Integrand size = 26, antiderivative size = 245 \[ \int \frac {A+B x}{(d+e x)^2 \left (b x+c x^2\right )^{3/2}} \, dx=-\frac {2 (A b (c d-b e)+c (2 A c d-b (B d+A e)) x)}{b^2 d (c d-b e) (d+e x) \sqrt {b x+c x^2}}-\frac {e \left (4 A c^2 d^2-b^2 e (B d-3 A e)-2 b c d (B d+2 A e)\right ) \sqrt {b x+c x^2}}{b^2 d^2 (c d-b e)^2 (d+e x)}+\frac {e (3 A e (2 c d-b e)-B d (4 c d-b e)) \text {arctanh}\left (\frac {b d+(2 c d-b e) x}{2 \sqrt {d} \sqrt {c d-b e} \sqrt {b x+c x^2}}\right )}{2 d^{5/2} (c d-b e)^{5/2}} \]
1/2*e*(3*A*e*(-b*e+2*c*d)-B*d*(-b*e+4*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* (A*b*(-b*e+c*d)+c*(2*A*c*d-b*(A*e+B*d))*x)/b^2/d/(-b*e+c*d)/(e*x+d)/(c*x^2 +b*x)^(1/2)-e*(4*A*c^2*d^2-b^2*e*(-3*A*e+B*d)-2*b*c*d*(2*A*e+B*d))*(c*x^2+ b*x)^(1/2)/b^2/d^2/(-b*e+c*d)^2/(e*x+d)
Time = 1.13 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.08 \[ \int \frac {A+B x}{(d+e x)^2 \left (b x+c x^2\right )^{3/2}} \, dx=\frac {x \left (\frac {\sqrt {d} (b+c x) \left (b B d x \left (b^2 e^2+b c e^2 x+2 c^2 d (d+e x)\right )-A \left (4 c^3 d^2 x (d+e x)+b^3 e^2 (2 d+3 e x)+2 b c^2 d \left (d^2-d e x-2 e^2 x^2\right )+b^2 c e \left (-4 d^2-2 d e x+3 e^2 x^2\right )\right )\right )}{b^2 (c d-b e)^2 (d+e x)}+\frac {e (B d (4 c d-b e)+3 A e (-2 c d+b e)) \sqrt {x} (b+c x)^{3/2} \arctan \left (\frac {-e \sqrt {x} \sqrt {b+c x}+\sqrt {c} (d+e x)}{\sqrt {d} \sqrt {-c d+b e}}\right )}{(-c d+b e)^{5/2}}\right )}{d^{5/2} (x (b+c x))^{3/2}} \]
(x*((Sqrt[d]*(b + c*x)*(b*B*d*x*(b^2*e^2 + b*c*e^2*x + 2*c^2*d*(d + e*x)) - A*(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)) + (e*(B*d*(4*c*d - b*e) + 3*A*e*(-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 = 0.46 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.07, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {1235, 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.
| \(\displaystyle \int \frac {A+B x}{\left (b x+c x^2\right )^{3/2} (d+e x)^2} \, dx\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle -\frac {2 \int \frac {e (b (b B d+2 A c d-3 A b e)-2 c (b B d-2 A c d+A b e) x)}{2 (d+e x)^2 \sqrt {c x^2+b x}}dx}{b^2 d (c d-b e)}-\frac {2 (c x (2 A c d-b (A e+B d))+A b (c d-b e))}{b^2 d \sqrt {b x+c x^2} (d+e x) (c d-b e)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {e \int \frac {b (b B d+2 A c d-3 A b e)-2 c (b B d-2 A c d+A b e) x}{(d+e x)^2 \sqrt {c x^2+b x}}dx}{b^2 d (c d-b e)}-\frac {2 (c x (2 A c d-b (A e+B d))+A b (c d-b e))}{b^2 d \sqrt {b x+c x^2} (d+e x) (c d-b e)}\) |
| \(\Big \downarrow \) 1228 |
| \(\displaystyle -\frac {e \left (\frac {\sqrt {b x+c x^2} \left (b^2 (-e) (B d-3 A e)-2 b c d (2 A e+B d)+4 A c^2 d^2\right )}{d (d+e x) (c d-b e)}-\frac {b^2 (3 A e (2 c d-b e)-B d (4 c d-b e)) \int \frac {1}{(d+e x) \sqrt {c x^2+b x}}dx}{2 d (c d-b e)}\right )}{b^2 d (c d-b e)}-\frac {2 (c x (2 A c d-b (A e+B d))+A b (c d-b e))}{b^2 d \sqrt {b x+c x^2} (d+e x) (c d-b e)}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle -\frac {e \left (\frac {b^2 (3 A e (2 c d-b e)-B d (4 c d-b e)) \int \frac {1}{4 d (c d-b e)-\frac {(b d+(2 c d-b e) x)^2}{c