Integrand size = 21, antiderivative size = 159 \[ \int \frac {(d+e x)^{5/2}}{\left (b x+c x^2\right )^2} \, dx=\frac {e (2 c d-b e) \sqrt {d+e x}}{b^2 c}-\frac {(d+e x)^{3/2} (b d+(2 c d-b e) x)}{b^2 \left (b x+c x^2\right )}+\frac {d^{3/2} (4 c d-5 b e) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b^3}-\frac {(c d-b e)^{3/2} (4 c d+b e) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 c^{3/2}} \]
-(e*x+d)^(3/2)*(b*d+(-b*e+2*c*d)*x)/b^2/(c*x^2+b*x)+d^(3/2)*(-5*b*e+4*c*d) *arctanh((e*x+d)^(1/2)/d^(1/2))/b^3-(-b*e+c*d)^(3/2)*(b*e+4*c*d)*arctanh(c ^(1/2)*(e*x+d)^(1/2)/(-b*e+c*d)^(1/2))/b^3/c^(3/2)+e*(-b*e+2*c*d)*(e*x+d)^ (1/2)/b^2/c
Time = 0.85 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.90 \[ \int \frac {(d+e x)^{5/2}}{\left (b x+c x^2\right )^2} \, dx=\frac {-\frac {b \sqrt {d+e x} \left (2 c^2 d^2 x+b^2 e^2 x+b c d (d-2 e x)\right )}{c x (b+c x)}+\frac {(-c d+b e)^{3/2} (4 c d+b e) \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {-c d+b e}}\right )}{c^{3/2}}+d^{3/2} (4 c d-5 b e) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b^3} \]
(-((b*Sqrt[d + e*x]*(2*c^2*d^2*x + b^2*e^2*x + b*c*d*(d - 2*e*x)))/(c*x*(b + c*x))) + ((-(c*d) + b*e)^(3/2)*(4*c*d + b*e)*ArcTan[(Sqrt[c]*Sqrt[d + e *x])/Sqrt[-(c*d) + b*e]])/c^(3/2) + d^(3/2)*(4*c*d - 5*b*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/b^3
Time = 0.45 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.13, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1164, 27, 1196, 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.
| \(\displaystyle \int \frac {(d+e x)^{5/2}}{\left (b x+c x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 1164 |
| \(\displaystyle -\frac {\int \frac {\sqrt {d+e x} (d (4 c d-5 b e)-e (2 c d-b e) x)}{2 \left (c x^2+b x\right )}dx}{b^2}-\frac {(d+e x)^{3/2} (x (2 c d-b e)+b d)}{b^2 \left (b x+c x^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {\sqrt {d+e x} (d (4 c d-5 b e)-e (2 c d-b e) x)}{c x^2+b x}dx}{2 b^2}-\frac {(d+e x)^{3/2} (x (2 c d-b e)+b d)}{b^2 \left (b x+c x^2\right )}\) |
| \(\Big \downarrow \) 1196 |
| \(\displaystyle -\frac {\frac {\int \frac {c (4 c d-5 b e) d^2+e \left (2 c^2 d^2-2 b c e d-b^2 e^2\right ) x}{\sqrt {d+e x} \left (c x^2+b x\right )}dx}{c}-\frac {2 e \sqrt {d+e x} (2 c d-b e)}{c}}{2 b^2}-\frac {(d+e x)^{3/2} (x (2 c d-b e)+b d)}{b^2 \left (b x+c x^2\right )}\) |
| \(\Big \downarrow \) 1197 |
