Integrand size = 40, antiderivative size = 755 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^7 (d+e x)^4} \, dx=-\frac {\left (75 c^5 d^{10}+295 a c^4 d^8 e^2+1230 a^2 c^3 d^6 e^4-45234 a^3 c^2 d^4 e^6+88935 a^4 c d^2 e^8-45045 a^5 e^{10}\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7680 a^3 d^7 e^2 (d+e x)}-\frac {a e \left (15 c d^2-13 a e^2\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{60 d^2 x^5 (d+e x)}-\frac {\left (135 c d^2-143 a e^2\right ) \left (c d^2-a e^2\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{480 d^3 x^4 (d+e x)}-\frac {\left (c d^2-a e^2\right ) \left (5 c^2 d^4-418 a c d^2 e^2+429 a^2 e^4\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{960 a d^4 e x^3 (d+e x)}+\frac {\left (c d^2-a e^2\right ) \left (25 c^3 d^6+105 a c^2 d^4 e^2-3069 a^2 c d^2 e^4+3003 a^3 e^6\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3840 a^2 d^5 e^2 x^2 (d+e x)}-\frac {\left (c d^2-a e^2\right ) \left (75 c^4 d^8+320 a c^3 d^6 e^2+1350 a^2 c^2 d^4 e^4-16632 a^3 c d^2 e^6+15015 a^4 e^8\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7680 a^3 d^6 e^3 x (d+e x)}-\frac {a e \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{6 d x^6 (d+e x)^2}+\frac {\left (c d^2-a e^2\right )^2 \left (5 c^4 d^8+28 a c^3 d^6 e^2+126 a^2 c^2 d^4 e^4+924 a^3 c d^2 e^6-3003 a^4 e^8\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {a} \sqrt {e} (d+e x)}\right )}{512 a^{7/2} d^{15/2} e^{7/2}} \] Output:
-1/7680*(-45045*a^5*e^10+88935*a^4*c*d^2*e^8-45234*a^3*c^2*d^4*e^6+1230*a^ 2*c^3*d^6*e^4+295*a*c^4*d^8*e^2+75*c^5*d^10)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e* x^2)^(1/2)/a^3/d^7/e^2/(e*x+d)-1/60*a*e*(-13*a*e^2+15*c*d^2)*(a*d*e+(a*e^2 +c*d^2)*x+c*d*e*x^2)^(1/2)/d^2/x^5/(e*x+d)-1/480*(-143*a*e^2+135*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^4/(e*x+d)-1/960 *(-a*e^2+c*d^2)*(429*a^2*e^4-418*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^3/(e*x+d)+1/3840*(-a*e^2+c*d^2)*(3003*a^3* e^6-3069*a^2*c*d^2*e^4+105*a*c^2*d^4*e^2+25*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^2/(e*x+d)-1/7680*(-a*e^2+c*d^2)*(15015*a^ 4*e^8-16632*a^3*c*d^2*e^6+1350*a^2*c^2*d^4*e^4+320*a*c^3*d^6*e^2+75*c^4*d^ 8)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/a^3/d^6/e^3/x/(e*x+d)-1/6*a*e*( a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/d/x^6/(e*x+d)^2+1/512*(-a*e^2+c*d^2 )^2*(-3003*a^4*e^8+924*a^3*c*d^2*e^6+126*a^2*c^2*d^4*e^4+28*a*c^3*d^6*e^2+ 5*c^4*d^8)*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^(7/2)/d^(15/2)/e^(7/2)
