Integrand size = 37, antiderivative size = 210 \[ \int \frac {(d+e x)^{13/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\frac {9 e \left (c d^2-a e^2\right )^3 \sqrt {d+e x}}{c^5 d^5}+\frac {3 e \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}{c^4 d^4}+\frac {9 e \left (c d^2-a e^2\right ) (d+e x)^{5/2}}{5 c^3 d^3}+\frac {9 e (d+e x)^{7/2}}{7 c^2 d^2}-\frac {(d+e x)^{9/2}}{c d (a e+c d x)}-\frac {9 e \left (c d^2-a e^2\right )^{7/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{c^{11/2} d^{11/2}} \] Output:
9*e*(-a*e^2+c*d^2)^3*(e*x+d)^(1/2)/c^5/d^5+3*e*(-a*e^2+c*d^2)^2*(e*x+d)^(3 /2)/c^4/d^4+9/5*e*(-a*e^2+c*d^2)*(e*x+d)^(5/2)/c^3/d^3+9/7*e*(e*x+d)^(7/2) /c^2/d^2-(e*x+d)^(9/2)/c/d/(c*d*x+a*e)-9*e*(-a*e^2+c*d^2)^(7/2)*arctanh(c^ (1/2)*d^(1/2)*(e*x+d)^(1/2)/(-a*e^2+c*d^2)^(1/2))/c^(11/2)/d^(11/2)
Time = 0.53 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.17 \[ \int \frac {(d+e x)^{13/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=-\frac {\sqrt {d+e x} \left (315 a^4 e^8-210 a^3 c d e^6 (5 d-e x)+42 a^2 c^2 d^2 e^4 \left (29 d^2-17 d e x-e^2 x^2\right )+6 a c^3 d^3 e^2 \left (-88 d^3+142 d^2 e x+23 d e^2 x^2+3 e^3 x^3\right )+c^4 d^4 \left (35 d^4-388 d^3 e x-156 d^2 e^2 x^2-58 d e^3 x^3-10 e^4 x^4\right )\right )}{35 c^5 d^5 (a e+c d x)}+\frac {9 e \left (-c d^2+a e^2\right )^{7/2} \arctan \left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {-c d^2+a e^2}}\right )}{c^{11/2} d^{11/2}} \] Input:
Integrate[(d + e*x)^(13/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]
Output:
-1/35*(Sqrt[d + e*x]*(315*a^4*e^8 - 210*a^3*c*d*e^6*(5*d - e*x) + 42*a^2*c ^2*d^2*e^4*(29*d^2 - 17*d*e*x - e^2*x^2) + 6*a*c^3*d^3*e^2*(-88*d^3 + 142* d^2*e*x + 23*d*e^2*x^2 + 3*e^3*x^3) + c^4*d^4*(35*d^4 - 388*d^3*e*x - 156* d^2*e^2*x^2 - 58*d*e^3*x^3 - 10*e^4*x^4)))/(c^5*d^5*(a*e + c*d*x)) + (9*e* (-(c*d^2) + a*e^2)^(7/2)*ArcTan[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/Sqrt[-(c*d ^2) + a*e^2]])/(c^(11/2)*d^(11/2))
Time = 0.51 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.17, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.216, Rules used = {1121, 51, 60, 60, 60, 60, 73, 221}
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.
