Integrand size = 21, antiderivative size = 398 \[ \int \frac {1}{(d+e x)^{3/2} \left (b x+c x^2\right )^3} \, dx=\frac {3 e \left (c^2 d^2-b c d e-b^2 e^2\right ) \left (8 c^2 d^2-8 b c d e+5 b^2 e^2\right )}{4 b^4 d^3 (c d-b e)^3 \sqrt {d+e x}}+\frac {c \left (12 c^2 d^2-5 b c d e-5 b^2 e^2\right )}{4 b^3 d^2 (c d-b e) (b+c x)^2 \sqrt {d+e x}}-\frac {1}{2 b d x^2 (b+c x)^2 \sqrt {d+e x}}+\frac {8 c d+5 b e}{4 b^2 d^2 x (b+c x)^2 \sqrt {d+e x}}+\frac {c (2 c d-b e) \left (12 c^2 d^2-12 b c d e-5 b^2 e^2\right )}{4 b^4 d^2 (c d-b e)^2 (b+c x) \sqrt {d+e x}}-\frac {3 \left (16 c^2 d^2+12 b c d e+5 b^2 e^2\right ) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5 d^{7/2}}+\frac {3 c^{7/2} \left (16 c^2 d^2-44 b c d e+33 b^2 e^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 (c d-b e)^{7/2}} \] Output:
3/4*e*(-b^2*e^2-b*c*d*e+c^2*d^2)*(5*b^2*e^2-8*b*c*d*e+8*c^2*d^2)/b^4/d^3/( -b*e+c*d)^3/(e*x+d)^(1/2)+1/4*c*(-5*b^2*e^2-5*b*c*d*e+12*c^2*d^2)/b^3/d^2/ (-b*e+c*d)/(c*x+b)^2/(e*x+d)^(1/2)-1/2/b/d/x^2/(c*x+b)^2/(e*x+d)^(1/2)+1/4 *(5*b*e+8*c*d)/b^2/d^2/x/(c*x+b)^2/(e*x+d)^(1/2)+1/4*c*(-b*e+2*c*d)*(-5*b^ 2*e^2-12*b*c*d*e+12*c^2*d^2)/b^4/d^2/(-b*e+c*d)^2/(c*x+b)/(e*x+d)^(1/2)-3/ 4*(5*b^2*e^2+12*b*c*d*e+16*c^2*d^2)*arctanh((e*x+d)^(1/2)/d^(1/2))/b^5/d^( 7/2)+3/4*c^(7/2)*(33*b^2*e^2-44*b*c*d*e+16*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)^(7/2)
Time = 1.66 (sec) , antiderivative size = 408, normalized size of antiderivative = 1.03 \[ \int \frac {1}{(d+e x)^{3/2} \left (b x+c x^2\right )^3} \, dx=\frac {\frac {b \left (24 c^6 d^4 x^3 (d+e x)+b^6 e^3 \left (2 d^2-5 d e x-15 e^2 x^2\right )-12 b c^5 d^3 x^2 \left (-3 d^2+d e x+4 e^2 x^2\right )+b^2 c^4 d^2 x \left (8 d^3-65 d^2 e x-58 d e^2 x^2+15 e^3 x^3\right )-b^5 c e^2 \left (6 d^3-7 d^2 e x+d e^2 x^2+30 e^3 x^3\right )+b^4 c^2 e \left (6 d^4+9 d^3 e x+23 d^2 e^2 x^2+13 d e^3 x^3-15 e^4 x^4\right )+b^3 c^3 d \left (-2 d^4-19 d^3 e x+7 d^2 e^2 x^2+33 d e^3 x^3+9 e^4 x^4\right )\right )}{d^3 (c d-b e)^3 x^2 (b+c x)^2 \sqrt {d+e x}}+\frac {3 c^{7/2} \left (16 c^2 d^2-44 b c d e+33 b^2 e^2\right ) \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {-c d+b e}}\right )}{(-c d+b e)^{7/2}}-\frac {3 \left (16 c^2 d^2+12 b c d e+5 b^2 e^2\right ) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{d^{7/2}}}{4 b^5} \] Input:
Integrate[1/((d + e*x)^(3/2)*(b*x + c*x^2)^3),x]
Output:
((b*(24*c^6*d^4*x^3*(d + e*x) + b^6*e^3*(2*d^2 - 5*d*e*x - 15*e^2*x^2) - 1 2*b*c^5*d^3*x^2*(-3*d^2 + d*e*x + 4*e^2*x^2) + b^2*c^4*d^2*x*(8*d^3 - 65*d ^2*e*x - 58*d*e^2*x^2 + 15*e^3*x^3) - b^5*c*e^2*(6*d^3 - 7*d^2*e*x + d*e^2 *x^2 + 30*e^3*x^3) + b^4*c^2*e*(6*d^4 + 9*d^3*e*x + 23*d^2*e^2*x^2 + 13*d* e^3*x^3 - 15*e^4*x^4) + b^3*c^3*d*(-2*d^4 - 19*d^3*e*x + 7*d^2*e^2*x^2 + 3 3*d*e^3*x^3 + 9*e^4*x^4)))/(d^3*(c*d - b*e)^3*x^2*(b + c*x)^2*Sqrt[d + e*x ]) + (3*c^(7/2)*(16*c^2*d^2 - 44*b*c*d*e + 33*b^2*e^2)*ArcTan[(Sqrt[c]*Sqr t[d + e*x])/Sqrt[-(c*d) + b*e]])/(-(c*d) + b*e)^(7/2) - (3*(16*c^2*d^2 + 1 2*b*c*d*e + 5*b^2*e^2)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/d^(7/2))/(4*b^5)
Time = 1.48 (sec) , antiderivative size = 436, normalized size of antiderivative = 1.10, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {1165, 27, 1235, 27, 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.
| \(\displaystyle \int \frac {1}{\left (b x+c x^2\right )^3 (d+e x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1165 |
| \(\displaystyle -\frac {\int \frac {12 c^2 d^2-5 b c e d-5 b^2 e^2+7 c e (2 c d-b e) x}{2 (d+e x)^{3/2} \left (c x^2+b x\right )^2}dx}{2 b^2 d (c d-b e)}-\frac {c x (2 c d-b e)+b (c d-b e)}{2 b^2 d \left (b x+c x^2\right )^2 \sqrt {d+e x} (c d-b e)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {12 c^2 d^2-5 b c e d-5 b^2 e^2+7 c e (2 c d-b e) x}{(d+e x)^{3/2} \left (c x^2+b x\right )^2}dx}{4 b^2 d (c d-b e)}-\frac {c x (2 c d-b e)+b (c d-b e)}{2 b^2 d \left (b x+c x^2\right )^2 \sqrt {d+e x} (c d-b e)}\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle -\frac {-\frac {\int \frac {3 \left (\left (16 c^2 d^2+12 b c e d+5 b^2 e^2\right ) (c d-b e)^2+c e (2 c d-b e) \left (12 c^2 d^2-12 b c e d-5 b^2 e^2\right ) x\right )}{2 (d+e x)^{3/2} \left (c x^2+b x\right )}dx}{b^2 d (c d-b e)}-\frac {b \left (5 b^3 e^3-17 b c^2 d^2 e+12 c^3 d^3\right )+c x (2 c d-b e) \left (-5 b^2 e^2-12 b c d e+12 c^2 d^2\right )}{b^2 d \left (b x+c x^2\right ) \sqrt {d+e x} (c d-b e)}}{4 b^2 d (c d-b e)}-\frac {c x (2 c d-b e)+b (c d-b e)}{2 b^2 d \left (b x+c x^2\right )^2 \sqrt {d+e x} (c d-b e)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {3 \int \frac {\left (16 c^2 d^2+12 b c e d+5 b^2 e^2\right ) (c d-b e)^2+c e (2 c d-b e) \left (12 c^2 d^2-12 b c e d-5 b^2 e^2\right ) x}{(d+e x)^{3/2} \left (c x^2+b x\right )}dx}{2 b^2 d (c d-b e)}-\frac {b \left (5 b^3 e^3-17 b c^2 d^2 e+12 c^3 d^3\right )+c x (2 c d-b e) \left (-5 b^2 e^2-12 b c d e+12 c^2 d^2\right )}{b^2 d \left (b x+c x^2\right ) \sqrt {d+e x} (c d-b e)}}{4 b^2 d (c d-b e)}-\frac {c x (2 c d-b e)+b (c d-b e)}{2 b^2 d \left (b x+c x^2\right )^2 \sqrt {d+e x} (c d-b e)}\) |
| \(\Big \downarrow \) 1198 |