Time = 2.84 (sec) , antiderivative size = 483, normalized size of antiderivative = 0.64 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^7 (d+e x)^4} \, dx=\frac {\sqrt {(a e+c d x) (d+e x)} \left (-\frac {\sqrt {a} \sqrt {d} \sqrt {e} \left (75 c^5 d^{10} x^5 (d+e x)+5 a c^4 d^8 e x^4 \left (-10 d^2+49 d e x+59 e^2 x^2\right )+10 a^2 c^3 d^6 e^2 x^3 \left (4 d^3-16 d^2 e x+103 d e^2 x^2+123 e^3 x^3\right )+6 a^3 c^2 d^4 e^3 x^2 \left (360 d^4-564 d^3 e x+1058 d^2 e^2 x^2-2997 d e^3 x^3-7539 e^4 x^4\right )+a^4 c d^2 e^4 x \left (3200 d^5-4448 d^4 e x+6776 d^3 e^2 x^2-12144 d^2 e^3 x^3+31647 d e^4 x^4+88935 e^5 x^5\right )+a^5 e^5 \left (1280 d^6-1664 d^5 e x+2288 d^4 e^2 x^2-3432 d^3 e^3 x^3+6006 d^2 e^4 x^4-15015 d e^5 x^5-45045 e^6 x^6\right )\right )}{x^6 (d+e x)}+\frac {15 \left (c d^2-a e^2\right )^2 \left (5 c^4 d^8+28 a c^3 d^6 e^2+126 a^2 c^2 d^4 e^4+924 a^3 c d^2 e^6-3003 a^4 e^8\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a e+c d x}}{\sqrt {a} \sqrt {e} \sqrt {d+e x}}\right )}{\sqrt {a e+c d x} \sqrt {d+e x}}\right )}{7680 a^{7/2} d^{15/2} e^{7/2}} \] Input:
Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(x^7*(d + e*x)^4), x]
Output:
(Sqrt[(a*e + c*d*x)*(d + e*x)]*(-((Sqrt[a]*Sqrt[d]*Sqrt[e]*(75*c^5*d^10*x^ 5*(d + e*x) + 5*a*c^4*d^8*e*x^4*(-10*d^2 + 49*d*e*x + 59*e^2*x^2) + 10*a^2 *c^3*d^6*e^2*x^3*(4*d^3 - 16*d^2*e*x + 103*d*e^2*x^2 + 123*e^3*x^3) + 6*a^ 3*c^2*d^4*e^3*x^2*(360*d^4 - 564*d^3*e*x + 1058*d^2*e^2*x^2 - 2997*d*e^3*x ^3 - 7539*e^4*x^4) + a^4*c*d^2*e^4*x*(3200*d^5 - 4448*d^4*e*x + 6776*d^3*e ^2*x^2 - 12144*d^2*e^3*x^3 + 31647*d*e^4*x^4 + 88935*e^5*x^5) + a^5*e^5*(1 280*d^6 - 1664*d^5*e*x + 2288*d^4*e^2*x^2 - 3432*d^3*e^3*x^3 + 6006*d^2*e^ 4*x^4 - 15015*d*e^5*x^5 - 45045*e^6*x^6)))/(x^6*(d + e*x))) + (15*(c*d^2 - a*e^2)^2*(5*c^4*d^8 + 28*a*c^3*d^6*e^2 + 126*a^2*c^2*d^4*e^4 + 924*a^3*c* d^2*e^6 - 3003*a^4*e^8)*ArcTanh[(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])))/(7680*a^(7/2)*d^(1 5/2)*e^(7/2))
Time = 5.63 (sec) , antiderivative size = 760, normalized size of antiderivative = 1.01, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {1214, 25, 2181, 27, 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.