| \(\displaystyle \int \frac {(d+e x)^{13/2}}{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 1121 |
| \(\displaystyle \int \frac {(d+e x)^{9/2}}{(a e+c d x)^2}dx\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {9 e \int \frac {(d+e x)^{7/2}}{a e+c d x}dx}{2 c d}-\frac {(d+e x)^{9/2}}{c d (a e+c d x)}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {9 e \left (\frac {\left (d^2-\frac {a e^2}{c}\right ) \int \frac {(d+e x)^{5/2}}{a e+c d x}dx}{d}+\frac {2 (d+e x)^{7/2}}{7 c d}\right )}{2 c d}-\frac {(d+e x)^{9/2}}{c d (a e+c d x)}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {9 e \left (\frac {\left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (d^2-\frac {a e^2}{c}\right ) \int \frac {(d+e x)^{3/2}}{a e+c d x}dx}{d}+\frac {2 (d+e x)^{5/2}}{5 c d}\right )}{d}+\frac {2 (d+e x)^{7/2}}{7 c d}\right )}{2 c d}-\frac {(d+e x)^{9/2}}{c d (a e+c d x)}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {9 e \left (\frac {\left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (d^2-\frac {a e^2}{c}\right ) \int \frac {\sqrt {d+e x}}{a e+c d x}dx}{d}+\frac {2 (d+e x)^{3/2}}{3 c d}\right )}{d}+\frac {2 (d+e x)^{5/2}}{5 c d}\right )}{d}+\frac {2 (d+e x)^{7/2}}{7 c d}\right )}{2 c d}-\frac {(d+e x)^{9/2}}{c d (a e+c d x)}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {9 e \left (\frac {\left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (d^2-\frac {a e^2}{c}\right ) \int \frac {1}{(a e+c d x) \sqrt {d+e x}}dx}{d}+\frac {2 \sqrt {d+e x}}{c d}\right )}{d}+\frac {2 (d+e x)^{3/2}}{3 c d}\right )}{d}+\frac {2 (d+e x)^{5/2}}{5 c d}\right )}{d}+\frac {2 (d+e x)^{7/2}}{7 c d}\right )}{2 c d}-\frac {(d+e x)^{9/2}}{c d (a e+c d x)}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {9 e \left (\frac {\left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (d^2-\frac {a e^2}{c}\right ) \left (\frac {2 \left (d^2-\frac {a e^2}{c}\right ) \int \frac {1}{-\frac {c d^2}{e}+\frac {c (d+e x) d}{e}+a e}d\sqrt {d+e x}}{d e}+\frac {2 \sqrt {d+e x}}{c d}\right )}{d}+\frac {2 (d+e x)^{3/2}}{3 c d}\right )}{d}+\frac {2 (d+e x)^{5/2}}{5 c d}\right )}{d}+\frac {2 (d+e x)^{7/2}}{7 c d}\right )}{2 c d}-\frac {(d+e x)^{9/2}}{c d (a e+c d x)}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {9 e \left (\frac {\left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (d^2-\frac {a e^2}{c}\right ) \left (\frac {2 \sqrt {d+e x}}{c d}-\frac {2 \left (d^2-\frac {a e^2}{c}\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{\sqrt {c} d^{3/2} \sqrt {c d^2-a e^2}}\right )}{d}+\frac {2 (d+e x)^{3/2}}{3 c d}\right )}{d}+\frac {2 (d+e x)^{5/2}}{5 c d}\right )}{d}+\frac {2 (d+e x)^{7/2}}{7 c d}\right )}{2 c d}-\frac {(d+e x)^{9/2}}{c d (a e+c d x)}\) |
Input:
Int[(d + e*x)^(13/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]
Output:
-((d + e*x)^(9/2)/(c*d*(a*e + c*d*x))) + (9*e*((2*(d + e*x)^(7/2))/(7*c*d) + ((d^2 - (a*e^2)/c)*((2*(d + e*x)^(5/2))/(5*c*d) + ((d^2 - (a*e^2)/c)*(( 2*(d + e*x)^(3/2))/(3*c*d) + ((d^2 - (a*e^2)/c)*((2*Sqrt[d + e*x])/(c*d) - (2*(d^2 - (a*e^2)/c)*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/Sqrt[c*d^2 - a*e^2]])/(Sqrt[c]*d^(3/2)*Sqrt[c*d^2 - a*e^2])))/d))/d))/d))/(2*c*d)
