Integrand size = 21, antiderivative size = 267 \[ \int \frac {1}{(d+e x)^{5/2} \left (b x+c x^2\right )^2} \, dx=-\frac {e \left (6 c^2 d^2-6 b c d e+5 b^2 e^2\right )}{3 b^2 d^2 (c d-b e)^2 (d+e x)^{3/2}}-\frac {e (2 c d-b e) \left (c^2 d^2-b c d e+5 b^2 e^2\right )}{b^2 d^3 (c d-b e)^3 \sqrt {d+e x}}-\frac {b (c d-b e)+c (2 c d-b e) x}{b^2 d (c d-b e) (d+e x)^{3/2} \left (b x+c x^2\right )}+\frac {(4 c d+5 b e) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b^3 d^{7/2}}-\frac {c^{7/2} (4 c d-9 b e) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 (c d-b e)^{7/2}} \]
-1/3*e*(5*b^2*e^2-6*b*c*d*e+6*c^2*d^2)/b^2/d^2/(-b*e+c*d)^2/(e*x+d)^(3/2)+ (-b*(-b*e+c*d)-c*(-b*e+2*c*d)*x)/b^2/d/(-b*e+c*d)/(e*x+d)^(3/2)/(c*x^2+b*x )+(5*b*e+4*c*d)*arctanh((e*x+d)^(1/2)/d^(1/2))/b^3/d^(7/2)-c^(7/2)*(-9*b*e +4*c*d)*arctanh(c^(1/2)*(e*x+d)^(1/2)/(-b*e+c*d)^(1/2))/b^3/(-b*e+c*d)^(7/ 2)-e*(-b*e+2*c*d)*(5*b^2*e^2-b*c*d*e+c^2*d^2)/b^2/d^3/(-b*e+c*d)^3/(e*x+d) ^(1/2)
Time = 1.74 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.02 \[ \int \frac {1}{(d+e x)^{5/2} \left (b x+c x^2\right )^2} \, dx=\frac {\frac {b \left (-6 c^4 d^3 x (d+e x)^2-3 b c^3 d^2 (d-3 e x) (d+e x)^2+b^4 e^3 \left (3 d^2+20 d e x+15 e^2 x^2\right )+b^2 c^2 d e \left (9 d^3+9 d^2 e x-35 d e^2 x^2-33 e^3 x^3\right )+b^3 c e^2 \left (-9 d^3-41 d^2 e x-13 d e^2 x^2+15 e^3 x^3\right )\right )}{d^3 (c d-b e)^3 x (b+c x) (d+e x)^{3/2}}-\frac {3 c^{7/2} (4 c d-9 b e) \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {-c d+b e}}\right )}{(-c d+b e)^{7/2}}+\frac {3 (4 c d+5 b e) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{d^{7/2}}}{3 b^3} \]
((b*(-6*c^4*d^3*x*(d + e*x)^2 - 3*b*c^3*d^2*(d - 3*e*x)*(d + e*x)^2 + b^4* e^3*(3*d^2 + 20*d*e*x + 15*e^2*x^2) + b^2*c^2*d*e*(9*d^3 + 9*d^2*e*x - 35* d*e^2*x^2 - 33*e^3*x^3) + b^3*c*e^2*(-9*d^3 - 41*d^2*e*x - 13*d*e^2*x^2 + 15*e^3*x^3)))/(d^3*(c*d - b*e)^3*x*(b + c*x)*(d + e*x)^(3/2)) - (3*c^(7/2) *(4*c*d - 9*b*e)*ArcTan[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[-(c*d) + b*e]])/(-(c* d) + b*e)^(7/2) + (3*(4*c*d + 5*b*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/d^(7/ 2))/(3*b^3)
Time = 0.71 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.25, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {1165, 27, 1198, 1198, 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.
