Integrand size = 37, antiderivative size = 388 \[ \int (d+e x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {21 \left (c d^2-a e^2\right )^4 \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{512 c^5 d^5 e}+\frac {7 \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{64 c^4 d^4}+\frac {21 \left (c d^2-a e^2\right )^2 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{160 c^3 d^3}+\frac {3 \left (c d^2-a e^2\right ) (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{20 c^2 d^2}+\frac {(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{6 c d}-\frac {21 \left (c d^2-a e^2\right )^6 \text {arctanh}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{1024 c^{11/2} d^{11/2} e^{3/2}} \]
7/64*(-a*e^2+c*d^2)^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/c^4/d^4+21/1 60*(-a*e^2+c*d^2)^2*(e*x+d)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/c^3/d^ 3+3/20*(-a*e^2+c*d^2)*(e*x+d)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/c^ 2/d^2+1/6*(e*x+d)^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/c/d-21/1024*(- a*e^2+c*d^2)^6*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^(11/2)/d^(11/2)/e^(3/2)+21/512 *(-a*e^2+c*d^2)^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^5/d^5/e
Time = 1.00 (sec) , antiderivative size = 344, normalized size of antiderivative = 0.89 \[ \int (d+e x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {\sqrt {(a e+c d x) (d+e x)} \left (\sqrt {c} \sqrt {d} \sqrt {e} \left (315 a^5 e^{10}-105 a^4 c d e^8 (17 d+2 e x)+42 a^3 c^2 d^2 e^6 \left (99 d^2+28 d e x+4 e^2 x^2\right )-18 a^2 c^3 d^3 e^4 \left (281 d^3+150 d^2 e x+52 d e^2 x^2+8 e^3 x^3\right )+a c^4 d^4 e^2 \left (3335 d^4+3224 d^3 e x+2136 d^2 e^2 x^2+800 d e^3 x^3+128 e^4 x^4\right )+c^5 d^5 \left (315 d^5+4910 d^4 e x+11432 d^3 e^2 x^2+12144 d^2 e^3 x^3+6272 d e^4 x^4+1280 e^5 x^5\right )\right )-\frac {315 \left (c d^2-a e^2\right )^6 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )}{\sqrt {a e+c d x} \sqrt {d+e x}}\right )}{7680 c^{11/2} d^{11/2} e^{3/2}} \]
(Sqrt[(a*e + c*d*x)*(d + e*x)]*(Sqrt[c]*Sqrt[d]*Sqrt[e]*(315*a^5*e^10 - 10 5*a^4*c*d*e^8*(17*d + 2*e*x) + 42*a^3*c^2*d^2*e^6*(99*d^2 + 28*d*e*x + 4*e ^2*x^2) - 18*a^2*c^3*d^3*e^4*(281*d^3 + 150*d^2*e*x + 52*d*e^2*x^2 + 8*e^3 *x^3) + a*c^4*d^4*e^2*(3335*d^4 + 3224*d^3*e*x + 2136*d^2*e^2*x^2 + 800*d* e^3*x^3 + 128*e^4*x^4) + c^5*d^5*(315*d^5 + 4910*d^4*e*x + 11432*d^3*e^2*x ^2 + 12144*d^2*e^3*x^3 + 6272*d*e^4*x^4 + 1280*e^5*x^5)) - (315*(c*d^2 - a *e^2)^6*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[a*e + c*d*x] )])/(Sqrt[a*e + c*d*x]*Sqrt[d + e*x])))/(7680*c^(11/2)*d^(11/2)*e^(3/2))
Time = 0.62 (sec) , antiderivative size = 418, normalized size of antiderivative = 1.08, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.189, Rules used = {1134, 1134, 1134, 1160, 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.
