Integrand size = 37, antiderivative size = 328 \[ \int (d+e x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {7 \left (c d^2-a e^2\right )^3 \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 c^4 d^4 e}+\frac {7 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{48 c^3 d^3}+\frac {7 \left (c d^2-a e^2\right ) (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{40 c^2 d^2}+\frac {(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{5 c d}-\frac {7 \left (c d^2-a e^2\right )^5 \text {arctanh}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{256 c^{9/2} d^{9/2} e^{3/2}} \]
7/48*(-a*e^2+c*d^2)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/c^3/d^3+7/40 *(-a*e^2+c*d^2)*(e*x+d)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/c^2/d^2+1/ 5*(e*x+d)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/c/d-7/256*(-a*e^2+c*d^ 2)^5*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^(9/2)/d^(9/2)/e^(3/2)+7/128*(-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)^(1/2)/c^4/d ^4/e
Time = 0.59 (sec) , antiderivative size = 278, normalized size of antiderivative = 0.85 \[ \int (d+e x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {\sqrt {(a e+c d x) (d+e x)} \left (\sqrt {c} \sqrt {d} \sqrt {e} \left (-105 a^4 e^8+70 a^3 c d e^6 (7 d+e x)-14 a^2 c^2 d^2 e^4 \left (64 d^2+23 d e x+4 e^2 x^2\right )+2 a c^3 d^3 e^2 \left (395 d^3+289 d^2 e x+128 d e^2 x^2+24 e^3 x^3\right )+c^4 d^4 \left (105 d^4+1210 d^3 e x+2104 d^2 e^2 x^2+1488 d e^3 x^3+384 e^4 x^4\right )\right )-\frac {105 \left (c d^2-a e^2\right )^5 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )}{\sqrt {a e+c d x} \sqrt {d+e x}}\right )}{1920 c^{9/2} d^{9/2} e^{3/2}} \]
(Sqrt[(a*e + c*d*x)*(d + e*x)]*(Sqrt[c]*Sqrt[d]*Sqrt[e]*(-105*a^4*e^8 + 70 *a^3*c*d*e^6*(7*d + e*x) - 14*a^2*c^2*d^2*e^4*(64*d^2 + 23*d*e*x + 4*e^2*x ^2) + 2*a*c^3*d^3*e^2*(395*d^3 + 289*d^2*e*x + 128*d*e^2*x^2 + 24*e^3*x^3) + c^4*d^4*(105*d^4 + 1210*d^3*e*x + 2104*d^2*e^2*x^2 + 1488*d*e^3*x^3 + 3 84*e^4*x^4)) - (105*(c*d^2 - a*e^2)^5*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])))/(192 0*c^(9/2)*d^(9/2)*e^(3/2))
Time = 0.51 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.07, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.162, Rules used = {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.
| \(\displaystyle \int (d+e x)^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \, dx\) |
| \(\Big \downarrow \) 1134 |
| \(\displaystyle \frac {7 \left (d^2-\frac {a e^2}{c}\right ) \int (d+e x)^2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}dx}{10 d}+\frac {(d+e x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 c d}\) |
| \(\Big \downarrow \) 1134 |
| \(\displaystyle \frac {7 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {5 \left (d^2-\frac {a e^2}{c}\right ) \int (d+e x) \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}dx}{8 d}+\frac {(d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 c d}\right )}{10 d}+\frac {(d+e x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 c d}\) |
| \(\Big \downarrow \) 1160 |
| \(\displaystyle \frac {7 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {5 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (d^2-\frac {a e^2}{c}\right ) \int \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}dx}{2 d}+\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 c d}\right )}{8 d}+\frac {(d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 c d}\right )}{10 d}+\frac {(d+e x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 c d}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {7 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {5 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c d e}-\frac {\left (c d^2-a e^2\right )^2 \int \frac {1}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{8 c d e}\right )}{2 d}+\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 c d}\right )}{8 d}+\frac {(d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 c d}\right )}{10 d}+\frac {(d+e x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 c d}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {7 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {5 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c d e}-\frac {\left (c d^2-a e^2\right )^2 \int \frac {1}{4 c d e-\frac {\left (c d^2+2 c e x d+a e^2\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {c d^2+2 c e x d+a e^2}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}}{4 c d e}\right )}{2 d}+\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 c d}\right )}{8 d}+\frac {(d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 c d}\right )}{10 d}+\frac {(d+e x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 c d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {7 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {5 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c d e}-\frac {\left (c d^2-a e^2\right )^2 \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 c^{3/2} d^{3/2} e^{3/2}}\right )}{2 d}+\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 c d}\right )}{8 d}+\frac {(d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 c d}\right )}{10 d}+\frac {(d+e x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 c d}\) |
((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)
3.20.8.3.1 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. \(1542\) vs. \(2(294)=588\).