| \(\displaystyle \int \frac {1}{\left (b x+c x^2\right )^2 (d+e x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1165 |
| \(\displaystyle -\frac {\int \frac {(c d-b e) (4 c d+5 b e)+5 c e (2 c d-b e) x}{2 (d+e x)^{5/2} \left (c x^2+b x\right )}dx}{b^2 d (c d-b e)}-\frac {c x (2 c d-b e)+b (c d-b e)}{b^2 d \left (b x+c x^2\right ) (d+e x)^{3/2} (c d-b e)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {(c d-b e) (4 c d+5 b e)+5 c e (2 c d-b e) x}{(d+e x)^{5/2} \left (c x^2+b x\right )}dx}{2 b^2 d (c d-b e)}-\frac {c x (2 c d-b e)+b (c d-b e)}{b^2 d \left (b x+c x^2\right ) (d+e x)^{3/2} (c d-b e)}\) |
| \(\Big \downarrow \) 1198 |
| \(\displaystyle -\frac {\frac {\int \frac {(4 c d+5 b e) (c d-b e)^2+c e \left (6 c^2 d^2-6 b c e d+5 b^2 e^2\right ) x}{(d+e x)^{3/2} \left (c x^2+b x\right )}dx}{d (c d-b e)}+\frac {2 e \left (5 b^2 e^2-6 b c d e+6 c^2 d^2\right )}{3 d (d+e x)^{3/2} (c d-b e)}}{2 b^2 d (c d-b e)}-\frac {c x (2 c d-b e)+b (c d-b e)}{b^2 d \left (b x+c x^2\right ) (d+e x)^{3/2} (c d-b e)}\) |
| \(\Big \downarrow \) 1198 |
| \(\displaystyle -\frac {\frac {\frac {\int \frac {(4 c d+5 b e) (c d-b e)^3+c e (2 c d-b e) \left (c^2 d^2-b c e d+5 b^2 e^2\right ) x}{\sqrt {d+e x} \left (c x^2+b x\right )}dx}{d (c d-b e)}+\frac {2 e (2 c d-b e) \left (5 b^2 e^2-b c d e+c^2 d^2\right )}{d \sqrt {d+e x} (c d-b e)}}{d (c d-b e)}+\frac {2 e \left (5 b^2 e^2-6 b c d e+6 c^2 d^2\right )}{3 d (d+e x)^{3/2} (c d-b e)}}{2 b^2 d (c d-b e)}-\frac {c x (2 c d-b e)+b (c d-b e)}{b^2 d \left (b x+c x^2\right ) (d+e x)^{3/2} (c d-b e)}\) |
| \(\Big \downarrow \) 1197 |
| \(\displaystyle -\frac {\frac {\frac {2 \int \frac {e \left (2 c^4 d^4-4 b c^3 e d^3-14 b^2 c^2 e^2 d^2+16 b^3 c e^3 d-5 b^4 e^4+c (2 c d-b e) \left (c^2 d^2-b c e d+5 b^2 e^2\right ) (d+e x)\right )}{c (d+e x)^2-(2 c d-b e) (d+e x)+d (c d-b e)}d\sqrt {d+e x}}{d (c d-b e)}+\frac {2 e (2 c d-b e) \left (5 b^2 e^2-b c d e+c^2 d^2\right )}{d \sqrt {d+e x} (c d-b e)}}{d (c d-b e)}+\frac {2 e \left (5 b^2 e^2-6 b c d e+6 c^2 d^2\right )}{3 d (d+e x)^{3/2} (c d-b e)}}{2 b^2 d (c d-b e)}-\frac {c x (2 c d-b e)+b (c d-b e)}{b^2 d \left (b x+c x^2\right ) (d+e x)^{3/2} (c d-b e)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\frac {2 e \int \frac {2 c^4 d^4-4 b c^3 e d^3-14 b^2 c^2 e^2 d^2+16 b^3 c e^3 d-5 b^4 e^4+c (2 c d-b e) \left (c^2 d^2-b c e d+5 b^2 e^2\right ) (d+e x)}{c (d+e x)^2-(2 c d-b e) (d+e x)+d (c d-b e)}d\sqrt {d+e x}}{d (c d-b e)}+\frac {2 e (2 c d-b e) \left (5 b^2 e^2-b c d e+c^2 d^2\right )}{d \sqrt {d+e x} (c d-b e)}}{d (c d-b e)}+\frac {2 e \left (5 b^2 e^2-6 b c d e+6 c^2 d^2\right )}{3 d (d+e x)^{3/2} (c d-b e)}}{2 b^2 d (c d-b e)}-\frac {c x (2 c d-b e)+b (c d-b e)}{b^2 d \left (b x+c x^2\right ) (d+e x)^{3/2} (c d-b e)}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle -\frac {\frac {\frac {2 e \left (\frac {c (c d-b e)^3 (5 b e+4 c d) \int \frac {1}{c (d+e x)-c d}d\sqrt {d+e x}}{b e}-\frac {c^4 d^3 (4 c d-9 b e) \int \frac {1}{-c d+b e+c (d+e x)}d\sqrt {d+e x}}{b e}\right )}{d (c d-b e)}+\frac {2 e (2 c d-b e) \left (5 b^2 e^2-b c d e+c^2 d^2\right )}{d \sqrt {d+e x} (c d-b e)}}{d (c d-b e)}+\frac {2 e \left (5 b^2 e^2-6 b c d e+6 c^2 d^2\right )}{3 d (d+e x)^{3/2} (c d-b e)}}{2 b^2 d (c d-b e)}-\frac {c x (2 c d-b e)+b (c d-b e)}{b^2 d \left (b x+c x^2\right ) (d+e x)^{3/2} (c d-b e)}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {\frac {\frac {2 e \left (\frac {c^{7/2} d^3 (4 c d-9 b e) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b e \sqrt {c d-b e}}-\frac {\text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) (c d-b e)^3 (5 b e+4 c d)}{b \sqrt {d} e}\right )}{d (c d-b e)}+\frac {2 e (2 c d-b e) \left (5 b^2 e^2-b c d e+c^2 d^2\right )}{d \sqrt {d+e x} (c d-b e)}}{d (c d-b e)}+\frac {2 e \left (5 b^2 e^2-6 b c d e+6 c^2 d^2\right )}{3 d (d+e x)^{3/2} (c d-b e)}}{2 b^2 d (c d-b e)}-\frac {c x (2 c d-b e)+b (c d-b e)}{b^2 d \left (b x+c x^2\right ) (d+e x)^{3/2} (c d-b e)}\) |
-((b*(c*d - b*e) + c*(2*c*d - b*e)*x)/(b^2*d*(c*d - b*e)*(d + e*x)^(3/2)*( b*x + c*x^2))) - ((2*e*(6*c^2*d^2 - 6*b*c*d*e + 5*b^2*e^2))/(3*d*(c*d - b* e)*(d + e*x)^(3/2)) + ((2*e*(2*c*d - b*e)*(c^2*d^2 - b*c*d*e + 5*b^2*e^2)) /(d*(c*d - b*e)*Sqrt[d + e*x]) + (2*e*(-(((c*d - b*e)^3*(4*c*d + 5*b*e)*Ar cTanh[Sqrt[d + e*x]/Sqrt[d]])/(b*Sqrt[d]*e)) + (c^(7/2)*d^3*(4*c*d - 9*b*e )*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b*e*Sqrt[c*d - b*e])) )/(d*(c*d - b*e)))/(d*(c*d - b*e)))/(2*b^2*d*(c*d - b*e))
3.4.76.3.1 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)*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e) *x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^ 2))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^m*Simp[b*c*d*e*(2*p - m + 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && LtQ[p, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]
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_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(e*f - d*g)*((d + e*x)^(m + 1)/((m + 1)*(c *d^2 - b*d*e + a*e^2))), x] + Simp[1/(c*d^2 - b*d*e + a*e^2) Int[(d + e*x )^(m + 1)*(Simp[c*d*f - f*b*e + a*e*g - c*(e*f - d*g)*x, x]/(a + b*x + c*x^ 2)), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && FractionQ[m] && LtQ[m, -1 ]
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 = 2.11 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.76
| method | result | size |
| derivativedivides | \(2 e^{3} \left (\frac {-\frac {b \sqrt {e x +d}}{2 x}+\frac {\left (5 b e +4 c d \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{2 \sqrt {d}}}{d^{3} b^{3} e^{3}}+\frac {c^{4} \left (\frac {b e \sqrt {e x +d}}{2 c \left (e x +d \right )+2 b e -2 c d}+\frac {\left (9 b e -4 c d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{2 \sqrt {\left (b e -c d \right ) c}}\right )}{b^{3} e^{3} \left (b e -c d \right )^{3}}-\frac {1}{3 d^{2} \left (b e -c d \right )^{2} \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 b e -4 c d}{d^{3} \left (b e -c d \right )^{3} \sqrt {e x +d}}\right )\) | \(204\) |
