Integrand size = 21, antiderivative size = 251 \[ \int \frac {(d+e x)^{9/2}}{\left (b x+c x^2\right )^2} \, dx=\frac {e (2 c d-b e) \left (c^2 d^2-b c d e+5 b^2 e^2\right ) \sqrt {d+e x}}{b^2 c^3}+\frac {e \left (6 c^2 d^2-6 b c d e+5 b^2 e^2\right ) (d+e x)^{3/2}}{3 b^2 c^2}+\frac {e (2 c d-b e) (d+e x)^{5/2}}{b^2 c}-\frac {(d+e x)^{7/2} (b d+(2 c d-b e) x)}{b^2 \left (b x+c x^2\right )}+\frac {d^{7/2} (4 c d-9 b e) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b^3}-\frac {(c d-b e)^{7/2} (4 c d+5 b e) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 c^{7/2}} \]
1/3*e*(5*b^2*e^2-6*b*c*d*e+6*c^2*d^2)*(e*x+d)^(3/2)/b^2/c^2+e*(-b*e+2*c*d) *(e*x+d)^(5/2)/b^2/c-(e*x+d)^(7/2)*(b*d+(-b*e+2*c*d)*x)/b^2/(c*x^2+b*x)+d^ (7/2)*(-9*b*e+4*c*d)*arctanh((e*x+d)^(1/2)/d^(1/2))/b^3-(-b*e+c*d)^(7/2)*( 5*b*e+4*c*d)*arctanh(c^(1/2)*(e*x+d)^(1/2)/(-b*e+c*d)^(1/2))/b^3/c^(7/2)+e *(-b*e+2*c*d)*(5*b^2*e^2-b*c*d*e+c^2*d^2)*(e*x+d)^(1/2)/b^2/c^3
Time = 0.87 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.81 \[ \int \frac {(d+e x)^{9/2}}{\left (b x+c x^2\right )^2} \, dx=\frac {-\frac {b \sqrt {d+e x} \left (6 c^4 d^4 x+15 b^4 e^4 x+3 b c^3 d^3 (d-4 e x)+2 b^3 c e^3 x (-19 d+5 e x)-2 b^2 c^2 e^2 x \left (-9 d^2+13 d e x+e^2 x^2\right )\right )}{c^3 x (b+c x)}+\frac {3 (-c d+b e)^{7/2} (4 c d+5 b e) \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {-c d+b e}}\right )}{c^{7/2}}+3 d^{7/2} (4 c d-9 b e) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{3 b^3} \]
(-((b*Sqrt[d + e*x]*(6*c^4*d^4*x + 15*b^4*e^4*x + 3*b*c^3*d^3*(d - 4*e*x) + 2*b^3*c*e^3*x*(-19*d + 5*e*x) - 2*b^2*c^2*e^2*x*(-9*d^2 + 13*d*e*x + e^2 *x^2)))/(c^3*x*(b + c*x))) + (3*(-(c*d) + b*e)^(7/2)*(4*c*d + 5*b*e)*ArcTa n[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[-(c*d) + b*e]])/c^(7/2) + 3*d^(7/2)*(4*c*d - 9*b*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(3*b^3)
Time = 0.72 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.11, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {1164, 27, 1196, 1196, 1196, 1197, 27, 1480, 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.
