Integrand size = 21, antiderivative size = 245 \[ \int \frac {\sqrt {d+e x}}{\left (b x+c x^2\right )^3} \, dx=-\frac {(b+2 c x) \sqrt {d+e x}}{2 b^2 \left (b x+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (b (c d-b e) (12 c d-b e)+c \left (24 c^2 d^2-24 b c d e+b^2 e^2\right ) x\right )}{4 b^4 d (c d-b e) \left (b x+c x^2\right )}-\frac {\left (48 c^2 d^2-12 b c d e-b^2 e^2\right ) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5 d^{3/2}}+\frac {c^{3/2} \left (48 c^2 d^2-84 b c d e+35 b^2 e^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 (c d-b e)^{3/2}} \]
-1/4*(-b^2*e^2-12*b*c*d*e+48*c^2*d^2)*arctanh((e*x+d)^(1/2)/d^(1/2))/b^5/d ^(3/2)+1/4*c^(3/2)*(35*b^2*e^2-84*b*c*d*e+48*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)^(3/2)-1/2*(2*c*x+b)*(e*x+d)^(1/ 2)/b^2/(c*x^2+b*x)^2+1/4*(b*(-b*e+c*d)*(-b*e+12*c*d)+c*(b^2*e^2-24*b*c*d*e +24*c^2*d^2)*x)*(e*x+d)^(1/2)/b^4/d/(-b*e+c*d)/(c*x^2+b*x)
Time = 2.01 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.01 \[ \int \frac {\sqrt {d+e x}}{\left (b x+c x^2\right )^3} \, dx=\frac {\frac {b \sqrt {d+e x} \left (24 c^4 d^2 x^3+12 b c^3 d x^2 (3 d-2 e x)+b^4 e (2 d+e x)+b^2 c^2 x \left (8 d^2-37 d e x+e^2 x^2\right )+b^3 c \left (-2 d^2-9 d e x+2 e^2 x^2\right )\right )}{d (c d-b e) x^2 (b+c x)^2}+\frac {c^{3/2} \left (48 c^2 d^2-84 b c d e+35 b^2 e^2\right ) \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {-c d+b e}}\right )}{(-c d+b e)^{3/2}}+\frac {\left (-48 c^2 d^2+12 b c d e+b^2 e^2\right ) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{d^{3/2}}}{4 b^5} \]
((b*Sqrt[d + e*x]*(24*c^4*d^2*x^3 + 12*b*c^3*d*x^2*(3*d - 2*e*x) + b^4*e*( 2*d + e*x) + b^2*c^2*x*(8*d^2 - 37*d*e*x + e^2*x^2) + b^3*c*(-2*d^2 - 9*d* e*x + 2*e^2*x^2)))/(d*(c*d - b*e)*x^2*(b + c*x)^2) + (c^(3/2)*(48*c^2*d^2 - 84*b*c*d*e + 35*b^2*e^2)*ArcTan[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[-(c*d) + b* e]])/(-(c*d) + b*e)^(3/2) + ((-48*c^2*d^2 + 12*b*c*d*e + b^2*e^2)*ArcTanh[ Sqrt[d + e*x]/Sqrt[d]])/d^(3/2))/(4*b^5)
Time = 0.52 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.15, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {1163, 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.
