Integrand size = 21, antiderivative size = 417 \[ \int \frac {1}{(d+e x) \left (b x+c x^2\right )^{7/2}} \, dx=-\frac {2}{5 b d \left (b x+c x^2\right )^{5/2}}+\frac {2 (2 c d+b e) x}{3 b^2 d^2 \left (b x+c x^2\right )^{5/2}}-\frac {2 \left (16 c^2 d^2+8 b c d e+3 b^2 e^2\right ) x^2}{3 b^3 d^3 \left (b x+c x^2\right )^{5/2}}-\frac {2 c \left (32 c^3 d^3-16 b c^2 d^2 e-10 b^2 c d e^2-5 b^3 e^3\right ) x^3}{5 b^4 d^3 (c d-b e) \left (b x+c x^2\right )^{5/2}}-\frac {2 c \left (128 c^4 d^4-192 b c^3 d^3 e+24 b^2 c^2 d^2 e^2+20 b^3 c d e^3+15 b^4 e^4\right ) x^2}{15 b^5 d^3 (c d-b e)^2 \left (b x+c x^2\right )^{3/2}}-\frac {2 c (2 c d-b e) \left (128 c^4 d^4-256 b c^3 d^3 e+88 b^2 c^2 d^2 e^2+40 b^3 c d e^3+15 b^4 e^4\right ) x}{15 b^6 d^3 (c d-b e)^3 \sqrt {b x+c x^2}}+\frac {2 e^6 \text {arctanh}\left (\frac {\sqrt {c d-b e} x}{\sqrt {d} \sqrt {b x+c x^2}}\right )}{d^{7/2} (c d-b e)^{7/2}} \] Output:
-2/5/b/d/(c*x^2+b*x)^(5/2)+2/3*(b*e+2*c*d)*x/b^2/d^2/(c*x^2+b*x)^(5/2)-2/3 *(3*b^2*e^2+8*b*c*d*e+16*c^2*d^2)*x^2/b^3/d^3/(c*x^2+b*x)^(5/2)-2/5*c*(-5* b^3*e^3-10*b^2*c*d*e^2-16*b*c^2*d^2*e+32*c^3*d^3)*x^3/b^4/d^3/(-b*e+c*d)/( c*x^2+b*x)^(5/2)-2/15*c*(15*b^4*e^4+20*b^3*c*d*e^3+24*b^2*c^2*d^2*e^2-192* b*c^3*d^3*e+128*c^4*d^4)*x^2/b^5/d^3/(-b*e+c*d)^2/(c*x^2+b*x)^(3/2)-2/15*c *(-b*e+2*c*d)*(15*b^4*e^4+40*b^3*c*d*e^3+88*b^2*c^2*d^2*e^2-256*b*c^3*d^3* e+128*c^4*d^4)*x/b^6/d^3/(-b*e+c*d)^3/(c*x^2+b*x)^(1/2)+2*e^6*arctanh((-b* e+c*d)^(1/2)*x/d^(1/2)/(c*x^2+b*x)^(1/2))/d^(7/2)/(-b*e+c*d)^(7/2)
Time = 2.14 (sec) , antiderivative size = 452, normalized size of antiderivative = 1.08 \[ \int \frac {1}{(d+e x) \left (b x+c x^2\right )^{7/2}} \, dx=\frac {2 x \left (-\frac {\sqrt {d} (b+c x) \left (-256 c^8 d^5 x^5-640 b c^7 d^4 x^4 (d-e x)+b^8 e^3 \left (3 d^2-5 d e x+15 e^2 x^2\right )-16 b^2 c^6 d^3 x^3 \left (30 d^2-100 d e x+27 e^2 x^2\right )+8 b^3 c^5 d^2 x^2 \left (-10 d^3+150 d^2 e x-135 d e^2 x^2+e^3 x^3\right )+b^7 c e^2 \left (-9 d^3+5 d^2 e x-5 d e^2 x^2+45 e^3 x^3\right )+10 b^4 c^4 d x \left (d^4+20 d^3 e x-81 d^2 e^2 x^2+2 d e^3 x^3+e^4 x^4\right )+b^6 c^2 e \left (9 d^4+15 d^3 e x+5 d^2 e^2 x^2+15 d e^3 x^3+45 e^4 x^4\right )+b^5 c^3 \left (-3 d^5-25 d^4 e x-135 d^3 e^2 x^2+15 d^2 e^3 x^3+25 d e^4 x^4+15 e^5 x^5\right )\right )}{b^6 (-c d+b e)^3}+\frac {15 e^6 x^{5/2} (b+c x)^{7/2} \arctan \left (\frac {-e \sqrt {x} \sqrt {b+c x}+\sqrt {c} (d+e x)}{\sqrt {d} \sqrt {-c d+b e}}\right )}{(-c d+b e)^{7/2}}\right )}{15 d^{7/2} (x (b+c x))^{7/2}} \] Input:
Integrate[1/((d + e*x)*(b*x + c*x^2)^(7/2)),x]
