Integrand size = 37, antiderivative size = 474 \[ \int (d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=\frac {55 \left (c d^2-a e^2\right )^7 \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{32768 c^6 d^6 e^3}-\frac {55 \left (c d^2-a e^2\right )^5 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{12288 c^5 d^5 e^2}+\frac {11 \left (c d^2-a e^2\right )^3 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{768 c^4 d^4 e}+\frac {11 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{224 c^3 d^3}+\frac {11 \left (c d^2-a e^2\right ) (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{144 c^2 d^2}+\frac {(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{9 c d}-\frac {55 \left (c d^2-a e^2\right )^9 \text {arctanh}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{65536 c^{13/2} d^{13/2} e^{7/2}} \]
-55/12288*(-a*e^2+c*d^2)^5*(2*c*d*e*x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+ c*d*e*x^2)^(3/2)/c^5/d^5/e^2+11/768*(-a*e^2+c*d^2)^3*(2*c*d*e*x+a*e^2+c*d^ 2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/c^4/d^4/e+11/224*(-a*e^2+c*d^2) ^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/c^3/d^3+11/144*(-a*e^2+c*d^2)*( e*x+d)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/c^2/d^2+1/9*(e*x+d)^2*(a*d* e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/c/d-55/65536*(-a*e^2+c*d^2)^9*arctanh(1 /2*(2*c*d*e*x+a*e^2+c*d^2)/c^(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))/c^(13/2)/d^(13/2)/e^(7/2)+55/32768*(-a*e^2+c*d^2)^7*(2*c *d*e*x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^6/d^6/e^3
Time = 1.66 (sec) , antiderivative size = 625, normalized size of antiderivative = 1.32 \[ \int (d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=\frac {((a e+c d x) (d+e x))^{5/2} \left (\frac {\sqrt {c} \sqrt {d} \sqrt {e} \left (-3465 a^8 e^{16}+2310 a^7 c d e^{14} (13 d+e x)-462 a^6 c^2 d^2 e^{12} \left (249 d^2+43 d e x+4 e^2 x^2\right )+198 a^5 c^3 d^3 e^{10} \left (1289 d^3+381 d^2 e x+80 d e^2 x^2+8 e^3 x^3\right )-22 a^4 c^4 d^4 e^8 \left (16384 d^4+7531 d^3 e x+2724 d^2 e^2 x^2+616 d e^3 x^3+64 e^4 x^4\right )+2 a^3 c^5 d^5 e^6 \left (167301 d^5+115609 d^4 e x+65536 d^3 e^2 x^2+25584 d^2 e^3 x^3+6016 d e^4 x^4+640 e^5 x^5\right )+6 a^2 c^6 d^6 e^4 \left (19173 d^6+282339 d^5 e x+763652 d^4 e^2 x^2+1040048 d^3 e^3 x^3+786432 d^2 e^4 x^4+315776 d e^5 x^5+52736 e^6 x^6\right )+2 a c^7 d^7 e^2 \left (-15015 d^7+9933 d^6 e x+876816 d^5 e^2 x^2+2988664 d^4 e^3 x^3+4548736 d^3 e^4 x^4+3672960 d^2 e^5 x^5+1540096 d e^6 x^6+265216 e^7 x^7\right )+c^8 d^8 \left (3465 d^8-2310 d^7 e x+1848 d^6 e^2 x^2+588240 d^5 e^3 x^3+2229632 d^4 e^4 x^4+3603200 d^3 e^5 x^5+3025920 d^2 e^6 x^6+1304576 d e^7 x^7+229376 e^8 x^8\right )\right )}{(a e+c d x)^2 (d+e x)^2}-\frac {3465 \left (c d^2-a e^2\right )^9 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )}{(a e+c d x)^{5/2} (d+e x)^{5/2}}\right )}{2064384 c^{13/2} d^{13/2} e^{7/2}} \]
(((a*e + c*d*x)*(d + e*x))^(5/2)*((Sqrt[c]*Sqrt[d]*Sqrt[e]*(-3465*a^8*e^16 + 2310*a^7*c*d*e^14*(13*d + e*x) - 462*a^6*c^2*d^2*e^12*(249*d^2 + 43*d*e *x + 4*e^2*x^2) + 198*a^5*c^3*d^3*e^10*(1289*d^3 + 381*d^2*e*x + 80*d*e^2* x^2 + 8*e^3*x^3) - 22*a^4*c^4*d^4*e^8*(16384*d^4 + 7531*d^3*e*x + 2724*d^2 *e^2*x^2 + 616*d*e^3*x^3 + 64*e^4*x^4) + 2*a^3*c^5*d^5*e^6*(167301*d^5 + 1 15609*d^4*e*x + 65536*d^3*e^2*x^2 + 25584*d^2*e^3*x^3 + 6016*d*e^4*x^4 + 6 40*e^5*x^5) + 6*a^2*c^6*d^6*e^4*(19173*d^6 + 282339*d^5*e*x + 763652*d^4*e ^2*x^2 + 1040048*d^3*e^3*x^3 + 786432*d^2*e^4*x^4 + 315776*d*e^5*x^5 + 527 36*e^6*x^6) + 2*a*c^7*d^7*e^2*(-15015*d^7 + 9933*d^6*e*x + 876816*d^5*e^2* x^2 + 2988664*d^4*e^3*x^3 + 4548736*d^3*e^4*x^4 + 3672960*d^2*e^5*x^5 + 15 40096*d*e^6*x^6 + 265216*e^7*x^7) + c^8*d^8*(3465*d^8 - 2310*d^7*e*x + 184 8*d^6*e^2*x^2 + 588240*d^5*e^3*x^3 + 2229632*d^4*e^4*x^4 + 3603200*d^3*e^5 *x^5 + 3025920*d^2*e^6*x^6 + 1304576*d*e^7*x^7 + 229376*e^8*x^8)))/((a*e + c*d*x)^2*(d + e*x)^2) - (3465*(c*d^2 - a*e^2)^9*ArcTanh[(Sqrt[c]*Sqrt[d]* Sqrt[d + e*x])/(Sqrt[e]*Sqrt[a*e + c*d*x])])/((a*e + c*d*x)^(5/2)*(d + e*x )^(5/2))))/(2064384*c^(13/2)*d^(13/2)*e^(7/2))