| \(\displaystyle \int \frac {(d+e x)^{9/2}}{\left (b x+c x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 1164 |
| \(\displaystyle -\frac {\int \frac {(d+e x)^{5/2} (d (4 c d-9 b e)-5 e (2 c d-b e) x)}{2 \left (c x^2+b x\right )}dx}{b^2}-\frac {(d+e x)^{7/2} (x (2 c d-b e)+b d)}{b^2 \left (b x+c x^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {(d+e x)^{5/2} (d (4 c d-9 b e)-5 e (2 c d-b e) x)}{c x^2+b x}dx}{2 b^2}-\frac {(d+e x)^{7/2} (x (2 c d-b e)+b d)}{b^2 \left (b x+c x^2\right )}\) |
| \(\Big \downarrow \) 1196 |
| \(\displaystyle -\frac {\frac {\int \frac {(d+e x)^{3/2} \left (c d^2 (4 c d-9 b e)-e \left (6 c^2 d^2-6 b c e d+5 b^2 e^2\right ) x\right )}{c x^2+b x}dx}{c}-\frac {2 e (d+e x)^{5/2} (2 c d-b e)}{c}}{2 b^2}-\frac {(d+e x)^{7/2} (x (2 c d-b e)+b d)}{b^2 \left (b x+c x^2\right )}\) |
| \(\Big \downarrow \) 1196 |
| \(\displaystyle -\frac {\frac {\frac {\int \frac {\sqrt {d+e x} \left (c^2 d^3 (4 c d-9 b e)-e (2 c d-b e) \left (c^2 d^2-b c e d+5 b^2 e^2\right ) x\right )}{c x^2+b x}dx}{c}-\frac {2 e (d+e x)^{3/2} \left (5 b^2 e^2-6 b c d e+6 c^2 d^2\right )}{3 c}}{c}-\frac {2 e (d+e x)^{5/2} (2 c d-b e)}{c}}{2 b^2}-\frac {(d+e x)^{7/2} (x (2 c d-b e)+b d)}{b^2 \left (b x+c x^2\right )}\) |
| \(\Big \downarrow \) 1196 |
| \(\displaystyle -\frac {\frac {\frac {\frac {\int \frac {c^3 (4 c d-9 b e) d^4+e \left (2 c^4 d^4-4 b c^3 e d^3-14 b^2 c^2 e^2 d^2+16 b^3 c e^3 d-5 b^4 e^4\right ) x}{\sqrt {d+e x} \left (c x^2+b x\right )}dx}{c}-\frac {2 e \sqrt {d+e x} (2 c d-b e) \left (5 b^2 e^2-b c d e+c^2 d^2\right )}{c}}{c}-\frac {2 e (d+e x)^{3/2} \left (5 b^2 e^2-6 b c d e+6 c^2 d^2\right )}{3 c}}{c}-\frac {2 e (d+e x)^{5/2} (2 c d-b e)}{c}}{2 b^2}-\frac {(d+e x)^{7/2} (x (2 c d-b e)+b d)}{b^2 \left (b x+c x^2\right )}\) |
| \(\Big \downarrow \) 1197 |
| \(\displaystyle -\frac {\frac {\frac {\frac {2 \int \frac {e \left (d (c d-b e) (2 c d-b e) \left (c^2 d^2-b c e d+5 b^2 e^2\right )+\left (2 c^4 d^4-4 b c^3 e d^3-14 b^2 c^2 e^2 d^2+16 b^3 c e^3 d-5 b^4 e^4\right ) (d+e x)\right )}{c (d+e x)^2-(2 c d-b e) (d+e x)+d (c d-b e)}d\sqrt {d+e x}}{c}-\frac {2 e \sqrt {d+e x} (2 c d-b e) \left (5 b^2 e^2-b c d e+c^2 d^2\right )}{c}}{c}-\frac {2 e (d+e x)^{3/2} \left (5 b^2 e^2-6 b c d e+6 c^2 d^2\right )}{3 c}}{c}-\frac {2 e (d+e x)^{5/2} (2 c d-b e)}{c}}{2 b^2}-\frac {(d+e x)^{7/2} (x (2 c d-b e)+b d)}{b^2 \left (b x+c x^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\frac {\frac {2 e \int \frac {d (c d-b e) (2 c d-b e) \left (c^2 d^2-b c e d+5 b^2 e^2\right )+\left (2 c^4 d^4-4 b c^3 e d^3-14 b^2 c^2 e^2 d^2+16 b^3 c e^3 d-5 b^4 e^4\right ) (d+e x)}{c (d+e x)^2-(2 c d-b e) (d+e x)+d (c d-b e)}d\sqrt {d+e x}}{c}-\frac {2 e \sqrt {d+e x} (2 c d-b e) \left (5 b^2 e^2-b c d