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x ] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_ Symbol] :> Int[ExpandIntegrand[(d + e*x)^(m + p)*(a/d + (c/e)*x)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && (Int egerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && LtQ[c, 0]))
Time = 4.24 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.22
| method | result | size |
| pseudoelliptic | \(\frac {9 e \left (a \,e^{2}-c \,d^{2}\right )^{4} \left (c d x +a e \right ) \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {c d \left (a \,e^{2}-c \,d^{2}\right )}}\right )-9 \sqrt {e x +d}\, \sqrt {c d \left (a \,e^{2}-c \,d^{2}\right )}\, \left (\frac {\left (-\frac {2}{7} e^{4} x^{4}-\frac {58}{35} d \,e^{3} x^{3}-\frac {156}{35} d^{2} e^{2} x^{2}-\frac {388}{35} d^{3} e x +d^{4}\right ) d^{4} c^{4}}{9}-\frac {176 e^{2} \left (-\frac {3}{88} e^{3} x^{3}-\frac {23}{88} d \,e^{2} x^{2}-\frac {71}{44} d^{2} e x +d^{3}\right ) a \,d^{3} c^{3}}{105}+\frac {58 e^{4} a^{2} d^{2} \left (-\frac {1}{29} e^{2} x^{2}-\frac {17}{29} d e x +d^{2}\right ) c^{2}}{15}-\frac {10 e^{6} a^{3} \left (-\frac {e x}{5}+d \right ) d c}{3}+a^{4} e^{8}\right )}{d^{5} c^{5} \left (c d x +a e \right ) \sqrt {c d \left (a \,e^{2}-c \,d^{2}\right )}}\) | \(257\) |
| risch | \(-\frac {2 e \left (-5 c^{3} d^{3} e^{3} x^{3}+14 x^{2} a \,c^{2} d^{2} e^{4}-29 c^{3} d^{4} e^{2} x^{2}-35 x \,a^{2} c d \,e^{5}+98 x a \,c^{2} d^{3} e^{3}-78 c^{3} d^{5} e x +140 e^{6} a^{3}-455 d^{2} e^{4} a^{2} c +504 d^{4} e^{2} a \,c^{2}-194 d^{6} c^{3}\right ) \sqrt {e x +d}}{35 d^{5} c^{5}}+\frac {\left (2 a^{4} e^{8}-8 a^{3} c \,d^{2} e^{6}+12 a^{2} c^{2} d^{4} e^{4}-8 a \,c^{3} d^{6} e^{2}+2 c^{4} d^{8}\right ) e \left (-\frac {\sqrt {e x +d}}{2 \left (c d \left (e x +d \right )+a \,e^{2}-c \,d^{2}\right )}+\frac {9 \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {c d \left (a \,e^{2}-c \,d^{2}\right )}}\right )}{2 \sqrt {c d \left (a \,e^{2}-c \,d^{2}\right )}}\right )}{c^{5} d^{5}}\) | \(278\) |
| derivativedivides | \(2 e \left (-\frac {-\frac {c^{3} d^{3} \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (e x +d \right )^{\frac {5}{2}} a \,c^{2} d^{2} e^{2}}{5}-\frac {2 \left (e x +d \right )^{\frac {5}{2}} c^{3} d^{4}}{5}-\left (e x +d \right )^{\frac {3}{2}} a^{2} c d \,e^{4}+2 \left (e x +d \right )^{\frac {3}{2}} a \,c^{2} d^{3} e^{2}-\left (e x +d \right )^{\frac {3}{2}} c^{3} d^{5}+4 \sqrt {e x +d}\, e^{6} a^{3}-12 \sqrt {e x +d}\, d^{2} e^{4} a^{2} c +12 \sqrt {e x +d}\, d^{4} e^{2} a \,c^{2}-4 \sqrt {e x +d}\, d^{6} c^{3}}{c^{5} d^{5}}+\frac {\frac {\left (-\frac {1}{2} a^{4} e^{8}+2 a^{3} c \,d^{2} e^{6}-3 a^{2} c^{2} d^{4} e^{4}+2 a \,c^{3} d^{6} e^{2}-\frac {1}{2} c^{4} d^{8}\right ) \sqrt {e x +d}}{c d \left (e x +d \right )+a \,e^{2}-c \,d^{2}}+\frac {9 \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right ) \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {c d \left (a \,e^{2}-c \,d^{2}\right )}}\right )}{2 \sqrt {c d \left (a \,e^{2}-c \,d^{2}\right )}}}{d^{5} c^{5}}\right )\) | \(375\) |