Time = 0.70 (sec) , antiderivative size = 525, normalized size of antiderivative = 1.11, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.216, Rules used = {1134, 1134, 1160, 1087, 1087, 1087, 1092, 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.
| \(\displaystyle \int (d+e x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2} \, dx\) |
| \(\Big \downarrow \) 1134 |
| \(\displaystyle \frac {11 \left (d^2-\frac {a e^2}{c}\right ) \int (d+e x)^2 \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{5/2}dx}{18 d}+\frac {(d+e x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{9 c d}\) |
| \(\Big \downarrow \) 1134 |
| \(\displaystyle \frac {11 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {9 \left (d^2-\frac {a e^2}{c}\right ) \int (d+e x) \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{5/2}dx}{16 d}+\frac {(d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{8 c d}\right )}{18 d}+\frac {(d+e x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{9 c d}\) |
| \(\Big \downarrow \) 1160 |
| \(\displaystyle \frac {11 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {9 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (d^2-\frac {a e^2}{c}\right ) \int \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{5/2}dx}{2 d}+\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{7 c d}\right )}{16 d}+\frac {(d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{8 c d}\right )}{18 d}+\frac {(d+e x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{9 c d}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {11 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {9 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{12 c d e}-\frac {5 \left (c d^2-a e^2\right )^2 \int \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}dx}{24 c d e}\right )}{2 d}+\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{7 c d}\right )}{16 d}+\frac {(d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{8 c d}\right )}{18 d}+\frac {(d+e x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{9 c d}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {11 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {9 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{12 c d e}-\frac {5 \left (c d^2-a e^2\right )^2 \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{8 c d e}-\frac {3 \left (c d^2-a e^2\right )^2 \int \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}dx}{16 c d e}\right )}{24 c d e}\right )}{2 d}+\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{7 c d}\right )}{16 d}+\frac {(d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{8 c d}\right )}{18 d}+\frac {(d+e x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{9 c d}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {11 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {9 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{12 c d e}-\frac {5 \left (c d^2-a e^2\right )^2 \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{8 c d e}-\frac {3 \left (c d^2-a e^2\right )^2 \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c d e}-\frac {\left (c d^2-a e^2\right )^2 \int \frac {1}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{8 c d e}\right )}{16 c d e}\right )}{24 c d e}\right )}{2 d}+\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{7 