Integrand size = 21, antiderivative size = 299 \[ \int \frac {1}{\sqrt {d+e x} \left (b x+c x^2\right )^3} \, dx=-\frac {\sqrt {d+e x} (b (c d-b e)+c (2 c d-b e) x)}{2 b^2 d (c d-b e) \left (b x+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (b (c d-b e) \left (12 c^2 d^2-7 b c d e-3 b^2 e^2\right )+3 c (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) x\right )}{4 b^4 d^2 (c d-b e)^2 \left (b x+c x^2\right )}-\frac {3 \left (16 c^2 d^2+4 b c d e+b^2 e^2\right ) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5 d^{5/2}}+\frac {3 c^{5/2} \left (16 c^2 d^2-36 b c d e+21 b^2 e^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 (c d-b e)^{5/2}} \]
-3/4*(b^2*e^2+4*b*c*d*e+16*c^2*d^2)*arctanh((e*x+d)^(1/2)/d^(1/2))/b^5/d^( 5/2)+3/4*c^(5/2)*(21*b^2*e^2-36*b*c*d*e+16*c^2*d^2)*arctanh(c^(1/2)*(e*x+d )^(1/2)/(-b*e+c*d)^(1/2))/b^5/(-b*e+c*d)^(5/2)-1/2*(b*(-b*e+c*d)+c*(-b*e+2 *c*d)*x)*(e*x+d)^(1/2)/b^2/d/(-b*e+c*d)/(c*x^2+b*x)^2+1/4*(b*(-b*e+c*d)*(- 3*b^2*e^2-7*b*c*d*e+12*c^2*d^2)+3*c*(-b*e+2*c*d)*(-b^2*e^2-4*b*c*d*e+4*c^2 *d^2)*x)*(e*x+d)^(1/2)/b^4/d^2/(-b*e+c*d)^2/(c*x^2+b*x)
Time = 2.77 (sec) , antiderivative size = 292, normalized size of antiderivative = 0.98 \[ \int \frac {1}{\sqrt {d+e x} \left (b x+c x^2\right )^3} \, dx=\frac {\frac {b \sqrt {d+e x} \left (24 c^5 d^3 x^3+36 b c^4 d^2 x^2 (d-e x)+b^5 e^2 (-2 d+3 e x)+2 b^4 c e \left (2 d^2+d e x+3 e^2 x^2\right )+b^2 c^3 d x \left (8 d^2-55 d e x+6 e^2 x^2\right )+b^3 c^2 \left (-2 d^3-13 d^2 e x+10 d e^2 x^2+3 e^3 x^3\right )\right )}{d^2 (c d-b e)^2 x^2 (b+c x)^2}-\frac {3 c^{5/2} \left (16 c^2 d^2-36 b c d e+21 b^2 e^2\right ) \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {-c d+b e}}\right )}{(-c d+b e)^{5/2}}-\frac {3 \left (16 c^2 d^2+4 b c d e+b^2 e^2\right ) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{d^{5/2}}}{4 b^5} \]
((b*Sqrt[d + e*x]*(24*c^5*d^3*x^3 + 36*b*c^4*d^2*x^2*(d - e*x) + b^5*e^2*( -2*d + 3*e*x) + 2*b^4*c*e*(2*d^2 + d*e*x + 3*e^2*x^2) + b^2*c^3*d*x*(8*d^2 - 55*d*e*x + 6*e^2*x^2) + b^3*c^2*(-2*d^3 - 13*d^2*e*x + 10*d*e^2*x^2 + 3 *e^3*x^3)))/(d^2*(c*d - b*e)^2*x^2*(b + c*x)^2) - (3*c^(5/2)*(16*c^2*d^2 - 36*b*c*d*e + 21*b^2*e^2)*ArcTan[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[-(c*d) + b*e ]])/(-(c*d) + b*e)^(5/2) - (3*(16*c^2*d^2 + 4*b*c*d*e + b^2*e^2)*ArcTanh[S qrt[d + e*x]/Sqrt[d]])/d^(5/2))/(4*b^5)
Time = 0.69 (sec) , antiderivative size = 352, normalized size of antiderivative = 1.18, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {1165, 27, 1235, 27, 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.
