Integrand size = 21, antiderivative size = 349 \[ \int \frac {1}{(d+e x)^{7/2} \left (b x+c x^2\right )^2} \, dx=-\frac {e \left (10 c^2 d^2-10 b c d e+7 b^2 e^2\right )}{5 b^2 d^2 (c d-b e)^2 (d+e x)^{5/2}}-\frac {e (2 c d-b e) \left (3 c^2 d^2-3 b c d e+7 b^2 e^2\right )}{3 b^2 d^3 (c d-b e)^3 (d+e x)^{3/2}}-\frac {e \left (2 c^4 d^4-4 b c^3 d^3 e+26 b^2 c^2 d^2 e^2-24 b^3 c d e^3+7 b^4 e^4\right )}{b^2 d^4 (c d-b e)^4 \sqrt {d+e x}}-\frac {b (c d-b e)+c (2 c d-b e) x}{b^2 d (c d-b e) (d+e x)^{5/2} \left (b x+c x^2\right )}+\frac {(4 c d+7 b e) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b^3 d^{9/2}}-\frac {c^{9/2} (4 c d-11 b e) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 (c d-b e)^{9/2}} \]
-1/5*e*(7*b^2*e^2-10*b*c*d*e+10*c^2*d^2)/b^2/d^2/(-b*e+c*d)^2/(e*x+d)^(5/2 )-1/3*e*(-b*e+2*c*d)*(7*b^2*e^2-3*b*c*d*e+3*c^2*d^2)/b^2/d^3/(-b*e+c*d)^3/ (e*x+d)^(3/2)+(-b*(-b*e+c*d)-c*(-b*e+2*c*d)*x)/b^2/d/(-b*e+c*d)/(e*x+d)^(5 /2)/(c*x^2+b*x)+(7*b*e+4*c*d)*arctanh((e*x+d)^(1/2)/d^(1/2))/b^3/d^(9/2)-c ^(9/2)*(-11*b*e+4*c*d)*arctanh(c^(1/2)*(e*x+d)^(1/2)/(-b*e+c*d)^(1/2))/b^3 /(-b*e+c*d)^(9/2)-e*(7*b^4*e^4-24*b^3*c*d*e^3+26*b^2*c^2*d^2*e^2-4*b*c^3*d ^3*e+2*c^4*d^4)/b^2/d^4/(-b*e+c*d)^4/(e*x+d)^(1/2)
Time = 2.34 (sec) , antiderivative size = 363, normalized size of antiderivative = 1.04 \[ \int \frac {1}{(d+e x)^{7/2} \left (b x+c x^2\right )^2} \, dx=\frac {-\frac {b \left (30 c^5 d^4 x (d+e x)^3+15 b c^4 d^3 (d-4 e x) (d+e x)^3+b^5 e^4 \left (15 d^3+161 d^2 e x+245 d e^2 x^2+105 e^3 x^3\right )+2 b^3 c^2 d e^2 \left (45 d^4+278 d^3 e x+179 d^2 e^2 x^2-225 d e^3 x^3-180 e^4 x^4\right )-b^4 c e^3 \left (60 d^4+537 d^3 e x+679 d^2 e^2 x^2+115 d e^3 x^3-105 e^4 x^4\right )+2 b^2 c^3 d^2 e \left (-30 d^4-45 d^3 e x+218 d^2 e^2 x^2+425 d e^3 x^3+195 e^4 x^4\right )\right )}{d^4 (c d-b e)^4 x (b+c x) (d+e x)^{5/2}}+\frac {15 c^{9/2} (4 c d-11 b e) \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {-c d+b e}}\right )}{(-c d+b e)^{9/2}}+\frac {15 (4 c d+7 b e) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{d^{9/2}}}{15 b^3} \]