x^2+b x}}d\left (-\frac {b d+(2 c d-b e) x}{\sqrt {c x^2+b x}}\right )}{d (c d-b e)}+\frac {\sqrt {b x+c x^2} \left (b^2 (-e) (B d-3 A e)-2 b c d (2 A e+B d)+4 A c^2 d^2\right )}{d (d+e x) (c d-b e)}\right )}{b^2 d (c d-b e)}-\frac {2 (c x (2 A c d-b (A e+B d))+A b (c d-b e))}{b^2 d \sqrt {b x+c x^2} (d+e x) (c d-b e)}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {e \left (\frac {\sqrt {b x+c x^2} \left (b^2 (-e) (B d-3 A e)-2 b c d (2 A e+B d)+4 A c^2 d^2\right )}{d (d+e x) (c d-b e)}-\frac {b^2 (3 A e (2 c d-b e)-B d (4 c d-b e)) \text {arctanh}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{2 d^{3/2} (c d-b e)^{3/2}}\right )}{b^2 d (c d-b e)}-\frac {2 (c x (2 A c d-b (A e+B d))+A b (c d-b e))}{b^2 d \sqrt {b x+c x^2} (d+e x) (c d-b e)}\) |
(-2*(A*b*(c*d - b*e) + c*(2*A*c*d - b*(B*d + A*e))*x))/(b^2*d*(c*d - b*e)* (d + e*x)*Sqrt[b*x + c*x^2]) - (e*(((4*A*c^2*d^2 - b^2*e*(B*d - 3*A*e) - 2 *b*c*d*(B*d + 2*A*e))*Sqrt[b*x + c*x^2])/(d*(c*d - b*e)*(d + e*x)) - (b^2* (3*A*e*(2*c*d - b*e) - B*d*(4*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))
3.13.4.3.1 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_)*((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]
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] )
Time = 0.47 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.93
| method | result | size |
| pseudoelliptic | \(-\frac {2 \left (-\frac {3 \sqrt {x \left (c x +b \right )}\, \left (e x +d \right ) e \,b^{2} \left (\frac {4 B c \,d^{2}}{3}-2 \left (A c +\frac {B b}{6}\right ) e d +A b \,e^{2}\right ) \arctan \left (\frac {\sqrt {x \left (c x +b \right )}\, d}{x \sqrt {d \left (b e -c d \right )}}\right )}{2}+\left (c^{2} \left (\left (-B x +A \right ) b +2 A c x \right ) d^{3}-2 c e \left (A \,b^{2}+\frac {c x \left (B x +A \right ) b}{2}-A \,c^{2} x^{2}\right ) d^{2}+\left (\left (-\frac {B x}{2}+A \right ) b -2 A c x \right ) e^{2} \left (c x +b \right ) b d +\frac {3 A \,b^{2} e^{3} x \left (c x +b \right )}{2}\right ) \sqrt {d \left (b e -c d \right )}\right )}{\sqrt {d \left (b e -c d \right )}\, \sqrt {x \left (c x +b \right )}\, d^{2} \left (e x +d \right ) \left (b e -c d \right )^{2} b^{2}}\) | \(228\) |
| risch | \(-\frac {2 A \left (c x +b \right )}{b^{2} d^{2} \sqrt {x \left (c x +b \right )}}-\frac {\frac {2 d^{2} \left (A c -B b \right ) c \sqrt {c \left (x +\frac {b}{c}\right )^{2}-b \left (x +\frac {b}{c}\right )}}{\left (b e -c d \right )^{2} b \left (x +\frac {b}{c}\right )}+\frac {b d \left (A e -B d \right ) \left (\frac {e^{2} \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{d \left (b e -c d \right ) \left (x +\frac {d}{e}\right )}-\frac {\left (b e -2 c d \right ) e \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right )}{2 d \left (b e -c d \right ) \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}\right )}{e \left (b e -c d \right )}-\frac {b \left (A b \,e^{2}-2 A c d e +B c \,d^{2}\right ) \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (b e -c d \right )^{2} \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}}{d^{2} b}\) | \(500\) |
