Integrand size = 38, antiderivative size = 381 \[ \int \frac {x \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{d+e x} \, dx=-\frac {\left (c d^2-a e^2\right )^3 \left (7 c d^2+5 a e^2\right ) \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{512 c^3 d^3 e^4}+\frac {\left (c d^2-a e^2\right ) \left (7 c d^2+5 a e^2\right ) \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{192 c^2 d^2 e^3}-\frac {1}{60} \left (\frac {5 a}{c d}+\frac {7 d}{e^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}+\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{6 c d e (d+e x)}+\frac {\left (c d^2-a e^2\right )^5 \left (7 c d^2+5 a e^2\right ) \text {arctanh}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{1024 c^{7/2} d^{7/2} e^{9/2}} \]
1/192*(-a*e^2+c*d^2)*(5*a*e^2+7*c*d^2)*(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^2/d^2/e^3-1/60*(5*a/c/d+7*d/e^2)*(a*d*e+(a* e^2+c*d^2)*x+c*d*e*x^2)^(5/2)+1/6*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/ c/d/e/(e*x+d)+1/1024*(-a*e^2+c*d^2)^5*(5*a*e^2+7*c*d^2)*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^(7/2)/d^(7/2)/e^(9/2)-1/512*(-a*e^2+c*d^2)^3*(5*a*e^2+7*c*d^2)* (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^3/d^3/e^ 4
Time = 0.89 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.02 \[ \int \frac {x \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{d+e x} \, dx=\frac {\left (c d^2-a e^2\right )^5 ((a e+c d x) (d+e x))^{3/2} \left (\frac {\sqrt {c} \sqrt {d} \sqrt {e} \left (75 a^5 e^{10}-5 a^4 c d e^8 (49 d+10 e x)+10 a^3 c^2 d^2 e^6 \left (15 d^2+16 d e x+4 e^2 x^2\right )-6 a^2 c^3 d^3 e^4 \left (91 d^3-58 d^2 e x-564 d e^2 x^2-360 e^3 x^3\right )+a c^4 d^4 e^2 \left (415 d^4-272 d^3 e x+216 d^2 e^2 x^2+4448 d e^3 x^3+3200 e^4 x^4\right )+c^5 d^5 \left (-105 d^5+70 d^4 e x-56 d^3 e^2 x^2+48 d^2 e^3 x^3+1664 d e^4 x^4+1280 e^5 x^5\right )\right )}{\left (c d^2-a e^2\right )^5 (a e+c d x) (d+e x)}+\frac {15 \left (7 c d^2+5 a e^2\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )}{(a e+c d x)^{3/2} (d+e x)^{3/2}}\right )}{7680 c^{7/2} d^{7/2} e^{9/2}} \]
((c*d^2 - a*e^2)^5*((a*e + c*d*x)*(d + e*x))^(3/2)*((Sqrt[c]*Sqrt[d]*Sqrt[ e]*(75*a^5*e^10 - 5*a^4*c*d*e^8*(49*d + 10*e*x) + 10*a^3*c^2*d^2*e^6*(15*d ^2 + 16*d*e*x + 4*e^2*x^2) - 6*a^2*c^3*d^3*e^4*(91*d^3 - 58*d^2*e*x - 564* d*e^2*x^2 - 360*e^3*x^3) + a*c^4*d^4*e^2*(415*d^4 - 272*d^3*e*x + 216*d^2* e^2*x^2 + 4448*d*e^3*x^3 + 3200*e^4*x^4) + c^5*d^5*(-105*d^5 + 70*d^4*e*x - 56*d^3*e^2*x^2 + 48*d^2*e^3*x^3 + 1664*d*e^4*x^4 + 1280*e^5*x^5)))/((c*d ^2 - a*e^2)^5*(a*e + c*d*x)*(d + e*x)) + (15*(7*c*d^2 + 5*a*e^2)*ArcTanh[( Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])])/((a*e + c*d*x )^(3/2)*(d + e*x)^(3/2))))/(7680*c^(7/2)*d^(7/2)*e^(9/2))
Time = 0.48 (sec) , antiderivative size = 342, normalized size of antiderivative = 0.90, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {1215, 1225, 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.
