*e^4+14*a*c*d^2*e^2+c^2*d^4)*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^(3/2)/d^(11/2)/e^(3/2)
Time = 1.38 (sec) , antiderivative size = 315, normalized size of antiderivative = 0.67 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^5 (d+e x)^4} \, dx=\frac {((a e+c d x) (d+e x))^{5/2} \left (-\frac {\sqrt {a} \sqrt {d} \sqrt {e} \left (15 c^3 d^6 x^3 (d+e x)+a c^2 d^4 e x^2 \left (118 d^2-337 d e x-839 e^2 x^2\right )+a^2 c d^2 e^2 x \left (136 d^3-244 d^2 e x+637 d e^2 x^2+1785 e^3 x^3\right )-3 a^3 e^3 \left (-16 d^4+24 d^3 e x-42 d^2 e^2 x^2+105 d e^3 x^3+315 e^4 x^4\right )\right )}{x^4 (a e+c d x)^2 (d+e x)^3}+\frac {15 \left (c d^2-a e^2\right )^2 \left (c^2 d^4+14 a c d^2 e^2-63 a^2 e^4\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a e+c d x}}{\sqrt {a} \sqrt {e} \sqrt {d+e x}}\right )}{(a e+c d x)^{5/2} (d+e x)^{5/2}}\right )}{192 a^{3/2} d^{11/2} e^{3/2}} \] Input:
Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(x^5*(d + e*x)^4), x]
Output:
(((a*e + c*d*x)*(d + e*x))^(5/2)*(-((Sqrt[a]*Sqrt[d]*Sqrt[e]*(15*c^3*d^6*x ^3*(d + e*x) + a*c^2*d^4*e*x^2*(118*d^2 - 337*d*e*x - 839*e^2*x^2) + a^2*c *d^2*e^2*x*(136*d^3 - 244*d^2*e*x + 637*d*e^2*x^2 + 1785*e^3*x^3) - 3*a^3* e^3*(-16*d^4 + 24*d^3*e*x - 42*d^2*e^2*x^2 + 105*d*e^3*x^3 + 315*e^4*x^4)) )/(x^4*(a*e + c*d*x)^2*(d + e*x)^3)) + (15*(c*d^2 - a*e^2)^2*(c^2*d^4 + 14 *a*c*d^2*e^2 - 63*a^2*e^4)*ArcTanh[(Sqrt[d]*Sqrt[a*e + c*d*x])/(Sqrt[a]*Sq rt[e]*Sqrt[d + e*x])])/((a*e + c*d*x)^(5/2)*(d + e*x)^(5/2))))/(192*a^(3/2 )*d^(11/2)*e^(3/2))
Time = 3.00 (sec) , antiderivative size = 492, normalized size of antiderivative = 1.04, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.275, Rules used = {1214, 25, 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.
| \(\displaystyle \int \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{x^5 (d+e x)^4} \, dx\) |
| \(\Big \downarrow \) 1214 |
| \(\displaystyle \frac {2 e^2 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^5 (d+e x)}-\frac {\int -\frac {\frac {a^3 e^9}{d}+\frac {a^2 \left (3 c d^2-a e^2\right ) x e^8}{d^2}-\frac {\left (c d^2-a e^2\right )^3 x^4 e^7}{d^5}+\frac {a \left (3 c^2 d^4-3 a c e^2 d^2+a^2 e^4\right ) x^2 e^7}{d^3}+\frac {\left (c d^2-a e^2\right )^3 x^3 e^6}{d^4}}{x^5 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{e^6}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\frac {a^3 e^9}{d}+\frac {a^2 \left (3 c d^2-a e^2\right ) x e^8}{d^2}-\frac {\left (c d^2-a e^2\right )^3 x^4 e^7}{d^5}+\frac {a \left (3 c^2 d^4-3 a c e^2 d^2+a^2 e^4\right ) x^2 e^7}{d^3}+\frac {\left (c d^2-a e^2\right )^3 x^3 e^6}{d^4}}{x^5 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{e^6}+\frac {2 e^2 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^5 (d+e x)}\) |
| \(\Big \downarrow \) 2181 |
| \(\displaystyle \frac {-\frac {\int -\frac {\frac {a^3 \left (17 c d^2-15 a e^2\right ) e^9}{d}-\frac {8 a \left (c d^2-a e^2\right )^3 x^3 e^8}{d^4}+2 a^2 \left (\frac {4 a^2 e^4}{d^2}-15 a c e^2+12 c^2 d^2\right ) x e^8+\frac {8 a \left (c d^2-a e^2\right )^3 x^2 e^7}{d^3}}{2 x^4 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{4 a d e}-\frac {a^2 e^8 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 d^2 x^4}}{e^6}+\frac {2 e^2 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^5 (d+e