e+c^2 d^2\right )}{c}}{c}-\frac {2 e (d+e x)^{3/2} \left (5 b^2 e^2-6 b c d e+6 c^2 d^2\right )}{3 c}}{c}-\frac {2 e (d+e x)^{5/2} (2 c d-b e)}{c}}{2 b^2}-\frac {(d+e x)^{7/2} (x (2 c d-b e)+b d)}{b^2 \left (b x+c x^2\right )}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle -\frac {\frac {\frac {\frac {2 e \left (\frac {c^4 d^4 (4 c d-9 b e) \int \frac {1}{c (d+e x)-c d}d\sqrt {d+e x}}{b e}-\frac {(c d-b e)^4 (5 b e+4 c d) \int \frac {1}{-c d+b e+c (d+e x)}d\sqrt {d+e x}}{b e}\right )}{c}-\frac {2 e \sqrt {d+e x} (2 c d-b e) \left (5 b^2 e^2-b c d e+c^2 d^2\right )}{c}}{c}-\frac {2 e (d+e x)^{3/2} \left (5 b^2 e^2-6 b c d e+6 c^2 d^2\right )}{3 c}}{c}-\frac {2 e (d+e x)^{5/2} (2 c d-b e)}{c}}{2 b^2}-\frac {(d+e x)^{7/2} (x (2 c d-b e)+b d)}{b^2 \left (b x+c x^2\right )}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {\frac {\frac {\frac {2 e \left (\frac {(c d-b e)^{7/2} (5 b e+4 c d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b \sqrt {c} e}-\frac {c^3 d^{7/2} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) (4 c d-9 b e)}{b e}\right )}{c}-\frac {2 e \sqrt {d+e x} (2 c d-b e) \left (5 b^2 e^2-b c d e+c^2 d^2\right )}{c}}{c}-\frac {2 e (d+e x)^{3/2} \left (5 b^2 e^2-6 b c d e+6 c^2 d^2\right )}{3 c}}{c}-\frac {2 e (d+e x)^{5/2} (2 c d-b e)}{c}}{2 b^2}-\frac {(d+e x)^{7/2} (x (2 c d-b e)+b d)}{b^2 \left (b x+c x^2\right )}\) |
-(((d + e*x)^(7/2)*(b*d + (2*c*d - b*e)*x))/(b^2*(b*x + c*x^2))) - ((-2*e* (2*c*d - b*e)*(d + e*x)^(5/2))/c + ((-2*e*(6*c^2*d^2 - 6*b*c*d*e + 5*b^2*e ^2)*(d + e*x)^(3/2))/(3*c) + ((-2*e*(2*c*d - b*e)*(c^2*d^2 - b*c*d*e + 5*b ^2*e^2)*Sqrt[d + e*x])/c + (2*e*(-((c^3*d^(7/2)*(4*c*d - 9*b*e)*ArcTanh[Sq rt[d + e*x]/Sqrt[d]])/(b*e)) + ((c*d - b*e)^(7/2)*(4*c*d + 5*b*e)*ArcTanh[ (Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b*Sqrt[c]*e)))/c)/c)/c)/(2*b^2)
3.4.69.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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_S ymbol] :> Simp[(d + e*x)^(m - 1)*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Simp[1/((p + 1)*(b^2 - 4*a* c)) Int[(d + e*x)^(m - 2)*Simp[e*(2*a*e*(m - 1) + b*d*(2*p - m + 4)) - 2* c*d^2*(2*p + 3) + e*(b*e - 2*d*c)*(m + 2*p + 2)*x, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[p, -1] && GtQ[m, 1] && Int QuadraticQ[a, b, c, d, e, m, p, x]
Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[g*((d + e*x)^m/(c*m)), x] + Simp[1/c Int [(d + e*x)^(m - 1)*(Simp[c*d*f - a*e*g + (g*c*d - b*e*g + c*e*f)*x, x]/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && FractionQ[m] & & GtQ[m, 0]
Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)), x_Symbol] :> Simp[2 Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; Fr eeQ[{a, b, c, d, e, f, g}, x]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q)) Int[1/( b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q)) Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