(-((b*(30*c^5*d^4*x*(d + e*x)^3 + 15*b*c^4*d^3*(d - 4*e*x)*(d + e*x)^3 + b ^5*e^4*(15*d^3 + 161*d^2*e*x + 245*d*e^2*x^2 + 105*e^3*x^3) + 2*b^3*c^2*d* e^2*(45*d^4 + 278*d^3*e*x + 179*d^2*e^2*x^2 - 225*d*e^3*x^3 - 180*e^4*x^4) - b^4*c*e^3*(60*d^4 + 537*d^3*e*x + 679*d^2*e^2*x^2 + 115*d*e^3*x^3 - 105 *e^4*x^4) + 2*b^2*c^3*d^2*e*(-30*d^4 - 45*d^3*e*x + 218*d^2*e^2*x^2 + 425* d*e^3*x^3 + 195*e^4*x^4)))/(d^4*(c*d - b*e)^4*x*(b + c*x)*(d + e*x)^(5/2)) ) + (15*c^(9/2)*(4*c*d - 11*b*e)*ArcTan[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[-(c*d ) + b*e]])/(-(c*d) + b*e)^(9/2) + (15*(4*c*d + 7*b*e)*ArcTanh[Sqrt[d + e*x ]/Sqrt[d]])/d^(9/2))/(15*b^3)
Time = 0.89 (sec) , antiderivative size = 427, normalized size of antiderivative = 1.22, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {1165, 27, 1198, 1198, 1198, 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.
| \(\displaystyle \int \frac {1}{\left (b x+c x^2\right )^2 (d+e x)^{7/2}} \, dx\) |
| \(\Big \downarrow \) 1165 |
| \(\displaystyle -\frac {\int \frac {(c d-b e) (4 c d+7 b e)+7 c e (2 c d-b e) x}{2 (d+e x)^{7/2} \left (c x^2+b x\right )}dx}{b^2 d (c d-b e)}-\frac {c x (2 c d-b e)+b (c d-b e)}{b^2 d \left (b x+c x^2\right ) (d+e x)^{5/2} (c d-b e)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {(c d-b e) (4 c d+7 b e)+7 c e (2 c d-b e) x}{(d+e x)^{7/2} \left (c x^2+b x\right )}dx}{2 b^2 d (c d-b e)}-\frac {c x (2 c d-b e)+b (c d-b e)}{b^2 d \left (b x+c x^2\right ) (d+e x)^{5/2} (c d-b e)}\) |
| \(\Big \downarrow \) 1198 |
| \(\displaystyle -\frac {\frac {\int \frac {(4 c d+7 b e) (c d-b e)^2+c e \left (10 c^2 d^2-10 b c e d+7 b^2 e^2\right ) x}{(d+e x)^{5/2} \left (c x^2+b x\right )}dx}{d (c d-b e)}+\frac {2 e \left (7 b^2 e^2-10 b c d e+10 c^2 d^2\right )}{5 d (d+e x)^{5/2} (c d-b e)}}{2 b^2 d (c d-b e)}-\frac {c x (2 c d-b e)+b (c d-b e)}{b^2 d \left (b x+c x^2\right ) (d+e x)^{5/2} (c d-b e)}\) |
| \(\Big \downarrow \) 1198 |
| \(\displaystyle -\frac {\frac {\frac {\int \frac {(4 c d+7 b e) (c d-b e)^3+c e (2 c d-b e) \left (3 c^2 d^2-3 b c e d+7 b^2 e^2\right ) x}{(d+e x)^{3/2} \left (c x^2+b x\right )}dx}{d (c d-b e)}+\frac {2 e (2 c d-b e) \left (7 b^2 e^2-3 b c d e+3 c^2 d^2\right )}{3 d (d+e x)^{3/2} (c d-b e)}}{d (c d-b e)}+\frac {2 e \left (7 b^2 e^2-10 b c d e+10 c^2 d^2\right )}{5 d (d+e x)^{5/2} (c d-b e)}}{2 b^2 d (c d-b e)}-\frac {c x (2 c d-b e)+b (c d-b e)}{b^2 d \left (b x+c x^2\right ) (d+e x)^{5/2} (c d-b e)}\) |
| \(\Big \downarrow \) 1198 |