x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {\frac {a^3 \left (17 c d^2-15 a e^2\right ) e^9}{d}-\frac {8 a \left (c d^2-a e^2\right )^3 x^3 e^8}{d^4}+2 a^2 \left (\frac {4 a^2 e^4}{d^2}-15 a c e^2+12 c^2 d^2\right ) x e^8+\frac {8 a \left (c d^2-a e^2\right )^3 x^2 e^7}{d^3}}{x^4 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{8 a d e}-\frac {a^2 e^8 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 d^2 x^4}}{e^6}+\frac {2 e^2 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^5 (d+e x)}\) |
| \(\Big \downarrow \) 2181 |
| \(\displaystyle \frac {\frac {-\frac {\int -\frac {-\frac {48 a^2 \left (c d^2-a e^2\right )^3 x^2 e^9}{d^3}+\frac {a^3 \left (59 c^2 d^4-190 a c e^2 d^2+123 a^2 e^4\right ) e^9}{d}+4 a^2 \left (-\frac {12 a^3 e^6}{d^2}+51 a^2 c e^4-53 a c^2 d^2 e^2+12 c^3 d^4\right ) x e^8}{2 x^3 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{3 a d e}-\frac {a^2 e^8 \left (17 c d^2-15 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 d^2 x^3}}{8 a d e}-\frac {a^2 e^8 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 d^2 x^4}}{e^6}+\frac {2 e^2 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^5 (d+e x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {-\frac {48 a^2 \left (c d^2-a e^2\right )^3 x^2 e^9}{d^3}+\frac {a^3 \left (59 c^2 d^4-190 a c e^2 d^2+123 a^2 e^4\right ) e^9}{d}+4 a^2 \left (-\frac {12 a^3 e^6}{d^2}+51 a^2 c e^4-53 a c^2 d^2 e^2+12 c^3 d^4\right ) x e^8}{x^3 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{6 a d e}-\frac {a^2 e^8 \left (17 c d^2-15 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 d^2 x^3}}{8 a d e}-\frac {a^2 e^8 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 d^2 x^4}}{e^6}+\frac {2 e^2 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^5 (d+e x)}\) |
| \(\Big \downarrow \) 2181 |
| \(\displaystyle \frac {\frac {\frac {-\frac {\int -\frac {a^3 e^9 \left (15 c^3 d^6-455 a c^2 e^2 d^4+1017 a^2 c e^4 d^2-2 e \left (-\frac {96 a^3 e^6}{d^2}+411 a^2 c e^4-478 a c^2 d^2 e^2+155 c^3 d^4\right ) x d-561 a^3 e^6\right )}{2 d x^2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 a d e}-\frac {a^2 e^8 \left (123 a^2 e^4-190 a c d^2 e^2+59 c^2 d^4\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 d^2 x^2}}{6 a d e}-\frac {a^2 e^8 \left (17 c d^2-15 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 d^2 x^3}}{8 a d e}-\frac {a^2 e^8 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 d^2 x^4}}{e^6}+\frac {2 e^2 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^5 (d+e x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\frac {a^2 e^8 \int \frac {15 c^3 d^6-455 a c^2 e^2 d^4+1017 a^2 c e^4 d^2-2 e \left (-\frac {96 a^3 e^6}{d^2}+411 a^2 c e^4-478 a c^2 d^2 e^2+155 c^3 d^4\right ) x d-561 a^3 e^6}{x^2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{4 d^2}-\frac {a^2 e^8 \left (123 a^2 e^4-190 a c d^2 e^2+59 c^2 d^4\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 d^2 x^2}}{6 a d e}-\frac {a^2 e^8 \left (17 c d^2-15 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 d^2 x^3}}{8 a d e}-\frac {a^2 e^8 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 d^2 x^4}}{e^6}+\frac {2 e^2 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^5 (d+e x)}\) |
| \(\Big \downarrow \) 1228 |
