Integrand size = 21, antiderivative size = 207 \[ \int \frac {1}{(d+e x)^2 \left (b x+c x^2\right )^{3/2}} \, dx=-\frac {2 (b (c d-b e)+c (2 c d-b e) x)}{b^2 d (c d-b e) (d+e x) \sqrt {b x+c x^2}}-\frac {e \left (4 c^2 d^2-4 b c d e+3 b^2 e^2\right ) \sqrt {b x+c x^2}}{b^2 d^2 (c d-b e)^2 (d+e x)}+\frac {3 e^2 (2 c d-b e) \text {arctanh}\left (\frac {b d+(2 c d-b e) x}{2 \sqrt {d} \sqrt {c d-b e} \sqrt {b x+c x^2}}\right )}{2 d^{5/2} (c d-b e)^{5/2}} \]
3/2*e^2*(-b*e+2*c*d)*arctanh(1/2*(b*d+(-b*e+2*c*d)*x)/d^(1/2)/(-b*e+c*d)^( 1/2)/(c*x^2+b*x)^(1/2))/d^(5/2)/(-b*e+c*d)^(5/2)-2*(b*(-b*e+c*d)+c*(-b*e+2 *c*d)*x)/b^2/d/(-b*e+c*d)/(e*x+d)/(c*x^2+b*x)^(1/2)-e*(3*b^2*e^2-4*b*c*d*e +4*c^2*d^2)*(c*x^2+b*x)^(1/2)/b^2/d^2/(-b*e+c*d)^2/(e*x+d)
Time = 1.20 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.05 \[ \int \frac {1}{(d+e x)^2 \left (b x+c x^2\right )^{3/2}} \, dx=\frac {x \left (-\frac {\sqrt {d} (b+c x) \left (4 c^3 d^2 x (d+e x)+b^3 e^2 (2 d+3 e x)+2 b c^2 d \left (d^2-d e x-2 e^2 x^2\right )+b^2 c e \left (-4 d^2-2 d e x+3 e^2 x^2\right )\right )}{b^2 (c d-b e)^2 (d+e x)}-\frac {3 e^2 (2 c d-b e) \sqrt {x} (b+c x)^{3/2} \arctan \left (\frac {-e \sqrt {x} \sqrt {b+c x}+\sqrt {c} (d+e x)}{\sqrt {d} \sqrt {-c d+b e}}\right )}{(-c d+b e)^{5/2}}\right )}{d^{5/2} (x (b+c x))^{3/2}} \]
(x*(-((Sqrt[d]*(b + c*x)*(4*c^3*d^2*x*(d + e*x) + b^3*e^2*(2*d + 3*e*x) + 2*b*c^2*d*(d^2 - d*e*x - 2*e^2*x^2) + b^2*c*e*(-4*d^2 - 2*d*e*x + 3*e^2*x^ 2)))/(b^2*(c*d - b*e)^2*(d + e*x))) - (3*e^2*(2*c*d - b*e)*Sqrt[x]*(b + c* x)^(3/2)*ArcTan[(-(e*Sqrt[x]*Sqrt[b + c*x]) + Sqrt[c]*(d + e*x))/(Sqrt[d]* Sqrt[-(c*d) + b*e])])/(-(c*d) + b*e)^(5/2)))/(d^(5/2)*(x*(b + c*x))^(3/2))
Time = 0.42 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.08, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {1165, 27, 1228, 1154, 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.