| \(\displaystyle \int \frac {x \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{d+e x} \, dx\) |
| \(\Big \downarrow \) 1215 |
| \(\displaystyle \int x (a e+c d x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}dx\) |
| \(\Big \downarrow \) 1225 |
| \(\displaystyle -\frac {\left (\frac {5 a^2 e^2}{c}+2 a d^2-\frac {7 c d^4}{e^2}\right ) \int \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}dx}{24 d}-\frac {\left (-5 a e^2+7 c d^2-10 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{60 c d e^2}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle -\frac {\left (\frac {5 a^2 e^2}{c}+2 a d^2-\frac {7 c d^4}{e^2}\right ) \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{8 c d e}-\frac {3 \left (c d^2-a e^2\right )^2 \int \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}dx}{16 c d e}\right )}{24 d}-\frac {\left (-5 a e^2+7 c d^2-10 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{60 c d e^2}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle -\frac {\left (\frac {5 a^2 e^2}{c}+2 a d^2-\frac {7 c d^4}{e^2}\right ) \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{8 c d e}-\frac {3 \left (c d^2-a e^2\right )^2 \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c d e}-\frac {\left (c d^2-a e^2\right )^2 \int \frac {1}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{8 c d e}\right )}{16 c d e}\right )}{24 d}-\frac {\left (-5 a e^2+7 c d^2-10 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{60 c d e^2}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle -\frac {\left (\frac {5 a^2 e^2}{c}+2 a d^2-\frac {7 c d^4}{e^2}\right ) \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{8 c d e}-\frac {3 \left (c d^2-a e^2\right )^2 \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c d e}-\frac {\left (c d^2-a e^2\right )^2 \int \frac {1}{4 c d e-\frac {\left (c d^2+2 c e x d+a e^2\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {c d^2+2 c e x d+a e^2}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}}{4 c d e}\right )}{16 c d e}\right )}{24 d}-\frac {\left (-5 a e^2+7 c d^2-10 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{60 c d e^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {\left (\frac {5 a^2 e^2}{c}+2 a d^2-\frac {7 c d^4}{e^2}\right ) \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{8 c d e}-\frac {3 \left (c d^2-a e^2\right )^2 \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c d e}-\frac {\left (c d^2-a e^2\right )^2 \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 c^{3/2} d^{3/2} e^{3/2}}\right )}{16 c d e}\right )}{24 d}-\frac {\left (-5 a e^2+7 c d^2-10 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{60 c d e^2}\) |
-1/60*((7*c*d^2 - 5*a*e^2 - 10*c*d*e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e *x^2)^(5/2))/(c*d*e^2) - ((2*a*d^2 - (7*c*d^4)/e^2 + (5*a^2*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)^(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)))/(24*d)
3.5.59.3.1 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[(((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_))/( (d_) + (e_.)*(x_)), x_Symbol] :> Int[(a/d + c*(x/e))*(f + g*x)^n*(a + b*x + c*x^2)^(p - 1), x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0]
Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*( x_)^2)^(p_), x_Symbol] :> Simp[(-(b*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 2*c*e*g*(p + 1)*x))*((a + b*x + c*x^2)^(p + 1)/(2*c^2*(p + 1)*(2*p + 3))), x] + Simp[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(2*c^2*(2*p + 3)) Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c , d, e, f, g, p}, x] && !LeQ[p, -1]
Time = 0.55 (sec) , antiderivative size = 673, normalized size of antiderivative = 1.77
| method | result | size |
| default | \(\frac {\frac {\left (2 c d e x +e^{2} a +c \,d^{2}\right ) {\left (a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}\right )}^{\frac {5}{2}}}{12 c d e}+\frac {5 \left (4 a c \,d^{2} e^{2}-\left (e^{2} a +c \,d^{2}\right )^{2}\right ) \left (\frac {\left (2 c d e x +e^{2} a +c \,d^{2}\right ) {\left (a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}\right )}^{\frac {3}{2}}}{8 c d e}+\frac {3 \left (4 a c \,d^{2} e^{2}-\left (e^{2} a +c \,d^{2}\right )^{2}\right ) \left (\frac {\left (2 c d e x +e^{2} a +c \,d^{2}\right ) \sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}}{4 c d e}+\frac {\left (4 a c \,d^{2} e^{2}-\left (e^{2} a +c \,d^{2}\right )^{2}\right ) \ln \left (\frac {\frac {1}{2} e^{2} a +\frac {1}{2} c \,d^{2}+c d e x}{\sqrt {c d e}}+\sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}\right )}{8 c d e \sqrt {c d e}}\right )}{16 c d e}\right )}{24 c d e}}{e}-\frac {d \left (\frac {\left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{5}+\frac {\left (e^{2} a -c \,d^{2}\right ) \left (\frac {\left (2 c d e \left (x +\frac {d}{e}\right )+e^{2} a -c \,d^{2}\right ) \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{8 c d e}-\frac {3 \left (e^{2} a -c \,d^{2}\right )^{2} \left (\frac {\left (2 c d e \left (x +\frac {d}{e}\right )+e^{2} a -c \,d^{2}\right ) \sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}{4 c d e}-\frac {\left (e^{2} a -c \,d^{2}\right )^{2} \ln \left (\frac {\frac {e^{2} a}{2}-\frac {c \,d^{2}}{2}+c d e \left (x +\frac {d}{e}\right )}{\sqrt {c d e}}+\sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}\right )}{8 c d e \sqrt {c d e}}\right )}{16 c d e}\right )}{2}\right )}{e^{2}}\) | \(673\) |
1/e*(1/12*(2*c*d*e*x+a*e^2+c*d^2)/c/d/e*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^ (5/2)+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)/c/d/e*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+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)/c/d/e*(a*d*e+(a*e^2+c*d^2) *x+c*d*e*x^2)^(1/2)+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+c*d*e*x)/(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))))-d/e^2*(1/5*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(5/2 )+1/2*(a*e^2-c*d^2)*(1/8*(2*c*d*e*(x+d/e)+e^2*a-c*d^2)/c/d/e*(c*d*e*(x+d/e )^2+(a*e^2-c*d^2)*(x+d/e))^(3/2)-3/16*(a*e^2-c*d^2)^2/c/d/e*(1/4*(2*c*d*e* (x+d/e)+e^2*a-c*d^2)/c/d/e*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)-1 /8*(a*e^2-c*d^2)^2/c/d/e*ln((1/2*e^2*a-1/2*c*d^2+c*d*e*(x+d/e))/(c*d*e)^(1 /2)+(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2))/(c*d*e)^(1/2))))
