Integrand size = 21, antiderivative size = 337 \[ \int \frac {\sqrt {b x+c x^2}}{(d+e x)^6} \, dx=\frac {(2 c d-b e) \left (16 c^2 d^2-16 b c d e+7 b^2 e^2\right ) (b d+(2 c d-b e) x) \sqrt {b x+c x^2}}{128 d^4 (c d-b e)^4 (d+e x)^2}-\frac {e \left (b x+c x^2\right )^{3/2}}{5 d (c d-b e) (d+e x)^5}-\frac {7 e (2 c d-b e) \left (b x+c x^2\right )^{3/2}}{40 d^2 (c d-b e)^2 (d+e x)^4}-\frac {e \left (108 c^2 d^2-108 b c d e+35 b^2 e^2\right ) \left (b x+c x^2\right )^{3/2}}{240 d^3 (c d-b e)^3 (d+e x)^3}-\frac {b^2 (2 c d-b e) \left (16 c^2 d^2-16 b c d e+7 b^2 e^2\right ) \text {arctanh}\left (\frac {b d+(2 c d-b e) x}{2 \sqrt {d} \sqrt {c d-b e} \sqrt {b x+c x^2}}\right )}{256 d^{9/2} (c d-b e)^{9/2}} \]
-1/5*e*(c*x^2+b*x)^(3/2)/d/(-b*e+c*d)/(e*x+d)^5-7/40*e*(-b*e+2*c*d)*(c*x^2 +b*x)^(3/2)/d^2/(-b*e+c*d)^2/(e*x+d)^4-1/240*e*(35*b^2*e^2-108*b*c*d*e+108 *c^2*d^2)*(c*x^2+b*x)^(3/2)/d^3/(-b*e+c*d)^3/(e*x+d)^3-1/256*b^2*(-b*e+2*c *d)*(7*b^2*e^2-16*b*c*d*e+16*c^2*d^2)*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^(9/2)/(-b*e+c*d)^(9/2)+1/128*(- b*e+2*c*d)*(7*b^2*e^2-16*b*c*d*e+16*c^2*d^2)*(b*d+(-b*e+2*c*d)*x)*(c*x^2+b *x)^(1/2)/d^4/(-b*e+c*d)^4/(e*x+d)^2
Time = 11.12 (sec) , antiderivative size = 308, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt {b x+c x^2}}{(d+e x)^6} \, dx=\frac {\sqrt {x (b+c x)} \left (384 e x^{3/2} (b+c x)+\frac {336 e (2 c d-b e) x^{3/2} (b+c x) (d+e x)}{d (c d-b e)}+\frac {8 e \left (108 c^2 d^2-108 b c d e+35 b^2 e^2\right ) x^{3/2} (b+c x) (d+e x)^2}{d^2 (c d-b e)^2}+\frac {15 (2 c d-b e) \left (16 c^2 d^2-16 b c d e+7 b^2 e^2\right ) (d+e x)^3 \left (\sqrt {d} \sqrt {c d-b e} \sqrt {x} \sqrt {b+c x} (-b d-2 c d x+b e x)+b^2 (d+e x)^2 \text {arctanh}\left (\frac {\sqrt {c d-b e} \sqrt {x}}{\sqrt {d} \sqrt {b+c x}}\right )\right )}{d^{7/2} (c d-b e)^{7/2} \sqrt {b+c x}}\right )}{1920 d (-c d+b e) \sqrt {x} (d+e x)^5} \]
(Sqrt[x*(b + c*x)]*(384*e*x^(3/2)*(b + c*x) + (336*e*(2*c*d - b*e)*x^(3/2) *(b + c*x)*(d + e*x))/(d*(c*d - b*e)) + (8*e*(108*c^2*d^2 - 108*b*c*d*e + 35*b^2*e^2)*x^(3/2)*(b + c*x)*(d + e*x)^2)/(d^2*(c*d - b*e)^2) + (15*(2*c* d - b*e)*(16*c^2*d^2 - 16*b*c*d*e + 7*b^2*e^2)*(d + e*x)^3*(Sqrt[d]*Sqrt[c *d - b*e]*Sqrt[x]*Sqrt[b + c*x]*(-(b*d) - 2*c*d*x + b*e*x) + b^2*(d + e*x) ^2*ArcTanh[(Sqrt[c*d - b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])]))/(d^(7/2)*( c*d - b*e)^(7/2)*Sqrt[b + c*x])))/(1920*d*(-(c*d) + b*e)*Sqrt[x]*(d + e*x) ^5)
Time = 0.60 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.07, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {1167, 27, 1237, 27, 1228, 1152, 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.