c d}\right )}{16 d}+\frac {(d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{8 c d}\right )}{18 d}+\frac {(d+e x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{9 c d}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {11 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {9 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{12 c d e}-\frac {5 \left (c d^2-a e^2\right )^2 \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{8 c d e}-\frac {3 \left (c d^2-a e^2\right )^2 \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c d e}-\frac {\left (c d^2-a e^2\right )^2 \int \frac {1}{4 c d e-\frac {\left (c d^2+2 c e x d+a e^2\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {c d^2+2 c e x d+a e^2}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}}{4 c d e}\right )}{16 c d e}\right )}{24 c d e}\right )}{2 d}+\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{7 c d}\right )}{16 d}+\frac {(d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{8 c d}\right )}{18 d}+\frac {(d+e x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{9 c d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {11 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {9 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{12 c d e}-\frac {5 \left (c d^2-a e^2\right )^2 \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{8 c d e}-\frac {3 \left (c d^2-a e^2\right )^2 \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c d e}-\frac {\left (c d^2-a e^2\right )^2 \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 c^{3/2} d^{3/2} e^{3/2}}\right )}{16 c d e}\right )}{24 c d e}\right )}{2 d}+\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{7 c d}\right )}{16 d}+\frac {(d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{8 c d}\right )}{18 d}+\frac {(d+e x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{9 c d}\) |
((d + e*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(9*c*d) + (11* (d^2 - (a*e^2)/c)*(((d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2 ))/(8*c*d) + (9*(d^2 - (a*e^2)/c)*((a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2) ^(7/2)/(7*c*d) + ((d^2 - (a*e^2)/c)*(((c*d^2 + a*e^2 + 2*c*d*e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(12*c*d*e) - (5*(c*d^2 - a*e^2)^2*( ((c*d^2 + a*e^2 + 2*c*d*e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2) )/(8*c*d*e) - (3*(c*d^2 - a*e^2)^2*(((c*d^2 + a*e^2 + 2*c*d*e*x)*Sqrt[a*d* e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(4*c*d*e) - ((c*d^2 - a*e^2)^2*ArcTanh [(c*d^2 + a*e^2 + 2*c*d*e*x)/(2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^ 2 + a*e^2)*x + c*d*e*x^2])])/(8*c^(3/2)*d^(3/2)*e^(3/2))))/(16*c*d*e)))/(2 4*c*d*e)))/(2*d)))/(16*d)))/(18*d)
3.20.32.3.1 Defintions of rubi rules used
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[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) *((a + b*x + c*x^2)^p/(2*c*(2*p + 1))), x] - Simp[p*((b^2 - 4*a*c)/(2*c*(2* p + 1))) Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[I nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a , b, c}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[e*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 1))), x] + Simp[(m + p)*((2*c*d - b*e)/(c*(m + 2*p + 1))) Int[(d + e*x)^ (m - 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[ c*d^2 - b*d*e + a*e^2, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && IntegerQ[2 *p]
Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol ] :> Simp[e*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + Simp[(2*c*d - b *e)/(2*c) Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[p, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(2844\) vs. \(2(432)=864\).