| \(\displaystyle \int \frac {1}{\left (b x+c x^2\right )^3 \sqrt {d+e x}} \, dx\) |
| \(\Big \downarrow \) 1165 |
| \(\displaystyle -\frac {\int \frac {12 c^2 d^2-7 b c e d-3 b^2 e^2+5 c e (2 c d-b e) x}{2 \sqrt {d+e x} \left (c x^2+b x\right )^2}dx}{2 b^2 d (c d-b e)}-\frac {\sqrt {d+e x} (c x (2 c d-b e)+b (c d-b e))}{2 b^2 d \left (b x+c x^2\right )^2 (c d-b e)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {12 c^2 d^2-7 b c e d-3 b^2 e^2+5 c e (2 c d-b e) x}{\sqrt {d+e x} \left (c x^2+b x\right )^2}dx}{4 b^2 d (c d-b e)}-\frac {\sqrt {d+e x} (c x (2 c d-b e)+b (c d-b e))}{2 b^2 d \left (b x+c x^2\right )^2 (c d-b e)}\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle -\frac {-\frac {\int \frac {3 \left (\left (16 c^2 d^2+4 b c e d+b^2 e^2\right ) (c d-b e)^2+c e (2 c d-b e) \left (4 c^2 d^2-4 b c e d-b^2 e^2\right ) x\right )}{2 \sqrt {d+e x} \left (c x^2+b x\right )}dx}{b^2 d (c d-b e)}-\frac {\sqrt {d+e x} \left (3 c x (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right )+b (c d-b e) \left (-3 b^2 e^2-7 b c d e+12 c^2 d^2\right )\right )}{b^2 d \left (b x+c x^2\right ) (c d-b e)}}{4 b^2 d (c d-b e)}-\frac {\sqrt {d+e x} (c x (2 c d-b e)+b (c d-b e))}{2 b^2 d \left (b x+c x^2\right )^2 (c d-b e)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {3 \int \frac {\left (16 c^2 d^2+4 b c e d+b^2 e^2\right ) (c d-b e)^2+c e (2 c d-b e) \left (4 c^2 d^2-4 b c e d-b^2 e^2\right ) x}{\sqrt {d+e x} \left (c x^2+b x\right )}dx}{2 b^2 d (c d-b e)}-\frac {\sqrt {d+e x} \left (3 c x (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right )+b (c d-b e) \left (-3 b^2 e^2-7 b c d e+12 c^2 d^2\right )\right )}{b^2 d \left (b x+c x^2\right ) (c d-b e)}}{4 b^2 d (c d-b e)}-\frac {\sqrt {d+e x} (c x (2 c d-b e)+b (c d-b e))}{2 b^2 d \left (b x+c x^2\right )^2 (c d-b e)}\) |
| \(\Big \downarrow \) 1197 |
| \(\displaystyle -\frac {-\frac {3 \int \frac {e \left (8 c^4 d^4-16 b c^3 e d^3+7 b^2 c^2 e^2 d^2+b^3 c e^3 d+b^4 e^4+c (2 c d-b e) \left (4 c^2 d^2-4 b c e d-b^2 e^2\right ) (d+e x)\right )}{c (d+e x)^2-(2 c d-b e) (d+e x)+d (c d-b e)}d\sqrt {d+e x}}{b^2 d (c d-b e)}-\frac {\sqrt {d+e x} \left (3 c x (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right )+b (c d-b e) \left (-3 b^2 e^2-7 b c d e+12 c^2 d^2\right )\right )}{b^2 d \left (b x+c x^2\right ) (c d-b e)}}{4 b^2 d (c d-b e)}-\frac {\sqrt {d+e x} (c x (2 c d-b e)+b (c d-b