| \(\displaystyle \int \frac {\sqrt {b x+c x^2}}{(d+e x)^6} \, dx\) |
| \(\Big \downarrow \) 1167 |
| \(\displaystyle -\frac {\int -\frac {(10 c d-7 b e-4 c e x) \sqrt {c x^2+b x}}{2 (d+e x)^5}dx}{5 d (c d-b e)}-\frac {e \left (b x+c x^2\right )^{3/2}}{5 d (d+e x)^5 (c d-b e)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(10 c d-7 b e-4 c e x) \sqrt {c x^2+b x}}{(d+e x)^5}dx}{10 d (c d-b e)}-\frac {e \left (b x+c x^2\right )^{3/2}}{5 d (d+e x)^5 (c d-b e)}\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle \frac {-\frac {\int -\frac {\left (80 c^2 d^2-94 b c e d+35 b^2 e^2-14 c e (2 c d-b e) x\right ) \sqrt {c x^2+b x}}{2 (d+e x)^4}dx}{4 d (c d-b e)}-\frac {7 e \left (b x+c x^2\right )^{3/2} (2 c d-b e)}{4 d (d+e x)^4 (c d-b e)}}{10 d (c d-b e)}-\frac {e \left (b x+c x^2\right )^{3/2}}{5 d (d+e x)^5 (c d-b e)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {\left (80 c^2 d^2-94 b c e d+35 b^2 e^2-14 c e (2 c d-b e) x\right ) \sqrt {c x^2+b x}}{(d+e x)^4}dx}{8 d (c d-b e)}-\frac {7 e \left (b x+c x^2\right )^{3/2} (2 c d-b e)}{4 d (d+e x)^4 (c d-b e)}}{10 d (c d-b e)}-\frac {e \left (b x+c x^2\right )^{3/2}}{5 d (d+e x)^5 (c d-b e)}\) |
| \(\Big \downarrow \) 1228 |
| \(\displaystyle \frac {\frac {\frac {5 (2 c d-b e) \left (7 b^2 e^2-16 b c d e+16 c^2 d^2\right ) \int \frac {\sqrt {c x^2+b x}}{(d+e x)^3}dx}{2 d (c d-b e)}-\frac {e \left (b x+c x^2\right )^{3/2} \left (35 b^2 e^2-108 b c d e+108 c^2 d^2\right )}{3 d (d+e x)^3 (c d-b e)}}{8 d (c d-b e)}-\frac {7 e \left (b x+c x^2\right )^{3/2} (2 c d-b e)}{4 d (d+e x)^4 (c d-b e)}}{10 d (c d-b e)}-\frac {e \left (b x+c x^2\right )^{3/2}}{5 d (d+e x)^5 (c d-b e)}\) |
| \(\Big \downarrow \) 1152 |
| \(\displaystyle \frac {\frac {\frac {5 (2 c d-b e) \left (7 b^2 e^2-16 b c d e+16 c^2 d^2\right ) \left (\frac {\sqrt {b x+c x^2} (x (2 c d-b e)+b d)}{4 d (d+e x)^2 (c d-b e)}-\frac {b^2 \int \frac {1}{(d+e x) \sqrt {c x^2+b x}}dx}{8 d (c d-b e)}\right )}{2 d (c d-b e)}-\frac {e \left (b x+c x^2\right )^{3/2} \left (35 b^2 e^2-108 b c d e+108 c^2 d^2\right )}{3 d (d+e x)^3 (c d-b e)}}{8 d (c d-b e)}-\frac {7 e \left (b x+c x^2\right )^{3/2} (2 c d-b e)}{4 d (d+e x)^4 (c d-b e)}}{10 d (c d-b e)}-\frac {e \left (b x+c x^2\right )^{3/2}}{5 d (d+e x)^5 (c d-b e)}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {\frac {\frac {5 (2 c d-b e) \left (7 b^2 e^2-16 b c d e+16 c^2 d^2\right ) \left (\frac {b^2 \int \frac {1}{4 d (c d-b e)-\frac {(b d+(2 c d-b e) x)^2}{c x^2+b x}}d\left (-\frac {b d+(2 c d-b e) x}{\sqrt {c x^2+b x}}\right )}{4 d (c d-b e)}+\frac {\sqrt {b x+c x^2} (x (2 c d-b e)+b d)}{4 d (d+e x)^2 (c d-b e)}\right )}{2 d (c d-b e)}-\frac {e \left (b x+c x^2\right )^{3/2} \left (35 b^2 e^2-108 b c d e+108 c^2 d^2\right )}{3 d (d+e x)^3 (c d-b e)}}{8 d (c d-b e)}-\frac {7 e \left (b x+c x^2\right )^{3/2} (2 c d-b e)}{4 d (d+e x)^4 (c d-b e)}}{10 d (c d-b