| \(\displaystyle \int \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{x^7 (d+e x)^4} \, dx\) |
| \(\Big \downarrow \) 1214 |
| \(\displaystyle \frac {2 e^4 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^7 (d+e x)}-\frac {\int -\frac {-\frac {\left (c d^2-a e^2\right )^3 x^6 e^9}{d^7}+\frac {a^3 e^9}{d}+\frac {\left (c d^2-a e^2\right )^3 x^5 e^8}{d^6}+\frac {a^2 \left (3 c d^2-a e^2\right ) x e^8}{d^2}-\frac {\left (c d^2-a e^2\right )^3 x^4 e^7}{d^5}+\frac {a \left (3 c^2 d^4-3 a c e^2 d^2+a^2 e^4\right ) x^2 e^7}{d^3}+\frac {\left (c d^2-a e^2\right )^3 x^3 e^6}{d^4}}{x^7 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{e^6}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {-\frac {\left (c d^2-a e^2\right )^3 x^6 e^9}{d^7}+\frac {a^3 e^9}{d}+\frac {\left (c d^2-a e^2\right )^3 x^5 e^8}{d^6}+\frac {a^2 \left (3 c d^2-a e^2\right ) x e^8}{d^2}-\frac {\left (c d^2-a e^2\right )^3 x^4 e^7}{d^5}+\frac {a \left (3 c^2 d^4-3 a c e^2 d^2+a^2 e^4\right ) x^2 e^7}{d^3}+\frac {\left (c d^2-a e^2\right )^3 x^3 e^6}{d^4}}{x^7 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{e^6}+\frac {2 e^4 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^7 (d+e x)}\) |
| \(\Big \downarrow \) 2181 |
| \(\displaystyle \frac {-\frac {\int -\frac {-\frac {12 a \left (c d^2-a e^2\right )^3 x^5 e^{10}}{d^6}+\frac {12 a \left (c d^2-a e^2\right )^3 x^4 e^9}{d^5}+\frac {a^3 \left (25 c d^2-23 a e^2\right ) e^9}{d}-\frac {12 a \left (c d^2-a e^2\right )^3 x^3 e^8}{d^4}+2 a^2 \left (\frac {6 a^2 e^4}{d^2}-23 a c e^2+18 c^2 d^2\right ) x e^8+\frac {12 a \left (c d^2-a e^2\right )^3 x^2 e^7}{d^3}}{2 x^6 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{6 a d e}-\frac {a^2 e^8 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{6 d^2 x^6}}{e^6}+\frac {2 e^4 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^7 (d+e x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {-\frac {12 a \left (c d^2-a e^2\right )^3 x^5 e^{10}}{d^6}+\frac {12 a \left (c d^2-a e^2\right )^3 x^4 e^9}{d^5}+\frac {a^3 \left (25 c d^2-23 a e^2\right ) e^9}{d}-\frac {12 a \left (c d^2-a e^2\right )^3 x^3 e^8}{d^4}+2 a^2 \left (\frac {6 a^2 e^4}{d^2}-23 a c e^2+18 c^2 d^2\right ) x e^8+\frac {12 a \left (c d^2-a e^2\right )^3 x^2 e^7}{d^3}}{x^6 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{12 a d e}-\frac {a^2 e^8 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{6 d^2 x^6}}{e^6}+\frac {2 e^4 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^7 (d+e x)}\) |
| \(\Big \downarrow \) 2181 |
| \(\displaystyle \frac {\frac {-\frac {\int -\frac {-\frac {120 a^2 \left (c d^2-a e^2\right )^3 x^4 e^{11}}{d^5}+\frac {120 a^2 \left (c d^2-a e^2\right )^3 x^3 e^{10}}{d^4}-\frac {120 a^2 \left (c d^2-a e^2\right )^3 x^2 e^9}{d^3}+\frac {a^3 \left (135 c^2 d^4-478 a c e^2 d^2+327 a^2 e^4\right ) e^9}{d}+8 a^2 \left (-\frac {15 a^3 e^6}{d^2}+68 a^2 c e^4-70 a c^2 d^2 e^2+15 c^3 d^4\right ) x e^8}{2 x^5 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{5 a d e}-\frac {a^2 e^8 \left (25 c d^2-23 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 d^2 x^5}}{12 a d