Time = 0.36 (sec) , antiderivative size = 1046, normalized size of antiderivative = 2.75 \[ \int \frac {x \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{d+e x} \, dx=\left [-\frac {15 \, {\left (7 \, c^{6} d^{12} - 30 \, a c^{5} d^{10} e^{2} + 45 \, 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} + 18 \, a^{5} c d^{2} e^{10} - 5 \, 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} - 105 \, c^{6} d^{11} e + 415 \, a c^{5} d^{9} e^{3} - 546 \, a^{2} c^{4} d^{7} e^{5} + 150 \, a^{3} c^{3} d^{5} e^{7} - 245 \, a^{4} c^{2} d^{3} e^{9} + 75 \, a^{5} c d e^{11} + 128 \, {\left (13 \, c^{6} d^{7} e^{5} + 25 \, a c^{5} d^{5} e^{7}\right )} x^{4} + 16 \, {\left (3 \, c^{6} d^{8} e^{4} + 278 \, a c^{5} d^{6} e^{6} + 135 \, a^{2} c^{4} d^{4} e^{8}\right )} x^{3} - 8 \, {\left (7 \, c^{6} d^{9} e^{3} - 27 \, a c^{5} d^{7} e^{5} - 423 \, a^{2} c^{4} d^{5} e^{7} - 5 \, a^{3} c^{3} d^{3} e^{9}\right )} x^{2} + 2 \, {\left (35 \, c^{6} d^{10} e^{2} - 136 \, a c^{5} d^{8} e^{4} + 174 \, a^{2} c^{4} d^{6} e^{6} + 80 \, a^{3} c^{3} d^{4} e^{8} - 25 \, 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^{4} d^{4} e^{5}}, -\frac {15 \, {\left (7 \, c^{6} d^{12} - 30 \, a c^{5} d^{10} e^{2} + 45 \, 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} + 18 \, a^{5} c d^{2} e^{10} - 5 \, 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} - 105 \, c^{6} d^{11} e + 415 \, a c^{5} d^{9} e^{3} - 546 \, a^{2} c^{4} d^{7} e^{5} + 150 \, a^{3} c^{3} d^{5} e^{7} - 245 \, a^{4} c^{2} d^{3} e^{9} + 75 \, a^{5} c d e^{11} + 128 \, {\left (13 \, c^{6} d^{7} e^{5} + 25 \, a c^{5} d^{5} e^{7}\right )} x^{4} + 16 \, {\left (3 \, c^{6} d^{8} e^{4} + 278 \, a c^{5} d^{6} e^{6} + 135 \, a^{2} c^{4} d^{4} e^{8}\right )} x^{3} - 8 \, {\left (7 \, c^{6} d^{9} e^{3} - 27 \, a c^{5} d^{7} e^{5} - 423 \, a^{2} c^{4} d^{5} e^{7} - 5 \, a^{3} c^{3} d^{3} e^{9}\right )} x^{2} + 2 \, {\left (35 \, c^{6} d^{10} e^{2} - 136 \, a c^{5} d^{8} e^{4} + 174 \, a^{2} c^{4} d^{6} e^{6} + 80 \, a^{3} c^{3} d^{4} e^{8} - 25 \, 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^{4} d^{4} e^{5}}\right ] \]
[-1/30720*(15*(7*c^6*d^12 - 30*a*c^5*d^10*e^2 + 45*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 + 18*a^5*c*d^2*e^10 - 5*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 - 105*c^6*d^ 11*e + 415*a*c^5*d^9*e^3 - 546*a^2*c^4*d^7*e^5 + 150*a^3*c^3*d^5*e^7 - 245 *a^4*c^2*d^3*e^9 + 75*a^5*c*d*e^11 + 128*(13*c^6*d^7*e^5 + 25*a*c^5*d^5*e^ 7)*x^4 + 16*(3*c^6*d^8*e^4 + 278*a*c^5*d^6*e^6 + 135*a^2*c^4*d^4*e^8)*x^3 - 8*(7*c^6*d^9*e^3 - 27*a*c^5*d^7*e^5 - 423*a^2*c^4*d^5*e^7 - 5*a^3*c^3*d^ 3*e^9)*x^2 + 2*(35*c^6*d^10*e^2 - 136*a*c^5*d^8*e^4 + 174*a^2*c^4*d^6*e^6 + 80*a^3*c^3*d^4*e^8 - 25*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^4*d^4*e^5), -1/15360*(15*(7*c^6*d^12 - 30*a*c^5*d^10* e^2 + 45*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 + 18*a^ 5*c*d^2*e^10 - 5*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 - 105*c^6*d^11*e + 415*a*c^5*d^9*e^3 - 546*a^2*c^4*d^7*e^5 + 150*a^3*c^3* d^5*e^7 - 245*a^4*c^2*d^3*e^9 + 75*a^5*c*d*e^11 + 128*(13*c^6*d^7*e^5 + 25 *a*c^5*d^5*e^7)*x^4 + 16*(3*c^6*d^8*e^4 + 278*a*c^5*d^6*e^6 + 135*a^2*c^4* d^4*e^8)*x^3 - 8*(7*c^6*d^9*e^3 - 27*a*c^5*d^7*e^5 - 423*a^2*c^4*d^5*e^...
