Integrand size = 37, antiderivative size = 414 \[ \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=\frac {45 \left (c d^2-a e^2\right )^6 \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16384 c^5 d^5 e^3}-\frac {15 \left (c d^2-a e^2\right )^4 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{2048 c^4 d^4 e^2}+\frac {3 \left (c d^2-a e^2\right )^2 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{128 c^3 d^3 e}+\frac {9 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{112 c^2 d^2}+\frac {(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{8 c d}-\frac {45 \left (c d^2-a e^2\right )^8 \text {arctanh}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{32768 c^{11/2} d^{11/2} e^{7/2}} \]
-15/2048*(-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)^(3/2)/c^4/d^4/e^2+3/128*(-a*e^2+c*d^2)^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)^(5/2)/c^3/d^3/e+9/112*(-a*e^2+c*d^2)*(a *d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/c^2/d^2+1/8*(e*x+d)*(a*d*e+(a*e^2+c* d^2)*x+c*d*e*x^2)^(7/2)/c/d-45/32768*(-a*e^2+c*d^2)^8*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^(7/2)+45/16384*(-a*e^2+c*d^2)^6*(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^3
Time = 1.30 (sec) , antiderivative size = 526, normalized size of antiderivative = 1.27 \[ \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=\frac {((a e+c d x) (d+e x))^{5/2} \left (\frac {\sqrt {c} \sqrt {d} \sqrt {e} \left (315 a^7 e^{14}-105 a^6 c d e^{12} (23 d+2 e x)+21 a^5 c^2 d^2 e^{10} \left (383 d^2+76 d e x+8 e^2 x^2\right )-3 a^4 c^3 d^3 e^8 \left (5053 d^3+1754 d^2 e x+424 d e^2 x^2+48 e^3 x^3\right )+a^3 c^4 d^4 e^6 \left (17609 d^4+9800 d^3 e x+4176 d^2 e^2 x^2+1088 d e^3 x^3+128 e^4 x^4\right )+3 a^2 c^5 d^5 e^4 \left (2681 d^5+31014 d^4 e x+66928 d^3 e^2 x^2+68320 d^2 e^3 x^3+34432 d e^4 x^4+6912 e^5 x^5\right )+3 a c^6 d^6 e^2 \left (-805 d^6+532 d^5 e x+32344 d^4 e^2 x^2+87744 d^3 e^3 x^3+99968 d^2 e^4 x^4+53760 d e^5 x^5+11264 e^6 x^6\right )+c^7 d^7 \left (315 d^7-210 d^6 e x+168 d^5 e^2 x^2+32624 d^4 e^3 x^3+98432 d^3 e^4 x^4+119040 d^2 e^5 x^5+66560 d e^6 x^6+14336 e^7 x^7\right )\right )}{(a e+c d x)^2 (d+e x)^2}-\frac {315 \left (c d^2-a e^2\right )^8 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )}{(a e+c d x)^{5/2} (d+e x)^{5/2}}\right )}{114688 c^{11/2} d^{11/2} e^{7/2}} \]
(((a*e + c*d*x)*(d + e*x))^(5/2)*((Sqrt[c]*Sqrt[d]*Sqrt[e]*(315*a^7*e^14 - 105*a^6*c*d*e^12*(23*d + 2*e*x) + 21*a^5*c^2*d^2*e^10*(383*d^2 + 76*d*e*x + 8*e^2*x^2) - 3*a^4*c^3*d^3*e^8*(5053*d^3 + 1754*d^2*e*x + 424*d*e^2*x^2 + 48*e^3*x^3) + a^3*c^4*d^4*e^6*(17609*d^4 + 9800*d^3*e*x + 4176*d^2*e^2* x^2 + 1088*d*e^3*x^3 + 128*e^4*x^4) + 3*a^2*c^5*d^5*e^4*(2681*d^5 + 31014* d^4*e*x + 66928*d^3*e^2*x^2 + 68320*d^2*e^3*x^3 + 34432*d*e^4*x^4 + 6912*e ^5*x^5) + 3*a*c^6*d^6*e^2*(-805*d^6 + 532*d^5*e*x + 32344*d^4*e^2*x^2 + 87 744*d^3*e^3*x^3 + 99968*d^2*e^4*x^4 + 53760*d*e^5*x^5 + 11264*e^6*x^6) + c ^7*d^7*(315*d^7 - 210*d^6*e*x + 168*d^5*e^2*x^2 + 32624*d^4*e^3*x^3 + 9843 2*d^3*e^4*x^4 + 119040*d^2*e^5*x^5 + 66560*d*e^6*x^6 + 14336*e^7*x^7)))/(( a*e + c*d*x)^2*(d + e*x)^2) - (315*(c*d^2 - a*e^2)^8*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))))/(114688*c^(11/2)*d^(11/2)*e^(7/2))
Time = 0.57 (sec) , antiderivative size = 458, normalized size of antiderivative = 1.11, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.189, Rules used = {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.