| \(\displaystyle \int \frac {\sqrt {d+e x}}{\left (b x+c x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 1163 |
| \(\displaystyle \frac {\int -\frac {12 c d-b e+10 c e x}{2 \sqrt {d+e x} \left (c x^2+b x\right )^2}dx}{2 b^2}-\frac {(b+2 c x) \sqrt {d+e x}}{2 b^2 \left (b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {12 c d-b e+10 c e x}{\sqrt {d+e x} \left (c x^2+b x\right )^2}dx}{4 b^2}-\frac {(b+2 c x) \sqrt {d+e x}}{2 b^2 \left (b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle -\frac {-\frac {\int \frac {(c d-b e) \left (48 c^2 d^2-12 b c e d-b^2 e^2\right )+c e \left (24 c^2 d^2-24 b c e d+b^2 e^2\right ) x}{2 \sqrt {d+e x} \left (c x^2+b x\right )}dx}{b^2 d (c d-b e)}-\frac {\sqrt {d+e x} \left (c x \left (b^2 e^2-24 b c d e+24 c^2 d^2\right )+b (c d-b e) (12 c d-b e)\right )}{b^2 d \left (b x+c x^2\right ) (c d-b e)}}{4 b^2}-\frac {(b+2 c x) \sqrt {d+e x}}{2 b^2 \left (b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\int \frac {(c d-b e) \left (48 c^2 d^2-12 b c e d-b^2 e^2\right )+c e \left (24 c^2 d^2-24 b c e d+b^2 e^2\right ) x}{\sqrt {d+e x} \left (c x^2+b x\right )}dx}{2 b^2 d (c d-b e)}-\frac {\sqrt {d+e x} \left (c x \left (b^2 e^2-24 b c d e+24 c^2 d^2\right )+b (c d-b e) (12 c d-b e)\right )}{b^2 d \left (b x+c x^2\right ) (c d-b e)}}{4 b^2}-\frac {(b+2 c x) \sqrt {d+e x}}{2 b^2 \left (b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1197 |
| \(\displaystyle -\frac {-\frac {\int \frac {e \left ((2 c d-b e) \left (12 c^2 d^2-12 b c e d-b^2 e^2\right )+c \left (24 c^2 d^2-24 b c e d+b^2 e^2\right ) (d+e x)\right )}{c (d+e x)^2-(2 c d-b e) (d+e x)+d (c d-b e)}d\sqrt {d+e x}}{b^2 d (c d-b e)}-\frac {\sqrt {d+e x} \left (c x \left (b^2 e^2-24 b c d e+24 c^2 d^2\right )+b (c d-b e) (12 c d-b e)\right )}{b^2 d \left (b x+c x^2\right ) (c d-b e)}}{4 b^2}-\frac {(b+2 c x) \sqrt {d+e x}}{2 b^2 \left (b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {e \int \frac {(2 c d-b e) \left (12 c^2 d^2-12 b c e d-b^2 e^2\right )+c \left (24 c^2 d^2-24 b c e d+b^2 e^2\right ) (d+e x)}{c (d+e x)^2-(2 c d-b e) (d+e x)+d (c d-b e)}d\sqrt {d+e x}}{b^2 d (c d-b e)}-\frac {\sqrt {d+e x} \left (c x \left (b^2 e^2-24 b c d e+24 c^2 d^2\right )+b (c d-b e) (12 c d-b e)\right )}{b^2 d \left (b x+c x^2\right ) (c d-b e)}}{4 b^2}-\frac {(b+2 c x) \sqrt {d+e x}}{2 b^2 \left (b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle -\frac {-\frac {e \left (\frac {c (c d-b e) \left (-b^2 e^2-12 b c d e+48 c^2 d^2\right ) \int \frac {1}{c (d+e x)-c d}d\sqrt {d+e x}}{b e}-\frac {c^2 d \left (35 b^2 e^2-84 b c d e+48 c^2 d^2\right ) \int \frac {1}{-c d+b e+c (d+e x)}d\sqrt {d+e x}}{b e}\right )}{b^2 d (c d-b e)}-\frac {\sqrt {d+e x} \left (c x \left (b^2 e^2-24 b c d e+24 c^2 d^2\right )+b (c d-b e) (12 c d-b e)\right )}{b^2 d \left (b x+c x^2\right ) (c d-b e)}}{4 b^2}-\frac {(b+2 c x) \sqrt {d+e x}}{2 b^2 \left (b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {-\frac {e \left (\frac {c^{3/2} d \left (35 b^2 e^2-84 b c d e+48 c^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b e \sqrt {c d-b e}}-\frac {\text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) (c d-b e) \left (-b^2 e^2-12 b c d e+48 c^2 d^2\right )}{b \sqrt {d} e}\right )}{b^2 d (c d-b e)}-\frac {\sqrt {d+e x} \left (c x \left (b^2 e^2-24 b c d e+24 c^2 d^2\right )+b (c d-b e) (12 c d-b e)\right )}{b^2 d \left (b x+c x^2\right ) (c d-b e)}}{4 b^2}-\frac {(b+2 c x) \sqrt {d+e x}}{2 b^2 \left (b x+c x^2\right )^2}\) |