| \(\displaystyle -\frac {\frac {2 \int \frac {e \left (d (c d-b e) (2 c d-b e)+\left (2 c^2 d^2-2 b c e d-b^2 e^2\right ) (d+e x)\right )}{c (d+e x)^2-(2 c d-b e) (d+e x)+d (c d-b e)}d\sqrt {d+e x}}{c}-\frac {2 e \sqrt {d+e x} (2 c d-b e)}{c}}{2 b^2}-\frac {(d+e x)^{3/2} (x (2 c d-b e)+b d)}{b^2 \left (b x+c x^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {2 e \int \frac {d (c d-b e) (2 c d-b e)+\left (2 c^2 d^2-2 b c e d-b^2 e^2\right ) (d+e x)}{c (d+e x)^2-(2 c d-b e) (d+e x)+d (c d-b e)}d\sqrt {d+e x}}{c}-\frac {2 e \sqrt {d+e x} (2 c d-b e)}{c}}{2 b^2}-\frac {(d+e x)^{3/2} (x (2 c d-b e)+b d)}{b^2 \left (b x+c x^2\right )}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle -\frac {\frac {2 e \left (\frac {c^2 d^2 (4 c d-5 b e) \int \frac {1}{c (d+e x)-c d}d\sqrt {d+e x}}{b e}-\frac {(c d-b e)^2 (b e+4 c d) \int \frac {1}{-c d+b e+c (d+e x)}d\sqrt {d+e x}}{b e}\right )}{c}-\frac {2 e \sqrt {d+e x} (2 c d-b e)}{c}}{2 b^2}-\frac {(d+e x)^{3/2} (x (2 c d-b e)+b d)}{b^2 \left (b x+c x^2\right )}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {\frac {2 e \left (\frac {(c d-b e)^{3/2} (b e+4 c d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b \sqrt {c} e}-\frac {c d^{3/2} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) (4 c d-5 b e)}{b e}\right )}{c}-\frac {2 e \sqrt {d+e x} (2 c d-b e)}{c}}{2 b^2}-\frac {(d+e x)^{3/2} (x (2 c d-b e)+b d)}{b^2 \left (b x+c x^2\right )}\) |
-(((d + e*x)^(3/2)*(b*d + (2*c*d - b*e)*x))/(b^2*(b*x + c*x^2))) - ((-2*e* (2*c*d - b*e)*Sqrt[d + e*x])/c + (2*e*(-((c*d^(3/2)*(4*c*d - 5*b*e)*ArcTan h[Sqrt[d + e*x]/Sqrt[d]])/(b*e)) + ((c*d - b*e)^(3/2)*(4*c*d + b*e)*ArcTan h[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b*Sqrt[c]*e)))/c)/(2*b^2)
3.4.71.3.1 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)*(d*b - 2*a*e + (2*c*d - b*e)*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 - 2)*Simp[e*(2*a*e*(m - 1) + b*d*(2*p - m + 4)) - 2* c*d^2*(2*p + 3) + e*(b*e - 2*d*c)*(m + 2*p + 2)*x, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[p, -1] && GtQ[m, 1] && Int QuadraticQ[a, b, c, d, e, m, p, x]
Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[g*((d + e*x)^m/(c*m)), x] + Simp[1/c Int [(d + e*x)^(m - 1)*(Simp[c*d*f - a*e*g + (g*c*d - b*e*g + c*e*f)*x, x]/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && FractionQ[m] & & GtQ[m, 0]
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_)^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 = 2.05 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.97
| method | result | size |