| \(\displaystyle -\frac {\frac {\frac {\frac {\int \frac {(4 c d+7 b e) (c d-b e)^4+c e \left (2 c^4 d^4-4 b c^3 e d^3+26 b^2 c^2 e^2 d^2-24 b^3 c e^3 d+7 b^4 e^4\right ) x}{\sqrt {d+e x} \left (c x^2+b x\right )}dx}{d (c d-b e)}+\frac {2 e \left (7 b^4 e^4-24 b^3 c d e^3+26 b^2 c^2 d^2 e^2-4 b c^3 d^3 e+2 c^4 d^4\right )}{d \sqrt {d+e x} (c d-b e)}}{d (c d-b e)}+\frac {2 e (2 c d-b e) \left (7 b^2 e^2-3 b c d e+3 c^2 d^2\right )}{3 d (d+e x)^{3/2} (c d-b e)}}{d (c d-b e)}+\frac {2 e \left (7 b^2 e^2-10 b c d e+10 c^2 d^2\right )}{5 d (d+e x)^{5/2} (c d-b e)}}{2 b^2 d (c d-b e)}-\frac {c x (2 c d-b e)+b (c d-b e)}{b^2 d \left (b x+c x^2\right ) (d+e x)^{5/2} (c d-b e)}\) |
| \(\Big \downarrow \) 1197 |
| \(\displaystyle -\frac {\frac {\frac {\frac {2 \int \frac {e \left ((2 c d-b e) \left (c^4 d^4-2 b c^3 e d^3-16 b^2 c^2 e^2 d^2+17 b^3 c e^3 d-7 b^4 e^4\right )+c \left (2 c^4 d^4-4 b c^3 e d^3+26 b^2 c^2 e^2 d^2-24 b^3 c e^3 d+7 b^4 e^4\right ) (d+e x)\right )}{c (d+e x)^2-(2 c d-b e) (d+e x)+d (c d-b e)}d\sqrt {d+e x}}{d (c d-b e)}+\frac {2 e \left (7 b^4 e^4-24 b^3 c d e^3+26 b^2 c^2 d^2 e^2-4 b c^3 d^3 e+2 c^4 d^4\right )}{d \sqrt {d+e x} (c d-b e)}}{d (c d-b e)}+\frac {2 e (2 c d-b e) \left (7 b^2 e^2-3 b c d e+3 c^2 d^2\right )}{3 d (d+e x)^{3/2} (c d-b e)}}{d (c d-b e)}+\frac {2 e \left (7 b^2 e^2-10 b c d e+10 c^2 d^2\right )}{5 d (d+e x)^{5/2} (c d-b e)}}{2 b^2 d (c d-b e)}-\frac {c x (2 c d-b e)+b (c d-b e)}{b^2 d \left (b x+c x^2\right ) (d+e x)^{5/2} (c d-b e)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\frac {\frac {2 e \int \frac {(2 c d-b e) \left (c^4 d^4-2 b c^3 e d^3-16 b^2 c^2 e^2 d^2+17 b^3 c e^3 d-7 b^4 e^4\right )+c \left (2 c^4 d^4-4 b c^3 e d^3+26 b^2 c^2 e^2 d^2-24 b^3 c e^3 d+7 b^4 e^4\right ) (d+e x)}{c (d+e x)^2-(2 c d-b e) (d+e x)+d (c d-b e)}d\sqrt {d+e x}}{d (c d-b e)}+\frac {2 e \left (7 b^4 e^4-24 b^3 c d e^3+26 b^2 c^2 d^2 e^2-4 b c^3 d^3 e+2 c^4 d^4\right )}{d \sqrt {d+e x} (c d-b e)}}{d (c d-b e)}+\frac {2 e (2 c d-b e) \left (7 b^2 e^2-3 b c d e+3 c^2 d^2\right )}{3 d (d+e x)^{3/2} (c d-b e)}}{d (c d-b e)}+\frac {2 e \left (7 b^2 e^2-10 b c d e+10 c^2 d^2\right )}{5 d (d+e x)^{5/2} (c d-b e)}}{2 b^2 d (c d-b e)}-\frac {c x (2 c d-b e)+b (c d-b e)}{b^2 d \left (b x+c x^2\right ) (d+e x)^{5/2} (c d-b e)}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle -\frac {\frac {\frac {\frac {2 e \left (\frac {c (c d-b e)^4 (7 b e+4 c d) \int \frac {1}{c (d+e x)-c d}d\sqrt {d+e x}}{b e}-\frac {c^5 d^4 (4 c d-11 b e) \int \frac {1}{-c d+b e+c (d+e x)}d\sqrt {d+e x}}{b e}\right )}{d (c d-b e)}+\frac {2 e \left (7 b^4 e^4-24 b^3 c d e^3+26 b^2 c^2 d^2 e^2-4 b c^3 d^3 e+2 c^4 d^4\right )}{d \sqrt {d+e x} (c d-b e)}}{d (c d-b e)}+\frac {2 e (2 c d-b e) \left (7 b^2 e^2-3 b c d e+3 c^2 d^2\right )}{3 d (d+e x)^{3/2} (c d-b e)}}{d (c d-b e)}+\frac {2 e \left (7 b^2 e^2-10 b c d e+10 c^2 d^2\right )}{5 d (d+e x)^{5/2} (c d-b e)}}{2 b^2 d (c d-b e)}-\frac {c x (2 c d-b e)+b (c d-b e)}{b^2 d \left (b x+c x^2\right ) (d+e x)^{5/2} (c d-b e)}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {\frac {\frac {\frac {2 e \left (\frac {c^{9/2} d^4 (4 c d-11 b e) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b e \sqrt {c d-b e}}-\frac {\text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) (c d-b e)^4 (7 b e+4 c d)}{b \sqrt {d} e}\right )}{d (c d-b e)}+\frac {2 e \left (7 b^4 e^4-24 b^3 c d e^3+26 b^2 c^2 d^2 e^2-4 b c^3 d^3 e+2 c^4 d^4\right )}{d \sqrt {d+e x} (c d-b e)}}{d (c d-b e)}+\frac {2 e (2 c d-b e) \left (7 b^2 e^2-3 b c d e+3 c^2 d^2\right )}{3 d (d+e x)^{3/2} (c d-b e)}}{d (c d-b e)}+\frac {2 e \left (7 b^2 e^2-10 b c d e+10 c^2 d^2\right )}{5 d (d+e x)^{5/2} (c d-b e)}}{2 b^2 d (c d-b e)}-\frac {c x (2 c d-b e)+b (c d-b e)}{b^2 d \left (b x+c x^2\right ) (d+e x)^{5/2} (c d-b e)}\) |
-((b*(c*d - b*e) + c*(2*c*d - b*e)*x)/(b^2*d*(c*d - b*e)*(d + e*x)^(5/2)*( b*x + c*x^2))) - ((2*e*(10*c^2*d^2 - 10*b*c*d*e + 7*b^2*e^2))/(5*d*(c*d - b*e)*(d + e*x)^(5/2)) + ((2*e*(2*c*d - b*e)*(3*c^2*d^2 - 3*b*c*d*e + 7*b^2 *e^2))/(3*d*(c*d - b*e)*(d + e*x)^(3/2)) + ((2*e*(2*c^4*d^4 - 4*b*c^3*d^3* e + 26*b^2*c^2*d^2*e^2 - 24*b^3*c*d*e^3 + 7*b^4*e^4))/(d*(c*d - b*e)*Sqrt[ d + e*x]) + (2*e*(-(((c*d - b*e)^4*(4*c*d + 7*b*e)*ArcTanh[Sqrt[d + e*x]/S qrt[d]])/(b*Sqrt[d]*e)) + (c^(9/2)*d^4*(4*c*d - 11*b*e)*ArcTanh[(Sqrt[c]*S qrt[d + e*x])/Sqrt[c*d - b*e]])/(b*e*Sqrt[c*d - b*e])))/(d*(c*d - b*e)))/( d*(c*d - b*e)))/(d*(c*d - b*e)))/(2*b^2*d*(c*d - b*e))
3.4.77.3.1 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)*(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[((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), x_Symbol] :> Simp[(e*f - d*g)*((d + e*x)^(m + 1)/((m + 1)*(c *d^2 - b*d*e + a*e^2))), x] + Simp[1/(c*d^2 - b*d*e + a*e^2) Int[(d + e*x )^(m + 1)*(Simp[c*d*f - f*b*e + a*e*g - c*(e*f - d*g)*x, x]/(a + b*x + c*x^ 2)), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && FractionQ[m] && LtQ[m, -1 ]