| \(\displaystyle -\frac {-\frac {3 \left (\frac {\int \frac {\left (16 c^2 d^2+12 b c e d+5 b^2 e^2\right ) (c d-b e)^3+c e \left (c^2 d^2-b c e d-b^2 e^2\right ) \left (8 c^2 d^2-8 b c e d+5 b^2 e^2\right ) x}{\sqrt {d+e x} \left (c x^2+b x\right )}dx}{d (c d-b e)}+\frac {2 e \left (-b^2 e^2-b c d e+c^2 d^2\right ) \left (5 b^2 e^2-8 b c d e+8 c^2 d^2\right )}{d \sqrt {d+e x} (c d-b e)}\right )}{2 b^2 d (c d-b e)}-\frac {b \left (5 b^3 e^3-17 b c^2 d^2 e+12 c^3 d^3\right )+c x (2 c d-b e) \left (-5 b^2 e^2-12 b c d e+12 c^2 d^2\right )}{b^2 d \left (b x+c x^2\right ) \sqrt {d+e x} (c d-b e)}}{4 b^2 d (c d-b e)}-\frac {c x (2 c d-b e)+b (c d-b e)}{2 b^2 d \left (b x+c x^2\right )^2 \sqrt {d+e x} (c d-b e)}\) |
| \(\Big \downarrow \) 1197 |
| \(\displaystyle -\frac {-\frac {3 \left (\frac {2 \int \frac {e \left ((2 c d-b e) \left (4 c^4 d^4-8 b c^3 e d^3+2 b^2 c^2 e^2 d^2+2 b^3 c e^3 d+5 b^4 e^4\right )+c \left (c^2 d^2-b c e d-b^2 e^2\right ) \left (8 c^2 d^2-8 b c e d+5 b^2 e^2\right ) (d+e x)\right )}{c (d+e x)^2-(2 c d-b e) (d+e x)+d (c d-b e)}d\sqrt {d+e x}}{d (c d-b e)}+\frac {2 e \left (-b^2 e^2-b c d e+c^2 d^2\right ) \left (5 b^2 e^2-8 b c d e+8 c^2 d^2\right )}{d \sqrt {d+e x} (c d-b e)}\right )}{2 b^2 d (c d-b e)}-\frac {b \left (5 b^3 e^3-17 b c^2 d^2 e+12 c^3 d^3\right )+c x (2 c d-b e) \left (-5 b^2 e^2-12 b c d e+12 c^2 d^2\right )}{b^2 d \left (b x+c x^2\right ) \sqrt {d+e x} (c d-b e)}}{4 b^2 d (c d-b e)}-\frac {c x (2 c d-b e)+b (c d-b e)}{2 b^2 d \left (b x+c x^2\right )^2 \sqrt {d+e x} (c d-b e)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {3 \left (\frac {2 e \int \frac {(2 c d-b e) \left (4 c^4 d^4-8 b c^3 e d^3+2 b^2 c^2 e^2 d^2+2 b^3 c e^3 d+5 b^4 e^4\right )+c \left (c^2 d^2-b c e d-b^2 e^2\right ) \left (8 c^2 d^2-8 b c e d+5 b^2 e^2\right ) (d+e x)}{c (d+e x)^2-(2 c d-b e) (d+e x)+d (c d-b e)}d\sqrt {d+e x}}{d (c d-b e)}+\frac {2 e \left (-b^2 e^2-b c d e+c^2 d^2\right ) \left (5 b^2 e^2-8 b c d e+8 c^2 d^2\right )}{d \sqrt {d+e x} (c d-b e)}\right )}{2 b^2 d (c d-b e)}-\frac {b \left (5 b^3 e^3-17 b c^2 d^2 e+12 c^3 d^3\right )+c x (2 c d-b e) \left (-5 b^2 e^2-12 b c d e+12 c^2 d^2\right )}{b^2 d \left (b x+c x^2\right ) \sqrt {d+e x} (c d-b e)}}{4 b^2 d (c d-b e)}-\frac {c x (2 c d-b e)+b (c d-b e)}{2 b^2 d \left (b x+c x^2\right )^2 \sqrt {d+e x} (c d-b e)}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle -\frac {-\frac {3 \left (\frac {2 e \left (\frac {c (c d-b e)^3 \left (5 b^2 e^2+12 b c d e+16 c^2 d^2\right ) \int \frac {1}{c (d+e x)-c d}d\sqrt {d+e x}}{b e}-\frac {c^4 d^3 \left (33 b^2 e^2-44 b c d e+16 c^2 d^2\right ) \int \frac {1}{-c d+b e+c (d+e x)}d\sqrt {d+e x}}{b e}\right )}{d (c d-b e)}+\frac {2 e \left (-b^2 e^2-b c d e+c^2 d^2\right ) \left (5 b^2 e^2-8 b c d e+8 c^2 d^2\right )}{d \sqrt {d+e x} (c d-b e)}\right )}{2 b^2 d (c d-b e)}-\frac {b \left (5 b^3 e^3-17 b c^2 d^2 e+12 c^3 d^3\right )+c x (2 c d-b e) \left (-5 b^2 e^2-12 b c d e+12 c^2 d^2\right )}{b^2 d \left (b x+c x^2\right ) \sqrt {d+e x} (c d-b e)}}{4 b^2 d (c d-b e)}-\frac {c x (2 c d-b e)+b (c d-b e)}{2 b^2 d \left (b x+c x^2\right )^2 \sqrt {d+e x} (c d-b e)}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {-\frac {3 \left (\frac {2 e \left (\frac {c^{7/2} d^3 \left (33 b^2 e^2-44 b c d e+16 c^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b e \sqrt {c d-b e}}-\frac {\text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) (c d-b e)^3 \left (5 b^2 e^2+12 b c d e+16 c^2 d^2\right )}{b \sqrt {d} e}\right )}{d (c d-b e)}+\frac {2 e \left (-b^2 e^2-b c d e+c^2 d^2\right ) \left (5 b^2 e^2-8 b c d e+8 c^2 d^2\right )}{d \sqrt {d+e x} (c d-b e)}\right )}{2 b^2 d (c d-b e)}-\frac {b \left (5 b^3 e^3-17 b c^2 d^2 e+12 c^3 d^3\right )+c x (2 c d-b e) \left (-5 b^2 e^2-12 b c d e+12 c^2 d^2\right )}{b^2 d \left (b x+c x^2\right ) \sqrt {d+e x} (c d-b e)}}{4 b^2 d (c d-b e)}-\frac {c x (2 c d-b e)+b (c d-b e)}{2 b^2 d \left (b x+c x^2\right )^2 \sqrt {d+e x} (c d-b e)}\) |
Input:
Int[1/((d + e*x)^(3/2)*(b*x + c*x^2)^3),x]
Output:
-1/2*(b*(c*d - b*e) + c*(2*c*d - b*e)*x)/(b^2*d*(c*d - b*e)*Sqrt[d + e*x]* (b*x + c*x^2)^2) - (-((b*(12*c^3*d^3 - 17*b*c^2*d^2*e + 5*b^3*e^3) + c*(2* c*d - b*e)*(12*c^2*d^2 - 12*b*c*d*e - 5*b^2*e^2)*x)/(b^2*d*(c*d - b*e)*Sqr t[d + e*x]*(b*x + c*x^2))) - (3*((2*e*(c^2*d^2 - b*c*d*e - b^2*e^2)*(8*c^2 *d^2 - 8*b*c*d*e + 5*b^2*e^2))/(d*(c*d - b*e)*Sqrt[d + e*x]) + (2*e*(-(((c *d - b*e)^3*(16*c^2*d^2 + 12*b*c*d*e + 5*b^2*e^2)*ArcTanh[Sqrt[d + e*x]/Sq rt[d]])/(b*Sqrt[d]*e)) + (c^(7/2)*d^3*(16*c^2*d^2 - 44*b*c*d*e + 33*b^2*e^ 2)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b*e*Sqrt[c*d - b*e]) ))/(d*(c*d - b*e))))/(2*b^2*d*(c*d - b*e)))/(4*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[(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_))^(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 = 0.76 (sec) , antiderivative size = 279, normalized size of antiderivative = 0.70
| method | result | size |
| risch | \(-\frac {\sqrt {e x +d}\, \left (-7 b e x -12 c d x +2 b d \right )}{4 d^{3} b^{4} x^{2}}+\frac {e \left (-\frac {\left (15 b^{2} e^{2}+36 b c d e +48 c^{2} d^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b e \sqrt {d}}+\frac {8 b^{4} e^{4}}{\left (b e -c d \right )^{3} \sqrt {e x +d}}+\frac {8 c^{4} d^{3} \left (\frac {\left (\frac {19}{8} e^{2} b^{2} c -\frac {3}{2} c^{2} d e b \right ) \left (e x +d \right )^{\frac {3}{2}}+\frac {3 