| \(\displaystyle \int \frac {1}{\left (b x+c x^2\right )^{3/2} (d+e x)^2} \, dx\) |
| \(\Big \downarrow \) 1165 |
| \(\displaystyle -\frac {2 \int \frac {e (b (2 c d-3 b e)+2 c (2 c d-b e) x)}{2 (d+e x)^2 \sqrt {c x^2+b x}}dx}{b^2 d (c d-b e)}-\frac {2 (c x (2 c d-b e)+b (c d-b e))}{b^2 d \sqrt {b x+c x^2} (d+e x) (c d-b e)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {e \int \frac {b (2 c d-3 b e)+2 c (2 c d-b e) x}{(d+e x)^2 \sqrt {c x^2+b x}}dx}{b^2 d (c d-b e)}-\frac {2 (c x (2 c d-b e)+b (c d-b e))}{b^2 d \sqrt {b x+c x^2} (d+e x) (c d-b e)}\) |
| \(\Big \downarrow \) 1228 |
| \(\displaystyle -\frac {e \left (\frac {\sqrt {b x+c x^2} \left (3 b^2 e^2-4 b c d e+4 c^2 d^2\right )}{d (d+e x) (c d-b e)}-\frac {3 b^2 e (2 c d-b e) \int \frac {1}{(d+e x) \sqrt {c x^2+b x}}dx}{2 d (c d-b e)}\right )}{b^2 d (c d-b e)}-\frac {2 (c x (2 c d-b e)+b (c d-b e))}{b^2 d \sqrt {b x+c x^2} (d+e x) (c d-b e)}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle -\frac {e \left (\frac {3 b^2 e (2 c d-b e) \int \frac {1}{4 d (c d-b e)-\frac {(b d+(2 c d-b e) x)^2}{c x^2+b x}}d\left (-\frac {b d+(2 c d-b e) x}{\sqrt {c x^2+b x}}\right )}{d (c d-b e)}+\frac {\sqrt {b x+c x^2} \left (3 b^2 e^2-4 b c d e+4 c^2 d^2\right )}{d (d+e x) (c d-b e)}\right )}{b^2 d (c d-b e)}-\frac {2 (c x (2 c d-b e)+b (c d-b e))}{b^2 d \sqrt {b x+c x^2} (d+e x) (c d-b e)}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {e \left (\frac {\sqrt {b x+c x^2} \left (3 b^2 e^2-4 b c d e+4 c^2 d^2\right )}{d (d+e x) (c d-b e)}-\frac {3 b^2 e (2 c d-b e) \text {arctanh}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{2 d^{3/2} (c d-b e)^{3/2}}\right )}{b^2 d (c d-b e)}-\frac {2 (c x (2 c d-b e)+b (c d-b e))}{b^2 d \sqrt {b x+c x^2} (d+e x) (c d-b e)}\) |
(-2*(b*(c*d - b*e) + c*(2*c*d - b*e)*x))/(b^2*d*(c*d - b*e)*(d + e*x)*Sqrt [b*x + c*x^2]) - (e*(((4*c^2*d^2 - 4*b*c*d*e + 3*b^2*e^2)*Sqrt[b*x + c*x^2 ])/(d*(c*d - b*e)*(d + e*x)) - (3*b^2*e*(2*c*d - b*e)*ArcTanh[(b*d + (2*c* d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/(2*d^(3/2)*(c* d - b*e)^(3/2))))/(b^2*d*(c*d - b*e))
3.4.27.3.1 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[(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[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym bol] :> Simp[-2 Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c , d, e}, x]
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[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Simp[(b*(e *f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^ (m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x ] && EqQ[Simplify[m + 2*p + 3], 0]
Time = 2.49 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.90
| method | result | size |