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 = 2.44 (sec) , antiderivative size = 250, normalized size of antiderivative = 0.72
| method | result | size |
| derivativedivides | \(2 e^{3} \left (-\frac {c^{5} \left (\frac {b e \sqrt {e x +d}}{2 c \left (e x +d \right )+2 b e -2 c d}+\frac {\left (11 b e -4 c d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{2 \sqrt {\left (b e -c d \right ) c}}\right )}{b^{3} e^{3} \left (b e -c d \right )^{4}}+\frac {-\frac {b \sqrt {e x +d}}{2 x}+\frac {\left (7 b e +4 c d \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{2 \sqrt {d}}}{d^{4} e^{3} b^{3}}-\frac {1}{5 d^{2} \left (b e -c d \right )^{2} \left (e x +d \right )^{\frac {5}{2}}}-\frac {2 b e -4 c d}{3 d^{3} \left (b e -c d \right )^{3} \left (e x +d \right )^{\frac {3}{2}}}-\frac {3 b^{2} e^{2}-10 b c d e +10 c^{2} d^{2}}{d^{4} \left (b e -c d \right )^{4} \sqrt {e x +d}}\right )\) | \(250\) |
| default | \(2 e^{3} \left (-\frac {c^{5} \left (\frac {b e \sqrt {e x +d}}{2 c \left (e x +d \right )+2 b e -2 c d}+\frac {\left (11 b e -4 c d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{2 \sqrt {\left (b e -c d \right ) c}}\right )}{b^{3} e^{3} \left (b e -c d \right )^{4}}+\frac {-\frac {b \sqrt {e x +d}}{2 x}+\frac {\left (7 b e +4 c d \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{2 \sqrt {d}}}{d^{4} e^{3} b^{3}}-\frac {1}{5 d^{2} \left (b e -c d \right )^{2} \left (e x +d \right )^{\frac {5}{2}}}-\frac {2 b e -4 c d}{3 d^{3} \left (b e -c d \right )^{3} \left (e x +d \right )^{\frac {3}{2}}}-\frac {3 b^{2} e^{2}-10 b c d e +10 c^{2} d^{2}}{d^{4} \left (b e -c d \right )^{4} \sqrt {e x +d}}\right )\) | \(250\) |
| risch | \(-\frac {\sqrt {e x +d}}{d^{4} b^{2} x}-\frac {e \left (-\frac {\left (7 b e +4 c d \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b e \sqrt {d}}+\frac {2 b^{2} d^{2} e^{2}}{5 \left (b e -c d \right )^{2} \left (e x +d \right )^{\frac {5}{2}}}+\frac {2 b^{2} e^{2} \left (3 b^{2} e^{2}-10 b c d e +10 c^{2} d^{2}\right )}{\left (b e -c d \right )^{4} \sqrt {e x +d}}+\frac {4 b^{2} d \,e^{2} \left (b e -2 c d \right )}{3 \left (b e -c d \right )^{3} \left (e x +d \right )^{\frac {3}{2}}}+\frac {2 c^{5} d^{4} \left (\frac {b e \sqrt {e x +d}}{2 c \left (e x +d \right )+2 b e -2 c d}+\frac {\left (11 b e -4 c d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{2 \sqrt {\left (b e -c d \right ) c}}\right )}{\left (b e -c d \right )^{4} b e}\right )}{b^{2} d^{4}}\) | \(270\) |