Output:
(2*x*(-((Sqrt[d]*(b + c*x)*(-256*c^8*d^5*x^5 - 640*b*c^7*d^4*x^4*(d - e*x) + b^8*e^3*(3*d^2 - 5*d*e*x + 15*e^2*x^2) - 16*b^2*c^6*d^3*x^3*(30*d^2 - 1 00*d*e*x + 27*e^2*x^2) + 8*b^3*c^5*d^2*x^2*(-10*d^3 + 150*d^2*e*x - 135*d* e^2*x^2 + e^3*x^3) + b^7*c*e^2*(-9*d^3 + 5*d^2*e*x - 5*d*e^2*x^2 + 45*e^3* x^3) + 10*b^4*c^4*d*x*(d^4 + 20*d^3*e*x - 81*d^2*e^2*x^2 + 2*d*e^3*x^3 + e ^4*x^4) + b^6*c^2*e*(9*d^4 + 15*d^3*e*x + 5*d^2*e^2*x^2 + 15*d*e^3*x^3 + 4 5*e^4*x^4) + b^5*c^3*(-3*d^5 - 25*d^4*e*x - 135*d^3*e^2*x^2 + 15*d^2*e^3*x ^3 + 25*d*e^4*x^4 + 15*e^5*x^5)))/(b^6*(-(c*d) + b*e)^3)) + (15*e^6*x^(5/2 )*(b + c*x)^(7/2)*ArcTan[(-(e*Sqrt[x]*Sqrt[b + c*x]) + Sqrt[c]*(d + e*x))/ (Sqrt[d]*Sqrt[-(c*d) + b*e])])/(-(c*d) + b*e)^(7/2)))/(15*d^(7/2)*(x*(b + c*x))^(7/2))
Time = 1.23 (sec) , antiderivative size = 426, normalized size of antiderivative = 1.02, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {1165, 27, 1235, 27, 1235, 27, 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.
| \(\displaystyle \int \frac {1}{\left (b x+c x^2\right )^{7/2} (d+e x)} \, dx\) |
| \(\Big \downarrow \) 1165 |
| \(\displaystyle -\frac {2 \int \frac {16 c^2 d^2-8 b c e d-5 b^2 e^2+8 c e (2 c d-b e) x}{2 (d+e x) \left (c x^2+b x\right )^{5/2}}dx}{5 b^2 d (c d-b e)}-\frac {2 (c x (2 c d-b e)+b (c d-b e))}{5 b^2 d \left (b x+c x^2\right )^{5/2} (c d-b e)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {16 c^2 d^2-8 b c e d-5 b^2 e^2+8 c e (2 c d-b e) x}{(d+e x) \left (c x^2+b x\right )^{5/2}}dx}{5 b^2 d (c d-b e)}-\frac {2 (c x (2 c d-b e)+b (c d-b e))}{5 b^2 d \left (b x+c x^2\right )^{5/2} (c d-b e)}\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle -\frac {-\frac {2 \int \frac {128 c^4 d^4-192 b c^3 e d^3+24 b^2 c^2 e^2 d^2+20 b^3 c e^3 d+15 b^4 e^4+4 c e (4 c d-5 b e) (2 c d-b e) (4 c d+b e) x}{2 (d+e x) \left (c x^2+b x\right )^{3/2}}dx}{3 b^2 d (c d-b e)}-\frac {2 \left (b (c d-b e) \left (-5 b^2 e^2-8 b c d e+16 c^2 d^2\right )+c x (4 c d-5 b e) (2 c d-b e) (b e+4 c d)\right )}{3 b^2 d \left (b x+c x^2\right )^{3/2} (c d-b e)}}{5 b^2 d (c d-b e)}-\frac {2 (c x (2 c d-b e)+b (c d-b e))}{5 b^2 d \left (b x+c x^2\right )^{5/2} (c d-b e)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\int \frac {128 c^4 d^4-192 b c^3 e d^3+24 b^2 c^2 e^2 d^2+20 b^3 c e^3 d+15 b^4 e^4+4 c e (4 c d-5 b e) (2 c d-b e) (4 c d+b e) x}{(d+e x) \left (c x^2+b x\right )^{3/2}}dx}{3 b^2 d (c d-b e)}-\frac {2 \left (b (c d-b e) \left (-5 b^2 e^2-8 b c d e+16 c^2 d^2\right )+c x (4 c d-5 b e) (2 c d-b e) (b e+4 c d)\right )}{3 b^2 d \left (b x+c x^2\right )^{3/2} (c d-b e)}}{5 b^2 d (c d-b e)}-\frac {2 (c x (2 c d-b e)+b (c d-b e))}{5 b^2 d \left (b x+c x^2\right )^{5/2} (c d-b e)}\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle -\frac {-\frac {-\frac {2 \int -\frac {15 b^6 e^6}{2 (d+e x) \sqrt {c x^2+b x}}dx}{b^2 d (c d-b e)}-\frac {2 \left (c x (2 c d-b e) \left (15 b^4 e^4+40 b^3 c d e^3+88 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right )+b (c d-b e) \left (15 b^4 e^4+20 b^3 c d e^3+24 b^2 c^2 d^2 e^2-192 b c^3 d^3 e+128 c^4 d^4\right )\right )}{b^2 d \sqrt {b x+c x^2} (c d-b e)}}{3 b^2 d (c d-b e)}-\frac {2 \left (b (c d-b e) \left (-5 b^2 e^2-8 b c d e+16 c^2 d^2\right )+c x (4 c d-5 b e) (2 c d-b e) (b e+4 c d)\right )}{3 b^2 d \left (b x+c x^2\right )^{3/2} (c d-b e)}}{5 b^2 d (c d-b e)}-\frac {2 (c x (2 c d-b e)+b (c d-b e))}{5 b^2 d \left (b x+c x^2\right )^{5/2} (c d-b e)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\frac {15 b^4 e^6 \int \frac {1}{(d+e x) \sqrt {c x^2+b x}}dx}{d (c d-b e)}-\frac {2 \left (c x (2 c d-b e) \left (15 b^4 e^4+40 b^3 c d e^3+88 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right )+b (c d-b e) \left (15 b^4 e^4+20 b^3 c d e^3+24 b^2 c^2 d^2 e^2-192 b c^3 d^3 e+128 c^4 d^4\right )\right )}{b^2 d \sqrt {b x+c x^2} (c d-b e)}}{3 b^2 d (c d-b e)}-\frac {2 \left (b (c d-b e) \left (-5 b^2 e^2-8 b c d e+16 c^2 d^2\right )+c x (4 c d-5 b e) (2 c d-b e) (b e+4 c d)\right )}{3 b^2 d \left (b x+c x^2\right )^{3/2} (c d-b e)}}{5 b^2 d (c d-b e)}-\frac {2 (c x (2 c d-b e)+b (c d-b e))}{5 b^2 d \left (b x+c x^2\right )^{5/2} (c d-b e)}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle -\frac {-\frac {-\frac {30 b^4 e^6 \int \frac {1}{4 d (c d-b e)-\frac {(b d+(2 c d-b e) x)^2}{c x^2+b x}}d\left (-\frac {b d+(2 c d-b e) x}{\sqrt {c x^2+b x}}\right )}{d (c d-b e)}-\frac {2 \left (c x (2 c d-b e) \left (15 b^4 e^4+40 b^3 c d e^3+88 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right )+b (c d-b e) \left (15 b^4 e^4+20 b^3 c d e^3+24 b^2 c^2 d^2 e^2-192 b c^3 d^3 e+128 c^4 d^4\right )\right )}{b^2 d \sqrt {b x+c x^2} (c d-b e)}}{3 b^2 d (c d-b e)}-\frac {2 \left (b (c d-b e) \left (-5 b^2 e^2-8 b c d e+16 c^2 d^2\right )+c x (4 c d-5 b e) (2 c d-b e) (b e+4 c d)\right )}{3 b^2 d \left (b x+c x^2\right )^{3/2} (c d-b e)}}{5 b^2 d (c d-b e)}-\frac {2 (c x (2 c d-b e)+b (c d-b e))}{5 b^2 d \left (b x+c x^2\right )^{5/2} (c d-b e)}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {-\frac {\frac {15 b^4 e^6 \text {arctanh}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{d^{3/2} (c