| pseudoelliptic | \(-\frac {2 \left (-\frac {3 \sqrt {x \left (c x +b \right )}\, b^{2} e^{2} \left (e x +d \right ) \left (b e -2 c d \right ) \arctan \left (\frac {\sqrt {x \left (c x +b \right )}\, d}{x \sqrt {d \left (b e -c d \right )}}\right )}{2}+\left (c^{2} \left (2 c x +b \right ) d^{3}-2 \left (-c^{2} x^{2}+\frac {1}{2} b c x +b^{2}\right ) c e \,d^{2}+e^{2} b \left (c x +b \right ) \left (-2 c x +b \right ) d +\frac {3 b^{2} e^{3} x \left (c x +b \right )}{2}\right ) \sqrt {d \left (b e -c d \right )}\right )}{\sqrt {x \left (c x +b \right )}\, \sqrt {d \left (b e -c d \right )}\, d^{2} \left (e x +d \right ) \left (b e -c d \right )^{2} b^{2}}\) | \(187\) |
| risch | \(-\frac {2 \left (c x +b \right )}{b^{2} d^{2} \sqrt {x \left (c x +b \right )}}-\frac {2 c^{2} \sqrt {c \left (\frac {b}{c}+x \right )^{2}-b \left (\frac {b}{c}+x \right )}}{b^{2} \left (b e -c d \right )^{2} \left (\frac {b}{c}+x \right )}-\frac {e^{2} \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{d^{2} \left (b e -c d \right )^{2} \left (x +\frac {d}{e}\right )}+\frac {3 b \,e^{2} \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right )}{2 d^{2} \left (b e -c d \right )^{2} \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}-\frac {3 e \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right ) c}{d \left (b e -c d \right )^{2} \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}\) | \(437\) |
| default | \(\frac {\frac {e^{2}}{d \left (b e -c d \right ) \left (x +\frac {d}{e}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}+\frac {3 e \left (b e -2 c d \right ) \left (-\frac {e^{2}}{d \left (b e -c d \right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}+\frac {e \left (b e -2 c d \right ) \left (2 c \left (x +\frac {d}{e}\right )+\frac {b e -2 c d}{e}\right )}{d \left (b e -c d \right ) \left (-\frac {4 c d \left (b e -c d \right )}{e^{2}}-\frac {\left (b e -2 c d \right )^{2}}{e^{2}}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}+\frac {e^{2} \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right )}{d \left (b e -c d \right ) \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}\right )}{2 d \left (b e -c d \right )}+\frac {4 c \,e^{2} \left (2 c \left (x +\frac {d}{e}\right )+\frac {b e -2 c d}{e}\right )}{d \left (b e -c d \right ) \left (-\frac {4 c d \left (b e -c d \right )}{e^{2}}-\frac {\left (b e -2 c d \right )^{2}}{e^{2}}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}}{e^{2}}\) | \(560\) |
-2/(x*(c*x+b))^(1/2)*(-3/2*(x*(c*x+b))^(1/2)*b^2*e^2*(e*x+d)*(b*e-2*c*d)*a rctan((x*(c*x+b))^(1/2)/x*d/(d*(b*e-c*d))^(1/2))+(c^2*(2*c*x+b)*d^3-2*(-c^ 2*x^2+1/2*b*c*x+b^2)*c*e*d^2+e^2*b*(c*x+b)*(-2*c*x+b)*d+3/2*b^2*e^3*x*(c*x +b))*(d*(b*e-c*d))^(1/2))/(d*(b*e-c*d))^(1/2)/d^2/(e*x+d)/(b*e-c*d)^2/b^2
Leaf count of result is larger than twice the leaf count of optimal. 451 vs. \(2 (189) = 378\).