-1/2*((b + 2*c*x)*Sqrt[d + e*x])/(b^2*(b*x + c*x^2)^2) - (-((Sqrt[d + e*x] *(b*(c*d - b*e)*(12*c*d - b*e) + c*(24*c^2*d^2 - 24*b*c*d*e + b^2*e^2)*x)) /(b^2*d*(c*d - b*e)*(b*x + c*x^2))) - (e*(-(((c*d - b*e)*(48*c^2*d^2 - 12* b*c*d*e - b^2*e^2)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(b*Sqrt[d]*e)) + (c^(3/ 2)*d*(48*c^2*d^2 - 84*b*c*d*e + 35*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)
3.4.82.3.1 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*(b + 2*c*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 - 1 )*(b*e*m + 2*c*d*(2*p + 3) + 2*c*e*(m + 2*p + 3)*x)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[p, -1] && GtQ[m, 0] && (LtQ[ m, 1] || (ILtQ[m + 2*p + 3, 0] && NeQ[m, 2])) && 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 = 2.09 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.98
| method | result | size |
| risch | \(-\frac {\sqrt {e x +d}\, \left (b e x -12 c d x +2 b d \right )}{4 b^{4} x^{2} d}-\frac {e \left (-\frac {\left (b^{2} e^{2}+12 b c d e -48 c^{2} d^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b e \sqrt {d}}-\frac {8 c^{2} d \left (\frac {\frac {c b e \left (11 b e -12 c d \right ) \left (e x +d \right )^{\frac {3}{2}}}{8 b e -8 c d}+\frac {b e \left (13 b e -12 c d \right ) \sqrt {e x +d}}{8}}{\left (c \left (e x +d \right )+b e -c d \right )^{2}}+\frac {\left (35 b^{2} e^{2}-84 b c d e +48 c^{2} d^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{8 \left (b e -c d \right ) \sqrt {\left (b e -c d \right ) c}}\right )}{b e}\right )}{4 b^{4} d}\) | \(241\) |
| derivativedivides | \(2 e^{5} \left (-\frac {\frac {\frac {b e \left (b e -12 c d \right ) \left (e x +d \right )^{\frac {3}{2}}}{8 d}+\left (\frac {1}{8} b^{2} e^{2}+\frac {3}{2} b c d e \right ) \sqrt {e x +d}}{e^{2} x^{2}}-\frac {\left (b^{2} e^{2}+12 b c d e -48 c^{2} d^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{8 d^{\frac {3}{2}}}}{b^{5} e^{5}}+\frac {c^{2} \left (\frac {\frac {c b e \left (11 b e -12 c d \right ) \left (e x +d \right )^{\frac {3}{2}}}{8 b e -8 c d}+\frac {b e \left (13 b e -12 c d \right ) \sqrt {e x +d}}{8}}{\left (c \left (e x +d \right )+b e -c d \right )^{2}}+\frac {\left (35 b^{2} e^{2}-84 b c d e +48 c^{2} d^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{8 \left (b e -c d \right ) \sqrt {\left (b e -c d \right ) c}}\right )}{b^{5} e^{5}}\right )\) | \(258\) |
| default | \(2 e^{5} \left (-\frac {\frac {\frac {b e \left (b e -12 c d \right ) \left (e x +d \right )^{\frac {3}{2}}}{8 d}+\left (\frac {1}{8} b^{2} e^{2}+\frac {3}{2} b c d e \right ) \sqrt {e x +d}}{e^{2} x^{2}}-\frac {\left (b^{2} e^{2}+12 b c d e -48 c^{2} d^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{8 d^{\frac {3}{2}}}}{b^{5} e^{5}}+\frac {c^{2} \left (\frac {\frac {c b e \left (11 b e -12 c d \right ) \left (e x +d \right )^{\frac {3}{2}}}{8 b e -8 c d}+\frac {b e \left (13 b e -12 c d \right ) \sqrt {e x +d}}{8}}{\left (c \left (e x +d \right )+b e -c d \right )^{2}}+\frac {\left (35 b^{2} e^{2}-84 b c d e +48 c^{2} d^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{8 \left (b e -c d \right ) \sqrt {\left (b e -c d \right ) c}}\right )}{b^{5} e^{5}}\right )\) | \(258\) |