| default | \(2 e \left (-\frac {-\frac {c^{3} d^{3} \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (e x +d \right )^{\frac {5}{2}} a \,c^{2} d^{2} e^{2}}{5}-\frac {2 \left (e x +d \right )^{\frac {5}{2}} c^{3} d^{4}}{5}-\left (e x +d \right )^{\frac {3}{2}} a^{2} c d \,e^{4}+2 \left (e x +d \right )^{\frac {3}{2}} a \,c^{2} d^{3} e^{2}-\left (e x +d \right )^{\frac {3}{2}} c^{3} d^{5}+4 \sqrt {e x +d}\, e^{6} a^{3}-12 \sqrt {e x +d}\, d^{2} e^{4} a^{2} c +12 \sqrt {e x +d}\, d^{4} e^{2} a \,c^{2}-4 \sqrt {e x +d}\, d^{6} c^{3}}{c^{5} d^{5}}+\frac {\frac {\left (-\frac {1}{2} a^{4} e^{8}+2 a^{3} c \,d^{2} e^{6}-3 a^{2} c^{2} d^{4} e^{4}+2 a \,c^{3} d^{6} e^{2}-\frac {1}{2} c^{4} d^{8}\right ) \sqrt {e x +d}}{c d \left (e x +d \right )+a \,e^{2}-c \,d^{2}}+\frac {9 \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right ) \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {c d \left (a \,e^{2}-c \,d^{2}\right )}}\right )}{2 \sqrt {c d \left (a \,e^{2}-c \,d^{2}\right )}}}{d^{5} c^{5}}\right )\) | \(375\) |
Input:
int((e*x+d)^(13/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^2,x,method=_RETURNVER BOSE)
Output:
9*(e*(a*e^2-c*d^2)^4*(c*d*x+a*e)*arctan(c*d*(e*x+d)^(1/2)/(c*d*(a*e^2-c*d^ 2))^(1/2))-(e*x+d)^(1/2)*(c*d*(a*e^2-c*d^2))^(1/2)*(1/9*(-2/7*e^4*x^4-58/3 5*d*e^3*x^3-156/35*d^2*e^2*x^2-388/35*d^3*e*x+d^4)*d^4*c^4-176/105*e^2*(-3 /88*e^3*x^3-23/88*d*e^2*x^2-71/44*d^2*e*x+d^3)*a*d^3*c^3+58/15*e^4*a^2*d^2 *(-1/29*e^2*x^2-17/29*d*e*x+d^2)*c^2-10/3*e^6*a^3*(-1/5*e*x+d)*d*c+a^4*e^8 ))/(c*d*(a*e^2-c*d^2))^(1/2)/d^5/c^5/(c*d*x+a*e)
Leaf count of result is larger than twice the leaf count of optimal. 387 vs. \(2 (182) = 364\).
Time = 0.10 (sec) , antiderivative size = 788, normalized size of antiderivative = 3.75 \[ \int \frac {(d+e x)^{13/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\left [-\frac {315 \, {\left (a c^{3} d^{6} e^{2} - 3 \, a^{2} c^{2} d^{4} e^{4} + 3 \, a^{3} c d^{2} e^{6} - a^{4} e^{8} + {\left (c^{4} d^{7} e - 3 \, a c^{3} d^{5} e^{3} + 3 \, a^{2} c^{2} d^{3} e^{5} - a^{3} c d e^{7}\right )} x\right )} \sqrt {\frac {c d^{2} - a e^{2}}{c d}} \log \left (\frac {c d e x + 2 \, c d^{2} - a e^{2} + 2 \, \sqrt {e x + d} c d \sqrt {\frac {c d^{2} - a e^{2}}{c d}}}{c d x + a e}\right ) - 2 \, {\left (10 \, c^{4} d^{4} e^{4} x^{4} - 35 \, c^{4} d^{8} + 528 \, a c^{3} d^{6} e^{2} - 1218 \, a^{2} c^{2} d^{4} e^{4} + 1050 \, a^{3} c d^{2} e^{6} - 315 \, a^{4} e^{8} + 2 \, {\left (29 \, c^{4} d^{5} e^{3} - 9 \, a c^{3} d^{3} e^{5}\right )} x^{3} + 6 \, {\left (26 \, c^{4} d^{6} e^{2} - 23 \, a c^{3} d^{4} e^{4} + 7 \, a^{2} c^{2} d^{2} e^{6}\right )} x^{2} + 2 \, {\left (194 \, c^{4} d^{7} e - 426 \, a c^{3} d^{5} e^{3} + 357 \, a^{2} c^{2} d^{3} e^{5} - 105 \, a^{3} c d e^{7}\right )} x\right )} \sqrt {e x + d}}{70 \, {\left (c^{6} d^{6} x + a c^{5} d^{5} e\right )}}, -\frac {315 \, {\left (a c^{3} d^{6} e^{2} - 3 \, a^{2} c^{2} d^{4} e^{4} + 3 \, a^{3} c d^{2} e^{6} - a^{4} e^{8} + {\left (c^{4} d^{7} e - 3 \, a c^{3} d^{5} e^{3} + 3 \, a^{2} c^{2} d^{3} e^{5} - a^{3} c d e^{7}\right )} x\right )} \sqrt {-\frac {c d^{2} - a e^{2}}{c d}} \arctan \left (-\frac {\sqrt {e x + d} c d \sqrt {-\frac {c d^{2} - a e^{2}}{c d}}}{c d^{2} - a e^{2}}\right ) - {\left (10 \, c^{4} d^{4} e^{4} x^{4} - 35 \, c^{4} d^{8} + 528 \, a c^{3} d^{6} e^{2} - 1218 \, a^{2} c^{2} d^{4} e^{4} + 1050 \, a^{3} c d^{2} e^{6} - 315 \, a^{4} e^{8} + 2 \, {\left (29 \, c^{4} d^{5} e^{3} - 9 \, a c^{3} d^{3} e^{5}\right )} x^{3} + 6 \, {\left (26 \, c^{4} d^{6} e^{2} - 23 \, a c^{3} d^{4} e^{4} + 7 \, a^{2} c^{2} d^{2} e^{6}\right )} x^{2} + 2 \, {\left (194 \, c^{4} d^{7} e - 426 \, a c^{3} d^{5} e^{3} + 357 \, a^{2} c^{2} d^{3} e^{5} - 105 \, a^{3} c d e^{7}\right )} x\right )} \sqrt {e x + d}}{35 \, {\left (c^{6} d^{6} x + a c^{5} d^{5} e\right )}}\right ] \] Input:
integrate((e*x+d)^(13/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x, algorithm= "fricas")
Output:
[-1/70*(315*(a*c^3*d^6*e^2 - 3*a^2*c^2*d^4*e^4 + 3*a^3*c*d^2*e^6 - a^4*e^8 + (c^4*d^7*e - 3*a*c^3*d^5*e^3 + 3*a^2*c^2*d^3*e^5 - a^3*c*d*e^7)*x)*sqrt ((c*d^2 - a*e^2)/(c*d))*log((c*d*e*x + 2*c*d^2 - a*e^2 + 2*sqrt(e*x + d)*c *d*sqrt((c*d^2 - a*e^2)/(c*d)))/(c*d*x + a*e)) - 2*(10*c^4*d^4*e^4*x^4 - 3 5*c^4*d^8 + 528*a*c^3*d^6*e^2 - 1218*a^2*c^2*d^4*e^4 + 1050*a^3*c*d^2*e^6 - 315*a^4*e^8 + 2*(29*c^4*d^5*e^3 - 9*a*c^3*d^3*e^5)*x^3 + 6*(26*c^4*d^6*e ^2 - 23*a*c^3*d^4*e^4 + 7*a^2*c^2*d^2*e^6)*x^2 + 2*(194*c^4*d^7*e - 426*a* c^3*d^5*e^3 + 357*a^2*c^2*d^3*e^5 - 105*a^3*c*d*e^7)*x)*sqrt(e*x + d))/(c^ 6*d^6*x + a*c^5*d^5*e), -1/35*(315*(a*c^3*d^6*e^2 - 3*a^2*c^2*d^4*e^4 + 3* a^3*c*d^2*e^6 - a^4*e^8 + (c^4*d^7*e - 3*a*c^3*d^5*e^3 + 3*a^2*c^2*d^3*e^5 - a^3*c*d*e^7)*x)*sqrt(-(c*d^2 - a*e^2)/(c*d))*arctan(-sqrt(e*x + d)*c*d* sqrt(-(c*d^2 - a*e^2)/(c*d))/(c*d^2 - a*e^2)) - (10*c^4*d^4*e^4*x^4 - 35*c ^4*d^8 + 528*a*c^3*d^6*e^2 - 1218*a^2*c^2*d^4*e^4 + 1050*a^3*c*d^2*e^6 - 3 15*a^4*e^8 + 2*(29*c^4*d^5*e^3 - 9*a*c^3*d^3*e^5)*x^3 + 6*(26*c^4*d^6*e^2 - 23*a*c^3*d^4*e^4 + 7*a^2*c^2*d^2*e^6)*x^2 + 2*(194*c^4*d^7*e - 426*a*c^3 *d^5*e^3 + 357*a^2*c^2*d^3*e^5 - 105*a^3*c*d*e^7)*x)*sqrt(e*x + d))/(c^6*d ^6*x + a*c^5*d^5*e)]
Timed out. \[ \int \frac {(d+e x)^{13/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\text {Timed out} \] Input:
integrate((e*x+d)**(13/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**2,x)
Output:
Timed out
Exception generated. \[ \int \frac {(d+e x)^{13/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((e*x+d)^(13/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x, algorithm= "maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a*e^2-c*d^2>0)', see `assume?` f or more de
Leaf count of result is larger than twice the leaf count of optimal. 422 vs. \(2 (182) = 364\).