Time = 0.44 (sec) , antiderivative size = 914, normalized size of antiderivative = 4.42 \[ \int \frac {1}{(d+e x)^2 \left (b x+c x^2\right )^{3/2}} \, dx=\left [-\frac {3 \, {\left ({\left (2 \, b^{2} c^{2} d e^{3} - b^{3} c e^{4}\right )} x^{3} + {\left (2 \, b^{2} c^{2} d^{2} e^{2} + b^{3} c d e^{3} - b^{4} e^{4}\right )} x^{2} + {\left (2 \, b^{3} c d^{2} e^{2} - b^{4} d e^{3}\right )} x\right )} \sqrt {c d^{2} - b d e} \log \left (\frac {b d + {\left (2 \, c d - b e\right )} x - 2 \, \sqrt {c d^{2} - b d e} \sqrt {c x^{2} + b x}}{e x + d}\right ) + 2 \, {\left (2 \, b c^{3} d^{5} - 6 \, b^{2} c^{2} d^{4} e + 6 \, b^{3} c d^{3} e^{2} - 2 \, b^{4} d^{2} e^{3} + {\left (4 \, c^{4} d^{4} e - 8 \, b c^{3} d^{3} e^{2} + 7 \, b^{2} c^{2} d^{2} e^{3} - 3 \, b^{3} c d e^{4}\right )} x^{2} + {\left (4 \, c^{4} d^{5} - 6 \, b c^{3} d^{4} e + 5 \, b^{3} c d^{2} e^{3} - 3 \, b^{4} d e^{4}\right )} x\right )} \sqrt {c x^{2} + b x}}{2 \, {\left ({\left (b^{2} c^{4} d^{6} e - 3 \, b^{3} c^{3} d^{5} e^{2} + 3 \, b^{4} c^{2} d^{4} e^{3} - b^{5} c d^{3} e^{4}\right )} x^{3} + {\left (b^{2} c^{4} d^{7} - 2 \, b^{3} c^{3} d^{6} e + 2 \, b^{5} c d^{4} e^{3} - b^{6} d^{3} e^{4}\right )} x^{2} + {\left (b^{3} c^{3} d^{7} - 3 \, b^{4} c^{2} d^{6} e + 3 \, b^{5} c d^{5} e^{2} - b^{6} d^{4} e^{3}\right )} x\right )}}, \frac {3 \, {\left ({\left (2 \, b^{2} c^{2} d e^{3} - b^{3} c e^{4}\right )} x^{3} + {\left (2 \, b^{2} c^{2} d^{2} e^{2} + b^{3} c d e^{3} - b^{4} e^{4}\right )} x^{2} + {\left (2 \, b^{3} c d^{2} e^{2} - b^{4} d e^{3}\right )} x\right )} \sqrt {-c d^{2} + b d e} \arctan \left (-\frac {\sqrt {-c d^{2} + b d e} \sqrt {c x^{2} + b x}}{{\left (c d - b e\right )} x}\right ) - {\left (2 \, b c^{3} d^{5} - 6 \, b^{2} c^{2} d^{4} e + 6 \, b^{3} c d^{3} e^{2} - 2 \, b^{4} d^{2} e^{3} + {\left (4 \, c^{4} d^{4} e - 8 \, b c^{3} d^{3} e^{2} + 7 \, b^{2} c^{2} d^{2} e^{3} - 3 \, b^{3} c d e^{4}\right )} x^{2} + {\left (4 \, c^{4} d^{5} - 6 \, b c^{3} d^{4} e + 5 \, b^{3} c d^{2} e^{3} - 3 \, b^{4} d e^{4}\right )} x\right )} \sqrt {c x^{2} + b x}}{{\left (b^{2} c^{4} d^{6} e - 3 \, b^{3} c^{3} d^{5} e^{2} + 3 \, b^{4} c^{2} d^{4} e^{3} - b^{5} c d^{3} e^{4}\right )} x^{3} + {\left (b^{2} c^{4} d^{7} - 2 \, b^{3} c^{3} d^{6} e + 2 \, b^{5} c d^{4} e^{3} - b^{6} d^{3} e^{4}\right )} x^{2} + {\left (b^{3} c^{3} d^{7} - 3 \, b^{4} c^{2} d^{6} e + 3 \, b^{5} c d^{5} e^{2} - b^{6} d^{4} e^{3}\right )} x}\right ] \]
[-1/2*(3*((2*b^2*c^2*d*e^3 - b^3*c*e^4)*x^3 + (2*b^2*c^2*d^2*e^2 + b^3*c*d *e^3 - b^4*e^4)*x^2 + (2*b^3*c*d^2*e^2 - b^4*d*e^3)*x)*sqrt(c*d^2 - b*d*e) *log((b*d + (2*c*d - b*e)*x - 2*sqrt(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x))/(e* x + d)) + 2*(2*b*c^3*d^5 - 6*b^2*c^2*d^4*e + 6*b^3*c*d^3*e^2 - 2*b^4*d^2*e ^3 + (4*c^4*d^4*e - 8*b*c^3*d^3*e^2 + 7*b^2*c^2*d^2*e^3 - 3*b^3*c*d*e^4)*x ^2 + (4*c^4*d^5 - 6*b*c^3*d^4*e + 5*b^3*c*d^2*e^3 - 3*b^4*d*e^4)*x)*sqrt(c *x^2 + b*x))/((b^2*c^4*d^6*e - 3*b^3*c^3*d^5*e^2 + 3*b^4*c^2*d^4*e^3 - b^5 *c*d^3*e^4)*x^3 + (b^2*c^4*d^7 - 2*b^3*c^3*d^6*e + 2*b^5*c*d^4*e^3 - b^6*d ^3*e^4)*x^2 + (b^3*c^3*d^7 - 3*b^4*c^2*d^6*e + 3*b^5*c*d^5*e^2 - b^6*d^4*e ^3)*x), (3*((2*b^2*c^2*d*e^3 - b^3*c*e^4)*x^3 + (2*b^2*c^2*d^2*e^2 + b^3*c *d*e^3 - b^4*e^4)*x^2 + (2*b^3*c*d^2*e^2 - b^4*d*e^3)*x)*sqrt(-c*d^2 + b*d *e)*arctan(-sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x)/((c*d - b*e)*x)) - (2*b *c^3*d^5 - 6*b^2*c^2*d^4*e + 6*b^3*c*d^3*e^2 - 2*b^4*d^2*e^3 + (4*c^4*d^4* e - 8*b*c^3*d^3*e^2 + 7*b^2*c^2*d^2*e^3 - 3*b^3*c*d*e^4)*x^2 + (4*c^4*d^5 - 6*b*c^3*d^4*e + 5*b^3*c*d^2*e^3 - 3*b^4*d*e^4)*x)*sqrt(c*x^2 + b*x))/((b ^2*c^4*d^6*e - 3*b^3*c^3*d^5*e^2 + 3*b^4*c^2*d^4*e^3 - b^5*c*d^3*e^4)*x^3 + (b^2*c^4*d^7 - 2*b^3*c^3*d^6*e + 2*b^5*c*d^4*e^3 - b^6*d^3*e^4)*x^2 + (b ^3*c^3*d^7 - 3*b^4*c^2*d^6*e + 3*b^5*c*d^5*e^2 - b^6*d^4*e^3)*x)]
\[ \int \frac {1}{(d+e x)^2 \left (b x+c x^2\right )^{3/2}} \, dx=\int \frac {1}{\left (x \left (b + c x\right )\right )^{\frac {3}{2}} \left (d + e x\right )^{2}}\, dx \]
Exception generated. \[ \int \frac {1}{(d+e x)^2 \left (b x+c x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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
Leaf count of result is larger than twice the leaf count of optimal. 821 vs. \(2 (189) = 378\).
Time = 0.60 (sec) , antiderivative size = 821, normalized size of antiderivative = 3.97 \[ \int \frac {1}{(d+e x)^2 \left (b x+c x^2\right )^{3/2}} \, dx=\frac {\frac {{\left (6 \, b^{2} c d e^{5} \log \left ({\left | 2 \, c d e - b e^{2} - 2 \, \sqrt {c d^{2} - b d e} \sqrt {c} {\left | e \right |} \right |}\right ) - 3 \, b^{3} e^{6} \log \left ({\left | 2 \, c d e - b e^{2} - 2 \, \sqrt {c d^{2} - b d e} \sqrt {c} {\left | e \right |} \right |}\right ) + 8 \, \sqrt {c d^{2} - b d e} c^{\frac {5}{2}} d^{2} e^{2} {\left | e \right |} - 8 \, \sqrt {c d^{2} - b d e} b c^{\frac {3}{2}} d e^{3} {\left | e \right |} + 6 \, \sqrt {c d^{2} - b d e} b^{2} \sqrt {c} e^{4} {\left | e \right |}\right )} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )}{\sqrt {c d^{2} - b d e} b^{2} c^{2} d^{4} {\left | e \right |} - 2 \, \sqrt {c d^{2} - b d e} b^{3} c d^{3} e {\left | e \right |} + \sqrt {c d^{2} - b d e} b^{4} d^{2} e^{2} {\left | e \right |}} - \frac {2 \, {\left (\frac {4 \, c^{3} d^{2} e^{7} - 4 \, b c^{2} d e^{8} + 3 \, b^{2} c e^{9}}{b^{2} c^{2} d^{4} e^{5} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) - 2 \, b^{3} c d^{3} e^{6} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) + b^{4} d^{2} e^{7} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )} - \frac {\frac {4 \, c^{3} d^{3} e^{8} - 6 \, b c^{2} d^{2} e^{9} + 8 \, b^{2} c d e^{10} - 3 \, b^{3} e^{11}}{b^{2} c^{2} d^{4} e^{5} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) - 2 \, b^{3} c