| pseudoelliptic | \(\frac {12 x^{2} \left (\frac {35 b^{2} e^{2} d^{\frac {5}{2}}}{48}+\left (c d -\frac {7 b e}{4}\right ) c \,d^{\frac {7}{2}}\right ) \left (c x +b \right )^{2} c^{2} \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )+\frac {\sqrt {\left (b e -c d \right ) c}\, \left (\frac {d \,x^{2} \left (c x +b \right )^{2} \left (b e -c d \right ) \left (b^{2} e^{2}+12 b c d e -48 c^{2} d^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{2}+\left (-e b \left (-12 c^{3} x^{3}-\frac {37}{2} b \,c^{2} x^{2}-\frac {9}{2} b^{2} c x +b^{3}\right ) d^{\frac {5}{2}}-\frac {b^{2} e^{2} x \left (c x +b \right )^{2} d^{\frac {3}{2}}}{2}+\left (-6 c^{2} x^{2}-6 b c x +b^{2}\right ) c \,d^{\frac {7}{2}} \left (2 c x +b \right )\right ) \sqrt {e x +d}\, b \right )}{2}}{\sqrt {\left (b e -c d \right ) c}\, d^{\frac {5}{2}} x^{2} b^{5} \left (c x +b \right )^{2} \left (b e -c d \right )}\) | \(265\) |
-1/4*(e*x+d)^(1/2)*(b*e*x-12*c*d*x+2*b*d)/b^4/x^2/d-1/4/b^4/d*e*(-1/b/e*(b ^2*e^2+12*b*c*d*e-48*c^2*d^2)/d^(1/2)*arctanh((e*x+d)^(1/2)/d^(1/2))-8*c^2 *d/b/e*((1/8*c*b*e*(11*b*e-12*c*d)/(b*e-c*d)*(e*x+d)^(3/2)+1/8*b*e*(13*b*e -12*c*d)*(e*x+d)^(1/2))/(c*(e*x+d)+b*e-c*d)^2+1/8*(35*b^2*e^2-84*b*c*d*e+4 8*c^2*d^2)/(b*e-c*d)/((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. 558 vs. \(2 (217) = 434\).
Time = 0.49 (sec) , antiderivative size = 2265, normalized size of antiderivative = 9.24 \[ \int \frac {\sqrt {d+e x}}{\left (b x+c x^2\right )^3} \, dx=\text {Too large to display} \]
[-1/8*(((48*c^5*d^4 - 84*b*c^4*d^3*e + 35*b^2*c^3*d^2*e^2)*x^4 + 2*(48*b*c ^4*d^4 - 84*b^2*c^3*d^3*e + 35*b^3*c^2*d^2*e^2)*x^3 + (48*b^2*c^3*d^4 - 84 *b^3*c^2*d^3*e + 35*b^4*c*d^2*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)) + ((48*c^5*d^3 - 60*b*c^4*d^2*e + 11*b^2*c^3*d*e^2 + b^3*c^2*e^3)*x^4 + 2*( 48*b*c^4*d^3 - 60*b^2*c^3*d^2*e + 11*b^3*c^2*d*e^2 + b^4*c*e^3)*x^3 + (48* b^2*c^3*d^3 - 60*b^3*c^2*d^2*e + 11*b^4*c*d*e^2 + b^5*e^3)*x^2)*sqrt(d)*lo g((e*x + 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) + 2*(2*b^4*c*d^3 - 2*b^5*d^2*e - (24*b*c^4*d^3 - 24*b^2*c^3*d^2*e + b^3*c^2*d*e^2)*x^3 - (36*b^2*c^3*d^3 - 37*b^3*c^2*d^2*e + 2*b^4*c*d*e^2)*x^2 - (8*b^3*c^2*d^3 - 9*b^4*c*d^2*e + b^5*d*e^2)*x)*sqrt(e*x + d))/((b^5*c^3*d^3 - b^6*c^2*d^2*e)*x^4 + 2*(b^6* c^2*d^3 - b^7*c*d^2*e)*x^3 + (b^7*c*d^3 - b^8*d^2*e)*x^2), 1/8*(2*((48*c^5 *d^4 - 84*b*c^4*d^3*e + 35*b^2*c^3*d^2*e^2)*x^4 + 2*(48*b*c^4*d^4 - 84*b^2 *c^3*d^3*e + 35*b^3*c^2*d^2*e^2)*x^3 + (48*b^2*c^3*d^4 - 84*b^3*c^2*d^3*e + 35*b^4*c*d^2*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)) - ((48*c^5*d^3 - 60*b*c^4*d^2*e + 11*b^2*c^3*d*e^2 + b^3*c^2*e^3)*x^4 + 2*(48*b*c^4*d^3 - 60*b^2*c^3*d^2*e + 11*b^3*c^2*d*e^2 + b^4*c*e^3)*x^3 + (48*b^2*c^3*d^3 - 60*b^3*c^2*d^2*e + 11*b^4*c*d*e^2 + b^5*e^3)*x^2)*sqrt(d)*log((e*x + 2*sqrt(e*x + d)*sqrt(d ) + 2*d)/x) - 2*(2*b^4*c*d^3 - 2*b^5*d^2*e - (24*b*c^4*d^3 - 24*b^2*c^3...