| default | \(2 e^{3} \left (\frac {-\frac {b \sqrt {e x +d}}{2 x}+\frac {\left (5 b e +4 c d \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{2 \sqrt {d}}}{d^{3} b^{3} e^{3}}+\frac {c^{4} \left (\frac {b e \sqrt {e x +d}}{2 c \left (e x +d \right )+2 b e -2 c d}+\frac {\left (9 b e -4 c d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{2 \sqrt {\left (b e -c d \right ) c}}\right )}{b^{3} e^{3} \left (b e -c d \right )^{3}}-\frac {1}{3 d^{2} \left (b e -c d \right )^{2} \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 b e -4 c d}{d^{3} \left (b e -c d \right )^{3} \sqrt {e x +d}}\right )\) | \(204\) |
| risch | \(-\frac {\sqrt {e x +d}}{d^{3} b^{2} x}-\frac {e \left (-\frac {\left (5 b e +4 c d \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b e \sqrt {d}}+\frac {2 b^{2} d \,e^{2}}{3 \left (b e -c d \right )^{2} \left (e x +d \right )^{\frac {3}{2}}}+\frac {4 b^{2} e^{2} \left (b e -2 c d \right )}{\left (b e -c d \right )^{3} \sqrt {e x +d}}-\frac {2 c^{4} d^{3} \left (\frac {b e \sqrt {e x +d}}{2 c \left (e x +d \right )+2 b e -2 c d}+\frac {\left (9 b e -4 c d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{2 \sqrt {\left (b e -c d \right ) c}}\right )}{\left (b e -c d \right )^{3} b e}\right )}{b^{2} d^{3}}\) | \(219\) |
| pseudoelliptic | \(e^{3} \left (-\frac {-5 \,\operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right ) b e x -4 \,\operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right ) c d x +b \sqrt {e x +d}\, \sqrt {d}}{x \,d^{\frac {7}{2}} b^{3} e^{3}}-\frac {2}{3 d^{2} \left (b e -c d \right )^{2} \left (e x +d \right )^{\frac {3}{2}}}-\frac {4 \left (b e -2 c d \right )}{d^{3} \left (b e -c d \right )^{3} \sqrt {e x +d}}+\frac {\sqrt {e x +d}\, c^{4}}{b^{2} e^{3} \left (c x +b \right ) \left (b e -c d \right )^{3}}+\frac {9 \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right ) c^{4}}{\sqrt {\left (b e -c d \right ) c}\, b^{2} e^{2} \left (b e -c d \right )^{3}}-\frac {4 \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right ) c^{5} d}{\sqrt {\left (b e -c d \right ) c}\, b^{3} e^{3} \left (b e -c d \right )^{3}}\right )\) | \(264\) |
2*e^3*(1/d^3/b^3/e^3*(-1/2*b*(e*x+d)^(1/2)/x+1/2*(5*b*e+4*c*d)/d^(1/2)*arc tanh((e*x+d)^(1/2)/d^(1/2)))+c^4/b^3/e^3/(b*e-c*d)^3*(1/2*b*e*(e*x+d)^(1/2 )/(c*(e*x+d)+b*e-c*d)+1/2*(9*b*e-4*c*d)/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+ d)^(1/2)/((b*e-c*d)*c)^(1/2)))-1/3/d^2/(b*e-c*d)^2/(e*x+d)^(3/2)-(2*b*e-4* c*d)/d^3/(b*e-c*d)^3/(e*x+d)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 960 vs. \(2 (243) = 486\).