d^{3} e^{6} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) + b^{4} d^{2} e^{7} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )} - \frac {b^{2} c d^{2} e^{11} - b^{3} d e^{12}}{{\left (b^{2} c^{2} d^{4} e^{5} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) - 2 \, b^{3} c d^{3} e^{6} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) + b^{4} d^{2} e^{7} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )\right )} {\left (e x + d\right )} e}}{{\left (e x + d\right )} e}\right )}}{\sqrt {c - \frac {2 \, c d}{e x + d} + \frac {c d^{2}}{{\left (e x + d\right )}^{2}} + \frac {b e}{e x + d} - \frac {b d e}{{\left (e x + d\right )}^{2}}}} - \frac {3 \, {\left (2 \, c d e^{5} - b e^{6}\right )} \log \left ({\left | 2 \, c d e - b e^{2} - 2 \, \sqrt {c d^{2} - b d e} {\left (\sqrt {c - \frac {2 \, c d}{e x + d} + \frac {c d^{2}}{{\left (e x + d\right )}^{2}} + \frac {b e}{e x + d} - \frac {b d e}{{\left (e x + d\right )}^{2}}} + \frac {\sqrt {c d^{2} e^{2} - b d e^{3}}}{{\left (e x + d\right )} e}\right )} {\left | e \right |} \right |}\right )}{{\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )} \sqrt {c d^{2} - b d e} {\left | e \right |} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )}}{2 \, e^{2}} \]
1/2*((6*b^2*c*d*e^5*log(abs(2*c*d*e - b*e^2 - 2*sqrt(c*d^2 - b*d*e)*sqrt(c )*abs(e))) - 3*b^3*e^6*log(abs(2*c*d*e - b*e^2 - 2*sqrt(c*d^2 - b*d*e)*sqr t(c)*abs(e))) + 8*sqrt(c*d^2 - b*d*e)*c^(5/2)*d^2*e^2*abs(e) - 8*sqrt(c*d^ 2 - b*d*e)*b*c^(3/2)*d*e^3*abs(e) + 6*sqrt(c*d^2 - b*d*e)*b^2*sqrt(c)*e^4* abs(e))*sgn(1/(e*x + d))*sgn(e)/(sqrt(c*d^2 - b*d*e)*b^2*c^2*d^4*abs(e) - 2*sqrt(c*d^2 - b*d*e)*b^3*c*d^3*e*abs(e) + sqrt(c*d^2 - b*d*e)*b^4*d^2*e^2 *abs(e)) - 2*((4*c^3*d^2*e^7 - 4*b*c^2*d*e^8 + 3*b^2*c*e^9)/(b^2*c^2*d^4*e ^5*sgn(1/(e*x + d))*sgn(e) - 2*b^3*c*d^3*e^6*sgn(1/(e*x + d))*sgn(e) + b^4 *d^2*e^7*sgn(1/(e*x + d))*sgn(e)) - ((4*c^3*d^3*e^8 - 6*b*c^2*d^2*e^9 + 8* b^2*c*d*e^10 - 3*b^3*e^11)/(b^2*c^2*d^4*e^5*sgn(1/(e*x + d))*sgn(e) - 2*b^ 3*c*d^3*e^6*sgn(1/(e*x + d))*sgn(e) + b^4*d^2*e^7*sgn(1/(e*x + d))*sgn(e)) - (b^2*c*d^2*e^11 - b^3*d*e^12)/((b^2*c^2*d^4*e^5*sgn(1/(e*x + d))*sgn(e) - 2*b^3*c*d^3*e^6*sgn(1/(e*x + d))*sgn(e) + b^4*d^2*e^7*sgn(1/(e*x + d))* sgn(e))*(e*x + d)*e))/((e*x + d)*e))/sqrt(c - 2*c*d/(e*x + d) + c*d^2/(e*x + d)^2 + b*e/(e*x + d) - b*d*e/(e*x + d)^2) - 3*(2*c*d*e^5 - b*e^6)*log(a bs(2*c*d*e - b*e^2 - 2*sqrt(c*d^2 - b*d*e)*(sqrt(c - 2*c*d/(e*x + d) + c*d ^2/(e*x + d)^2 + b*e/(e*x + d) - b*d*e/(e*x + d)^2) + sqrt(c*d^2*e^2 - b*d *e^3)/((e*x + d)*e))*abs(e)))/((c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2)*sqrt( c*d^2 - b*d*e)*abs(e)*sgn(1/(e*x + d))*sgn(e)))/e^2
Timed out. \[ \int \frac {1}{(d+e x)^2 \left (b x+c x^2\right )^{3/2}} \, dx=\int \frac {1}{{\left (c\,x^2+b\,x\right )}^{3/2}\,{\left (d+e\,x\right )}^2} \,d x \]