| \(\displaystyle \int (d+e x)^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \, dx\) |
| \(\Big \downarrow \) 1134 |
| \(\displaystyle \frac {3 \left (d^2-\frac {a e^2}{c}\right ) \int (d+e x)^3 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}dx}{4 d}+\frac {(d+e x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{6 c d}\) |
| \(\Big \downarrow \) 1134 |
| \(\displaystyle \frac {3 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {7 \left (d^2-\frac {a e^2}{c}\right ) \int (d+e x)^2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}dx}{10 d}+\frac {(d+e x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 c d}\right )}{4 d}+\frac {(d+e x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{6 c d}\) |
| \(\Big \downarrow \) 1134 |
| \(\displaystyle \frac {3 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {7 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {5 \left (d^2-\frac {a e^2}{c}\right ) \int (d+e x) \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}dx}{8 d}+\frac {(d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 c d}\right )}{10 d}+\frac {(d+e x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 c d}\right )}{4 d}+\frac {(d+e x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{6 c d}\) |
| \(\Big \downarrow \) 1160 |
| \(\displaystyle \frac {3 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {7 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {5 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (d^2-\frac {a e^2}{c}\right ) \int \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}dx}{2 d}+\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 c d}\right )}{8 d}+\frac {(d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 c d}\right )}{10 d}+\frac {(d+e x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 c d}\right )}{4 d}+\frac {(d+e x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{6 c d}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {3 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {7 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {5 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c d e}-\frac {\left (c d^2-a e^2\right )^2 \int \frac {1}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{8 c d e}\right )}{2 d}+\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 c d}\right )}{8 d}+\frac {(d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 c d}\right )}{10 d}+\frac {(d+e x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 c d}\right )}{4 d}+\frac {(d+e x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{6 c d}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {3 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {7 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {5 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c d e}-\frac {\left (c d^2-a e^2\right )^2 \int \frac {1}{4 c d e-\frac {\left (c d^2+2 c e x d+a e^2\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {c d^2+2 c e x d+a e^2}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}}{4 c d e}\right )}{2 d}+\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 c d}\right )}{8 d}+\frac {(d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 c d}\right )}{10 d}+\frac {(d+e x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 c d}\right )}{4 d}+\frac {(d+e x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{6 c d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {3 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {7 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {5 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c d e}-\frac {\left (c d^2-a e^2\right )^2 \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 c^{3/2} d^{3/2} e^{3/2}}\right )}{2 d}+\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 c d}\right )}{8 d}+\frac {(d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 c d}\right )}{10 d}+\frac {(d+e x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 c d}\right )}{4 d}+\frac {(d+e x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{6 c d}\) |
((d + e*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(6*c*d) + (3*( d^2 - (a*e^2)/c)*(((d + e*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/ 2))/(5*c*d) + (7*(d^2 - (a*e^2)/c)*(((d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(4*c*d) + (5*(d^2 - (a*e^2)/c)*((a*d*e + (c*d^2 + a*e^ 2)*x + c*d*e*x^2)^(3/2)/(3*c*d) + ((d^2 - (a*e^2)/c)*(((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)) ))/(2*d)))/(8*d)))/(10*d)))/(4*d)
3.20.7.3.1 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. \(2774\) vs. \(2(350)=700\).
Time = 3.18 (sec) , antiderivative size = 2775, normalized size of antiderivative = 7.15
d^4*(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^4*(1/6*x^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/c/d/e-3/4*(a*e^2+c* d^2)/c/d/e*(1/5*x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/c/d/e-7/10*(a* e^2+c*d^2)/c/d/e*(1/4*x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/c/d/e-5/8* (a*e^2+c*d^2)/c/d/e*(1/3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/c/d/e-1/2 *(a*e^2+c*d^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)))-1/4*a/c*(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)))-2/5*a/c*(1/3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^ (3/2)/c/d/e-1/2*(a*e^2+c*d^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))))-1/2*a/c*(1/4*x*(a*d*e+(a*e^2+c*d^2)*x+ c*d*e*x^2)^(3/2)/c/d/e-5/8*(a*e^2+c*d^2)/c/d/e*(1/3*(a*d*e+(a*e^2+c*d^2)*x +c*d*e*x^2)^(3/2)/c/d/e-1/2*(a*e^2+c*d^2)/c/d/e*(1/4*(2*c*d*e*x+a*e^2+c...