| \(\displaystyle \frac {\frac {\frac {\frac {a^2 e^8 \left (-\frac {15 \left (-63 a^2 e^4+14 a c d^2 e^2+c^2 d^4\right ) \left (c d^2-a e^2\right )^2 \int \frac {1}{x \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 a d e}-\frac {\left (-561 a^3 e^6+1017 a^2 c d^2 e^4-455 a c^2 d^4 e^2+15 c^3 d^6\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{a d e x}\right )}{4 d^2}-\frac {a^2 e^8 \left (123 a^2 e^4-190 a c d^2 e^2+59 c^2 d^4\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 d^2 x^2}}{6 a d e}-\frac {a^2 e^8 \left (17 c d^2-15 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 d^2 x^3}}{8 a d e}-\frac {a^2 e^8 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 d^2 x^4}}{e^6}+\frac {2 e^2 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^5 (d+e x)}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {\frac {\frac {\frac {a^2 e^8 \left (\frac {15 \left (c d^2-a e^2\right )^2 \left (-63 a^2 e^4+14 a c d^2 e^2+c^2 d^4\right ) \int \frac {1}{4 a d e-\frac {\left (2 a d e+\left (c d^2+a e^2\right ) x\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {2 a d e+\left (c d^2+a e^2\right ) x}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}}{a d e}-\frac {\left (-561 a^3 e^6+1017 a^2 c d^2 e^4-455 a c^2 d^4 e^2+15 c^3 d^6\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{a d e x}\right )}{4 d^2}-\frac {a^2 e^8 \left (123 a^2 e^4-190 a c d^2 e^2+59 c^2 d^4\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 d^2 x^2}}{6 a d e}-\frac {a^2 e^8 \left (17 c d^2-15 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 d^2 x^3}}{8 a d e}-\frac {a^2 e^8 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 d^2 x^4}}{e^6}+\frac {2 e^2 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^5 (d+e x)}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\frac {\frac {a^2 e^8 \left (\frac {15 \left (c d^2-a e^2\right )^2 \left (-63 a^2 e^4+14 a c d^2 e^2+c^2 d^4\right ) \text {arctanh}\left (\frac {x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 a^{3/2} d^{3/2} e^{3/2}}-\frac {\left (-561 a^3 e^6+1017 a^2 c d^2 e^4-455 a c^2 d^4 e^2+15 c^3 d^6\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{a d e x}\right )}{4 d^2}-\frac {a^2 e^8 \left (123 a^2 e^4-190 a c d^2 e^2+59 c^2 d^4\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 d^2 x^2}}{6 a d e}-\frac {a^2 e^8 \left (17 c d^2-15 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 d^2 x^3}}{8 a d e}-\frac {a^2 e^8 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 d^2 x^4}}{e^6}+\frac {2 e^2 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^5 (d+e x)}\) |
Input:
Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(x^5*(d + e*x)^4),x]
Output:
(2*e^2*(c*d^2 - a*e^2)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(d^5 *(d + e*x)) + (-1/4*(a^2*e^8*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/ (d^2*x^4) + (-1/3*(a^2*e^8*(17*c*d^2 - 15*a*e^2)*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*(59*c^2*d^4 - 190*a*c*d^2*e ^2 + 123*a^2*e^4)*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^3*d^6 - 455*a*c^2*d^4*e^2 + 1017*a^2*c*d^2*e^4 - 561*a ^3*e^6)*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)^2*(c^2*d^4 + 14*a*c*d^2*e^2 - 63*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*d^2))/(6*a*d*e))/(8*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. \(13160\) vs. \(2(435)=870\).