e))}{2 b^2 d \left (b x+c x^2\right )^2 (c d-b e)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {3 e \int \frac {8 c^4 d^4-16 b c^3 e d^3+7 b^2 c^2 e^2 d^2+b^3 c e^3 d+b^4 e^4+c (2 c d-b e) \left (4 c^2 d^2-4 b c e d-b^2 e^2\right ) (d+e x)}{c (d+e x)^2-(2 c d-b e) (d+e x)+d (c d-b e)}d\sqrt {d+e x}}{b^2 d (c d-b e)}-\frac {\sqrt {d+e x} \left (3 c x (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right )+b (c d-b e) \left (-3 b^2 e^2-7 b c d e+12 c^2 d^2\right )\right )}{b^2 d \left (b x+c x^2\right ) (c d-b e)}}{4 b^2 d (c d-b e)}-\frac {\sqrt {d+e x} (c x (2 c d-b e)+b (c d-b e))}{2 b^2 d \left (b x+c x^2\right )^2 (c d-b e)}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle -\frac {-\frac {3 e \left (\frac {c (c d-b e)^2 \left (b^2 e^2+4 b c d e+16 c^2 d^2\right ) \int \frac {1}{c (d+e x)-c d}d\sqrt {d+e x}}{b e}-\frac {c^3 d^2 \left (21 b^2 e^2-36 b c d e+16 c^2 d^2\right ) \int \frac {1}{-c d+b e+c (d+e x)}d\sqrt {d+e x}}{b e}\right )}{b^2 d (c d-b e)}-\frac {\sqrt {d+e x} \left (3 c x (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right )+b (c d-b e) \left (-3 b^2 e^2-7 b c d e+12 c^2 d^2\right )\right )}{b^2 d \left (b x+c x^2\right ) (c d-b e)}}{4 b^2 d (c d-b e)}-\frac {\sqrt {d+e x} (c x (2 c d-b e)+b (c d-b e))}{2 b^2 d \left (b x+c x^2\right )^2 (c d-b e)}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {-\frac {3 e \left (\frac {c^{5/2} d^2 \left (21 b^2 e^2-36 b c d e+16 c^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b e \sqrt {c d-b e}}-\frac {\text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) (c d-b e)^2 \left (b^2 e^2+4 b c d e+16 c^2 d^2\right )}{b \sqrt {d} e}\right )}{b^2 d (c d-b e)}-\frac {\sqrt {d+e x} \left (3 c x (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right )+b (c d-b e) \left (-3 b^2 e^2-7 b c d e+12 c^2 d^2\right )\right )}{b^2 d \left (b x+c x^2\right ) (c d-b e)}}{4 b^2 d (c d-b e)}-\frac {\sqrt {d+e x} (c x (2 c d-b e)+b (c d-b e))}{2 b^2 d \left (b x+c x^2\right )^2 (c d-b e)}\) |
-1/2*(Sqrt[d + e*x]*(b*(c*d - b*e) + c*(2*c*d - b*e)*x))/(b^2*d*(c*d - b*e )*(b*x + c*x^2)^2) - (-((Sqrt[d + e*x]*(b*(c*d - b*e)*(12*c^2*d^2 - 7*b*c* d*e - 3*b^2*e^2) + 3*c*(2*c*d - b*e)*(4*c^2*d^2 - 4*b*c*d*e - b^2*e^2)*x)) /(b^2*d*(c*d - b*e)*(b*x + c*x^2))) - (3*e*(-(((c*d - b*e)^2*(16*c^2*d^2 + 4*b*c*d*e + b^2*e^2)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(b*Sqrt[d]*e)) + (c^ (5/2)*d^2*(16*c^2*d^2 - 36*b*c*d*e + 21*b^2*e^2)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b*e*Sqrt[c*d - b*e])))/(b^2*d*(c*d - b*e)))/(4*b ^2*d*(c*d - b*e))