| \(\displaystyle \int (d+e x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2} \, dx\) |
| \(\Big \downarrow \) 1134 |
| \(\displaystyle \frac {9 \left (d^2-\frac {a e^2}{c}\right ) \int (d+e x) \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{5/2}dx}{16 d}+\frac {(d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{8 c d}\) |
| \(\Big \downarrow \) 1160 |
| \(\displaystyle \frac {9 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (d^2-\frac {a e^2}{c}\right ) \int \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{5/2}dx}{2 d}+\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{7 c d}\right )}{16 d}+\frac {(d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{8 c d}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {9 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{12 c d e}-\frac {5 \left (c d^2-a e^2\right )^2 \int \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}dx}{24 c d e}\right )}{2 d}+\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{7 c d}\right )}{16 d}+\frac {(d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{8 c d}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {9 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{12 c d e}-\frac {5 \left (c d^2-a e^2\right )^2 \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{8 c d e}-\frac {3 \left (c d^2-a e^2\right )^2 \int \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}dx}{16 c d e}\right )}{24 c d e}\right )}{2 d}+\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{7 c d}\right )}{16 d}+\frac {(d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{8 c d}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {9 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{12 c d e}-\frac {5 \left (c d^2-a e^2\right )^2 \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{8 c d e}-\frac {3 \left (c d^2-a e^2\right )^2 \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c d e}-\frac {\left (c d^2-a e^2\right )^2 \int \frac {1}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{8 c d e}\right )}{16 c d e}\right )}{24 c d e}\right )}{2 d}+\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{7 c d}\right )}{16 d}+\frac {(d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{8 c d}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {9 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{12 c d e}-\frac {5 \left (c d^2-a e^2\right )^2 \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{8 c d e}-\frac {3 \left (c d^2-a e^2\right )^2 \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c d e}-\frac {\left (c d^2-a e^2\right )^2 \int \frac {1}{4 c d e-\frac {\left (c d^2+2 c e x d+a e^2\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {c d^2+2 c e x d+a e^2}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}}{4 c d e}\right )}{16 c d e}\right )}{24 c d e}\right )}{2 d}+\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{7 c d}\right )}{16 d}+\frac {(d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{8 c d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {9 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{12 c d e}-\frac {5 \left (c d^2-a e^2\right )^2 \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{8 c d e}-\frac {3 \left (c d^2-a e^2\right )^2 \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c d e}-\frac {\left (c d^2-a e^2\right )^2 \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 c^{3/2} d^{3/2} e^{3/2}}\right )}{16 c d e}\right )}{24 c d e}\right )}{2 d}+\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{7 c d}\right )}{16 d}+\frac {(d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{8 c d}\) |
((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)))/(24*c*d*e)))/(2*d)))/ (16*d)
3.20.33.3.1 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. \(1559\) vs. \(2(376)=752\).
Time = 2.58 (sec) , antiderivative size = 1560, normalized size of antiderivative = 3.77
d^2*(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^2*(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/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...
Leaf count of result is larger than twice the leaf count of optimal. 753 vs. \(2 (376) = 752\).