\[ \int \frac {\sqrt {d+e x}}{\left (b x+c x^2\right )^3} \, dx=\int \frac {\sqrt {d + e x}}{x^{3} \left (b + c x\right )^{3}}\, dx \]
Exception generated. \[ \int \frac {\sqrt {d+e x}}{\left (b x+c x^2\right )^3} \, dx=\text {Exception raised: ValueError} \]
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. 489 vs. \(2 (217) = 434\).
Time = 0.32 (sec) , antiderivative size = 489, normalized size of antiderivative = 2.00 \[ \int \frac {\sqrt {d+e x}}{\left (b x+c x^2\right )^3} \, dx=-\frac {{\left (48 \, c^{4} d^{2} - 84 \, b c^{3} d e + 35 \, b^{2} c^{2} e^{2}\right )} \arctan \left (\frac {\sqrt {e x + d} c}{\sqrt {-c^{2} d + b c e}}\right )}{4 \, {\left (b^{5} c d - b^{6} e\right )} \sqrt {-c^{2} d + b c e}} + \frac {24 \, {\left (e x + d\right )}^{\frac {7}{2}} c^{4} d^{2} e - 72 \, {\left (e x + d\right )}^{\frac {5}{2}} c^{4} d^{3} e + 72 \, {\left (e x + d\right )}^{\frac {3}{2}} c^{4} d^{4} e - 24 \, \sqrt {e x + d} c^{4} d^{5} e - 24 \, {\left (e x + d\right )}^{\frac {7}{2}} b c^{3} d e^{2} + 108 \, {\left (e x + d\right )}^{\frac {5}{2}} b c^{3} d^{2} e^{2} - 144 \, {\left (e x + d\right )}^{\frac {3}{2}} b c^{3} d^{3} e^{2} + 60 \, \sqrt {e x + d} b c^{3} d^{4} e^{2} + {\left (e x + d\right )}^{\frac {7}{2}} b^{2} c^{2} e^{3} - 40 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{2} c^{2} d e^{3} + 85 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{2} c^{2} d^{2} e^{3} - 46 \, \sqrt {e x + d} b^{2} c^{2} d^{3} e^{3} + 2 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{3} c e^{4} - 13 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{3} c d e^{4} + 9 \, \sqrt {e x + d} b^{3} c d^{2} e^{4} + {\left (e x + d\right )}^{\frac {3}{2}} b^{4} e^{5} + \sqrt {e x + d} b^{4} d e^{5}}{4 \, {\left (b^{4} c d^{2} - b^{5} d e\right )} {\left ({\left (e x + d\right )}^{2} c - 2 \, {\left (e x + d\right )} c d + c d^{2} + {\left (e x + d\right )} b e - b d e\right )}^{2}} + \frac {{\left (48 \, c^{2} d^{2} - 12 \, b c d e - b^{2} e^{2}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-d}}\right )}{4 \, b^{5} \sqrt {-d} d} \]
