Integrand size = 20, antiderivative size = 277 \[ \int \frac {1}{x^2 (a+b x)^2 (c+d x)^{5/2}} \, dx=-\frac {d \left (6 b^2 c^2-6 a b c d+5 a^2 d^2\right )}{3 a^2 c^2 (b c-a d)^2 (c+d x)^{3/2}}-\frac {b (2 b c-a d)}{a^2 c (b c-a d) (a+b x) (c+d x)^{3/2}}-\frac {1}{a c x (a+b x) (c+d x)^{3/2}}-\frac {d (2 b c-a d) \left (b^2 c^2-a b c d+5 a^2 d^2\right )}{a^2 c^3 (b c-a d)^3 \sqrt {c+d x}}+\frac {(4 b c+5 a d) \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^3 c^{7/2}}-\frac {b^{7/2} (4 b c-9 a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{a^3 (b c-a d)^{7/2}} \]
-1/3*d*(5*a^2*d^2-6*a*b*c*d+6*b^2*c^2)/a^2/c^2/(-a*d+b*c)^2/(d*x+c)^(3/2)- b*(-a*d+2*b*c)/a^2/c/(-a*d+b*c)/(b*x+a)/(d*x+c)^(3/2)-1/a/c/x/(b*x+a)/(d*x +c)^(3/2)+(5*a*d+4*b*c)*arctanh((d*x+c)^(1/2)/c^(1/2))/a^3/c^(7/2)-b^(7/2) *(-9*a*d+4*b*c)*arctanh(b^(1/2)*(d*x+c)^(1/2)/(-a*d+b*c)^(1/2))/a^3/(-a*d+ b*c)^(7/2)-d*(-a*d+2*b*c)*(5*a^2*d^2-a*b*c*d+b^2*c^2)/a^2/c^3/(-a*d+b*c)^3 /(d*x+c)^(1/2)
Time = 1.10 (sec) , antiderivative size = 272, normalized size of antiderivative = 0.98 \[ \int \frac {1}{x^2 (a+b x)^2 (c+d x)^{5/2}} \, dx=\frac {\frac {a \left (-6 b^4 c^3 x (c+d x)^2-3 a b^3 c^2 (c-3 d x) (c+d x)^2+a^4 d^3 \left (3 c^2+20 c d x+15 d^2 x^2\right )+a^2 b^2 c d \left (9 c^3+9 c^2 d x-35 c d^2 x^2-33 d^3 x^3\right )+a^3 b d^2 \left (-9 c^3-41 c^2 d x-13 c d^2 x^2+15 d^3 x^3\right )\right )}{c^3 (b c-a d)^3 x (a+b x) (c+d x)^{3/2}}-\frac {3 b^{7/2} (4 b c-9 a d) \arctan \left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{(-b c+a d)^{7/2}}+\frac {3 (4 b c+5 a d) \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{c^{7/2}}}{3 a^3} \]
((a*(-6*b^4*c^3*x*(c + d*x)^2 - 3*a*b^3*c^2*(c - 3*d*x)*(c + d*x)^2 + a^4* d^3*(3*c^2 + 20*c*d*x + 15*d^2*x^2) + a^2*b^2*c*d*(9*c^3 + 9*c^2*d*x - 35* c*d^2*x^2 - 33*d^3*x^3) + a^3*b*d^2*(-9*c^3 - 41*c^2*d*x - 13*c*d^2*x^2 + 15*d^3*x^3)))/(c^3*(b*c - a*d)^3*x*(a + b*x)*(c + d*x)^(3/2)) - (3*b^(7/2) *(4*b*c - 9*a*d)*ArcTan[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[-(b*c) + a*d]])/(-(b* c) + a*d)^(7/2) + (3*(4*b*c + 5*a*d)*ArcTanh[Sqrt[c + d*x]/Sqrt[c]])/c^(7/ 2))/(3*a^3)
Time = 0.51 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.22, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {114, 27, 168, 169, 27, 169, 27, 174, 73, 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.