Time = 2.69 (sec) , antiderivative size = 1543, normalized size of antiderivative = 4.70
d^3*(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*(1/5*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))))+3*d*e^2*(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...
Time = 0.46 (sec) , antiderivative size = 844, normalized size of antiderivative = 2.57 \[ \int (d+e x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\left [\frac {105 \, {\left (c^{5} d^{10} - 5 \, a c^{4} d^{8} e^{2} + 10 \, a^{2} c^{3} d^{6} e^{4} - 10 \, a^{3} c^{2} d^{4} e^{6} + 5 \, a^{4} c d^{2} e^{8} - a^{5} e^{10}\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 (384 \, c^{5} d^{5} e^{5} x^{4} + 105 \, c^{5} d^{9} e + 790 \, a c^{4} d^{7} e^{3} - 896 \, a^{2} c^{3} d^{5} e^{5} + 490 \, a^{3} c^{2} d^{3} e^{7} - 105 \, a^{4} c d e^{9} + 48 \, {\left (31 \, c^{5} d^{6} e^{4} + a c^{4} d^{4} e^{6}\right )} x^{3} + 8 \, {\left (263 \, c^{5} d^{7} e^{3} + 32 \, a c^{4} d^{5} e^{5} - 7 \, a^{2} c^{3} d^{3} e^{7}\right )} x^{2} + 2 \, {\left (605 \, c^{5} d^{8} e^{2} + 289 \, a c^{4} d^{6} e^{4} - 161 \, a^{2} c^{3} d^{4} e^{6} + 35 \, a^{3} c^{2} d^{2} e^{8}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{7680 \, c^{5} d^{5} e^{2}}, \frac {105 \, {\left (c^{5} d^{10} - 5 \, a c^{4} d^{8} e^{2} + 10 \, a^{2} c^{3} d^{6} e^{4} - 10 \, a^{3} c^{2} d^{4} e^{6} + 5 \, a^{4} c d^{2} e^{8} - a^{5} e^{10}\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 (384 \, c^{5} d^{5} e^{5} x^{4} + 105 \, c^{5} d^{9} e + 790 \, a c^{4} d^{7} e^{3} - 896 \, a^{2} c^{3} d^{5} e^{5} + 490 \, a^{3} c^{2} d^{3} e^{7} - 105 \, a^{4} c d e^{9} + 48 \, {\left (31 \, c^{5} d^{6} e^{4} + a c^{4} d^{4} e^{6}\right )} x^{3} + 8 \, {\left (263 \, c^{5} d^{7} e^{3} + 32 \, a c^{4} d^{5} e^{5} - 7 \, a^{2} c^{3} d^{3} e^{7}\right )} x^{2} + 2 \, {\left (605 \, c^{5} d^{8} e^{2} + 289 \, a c^{4} d^{6} e^{4} - 161 \, a^{2} c^{3} d^{4} e^{6} + 35 \, a^{3} c^{2} d^{2} e^{8}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{3840 \, c^{5} d^{5} e^{2}}\right ] \]
[1/7680*(105*(c^5*d^10 - 5*a*c^4*d^8*e^2 + 10*a^2*c^3*d^6*e^4 - 10*a^3*c^2 *d^4*e^6 + 5*a^4*c*d^2*e^8 - a^5*e^10)*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*(384*c^5*d^5*e^5*x^4 + 105*c^5*d^9*e + 790*a*c^4*d^7*e^3 - 896*a ^2*c^3*d^5*e^5 + 490*a^3*c^2*d^3*e^7 - 105*a^4*c*d*e^9 + 48*(31*c^5*d^6*e^ 4 + a*c^4*d^4*e^6)*x^3 + 8*(263*c^5*d^7*e^3 + 32*a*c^4*d^5*e^5 - 7*a^2*c^3 *d^3*e^7)*x^2 + 2*(605*c^5*d^8*e^2 + 289*a*c^4*d^6*e^4 - 161*a^2*c^3*d^4*e ^6 + 35*a^3*c^2*d^2*e^8)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/( c^5*d^5*e^2), 1/3840*(105*(c^5*d^10 - 5*a*c^4*d^8*e^2 + 10*a^2*c^3*d^6*e^4 - 10*a^3*c^2*d^4*e^6 + 5*a^4*c*d^2*e^8 - a^5*e^10)*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*(384*c^5*d^5*e^5*x^4 + 105*c^5*d^9*e + 790*a*c^4*d^7*e^3 - 896*a^2*c^3 *d^5*e^5 + 490*a^3*c^2*d^3*e^7 - 105*a^4*c*d*e^9 + 48*(31*c^5*d^6*e^4 + a* c^4*d^4*e^6)*x^3 + 8*(263*c^5*d^7*e^3 + 32*a*c^4*d^5*e^5 - 7*a^2*c^3*d^3*e ^7)*x^2 + 2*(605*c^5*d^8*e^2 + 289*a*c^4*d^6*e^4 - 161*a^2*c^3*d^4*e^6 + 3 5*a^3*c^2*d^2*e^8)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^5*d^ 5*e^2)]
Leaf count of result is larger than twice the leaf count of optimal. 1646 vs. \(2 (316) = 632\).
Time = 0.99 (sec) , antiderivative size = 1646, normalized size of antiderivative = 5.02 \[ \int (d+e x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\text {Too large to display} \]
Piecewise((sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))*(e**3*x**4/5 + x **3*(a*e**5 + 4*c*d**2*e**3 - e**3*(9*a*e**2/2 + 9*c*d**2/2)/5)/(4*c*d*e) + x**2*(16*a*d*e**4/5 + 6*c*d**3*e**2 - (7*a*e**2/2 + 7*c*d**2/2)*(a*e**5 + 4*c*d**2*e**3 - e**3*(9*a*e**2/2 + 9*c*d**2/2)/5)/(4*c*d*e))/(3*c*d*e) + x*(6*a*d**2*e**3 - 3*a*(a*e**5 + 4*c*d**2*e**3 - e**3*(9*a*e**2/2 + 9*c*d **2/2)/5)/(4*c) + 4*c*d**4*e - (5*a*e**2/2 + 5*c*d**2/2)*(16*a*d*e**4/5 + 6*c*d**3*e**2 - (7*a*e**2/2 + 7*c*d**2/2)*(a*e**5 + 4*c*d**2*e**3 - e**3*( 9*a*e**2/2 + 9*c*d**2/2)/5)/(4*c*d*e))/(3*c*d*e))/(2*c*d*e) + (4*a*d**3*e* *2 - 2*a*(16*a*d*e**4/5 + 6*c*d**3*e**2 - (7*a*e**2/2 + 7*c*d**2/2)*(a*e** 5 + 4*c*d**2*e**3 - e**3*(9*a*e**2/2 + 9*c*d**2/2)/5)/(4*c*d*e))/(3*c) + c *d**5 - (3*a*e**2/2 + 3*c*d**2/2)*(6*a*d**2*e**3 - 3*a*(a*e**5 + 4*c*d**2* e**3 - e**3*(9*a*e**2/2 + 9*c*d**2/2)/5)/(4*c) + 4*c*d**4*e - (5*a*e**2/2 + 5*c*d**2/2)*(16*a*d*e**4/5 + 6*c*d**3*e**2 - (7*a*e**2/2 + 7*c*d**2/2)*( a*e**5 + 4*c*d**2*e**3 - e**3*(9*a*e**2/2 + 9*c*d**2/2)/5)/(4*c*d*e))/(3*c *d*e))/(2*c*d*e))/(c*d*e)) + (a*d**4*e - a*(6*a*d**2*e**3 - 3*a*(a*e**5 + 4*c*d**2*e**3 - e**3*(9*a*e**2/2 + 9*c*d**2/2)/5)/(4*c) + 4*c*d**4*e - (5* a*e**2/2 + 5*c*d**2/2)*(16*a*d*e**4/5 + 6*c*d**3*e**2 - (7*a*e**2/2 + 7*c* d**2/2)*(a*e**5 + 4*c*d**2*e**3 - e**3*(9*a*e**2/2 + 9*c*d**2/2)/5)/(4*c*d *e))/(3*c*d*e))/(2*c) - (a*e**2 + c*d**2)*(4*a*d**3*e**2 - 2*a*(16*a*d*e** 4/5 + 6*c*d**3*e**2 - (7*a*e**2/2 + 7*c*d**2/2)*(a*e**5 + 4*c*d**2*e**3...