Time = 2.05 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.86
| method | result | size |
| pseudoelliptic | \(\frac {5 x \left (b e +\frac {4 c d}{5}\right ) \left (c x +b \right ) \left (b e -c d \right )^{4} \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )+4 \sqrt {\left (b e -c d \right ) c}\, \left (x \left (c x +b \right ) \left (d^{\frac {9}{2}} c -\frac {9 e b \,d^{\frac {7}{2}}}{4}\right ) c^{3} \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )-\frac {5 \sqrt {e x +d}\, \left (\frac {2 c^{4} d^{4} x}{5}+\frac {d^{3} b \left (-4 e x +d \right ) c^{3}}{5}+\frac {6 x \,e^{2} \left (-\frac {1}{9} x^{2} e^{2}-\frac {13}{9} d e x +d^{2}\right ) b^{2} c^{2}}{5}-\frac {38 x \left (-\frac {5 e x}{19}+d \right ) e^{3} b^{3} c}{15}+b^{4} e^{4} x \right ) b}{4}\right )}{c^{3} b^{3} x \left (c x +b \right ) \sqrt {\left (b e -c d \right ) c}}\) | \(217\) |
| derivativedivides | \(2 e^{3} \left (-\frac {-\frac {c \left (e x +d \right )^{\frac {3}{2}}}{3}+2 b e \sqrt {e x +d}-4 c d \sqrt {e x +d}}{c^{3}}-\frac {d^{4} \left (\frac {b \sqrt {e x +d}}{2 x}+\frac {\left (9 b e -4 c d \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{2 \sqrt {d}}\right )}{b^{3} e^{3}}+\frac {\frac {\left (-\frac {1}{2} b^{5} e^{5}+2 b^{4} c d \,e^{4}-3 b^{3} c^{2} d^{2} e^{3}+2 b^{2} c^{3} d^{3} e^{2}-\frac {1}{2} b \,c^{4} d^{4} e \right ) \sqrt {e x +d}}{c \left (e x +d \right )+b e -c d}+\frac {\left (5 b^{5} e^{5}-16 b^{4} c d \,e^{4}+14 b^{3} c^{2} d^{2} e^{3}+4 b^{2} c^{3} d^{3} e^{2}-11 b \,c^{4} d^{4} e +4 c^{5} d^{5}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{2 \sqrt {\left (b e -c d \right ) c}}}{c^{3} b^{3} e^{3}}\right )\) | \(290\) |
| default | \(2 e^{3} \left (-\frac {-\frac {c \left (e x +d \right )^{\frac {3}{2}}}{3}+2 b e \sqrt {e x +d}-4 c d \sqrt {e x +d}}{c^{3}}-\frac {d^{4} \left (\frac {b \sqrt {e x +d}}{2 x}+\frac {\left (9 b e -4 c d \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{2 \sqrt {d}}\right )}{b^{3} e^{3}}+\frac {\frac {\left (-\frac {1}{2} b^{5} e^{5}+2 b^{4} c d \,e^{4}-3 b^{3} c^{2} d^{2} e^{3}+2 b^{2} c^{3} d^{3} e^{2}-\frac {1}{2} b \,c^{4} d^{4} e \right ) \sqrt {e x +d}}{c \left (e x +d \right )+b e -c d}+\frac {\left (5 b^{5} e^{5}-16 b^{4} c d \,e^{4}+14 b^{3} c^{2} d^{2} e^{3}+4 b^{2} c^{3} d^{3} e^{2}-11 b \,c^{4} d^{4} e +4 c^{5} d^{5}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{2 \sqrt {\left (b e -c d \right ) c}}}{c^{3} b^{3} e^{3}}\right )\) | \(290\) |