| pseudoelliptic | \(e^{3} \left (\frac {7 \,\operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right ) b e x +4 \,\operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right ) c d x -b \sqrt {e x +d}\, \sqrt {d}}{x \,d^{\frac {9}{2}} b^{3} e^{3}}-\frac {2}{5 d^{2} \left (b e -c d \right )^{2} \left (e x +d \right )^{\frac {5}{2}}}-\frac {4 \left (b e -2 c d \right )}{3 d^{3} \left (b e -c d \right )^{3} \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 \left (3 b^{2} e^{2}-10 b c d e +10 c^{2} d^{2}\right )}{d^{4} \left (b e -c d \right )^{4} \sqrt {e x +d}}-\frac {\sqrt {e x +d}\, c^{5}}{b^{2} e^{3} \left (c x +b \right ) \left (b e -c d \right )^{4}}-\frac {11 \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right ) c^{5}}{\sqrt {\left (b e -c d \right ) c}\, b^{2} e^{2} \left (b e -c d \right )^{4}}+\frac {4 \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right ) c^{6} d}{\sqrt {\left (b e -c d \right ) c}\, b^{3} e^{3} \left (b e -c d \right )^{4}}\right )\) | \(310\) |
2*e^3*(-c^5/b^3/e^3/(b*e-c*d)^4*(1/2*b*e*(e*x+d)^(1/2)/(c*(e*x+d)+b*e-c*d) +1/2*(11*b*e-4*c*d)/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)* c)^(1/2)))+1/d^4/e^3/b^3*(-1/2*b*(e*x+d)^(1/2)/x+1/2*(7*b*e+4*c*d)/d^(1/2) *arctanh((e*x+d)^(1/2)/d^(1/2)))-1/5/d^2/(b*e-c*d)^2/(e*x+d)^(5/2)-1/3*(2* b*e-4*c*d)/d^3/(b*e-c*d)^3/(e*x+d)^(3/2)-(3*b^2*e^2-10*b*c*d*e+10*c^2*d^2) /d^4/(b*e-c*d)^4/(e*x+d)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 1430 vs. \(2 (321) = 642\).
Time = 4.24 (sec) , antiderivative size = 5752, normalized size of antiderivative = 16.48 \[ \int \frac {1}{(d+e x)^{7/2} \left (b x+c x^2\right )^2} \, dx=\text {Too large to display} \]
\[ \int \frac {1}{(d+e x)^{7/2} \left (b x+c x^2\right )^2} \, dx=\int \frac {1}{x^{2} \left (b + c x\right )^{2} \left (d + e x\right )^{\frac {7}{2}}}\, dx \]
Exception generated. \[ \int \frac {1}{(d+e x)^{7/2} \left (b x+c x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]