Time = 2.94 (sec) , antiderivative size = 2845, normalized size of antiderivative = 6.00
d^3*(1/12*(2*c*d*e*x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/ c/d/e+5/24*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/c/d/e*(1/8*(2*c*d*e*x+a*e^2+c*d ^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/c/d/e+3/16*(4*a*c*d^2*e^2-(a*e ^2+c*d^2)^2)/c/d/e*(1/4*(2*c*d*e*x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d *e*x^2)^(1/2)/c/d/e+1/8*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/c/d/e*ln((1/2*e^2* a+1/2*c*d^2+x*c*d*e)/(c*d*e)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2) )/(c*d*e)^(1/2))))+e^3*(1/9*x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/c/ d/e-11/18*(a*e^2+c*d^2)/c/d/e*(1/8*x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/ 2)/c/d/e-9/16*(a*e^2+c*d^2)/c/d/e*(1/7*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^( 7/2)/c/d/e-1/2*(a*e^2+c*d^2)/c/d/e*(1/12*(2*c*d*e*x+a*e^2+c*d^2)*(a*d*e+(a *e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/c/d/e+5/24*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/ c/d/e*(1/8*(2*c*d*e*x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2) /c/d/e+3/16*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/c/d/e*(1/4*(2*c*d*e*x+a*e^2+c* d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c/d/e+1/8*(4*a*c*d^2*e^2-(a*e ^2+c*d^2)^2)/c/d/e*ln((1/2*e^2*a+1/2*c*d^2+x*c*d*e)/(c*d*e)^(1/2)+(a*d*e+( a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(c*d*e)^(1/2)))))-1/8*a/c*(1/12*(2*c*d*e* x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/c/d/e+5/24*(4*a*c*d ^2*e^2-(a*e^2+c*d^2)^2)/c/d/e*(1/8*(2*c*d*e*x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c *d^2)*x+c*d*e*x^2)^(3/2)/c/d/e+3/16*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/c/d/e* (1/4*(2*c*d*e*x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c/...
Leaf count of result is larger than twice the leaf count of optimal. 896 vs. \(2 (432) = 864\).
Time = 0.72 (sec) , antiderivative size = 1806, normalized size of antiderivative = 3.81 \[ \int (d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=\text {Too large to display} \]
[1/8257536*(3465*(c^9*d^18 - 9*a*c^8*d^16*e^2 + 36*a^2*c^7*d^14*e^4 - 84*a ^3*c^6*d^12*e^6 + 126*a^4*c^5*d^10*e^8 - 126*a^5*c^4*d^8*e^10 + 84*a^6*c^3 *d^6*e^12 - 36*a^7*c^2*d^4*e^14 + 9*a^8*c*d^2*e^16 - a^9*e^18)*sqrt(c*d*e) *log(8*c^2*d^2*e^2*x^2 + c^2*d^4 + 6*a*c*d^2*e^2 + a^2*e^4 - 4*sqrt(c*d*e* x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*c*d*e*x + c*d^2 + a*e^2)*sqrt(c*d*e) + 8*(c^2*d^3*e + a*c*d*e^3)*x) + 4*(229376*c^9*d^9*e^9*x^8 + 3465*c^9*d^17* e - 30030*a*c^8*d^15*e^3 + 115038*a^2*c^7*d^13*e^5 + 334602*a^3*c^6*d^11*e ^7 - 360448*a^4*c^5*d^9*e^9 + 255222*a^5*c^4*d^7*e^11 - 115038*a^6*c^3*d^5 *e^13 + 30030*a^7*c^2*d^3*e^15 - 3465*a^8*c*d*e^17 + 14336*(91*c^9*d^10*e^ 8 + 37*a*c^8*d^8*e^10)*x^7 + 1024*(2955*c^9*d^11*e^7 + 3008*a*c^8*d^9*e^9 + 309*a^2*c^7*d^7*e^11)*x^6 + 256*(14075*c^9*d^12*e^6 + 28695*a*c^8*d^10*e ^8 + 7401*a^2*c^7*d^8*e^10 + 5*a^3*c^6*d^6*e^12)*x^5 + 128*(17419*c^9*d^13 *e^5 + 71074*a*c^8*d^11*e^7 + 36864*a^2*c^7*d^9*e^9 + 94*a^3*c^6*d^7*e^11 - 11*a^4*c^5*d^5*e^13)*x^4 + 16*(36765*c^9*d^14*e^4 + 373583*a*c^8*d^12*e^ 6 + 390018*a^2*c^7*d^10*e^8 + 3198*a^3*c^6*d^8*e^10 - 847*a^4*c^5*d^6*e^12 + 99*a^5*c^4*d^4*e^14)*x^3 + 8*(231*c^9*d^15*e^3 + 219204*a*c^8*d^13*e^5 + 572739*a^2*c^7*d^11*e^7 + 16384*a^3*c^6*d^9*e^9 - 7491*a^4*c^5*d^7*e^11 + 1980*a^5*c^4*d^5*e^13 - 231*a^6*c^3*d^3*e^15)*x^2 - 2*(1155*c^9*d^16*e^2 - 9933*a*c^8*d^14*e^4 - 847017*a^2*c^7*d^12*e^6 - 115609*a^3*c^6*d^10*e^8 + 82841*a^4*c^5*d^8*e^10 - 37719*a^5*c^4*d^6*e^12 + 9933*a^6*c^3*d^4*e...