| risch | \(-\frac {d^{4} \sqrt {e x +d}}{b^{2} x}+\frac {e \left (\frac {2 b^{2} e^{2} \left (\frac {c \left (e x +d \right )^{\frac {3}{2}}}{3}-2 b e \sqrt {e x +d}+4 c d \sqrt {e x +d}\right )}{c^{3}}-\frac {d^{\frac {7}{2}} \left (9 b e -4 c d \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b e}+\frac {\frac {2 \left (-\frac {1}{2} b^{5} e^{5}+2 b^{4} c d \,e^{4}-3 b^{3} c^{2} d^{2} e^{3}+2 b^{2} c^{3} d^{3} e^{2}-\frac {1}{2} b \,c^{4} d^{4} e \right ) \sqrt {e x +d}}{c \left (e x +d \right )+b e -c d}+\frac {\left (5 b^{5} e^{5}-16 b^{4} c d \,e^{4}+14 b^{3} c^{2} d^{2} e^{3}+4 b^{2} c^{3} d^{3} e^{2}-11 b \,c^{4} d^{4} e +4 c^{5} d^{5}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{\sqrt {\left (b e -c d \right ) c}}}{b \,c^{3} e}\right )}{b^{2}}\) | \(297\) |
4*(5/4*x*(b*e+4/5*c*d)*(c*x+b)*(b*e-c*d)^4*arctan(c*(e*x+d)^(1/2)/((b*e-c* d)*c)^(1/2))+((b*e-c*d)*c)^(1/2)*(x*(c*x+b)*(d^(9/2)*c-9/4*e*b*d^(7/2))*c^ 3*arctanh((e*x+d)^(1/2)/d^(1/2))-5/4*(e*x+d)^(1/2)*(2/5*c^4*d^4*x+1/5*d^3* b*(-4*e*x+d)*c^3+6/5*x*e^2*(-1/9*x^2*e^2-13/9*d*e*x+d^2)*b^2*c^2-38/15*x*( -5/19*e*x+d)*e^3*b^3*c+b^4*e^4*x)*b))/((b*e-c*d)*c)^(1/2)/c^3/b^3/x/(c*x+b )
Time = 1.92 (sec) , antiderivative size = 1561, normalized size of antiderivative = 6.22 \[ \int \frac {(d+e x)^{9/2}}{\left (b x+c x^2\right )^2} \, dx=\text {Too large to display} \]
[-1/6*(3*((4*c^5*d^4 - 7*b*c^4*d^3*e - 3*b^2*c^3*d^2*e^2 + 11*b^3*c^2*d*e^ 3 - 5*b^4*c*e^4)*x^2 + (4*b*c^4*d^4 - 7*b^2*c^3*d^3*e - 3*b^3*c^2*d^2*e^2 + 11*b^4*c*d*e^3 - 5*b^5*e^4)*x)*sqrt((c*d - b*e)/c)*log((c*e*x + 2*c*d - b*e + 2*sqrt(e*x + d)*c*sqrt((c*d - b*e)/c))/(c*x + b)) + 3*((4*c^5*d^4 - 9*b*c^4*d^3*e)*x^2 + (4*b*c^4*d^4 - 9*b^2*c^3*d^3*e)*x)*sqrt(d)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) - 2*(2*b^3*c^2*e^4*x^3 - 3*b^2*c^3*d^4 + 2*(13*b^3*c^2*d*e^3 - 5*b^4*c*e^4)*x^2 - (6*b*c^4*d^4 - 12*b^2*c^3*d^3*e + 18*b^3*c^2*d^2*e^2 - 38*b^4*c*d*e^3 + 15*b^5*e^4)*x)*sqrt(e*x + d))/(b^ 3*c^4*x^2 + b^4*c^3*x), -1/6*(6*((4*c^5*d^4 - 7*b*c^4*d^3*e - 3*b^2*c^3*d^ 2*e^2 + 11*b^3*c^2*d*e^3 - 5*b^4*c*e^4)*x^2 + (4*b*c^4*d^4 - 7*b^2*c^3*d^3 *e - 3*b^3*c^2*d^2*e^2 + 11*b^4*c*d*e^3 - 5*b^5*e^4)*x)*sqrt(-(c*d - b*e)/ c)*arctan(-sqrt(e*x + d)*c*sqrt(-(c*d - b*e)/c)/(c*d - b*e)) + 3*((4*c^5*d ^4 - 9*b*c^4*d^3*e)*x^2 + (4*b*c^4*d^4 - 9*b^2*c^3*d^3*e)*x)*sqrt(d)*log(( e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) - 2*(2*b^3*c^2*e^4*x^3 - 3*b^2*c^3 *d^4 + 2*(13*b^3*c^2*d*e^3 - 5*b^4*c*e^4)*x^2 - (6*b*c^4*d^4 - 12*b^2*c^3* d^3*e + 18*b^3*c^2*d^2*e^2 - 38*b^4*c*d*e^3 + 15*b^5*e^4)*x)*sqrt(e*x + d) )/(b^3*c^4*x^2 + b^4*c^3*x), -1/6*(6*((4*c^5*d^4 - 9*b*c^4*d^3*e)*x^2 + (4 *b*c^4*d^4 - 9*b^2*c^3*d^3*e)*x)*sqrt(-d)*arctan(sqrt(e*x + d)*sqrt(-d)/d) + 3*((4*c^5*d^4 - 7*b*c^4*d^3*e - 3*b^2*c^3*d^2*e^2 + 11*b^3*c^2*d*e^3 - 5*b^4*c*e^4)*x^2 + (4*b*c^4*d^4 - 7*b^2*c^3*d^3*e - 3*b^3*c^2*d^2*e^2 +...