3.4.83.3.1 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)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2 *a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*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 *(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d* m + b*e*m) - b*d*(3*c*d - b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p] )
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 = 2.44 (sec) , antiderivative size = 281, normalized size of antiderivative = 0.94
| method | result | size |
| risch | \(-\frac {\sqrt {e x +d}\, \left (-3 b e x -12 c d x +2 b d \right )}{4 d^{2} b^{4} x^{2}}+\frac {e \left (-\frac {\left (3 b^{2} e^{2}+12 b c d e +48 c^{2} d^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b e \sqrt {d}}-\frac {8 c^{3} d^{2} \left (\frac {\frac {3 c b e \left (5 b e -4 c d \right ) \left (e x +d \right )^{\frac {3}{2}}}{8 \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right )}+\frac {\left (17 b e -12 c d \right ) b e \sqrt {e x +d}}{8 b e -8 c d}}{\left (c \left (e x +d \right )+b e -c d \right )^{2}}+\frac {3 \left (21 b^{2} e^{2}-36 b c d e +16 c^{2} d^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{8 \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right ) \sqrt {\left (b e -c d \right ) c}}\right )}{b e}\right )}{4 b^{4} d^{2}}\) | \(281\) |
| derivativedivides | \(2 e^{5} \left (-\frac {\frac {-\frac {3 b e \left (b e +4 c d \right ) \left (e x +d \right )^{\frac {3}{2}}}{8 d^{2}}+\frac {b e \left (5 b e +12 c d \right ) \sqrt {e x +d}}{8 d}}{e^{2} x^{2}}+\frac {3 \left (b^{2} e^{2}+4 b c d e +16 c^{2} d^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{8 d^{\frac {5}{2}}}}{b^{5} e^{5}}-\frac {c^{3} \left (\frac {\frac {3 c b e \left (5 b e -4 c d \right ) \left (e x +d \right )^{\frac {3}{2}}}{8 \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right )}+\frac {\left (17 b e -12 c d \right ) b e \sqrt {e x +d}}{8 b e -8 c d}}{\left (c \left (e x +d \right )+b e -c d \right )^{2}}+\frac {3 \left (21 b^{2} e^{2}-36 b c d e +16 c^{2} d^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{8 \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right ) \sqrt {\left (b e -c d \right ) c}}\right )}{b^{5} e^{5}}\right )\) | \(295\) |