| default | \(\frac {B \left (-\frac {e^{2}}{d \left (b e -c d \right ) \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\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 {\left (b e -2 c d \right ) e \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 {\left (x +\frac {d}{e}\right )^{2} c +\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 {\left (x +\frac {d}{e}\right )^{2} c +\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 )}{e^{2}}+\frac {\left (A e -B d \right ) \left (\frac {e^{2}}{d \left (b e -c d \right ) \left (x +\frac {d}{e}\right ) \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\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 \left (b e -2 c d \right ) e \left (-\frac {e^{2}}{d \left (b e -c d \right ) \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\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 {\left (b e -2 c d \right ) e \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 {\left (x +\frac {d}{e}\right )^{2} c +\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 {\left (x +\frac {d}{e}\right )^{2} c +\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 {\left (x +\frac {d}{e}\right )^{2} c +\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}}}}\right )}{e^{3}}\) | \(909\) |
-2/(d*(b*e-c*d))^(1/2)*(-3/2*(x*(c*x+b))^(1/2)*(e*x+d)*e*b^2*(4/3*B*c*d^2- 2*(A*c+1/6*B*b)*e*d+A*b*e^2)*arctan((x*(c*x+b))^(1/2)/x*d/(d*(b*e-c*d))^(1 /2))+(c^2*((-B*x+A)*b+2*A*c*x)*d^3-2*c*e*(A*b^2+1/2*c*x*(B*x+A)*b-A*c^2*x^ 2)*d^2+((-1/2*B*x+A)*b-2*A*c*x)*e^2*(c*x+b)*b*d+3/2*A*b^2*e^3*x*(c*x+b))*( d*(b*e-c*d))^(1/2))/(x*(c*x+b))^(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. 614 vs. \(2 (226) = 452\).
Time = 0.33 (sec) , antiderivative size = 1240, normalized size of antiderivative = 5.06 \[ \int \frac {A+B x}{(d+e x)^2 \left (b x+c x^2\right )^{3/2}} \, dx=\text {Too large to display} \]
[1/2*(((4*B*b^2*c^2*d^2*e^2 + 3*A*b^3*c*e^4 - (B*b^3*c + 6*A*b^2*c^2)*d*e^ 3)*x^3 + (4*B*b^2*c^2*d^3*e + 3*A*b^4*e^4 + 3*(B*b^3*c - 2*A*b^2*c^2)*d^2* e^2 - (B*b^4 + 3*A*b^3*c)*d*e^3)*x^2 + (4*B*b^3*c*d^3*e + 3*A*b^4*d*e^3 - (B*b^4 + 6*A*b^3*c)*d^2*e^2)*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*A*b*c^3* d^5 - 6*A*b^2*c^2*d^4*e + 6*A*b^3*c*d^3*e^2 - 2*A*b^4*d^2*e^3 - (3*A*b^3*c *d*e^4 + 2*(B*b*c^3 - 2*A*c^4)*d^4*e - (B*b^2*c^2 - 8*A*b*c^3)*d^3*e^2 - ( B*b^3*c + 7*A*b^2*c^2)*d^2*e^3)*x^2 - (B*b^3*c*d^3*e^2 + 3*A*b^4*d*e^4 + 2 *(B*b*c^3 - 2*A*c^4)*d^5 - 2*(B*b^2*c^2 - 3*A*b*c^3)*d^4*e - (B*b^4 + 5*A* b^3*c)*d^2*e^3)*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), -(((4*B*b^2*c^2*d^2*e^2 + 3*A*b^3*c*e^4 - (B*b^3*c + 6*A*b^2*c^2)*d*e^3)*x^3 + (4*B*b^2*c^2*d^3*e + 3*A*b^4*e^4 + 3 *(B*b^3*c - 2*A*b^2*c^2)*d^2*e^2 - (B*b^4 + 3*A*b^3*c)*d*e^3)*x^2 + (4*B*b ^3*c*d^3*e + 3*A*b^4*d*e^3 - (B*b^4 + 6*A*b^3*c)*d^2*e^2)*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*A*b*c^3*d^5 - 6*A*b^2*c^2*d^4*e + 6*A*b^3*c*d^3*e^2 - 2*A*b^4*d^2*e^3 - (3*A*b^3*c*d*e^4 + 2*(B*b*c^3 - 2*A*c^4)*d^4*e - (B*b^2*c^2 - 8*A*b*c^3)* d^3*e^2 - (B*b^3*c + 7*A*b^2*c^2)*d^2*e^3)*x^2 - (B*b^3*c*d^3*e^2 + 3*A...