Time = 1.64 (sec) , antiderivative size = 3872, normalized size of antiderivative = 14.50 \[ \int \frac {1}{(d+e x)^{5/2} \left (b x+c x^2\right )^2} \, dx=\text {Too large to display} \]
[1/6*(3*((4*c^5*d^5*e^2 - 9*b*c^4*d^4*e^3)*x^4 + (8*c^5*d^6*e - 14*b*c^4*d ^5*e^2 - 9*b^2*c^3*d^4*e^3)*x^3 + (4*c^5*d^7 - b*c^4*d^6*e - 18*b^2*c^3*d^ 5*e^2)*x^2 + (4*b*c^4*d^7 - 9*b^2*c^3*d^6*e)*x)*sqrt(c/(c*d - b*e))*log((c *e*x + 2*c*d - b*e - 2*(c*d - b*e)*sqrt(e*x + d)*sqrt(c/(c*d - b*e)))/(c*x + b)) + 3*((4*c^5*d^4*e^2 - 7*b*c^4*d^3*e^3 - 3*b^2*c^3*d^2*e^4 + 11*b^3* c^2*d*e^5 - 5*b^4*c*e^6)*x^4 + (8*c^5*d^5*e - 10*b*c^4*d^4*e^2 - 13*b^2*c^ 3*d^3*e^3 + 19*b^3*c^2*d^2*e^4 + b^4*c*d*e^5 - 5*b^5*e^6)*x^3 + (4*c^5*d^6 + b*c^4*d^5*e - 17*b^2*c^3*d^4*e^2 + 5*b^3*c^2*d^3*e^3 + 17*b^4*c*d^2*e^4 - 10*b^5*d*e^5)*x^2 + (4*b*c^4*d^6 - 7*b^2*c^3*d^5*e - 3*b^3*c^2*d^4*e^2 + 11*b^4*c*d^3*e^3 - 5*b^5*d^2*e^4)*x)*sqrt(d)*log((e*x + 2*sqrt(e*x + d)* sqrt(d) + 2*d)/x) - 2*(3*b^2*c^3*d^6 - 9*b^3*c^2*d^5*e + 9*b^4*c*d^4*e^2 - 3*b^5*d^3*e^3 + 3*(2*b*c^4*d^4*e^2 - 3*b^2*c^3*d^3*e^3 + 11*b^3*c^2*d^2*e ^4 - 5*b^4*c*d*e^5)*x^3 + (12*b*c^4*d^5*e - 15*b^2*c^3*d^4*e^2 + 35*b^3*c^ 2*d^3*e^3 + 13*b^4*c*d^2*e^4 - 15*b^5*d*e^5)*x^2 + (6*b*c^4*d^6 - 3*b^2*c^ 3*d^5*e - 9*b^3*c^2*d^4*e^2 + 41*b^4*c*d^3*e^3 - 20*b^5*d^2*e^4)*x)*sqrt(e *x + d))/((b^3*c^4*d^7*e^2 - 3*b^4*c^3*d^6*e^3 + 3*b^5*c^2*d^5*e^4 - b^6*c *d^4*e^5)*x^4 + (2*b^3*c^4*d^8*e - 5*b^4*c^3*d^7*e^2 + 3*b^5*c^2*d^6*e^3 + b^6*c*d^5*e^4 - b^7*d^4*e^5)*x^3 + (b^3*c^4*d^9 - b^4*c^3*d^8*e - 3*b^5*c ^2*d^7*e^2 + 5*b^6*c*d^6*e^3 - 2*b^7*d^5*e^4)*x^2 + (b^4*c^3*d^9 - 3*b^5*c ^2*d^8*e + 3*b^6*c*d^7*e^2 - b^7*d^6*e^3)*x), -1/6*(6*((4*c^5*d^5*e^2 -...