Time = 0.50 (sec) , antiderivative size = 1520, normalized size of antiderivative = 3.67 \[ \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=\text {Too large to display} \]
[1/458752*(315*(c^8*d^16 - 8*a*c^7*d^14*e^2 + 28*a^2*c^6*d^12*e^4 - 56*a^3 *c^5*d^10*e^6 + 70*a^4*c^4*d^8*e^8 - 56*a^5*c^3*d^6*e^10 + 28*a^6*c^2*d^4* e^12 - 8*a^7*c*d^2*e^14 + a^8*e^16)*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*(14336*c^8*d^8*e^8*x^7 + 315*c^8*d^15*e - 2415*a*c^7*d^13*e^3 + 804 3*a^2*c^6*d^11*e^5 + 17609*a^3*c^5*d^9*e^7 - 15159*a^4*c^4*d^7*e^9 + 8043* a^5*c^3*d^5*e^11 - 2415*a^6*c^2*d^3*e^13 + 315*a^7*c*d*e^15 + 1024*(65*c^8 *d^9*e^7 + 33*a*c^7*d^7*e^9)*x^6 + 768*(155*c^8*d^10*e^6 + 210*a*c^7*d^8*e ^8 + 27*a^2*c^6*d^6*e^10)*x^5 + 128*(769*c^8*d^11*e^5 + 2343*a*c^7*d^9*e^7 + 807*a^2*c^6*d^7*e^9 + a^3*c^5*d^5*e^11)*x^4 + 16*(2039*c^8*d^12*e^4 + 1 6452*a*c^7*d^10*e^6 + 12810*a^2*c^6*d^8*e^8 + 68*a^3*c^5*d^6*e^10 - 9*a^4* c^4*d^4*e^12)*x^3 + 24*(7*c^8*d^13*e^3 + 4043*a*c^7*d^11*e^5 + 8366*a^2*c^ 6*d^9*e^7 + 174*a^3*c^5*d^7*e^9 - 53*a^4*c^4*d^5*e^11 + 7*a^5*c^3*d^3*e^13 )*x^2 - 2*(105*c^8*d^14*e^2 - 798*a*c^7*d^12*e^4 - 46521*a^2*c^6*d^10*e^6 - 4900*a^3*c^5*d^8*e^8 + 2631*a^4*c^4*d^6*e^10 - 798*a^5*c^3*d^4*e^12 + 10 5*a^6*c^2*d^2*e^14)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^6*d ^6*e^4), 1/229376*(315*(c^8*d^16 - 8*a*c^7*d^14*e^2 + 28*a^2*c^6*d^12*e^4 - 56*a^3*c^5*d^10*e^6 + 70*a^4*c^4*d^8*e^8 - 56*a^5*c^3*d^6*e^10 + 28*a^6* c^2*d^4*e^12 - 8*a^7*c*d^2*e^14 + a^8*e^16)*sqrt(-c*d*e)*arctan(1/2*sqr...
Leaf count of result is larger than twice the leaf count of optimal. 8918 vs. \(2 (406) = 812\).
Time = 8.45 (sec) , antiderivative size = 8918, normalized size of antiderivative = 21.54 \[ \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=\text {Too large to display} \]
Piecewise((sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))*(c**2*d**2*e**4* x**7/8 + x**6*(3*a*c**2*d**2*e**6 + 5*c**3*d**4*e**4 - c**2*d**2*e**4*(15* a*e**2/2 + 15*c*d**2/2)/8)/(7*c*d*e) + x**5*(3*a**2*c*d*e**7 + 113*a*c**2* d**3*e**5/8 + 10*c**3*d**5*e**3 - (13*a*e**2/2 + 13*c*d**2/2)*(3*a*c**2*d* *2*e**6 + 5*c**3*d**4*e**4 - c**2*d**2*e**4*(15*a*e**2/2 + 15*c*d**2/2)/8) /(7*c*d*e))/(6*c*d*e) + x**4*(a**3*e**8 + 15*a**2*c*d**2*e**6 + 30*a*c**2* d**4*e**4 - 6*a*(3*a*c**2*d**2*e**6 + 5*c**3*d**4*e**4 - c**2*d**2*e**4*(1 5*a*e**2/2 + 15*c*d**2/2)/8)/(7*c) + 10*c**3*d**6*e**2 - (11*a*e**2/2 + 11 *c*d**2/2)*(3*a**2*c*d*e**7 + 113*a*c**2*d**3*e**5/8 + 10*c**3*d**5*e**3 - (13*a*e**2/2 + 13*c*d**2/2)*(3*a*c**2*d**2*e**6 + 5*c**3*d**4*e**4 - c**2 *d**2*e**4*(15*a*e**2/2 + 15*c*d**2/2)/8)/(7*c*d*e))/(6*c*d*e))/(5*c*d*e) + x**3*(5*a**3*d*e**7 + 30*a**2*c*d**3*e**5 + 30*a*c**2*d**5*e**3 - 5*a*(3 *a**2*c*d*e**7 + 113*a*c**2*d**3*e**5/8 + 10*c**3*d**5*e**3 - (13*a*e**2/2 + 13*c*d**2/2)*(3*a*c**2*d**2*e**6 + 5*c**3*d**4*e**4 - c**2*d**2*e**4*(1 5*a*e**2/2 + 15*c*d**2/2)/8)/(7*c*d*e))/(6*c) + 5*c**3*d**7*e - (9*a*e**2/ 2 + 9*c*d**2/2)*(a**3*e**8 + 15*a**2*c*d**2*e**6 + 30*a*c**2*d**4*e**4 - 6 *a*(3*a*c**2*d**2*e**6 + 5*c**3*d**4*e**4 - c**2*d**2*e**4*(15*a*e**2/2 + 15*c*d**2/2)/8)/(7*c) + 10*c**3*d**6*e**2 - (11*a*e**2/2 + 11*c*d**2/2)*(3 *a**2*c*d*e**7 + 113*a*c**2*d**3*e**5/8 + 10*c**3*d**5*e**3 - (13*a*e**2/2 + 13*c*d**2/2)*(3*a*c**2*d**2*e**6 + 5*c**3*d**4*e**4 - c**2*d**2*e**4...
Exception generated. \[ \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=\text {Exception raised: ValueError} \]
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. 786 vs. \(2 (376) = 752\).
Time = 0.37 (sec) , antiderivative size = 786, normalized size of antiderivative = 1.90 \[ \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=\frac {1}{114688} \, \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 \, c^{2} d^{2} e^{4} x + \frac {65 \, c^{9} d^{10} e^{10} + 33 \, a c^{8} d^{8} e^{12}}{c^{7} d^{7} e^{7}}\right )} x + \frac {3 \, {\left (155 \, c^{9} d^{11} e^{9} + 210 \, a c^{8} d^{9} e^{11} + 27 \, a^{2} c^{7} d^{7} e^{13}\right )}}{c^{7} d^{7} e^{7}}\right )} x + \frac {769 \, c^{9} d^{12} e^{8} + 2343 \, a c^{8} d^{10} e^{10} + 807 \, a^{2} c^{7} d^{8} e^{12} + a^{3} c^{6} d^{6} e^{14}}{c^{7} d^{7} e^{7}}\right )} x + \frac {2039 \, c^{9} d^{13} e^{7} + 16452 \, a c^{8} d^{11} e^{9} + 12810 \, a^{2} c^{7} d^{9} e^{11} + 68 \, a^{3} c^{6} d^{7} e^{13} - 9 \, a^{4} c^{5} d^{5} e^{15}}{c^{7} d^{7} e^{7}}\right )} x + \frac {3 \, {\left (7 \, c^{9} d^{14} e^{6} + 4043 \, a c^{8} d^{12} e^{8} + 8366 \, a^{2} c^{7} d^{10} e^{10} + 174 \, a^{3} c^{6} d^{8} e^{12} - 53 \, a^{4} c^{5} d^{6} e^{14} + 7 \, a^{5} c^{4} d^{4} e^{16}\right )}}{c^{7} d^{7} e^{7}}\right )} x - \frac {105 \, c^{9} d^{15} e^{5} - 798 \, a c^{8} d^{13} e^{7} - 46521 \, a^{2} c^{7} d^{11} e^{9} - 4900 \, a^{3} c^{6} d^{9} e^{11} + 2631 \, a^{4} c^{5} d^{7} e^{13} - 798 \, a^{5} c^{4} d^{5} e^{15} + 105 \, a^{6} c^{3} d^{3} e^{17}}{c^{7} d^{7} e^{7}}\right )} x + \frac {315 \, c^{9} d^{16} e^{4} - 2415 \, a c^{8} d^{14} e^{6} + 8043 \, a^{2} c^{7} d^{12} e^{8} + 17609 \, a^{3} c^{6} d^{10} e^{10} - 15159 \, a^{4} c^{5} d^{8} e^{12} + 8043 \, a^{5} c^{4} d^{6} e^{14} - 2415 \, a^{6} c^{3} d^{4} e^{16} + 315 \, a^{7} c^{2} d^{2} e^{18}}{c^{7} d^{7} e^{7}}\right )} + \frac {45 \, {\left (c^{8} d^{16} - 8 \, a c^{7} d^{14} e^{2} + 28 \, a^{2} c^{6} d^{12} e^{4} - 56 \, a^{3} c^{5} d^{10} e^{6} + 70 \, a^{4} c^{4} d^{8} e^{8} - 56 \, a^{5} c^{3} d^{6} e^{10} + 28 \, a^{6} c^{2} d^{4} e^{12} - 8 \, a^{7} c d^{2} e^{14} + a^{8} e^{16}\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 )}{32768 \, \sqrt {c d e} c^{5} d^{5} e^{3}} \]
1/114688*sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e)*(2*(4*(2*(8*(2*(4*(14 *c^2*d^2*e^4*x + (65*c^9*d^10*e^10 + 33*a*c^8*d^8*e^12)/(c^7*d^7*e^7))*x + 3*(155*c^9*d^11*e^9 + 210*a*c^8*d^9*e^11 + 27*a^2*c^7*d^7*e^13)/(c^7*d^7* e^7))*x + (769*c^9*d^12*e^8 + 2343*a*c^8*d^10*e^10 + 807*a^2*c^7*d^8*e^12 + a^3*c^6*d^6*e^14)/(c^7*d^7*e^7))*x + (2039*c^9*d^13*e^7 + 16452*a*c^8*d^ 11*e^9 + 12810*a^2*c^7*d^9*e^11 + 68*a^3*c^6*d^7*e^13 - 9*a^4*c^5*d^5*e^15 )/(c^7*d^7*e^7))*x + 3*(7*c^9*d^14*e^6 + 4043*a*c^8*d^12*e^8 + 8366*a^2*c^ 7*d^10*e^10 + 174*a^3*c^6*d^8*e^12 - 53*a^4*c^5*d^6*e^14 + 7*a^5*c^4*d^4*e ^16)/(c^7*d^7*e^7))*x - (105*c^9*d^15*e^5 - 798*a*c^8*d^13*e^7 - 46521*a^2 *c^7*d^11*e^9 - 4900*a^3*c^6*d^9*e^11 + 2631*a^4*c^5*d^7*e^13 - 798*a^5*c^ 4*d^5*e^15 + 105*a^6*c^3*d^3*e^17)/(c^7*d^7*e^7))*x + (315*c^9*d^16*e^4 - 2415*a*c^8*d^14*e^6 + 8043*a^2*c^7*d^12*e^8 + 17609*a^3*c^6*d^10*e^10 - 15 159*a^4*c^5*d^8*e^12 + 8043*a^5*c^4*d^6*e^14 - 2415*a^6*c^3*d^4*e^16 + 315 *a^7*c^2*d^2*e^18)/(c^7*d^7*e^7)) + 45/32768*(c^8*d^16 - 8*a*c^7*d^14*e^2 + 28*a^2*c^6*d^12*e^4 - 56*a^3*c^5*d^10*e^6 + 70*a^4*c^4*d^8*e^8 - 56*a^5* c^3*d^6*e^10 + 28*a^6*c^2*d^4*e^12 - 8*a^7*c*d^2*e^14 + a^8*e^16)*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^3)
Timed out. \[ \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=\int {\left (d+e\,x\right )}^2\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2} \,d x \]