| derivativedivides | \(2 e^{3} \left (-\frac {d^{2} \left (\frac {b \sqrt {e x +d}}{2 x}+\frac {\left (5 b e -4 c d \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{2 \sqrt {d}}\right )}{b^{3} e^{3}}+\frac {\left (b e -c d \right )^{2} \left (-\frac {b e \sqrt {e x +d}}{2 c \left (c \left (e x +d \right )+b e -c d \right )}+\frac {\left (b e +4 c d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{2 c \sqrt {\left (b e -c d \right ) c}}\right )}{b^{3} e^{3}}\right )\) | \(154\) |
| default | \(2 e^{3} \left (-\frac {d^{2} \left (\frac {b \sqrt {e x +d}}{2 x}+\frac {\left (5 b e -4 c d \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{2 \sqrt {d}}\right )}{b^{3} e^{3}}+\frac {\left (b e -c d \right )^{2} \left (-\frac {b e \sqrt {e x +d}}{2 c \left (c \left (e x +d \right )+b e -c d \right )}+\frac {\left (b e +4 c d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{2 c \sqrt {\left (b e -c d \right ) c}}\right )}{b^{3} e^{3}}\right )\) | \(154\) |
| pseudoelliptic | \(\frac {x \left (b e +4 c d \right ) \left (b e -c d \right )^{2} \left (c x +b \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )+4 \left (x \left (c x +b \right ) c \left (d^{\frac {5}{2}} c -\frac {5 d^{\frac {3}{2}} b e}{4}\right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )-\frac {\left (2 c^{2} d^{2} x +b d \left (-2 e x +d \right ) c +b^{2} e^{2} x \right ) \sqrt {e x +d}\, b}{4}\right ) \sqrt {\left (b e -c d \right ) c}}{b^{3} x c \left (c x +b \right ) \sqrt {\left (b e -c d \right ) c}}\) | \(165\) |
| risch | \(-\frac {d^{2} \sqrt {e x +d}}{b^{2} x}+\frac {e \left (-\frac {d^{\frac {3}{2}} \left (5 b e -4 c d \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b e}+\frac {-\frac {b e \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right ) \sqrt {e x +d}}{c \left (c \left (e x +d \right )+b e -c d \right )}+\frac {\left (b^{3} e^{3}+2 b^{2} d \,e^{2} c -7 b \,c^{2} d^{2} e +4 c^{3} d^{3}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{c \sqrt {\left (b e -c d \right ) c}}}{b e}\right )}{b^{2}}\) | \(194\) |
2*e^3*(-d^2/b^3/e^3*(1/2*b*(e*x+d)^(1/2)/x+1/2*(5*b*e-4*c*d)/d^(1/2)*arcta nh((e*x+d)^(1/2)/d^(1/2)))+(b*e-c*d)^2/b^3/e^3*(-1/2*b*e/c*(e*x+d)^(1/2)/( c*(e*x+d)+b*e-c*d)+1/2*(b*e+4*c*d)/c/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^ (1/2)/((b*e-c*d)*c)^(1/2))))