\[ \int \frac {A+B x}{(d+e x)^2 \left (b x+c x^2\right )^{3/2}} \, dx=\int \frac {A + B x}{\left (x \left (b + c x\right )\right )^{\frac {3}{2}} \left (d + e x\right )^{2}}\, dx \]
Exception generated. \[ \int \frac {A+B x}{(d+e x)^2 \left (b x+c x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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. 1281 vs. \(2 (226) = 452\).
Time = 0.66 (sec) , antiderivative size = 1281, normalized size of antiderivative = 5.23 \[ \int \frac {A+B x}{(d+e x)^2 \left (b x+c x^2\right )^{3/2}} \, dx=\text {Too large to display} \]
-1/2*((4*B*b^2*c*d^2*e^4*log(abs(2*c*d*e - b*e^2 - 2*sqrt(c*d^2 - b*d*e)*s qrt(c)*abs(e))) - B*b^3*d*e^5*log(abs(2*c*d*e - b*e^2 - 2*sqrt(c*d^2 - b*d *e)*sqrt(c)*abs(e))) - 6*A*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*A*b^3*e^6*log(abs(2*c*d*e - b*e^2 - 2*sq rt(c*d^2 - b*d*e)*sqrt(c)*abs(e))) + 4*sqrt(c*d^2 - b*d*e)*B*b*c^(3/2)*d^2 *e^2*abs(e) - 8*sqrt(c*d^2 - b*d*e)*A*c^(5/2)*d^2*e^2*abs(e) + 2*sqrt(c*d^ 2 - b*d*e)*B*b^2*sqrt(c)*d*e^3*abs(e) + 8*sqrt(c*d^2 - b*d*e)*A*b*c^(3/2)* d*e^3*abs(e) - 6*sqrt(c*d^2 - b*d*e)*A*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*((2*B* b*c^2*d^2*e^7*sgn(1/(e*x + d))*sgn(e) - 4*A*c^3*d^2*e^7*sgn(1/(e*x + d))*s gn(e) + B*b^2*c*d*e^8*sgn(1/(e*x + d))*sgn(e) + 4*A*b*c^2*d*e^8*sgn(1/(e*x + d))*sgn(e) - 3*A*b^2*c*e^9*sgn(1/(e*x + d))*sgn(e))/(b^2*c^2*d^4*e^5*sg n(1/(e*x + d))^2*sgn(e)^2 - 2*b^3*c*d^3*e^6*sgn(1/(e*x + d))^2*sgn(e)^2 + b^4*d^2*e^7*sgn(1/(e*x + d))^2*sgn(e)^2) - ((2*B*b*c^2*d^3*e^8*sgn(1/(e*x + d))*sgn(e) - 4*A*c^3*d^3*e^8*sgn(1/(e*x + d))*sgn(e) + 2*B*b^2*c*d^2*e^9 *sgn(1/(e*x + d))*sgn(e) + 6*A*b*c^2*d^2*e^9*sgn(1/(e*x + d))*sgn(e) - B*b ^3*d*e^10*sgn(1/(e*x + d))*sgn(e) - 8*A*b^2*c*d*e^10*sgn(1/(e*x + d))*sgn( e) + 3*A*b^3*e^11*sgn(1/(e*x + d))*sgn(e))/(b^2*c^2*d^4*e^5*sgn(1/(e*x + d ))^2*sgn(e)^2 - 2*b^3*c*d^3*e^6*sgn(1/(e*x + d))^2*sgn(e)^2 + b^4*d^2*e...
Timed out. \[ \int \frac {A+B x}{(d+e x)^2 \left (b x+c x^2\right )^{3/2}} \, dx=\int \frac {A+B\,x}{{\left (c\,x^2+b\,x\right )}^{3/2}\,{\left (d+e\,x\right )}^2} \,d x \]