-1/4*(48*c^4*d^2 - 84*b*c^3*d*e + 35*b^2*c^2*e^2)*arctan(sqrt(e*x + d)*c/s qrt(-c^2*d + b*c*e))/((b^5*c*d - b^6*e)*sqrt(-c^2*d + b*c*e)) + 1/4*(24*(e *x + d)^(7/2)*c^4*d^2*e - 72*(e*x + d)^(5/2)*c^4*d^3*e + 72*(e*x + d)^(3/2 )*c^4*d^4*e - 24*sqrt(e*x + d)*c^4*d^5*e - 24*(e*x + d)^(7/2)*b*c^3*d*e^2 + 108*(e*x + d)^(5/2)*b*c^3*d^2*e^2 - 144*(e*x + d)^(3/2)*b*c^3*d^3*e^2 + 60*sqrt(e*x + d)*b*c^3*d^4*e^2 + (e*x + d)^(7/2)*b^2*c^2*e^3 - 40*(e*x + d )^(5/2)*b^2*c^2*d*e^3 + 85*(e*x + d)^(3/2)*b^2*c^2*d^2*e^3 - 46*sqrt(e*x + d)*b^2*c^2*d^3*e^3 + 2*(e*x + d)^(5/2)*b^3*c*e^4 - 13*(e*x + d)^(3/2)*b^3 *c*d*e^4 + 9*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)/((b^4*c*d^2 - b^5*d*e)*((e*x + d)^2*c - 2*(e*x + d)*c* d + c*d^2 + (e*x + d)*b*e - b*d*e)^2) + 1/4*(48*c^2*d^2 - 12*b*c*d*e - b^2 *e^2)*arctan(sqrt(e*x + d)/sqrt(-d))/(b^5*sqrt(-d)*d)
Time = 11.68 (sec) , antiderivative size = 4815, normalized size of antiderivative = 19.65 \[ \int \frac {\sqrt {d+e x}}{\left (b x+c x^2\right )^3} \, dx=\text {Too large to display} \]
(((d + e*x)^(3/2)*(b^4*e^5 + 72*c^4*d^4*e - 144*b*c^3*d^3*e^2 + 85*b^2*c^2 *d^2*e^3 - 13*b^3*c*d*e^4))/(4*b^4*(c*d^2 - b*d*e)) - ((d + e*x)^(1/2)*(b^ 3*e^4 + 24*c^3*d^3*e - 36*b*c^2*d^2*e^2 + 10*b^2*c*d*e^3))/(4*b^4) + ((b*e - 2*c*d)*(d + e*x)^(5/2)*(b^2*c*e^3 + 18*c^3*d^2*e - 18*b*c^2*d*e^2))/(2* b^4*(c*d^2 - b*d*e)) + (c*e*(d + e*x)^(7/2)*(24*c^3*d^2 + b^2*c*e^2 - 24*b *c^2*d*e))/(4*b^4*(c*d^2 - b*d*e)))/(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((((((d + e*x)^(1/2)*(b^6*c^3*e^8 + 4608*c^9*d^6*e^2 - 13824*b* c^8*d^5*e^3 + 22*b^5*c^4*d*e^7 + 15072*b^2*c^7*d^4*e^4 - 7104*b^3*c^6*d^3* e^5 + 1226*b^4*c^5*d^2*e^6))/(8*(b^8*c^2*d^4 + b^10*d^2*e^2 - 2*b^9*c*d^3* e)) + (((b^14*c^2*d*e^7 - 24*b^10*c^6*d^5*e^3 + 60*b^11*c^5*d^4*e^4 - 46*b ^12*c^4*d^3*e^5 + 9*b^13*c^3*d^2*e^6)/(b^12*c^2*d^4 + b^14*d^2*e^2 - 2*b^1 3*c*d^3*e) - ((d + e*x)^(1/2)*(b^2*e^2 - 48*c^2*d^2 + 12*b*c*d*e)*(128*b^1 0*c^5*d^5*e^2 - 320*b^11*c^4*d^4*e^3 + 256*b^12*c^3*d^3*e^4 - 64*b^13*c^2* d^2*e^5))/(64*b^5*(d^3)^(1/2)*(b^8*c^2*d^4 + b^10*d^2*e^2 - 2*b^9*c*d^3*e) ))*(b^2*e^2 - 48*c^2*d^2 + 12*b*c*d*e))/(8*b^5*(d^3)^(1/2)))*(b^2*e^2 - 48 *c^2*d^2 + 12*b*c*d*e)*1i)/(8*b^5*(d^3)^(1/2)) + ((((d + e*x)^(1/2)*(b^6*c ^3*e^8 + 4608*c^9*d^6*e^2 - 13824*b*c^8*d^5*e^3 + 22*b^5*c^4*d*e^7 + 15072 *b^2*c^7*d^4*e^4 - 7104*b^3*c^6*d^3*e^5 + 1226*b^4*c^5*d^2*e^6))/(8*(b^...