e}-\frac {a^2 e^8 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{6 d^2 x^6}}{e^6}+\frac {2 e^4 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^7 (d+e x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {-\frac {120 a^2 \left (c d^2-a e^2\right )^3 x^4 e^{11}}{d^5}+\frac {120 a^2 \left (c d^2-a e^2\right )^3 x^3 e^{10}}{d^4}-\frac {120 a^2 \left (c d^2-a e^2\right )^3 x^2 e^9}{d^3}+\frac {a^3 \left (135 c^2 d^4-478 a c e^2 d^2+327 a^2 e^4\right ) e^9}{d}+8 a^2 \left (-\frac {15 a^3 e^6}{d^2}+68 a^2 c e^4-70 a c^2 d^2 e^2+15 c^3 d^4\right ) x e^8}{x^5 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{10 a d e}-\frac {a^2 e^8 \left (25 c d^2-23 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 d^2 x^5}}{12 a d e}-\frac {a^2 e^8 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{6 d^2 x^6}}{e^6}+\frac {2 e^4 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^7 (d+e x)}\) |
| \(\Big \downarrow \) 2181 |
| \(\displaystyle \frac {\frac {\frac {-\frac {\int -\frac {3 \left (-\frac {320 a^3 \left (c d^2-a e^2\right )^3 x^3 e^{12}}{d^4}+\frac {320 a^3 \left (c d^2-a e^2\right )^3 x^2 e^{11}}{d^3}-2 a^3 \left (-\frac {160 a^3 e^6}{d^2}+807 a^2 c e^4-958 a c^2 d^2 e^2+295 c^3 d^4\right ) x e^{10}+\frac {a^3 \left (5 c^3 d^6-693 a c^2 e^2 d^4+1803 a^2 c e^4 d^2-1083 a^3 e^6\right ) e^9}{d}\right )}{2 x^4 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{4 a d e}-\frac {a^2 e^8 \left (327 a^2 e^4-478 a c d^2 e^2+135 c^2 d^4\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 d^2 x^4}}{10 a d e}-\frac {a^2 e^8 \left (25 c d^2-23 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 d^2 x^5}}{12 a d e}-\frac {a^2 e^8 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{6 d^2 x^6}}{e^6}+\frac {2 e^4 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^7 (d+e x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\frac {3 \int \frac {-\frac {320 a^3 \left (c d^2-a e^2\right )^3 x^3 e^{12}}{d^4}+\frac {320 a^3 \left (c d^2-a e^2\right )^3 x^2 e^{11}}{d^3}-2 a^3 \left (-\frac {160 a^3 e^6}{d^2}+807 a^2 c e^4-958 a c^2 d^2 e^2+295 c^3 d^4\right ) x e^{10}+\frac {a^3 \left (5 c^3 d^6-693 a c^2 e^2 d^4+1803 a^2 c e^4 d^2-1083 a^3 e^6\right ) e^9}{d}}{x^4 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{8 a d e}-\frac {a^2 e^8 \left (327 a^2 e^4-478 a c d^2 e^2+135 c^2 d^4\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 d^2 x^4}}{10 a d e}-\frac {a^2 e^8 \left (25 c d^2-23 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 d^2 x^5}}{12 a d e}-\frac {a^2 e^8 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{6 d^2 x^6}}{e^6}+\frac {2 e^4 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^7 (d+e x)}\) |
| \(\Big \downarrow \) 2181 |