Time = 0.18 (sec) , antiderivative size = 422, normalized size of antiderivative = 2.01 \[ \int \frac {(d+e x)^{13/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\frac {9 \, {\left (c^{4} d^{8} e - 4 \, a c^{3} d^{6} e^{3} + 6 \, a^{2} c^{2} d^{4} e^{5} - 4 \, a^{3} c d^{2} e^{7} + a^{4} e^{9}\right )} \arctan \left (\frac {\sqrt {e x + d} c d}{\sqrt {-c^{2} d^{3} + a c d e^{2}}}\right )}{\sqrt {-c^{2} d^{3} + a c d e^{2}} c^{5} d^{5}} - \frac {\sqrt {e x + d} c^{4} d^{8} e - 4 \, \sqrt {e x + d} a c^{3} d^{6} e^{3} + 6 \, \sqrt {e x + d} a^{2} c^{2} d^{4} e^{5} - 4 \, \sqrt {e x + d} a^{3} c d^{2} e^{7} + \sqrt {e x + d} a^{4} e^{9}}{{\left ({\left (e x + d\right )} c d - c d^{2} + a e^{2}\right )} c^{5} d^{5}} + \frac {2 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} c^{12} d^{12} e + 14 \, {\left (e x + d\right )}^{\frac {5}{2}} c^{12} d^{13} e + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} c^{12} d^{14} e + 140 \, \sqrt {e x + d} c^{12} d^{15} e - 14 \, {\left (e x + d\right )}^{\frac {5}{2}} a c^{11} d^{11} e^{3} - 70 \, {\left (e x + d\right )}^{\frac {3}{2}} a c^{11} d^{12} e^{3} - 420 \, \sqrt {e x + d} a c^{11} d^{13} e^{3} + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{2} c^{10} d^{10} e^{5} + 420 \, \sqrt {e x + d} a^{2} c^{10} d^{11} e^{5} - 140 \, \sqrt {e x + d} a^{3} c^{9} d^{9} e^{7}\right )}}{35 \, c^{14} d^{14}} \] Input:
integrate((e*x+d)^(13/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x, algorithm= "giac")
Output:
9*(c^4*d^8*e - 4*a*c^3*d^6*e^3 + 6*a^2*c^2*d^4*e^5 - 4*a^3*c*d^2*e^7 + a^4 *e^9)*arctan(sqrt(e*x + d)*c*d/sqrt(-c^2*d^3 + a*c*d*e^2))/(sqrt(-c^2*d^3 + a*c*d*e^2)*c^5*d^5) - (sqrt(e*x + d)*c^4*d^8*e - 4*sqrt(e*x + d)*a*c^3*d ^6*e^3 + 6*sqrt(e*x + d)*a^2*c^2*d^4*e^5 - 4*sqrt(e*x + d)*a^3*c*d^2*e^7 + sqrt(e*x + d)*a^4*e^9)/(((e*x + d)*c*d - c*d^2 + a*e^2)*c^5*d^5) + 2/35*( 5*(e*x + d)^(7/2)*c^12*d^12*e + 14*(e*x + d)^(5/2)*c^12*d^13*e + 35*(e*x + d)^(3/2)*c^12*d^14*e + 140*sqrt(e*x + d)*c^12*d^15*e - 14*(e*x + d)^(5/2) *a*c^11*d^11*e^3 - 70*(e*x + d)^(3/2)*a*c^11*d^12*e^3 - 420*sqrt(e*x + d)* a*c^11*d^13*e^3 + 35*(e*x + d)^(3/2)*a^2*c^10*d^10*e^5 + 420*sqrt(e*x + d) *a^2*c^10*d^11*e^5 - 140*sqrt(e*x + d)*a^3*c^9*d^9*e^7)/(c^14*d^14)