e)}-\frac {e \left (b x+c x^2\right )^{3/2}}{5 d (d+e x)^5 (c d-b e)}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\frac {5 (2 c d-b e) \left (7 b^2 e^2-16 b c d e+16 c^2 d^2\right ) \left (\frac {\sqrt {b x+c x^2} (x (2 c d-b e)+b d)}{4 d (d+e x)^2 (c d-b e)}-\frac {b^2 \text {arctanh}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{8 d^{3/2} (c d-b e)^{3/2}}\right )}{2 d (c d-b e)}-\frac {e \left (b x+c x^2\right )^{3/2} \left (35 b^2 e^2-108 b c d e+108 c^2 d^2\right )}{3 d (d+e x)^3 (c d-b e)}}{8 d (c d-b e)}-\frac {7 e \left (b x+c x^2\right )^{3/2} (2 c d-b e)}{4 d (d+e x)^4 (c d-b e)}}{10 d (c d-b e)}-\frac {e \left (b x+c x^2\right )^{3/2}}{5 d (d+e x)^5 (c d-b e)}\) |
-1/5*(e*(b*x + c*x^2)^(3/2))/(d*(c*d - b*e)*(d + e*x)^5) + ((-7*e*(2*c*d - b*e)*(b*x + c*x^2)^(3/2))/(4*d*(c*d - b*e)*(d + e*x)^4) + (-1/3*(e*(108*c ^2*d^2 - 108*b*c*d*e + 35*b^2*e^2)*(b*x + c*x^2)^(3/2))/(d*(c*d - b*e)*(d + e*x)^3) + (5*(2*c*d - b*e)*(16*c^2*d^2 - 16*b*c*d*e + 7*b^2*e^2)*(((b*d + (2*c*d - b*e)*x)*Sqrt[b*x + c*x^2])/(4*d*(c*d - b*e)*(d + e*x)^2) - (b^2 *ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x ^2])])/(8*d^(3/2)*(c*d - b*e)^(3/2))))/(2*d*(c*d - b*e)))/(8*d*(c*d - b*e) ))/(10*d*(c*d - b*e))
3.3.94.3.1 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[((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/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[p*((b^2 - 4*a *c)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))) Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[m + 2*p + 2, 0] && GtQ[p, 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[e*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/((m + 1)*(c*d ^2 - b*d*e + a*e^2))), x] + Simp[1/((m + 1)*(c*d^2 - b*d*e + a*e^2)) Int[ (d + e*x)^(m + 1)*Simp[c*d*(m + 1) - b*e*(m + p + 2) - c*e*(m + 2*p + 3)*x, x]*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[m , -1] && ((LtQ[m, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]) || (SumSimp lerQ[m, 1] && IntegerQ[p]) || ILtQ[Simplify[m + 2*p + 3], 0])
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[(e*f - d*g)*(d + e*x)^(m + 1)*((a + b* x + c*x^2)^(p + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/((m + 1) *(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[ (c*d*f - f*b*e + a*e*g)*(m + 1) + b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && LtQ[m, -1 ] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Time = 2.17 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.06
| method | result | size |