d-b e)^{3/2}}-\frac {2 \left (c x (2 c d-b e) \left (15 b^4 e^4+40 b^3 c d e^3+88 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right )+b (c d-b e) \left (15 b^4 e^4+20 b^3 c d e^3+24 b^2 c^2 d^2 e^2-192 b c^3 d^3 e+128 c^4 d^4\right )\right )}{b^2 d \sqrt {b x+c x^2} (c d-b e)}}{3 b^2 d (c d-b e)}-\frac {2 \left (b (c d-b e) \left (-5 b^2 e^2-8 b c d e+16 c^2 d^2\right )+c x (4 c d-5 b e) (2 c d-b e) (b e+4 c d)\right )}{3 b^2 d \left (b x+c x^2\right )^{3/2} (c d-b e)}}{5 b^2 d (c d-b e)}-\frac {2 (c x (2 c d-b e)+b (c d-b e))}{5 b^2 d \left (b x+c x^2\right )^{5/2} (c d-b e)}\) |
Input:
Int[1/((d + e*x)*(b*x + c*x^2)^(7/2)),x]
Output:
(-2*(b*(c*d - b*e) + c*(2*c*d - b*e)*x))/(5*b^2*d*(c*d - b*e)*(b*x + c*x^2 )^(5/2)) - ((-2*(b*(c*d - b*e)*(16*c^2*d^2 - 8*b*c*d*e - 5*b^2*e^2) + c*(4 *c*d - 5*b*e)*(2*c*d - b*e)*(4*c*d + b*e)*x))/(3*b^2*d*(c*d - b*e)*(b*x + c*x^2)^(3/2)) - ((-2*(b*(c*d - b*e)*(128*c^4*d^4 - 192*b*c^3*d^3*e + 24*b^ 2*c^2*d^2*e^2 + 20*b^3*c*d*e^3 + 15*b^4*e^4) + c*(2*c*d - b*e)*(128*c^4*d^ 4 - 256*b*c^3*d^3*e + 88*b^2*c^2*d^2*e^2 + 40*b^3*c*d*e^3 + 15*b^4*e^4)*x) )/(b^2*d*(c*d - b*e)*Sqrt[b*x + c*x^2]) + (15*b^4*e^6*ArcTanh[(b*d + (2*c* d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/(d^(3/2)*(c*d - b*e)^(3/2)))/(3*b^2*d*(c*d - b*e)))/(5*b^2*d*(c*d - b*e))
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_)*((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[((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 = 0.99 (sec) , antiderivative size = 369, normalized size of antiderivative = 0.88
| method | result | size |
| pseudoelliptic | \(\frac {2 b^{6} e^{6} x^{2} \sqrt {x \left (c x +b \right )}\, \left (c x +b \right )^{2} \arctan \left (\frac {\sqrt {x \left (c x +b \right )}\, d}{x \sqrt {d \left (b e -c d \right )}}\right )-\frac {2 \sqrt {d \left (b e -c d \right )}\, \left (-c^{3} \left (2 c x +b \right ) \left (\frac {128}{3} c^{4} x^{4}+\frac {256}{3} b \,c^{3} x^{3}+\frac {112}{3} b^{2} c^{2} x^{2}-\frac {16}{3} b^{3} c x +b^{4}\right ) d^{5}+3 e \left (\frac {640}{9} c^{5} x^{5}+\frac {1600}{9} b \,x^{4} c^{4}+\frac {400}{3} b^{2} c^{3} x^{3}+\frac {200}{9} c^{2} x^{2} b^{3}-\frac {25}{9} b^{4} c x +b^{5}\right ) c^{2} b \,d^{4}-3 e^{2} \left (48 c^{5} x^{5}+120 b \,x^{4} c^{4}+90 b^{2} c^{3} x^{3}+15 c^{2} x^{2} b^{3}-\frac {5}{3} b^{4} c x +b^{5}\right ) c \,b^{2} d^{3}+\left (\frac {8}{3} c^{2} x^{2}-\frac {4}{3} c b x +b^{2}\right ) e^{3} \left (c x +b \right )^{3} b^{3} d^{2}-\frac {5 b^{4} e^{4} x \left (-2 c x +b \right ) \left (c x +b \right )^{3} d}{3}+5 b^{5} e^{5} x^{2} \left (c x +b \right )^{3}\right )}{5}}{x^{2} d^{3} \left (b e -c d \right )^{3} \left (c x +b \right )^{2} \sqrt {x \left (c x +b \right )}\, \sqrt {d \left (b e -c d \right )}\, b^{6}}\) | \(369\) |