Time = 0.08 (sec) , antiderivative size = 443, normalized size of antiderivative = 2.11 \[ \int \frac {(d+e x)^{13/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\frac {2\,e\,{\left (d+e\,x\right )}^{7/2}}{7\,c^2\,d^2}-\left (\frac {\left (2\,c^2\,d^3-2\,a\,c\,d\,e^2\right )\,\left (\frac {2\,e\,{\left (a\,e^2-c\,d^2\right )}^2}{c^4\,d^4}-\frac {2\,e\,{\left (2\,c^2\,d^3-2\,a\,c\,d\,e^2\right )}^2}{c^6\,d^6}\right )}{c^2\,d^2}+\frac {2\,e\,{\left (a\,e^2-c\,d^2\right )}^2\,\left (2\,c^2\,d^3-2\,a\,c\,d\,e^2\right )}{c^6\,d^6}\right )\,\sqrt {d+e\,x}-\frac {\sqrt {d+e\,x}\,\left (a^4\,e^9-4\,a^3\,c\,d^2\,e^7+6\,a^2\,c^2\,d^4\,e^5-4\,a\,c^3\,d^6\,e^3+c^4\,d^8\,e\right )}{c^6\,d^6\,\left (d+e\,x\right )-c^6\,d^7+a\,c^5\,d^5\,e^2}-\left (\frac {2\,e\,{\left (a\,e^2-c\,d^2\right )}^2}{3\,c^4\,d^4}-\frac {2\,e\,{\left (2\,c^2\,d^3-2\,a\,c\,d\,e^2\right )}^2}{3\,c^6\,d^6}\right )\,{\left (d+e\,x\right )}^{3/2}+\frac {2\,e\,\left (2\,c^2\,d^3-2\,a\,c\,d\,e^2\right )\,{\left (d+e\,x\right )}^{5/2}}{5\,c^4\,d^4}+\frac {9\,e\,\mathrm {atan}\left (\frac {\sqrt {c}\,\sqrt {d}\,e\,{\left (a\,e^2-c\,d^2\right )}^{7/2}\,\sqrt {d+e\,x}}{a^4\,e^9-4\,a^3\,c\,d^2\,e^7+6\,a^2\,c^2\,d^4\,e^5-4\,a\,c^3\,d^6\,e^3+c^4\,d^8\,e}\right )\,{\left (a\,e^2-c\,d^2\right )}^{7/2}}{c^{11/2}\,d^{11/2}} \] Input:
int((d + e*x)^(13/2)/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^2,x)
Output:
(2*e*(d + e*x)^(7/2))/(7*c^2*d^2) - (((2*c^2*d^3 - 2*a*c*d*e^2)*((2*e*(a*e ^2 - c*d^2)^2)/(c^4*d^4) - (2*e*(2*c^2*d^3 - 2*a*c*d*e^2)^2)/(c^6*d^6)))/( c^2*d^2) + (2*e*(a*e^2 - c*d^2)^2*(2*c^2*d^3 - 2*a*c*d*e^2))/(c^6*d^6))*(d + e*x)^(1/2) - ((d + e*x)^(1/2)*(a^4*e^9 + c^4*d^8*e - 4*a*c^3*d^6*e^3 - 4*a^3*c*d^2*e^7 + 6*a^2*c^2*d^4*e^5))/(c^6*d^6*(d + e*x) - c^6*d^7 + a*c^5 *d^5*e^2) - ((2*e*(a*e^2 - c*d^2)^2)/(3*c^4*d^4) - (2*e*(2*c^2*d^3 - 2*a*c *d*e^2)^2)/(3*c^6*d^6))*(d + e*x)^(3/2) + (2*e*(2*c^2*d^3 - 2*a*c*d*e^2)*( d + e*x)^(5/2))/(5*c^4*d^4) + (9*e*atan((c^(1/2)*d^(1/2)*e*(a*e^2 - c*d^2) ^(7/2)*(d + e*x)^(1/2))/(a^4*e^9 + c^4*d^8*e - 4*a*c^3*d^6*e^3 - 4*a^3*c*d ^2*e^7 + 6*a^2*c^2*d^4*e^5))*(a*e^2 - c*d^2)^(7/2))/(c^(11/2)*d^(11/2))