| pseudoelliptic | \(-\frac {7 \left (\left (e^{3} \left (-e^{4} x^{4}+d^{4}-\frac {14}{3} d \,e^{3} x^{3}-\frac {128}{15} d^{2} e^{2} x^{2}-\frac {158}{21} d^{3} e x \right ) b^{4}-\frac {30 c \left (-\frac {38}{45} e^{4} x^{4}-\frac {889}{225} d \,e^{3} x^{3}-\frac {1631}{225} d^{2} e^{2} x^{2}-\frac {59}{9} d^{3} e x +d^{4}\right ) e^{2} d \,b^{3}}{7}+\frac {48 c^{2} e \left (-\frac {119}{180} e^{4} x^{4}-\frac {559}{180} d \,e^{3} x^{3}-\frac {1049}{180} d^{2} e^{2} x^{2}-\frac {67}{12} d^{3} e x +d^{4}\right ) d^{2} b^{2}}{7}-\frac {32 c^{3} \left (-\frac {2}{5} e^{4} x^{4}-\frac {21}{10} d \,e^{3} x^{3}-\frac {9}{2} d^{2} e^{2} x^{2}-5 d^{3} e x +d^{4}\right ) d^{3} b}{7}-\frac {64 \left (\frac {1}{10} e^{3} x^{3}+\frac {1}{2} d \,e^{2} x^{2}+d^{2} e x +d^{3}\right ) x \,c^{4} d^{4}}{7}\right ) \sqrt {d \left (b e -c d \right )}\, \sqrt {x \left (c x +b \right )}+\left (e x +d \right )^{5} \left (b^{2} e^{2}-\frac {16}{7} b c d e +\frac {16}{7} c^{2} d^{2}\right ) \left (b e -2 c d \right ) b^{2} \arctan \left (\frac {\sqrt {x \left (c x +b \right )}\, d}{x \sqrt {d \left (b e -c d \right )}}\right )\right )}{128 \sqrt {d \left (b e -c d \right )}\, \left (e x +d \right )^{5} \left (b e -c d \right )^{4} d^{4}}\) | \(357\) |
| default | \(\text {Expression too large to display}\) | \(3382\) |
-7/128*((e^3*(-e^4*x^4+d^4-14/3*d*e^3*x^3-128/15*d^2*e^2*x^2-158/21*d^3*e* x)*b^4-30/7*c*(-38/45*e^4*x^4-889/225*d*e^3*x^3-1631/225*d^2*e^2*x^2-59/9* d^3*e*x+d^4)*e^2*d*b^3+48/7*c^2*e*(-119/180*e^4*x^4-559/180*d*e^3*x^3-1049 /180*d^2*e^2*x^2-67/12*d^3*e*x+d^4)*d^2*b^2-32/7*c^3*(-2/5*e^4*x^4-21/10*d *e^3*x^3-9/2*d^2*e^2*x^2-5*d^3*e*x+d^4)*d^3*b-64/7*(1/10*e^3*x^3+1/2*d*e^2 *x^2+d^2*e*x+d^3)*x*c^4*d^4)*(d*(b*e-c*d))^(1/2)*(x*(c*x+b))^(1/2)+(e*x+d) ^5*(b^2*e^2-16/7*b*c*d*e+16/7*c^2*d^2)*(b*e-2*c*d)*b^2*arctan((x*(c*x+b))^ (1/2)/x*d/(d*(b*e-c*d))^(1/2)))/(d*(b*e-c*d))^(1/2)/(e*x+d)^5/(b*e-c*d)^4/ d^4
Leaf count of result is larger than twice the leaf count of optimal. 1219 vs. \(2 (307) = 614\).
Time = 0.47 (sec) , antiderivative size = 2449, normalized size of antiderivative = 7.27 \[ \int \frac {\sqrt {b x+c x^2}}{(d+e x)^6} \, dx=\text {Too large to display} \]
[-1/3840*(15*(32*b^2*c^3*d^8 - 48*b^3*c^2*d^7*e + 30*b^4*c*d^6*e^2 - 7*b^5 *d^5*e^3 + (32*b^2*c^3*d^3*e^5 - 48*b^3*c^2*d^2*e^6 + 30*b^4*c*d*e^7 - 7*b ^5*e^8)*x^5 + 5*(32*b^2*c^3*d^4*e^4 - 48*b^3*c^2*d^3*e^5 + 30*b^4*c*d^2*e^ 6 - 7*b^5*d*e^7)*x^4 + 10*(32*b^2*c^3*d^5*e^3 - 48*b^3*c^2*d^4*e^4 + 30*b^ 4*c*d^3*e^5 - 7*b^5*d^2*e^6)*x^3 + 10*(32*b^2*c^3*d^6*e^2 - 48*b^3*c^2*d^5 *e^3 + 30*b^4*c*d^4*e^4 - 7*b^5*d^3*e^5)*x^2 + 5*(32*b^2*c^3*d^7*e - 48*b^ 3*c^2*d^6*e^2 + 30*b^4*c*d^5*e^3 - 7*b^5*d^4*e^4)*x)*sqrt(c*d^2 - b*d*e)*l og((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*(480*b*c^4*d^9 - 1200*b^2*c^3*d^8*e + 1170*b^3*c^2*d^7*e^2 - 555 *b^4*c*d^6*e^3 + 105*b^5*d^5*e^4 + (96*c^5*d^6*e^3 - 288*b*c^4*d^5*e^4 + 6 68*b^2*c^3*d^4*e^5 - 856*b^3*c^2*d^3*e^6 + 485*b^4*c*d^2*e^7 - 105*b^5*d*e ^8)*x^4 + 2*(240*c^5*d^7*e^2 - 744*b*c^4*d^6*e^3 + 1622*b^2*c^3*d^5*e^4 - 2007*b^3*c^2*d^4*e^5 + 1134*b^4*c*d^3*e^6 - 245*b^5*d^2*e^7)*x^3 + 2*(480* c^5*d^8*e - 1560*b*c^4*d^7*e^2 + 3178*b^2*c^3*d^6*e^3 - 3729*b^3*c^2*d^5*e ^4 + 