Time = 0.34 (sec) , antiderivative size = 1002, normalized size of antiderivative = 6.30 \[ \int \frac {(d+e x)^{5/2}}{\left (b x+c x^2\right )^2} \, dx=\left [-\frac {{\left ({\left (4 \, c^{3} d^{2} - 3 \, b c^{2} d e - b^{2} c e^{2}\right )} x^{2} + {\left (4 \, b c^{2} d^{2} - 3 \, b^{2} c d e - b^{3} e^{2}\right )} x\right )} \sqrt {\frac {c d - b e}{c}} \log \left (\frac {c e x + 2 \, c d - b e + 2 \, \sqrt {e x + d} c \sqrt {\frac {c d - b e}{c}}}{c x + b}\right ) + {\left ({\left (4 \, c^{3} d^{2} - 5 \, b c^{2} d e\right )} x^{2} + {\left (4 \, b c^{2} d^{2} - 5 \, b^{2} c d e\right )} x\right )} \sqrt {d} \log \left (\frac {e x - 2 \, \sqrt {e x + d} \sqrt {d} + 2 \, d}{x}\right ) + 2 \, {\left (b^{2} c d^{2} + {\left (2 \, b c^{2} d^{2} - 2 \, b^{2} c d e + b^{3} e^{2}\right )} x\right )} \sqrt {e x + d}}{2 \, {\left (b^{3} c^{2} x^{2} + b^{4} c x\right )}}, -\frac {2 \, {\left ({\left (4 \, c^{3} d^{2} - 3 \, b c^{2} d e - b^{2} c e^{2}\right )} x^{2} + {\left (4 \, b c^{2} d^{2} - 3 \, b^{2} c d e - b^{3} e^{2}\right )} x\right )} \sqrt {-\frac {c d - b e}{c}} \arctan \left (-\frac {\sqrt {e x + d} c \sqrt {-\frac {c d - b e}{c}}}{c d - b e}\right ) + {\left ({\left (4 \, c^{3} d^{2} - 5 \, b c^{2} d e\right )} x^{2} + {\left (4 \, b c^{2} d^{2} - 5 \, b^{2} c d e\right )} x\right )} \sqrt {d} \log \left (\frac {e x - 2 \, \sqrt {e x + d} \sqrt {d} + 2 \, d}{x}\right ) + 2 \, {\left (b^{2} c d^{2} + {\left (2 \, b c^{2} d^{2} - 2 \, b^{2} c d e + b^{3} e^{2}\right )} x\right )} \sqrt {e x + d}}{2 \, {\left (b^{3} c^{2} x^{2} + b^{4} c x\right )}}, -\frac {2 \, {\left ({\left (4 \, c^{3} d^{2} - 5 \, b c^{2} d e\right )} x^{2} + {\left (4 \, b c^{2} d^{2} - 5 \, b^{2} c d e\right )} x\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {e x + d} \sqrt {-d}}{d}\right ) + {\left ({\left (4 \, c^{3} d^{2} - 3 \, b c^{2} d e - b^{2} c e^{2}\right )} x^{2} + {\left (4 \, b c^{2} d^{2} - 3 \, b^{2} c d e - b^{3} e^{2}\right )} x\right )} \sqrt {\frac {c d - b e}{c}} \log \left (\frac {c e x + 2 \, c d - b e + 2 \, \sqrt {e x + d} c \sqrt {\frac {c d - b e}{c}}}{c x + b}\right ) + 2 \, {\left (b^{2} c d^{2} + {\left (2 \, b c^{2} d^{2} - 2 \, b^{2} c d e + b^{3} e^{2}\right )} x\right )} \sqrt {e x + d}}{2 \, {\left (b^{3} c^{2} x^{2} + b^{4} c x\right )}}, -\frac {{\left ({\left (4 \, c^{3} d^{2} - 3 \, b c^{2} d e - b^{2} c e^{2}\right )} x^{2} + {\left (4 \, b c^{2} d^{2} - 3 \, b^{2} c d e - b^{3} e^{2}\right )} x\right )} \sqrt {-\frac {c d - b e}{c}} \arctan \left (-\frac {\sqrt {e x + d} c \sqrt {-\frac {c d - b e}{c}}}{c d - b e}\right ) + {\left ({\left (4 \, c^{3} d^{2} - 5 \, b c^{2} d e\right )} x^{2} + {\left (4 \, b c^{2} d^{2} - 5 \, b^{2} c d e\right )} x\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {e x + d} \sqrt {-d}}{d}\right ) + {\left (b^{2} c d^{2} + {\left (2 \, b c^{2} d^{2} - 2 \, b^{2} c d e + b^{3} e^{2}\right )} x\right )} \sqrt {e x + d}}{b^{3} c^{2} x^{2} + b^{4} c x}\right ] \]