Exception generated. \[ \int (d+e x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\text {Exception raised: ValueError} \]
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 = 0.34 (sec) , antiderivative size = 400, normalized size of antiderivative = 1.22 \[ \int (d+e x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {1}{1920} \, \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e} {\left (2 \, {\left (4 \, {\left (6 \, {\left (8 \, e^{3} x + \frac {31 \, c^{4} d^{5} e^{6} + a c^{3} d^{3} e^{8}}{c^{4} d^{4} e^{4}}\right )} x + \frac {263 \, c^{4} d^{6} e^{5} + 32 \, a c^{3} d^{4} e^{7} - 7 \, a^{2} c^{2} d^{2} e^{9}}{c^{4} d^{4} e^{4}}\right )} x + \frac {605 \, c^{4} d^{7} e^{4} + 289 \, a c^{3} d^{5} e^{6} - 161 \, a^{2} c^{2} d^{3} e^{8} + 35 \, a^{3} c d e^{10}}{c^{4} d^{4} e^{4}}\right )} x + \frac {105 \, c^{4} d^{8} e^{3} + 790 \, a c^{3} d^{6} e^{5} - 896 \, a^{2} c^{2} d^{4} e^{7} + 490 \, a^{3} c d^{2} e^{9} - 105 \, a^{4} e^{11}}{c^{4} d^{4} e^{4}}\right )} + \frac {7 \, {\left (c^{5} d^{10} - 5 \, a c^{4} d^{8} e^{2} + 10 \, a^{2} c^{3} d^{6} e^{4} - 10 \, a^{3} c^{2} d^{4} e^{6} + 5 \, a^{4} c d^{2} e^{8} - a^{5} e^{10}\right )} \log \left ({\left | -c d^{2} - a e^{2} - 2 \, \sqrt {c d e} {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} \right |}\right )}{256 \, \sqrt {c d e} c^{4} d^{4} e} \]
1/1920*sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e)*(2*(4*(6*(8*e^3*x + (31 *c^4*d^5*e^6 + a*c^3*d^3*e^8)/(c^4*d^4*e^4))*x + (263*c^4*d^6*e^5 + 32*a*c ^3*d^4*e^7 - 7*a^2*c^2*d^2*e^9)/(c^4*d^4*e^4))*x + (605*c^4*d^7*e^4 + 289* a*c^3*d^5*e^6 - 161*a^2*c^2*d^3*e^8 + 35*a^3*c*d*e^10)/(c^4*d^4*e^4))*x + (105*c^4*d^8*e^3 + 790*a*c^3*d^6*e^5 - 896*a^2*c^2*d^4*e^7 + 490*a^3*c*d^2 *e^9 - 105*a^4*e^11)/(c^4*d^4*e^4)) + 7/256*(c^5*d^10 - 5*a*c^4*d^8*e^2 + 10*a^2*c^3*d^6*e^4 - 10*a^3*c^2*d^4*e^6 + 5*a^4*c*d^2*e^8 - a^5*e^10)*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^4*d^4*e)
Timed out. \[ \int (d+e x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\int {\left (d+e\,x\right )}^3\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e} \,d x \]