b e \left (7 b^{2} e^{2}-11 b c d e +4 c^{2} d^{2}\right ) \sqrt {e x +d}}{8}}{\left (\left (e x +d \right ) c +b e -c d \right )^{2}}+\frac {3 \left (33 b^{2} e^{2}-44 b c d e +16 c^{2} d^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {c \left (b e -c d \right )}}\right )}{8 \sqrt {c \left (b e -c d \right )}}\right )}{\left (b e -c d \right )^{3} b e}\right )}{4 b^{4} d^{3}}\) | \(279\) |
| derivativedivides | \(2 e^{5} \left (-\frac {\frac {-\frac {b e \left (7 b e +12 c d \right ) \left (e x +d \right )^{\frac {3}{2}}}{8}+\left (\frac {9}{8} d \,e^{2} b^{2}+\frac {3}{2} d^{2} e b c \right ) \sqrt {e x +d}}{e^{2} x^{2}}+\frac {3 \left (5 b^{2} e^{2}+12 b c d e +16 c^{2} d^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{8 \sqrt {d}}}{d^{3} b^{5} e^{5}}+\frac {1}{d^{3} \left (b e -c d \right )^{3} \sqrt {e x +d}}+\frac {c^{4} \left (\frac {\left (\frac {19}{8} e^{2} b^{2} c -\frac {3}{2} c^{2} d e b \right ) \left (e x +d \right )^{\frac {3}{2}}+\frac {3 b e \left (7 b^{2} e^{2}-11 b c d e +4 c^{2} d^{2}\right ) \sqrt {e x +d}}{8}}{\left (\left (e x +d \right ) c +b e -c d \right )^{2}}+\frac {3 \left (33 b^{2} e^{2}-44 b c d e +16 c^{2} d^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {c \left (b e -c d \right )}}\right )}{8 \sqrt {c \left (b e -c d \right )}}\right )}{b^{5} e^{5} \left (b e -c d \right )^{3}}\right )\) | \(293\) |
| default | \(2 e^{5} \left (-\frac {\frac {-\frac {b e \left (7 b e +12 c d \right ) \left (e x +d \right )^{\frac {3}{2}}}{8}+\left (\frac {9}{8} d \,e^{2} b^{2}+\frac {3}{2} d^{2} e b c \right ) \sqrt {e x +d}}{e^{2} x^{2}}+\frac {3 \left (5 b^{2} e^{2}+12 b c d e +16 c^{2} d^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{8 \sqrt {d}}}{d^{3} b^{5} e^{5}}+\frac {1}{d^{3} \left (b e -c d \right )^{3} \sqrt {e x +d}}+\frac {c^{4} \left (\frac {\left (\frac {19}{8} e^{2} b^{2} c -\frac {3}{2} c^{2} d e b \right ) \left (e x +d \right )^{\frac {3}{2}}+\frac {3 b e \left (7 b^{2} e^{2}-11 b c d e +4 c^{2} d^{2}\right ) \sqrt {e x +d}}{8}}{\left (\left (e x +d \right ) c +b e -c d \right )^{2}}+\frac {3 \left (33 b^{2} e^{2}-44 b c d e +16 c^{2} d^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {c \left (b e -c d \right )}}\right )}{8 \sqrt {c \left (b e -c d \right )}}\right )}{b^{5} e^{5} \left (b e -c d \right )^{3}}\right )\) | \(293\) |