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 = 0.31 (sec) , antiderivative size = 636, normalized size of antiderivative = 1.82 \[ \int \frac {1}{(d+e x)^{7/2} \left (b x+c x^2\right )^2} \, dx=\frac {{\left (4 \, c^{6} d - 11 \, b c^{5} e\right )} \arctan \left (\frac {\sqrt {e x + d} c}{\sqrt {-c^{2} d + b c e}}\right )}{{\left (b^{3} c^{4} d^{4} - 4 \, b^{4} c^{3} d^{3} e + 6 \, b^{5} c^{2} d^{2} e^{2} - 4 \, b^{6} c d e^{3} + b^{7} e^{4}\right )} \sqrt {-c^{2} d + b c e}} - \frac {2 \, {\left (e x + d\right )}^{\frac {3}{2}} c^{5} d^{4} e - 2 \, \sqrt {e x + d} c^{5} d^{5} e - 4 \, {\left (e x + d\right )}^{\frac {3}{2}} b c^{4} d^{3} e^{2} + 5 \, \sqrt {e x + d} b c^{4} d^{4} e^{2} + 6 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{2} c^{3} d^{2} e^{3} - 10 \, \sqrt {e x + d} b^{2} c^{3} d^{3} e^{3} - 4 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{3} c^{2} d e^{4} + 10 \, \sqrt {e x + d} b^{3} c^{2} d^{2} e^{4} + {\left (e x + d\right )}^{\frac {3}{2}} b^{4} c e^{5} - 5 \, \sqrt {e x + d} b^{4} c d e^{5} + \sqrt {e x + d} b^{5} e^{6}}{{\left (b^{2} c^{4} d^{8} - 4 \, b^{3} c^{3} d^{7} e + 6 \, b^{4} c^{2} d^{6} e^{2} - 4 \, b^{5} c d^{5} e^{3} + b^{6} d^{4} e^{4}\right )} {\left ({\left (e x + d\right )}^{2} c - 2 \, {\left (e x + d\right )} c d + c d^{2} + {\left (e x + d\right )} b e - b d e\right )}} - \frac {2 \, {\left (150 \, {\left (e x + d\right )}^{2} c^{2} d^{2} e^{3} + 20 \, {\left (e x + d\right )} c^{2} d^{3} e^{3} + 3 \, c^{2} d^{4} e^{3} - 150 \, {\left (e x + d\right )}^{2} b c d e^{4} - 30 \, {\left (e x + d\right )} b c d^{2} e^{4} - 6 \, b c d^{3} e^{4} + 45 \, {\left (e x + d\right )}^{2} b^{2} e^{5} + 10 \, {\left (e x + d\right )} b^{2} d e^{5} + 3 \, b^{2} d^{2} e^{5}\right )}}{15 \, {\left (c^{4} d^{8} - 4 \, b c^{3} d^{7} e + 6 \, b^{2} c^{2} d^{6} e^{2} - 4 \, b^{3} c d^{5} e^{3} + b^{4} d^{4} e^{4}\right )} {\left (e x + d\right )}^{\frac {5}{2}}} - \frac {{\left (4 \, c d + 7 \, b e\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-d}}\right )}{b^{3} \sqrt {-d} d^{4}} \]
(4*c^6*d - 11*b*c^5*e)*arctan(sqrt(e*x + d)*c/sqrt(-c^2*d + b*c*e))/((b^3* c^4*d^4 - 4*b^4*c^3*d^3*e + 6*b^5*c^2*d^2*e^2 - 4*b^6*c*d*e^3 + b^7*e^4)*s qrt(-c^2*d + b*c*e)) - (2*(e*x + d)^(3/2)*c^5*d^4*e - 2*sqrt(e*x + d)*c^5* d^5*e - 4*(e*x + d)^(3/2)*b*c^4*d^3*e^2 + 5*sqrt(e*x + d)*b*c^4*d^4*e^2 + 6*(e*x + d)^(3/2)*b^2*c^3*d^2*e^3 - 10*sqrt(e*x + d)*b^2*c^3*d^3*e^3 - 4*( e*x + d)^(3/2)*b^3*c^2*d*e^4 + 10*sqrt(e*x + d)*b^3*c^2*d^2*e^4 + (e*x + d )^(3/2)*b^4*c*e^5 - 5*sqrt(e*x + d)*b^4*c*d*e^5 + sqrt(e*x + d)*b^5*e^6)/( (b^2*c^4*d^8 - 4*b^3*c^3*d^7*e + 6*b^4*c^2*d^6*e^2 - 4*b^5*c*d^5*e^3 + b^6 *d^4*e^4)*((e*x + d)^2*c - 2*(e*x + d)*c*d + c*d^2 + (e*x + d)*b*e - b*d*e )) - 2/15*(150*(e*x + d)^2*c^2*d^2*e^3 + 20*(e*x + d)*c^2*d^3*e^3 + 3*c^2* d^4*e^3 - 150*(e*x + d)^2*b*c*d*e^4 - 30*(e*x + d)*b*c*d^2*e^4 - 6*b*c*d^3 *e^4 + 45*(e*x + d)^2*b^2*e^5 + 10*(e*x + d)*b^2*d*e^5 + 3*b^2*d^2*e^5)/(( 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)*(e*x + d)^(5/2)) - (4*c*d + 7*b*e)*arctan(sqrt(e*x + d)/sqrt(-d))/(b^3* sqrt(-d)*d^4)
Time = 13.60 (sec) , antiderivative size = 7254, normalized size of antiderivative = 20.79 \[ \int \frac {1}{(d+e x)^{7/2} \left (b x+c x^2\right )^2} \, dx=\text {Too large to display} \]
(atan((((-c^9*(b*e - c*d)^9)^(1/2)*(11*b*e - 4*c*d)*((d + e*x)^(1/2)*(1088 *b^7*c^24*d^33*e^3 - 64*b^6*c^25*d^34*e^2 - 8404*b^8*c^23*d^32*e^4 + 38720 *b^9*c^22*d^31*e^5 - 116512*b^10*c^21*d^30*e^6 + 230912*b^11*c^20*d^29*e^7 - 267432*b^12*c^19*d^28*e^8 + 38544*b^13*c^18*d^27*e^9 + 473880*b^14*c^17 *d^26*e^10 - 851136*b^15*c^16*d^25*e^11 + 393646*b^16*c^15*d^24*e^12 + 120 7368*b^17*c^14*d^23*e^13 - 3343724*b^18*c^13*d^22*e^14 + 4835160*b^19*c^12 *d^21*e^15 - 4903382*b^20*c^11*d^20*e^16 + 3751968*b^21*c^10*d^19*e^17 - 2 217072*b^22*c^9*d^18*e^18 + 1013232*b^23*c^8*d^17*e^19 - 353210*b^24*c^7*d ^16*e^20 + 91080*b^25*c^6*d^15*e^21 - 16412*b^26*c^5*d^14*e^22 + 1848*b^27 *c^4*d^13*e^23 - 98*b^28*c^3*d^12*e^24) - ((-c^9*(b*e - c*d)^9)^(1/2)*(11* b*e - 4*c*d)*(8*b^10*c^23*d^37*e^3 - 148*b^11*c^22*d^36*e^4 + 1160*b^12*c^ 21*d^35*e^5 - 4760*b^13*c^20*d^34*e^6 + 8036*b^14*c^19*d^33*e^7 + 21868*b^ 15*c^18*d^32*e^8 - 194304*b^16*c^17*d^31*e^9 + 709280*b^17*c^16*d^30*e^10 - 1744160*b^18*c^15*d^29*e^11 + 3218072*b^19*c^14*d^28*e^12 - 4654832*b^20 *c^13*d^27*e^13 + 5394480*b^21*c^12*d^26*e^14 - 5063240*b^22*c^11*d^25*e^1 5 + 3863800*b^23*c^10*d^24*e^16 - 2393152*b^24*c^9*d^23*e^17 + 1194528*b^2 5*c^8*d^22*e^18 - 474056*b^26*c^7*d^21*e^19 + 146300*b^27*c^6*d^20*e^20 - 33880*b^28*c^5*d^19*e^21 + 5544*b^29*c^4*d^18*e^22 - 572*b^30*c^3*d^17*e^2 3 + 28*b^31*c^2*d^16*e^24 - ((-c^9*(b*e - c*d)^9)^(1/2)*(11*b*e - 4*c*d)*( d + e*x)^(1/2)*(16*b^12*c^23*d^41*e^2 - 328*b^13*c^22*d^40*e^3 + 3200*b...