[-1/2*(((4*c^3*d^2 - 3*b*c^2*d*e - b^2*c*e^2)*x^2 + (4*b*c^2*d^2 - 3*b^2*c *d*e - b^3*e^2)*x)*sqrt((c*d - b*e)/c)*log((c*e*x + 2*c*d - b*e + 2*sqrt(e *x + d)*c*sqrt((c*d - b*e)/c))/(c*x + b)) + ((4*c^3*d^2 - 5*b*c^2*d*e)*x^2 + (4*b*c^2*d^2 - 5*b^2*c*d*e)*x)*sqrt(d)*log((e*x - 2*sqrt(e*x + d)*sqrt( d) + 2*d)/x) + 2*(b^2*c*d^2 + (2*b*c^2*d^2 - 2*b^2*c*d*e + b^3*e^2)*x)*sqr t(e*x + d))/(b^3*c^2*x^2 + b^4*c*x), -1/2*(2*((4*c^3*d^2 - 3*b*c^2*d*e - b ^2*c*e^2)*x^2 + (4*b*c^2*d^2 - 3*b^2*c*d*e - b^3*e^2)*x)*sqrt(-(c*d - b*e) /c)*arctan(-sqrt(e*x + d)*c*sqrt(-(c*d - b*e)/c)/(c*d - b*e)) + ((4*c^3*d^ 2 - 5*b*c^2*d*e)*x^2 + (4*b*c^2*d^2 - 5*b^2*c*d*e)*x)*sqrt(d)*log((e*x - 2 *sqrt(e*x + d)*sqrt(d) + 2*d)/x) + 2*(b^2*c*d^2 + (2*b*c^2*d^2 - 2*b^2*c*d *e + b^3*e^2)*x)*sqrt(e*x + d))/(b^3*c^2*x^2 + b^4*c*x), -1/2*(2*((4*c^3*d ^2 - 5*b*c^2*d*e)*x^2 + (4*b*c^2*d^2 - 5*b^2*c*d*e)*x)*sqrt(-d)*arctan(sqr t(e*x + d)*sqrt(-d)/d) + ((4*c^3*d^2 - 3*b*c^2*d*e - b^2*c*e^2)*x^2 + (4*b *c^2*d^2 - 3*b^2*c*d*e - b^3*e^2)*x)*sqrt((c*d - b*e)/c)*log((c*e*x + 2*c* d - b*e + 2*sqrt(e*x + d)*c*sqrt((c*d - b*e)/c))/(c*x + b)) + 2*(b^2*c*d^2 + (2*b*c^2*d^2 - 2*b^2*c*d*e + b^3*e^2)*x)*sqrt(e*x + d))/(b^3*c^2*x^2 + b^4*c*x), -(((4*c^3*d^2 - 3*b*c^2*d*e - b^2*c*e^2)*x^2 + (4*b*c^2*d^2 - 3* b^2*c*d*e - b^3*e^2)*x)*sqrt(-(c*d - b*e)/c)*arctan(-sqrt(e*x + d)*c*sqrt( -(c*d - b*e)/c)/(c*d - b*e)) + ((4*c^3*d^2 - 5*b*c^2*d*e)*x^2 + (4*b*c^2*d ^2 - 5*b^2*c*d*e)*x)*sqrt(-d)*arctan(sqrt(e*x + d)*sqrt(-d)/d) + (b^2*c...
Timed out. \[ \int \frac {(d+e x)^{5/2}}{\left (b x+c x^2\right )^2} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {(d+e x)^{5/2}}{\left (b x+c x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]
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 = 0.28 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.64 \[ \int \frac {(d+e x)^{5/2}}{\left (b x+c x^2\right )^2} \, dx=-\frac {{\left (4 \, c d^{3} - 5 \, b d^{2} e\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-d}}\right )}{b^{3} \sqrt {-d}} + \frac {{\left (4 \, c^{3} d^{3} - 7 \, b c^{2} d^{2} e + 2 \, b^{2} c d e^{2} + b^{3} e^{3}\right )} \arctan \left (\frac {\sqrt {e x + d} c}{\sqrt {-c^{2} d + b c e}}\right )}{\sqrt {-c^{2} d + b c e} b^{3} c} - \frac {2 \, {\left (e x + d\right )}^{\frac {3}{2}} c^{2} d^{2} e - 2 \, \sqrt {e x + d} c^{2} d^{3} e - 2 \, {\left (e x + d\right )}^{\frac {3}{2}} b c d e^{2} + 3 \, \sqrt {e x + d} b c d^{2} e^{2} + {\left (e x + d\right )}^{\frac {3}{2}} b^{2} e^{3} - \sqrt {e x + d} b^{2} d e^{3}}{{\left ({\left (e x + d\right )}^{2} c - 2 \, {\left (e x + d\right )} c d + c d^{2} + {\left (e x + d\right )} b e - b d e\right )} b^{2} c} \]
-(4*c*d^3 - 5*b*d^2*e)*arctan(sqrt(e*x + d)/sqrt(-d))/(b^3*sqrt(-d)) + (4* c^3*d^3 - 7*b*c^2*d^2*e + 2*b^2*c*d*e^2 + b^3*e^3)*arctan(sqrt(e*x + d)*c/ sqrt(-c^2*d + b*c*e))/(sqrt(-c^2*d + b*c*e)*b^3*c) - (2*(e*x + d)^(3/2)*c^ 2*d^2*e - 2*sqrt(e*x + d)*c^2*d^3*e - 2*(e*x + d)^(3/2)*b*c*d*e^2 + 3*sqrt (e*x + d)*b*c*d^2*e^2 + (e*x + d)^(3/2)*b^2*e^3 - sqrt(e*x + d)*b^2*d*e^3) /(((e*x + d)^2*c - 2*(e*x + d)*c*d + c*d^2 + (e*x + d)*b*e - b*d*e)*b^2*c)
Time = 9.42 (sec) , antiderivative size = 1127, normalized size of antiderivative = 7.09 \[ \int \frac {(d+e x)^{5/2}}{\left (b x+c x^2\right )^2} \, dx=\frac {\frac {\sqrt {d+e\,x}\,\left (b^2\,d\,e^3-3\,b\,c\,d^2\,e^2+2\,c^2\,d^3\,e\right )}{b^2\,c}-\frac {e\,{\left (d+e\,x\right )}^{3/2}\,\left (b^2\,e^2-2\,b\,c\,d\,e+2\,c^2\,d^2\right )}{b^2\,c}}{\left (b\,e-2\,c\,d\right )\,\left (d+e\,x\right )+c\,{\left (d+e\,x\right )}^2+c\,d^2-b\,d\,e}-\frac {\mathrm {atanh}\left (\frac {10\,e^9\,\sqrt {d^3}\,\sqrt {d+e\,x}}{10\,d^2\,e^9+\frac {32\,c\,d^3\,e^8}{b}-\frac {132\,c^2\,d^4\,e^7}{b^2}+\frac {130\,c^3\,d^5\,e^6}{b^3}-\frac {40\,c^4\,d^6\,e^5}{b^4}}+\frac {32\,d\,e^8\,\sqrt {d^3}\,\sqrt {d+e\,x}}{32\,d^3\,e^8+\frac {10\,b\,d^2\,e^9}{c}-\frac {132\,c\,d^4\,e^7}{b}+\frac {130\,c^2\,d^5\,e^6}{b^2}-\frac {40\,c^3\,d^6\,e^5}{b^3}}-\frac {132\,c\,d^2\,e^7\,\sqrt {d^3}\,\sqrt {d+e\,x}}{32\,b\,d^3\,e^8-132\,c\,d^4\,e^7+\frac {130\,c^2\,d^5\,e^6}{b}+\frac {10\,b^2\,d^2\,e^9}{c}-\frac {40\,c^3\,d^6\,e^5}{b^2}}+\frac {130\,c^2\,d^3\,e^6\,\sqrt {d^3}\,\sqrt {d+e\,x}}{32\,b^2\,d^3\,e^8+130\,c^2\,d^5\,e^6-\frac {40\,c^3\,d^6\,e^5}{b}+\frac {10\,b^3\,d^2\,e^9}{c}-132\,b\,c\,d^4\,e^7}-\frac {40\,c^3\,d^4\,e^5\,\sqrt {d^3}\,\sqrt {d+e\,x}}{32\,b^3\,d^3\,e^8-40\,c^3\,d^6\,e^5+130\,b\,c^2\,d^5\,e^6-132\,b^2\,c\,d^4\,e^7+\frac {10\,b^4\,d^2\,e^9}{c}}\right )\,\left (5\,b\,e-4\,c\,d\right )\,\sqrt {d^3}}{b^3}-\frac {\mathrm {atanh}\left (\frac {30\,d^3\,e^6\,\sqrt {d+e\,x}\,\sqrt {-b^3\,c^3\,e^3+3\,b^2\,c^4\,d\,e^2-3\,b\,c^5\,d^2\,e+c^6\,d^3}}{14\,b^3\,d^2\,e^9+110\,c^3\,d^5\,e^6-82\,b\,c^2\,d^4\,e^7-4\,b^2\,c\,d^3\,e^8+\frac {2\,b^4\,d\,e^{10}}{c}-\frac {40\,c^4\,d^6\,e^5}{b}}-\frac {2\,d\,e^8\,\sqrt {d+e\,x}\,\sqrt {-b^3\,c^3\,e^3+3\,b^2\,c^4\,d\,e^2-3\,b\,c^5\,d^2\,e+c^6\,d^3}}{4\,c^3\,d^3\,e^8-14\,b\,c^2\,d^2\,e^9+\frac {82\,c^4\,d^4\,e^7}{b}-\frac {110\,c^5\,d^5\,e^6}{b^2}+\frac {40\,c^6\,d^6\,e^5}{b^3}-2\,b^2\,c\,d\,e^{10}}+\frac {18\,d^2\,e^7\,\sqrt {d+e\,x}\,\sqrt {-b^3\,c^3\,e^3+3\,b^2\,c^4\,d\,e^2-3\,b\,c^5\,d^2\,e+c^6\,d^3}}{2\,b^3\,d\,e^{10}-82\,c^3\,d^4\,e^7-4\,b\,c^2\,d^3\,e^8+14\,b^2\,c\,d^2\,e^9+\frac {110\,c^4\,d^5\,e^6}{b}-\frac {40\,c^5\,d^6\,e^5}{b^2}}+\frac {40\,d^4\,e^5\,\sqrt {d+e\,x}\,\sqrt {-b^3\,c^3\,e^3+3\,b^2\,c^4\,d\,e^2-3\,b\,c^5\,d^2\,e+c^6\,d^3}}{4\,b^3\,d^3\,e^8+40\,c^3\,d^6\,e^5-110\,b\,c^2\,d^5\,e^6+82\,b^2\,c\,d^4\,e^7-\frac {2\,b^5\,d\,e^{10}}{c^2}-\frac {14\,b^4\,d^2\,e^9}{c}}\right )\,\sqrt {-c^3\,{\left (b\,e-c\,d\right )}^3}\,\left (b\,e+4\,c\,d\right )}{b^3\,c^3} \]
(((d + e*x)^(1/2)*(b^2*d*e^3 + 2*c^2*d^3*e - 3*b*c*d^2*e^2))/(b^2*c) - (e* (d + e*x)^(3/2)*(b^2*e^2 + 2*c^2*d^2 - 2*b*c*d*e))/(b^2*c))/((b*e - 2*c*d) *(d + e*x) + c*(d + e*x)^2 + c*d^2 - b*d*e) - (atanh((10*e^9*(d^3)^(1/2)*( d + e*x)^(1/2))/(10*d^2*e^9 + (32*c*d^3*e^8)/b - (132*c^2*d^4*e^7)/b^2 + ( 130*c^3*d^5*e^6)/b^3 - (40*c^4*d^6*e^5)/b^4) + (32*d*e^8*(d^3)^(1/2)*(d + e*x)^(1/2))/(32*d^3*e^8 + (10*b*d^2*e^9)/c - (132*c*d^4*e^7)/b + (130*c^2* d^5*e^6)/b^2 - (40*c^3*d^6*e^5)/b^3) - (132*c*d^2*e^7*(d^3)^(1/2)*(d + e*x )^(1/2))/(32*b*d^3*e^8 - 132*c*d^4*e^7 + (130*c^2*d^5*e^6)/b + (10*b^2*d^2 *e^9)/c - (40*c^3*d^6*e^5)/b^2) + (130*c^2*d^3*e^6*(d^3)^(1/2)*(d + e*x)^( 1/2))/(32*b^2*d^3*e^8 + 130*c^2*d^5*e^6 - (40*c^3*d^6*e^5)/b + (10*b^3*d^2 *e^9)/c - 132*b*c*d^4*e^7) - (40*c^3*d^4*e^5*(d^3)^(1/2)*(d + e*x)^(1/2))/ (32*b^3*d^3*e^8 - 40*c^3*d^6*e^5 + 130*b*c^2*d^5*e^6 - 132*b^2*c*d^4*e^7 + (10*b^4*d^2*e^9)/c))*(5*b*e - 4*c*d)*(d^3)^(1/2))/b^3 - (atanh((30*d^3*e^ 6*(d + e*x)^(1/2)*(c^6*d^3 - b^3*c^3*e^3 + 3*b^2*c^4*d*e^2 - 3*b*c^5*d^2*e )^(1/2))/(14*b^3*d^2*e^9 + 110*c^3*d^5*e^6 - 82*b*c^2*d^4*e^7 - 4*b^2*c*d^ 3*e^8 + (2*b^4*d*e^10)/c - (40*c^4*d^6*e^5)/b) - (2*d*e^8*(d + e*x)^(1/2)* (c^6*d^3 - b^3*c^3*e^3 + 3*b^2*c^4*d*e^2 - 3*b*c^5*d^2*e)^(1/2))/(4*c^3*d^ 3*e^8 - 14*b*c^2*d^2*e^9 + (82*c^4*d^4*e^7)/b - (110*c^5*d^5*e^6)/b^2 + (4 0*c^6*d^6*e^5)/b^3 - 2*b^2*c*d*e^10) + (18*d^2*e^7*(d + e*x)^(1/2)*(c^6*d^ 3 - b^3*c^3*e^3 + 3*b^2*c^4*d*e^2 - 3*b*c^5*d^2*e)^(1/2))/(2*b^3*d*e^10...