Time = 0.39 (sec) , antiderivative size = 1040, normalized size of antiderivative = 2.68 \[ \int (d+e x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\left [\frac {315 \, {\left (c^{6} d^{12} - 6 \, a c^{5} d^{10} e^{2} + 15 \, a^{2} c^{4} d^{8} e^{4} - 20 \, a^{3} c^{3} d^{6} e^{6} + 15 \, a^{4} c^{2} d^{4} e^{8} - 6 \, a^{5} c d^{2} e^{10} + a^{6} e^{12}\right )} \sqrt {c d e} \log \left (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 + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {c d e} + 8 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right ) + 4 \, {\left (1280 \, c^{6} d^{6} e^{6} x^{5} + 315 \, c^{6} d^{11} e + 3335 \, a c^{5} d^{9} e^{3} - 5058 \, a^{2} c^{4} d^{7} e^{5} + 4158 \, a^{3} c^{3} d^{5} e^{7} - 1785 \, a^{4} c^{2} d^{3} e^{9} + 315 \, a^{5} c d e^{11} + 128 \, {\left (49 \, c^{6} d^{7} e^{5} + a c^{5} d^{5} e^{7}\right )} x^{4} + 16 \, {\left (759 \, c^{6} d^{8} e^{4} + 50 \, a c^{5} d^{6} e^{6} - 9 \, a^{2} c^{4} d^{4} e^{8}\right )} x^{3} + 8 \, {\left (1429 \, c^{6} d^{9} e^{3} + 267 \, a c^{5} d^{7} e^{5} - 117 \, a^{2} c^{4} d^{5} e^{7} + 21 \, a^{3} c^{3} d^{3} e^{9}\right )} x^{2} + 2 \, {\left (2455 \, c^{6} d^{10} e^{2} + 1612 \, a c^{5} d^{8} e^{4} - 1350 \, a^{2} c^{4} d^{6} e^{6} + 588 \, a^{3} c^{3} d^{4} e^{8} - 105 \, a^{4} c^{2} d^{2} e^{10}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{30720 \, c^{6} d^{6} e^{2}}, \frac {315 \, {\left (c^{6} d^{12} - 6 \, a c^{5} d^{10} e^{2} + 15 \, a^{2} c^{4} d^{8} e^{4} - 20 \, a^{3} c^{3} d^{6} e^{6} + 15 \, a^{4} c^{2} d^{4} e^{8} - 6 \, a^{5} c d^{2} e^{10} + a^{6} e^{12}\right )} \sqrt {-c d e} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {-c d e}}{2 \, {\left (c^{2} d^{2} e^{2} x^{2} + a c d^{2} e^{2} + {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right )}}\right ) + 2 \, {\left (1280 \, c^{6} d^{6} e^{6} x^{5} + 315 \, c^{6} d^{11} e + 3335 \, a c^{5} d^{9} e^{3} - 5058 \, a^{2} c^{4} d^{7} e^{5} + 4158 \, a^{3} c^{3} d^{5} e^{7} - 1785 \, a^{4} c^{2} d^{3} e^{9} + 315 \, a^{5} c d e^{11} + 128 \, {\left (49 \, c^{6} d^{7} e^{5} + a c^{5} d^{5} e^{7}\right )} x^{4} + 16 \, {\left (759 \, c^{6} d^{8} e^{4} + 50 \, a c^{5} d^{6} e^{6} - 9 \, a^{2} c^{4} d^{4} e^{8}\right )} x^{3} + 8 \, {\left (1429 \, c^{6} d^{9} e^{3} + 267 \, a c^{5} d^{7} e^{5} - 117 \, a^{2} c^{4} d^{5} e^{7} + 21 \, a^{3} c^{3} d^{3} e^{9}\right )} x^{2} + 2 \, {\left (2455 \, c^{6} d^{10} e^{2} + 1612 \, a c^{5} d^{8} e^{4} - 1350 \, a^{2} c^{4} d^{6} e^{6} + 588 \, a^{3} c^{3} d^{4} e^{8} - 105 \, a^{4} c^{2} d^{2} e^{10}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{15360 \, c^{6} d^{6} e^{2}}\right ] \]