| risch | \(-\frac {2 \left (c x +b \right ) \left (15 b^{2} e^{2} x^{2}+55 b c d e \,x^{2}+128 d^{2} c^{2} x^{2}-5 b^{2} d e x -19 x b c \,d^{2}+3 b^{2} d^{2}\right )}{15 b^{6} d^{3} \sqrt {x \left (c x +b \right )}\, x^{2}}-\frac {-\frac {b^{5} e^{5} \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (b e -c d \right )^{3} \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}-\frac {b^{2} c \,d^{3} \left (\frac {2 \sqrt {c \left (\frac {b}{c}+x \right )^{2}-\left (\frac {b}{c}+x \right ) b}}{5 b \left (\frac {b}{c}+x \right )^{3}}+\frac {4 c \left (\frac {2 \sqrt {c \left (\frac {b}{c}+x \right )^{2}-\left (\frac {b}{c}+x \right ) b}}{3 b \left (\frac {b}{c}+x \right )^{2}}+\frac {4 c \sqrt {c \left (\frac {b}{c}+x \right )^{2}-\left (\frac {b}{c}+x \right ) b}}{3 b^{2} \left (\frac {b}{c}+x \right )}\right )}{5 b}\right )}{b e -c d}-\frac {2 c^{3} d^{3} \left (10 b^{2} e^{2}-15 b c d e +6 c^{2} d^{2}\right ) \sqrt {c \left (\frac {b}{c}+x \right )^{2}-\left (\frac {b}{c}+x \right ) b}}{\left (b e -c d \right )^{3} b \left (\frac {b}{c}+x \right )}-\frac {b \,c^{2} d^{3} \left (4 b e -3 c d \right ) \left (\frac {2 \sqrt {c \left (\frac {b}{c}+x \right )^{2}-\left (\frac {b}{c}+x \right ) b}}{3 b \left (\frac {b}{c}+x \right )^{2}}+\frac {4 c \sqrt {c \left (\frac {b}{c}+x \right )^{2}-\left (\frac {b}{c}+x \right ) b}}{3 b^{2} \left (\frac {b}{c}+x \right )}\right )}{\left (b e -c d \right )^{2}}}{b^{5} d^{3}}\) | \(561\) |
| default | \(\text {Expression too large to display}\) | \(1122\) |
Input:
int(1/(e*x+d)/(c*x^2+b*x)^(7/2),x,method=_RETURNVERBOSE)
Output:
2*(b^6*e^6*x^2*(x*(c*x+b))^(1/2)*(c*x+b)^2*arctan((x*(c*x+b))^(1/2)/x*d/(d *(b*e-c*d))^(1/2))-1/5*(d*(b*e-c*d))^(1/2)*(-c^3*(2*c*x+b)*(128/3*c^4*x^4+ 256/3*b*c^3*x^3+112/3*b^2*c^2*x^2-16/3*b^3*c*x+b^4)*d^5+3*e*(640/9*c^5*x^5 +1600/9*b*x^4*c^4+400/3*b^2*c^3*x^3+200/9*c^2*x^2*b^3-25/9*b^4*c*x+b^5)*c^ 2*b*d^4-3*e^2*(48*c^5*x^5+120*b*x^4*c^4+90*b^2*c^3*x^3+15*c^2*x^2*b^3-5/3* b^4*c*x+b^5)*c*b^2*d^3+(8/3*c^2*x^2-4/3*c*b*x+b^2)*e^3*(c*x+b)^3*b^3*d^2-5 /3*b^4*e^4*x*(-2*c*x+b)*(c*x+b)^3*d+5*b^5*e^5*x^2*(c*x+b)^3))/(x*(c*x+b))^ (1/2)/(d*(b*e-c*d))^(1/2)/x^2/d^3/(b*e-c*d)^3/(c*x+b)^2/b^6
Leaf count of result is larger than twice the leaf count of optimal. 883 vs. \(2 (383) = 766\).