Time = 0.24 (sec) , antiderivative size = 804, normalized size of antiderivative = 3.83 \[ \int \frac {(d+e x)^{13/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx =\text {Too large to display} \] Input:
int((e*x+d)^(13/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x)
Output:
(315*sqrt(d)*sqrt(c)*sqrt(a*e**2 - c*d**2)*atan((sqrt(d + e*x)*c*d)/(sqrt( d)*sqrt(c)*sqrt(a*e**2 - c*d**2)))*a**4*e**8 - 945*sqrt(d)*sqrt(c)*sqrt(a* e**2 - c*d**2)*atan((sqrt(d + e*x)*c*d)/(sqrt(d)*sqrt(c)*sqrt(a*e**2 - c*d **2)))*a**3*c*d**2*e**6 + 315*sqrt(d)*sqrt(c)*sqrt(a*e**2 - c*d**2)*atan(( sqrt(d + e*x)*c*d)/(sqrt(d)*sqrt(c)*sqrt(a*e**2 - c*d**2)))*a**3*c*d*e**7* x + 945*sqrt(d)*sqrt(c)*sqrt(a*e**2 - c*d**2)*atan((sqrt(d + e*x)*c*d)/(sq rt(d)*sqrt(c)*sqrt(a*e**2 - c*d**2)))*a**2*c**2*d**4*e**4 - 945*sqrt(d)*sq rt(c)*sqrt(a*e**2 - c*d**2)*atan((sqrt(d + e*x)*c*d)/(sqrt(d)*sqrt(c)*sqrt (a*e**2 - c*d**2)))*a**2*c**2*d**3*e**5*x - 315*sqrt(d)*sqrt(c)*sqrt(a*e** 2 - c*d**2)*atan((sqrt(d + e*x)*c*d)/(sqrt(d)*sqrt(c)*sqrt(a*e**2 - c*d**2 )))*a*c**3*d**6*e**2 + 945*sqrt(d)*sqrt(c)*sqrt(a*e**2 - c*d**2)*atan((sqr t(d + e*x)*c*d)/(sqrt(d)*sqrt(c)*sqrt(a*e**2 - c*d**2)))*a*c**3*d**5*e**3* x - 315*sqrt(d)*sqrt(c)*sqrt(a*e**2 - c*d**2)*atan((sqrt(d + e*x)*c*d)/(sq rt(d)*sqrt(c)*sqrt(a*e**2 - c*d**2)))*c**4*d**7*e*x - 315*sqrt(d + e*x)*a* *4*c*d*e**8 + 1050*sqrt(d + e*x)*a**3*c**2*d**3*e**6 - 210*sqrt(d + e*x)*a **3*c**2*d**2*e**7*x - 1218*sqrt(d + e*x)*a**2*c**3*d**5*e**4 + 714*sqrt(d + e*x)*a**2*c**3*d**4*e**5*x + 42*sqrt(d + e*x)*a**2*c**3*d**3*e**6*x**2 + 528*sqrt(d + e*x)*a*c**4*d**7*e**2 - 852*sqrt(d + e*x)*a*c**4*d**6*e**3* x - 138*sqrt(d + e*x)*a*c**4*d**5*e**4*x**2 - 18*sqrt(d + e*x)*a*c**4*d**4 *e**5*x**3 - 35*sqrt(d + e*x)*c**5*d**9 + 388*sqrt(d + e*x)*c**5*d**8*e...