| pseudoelliptic | \(\frac {24 x^{2} \left (c x +b \right )^{2} \left (\frac {33 b^{2} e^{2} d^{\frac {7}{2}}}{16}+c \,d^{\frac {9}{2}} \left (c d -\frac {11 b e}{4}\right )\right ) c^{4} \sqrt {e x +d}\, \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {c \left (b e -c d \right )}}\right )+\sqrt {c \left (b e -c d \right )}\, \left (-\frac {15 \left (b^{2} e^{2}+\frac {12}{5} b c d e +\frac {16}{5} c^{2} d^{2}\right ) \sqrt {e x +d}\, x^{2} \left (c x +b \right )^{2} \left (b e -c d \right )^{3} \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{2}+b \left (3 e^{2} c b \left (8 c^{4} x^{4}+\frac {29}{3} b \,c^{3} x^{3}-\frac {7}{6} b^{2} c^{2} x^{2}-\frac {3}{2} b^{3} c x +b^{4}\right ) d^{\frac {7}{2}}+\frac {15 b^{4} e^{5} x^{2} \left (c x +b \right )^{2} \sqrt {d}}{2}+\left (\left (\frac {5}{2} e^{4} x -d \,e^{3}\right ) b^{6}-\frac {7 e^{3} x \left (-\frac {e x}{7}+d \right ) c \,b^{5}}{2}-3 \left (\frac {13}{6} e^{3} x^{3}+\frac {23}{6} d \,e^{2} x^{2}+d^{3}\right ) e \,c^{2} b^{4}+c^{3} \left (\frac {19}{2} d^{3} e x -\frac {33}{2} d \,e^{3} x^{3}-\frac {9}{2} e^{4} x^{4}+d^{4}\right ) b^{3}-4 \left (\frac {15}{8} e^{3} x^{3}-\frac {65}{8} d^{2} e x +d^{3}\right ) x \,c^{4} d \,b^{2}-18 x^{2} \left (-\frac {e x}{3}+d \right ) c^{5} d^{3} b -12 c^{6} d^{3} x^{3} \left (e x +d \right )\right ) d^{\frac {3}{2}}\right )\right )}{2 \sqrt {c \left (b e -c d \right )}\, d^{\frac {7}{2}} \sqrt {e x +d}\, x^{2} b^{5} \left (b e -c d \right )^{3} \left (c x +b \right )^{2}}\) | \(434\) |
Input:
int(1/(e*x+d)^(3/2)/(c*x^2+b*x)^3,x,method=_RETURNVERBOSE)
Output:
-1/4*(e*x+d)^(1/2)*(-7*b*e*x-12*c*d*x+2*b*d)/d^3/b^4/x^2+1/4/b^4/d^3*e*(-( 15*b^2*e^2+36*b*c*d*e+48*c^2*d^2)/b/e/d^(1/2)*arctanh((e*x+d)^(1/2)/d^(1/2 ))+8*b^4*e^4/(b*e-c*d)^3/(e*x+d)^(1/2)+8*c^4*d^3/(b*e-c*d)^3/b/e*(((19/8*e ^2*b^2*c-3/2*c^2*d*e*b)*(e*x+d)^(3/2)+3/8*b*e*(7*b^2*e^2-11*b*c*d*e+4*c^2* d^2)*(e*x+d)^(1/2))/((e*x+d)*c+b*e-c*d)^2+3/8*(33*b^2*e^2-44*b*c*d*e+16*c^ 2*d^2)/(c*(b*e-c*d))^(1/2)*arctan(c*(e*x+d)^(1/2)/(c*(b*e-c*d))^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 1167 vs. \(2 (358) = 716\).
Time = 2.75 (sec) , antiderivative size = 4744, normalized size of antiderivative = 11.92 \[ \int \frac {1}{(d+e x)^{3/2} \left (b x+c x^2\right )^3} \, dx=\text {Too large to display} \] Input:
integrate(1/(e*x+d)^(3/2)/(c*x^2+b*x)^3,x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {1}{(d+e x)^{3/2} \left (b x+c x^2\right )^3} \, dx=\int \frac {1}{x^{3} \left (b + c x\right )^{3} \left (d + e x\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(1/(e*x+d)**(3/2)/(c*x**2+b*x)**3,x)
Output:
Integral(1/(x**3*(b + c*x)**3*(d + e*x)**(3/2)), x)
Exception generated. \[ \int \frac {1}{(d+e x)^{3/2} \left (b x+c x^2\right )^3} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(1/(e*x+d)^(3/2)/(c*x^2+b*x)^3,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. 773 vs. \(2 (358) = 716\).