Leaf count of result is larger than twice the leaf count of optimal. 14834 vs. \(2 (462) = 924\).
Time = 19.60 (sec) , antiderivative size = 14834, normalized size of antiderivative = 31.30 \[ \int (d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=\text {Too large to display} \]
Piecewise((sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))*(c**2*d**2*e**5* x**8/9 + x**7*(3*a*c**2*d**2*e**7 + 6*c**3*d**4*e**5 - c**2*d**2*e**5*(17* a*e**2/2 + 17*c*d**2/2)/9)/(8*c*d*e) + x**6*(3*a**2*c*d*e**8 + 154*a*c**2* d**3*e**6/9 + 15*c**3*d**5*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a*c**2*d* *2*e**7 + 6*c**3*d**4*e**5 - c**2*d**2*e**5*(17*a*e**2/2 + 17*c*d**2/2)/9) /(8*c*d*e))/(7*c*d*e) + x**5*(a**3*e**9 + 18*a**2*c*d**2*e**7 + 45*a*c**2* d**4*e**5 - 7*a*(3*a*c**2*d**2*e**7 + 6*c**3*d**4*e**5 - c**2*d**2*e**5*(1 7*a*e**2/2 + 17*c*d**2/2)/9)/(8*c) + 20*c**3*d**6*e**3 - (13*a*e**2/2 + 13 *c*d**2/2)*(3*a**2*c*d*e**8 + 154*a*c**2*d**3*e**6/9 + 15*c**3*d**5*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a*c**2*d**2*e**7 + 6*c**3*d**4*e**5 - c**2 *d**2*e**5*(17*a*e**2/2 + 17*c*d**2/2)/9)/(8*c*d*e))/(7*c*d*e))/(6*c*d*e) + x**4*(6*a**3*d*e**8 + 45*a**2*c*d**3*e**6 + 60*a*c**2*d**5*e**4 - 6*a*(3 *a**2*c*d*e**8 + 154*a*c**2*d**3*e**6/9 + 15*c**3*d**5*e**4 - (15*a*e**2/2 + 15*c*d**2/2)*(3*a*c**2*d**2*e**7 + 6*c**3*d**4*e**5 - c**2*d**2*e**5*(1 7*a*e**2/2 + 17*c*d**2/2)/9)/(8*c*d*e))/(7*c) + 15*c**3*d**7*e**2 - (11*a* e**2/2 + 11*c*d**2/2)*(a**3*e**9 + 18*a**2*c*d**2*e**7 + 45*a*c**2*d**4*e* *5 - 7*a*(3*a*c**2*d**2*e**7 + 6*c**3*d**4*e**5 - c**2*d**2*e**5*(17*a*e** 2/2 + 17*c*d**2/2)/9)/(8*c) + 20*c**3*d**6*e**3 - (13*a*e**2/2 + 13*c*d**2 /2)*(3*a**2*c*d*e**8 + 154*a*c**2*d**3*e**6/9 + 15*c**3*d**5*e**4 - (15*a* e**2/2 + 15*c*d**2/2)*(3*a*c**2*d**2*e**7 + 6*c**3*d**4*e**5 - c**2*d**...