| \(\displaystyle \frac {\frac {\frac {\frac {3 \left (-\frac {\int \frac {\frac {1920 a^4 \left (c d^2-a e^2\right )^3 x^2 e^{13}}{d^3}+\frac {4 a^3 \left (5 c^4 d^8-1173 a c^3 e^2 d^6+3243 a^2 c^2 e^4 d^4-2523 a^3 c e^6 d^2+480 a^4 e^8\right ) x e^{10}}{d^2}+\frac {a^3 \left (25 c^4 d^8+100 a c^3 e^2 d^6-5946 a^2 c^2 e^4 d^4+13284 a^3 c e^6 d^2-7335 a^4 e^8\right ) e^9}{d}}{2 x^3 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{3 a d e}-\frac {a^2 e^8 \left (-1083 a^3 e^6+1803 a^2 c d^2 e^4-693 a c^2 d^4 e^2+5 c^3 d^6\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 d^2 x^3}\right )}{8 a d e}-\frac {a^2 e^8 \left (327 a^2 e^4-478 a c d^2 e^2+135 c^2 d^4\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 d^2 x^4}}{10 a d e}-\frac {a^2 e^8 \left (25 c d^2-23 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 d^2 x^5}}{12 a d e}-\frac {a^2 e^8 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{6 d^2 x^6}}{e^6}+\frac {2 e^4 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^7 (d+e x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\frac {3 \left (-\frac {\int \frac {\frac {1920 a^4 \left (c d^2-a e^2\right )^3 x^2 e^{13}}{d^3}+\frac {4 a^3 \left (5 c^4 d^8-1173 a c^3 e^2 d^6+3243 a^2 c^2 e^4 d^4-2523 a^3 c e^6 d^2+480 a^4 e^8\right ) x e^{10}}{d^2}+\frac {a^3 \left (25 c^4 d^8+100 a c^3 e^2 d^6-5946 a^2 c^2 e^4 d^4+13284 a^3 c e^6 d^2-7335 a^4 e^8\right ) e^9}{d}}{x^3 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{6 a d e}-\frac {a^2 e^8 \left (-1083 a^3 e^6+1803 a^2 c d^2 e^4-693 a c^2 d^4 e^2+5 c^3 d^6\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 d^2 x^3}\right )}{8 a d e}-\frac {a^2 e^8 \left (327 a^2 e^4-478 a c d^2 e^2+135 c^2 d^4\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 d^2 x^4}}{10 a d e}-\frac {a^2 e^8 \left (25 c d^2-23 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 d^2 x^5}}{12 a d e}-\frac {a^2 e^8 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{6 d^2 x^6}}{e^6}+\frac {2 e^4 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^7 (d+e x)}\) |
| \(\Big \downarrow \) 2181 |
| \(\displaystyle \frac {\frac {\frac {\frac {3 \left (-\frac {-\frac {\int \frac {a^3 e^9 \left (d \left (75 c^5 d^{10}+295 a c^4 e^2 d^8+1230 a^2 c^3 e^4 d^6-29874 a^3 c^2 e^6 d^4+58215 a^4 c e^8 d^2-29685 a^5 e^{10}\right )+2 e \left (25 c^5 d^{10}+100 a c^4 e^2 d^8-9786 a^2 c^3 e^4 d^6+24804 a^3 c^2 e^6 d^4-18855 a^4 c e^8 d^2+3840 a^5 e^{10}\right ) x\right )}{2 d^2 x^2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 a d e}-\frac {a^2 e^8 \left (-7335 a^4 e^8+13284 a^3 c d^2 e^6-5946 a^2 c^2 d^4 e^4+100 a c^3 d^6 e^2+25 c^4 d^8\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 d^2 x^2}}{6 a d e}-\frac {a^2 e^8 \left (-1083 a^3 e^6+1803 a^2 c d^2 e^4-693 a c^2 d^4 e^2+5 c^3 d^6\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 d^2 x^3}\right )}{8 a d e}-\frac {a^2 e^8 \left (327 a^2 e^4-478 a c d^2 e^2+135 c^2 d^4\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 d^2 x^4}}{10 a d e}-\frac {a^2 e^8 \left (25 c d^2-23 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 d^2 x^5}}{12 a d e}-\frac {a^2 e^8 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{6 d^2 x^6}}{e^6}+\frac {2 e^4 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^7 (d+e x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\frac {3 \left (-\frac {-\frac {a^2 e^8 \int \frac {75 c^5 d^{11}+295 a c^4 e^2 d^9+1230 a^2 c^3 e^4 d^7-29874 a^3 c^2 e^6 d^5+58215 a^4 c e^8 d^3-29685 a^5 e^{10} d+2 e \left (25 c^5 d^{10}+100 a c^4 e^2 d^8-9786 a^2 c^3 e^4 d^6+24804 a^3 c^2 e^6 d^4-18855 a^4 c e^8 d^2+3840 a^5 e^{10}\right ) x}{x^2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{4 d^3}-\frac {a^2 e^8 \left (-7335 a^4 e^8+13284 a^3 c d^2 e^6-5946 a^2 c^2 d^4 e^4+100 a c^3 d^6 e^2+25 c^4 d^8\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 d^2 x^2}}{6 a d e}-\frac {a^2 e^8 \left (-1083 a^3 e^6+1803 a^2 c d^2 e^4-693 a c^2 d^4 e^2+5 c^3 d^6\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 d^2 x^3}\right )}{8 a d e}-\frac {a^2 e^8 \left (327 a^2 e^4-478 a c d^2 e^2+135 c^2 d^4\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 d^2 x^4}}{10 a d e}-\frac {a^2 e^8 \left (25 c d^2-23 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 d^2 x^5}}{12 a d e}-\frac {a^2 e^8 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{6 d^2 x^6}}{e^6}+\frac {2 e^4 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^7 (d+e x)}\) |
| \(\Big \downarrow \) 1228 |
| \(\displaystyle \frac {\frac {\frac {\frac {3 \left (-\frac {-\frac {a^2 e^8 \left (-\frac {15 \left (-3003 a^4 e^8+924 a^3 c d^2 e^6+126 a^2 c^2 d^4 e^4+28 a c^3 d^6 e^2+5 c^4 d^8\right ) \left (c d^2-a e^2\right )^2 \int \frac {1}{x \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 a e}-\frac {\left (-29685 a^5 e^{10}+58215 a^4 c d^2 e^8-29874 a^3 c^2 d^4 e^6+1230 a^2 c^3 d^6 e^4+295 a c^4 d^8 e^2+75 c^5 d^{10}\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{a e x}\right )}{4 d^3}-\frac {a^2 e^8 \left (-7335 a^4 e^8+13284 a^3 c d^2 e^6-5946 a^2 c^2 d^4 e^4+100 a c^3 d^6 e^2+25 c^4 d^8\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 d^2 x^2}}{6 a d e}-\frac {a^2 e^8 \left (-1083 a^3 e^6+1803 a^2 c d^2 e^4-693 a c^2 d^4 e^2+5 c^3 d^6\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 d^2 x^3}\right )}{8 a d e}-\frac {a^2 e^8 \left (327 a^2 e^4-478 a c d^2 e^2+135 c^2 d^4\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 d^2 x^4}}{10 a d e}-\frac {a^2 e^8 \left (25 c d^2-23 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 d^2 x^5}}{12 a d e}-\frac {a^2 e^8 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{6 d^2 x^6}}{e^6}+\frac {2 e^4 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^7 (d+e x)}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {\frac {\frac {\frac {3 \left (-\frac {-\frac {a^2 e^8 \left (\frac {15 \left (c d^2-a e^2\right )^2 \left (-3003 a^4 e^8+924 a^3 c d^2 e^6+126 a^2 c^2 d^4 e^4+28 a c^3 d^6 e^2+5 c^4 d^8\right ) \int \frac {1}{4 a d e-\frac {\left (2 a d e+\left (c d^2+a e^2\right ) x\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {2 a d e+\left (c d^2+a e^2\right ) x}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}}{a e}-\frac {\left (-29685 a^5 e^{10}+58215 a^4 c d^2 e^8-29874 a^3 c^2 d^4 e^6+1230 a^2 c^3 d^6 e^4+295 a c^4 d^8 e^2+75 c^5 d^{10}\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{a e x}\right )}{4 d^3}-\frac {a^2 e^8 \left (-7335 a^4 e^8+13284 a^3 c d^2 e^6-5946 a^2 c^2 d^4 e^4+100 a c^3 d^6 e^2+25 c^4 d^8\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 d^2 x^2}}{6 a d e}-\frac {a^2 e^8 \left (-1083 a^3 e^6+1803 a^2 c d^2 e^4-693 a c^2 d^4 e^2+5 c^3 d^6\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 d^2 x^3}\right )}{8 a d e}-\frac {a^2 e^8 \left (327 a^2 e^4-478 a c d^2 e^2+135 c^2 d^4\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 d^2 x^4}}{10 a d e}-\frac {a^2 e^8 \left (25 c d^2-23 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 d^2 x^5}}{12 a d e}-\frac {a^2 e^8 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{6 d^2 x^6}}{e^6}+\frac {2 e^4 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^7 (d+e x)}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\frac {\frac {3 \left (-\frac {a^2 e^8 \left (-1083 a^3 e^6+1803 a^2 c d^2 e^4-693 a c^2 d^4 e^2+5 c^3 d^6\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 d^2 x^3}-\frac {-\frac {a^2 e^8 \left (-7335 a^4 e^8+13284 a^3 c d^2 e^6-5946 a^2 c^2 d^4 e^4+100 a c^3 d^6 e^2+25 c^4 d^8\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 d^2 x^2}-\frac {a^2 e^8 \left (\frac {15 \left (c d^2-a e^2\right )^2 \left (-3003 a^4 e^8+924 a^3 c d^2 e^6+126 a^2 c^2 d^4 e^4+28 a c^3 d^6 e^2+5 c^4 d^8\right ) \text {arctanh}\left (\frac {x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 a^{3/2} \sqrt {d} e^{3/2}}-\frac {\left (-29685 a^5 e^{10}+58215 a^4 c d^2 e^8-29874 a^3 c^2 d^4 e^6+1230 a^2 c^3 d^6 e^4+295 a c^4 d^8 e^2+75 c^5 d^{10}\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{a e x}\right )}{4 d^3}}{6 a d e}\right )}{8 a d e}-\frac {a^2 e^8 \left (327 a^2 e^4-478 a c d^2 e^2+135 c^2 d^4\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 d^2 x^4}}{10 a d e}-\frac {a^2 e^8 \left (25 c d^2-23 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 d^2 x^5}}{12 a d e}-\frac {a^2 e^8 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{6 d^2 x^6}}{e^6}+\frac {2 e^4 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^7 (d+e x)}\) |
Input:
Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(x^7*(d + e*x)^4),x]
Output:
(2*e^4*(c*d^2 - a*e^2)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(d^7 *(d + e*x)) + (-1/6*(a^2*e^8*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/ (d^2*x^6) + (-1/5*(a^2*e^8*(25*c*d^2 - 23*a*e^2)*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*(135*c^2*d^4 - 478*a*c*d^2* e^2 + 327*a^2*e^4)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(d^2*x^4) + (3*(-1/3*(a^2*e^8*(5*c^3*d^6 - 693*a*c^2*d^4*e^2 + 1803*a^2*c*d^2*e^4 - 1083*a^3*e^6)*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*(25*c^4*d^8 + 100*a*c^3*d^6*e^2 - 5946*a^2*c^2*d^4*e^4 + 13284 *a^3*c*d^2*e^6 - 7335*a^4*e^8)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2] )/(d^2*x^2) - (a^2*e^8*(-(((75*c^5*d^10 + 295*a*c^4*d^8*e^2 + 1230*a^2*c^3 *d^6*e^4 - 29874*a^3*c^2*d^4*e^6 + 58215*a^4*c*d^2*e^8 - 29685*a^5*e^10)*S qrt[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*(5*c^4*d^8 + 28*a*c^3*d^6*e^2 + 126*a^2*c^2*d^4*e^4 + 924*a^3*c*d^2*e^6 - 3003*a^4*e^8)*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))/(6*a*d*e)))/(8*a*d*e))/(10*a*d*e))/(12*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. \(33769\) vs. \(2(711)=1422\).