Time = 0.15 (sec) , antiderivative size = 773, normalized size of antiderivative = 1.94 \[ \int \frac {1}{(d+e x)^{3/2} \left (b x+c x^2\right )^3} \, dx =\text {Too large to display} \] Input:
integrate(1/(e*x+d)^(3/2)/(c*x^2+b*x)^3,x, algorithm="giac")
Output:
-2*e^5/((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(e*x + d)) - 3/4*(16*c^6*d^2 - 44*b*c^5*d*e + 33*b^2*c^4*e^2)*arctan(sqrt(e*x + d)*c/sqrt(-c^2*d + b*c*e))/((b^5*c^3*d^3 - 3*b^6*c^2*d^2*e + 3*b^7*c*d*e ^2 - b^8*e^3)*sqrt(-c^2*d + b*c*e)) + 1/4*(24*(e*x + d)^(7/2)*c^6*d^4*e - 72*(e*x + d)^(5/2)*c^6*d^5*e + 72*(e*x + d)^(3/2)*c^6*d^6*e - 24*sqrt(e*x + d)*c^6*d^7*e - 48*(e*x + d)^(7/2)*b*c^5*d^3*e^2 + 180*(e*x + d)^(5/2)*b* c^5*d^4*e^2 - 216*(e*x + d)^(3/2)*b*c^5*d^5*e^2 + 84*sqrt(e*x + d)*b*c^5*d ^6*e^2 + 15*(e*x + d)^(7/2)*b^2*c^4*d^2*e^3 - 118*(e*x + d)^(5/2)*b^2*c^4* d^3*e^3 + 199*(e*x + d)^(3/2)*b^2*c^4*d^4*e^3 - 96*sqrt(e*x + d)*b^2*c^4*d ^5*e^3 + 9*(e*x + d)^(7/2)*b^3*c^3*d*e^4 - 3*(e*x + d)^(5/2)*b^3*c^3*d^2*e ^4 - 38*(e*x + d)^(3/2)*b^3*c^3*d^3*e^4 + 30*sqrt(e*x + d)*b^3*c^3*d^4*e^4 - 7*(e*x + d)^(7/2)*b^4*c^2*e^5 + 41*(e*x + d)^(5/2)*b^4*c^2*d*e^5 - 58*( e*x + d)^(3/2)*b^4*c^2*d^2*e^5 + 30*sqrt(e*x + d)*b^4*c^2*d^3*e^5 - 14*(e* x + d)^(5/2)*b^5*c*e^6 + 41*(e*x + d)^(3/2)*b^5*c*d*e^6 - 33*sqrt(e*x + d) *b^5*c*d^2*e^6 - 7*(e*x + d)^(3/2)*b^6*e^7 + 9*sqrt(e*x + d)*b^6*d*e^7)/(( b^4*c^3*d^6 - 3*b^5*c^2*d^5*e + 3*b^6*c*d^4*e^2 - b^7*d^3*e^3)*((e*x + d)^ 2*c - 2*(e*x + d)*c*d + c*d^2 + (e*x + d)*b*e - b*d*e)^2) + 3/4*(16*c^2*d^ 2 + 12*b*c*d*e + 5*b^2*e^2)*arctan(sqrt(e*x + d)/sqrt(-d))/(b^5*sqrt(-d)*d ^3)
Time = 7.93 (sec) , antiderivative size = 9635, normalized size of antiderivative = 24.21 \[ \int \frac {1}{(d+e x)^{3/2} \left (b x+c x^2\right )^3} \, dx=\text {Too large to display} \] Input:
int(1/((b*x + c*x^2)^3*(d + e*x)^(3/2)),x)
Output:
- ((2*e^5)/(c*d^2 - b*d*e) + (e*(d + e*x)^2*(15*b^6*e^6 - 72*c^6*d^6 - 199 *b^2*c^4*d^4*e^2 + 38*b^3*c^3*d^3*e^3 + 106*b^4*c^2*d^2*e^4 + 216*b*c^5*d^ 5*e - 89*b^5*c*d*e^5))/(4*b^4*(c*d^2 - b*d*e)^3) + (e*(d + e*x)*(25*b^5*e^ 5 + 24*c^5*d^5 + 36*b^2*c^3*d^3*e^2 + 6*b^3*c^2*d^2*e^3 - 60*b*c^4*d^4*e - 56*b^4*c*d*e^4))/(4*b^4*(c*d^2 - b*d*e)^2) - (3*e*(d + e*x)^4*(8*c^6*d^4 - 5*b^4*c^2*e^4 + 3*b^3*c^3*d*e^3 + 5*b^2*c^4*d^2*e^2 - 16*b*c^5*d^3*e))/( 4*b^4*(c*d^2 - b*d*e)^3) + (e*(d + e*x)^3*(72*c^6*d^5 + 30*b^5*c*e^5 - 73* b^4*c^2*d*e^4 + 118*b^2*c^4*d^3*e^2 + 3*b^3*c^3*d^2*e^3 - 180*b*c^5*d^4*e) )/(4*b^4*(c*d^2 - b*d*e)^3))/(c^2*(d + e*x)^(9/2) - (4*c^2*d - 2*b*c*e)*(d + e*x)^(7/2) - (d + e*x)^(3/2)*(4*c^2*d^3 + 2*b^2*d*e^2 - 6*b*c*d^2*e) + (d + e*x)^(5/2)*(b^2*e^2 + 6*c^2*d^2 - 6*b*c*d*e) + (d + e*x)^(1/2)*(c^2*d ^4 + b^2*d^2*e^2 - 2*b*c*d^3*e)) - (atan((((-c^7*(b*e - c*d)^7)^(1/2)*((d + e*x)^(1/2)*(589824*b^12*c^22*d^28*e^2 - 8257536*b^13*c^21*d^27*e^3 + 533 42208*b^14*c^20*d^26*e^4 - 210382848*b^15*c^19*d^25*e^5 + 564860160*b^16*c ^18*d^24*e^6 - 1089838080*b^17*c^17*d^23*e^7 + 1555380864*b^18*c^16*d^22*e ^8 - 1667850624*b^19*c^15*d^21*e^9 + 1358257536*b^20*c^14*d^20*e^10 - 8556 42240*b^21*c^13*d^19*e^11 + 438185088*b^22*c^12*d^18*e^12 - 201386880*b^23 *c^11*d^17*e^13 + 90100224*b^24*c^10*d^16*e^14 - 37986048*b^25*c^9*d^15*e^ 15 + 15108480*b^26*c^8*d^14*e^16 - 6844032*b^27*c^7*d^13*e^17 + 3399552*b^ 28*c^6*d^12*e^18 - 1300608*b^29*c^5*d^11*e^19 + 293760*b^30*c^4*d^10*e^...
Time = 0.32 (sec) , antiderivative size = 2580, normalized size of antiderivative = 6.48 \[ \int \frac {1}{(d+e x)^{3/2} \left (b x+c x^2\right )^3} \, dx =\text {Too large to display} \] Input:
int(1/(e*x+d)^(3/2)/(c*x^2+b*x)^3,x)
Output:
(198*sqrt(c)*sqrt(d + e*x)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c) *sqrt(b*e - c*d)))*b**4*c**3*d**4*e**2*x**2 - 264*sqrt(c)*sqrt(d + e*x)*sq rt(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - c*d)))*b**3*c**4* d**5*e*x**2 + 396*sqrt(c)*sqrt(d + e*x)*sqrt(b*e - c*d)*atan((sqrt(d + e*x )*c)/(sqrt(c)*sqrt(b*e - c*d)))*b**3*c**4*d**4*e**2*x**3 + 96*sqrt(c)*sqrt (d + e*x)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - c*d)) )*b**2*c**5*d**6*x**2 - 528*sqrt(c)*sqrt(d + e*x)*sqrt(b*e - c*d)*atan((sq rt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - c*d)))*b**2*c**5*d**5*e*x**3 + 198*sqrt (c)*sqrt(d + e*x)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - c*d)))*b**2*c**5*d**4*e**2*x**4 + 192*sqrt(c)*sqrt(d + e*x)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - c*d)))*b*c**6*d**6*x**3 - 264*sqrt(c)*sqrt(d + e*x)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)* sqrt(b*e - c*d)))*b*c**6*d**5*e*x**4 + 96*sqrt(c)*sqrt(d + e*x)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - c*d)))*c**7*d**6*x**4 + 1 5*sqrt(d)*sqrt(d + e*x)*log(sqrt(d + e*x) - sqrt(d))*b**8*e**6*x**2 - 24*s qrt(d)*sqrt(d + e*x)*log(sqrt(d + e*x) - sqrt(d))*b**7*c*d*e**5*x**2 + 30* sqrt(d)*sqrt(d + e*x)*log(sqrt(d + e*x) - sqrt(d))*b**7*c*e**6*x**3 - 6*sq rt(d)*sqrt(d + e*x)*log(sqrt(d + e*x) - sqrt(d))*b**6*c**2*d**2*e**4*x**2 - 48*sqrt(d)*sqrt(d + e*x)*log(sqrt(d + e*x) - sqrt(d))*b**6*c**2*d*e**5*x **3 + 15*sqrt(d)*sqrt(d + e*x)*log(sqrt(d + e*x) - sqrt(d))*b**6*c**2*e...