Time = 0.19 (sec) , antiderivative size = 1782, normalized size of antiderivative = 4.27 \[ \int \frac {1}{(d+e x) \left (b x+c x^2\right )^{7/2}} \, dx=\text {Too large to display} \] Input:
integrate(1/(e*x+d)/(c*x^2+b*x)^(7/2),x, algorithm="fricas")
Output:
[-1/15*(15*(b^6*c^3*e^6*x^6 + 3*b^7*c^2*e^6*x^5 + 3*b^8*c*e^6*x^4 + b^9*e^ 6*x^3)*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*(3*b^5*c^4*d^7 - 12*b^6*c^3*d^6*e + 18*b^7*c^2*d^5*e^2 - 12*b^8*c*d^4*e^3 + 3*b^9*d^3*e^4 + (256*c^9*d^7 - 896 *b*c^8*d^6*e + 1072*b^2*c^7*d^5*e^2 - 440*b^3*c^6*d^4*e^3 - 2*b^4*c^5*d^3* e^4 - 5*b^5*c^4*d^2*e^5 + 15*b^6*c^3*d*e^6)*x^5 + 5*(128*b*c^8*d^7 - 448*b ^2*c^7*d^6*e + 536*b^3*c^6*d^5*e^2 - 220*b^4*c^5*d^4*e^3 - b^5*c^4*d^3*e^4 - 4*b^6*c^3*d^2*e^5 + 9*b^7*c^2*d*e^6)*x^4 + 15*(32*b^2*c^7*d^7 - 112*b^3 *c^6*d^6*e + 134*b^4*c^5*d^5*e^2 - 55*b^5*c^4*d^4*e^3 - 2*b^7*c^2*d^2*e^5 + 3*b^8*c*d*e^6)*x^3 + 5*(16*b^3*c^6*d^7 - 56*b^4*c^5*d^6*e + 67*b^5*c^4*d ^5*e^2 - 28*b^6*c^3*d^4*e^3 + 2*b^7*c^2*d^3*e^4 - 4*b^8*c*d^2*e^5 + 3*b^9* d*e^6)*x^2 - 5*(2*b^4*c^5*d^7 - 7*b^5*c^4*d^6*e + 8*b^6*c^3*d^5*e^2 - 2*b^ 7*c^2*d^4*e^3 - 2*b^8*c*d^3*e^4 + b^9*d^2*e^5)*x)*sqrt(c*x^2 + b*x))/((b^6 *c^7*d^8 - 4*b^7*c^6*d^7*e + 6*b^8*c^5*d^6*e^2 - 4*b^9*c^4*d^5*e^3 + b^10* c^3*d^4*e^4)*x^6 + 3*(b^7*c^6*d^8 - 4*b^8*c^5*d^7*e + 6*b^9*c^4*d^6*e^2 - 4*b^10*c^3*d^5*e^3 + b^11*c^2*d^4*e^4)*x^5 + 3*(b^8*c^5*d^8 - 4*b^9*c^4*d^ 7*e + 6*b^10*c^3*d^6*e^2 - 4*b^11*c^2*d^5*e^3 + b^12*c*d^4*e^4)*x^4 + (b^9 *c^4*d^8 - 4*b^10*c^3*d^7*e + 6*b^11*c^2*d^6*e^2 - 4*b^12*c*d^5*e^3 + b^13 *d^4*e^4)*x^3), -2/15*(15*(b^6*c^3*e^6*x^6 + 3*b^7*c^2*e^6*x^5 + 3*b^8*c*e ^6*x^4 + b^9*e^6*x^3)*sqrt(-c*d^2 + b*d*e)*arctan(sqrt(-c*d^2 + b*d*e)*...