\[ \int \frac {1}{(d+e x)^{5/2} \left (b x+c x^2\right )^2} \, dx=\int \frac {1}{x^{2} \left (b + c x\right )^{2} \left (d + e x\right )^{\frac {5}{2}}}\, dx \]
Exception generated. \[ \int \frac {1}{(d+e x)^{5/2} \left (b x+c x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]
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 = 0.30 (sec) , antiderivative size = 470, normalized size of antiderivative = 1.76 \[ \int \frac {1}{(d+e x)^{5/2} \left (b x+c x^2\right )^2} \, dx=\frac {{\left (4 \, c^{5} d - 9 \, b c^{4} e\right )} \arctan \left (\frac {\sqrt {e x + d} c}{\sqrt {-c^{2} d + b c e}}\right )}{{\left (b^{3} c^{3} d^{3} - 3 \, b^{4} c^{2} d^{2} e + 3 \, b^{5} c d e^{2} - b^{6} e^{3}\right )} \sqrt {-c^{2} d + b c e}} - \frac {2 \, {\left (e x + d\right )}^{\frac {3}{2}} c^{4} d^{3} e - 2 \, \sqrt {e x + d} c^{4} d^{4} e - 3 \, {\left (e x + d\right )}^{\frac {3}{2}} b c^{3} d^{2} e^{2} + 4 \, \sqrt {e x + d} b c^{3} d^{3} e^{2} + 3 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{2} c^{2} d e^{3} - 6 \, \sqrt {e x + d} b^{2} c^{2} d^{2} e^{3} - {\left (e x + d\right )}^{\frac {3}{2}} b^{3} c e^{4} + 4 \, \sqrt {e x + d} b^{3} c d e^{4} - \sqrt {e x + d} b^{4} e^{5}}{{\left (b^{2} c^{3} d^{6} - 3 \, b^{3} c^{2} d^{5} e + 3 \, b^{4} c d^{4} e^{2} - b^{5} d^{3} e^{3}\right )} {\left ({\left (e x + d\right )}^{2} c - 2 \, {\left (e x + d\right )} c d + c d^{2} + {\left (e x + d\right )} b e - b d e\right )}} - \frac {2 \, {\left (12 \, {\left (e x + d\right )} c d e^{3} + c d^{2} e^{3} - 6 \, {\left (e x + d\right )} b e^{4} - b d e^{4}\right )}}{3 \, {\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}\right )} {\left (e x + d\right )}^{\frac {3}{2}}} - \frac {{\left (4 \, c d + 5 \, b e\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-d}}\right )}{b^{3} \sqrt {-d} d^{3}} \]