Timed out. \[ \int \frac {(d+e x)^{9/2}}{\left (b x+c x^2\right )^2} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {(d+e x)^{9/2}}{\left (b x+c x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(b*e-c*d>0)', see `assume?` for m ore detail
Time = 0.29 (sec) , antiderivative size = 425, normalized size of antiderivative = 1.69 \[ \int \frac {(d+e x)^{9/2}}{\left (b x+c x^2\right )^2} \, dx=-\frac {{\left (4 \, c d^{5} - 9 \, b d^{4} e\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-d}}\right )}{b^{3} \sqrt {-d}} + \frac {{\left (4 \, c^{5} d^{5} - 11 \, b c^{4} d^{4} e + 4 \, b^{2} c^{3} d^{3} e^{2} + 14 \, b^{3} c^{2} d^{2} e^{3} - 16 \, b^{4} c d e^{4} + 5 \, b^{5} e^{5}\right )} \arctan \left (\frac {\sqrt {e x + d} c}{\sqrt {-c^{2} d + b c e}}\right )}{\sqrt {-c^{2} d + b c e} b^{3} c^{3}} + \frac {2 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} c^{4} e^{3} + 12 \, \sqrt {e x + d} c^{4} d e^{3} - 6 \, \sqrt {e x + d} b c^{3} e^{4}\right )}}{3 \, c^{6}} - \frac {2 \, {\left (e x + d\right )}^{\frac {3}{2}} c^{4} d^{4} e - 2 \, \sqrt {e x + d} c^{4} d^{5} e - 4 \, {\left (e x + d\right )}^{\frac {3}{2}} b c^{3} d^{3} e^{2} + 5 \, \sqrt {e x + d} b c^{3} d^{4} e^{2} + 6 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{2} c^{2} d^{2} e^{3} - 6 \, \sqrt {e x + d} b^{2} c^{2} d^{3} e^{3} - 4 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{3} c d e^{4} + 4 \, \sqrt {e x + d} b^{3} c d^{2} e^{4} + {\left (e x + d\right )}^{\frac {3}{2}} b^{4} e^{5} - \sqrt {e x + d} b^{4} d e^{5}}{{\left ({\left (e x + d\right )}^{2} c - 2 \, {\left (e x + d\right )} c d + c d^{2} + {\left (e x + d\right )} b e - b d e\right )} b^{2} c^{3}} \]
-(4*c*d^5 - 9*b*d^4*e)*arctan(sqrt(e*x + d)/sqrt(-d))/(b^3*sqrt(-d)) + (4* c^5*d^5 - 11*b*c^4*d^4*e + 4*b^2*c^3*d^3*e^2 + 14*b^3*c^2*d^2*e^3 - 16*b^4 *c*d*e^4 + 5*b^5*e^5)*arctan(sqrt(e*x + d)*c/sqrt(-c^2*d + b*c*e))/(sqrt(- c^2*d + b*c*e)*b^3*c^3) + 2/3*((e*x + d)^(3/2)*c^4*e^3 + 12*sqrt(e*x + d)* c^4*d*e^3 - 6*sqrt(e*x + d)*b*c^3*e^4)/c^6 - (2*(e*x + d)^(3/2)*c^4*d^4*e - 2*sqrt(e*x + d)*c^4*d^5*e - 4*(e*x + d)^(3/2)*b*c^3*d^3*e^2 + 5*sqrt(e*x + d)*b*c^3*d^4*e^2 + 6*(e*x + d)^(3/2)*b^2*c^2*d^2*e^3 - 6*sqrt(e*x + d)* b^2*c^2*d^3*e^3 - 4*(e*x + d)^(3/2)*b^3*c*d*e^4 + 4*sqrt(e*x + d)*b^3*c*d^ 2*e^4 + (e*x + d)^(3/2)*b^4*e^5 - sqrt(e*x + d)*b^4*d*e^5)/(((e*x + d)^2*c - 2*(e*x + d)*c*d + c*d^2 + (e*x + d)*b*e - b*d*e)*b^2*c^3)
Time = 10.05 (sec) , antiderivative size = 3360, normalized size of antiderivative = 13.39 \[ \int \frac {(d+e x)^{9/2}}{\left (b x+c x^2\right )^2} \, dx=\text {Too large to display} \]
(((d + e*x)^(3/2)*(b^4*e^5 + 2*c^4*d^4*e - 4*b*c^3*d^3*e^2 + 6*b^2*c^2*d^2 *e^3 - 4*b^3*c*d*e^4))/b^2 - ((d + e*x)^(1/2)*(b^4*d*e^5 + 2*c^4*d^5*e - 5 *b*c^3*d^4*e^2 - 4*b^3*c*d^2*e^4 + 6*b^2*c^2*d^3*e^3))/b^2)/((2*c^4*d - b* c^3*e)*(d + e*x) - c^4*(d + e*x)^2 - c^4*d^2 + b*c^3*d*e) + (2*e^3*(d + e* x)^(3/2))/(3*c^2) + (2*e^3*(4*c^2*d - 2*b*c*e)*(d + e*x)^(1/2))/c^4 - (ata n(((((((20*b^10*c^4*d*e^7 + 8*b^6*c^8*d^5*e^3 - 20*b^7*c^7*d^4*e^4 + 56*b^ 8*c^6*d^3*e^5 - 64*b^9*c^5*d^2*e^6)/(b^6*c^5) + ((4*b^7*c^7*e^3 - 8*b^6*c^ 8*d*e^2)*(9*b*e - 4*c*d)*(d^7)^(1/2)*(d + e*x)^(1/2))/(b^7*c^5))*(9*b*e - 4*c*d)*(d^7)^(1/2))/(2*b^3) + (2*(d + e*x)^(1/2)*(25*b^10*e^12 + 32*c^10*d ^10*e^2 - 160*b*c^9*d^9*e^3 + 234*b^2*c^8*d^8*e^4 + 24*b^3*c^7*d^7*e^5 - 4 20*b^4*c^6*d^6*e^6 + 504*b^5*c^5*d^5*e^7 - 42*b^6*c^4*d^4*e^8 - 408*b^7*c^ 3*d^3*e^9 + 396*b^8*c^2*d^2*e^10 - 160*b^9*c*d*e^11))/(b^4*c^5))*(9*b*e - 4*c*d)*(d^7)^(1/2)*1i)/(2*b^3) - (((((20*b^10*c^4*d*e^7 + 8*b^6*c^8*d^5*e^ 3 - 20*b^7*c^7*d^4*e^4 + 56*b^8*c^6*d^3*e^5 - 64*b^9*c^5*d^2*e^6)/(b^6*c^5 ) - ((4*b^7*c^7*e^3 - 8*b^6*c^8*d*e^2)*(9*b*e - 4*c*d)*(d^7)^(1/2)*(d + e* x)^(1/2))/(b^7*c^5))*(9*b*e - 4*c*d)*(d^7)^(1/2))/(2*b^3) - (2*(d + e*x)^( 1/2)*(25*b^10*e^12 + 32*c^10*d^10*e^2 - 160*b*c^9*d^9*e^3 + 234*b^2*c^8*d^ 8*e^4 + 24*b^3*c^7*d^7*e^5 - 420*b^4*c^6*d^6*e^6 + 504*b^5*c^5*d^5*e^7 - 4 2*b^6*c^4*d^4*e^8 - 408*b^7*c^3*d^3*e^9 + 396*b^8*c^2*d^2*e^10 - 160*b^9*c *d*e^11))/(b^4*c^5))*(9*b*e - 4*c*d)*(d^7)^(1/2)*1i)/(2*b^3))/((((((20*...