Time = 7.42 (sec) , antiderivative size = 13161, normalized size of antiderivative = 27.94
Input:
int((a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(5/2)/x^5/(e*x+d)^4,x,method=_RETURN VERBOSE)
Output:
result too large to display
Time = 14.74 (sec) , antiderivative size = 968, normalized size of antiderivative = 2.06 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^5 (d+e x)^4} \, dx =\text {Too large to display} \] Input:
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/x^5/(e*x+d)^4,x, algorit hm="fricas")
Output:
[-1/768*(15*((c^4*d^8*e + 12*a*c^3*d^6*e^3 - 90*a^2*c^2*d^4*e^5 + 140*a^3* c*d^2*e^7 - 63*a^4*e^9)*x^5 + (c^4*d^9 + 12*a*c^3*d^7*e^2 - 90*a^2*c^2*d^5 *e^4 + 140*a^3*c*d^3*e^6 - 63*a^4*d*e^8)*x^4)*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*(48*a^4*d^5*e^4 + (15*a*c^3*d^7*e^2 - 839*a^2*c ^2*d^5*e^4 + 1785*a^3*c*d^3*e^6 - 945*a^4*d*e^8)*x^4 + (15*a*c^3*d^8*e - 3 37*a^2*c^2*d^6*e^3 + 637*a^3*c*d^4*e^5 - 315*a^4*d^2*e^7)*x^3 + 2*(59*a^2* c^2*d^7*e^2 - 122*a^3*c*d^5*e^4 + 63*a^4*d^3*e^6)*x^2 + 8*(17*a^3*c*d^6*e^ 3 - 9*a^4*d^4*e^5)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(a^2*d^ 6*e^3*x^5 + a^2*d^7*e^2*x^4), -1/384*(15*((c^4*d^8*e + 12*a*c^3*d^6*e^3 - 90*a^2*c^2*d^4*e^5 + 140*a^3*c*d^2*e^7 - 63*a^4*e^9)*x^5 + (c^4*d^9 + 12*a *c^3*d^7*e^2 - 90*a^2*c^2*d^5*e^4 + 140*a^3*c*d^3*e^6 - 63*a^4*d*e^8)*x^4) *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*(48*a^4*d^5*e^4 + (15*a*c^3*d^7*e^2 - 839*a^2 *c^2*d^5*e^4 + 1785*a^3*c*d^3*e^6 - 945*a^4*d*e^8)*x^4 + (15*a*c^3*d^8*e - 337*a^2*c^2*d^6*e^3 + 637*a^3*c*d^4*e^5 - 315*a^4*d^2*e^7)*x^3 + 2*(59*a^ 2*c^2*d^7*e^2 - 122*a^3*c*d^5*e^4 + 63*a^4*d^3*e^6)*x^2 + 8*(17*a^3*c*d^6* e^3 - 9*a^4*d^4*e^5)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(a...
Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^5 (d+e x)^4} \, dx=\text {Timed out} \] Input:
integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2)/x**5/(e*x+d)**4,x)
Output:
Timed out
\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^5 (d+e x)^4} \, dx=\int { \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{4} x^{5}} \,d x } \] Input:
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/x^5/(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^5), x)
Exception generated. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^5 (d+e x)^4} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/x^5/(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,[0,5,13]%%%},[2,6]%%%}+%%%{%%%{-6,[1,7,11]%%%},[2,5] %%%}+%%%{
Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^5 (d+e x)^4} \, dx=\int \frac {{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}}{x^5\,{\left (d+e\,x\right )}^4} \,d x \] Input:
int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)/(x^5*(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^5*(d + e*x)^4), x)
Time = 14.61 (sec) , antiderivative size = 3131, normalized size of antiderivative = 6.65 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^5 (d+e x)^4} \, dx =\text {Too large to display} \] Input:
int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/x^5/(e*x+d)^4,x)
Output:
( - 864*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**5*d**5*e**6 + 1296*sqrt(d + e*x )*sqrt(a*e + c*d*x)*a**5*d**4*e**7*x - 2268*sqrt(d + e*x)*sqrt(a*e + c*d*x )*a**5*d**3*e**8*x**2 + 5670*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**5*d**2*e** 9*x**3 + 17010*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**5*d*e**10*x**4 - 672*sqr t(d + e*x)*sqrt(a*e + c*d*x)*a**4*c*d**7*e**4 - 1440*sqrt(d + e*x)*sqrt(a* e + c*d*x)*a**4*c*d**6*e**5*x + 2628*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**4* c*d**5*e**6*x**2 - 7056*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**4*c*d**4*e**7*x **3 - 18900*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**4*c*d**3*e**8*x**4 - 1904*s qrt(d + e*x)*sqrt(a*e + c*d*x)*a**3*c**2*d**8*e**3*x + 1292*sqrt(d + e*x)* sqrt(a*e + c*d*x)*a**3*c**2*d**7*e**4*x**2 - 2852*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**3*c**2*d**6*e**5*x**3 - 9888*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a* *3*c**2*d**5*e**6*x**4 - 1652*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**2*c**3*d* *9*e**2*x**2 + 4448*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**2*c**3*d**8*e**3*x* *3 + 11476*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**2*c**3*d**7*e**4*x**4 - 210* sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c**4*d**10*e*x**3 - 210*sqrt(d + e*x)*sq rt(a*e + c*d*x)*a*c**4*d**9*e**2*x**4 + 8505*sqrt(e)*sqrt(d)*sqrt(a)*log(s qrt(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*d*e**10*x**4 + 8505*sqrt(e)*sqrt(d)*s qrt(a)*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*e**11*x**5 - 12285*sqr...