\[ \int \frac {1}{(d+e x) \left (b x+c x^2\right )^{7/2}} \, dx=\int \frac {1}{\left (x \left (b + c x\right )\right )^{\frac {7}{2}} \left (d + e x\right )}\, dx \] Input:
integrate(1/(e*x+d)/(c*x**2+b*x)**(7/2),x)
Output:
Integral(1/((x*(b + c*x))**(7/2)*(d + e*x)), x)
Exception generated. \[ \int \frac {1}{(d+e x) \left (b x+c x^2\right )^{7/2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(1/(e*x+d)/(c*x^2+b*x)^(7/2),x, algorithm="maxima")
Output:
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. 1851 vs. \(2 (383) = 766\).
Time = 0.18 (sec) , antiderivative size = 1851, normalized size of antiderivative = 4.44 \[ \int \frac {1}{(d+e x) \left (b x+c x^2\right )^{7/2}} \, dx=\text {Too large to display} \] Input:
integrate(1/(e*x+d)/(c*x^2+b*x)^(7/2),x, algorithm="giac")
Output:
-2*e^6*arctan(((sqrt(c)*x - sqrt(c*x^2 + b*x))*e + sqrt(c)*d)/sqrt(-c*d^2 + b*d*e))/((c^3*d^6 - 3*b*c^2*d^5*e + 3*b^2*c*d^4*e^2 - b^3*d^3*e^3)*sqrt( -c*d^2 + b*d*e)) - 2/15*((((((256*c^14*d^23 - 2176*b*c^13*d^22*e + 8112*b^ 2*c^12*d^21*e^2 - 17320*b^3*c^11*d^20*e^3 + 23158*b^4*c^10*d^19*e^4 - 1985 1*b^5*c^9*d^18*e^5 + 10676*b^6*c^8*d^17*e^6 - 3377*b^7*c^7*d^16*e^7 + 630* b^8*c^6*d^15*e^8 - 173*b^9*c^5*d^14*e^9 + 80*b^10*c^4*d^13*e^10 - 15*b^11* c^3*d^12*e^11)*x/(b^6*c^9*d^24 - 9*b^7*c^8*d^23*e + 36*b^8*c^7*d^22*e^2 - 84*b^9*c^6*d^21*e^3 + 126*b^10*c^5*d^20*e^4 - 126*b^11*c^4*d^19*e^5 + 84*b ^12*c^3*d^18*e^6 - 36*b^13*c^2*d^17*e^7 + 9*b^14*c*d^16*e^8 - b^15*d^15*e^ 9) + 5*(128*b*c^13*d^23 - 1088*b^2*c^12*d^22*e + 4056*b^3*c^11*d^21*e^2 - 8660*b^4*c^10*d^20*e^3 + 11579*b^5*c^9*d^19*e^4 - 9927*b^6*c^8*d^18*e^5 + 5347*b^7*c^7*d^17*e^6 - 1711*b^8*c^6*d^16*e^7 + 345*b^9*c^5*d^15*e^8 - 109 *b^10*c^4*d^14*e^9 + 49*b^11*c^3*d^13*e^10 - 9*b^12*c^2*d^12*e^11)/(b^6*c^ 9*d^24 - 9*b^7*c^8*d^23*e + 36*b^8*c^7*d^22*e^2 - 84*b^9*c^6*d^21*e^3 + 12 6*b^10*c^5*d^20*e^4 - 126*b^11*c^4*d^19*e^5 + 84*b^12*c^3*d^18*e^6 - 36*b^ 13*c^2*d^17*e^7 + 9*b^14*c*d^16*e^8 - b^15*d^15*e^9))*x + 15*(32*b^2*c^12* d^23 - 272*b^3*c^11*d^22*e + 1014*b^4*c^10*d^21*e^2 - 2165*b^5*c^9*d^20*e^ 3 + 2895*b^6*c^8*d^19*e^4 - 2484*b^7*c^7*d^18*e^5 + 1345*b^8*c^6*d^17*e^6 - 444*b^9*c^5*d^16*e^7 + 105*b^10*c^4*d^15*e^8 - 40*b^11*c^3*d^14*e^9 + 17 *b^12*c^2*d^13*e^10 - 3*b^13*c*d^12*e^11)/(b^6*c^9*d^24 - 9*b^7*c^8*d^2...