[1/30720*(315*(c^6*d^12 - 6*a*c^5*d^10*e^2 + 15*a^2*c^4*d^8*e^4 - 20*a^3*c ^3*d^6*e^6 + 15*a^4*c^2*d^4*e^8 - 6*a^5*c*d^2*e^10 + a^6*e^12)*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*(1280*c^6*d^6*e^6*x^5 + 315*c^6*d^11*e + 3335*a*c^5*d^9*e^3 - 5058*a^2*c^4*d^7*e^5 + 4158*a^3*c^3*d^5*e^7 - 1785*a ^4*c^2*d^3*e^9 + 315*a^5*c*d*e^11 + 128*(49*c^6*d^7*e^5 + a*c^5*d^5*e^7)*x ^4 + 16*(759*c^6*d^8*e^4 + 50*a*c^5*d^6*e^6 - 9*a^2*c^4*d^4*e^8)*x^3 + 8*( 1429*c^6*d^9*e^3 + 267*a*c^5*d^7*e^5 - 117*a^2*c^4*d^5*e^7 + 21*a^3*c^3*d^ 3*e^9)*x^2 + 2*(2455*c^6*d^10*e^2 + 1612*a*c^5*d^8*e^4 - 1350*a^2*c^4*d^6* e^6 + 588*a^3*c^3*d^4*e^8 - 105*a^4*c^2*d^2*e^10)*x)*sqrt(c*d*e*x^2 + a*d* e + (c*d^2 + a*e^2)*x))/(c^6*d^6*e^2), 1/15360*(315*(c^6*d^12 - 6*a*c^5*d^ 10*e^2 + 15*a^2*c^4*d^8*e^4 - 20*a^3*c^3*d^6*e^6 + 15*a^4*c^2*d^4*e^8 - 6* a^5*c*d^2*e^10 + a^6*e^12)*sqrt(-c*d*e)*arctan(1/2*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)/(c^2*d^2*e^2 *x^2 + a*c*d^2*e^2 + (c^2*d^3*e + a*c*d*e^3)*x)) + 2*(1280*c^6*d^6*e^6*x^5 + 315*c^6*d^11*e + 3335*a*c^5*d^9*e^3 - 5058*a^2*c^4*d^7*e^5 + 4158*a^3*c ^3*d^5*e^7 - 1785*a^4*c^2*d^3*e^9 + 315*a^5*c*d*e^11 + 128*(49*c^6*d^7*e^5 + a*c^5*d^5*e^7)*x^4 + 16*(759*c^6*d^8*e^4 + 50*a*c^5*d^6*e^6 - 9*a^2*c^4 *d^4*e^8)*x^3 + 8*(1429*c^6*d^9*e^3 + 267*a*c^5*d^7*e^5 - 117*a^2*c^4*d...
Leaf count of result is larger than twice the leaf count of optimal. 2686 vs. \(2 (374) = 748\).
Time = 1.25 (sec) , antiderivative size = 2686, normalized size of antiderivative = 6.92 \[ \int (d+e x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\text {Too large to display} \]
Piecewise((sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))*(e**4*x**5/6 + x **4*(a*e**6 + 5*c*d**2*e**4 - e**4*(11*a*e**2/2 + 11*c*d**2/2)/6)/(5*c*d*e ) + x**3*(25*a*d*e**5/6 + 10*c*d**3*e**3 - (9*a*e**2/2 + 9*c*d**2/2)*(a*e* *6 + 5*c*d**2*e**4 - e**4*(11*a*e**2/2 + 11*c*d**2/2)/6)/(5*c*d*e))/(4*c*d *e) + x**2*(10*a*d**2*e**4 - 4*a*(a*e**6 + 5*c*d**2*e**4 - e**4*(11*a*e**2 /2 + 11*c*d**2/2)/6)/(5*c) + 10*c*d**4*e**2 - (7*a*e**2/2 + 7*c*d**2/2)*(2 5*a*d*e**5/6 + 10*c*d**3*e**3 - (9*a*e**2/2 + 9*c*d**2/2)*(a*e**6 + 5*c*d* *2*e**4 - e**4*(11*a*e**2/2 + 11*c*d**2/2)/6)/(5*c*d*e))/(4*c*d*e))/(3*c*d *e) + x*(10*a*d**3*e**3 - 3*a*(25*a*d*e**5/6 + 10*c*d**3*e**3 - (9*a*e**2/ 2 + 9*c*d**2/2)*(a*e**6 + 5*c*d**2*e**4 - e**4*(11*a*e**2/2 + 11*c*d**2/2) /6)/(5*c*d*e))/(4*c) + 5*c*d**5*e - (5*a*e**2/2 + 5*c*d**2/2)*(10*a*d**2*e **4 - 4*a*(a*e**6 + 5*c*d**2*e**4 - e**4*(11*a*e**2/2 + 11*c*d**2/2)/6)/(5 *c) + 10*c*d**4*e**2 - (7*a*e**2/2 + 7*c*d**2/2)*(25*a*d*e**5/6 + 10*c*d** 3*e**3 - (9*a*e**2/2 + 9*c*d**2/2)*(a*e**6 + 5*c*d**2*e**4 - e**4*(11*a*e* *2/2 + 11*c*d**2/2)/6)/(5*c*d*e))/(4*c*d*e))/(3*c*d*e))/(2*c*d*e) + (5*a*d **4*e**2 - 2*a*(10*a*d**2*e**4 - 4*a*(a*e**6 + 5*c*d**2*e**4 - e**4*(11*a* e**2/2 + 11*c*d**2/2)/6)/(5*c) + 10*c*d**4*e**2 - (7*a*e**2/2 + 7*c*d**2/2 )*(25*a*d*e**5/6 + 10*c*d**3*e**3 - (9*a*e**2/2 + 9*c*d**2/2)*(a*e**6 + 5* c*d**2*e**4 - e**4*(11*a*e**2/2 + 11*c*d**2/2)/6)/(5*c*d*e))/(4*c*d*e))/(3 *c) + c*d**6 - (3*a*e**2/2 + 3*c*d**2/2)*(10*a*d**3*e**3 - 3*a*(25*a*d*...