(4*c^5*d - 9*b*c^4*e)*arctan(sqrt(e*x + d)*c/sqrt(-c^2*d + b*c*e))/((b^3*c ^3*d^3 - 3*b^4*c^2*d^2*e + 3*b^5*c*d*e^2 - b^6*e^3)*sqrt(-c^2*d + b*c*e)) - (2*(e*x + d)^(3/2)*c^4*d^3*e - 2*sqrt(e*x + d)*c^4*d^4*e - 3*(e*x + d)^( 3/2)*b*c^3*d^2*e^2 + 4*sqrt(e*x + d)*b*c^3*d^3*e^2 + 3*(e*x + d)^(3/2)*b^2 *c^2*d*e^3 - 6*sqrt(e*x + d)*b^2*c^2*d^2*e^3 - (e*x + d)^(3/2)*b^3*c*e^4 + 4*sqrt(e*x + d)*b^3*c*d*e^4 - sqrt(e*x + d)*b^4*e^5)/((b^2*c^3*d^6 - 3*b^ 3*c^2*d^5*e + 3*b^4*c*d^4*e^2 - b^5*d^3*e^3)*((e*x + d)^2*c - 2*(e*x + d)* c*d + c*d^2 + (e*x + d)*b*e - b*d*e)) - 2/3*(12*(e*x + d)*c*d*e^3 + c*d^2* e^3 - 6*(e*x + d)*b*e^4 - b*d*e^4)/((c^3*d^6 - 3*b*c^2*d^5*e + 3*b^2*c*d^4 *e^2 - b^3*d^3*e^3)*(e*x + d)^(3/2)) - (4*c*d + 5*b*e)*arctan(sqrt(e*x + d )/sqrt(-d))/(b^3*sqrt(-d)*d^3)
Time = 12.49 (sec) , antiderivative size = 5736, normalized size of antiderivative = 21.48 \[ \int \frac {1}{(d+e x)^{5/2} \left (b x+c x^2\right )^2} \, dx=\text {Too large to display} \]
((10*e^3*(b*e - 2*c*d)*(d + e*x))/(3*(c*d^2 - b*d*e)^2) - (2*e^3)/(3*(c*d^ 2 - b*d*e)) + (e*(d + e*x)^2*(15*b^4*e^4 + 6*c^4*d^4 + 64*b^2*c^2*d^2*e^2 - 12*b*c^3*d^3*e - 58*b^3*c*d*e^3))/(3*b^2*(c*d^2 - b*d*e)^3) + (e*(b*e - 2*c*d)*(d + e*x)^3*(c^3*d^2 + 5*b^2*c*e^2 - b*c^2*d*e))/(b^2*(c*d^2 - b*d* e)^3))/(c*(d + e*x)^(7/2) + (c*d^2 - b*d*e)*(d + e*x)^(3/2) + (b*e - 2*c*d )*(d + e*x)^(5/2)) + (atan((((-c^7*(b*e - c*d)^7)^(1/2)*(9*b*e - 4*c*d)*(( d + e*x)^(1/2)*(64*b^6*c^20*d^26*e^2 - 832*b^7*c^19*d^25*e^3 + 4820*b^8*c^ 18*d^24*e^4 - 16240*b^9*c^17*d^23*e^5 + 34490*b^10*c^16*d^22*e^6 - 45430*b ^11*c^15*d^21*e^7 + 29414*b^12*c^14*d^20*e^8 + 10670*b^13*c^13*d^19*e^9 - 39550*b^14*c^12*d^18*e^10 + 25730*b^15*c^11*d^17*e^11 + 19048*b^16*c^10*d^ 16*e^12 - 53852*b^17*c^9*d^15*e^13 + 55510*b^18*c^8*d^14*e^14 - 35210*b^19 *c^7*d^13*e^15 + 14830*b^20*c^6*d^12*e^16 - 4082*b^21*c^5*d^11*e^17 + 670* b^22*c^4*d^10*e^18 - 50*b^23*c^3*d^9*e^19) + ((-c^7*(b*e - c*d)^7)^(1/2)*( 9*b*e - 4*c*d)*(8*b^10*c^18*d^28*e^3 - 112*b^11*c^17*d^27*e^4 + 664*b^12*c ^16*d^26*e^5 - 2080*b^13*c^15*d^25*e^6 + 2996*b^14*c^14*d^24*e^7 + 2528*b^ 15*c^13*d^23*e^8 - 23056*b^16*c^12*d^22*e^9 + 59312*b^17*c^11*d^21*e^10 - 95700*b^18*c^10*d^20*e^11 + 109648*b^19*c^9*d^19*e^12 - 92840*b^20*c^8*d^1 8*e^13 + 58688*b^21*c^7*d^17*e^14 - 27476*b^22*c^6*d^16*e^15 + 9280*b^23*c ^5*d^15*e^16 - 2144*b^24*c^4*d^14*e^17 + 304*b^25*c^3*d^13*e^18 - 20*b^26* c^2*d^12*e^19 - ((-c^7*(b*e - c*d)^7)^(1/2)*(9*b*e - 4*c*d)*(d + e*x)^(...