2079*b^4*c*d^4*e^5 - 448*b^5*d^3*e^6)*x^2 + 10*(96*c^5*d^9 - 336*b*c^ 4*d^8*e + 642*b^2*c^3*d^7*e^2 - 697*b^3*c^2*d^6*e^3 + 374*b^4*c*d^5*e^4 - 79*b^5*d^4*e^5)*x)*sqrt(c*x^2 + b*x))/(c^5*d^15 - 5*b*c^4*d^14*e + 10*b^2* c^3*d^13*e^2 - 10*b^3*c^2*d^12*e^3 + 5*b^4*c*d^11*e^4 - b^5*d^10*e^5 + (c^ 5*d^10*e^5 - 5*b*c^4*d^9*e^6 + 10*b^2*c^3*d^8*e^7 - 10*b^3*c^2*d^7*e^8 + 5 *b^4*c*d^6*e^9 - b^5*d^5*e^10)*x^5 + 5*(c^5*d^11*e^4 - 5*b*c^4*d^10*e^5...
\[ \int \frac {\sqrt {b x+c x^2}}{(d+e x)^6} \, dx=\int \frac {\sqrt {x \left (b + c x\right )}}{\left (d + e x\right )^{6}}\, dx \]
Exception generated. \[ \int \frac {\sqrt {b x+c x^2}}{(d+e x)^6} \, dx=\text {Exception raised: ValueError} \]
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. 2144 vs. \(2 (307) = 614\).
Time = 0.33 (sec) , antiderivative size = 2144, normalized size of antiderivative = 6.36 \[ \int \frac {\sqrt {b x+c x^2}}{(d+e x)^6} \, dx=\text {Too large to display} \]
1/128*(32*b^2*c^3*d^3 - 48*b^3*c^2*d^2*e + 30*b^4*c*d*e^2 - 7*b^5*e^3)*arc tan(((sqrt(c)*x - sqrt(c*x^2 + b*x))*e + sqrt(c)*d)/sqrt(-c*d^2 + b*d*e))/ ((c^4*d^8 - 4*b*c^3*d^7*e + 6*b^2*c^2*d^6*e^2 - 4*b^3*c*d^5*e^3 + b^4*d^4* e^4)*sqrt(-c*d^2 + b*d*e)) + 1/1920*(480*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9 *b^2*c^3*d^3*e^6 - 720*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*b^3*c^2*d^2*e^7 + 450*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*b^4*c*d*e^8 - 105*(sqrt(c)*x - sqrt (c*x^2 + b*x))^9*b^5*e^9 + 4320*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*b^2*c^(7 /2)*d^4*e^5 - 6480*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*b^3*c^(5/2)*d^3*e^6 + 4050*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*b^4*c^(3/2)*d^2*e^7 - 945*(sqrt(c) *x - sqrt(c*x^2 + b*x))^8*b^5*sqrt(c)*d*e^8 + 15040*(sqrt(c)*x - sqrt(c*x^ 2 + b*x))^7*b^2*c^4*d^5*e^4 - 20320*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*b^3* c^3*d^4*e^5 + 10740*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*b^4*c^2*d^3*e^6 - 11 90*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*b^5*c*d^2*e^7 - 490*(sqrt(c)*x - sqrt (c*x^2 + b*x))^7*b^6*d*e^8 + 7680*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*c^(13/ 2)*d^8*e - 30720*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*b*c^(11/2)*d^7*e^2 + 70 720*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*b^2*c^(9/2)*d^6*e^3 - 52000*(sqrt(c) *x - sqrt(c*x^2 + b*x))^6*b^3*c^(7/2)*d^5*e^4 + 7260*(sqrt(c)*x - sqrt(c*x ^2 + b*x))^6*b^4*c^(5/2)*d^4*e^5 + 9310*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6* b^5*c^(3/2)*d^3*e^6 - 3430*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*b^6*sqrt(c)*d ^2*e^7 + 3072*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*c^7*d^9 + 9216*(sqrt(c)...
Timed out. \[ \int \frac {\sqrt {b x+c x^2}}{(d+e x)^6} \, dx=\int \frac {\sqrt {c\,x^2+b\,x}}{{\left (d+e\,x\right )}^6} \,d x \]