Time = 10.82 (sec) , antiderivative size = 33770, normalized size of antiderivative = 44.73
Input:
int((a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(5/2)/x^7/(e*x+d)^4,x,method=_RETURN VERBOSE)
Output:
result too large to display
Time = 82.46 (sec) , antiderivative size = 1452, normalized size of antiderivative = 1.92 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^7 (d+e x)^4} \, dx=\text {Too large to display} \] Input:
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/x^7/(e*x+d)^4,x, algorit hm="fricas")
Output:
[-1/30720*(15*((5*c^6*d^12*e + 18*a*c^5*d^10*e^3 + 75*a^2*c^4*d^8*e^5 + 70 0*a^3*c^3*d^6*e^7 - 4725*a^4*c^2*d^4*e^9 + 6930*a^5*c*d^2*e^11 - 3003*a^6* e^13)*x^7 + (5*c^6*d^13 + 18*a*c^5*d^11*e^2 + 75*a^2*c^4*d^9*e^4 + 700*a^3 *c^3*d^7*e^6 - 4725*a^4*c^2*d^5*e^8 + 6930*a^5*c*d^3*e^10 - 3003*a^6*d*e^1 2)*x^6)*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*(1280*a^6*d ^7*e^6 + (75*a*c^5*d^11*e^2 + 295*a^2*c^4*d^9*e^4 + 1230*a^3*c^3*d^7*e^6 - 45234*a^4*c^2*d^5*e^8 + 88935*a^5*c*d^3*e^10 - 45045*a^6*d*e^12)*x^6 + (7 5*a*c^5*d^12*e + 245*a^2*c^4*d^10*e^3 + 1030*a^3*c^3*d^8*e^5 - 17982*a^4*c ^2*d^6*e^7 + 31647*a^5*c*d^4*e^9 - 15015*a^6*d^2*e^11)*x^5 - 2*(25*a^2*c^4 *d^11*e^2 + 80*a^3*c^3*d^9*e^4 - 3174*a^4*c^2*d^7*e^6 + 6072*a^5*c*d^5*e^8 - 3003*a^6*d^3*e^10)*x^4 + 8*(5*a^3*c^3*d^10*e^3 - 423*a^4*c^2*d^8*e^5 + 847*a^5*c*d^6*e^7 - 429*a^6*d^4*e^9)*x^3 + 16*(135*a^4*c^2*d^9*e^4 - 278*a ^5*c*d^7*e^6 + 143*a^6*d^5*e^8)*x^2 + 128*(25*a^5*c*d^8*e^5 - 13*a^6*d^6*e ^7)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(a^4*d^8*e^5*x^7 + a^4 *d^9*e^4*x^6), -1/15360*(15*((5*c^6*d^12*e + 18*a*c^5*d^10*e^3 + 75*a^2*c^ 4*d^8*e^5 + 700*a^3*c^3*d^6*e^7 - 4725*a^4*c^2*d^4*e^9 + 6930*a^5*c*d^2*e^ 11 - 3003*a^6*e^13)*x^7 + (5*c^6*d^13 + 18*a*c^5*d^11*e^2 + 75*a^2*c^4*d^9 *e^4 + 700*a^3*c^3*d^7*e^6 - 4725*a^4*c^2*d^5*e^8 + 6930*a^5*c*d^3*e^10...
Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^7 (d+e x)^4} \, dx=\text {Timed out} \] Input:
integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2)/x**7/(e*x+d)**4,x)
Output:
Timed out
\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^7 (d+e x)^4} \, dx=\int { \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{4} x^{7}} \,d x } \] Input:
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/x^7/(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^7), x)
Exception generated. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^7 (d+e x)^4} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/x^7/(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,[0,7,17]%%%},[2,8]%%%}+%%%{%%%{-8,[1,9,15]%%%},[2,7] %%%}+%%%{
Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^7 (d+e x)^4} \, dx=\int \frac {{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}}{x^7\,{\left (d+e\,x\right )}^4} \,d x \] Input:
int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)/(x^7*(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^7*(d + e*x)^4), x)
\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^7 (d+e x)^4} \, dx=\int \frac {{\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )}^{\frac {5}{2}}}{x^{7} \left (e x +d \right )^{4}}d x \] Input:
int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/x^7/(e*x+d)^4,x)
Output:
int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/x^7/(e*x+d)^4,x)