Exception generated. \[ \int (d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
Leaf count of result is larger than twice the leaf count of optimal. 936 vs. \(2 (432) = 864\).
Time = 0.38 (sec) , antiderivative size = 936, normalized size of antiderivative = 1.97 \[ \int (d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=\frac {1}{2064384} \, \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (2 \, {\left (4 \, {\left (14 \, {\left (16 \, c^{2} d^{2} e^{5} x + \frac {91 \, c^{10} d^{11} e^{12} + 37 \, a c^{9} d^{9} e^{14}}{c^{8} d^{8} e^{8}}\right )} x + \frac {2955 \, c^{10} d^{12} e^{11} + 3008 \, a c^{9} d^{10} e^{13} + 309 \, a^{2} c^{8} d^{8} e^{15}}{c^{8} d^{8} e^{8}}\right )} x + \frac {14075 \, c^{10} d^{13} e^{10} + 28695 \, a c^{9} d^{11} e^{12} + 7401 \, a^{2} c^{8} d^{9} e^{14} + 5 \, a^{3} c^{7} d^{7} e^{16}}{c^{8} d^{8} e^{8}}\right )} x + \frac {17419 \, c^{10} d^{14} e^{9} + 71074 \, a c^{9} d^{12} e^{11} + 36864 \, a^{2} c^{8} d^{10} e^{13} + 94 \, a^{3} c^{7} d^{8} e^{15} - 11 \, a^{4} c^{6} d^{6} e^{17}}{c^{8} d^{8} e^{8}}\right )} x + \frac {36765 \, c^{10} d^{15} e^{8} + 373583 \, a c^{9} d^{13} e^{10} + 390018 \, a^{2} c^{8} d^{11} e^{12} + 3198 \, a^{3} c^{7} d^{9} e^{14} - 847 \, a^{4} c^{6} d^{7} e^{16} + 99 \, a^{5} c^{5} d^{5} e^{18}}{c^{8} d^{8} e^{8}}\right )} x + \frac {231 \, c^{10} d^{16} e^{7} + 219204 \, a c^{9} d^{14} e^{9} + 572739 \, a^{2} c^{8} d^{12} e^{11} + 16384 \, a^{3} c^{7} d^{10} e^{13} - 7491 \, a^{4} c^{6} d^{8} e^{15} + 1980 \, a^{5} c^{5} d^{6} e^{17} - 231 \, a^{6} c^{4} d^{4} e^{19}}{c^{8} d^{8} e^{8}}\right )} x - \frac {1155 \, c^{10} d^{17} e^{6} - 9933 \, a c^{9} d^{15} e^{8} - 847017 \, a^{2} c^{8} d^{13} e^{10} - 115609 \, a^{3} c^{7} d^{11} e^{12} + 82841 \, a^{4} c^{6} d^{9} e^{14} - 37719 \, a^{5} c^{5} d^{7} e^{16} + 9933 \, a^{6} c^{4} d^{5} e^{18} - 1155 \, a^{7} c^{3} d^{3} e^{20}}{c^{8} d^{8} e^{8}}\right )} x + \frac {3465 \, c^{10} d^{18} e^{5} - 30030 \, a c^{9} d^{16} e^{7} + 115038 \, a^{2} c^{8} d^{14} e^{9} + 334602 \, a^{3} c^{7} d^{12} e^{11} - 360448 \, a^{4} c^{6} d^{10} e^{13} + 255222 \, a^{5} c^{5} d^{8} e^{15} - 115038 \, a^{6} c^{4} d^{6} e^{17} + 30030 \, a^{7} c^{3} d^{4} e^{19} - 3465 \, a^{8} c^{2} d^{2} e^{21}}{c^{8} d^{8} e^{8}}\right )} + \frac {55 \, {\left (c^{9} d^{18} - 9 \, a c^{8} d^{16} e^{2} + 36 \, a^{2} c^{7} d^{14} e^{4} - 84 \, a^{3} c^{6} d^{12} e^{6} + 126 \, a^{4} c^{5} d^{10} e^{8} - 126 \, a^{5} c^{4} d^{8} e^{10} + 84 \, a^{6} c^{3} d^{6} e^{12} - 36 \, a^{7} c^{2} d^{4} e^{14} + 9 \, a^{8} c d^{2} e^{16} - a^{9} e^{18}\right )} \log \left ({\left | -c d^{2} - a e^{2} - 2 \, \sqrt {c d e} {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} \right |}\right )}{65536 \, \sqrt {c d e} c^{6} d^{6} e^{3}} \]
1/2064384*sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e)*(2*(4*(2*(8*(2*(4*(1 4*(16*c^2*d^2*e^5*x + (91*c^10*d^11*e^12 + 37*a*c^9*d^9*e^14)/(c^8*d^8*e^8 ))*x + (2955*c^10*d^12*e^11 + 3008*a*c^9*d^10*e^13 + 309*a^2*c^8*d^8*e^15) /(c^8*d^8*e^8))*x + (14075*c^10*d^13*e^10 + 28695*a*c^9*d^11*e^12 + 7401*a ^2*c^8*d^9*e^14 + 5*a^3*c^7*d^7*e^16)/(c^8*d^8*e^8))*x + (17419*c^10*d^14* e^9 + 71074*a*c^9*d^12*e^11 + 36864*a^2*c^8*d^10*e^13 + 94*a^3*c^7*d^8*e^1 5 - 11*a^4*c^6*d^6*e^17)/(c^8*d^8*e^8))*x + (36765*c^10*d^15*e^8 + 373583* a*c^9*d^13*e^10 + 390018*a^2*c^8*d^11*e^12 + 3198*a^3*c^7*d^9*e^14 - 847*a ^4*c^6*d^7*e^16 + 99*a^5*c^5*d^5*e^18)/(c^8*d^8*e^8))*x + (231*c^10*d^16*e ^7 + 219204*a*c^9*d^14*e^9 + 572739*a^2*c^8*d^12*e^11 + 16384*a^3*c^7*d^10 *e^13 - 7491*a^4*c^6*d^8*e^15 + 1980*a^5*c^5*d^6*e^17 - 231*a^6*c^4*d^4*e^ 19)/(c^8*d^8*e^8))*x - (1155*c^10*d^17*e^6 - 9933*a*c^9*d^15*e^8 - 847017* a^2*c^8*d^13*e^10 - 115609*a^3*c^7*d^11*e^12 + 82841*a^4*c^6*d^9*e^14 - 37 719*a^5*c^5*d^7*e^16 + 9933*a^6*c^4*d^5*e^18 - 1155*a^7*c^3*d^3*e^20)/(c^8 *d^8*e^8))*x + (3465*c^10*d^18*e^5 - 30030*a*c^9*d^16*e^7 + 115038*a^2*c^8 *d^14*e^9 + 334602*a^3*c^7*d^12*e^11 - 360448*a^4*c^6*d^10*e^13 + 255222*a ^5*c^5*d^8*e^15 - 115038*a^6*c^4*d^6*e^17 + 30030*a^7*c^3*d^4*e^19 - 3465* a^8*c^2*d^2*e^21)/(c^8*d^8*e^8)) + 55/65536*(c^9*d^18 - 9*a*c^8*d^16*e^2 + 36*a^2*c^7*d^14*e^4 - 84*a^3*c^6*d^12*e^6 + 126*a^4*c^5*d^10*e^8 - 126*a^ 5*c^4*d^8*e^10 + 84*a^6*c^3*d^6*e^12 - 36*a^7*c^2*d^4*e^14 + 9*a^8*c*d^...
Timed out. \[ \int (d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=\int {\left (d+e\,x\right )}^3\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2} \,d x \]