Timed out. \[ \int \frac {1}{(d+e x) \left (b x+c x^2\right )^{7/2}} \, dx=\int \frac {1}{{\left (c\,x^2+b\,x\right )}^{7/2}\,\left (d+e\,x\right )} \,d x \] Input:
int(1/((b*x + c*x^2)^(7/2)*(d + e*x)),x)
Output:
int(1/((b*x + c*x^2)^(7/2)*(d + e*x)), x)
Time = 0.70 (sec) , antiderivative size = 1726, normalized size of antiderivative = 4.14 \[ \int \frac {1}{(d+e x) \left (b x+c x^2\right )^{7/2}} \, dx =\text {Too large to display} \] Input:
int(1/(e*x+d)/(c*x^2+b*x)^(7/2),x)
Output:
(2*(15*sqrt(d)*sqrt(b + c*x)*sqrt(b*e - c*d)*atan((sqrt(b*e - c*d) - sqrt( e)*sqrt(b + c*x) - sqrt(x)*sqrt(e)*sqrt(c))/(sqrt(d)*sqrt(c)))*b**8*e**6*x **3 + 30*sqrt(d)*sqrt(b + c*x)*sqrt(b*e - c*d)*atan((sqrt(b*e - c*d) - sqr t(e)*sqrt(b + c*x) - sqrt(x)*sqrt(e)*sqrt(c))/(sqrt(d)*sqrt(c)))*b**7*c*e* *6*x**4 + 15*sqrt(d)*sqrt(b + c*x)*sqrt(b*e - c*d)*atan((sqrt(b*e - c*d) - sqrt(e)*sqrt(b + c*x) - sqrt(x)*sqrt(e)*sqrt(c))/(sqrt(d)*sqrt(c)))*b**6* c**2*e**6*x**5 + 15*sqrt(d)*sqrt(b + c*x)*sqrt(b*e - c*d)*atan((sqrt(b*e - c*d) + sqrt(e)*sqrt(b + c*x) + sqrt(x)*sqrt(e)*sqrt(c))/(sqrt(d)*sqrt(c)) )*b**8*e**6*x**3 + 30*sqrt(d)*sqrt(b + c*x)*sqrt(b*e - c*d)*atan((sqrt(b*e - c*d) + sqrt(e)*sqrt(b + c*x) + sqrt(x)*sqrt(e)*sqrt(c))/(sqrt(d)*sqrt(c )))*b**7*c*e**6*x**4 + 15*sqrt(d)*sqrt(b + c*x)*sqrt(b*e - c*d)*atan((sqrt (b*e - c*d) + sqrt(e)*sqrt(b + c*x) + sqrt(x)*sqrt(e)*sqrt(c))/(sqrt(d)*sq rt(c)))*b**6*c**2*e**6*x**5 + 9*sqrt(c)*sqrt(b + c*x)*b**8*d*e**6*x**3 + 1 3*sqrt(c)*sqrt(b + c*x)*b**7*c*d**2*e**5*x**3 + 18*sqrt(c)*sqrt(b + c*x)*b **7*c*d*e**6*x**4 - 14*sqrt(c)*sqrt(b + c*x)*b**6*c**2*d**3*e**4*x**3 + 26 *sqrt(c)*sqrt(b + c*x)*b**6*c**2*d**2*e**5*x**4 + 9*sqrt(c)*sqrt(b + c*x)* b**6*c**2*d*e**6*x**5 - 440*sqrt(c)*sqrt(b + c*x)*b**5*c**3*d**4*e**3*x**3 - 28*sqrt(c)*sqrt(b + c*x)*b**5*c**3*d**3*e**4*x**4 + 13*sqrt(c)*sqrt(b + c*x)*b**5*c**3*d**2*e**5*x**5 + 1072*sqrt(c)*sqrt(b + c*x)*b**4*c**4*d**5 *e**2*x**3 - 880*sqrt(c)*sqrt(b + c*x)*b**4*c**4*d**4*e**3*x**4 - 14*sq...