| default | \(2 e^{5} \left (-\frac {\frac {-\frac {3 b e \left (b e +4 c d \right ) \left (e x +d \right )^{\frac {3}{2}}}{8 d^{2}}+\frac {b e \left (5 b e +12 c d \right ) \sqrt {e x +d}}{8 d}}{e^{2} x^{2}}+\frac {3 \left (b^{2} e^{2}+4 b c d e +16 c^{2} d^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{8 d^{\frac {5}{2}}}}{b^{5} e^{5}}-\frac {c^{3} \left (\frac {\frac {3 c b e \left (5 b e -4 c d \right ) \left (e x +d \right )^{\frac {3}{2}}}{8 \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right )}+\frac {\left (17 b e -12 c d \right ) b e \sqrt {e x +d}}{8 b e -8 c d}}{\left (c \left (e x +d \right )+b e -c d \right )^{2}}+\frac {3 \left (21 b^{2} e^{2}-36 b c d e +16 c^{2} d^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{8 \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right ) \sqrt {\left (b e -c d \right ) c}}\right )}{b^{5} e^{5}}\right )\) | \(295\) |
| pseudoelliptic | \(\frac {12 x^{2} \left (c x +b \right )^{2} c^{3} \left (-\frac {21 b^{3} e^{3} d^{\frac {9}{2}}}{16}+d^{\frac {11}{2}} c \left (c^{2} d^{2}-\frac {13}{4} b c d e +\frac {57}{16} b^{2} e^{2}\right )\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )+\frac {\sqrt {\left (b e -c d \right ) c}\, \left (-\frac {3 d^{2} x^{2} \left (c x +b \right )^{2} \left (b^{2} e^{2}+4 b c d e +16 c^{2} d^{2}\right ) \left (b e -c d \right )^{3} \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{2}+\left (3 \left (-7 c^{3} x^{3}-\frac {65}{6} b \,c^{2} x^{2}-\frac {5}{2} b^{2} c x +b^{3}\right ) c \,e^{2} b^{2} d^{\frac {9}{2}}+\frac {3 b^{4} e^{4} x \left (c x +b \right )^{2} d^{\frac {5}{2}}}{2}+d^{\frac {7}{2}} \left (-b^{6} e^{3}-\frac {b^{5} c \,e^{3} x}{2}+\left (2 e^{3} x^{2}-3 d^{2} e \right ) c^{2} b^{4}+c^{3} \left (d^{3}+\frac {21}{2} d^{2} e x +\frac {3}{2} e^{3} x^{3}\right ) b^{3}-4 x \,c^{4} \left (-\frac {91 e x}{8}+d \right ) d^{2} b^{2}-18 x^{2} \left (-\frac {5 e x}{3}+d \right ) c^{5} d^{2} b -12 c^{6} d^{3} x^{3}\right )\right ) \sqrt {e x +d}\, b \right )}{2}}{\sqrt {\left (b e -c d \right ) c}\, x^{2} d^{\frac {9}{2}} \left (c x +b \right )^{2} b^{5} \left (b e -c d \right )^{3}}\) | \(378\) |