Exception generated. \[ \int (d+e x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\text {Exception raised: ValueError} \]
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
Time = 0.39 (sec) , antiderivative size = 507, normalized size of antiderivative = 1.31 \[ \int (d+e x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {1}{7680} \, \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, e^{4} x + \frac {49 \, c^{5} d^{6} e^{8} + a c^{4} d^{4} e^{10}}{c^{5} d^{5} e^{5}}\right )} x + \frac {759 \, c^{5} d^{7} e^{7} + 50 \, a c^{4} d^{5} e^{9} - 9 \, a^{2} c^{3} d^{3} e^{11}}{c^{5} d^{5} e^{5}}\right )} x + \frac {1429 \, c^{5} d^{8} e^{6} + 267 \, a c^{4} d^{6} e^{8} - 117 \, a^{2} c^{3} d^{4} e^{10} + 21 \, a^{3} c^{2} d^{2} e^{12}}{c^{5} d^{5} e^{5}}\right )} x + \frac {2455 \, c^{5} d^{9} e^{5} + 1612 \, a c^{4} d^{7} e^{7} - 1350 \, a^{2} c^{3} d^{5} e^{9} + 588 \, a^{3} c^{2} d^{3} e^{11} - 105 \, a^{4} c d e^{13}}{c^{5} d^{5} e^{5}}\right )} x + \frac {315 \, c^{5} d^{10} e^{4} + 3335 \, a c^{4} d^{8} e^{6} - 5058 \, a^{2} c^{3} d^{6} e^{8} + 4158 \, a^{3} c^{2} d^{4} e^{10} - 1785 \, a^{4} c d^{2} e^{12} + 315 \, a^{5} e^{14}}{c^{5} d^{5} e^{5}}\right )} + \frac {21 \, {\left (c^{6} d^{12} - 6 \, a c^{5} d^{10} e^{2} + 15 \, a^{2} c^{4} d^{8} e^{4} - 20 \, a^{3} c^{3} d^{6} e^{6} + 15 \, a^{4} c^{2} d^{4} e^{8} - 6 \, a^{5} c d^{2} e^{10} + a^{6} e^{12}\right )} \log \left ({\left | -c d^{2} - a e^{2} - 2 \, \sqrt {c d e} {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} \right |}\right )}{1024 \, \sqrt {c d e} c^{5} d^{5} e} \]
1/7680*sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e)*(2*(4*(2*(8*(10*e^4*x + (49*c^5*d^6*e^8 + a*c^4*d^4*e^10)/(c^5*d^5*e^5))*x + (759*c^5*d^7*e^7 + 5 0*a*c^4*d^5*e^9 - 9*a^2*c^3*d^3*e^11)/(c^5*d^5*e^5))*x + (1429*c^5*d^8*e^6 + 267*a*c^4*d^6*e^8 - 117*a^2*c^3*d^4*e^10 + 21*a^3*c^2*d^2*e^12)/(c^5*d^ 5*e^5))*x + (2455*c^5*d^9*e^5 + 1612*a*c^4*d^7*e^7 - 1350*a^2*c^3*d^5*e^9 + 588*a^3*c^2*d^3*e^11 - 105*a^4*c*d*e^13)/(c^5*d^5*e^5))*x + (315*c^5*d^1 0*e^4 + 3335*a*c^4*d^8*e^6 - 5058*a^2*c^3*d^6*e^8 + 4158*a^3*c^2*d^4*e^10 - 1785*a^4*c*d^2*e^12 + 315*a^5*e^14)/(c^5*d^5*e^5)) + 21/1024*(c^6*d^12 - 6*a*c^5*d^10*e^2 + 15*a^2*c^4*d^8*e^4 - 20*a^3*c^3*d^6*e^6 + 15*a^4*c^2*d ^4*e^8 - 6*a^5*c*d^2*e^10 + a^6*e^12)*log(abs(-c*d^2 - a*e^2 - 2*sqrt(c*d* e)*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))))/(sqrt(c *d*e)*c^5*d^5*e)
Timed out. \[ \int (d+e x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\int {\left (d+e\,x\right )}^4\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e} \,d x \]