Timed out. \[ \int \frac {x \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{d+e x} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {x \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{d+e x} \, dx=\text {Exception raised: ValueError} \]
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 = 0.38 (sec) , antiderivative size = 526, normalized size of antiderivative = 1.38 \[ \int \frac {x \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{d+e x} \, dx=\frac {1}{7680} \, \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, c^{2} d^{2} e x + \frac {13 \, c^{7} d^{8} e^{5} + 25 \, a c^{6} d^{6} e^{7}}{c^{5} d^{5} e^{5}}\right )} x + \frac {3 \, c^{7} d^{9} e^{4} + 278 \, a c^{6} d^{7} e^{6} + 135 \, a^{2} c^{5} d^{5} e^{8}}{c^{5} d^{5} e^{5}}\right )} x - \frac {7 \, c^{7} d^{10} e^{3} - 27 \, a c^{6} d^{8} e^{5} - 423 \, a^{2} c^{5} d^{6} e^{7} - 5 \, a^{3} c^{4} d^{4} e^{9}}{c^{5} d^{5} e^{5}}\right )} x + \frac {35 \, c^{7} d^{11} e^{2} - 136 \, a c^{6} d^{9} e^{4} + 174 \, a^{2} c^{5} d^{7} e^{6} + 80 \, a^{3} c^{4} d^{5} e^{8} - 25 \, a^{4} c^{3} d^{3} e^{10}}{c^{5} d^{5} e^{5}}\right )} x - \frac {105 \, c^{7} d^{12} e - 415 \, a c^{6} d^{10} e^{3} + 546 \, a^{2} c^{5} d^{8} e^{5} - 150 \, a^{3} c^{4} d^{6} e^{7} + 245 \, a^{4} c^{3} d^{4} e^{9} - 75 \, a^{5} c^{2} d^{2} e^{11}}{c^{5} d^{5} e^{5}}\right )} - \frac {{\left (7 \, c^{6} d^{12} - 30 \, a c^{5} d^{10} e^{2} + 45 \, a^{2} c^{4} d^{8} e^{4} - 20 \, a^{3} c^{3} d^{6} e^{6} - 15 \, a^{4} c^{2} d^{4} e^{8} + 18 \, a^{5} c d^{2} e^{10} - 5 \, a^{6} e^{12}\right )} \log \left ({\left | -c d^{2} - a e^{2} - 2 \, \sqrt {c d e} {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} \right |}\right )}{1024 \, \sqrt {c d e} c^{3} d^{3} e^{4}} \]
1/7680*sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e)*(2*(4*(2*(8*(10*c^2*d^2 *e*x + (13*c^7*d^8*e^5 + 25*a*c^6*d^6*e^7)/(c^5*d^5*e^5))*x + (3*c^7*d^9*e ^4 + 278*a*c^6*d^7*e^6 + 135*a^2*c^5*d^5*e^8)/(c^5*d^5*e^5))*x - (7*c^7*d^ 10*e^3 - 27*a*c^6*d^8*e^5 - 423*a^2*c^5*d^6*e^7 - 5*a^3*c^4*d^4*e^9)/(c^5* d^5*e^5))*x + (35*c^7*d^11*e^2 - 136*a*c^6*d^9*e^4 + 174*a^2*c^5*d^7*e^6 + 80*a^3*c^4*d^5*e^8 - 25*a^4*c^3*d^3*e^10)/(c^5*d^5*e^5))*x - (105*c^7*d^1 2*e - 415*a*c^6*d^10*e^3 + 546*a^2*c^5*d^8*e^5 - 150*a^3*c^4*d^6*e^7 + 245 *a^4*c^3*d^4*e^9 - 75*a^5*c^2*d^2*e^11)/(c^5*d^5*e^5)) - 1/1024*(7*c^6*d^1 2 - 30*a*c^5*d^10*e^2 + 45*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 + 18*a^5*c*d^2*e^10 - 5*a^6*e^12)*log(abs(-c*d^2 - a*e^2 - 2*sq rt(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^3*d^3*e^4)
Timed out. \[ \int \frac {x \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{d+e x} \, dx=\int \frac {x\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}}{d+e\,x} \,d x \]