-1/4*(e*x+d)^(1/2)*(-3*b*e*x-12*c*d*x+2*b*d)/d^2/b^4/x^2+1/4/b^4/d^2*e*(-( 3*b^2*e^2+12*b*c*d*e+48*c^2*d^2)/b/e/d^(1/2)*arctanh((e*x+d)^(1/2)/d^(1/2) )-8*c^3*d^2/b/e*((3/8*c*b*e*(5*b*e-4*c*d)/(b^2*e^2-2*b*c*d*e+c^2*d^2)*(e*x +d)^(3/2)+1/8*(17*b*e-12*c*d)*b*e/(b*e-c*d)*(e*x+d)^(1/2))/(c*(e*x+d)+b*e- c*d)^2+3/8*(21*b^2*e^2-36*b*c*d*e+16*c^2*d^2)/(b^2*e^2-2*b*c*d*e+c^2*d^2)/ ((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 700 vs. \(2 (271) = 542\).
Time = 1.01 (sec) , antiderivative size = 2829, normalized size of antiderivative = 9.46 \[ \int \frac {1}{\sqrt {d+e x} \left (b x+c x^2\right )^3} \, dx=\text {Too large to display} \]
[1/8*(3*((16*c^6*d^5 - 36*b*c^5*d^4*e + 21*b^2*c^4*d^3*e^2)*x^4 + 2*(16*b* c^5*d^5 - 36*b^2*c^4*d^4*e + 21*b^3*c^3*d^3*e^2)*x^3 + (16*b^2*c^4*d^5 - 3 6*b^3*c^3*d^4*e + 21*b^4*c^2*d^3*e^2)*x^2)*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*((16*c^6*d^4 - 28*b*c^5*d^3*e + 9*b^2*c^4*d^2*e^2 + 2*b^3*c^3*d*e^3 + b^4*c^2*e^4)*x^4 + 2*(16*b*c^5*d^4 - 28*b^2*c^4*d^3*e + 9*b^3*c^3*d^2*e^ 2 + 2*b^4*c^2*d*e^3 + b^5*c*e^4)*x^3 + (16*b^2*c^4*d^4 - 28*b^3*c^3*d^3*e + 9*b^4*c^2*d^2*e^2 + 2*b^5*c*d*e^3 + b^6*e^4)*x^2)*sqrt(d)*log((e*x - 2*s qrt(e*x + d)*sqrt(d) + 2*d)/x) - 2*(2*b^4*c^2*d^4 - 4*b^5*c*d^3*e + 2*b^6* d^2*e^2 - 3*(8*b*c^5*d^4 - 12*b^2*c^4*d^3*e + 2*b^3*c^3*d^2*e^2 + b^4*c^2* d*e^3)*x^3 - (36*b^2*c^4*d^4 - 55*b^3*c^3*d^3*e + 10*b^4*c^2*d^2*e^2 + 6*b ^5*c*d*e^3)*x^2 - (8*b^3*c^3*d^4 - 13*b^4*c^2*d^3*e + 2*b^5*c*d^2*e^2 + 3* b^6*d*e^3)*x)*sqrt(e*x + d))/((b^5*c^4*d^5 - 2*b^6*c^3*d^4*e + b^7*c^2*d^3 *e^2)*x^4 + 2*(b^6*c^3*d^5 - 2*b^7*c^2*d^4*e + b^8*c*d^3*e^2)*x^3 + (b^7*c ^2*d^5 - 2*b^8*c*d^4*e + b^9*d^3*e^2)*x^2), 1/8*(6*((16*c^6*d^5 - 36*b*c^5 *d^4*e + 21*b^2*c^4*d^3*e^2)*x^4 + 2*(16*b*c^5*d^5 - 36*b^2*c^4*d^4*e + 21 *b^3*c^3*d^3*e^2)*x^3 + (16*b^2*c^4*d^5 - 36*b^3*c^3*d^4*e + 21*b^4*c^2*d^ 3*e^2)*x^2)*sqrt(-c/(c*d - b*e))*arctan(-(c*d - b*e)*sqrt(e*x + d)*sqrt(-c /(c*d - b*e))/(c*e*x + c*d)) + 3*((16*c^6*d^4 - 28*b*c^5*d^3*e + 9*b^2*c^4 *d^2*e^2 + 2*b^3*c^3*d*e^3 + b^4*c^2*e^4)*x^4 + 2*(16*b*c^5*d^4 - 28*b^...
\[ \int \frac {1}{\sqrt {d+e x} \left (b x+c x^2\right )^3} \, dx=\int \frac {1}{x^{3} \left (b + c x\right )^{3} \sqrt {d + e x}}\, dx \]
Exception generated. \[ \int \frac {1}{\sqrt {d+e x} \left (b x+c x^2\right )^3} \, dx=\text {Exception raised: ValueError} \]
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
Leaf count of result is larger than twice the leaf count of optimal. 602 vs. \(2 (271) = 542\).
Time = 0.29 (sec) , antiderivative size = 602, normalized size of antiderivative = 2.01 \[ \int \frac {1}{\sqrt {d+e x} \left (b x+c x^2\right )^3} \, dx=-\frac {3 \, {\left (16 \, c^{5} d^{2} - 36 \, b c^{4} d e + 21 \, b^{2} c^{3} e^{2}\right )} \arctan \left (\frac {\sqrt {e x + d} c}{\sqrt {-c^{2} d + b c e}}\right )}{4 \, {\left (b^{5} c^{2} d^{2} - 2 \, b^{6} c d e + b^{7} e^{2}\right )} \sqrt {-c^{2} d + b c e}} + \frac {24 \, {\left (e x + d\right )}^{\frac {7}{2}} c^{5} d^{3} e - 72 \, {\left (e x + d\right )}^{\frac {5}{2}} c^{5} d^{4} e + 72 \, {\left (e x + d\right )}^{\frac {3}{2}} c^{5} d^{5} e - 24 \, \sqrt {e x + d} c^{5} d^{6} e - 36 \, {\left (e x + d\right )}^{\frac {7}{2}} b c^{4} d^{2} e^{2} + 144 \, {\left (e x + d\right )}^{\frac {5}{2}} b c^{4} d^{3} e^{2} - 180 \, {\left (e x + d\right )}^{\frac {3}{2}} b c^{4} d^{4} e^{2} + 72 \, \sqrt {e x + d} b c^{4} d^{5} e^{2} + 6 \, {\left (e x + d\right )}^{\frac {7}{2}} b^{2} c^{3} d e^{3} - 73 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{2} c^{3} d^{2} e^{3} + 136 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{2} c^{3} d^{3} e^{3} - 69 \, \sqrt {e x + d} b^{2} c^{3} d^{4} e^{3} + 3 \, {\left (e x + d\right )}^{\frac {7}{2}} b^{3} c^{2} e^{4} + {\left (e x + d\right )}^{\frac {5}{2}} b^{3} c^{2} d e^{4} - 24 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{3} c^{2} d^{2} e^{4} + 18 \, \sqrt {e x + d} b^{3} c^{2} d^{3} e^{4} + 6 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{4} c e^{5} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{4} c d e^{5} + 8 \, \sqrt {e x + d} b^{4} c d^{2} e^{5} + 3 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{5} e^{6} - 5 \, \sqrt {e x + d} b^{5} d e^{6}}{4 \, {\left (b^{4} c^{2} d^{4} - 2 \, b^{5} c d^{3} e + b^{6} d^{2} e^{2}\right )} {\left ({\left (e x + d\right )}^{2} c - 2 \, {\left (e x + d\right )} c d + c d^{2} + {\left (e x + d\right )} b e - b d e\right )}^{2}} + \frac {3 \, {\left (16 \, c^{2} d^{2} + 4 \, b c d e + b^{2} e^{2}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-d}}\right )}{4 \, b^{5} \sqrt {-d} d^{2}} \]
-3/4*(16*c^5*d^2 - 36*b*c^4*d*e + 21*b^2*c^3*e^2)*arctan(sqrt(e*x + d)*c/s qrt(-c^2*d + b*c*e))/((b^5*c^2*d^2 - 2*b^6*c*d*e + b^7*e^2)*sqrt(-c^2*d + b*c*e)) + 1/4*(24*(e*x + d)^(7/2)*c^5*d^3*e - 72*(e*x + d)^(5/2)*c^5*d^4*e + 72*(e*x + d)^(3/2)*c^5*d^5*e - 24*sqrt(e*x + d)*c^5*d^6*e - 36*(e*x + d )^(7/2)*b*c^4*d^2*e^2 + 144*(e*x + d)^(5/2)*b*c^4*d^3*e^2 - 180*(e*x + d)^ (3/2)*b*c^4*d^4*e^2 + 72*sqrt(e*x + d)*b*c^4*d^5*e^2 + 6*(e*x + d)^(7/2)*b ^2*c^3*d*e^3 - 73*(e*x + d)^(5/2)*b^2*c^3*d^2*e^3 + 136*(e*x + d)^(3/2)*b^ 2*c^3*d^3*e^3 - 69*sqrt(e*x + d)*b^2*c^3*d^4*e^3 + 3*(e*x + d)^(7/2)*b^3*c ^2*e^4 + (e*x + d)^(5/2)*b^3*c^2*d*e^4 - 24*(e*x + d)^(3/2)*b^3*c^2*d^2*e^ 4 + 18*sqrt(e*x + d)*b^3*c^2*d^3*e^4 + 6*(e*x + d)^(5/2)*b^4*c*e^5 - 10*(e *x + d)^(3/2)*b^4*c*d*e^5 + 8*sqrt(e*x + d)*b^4*c*d^2*e^5 + 3*(e*x + d)^(3 /2)*b^5*e^6 - 5*sqrt(e*x + d)*b^5*d*e^6)/((b^4*c^2*d^4 - 2*b^5*c*d^3*e + b ^6*d^2*e^2)*((e*x + d)^2*c - 2*(e*x + d)*c*d + c*d^2 + (e*x + d)*b*e - b*d *e)^2) + 3/4*(16*c^2*d^2 + 4*b*c*d*e + b^2*e^2)*arctan(sqrt(e*x + d)/sqrt( -d))/(b^5*sqrt(-d)*d^2)
Time = 12.53 (sec) , antiderivative size = 6715, normalized size of antiderivative = 22.46 \[ \int \frac {1}{\sqrt {d+e x} \left (b x+c x^2\right )^3} \, dx=\text {Too large to display} \]
((e*(d + e*x)^(3/2)*(3*b^5*e^5 + 72*c^5*d^5 + 136*b^2*c^3*d^3*e^2 - 24*b^3 *c^2*d^2*e^3 - 180*b*c^4*d^4*e - 10*b^4*c*d*e^4))/(4*b^4*(c*d^2 - b*d*e)^2 ) - ((d + e*x)^(1/2)*(24*c^4*d^4*e - 5*b^4*e^5 - 48*b*c^3*d^3*e^2 + 21*b^2 *c^2*d^2*e^3 + 3*b^3*c*d*e^4))/(4*b^4*(c*d^2 - b*d*e)) + (e*(d + e*x)^(5/2 )*(6*b^4*c*e^4 - 72*c^5*d^4 + b^3*c^2*d*e^3 - 73*b^2*c^3*d^2*e^2 + 144*b*c ^4*d^3*e))/(4*b^4*(c*d^2 - b*d*e)^2) + (3*c*e*(d + e*x)^(7/2)*(8*c^4*d^3 + b^3*c*e^3 + 2*b^2*c^2*d*e^2 - 12*b*c^3*d^2*e))/(4*b^4*(c*d^2 - b*d*e)^2)) /(c^2*(d + e*x)^4 - (d + e*x)*(4*c^2*d^3 + 2*b^2*d*e^2 - 6*b*c*d^2*e) - (4 *c^2*d - 2*b*c*e)*(d + e*x)^3 + (d + e*x)^2*(b^2*e^2 + 6*c^2*d^2 - 6*b*c*d *e) + c^2*d^4 + b^2*d^2*e^2 - 2*b*c*d^3*e) - (atan((((-c^5*(b*e - c*d)^5)^ (1/2)*(((d + e*x)^(1/2)*(9*b^8*c^3*e^10 + 4608*c^11*d^8*e^2 - 18432*b*c^10 *d^7*e^3 + 36*b^7*c^4*d*e^9 + 27360*b^2*c^9*d^6*e^4 - 17568*b^3*c^8*d^5*e^ 5 + 3978*b^4*c^7*d^4*e^6 - 180*b^5*c^6*d^3*e^7 + 198*b^6*c^5*d^2*e^8))/(8* (b^8*c^4*d^8 + b^12*d^4*e^4 - 4*b^9*c^3*d^7*e - 4*b^11*c*d^5*e^3 + 6*b^10* c^2*d^6*e^2)) + (3*(-c^5*(b*e - c*d)^5)^(1/2)*((24*b^10*c^8*d^8*e^3 - 96*b ^11*c^7*d^7*e^4 + 141*b^12*c^6*d^6*e^5 - 87*b^13*c^5*d^5*e^6 + 18*b^14*c^4 *d^4*e^7 - 3*b^15*c^3*d^3*e^8 + 3*b^16*c^2*d^2*e^9)/(b^12*c^4*d^8 + b^16*d ^4*e^4 - 4*b^13*c^3*d^7*e - 4*b^15*c*d^5*e^3 + 6*b^14*c^2*d^6*e^2) - (3*(- c^5*(b*e - c*d)^5)^(1/2)*(d + e*x)^(1/2)*(21*b^2*e^2 + 16*c^2*d^2 - 36*b*c *d*e)*(128*b^10*c^7*d^9*e^2 - 576*b^11*c^6*d^8*e^3 + 1024*b^12*c^5*d^7*...