| \(\displaystyle \int \frac {1}{x^2 (a+b x)^2 (c+d x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle -\frac {\int \frac {4 b c+5 a d+7 b d x}{2 x (a+b x)^2 (c+d x)^{5/2}}dx}{a c}-\frac {1}{a c x (a+b x) (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {4 b c+5 a d+7 b d x}{x (a+b x)^2 (c+d x)^{5/2}}dx}{2 a c}-\frac {1}{a c x (a+b x) (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle -\frac {\frac {\int \frac {(b c-a d) (4 b c+5 a d)+5 b d (2 b c-a d) x}{x (a+b x) (c+d x)^{5/2}}dx}{a (b c-a d)}+\frac {2 b (2 b c-a d)}{a (a+b x) (c+d x)^{3/2} (b c-a d)}}{2 a c}-\frac {1}{a c x (a+b x) (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {\frac {\frac {2 d \left (5 a^2 d^2-6 a b c d+6 b^2 c^2\right )}{3 c (c+d x)^{3/2} (b c-a d)}-\frac {2 \int -\frac {3 \left ((4 b c+5 a d) (b c-a d)^2+b d \left (6 b^2 c^2-6 a b d c+5 a^2 d^2\right ) x\right )}{2 x (a+b x) (c+d x)^{3/2}}dx}{3 c (b c-a d)}}{a (b c-a d)}+\frac {2 b (2 b c-a d)}{a (a+b x) (c+d x)^{3/2} (b c-a d)}}{2 a c}-\frac {1}{a c x (a+b x) (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\frac {\int \frac {(4 b c+5 a d) (b c-a d)^2+b d \left (6 b^2 c^2-6 a b d c+5 a^2 d^2\right ) x}{x (a+b x) (c+d x)^{3/2}}dx}{c (b c-a d)}+\frac {2 d \left (5 a^2 d^2-6 a b c d+6 b^2 c^2\right )}{3 c (c+d x)^{3/2} (b c-a d)}}{a (b c-a d)}+\frac {2 b (2 b c-a d)}{a (a+b x) (c+d x)^{3/2} (b c-a d)}}{2 a c}-\frac {1}{a c x (a+b x) (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {\frac {\frac {\frac {2 d (2 b c-a d) \left (5 a^2 d^2-a b c d+b^2 c^2\right )}{c \sqrt {c+d x} (b c-a d)}-\frac {2 \int -\frac {(4 b c+5 a d) (b c-a d)^3+b d (2 b c-a d) \left (b^2 c^2-a b d c+5 a^2 d^2\right ) x}{2 x (a+b x) \sqrt {c+d x}}dx}{c (b c-a d)}}{c (b c-a d)}+\frac {2 d \left (5 a^2 d^2-6 a b c d+6 b^2 c^2\right )}{3 c (c+d x)^{3/2} (b c-a d)}}{a (b c-a d)}+\frac {2 b (2 b c-a d)}{a (a+b x) (c+d x)^{3/2} (b c-a d)}}{2 a c}-\frac {1}{a c x (a+b x) (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\frac {\frac {\int \frac {(4 b c+5 a d) (b c-a d)^3+b d (2 b c-a d) \left (b^2 c^2-a b d c+5 a^2 d^2\right ) x}{x (a+b x) \sqrt {c+d x}}dx}{c (b c-a d)}+\frac {2 d (2 b c-a d) \left (5 a^2 d^2-a b c d+b^2 c^2\right )}{c \sqrt {c+d x} (b c-a d)}}{c (b c-a d)}+\frac {2 d \left (5 a^2 d^2-6 a b c d+6 b^2 c^2\right )}{3 c (c+d x)^{3/2} (b c-a d)}}{a (b c-a d)}+\frac {2 b (2 b c-a d)}{a (a+b x) (c+d x)^{3/2} (b c-a d)}}{2 a c}-\frac {1}{a c x (a+b x) (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle -\frac {\frac {\frac {\frac {\frac {(b c-a d)^3 (5 a d+4 b c) \int \frac {1}{x \sqrt {c+d x}}dx}{a}-\frac {b^4 c^3 (4 b c-9 a d) \int \frac {1}{(a+b x) \sqrt {c+d x}}dx}{a}}{c (b c-a d)}+\frac {2 d (2 b c-a d) \left (5 a^2 d^2-a b c d+b^2 c^2\right )}{c \sqrt {c+d x} (b c-a d)}}{c (b c-a d)}+\frac {2 d \left (5 a^2 d^2-6 a b c d+6 b^2 c^2\right )}{3 c (c+d x)^{3/2} (b c-a d)}}{a (b c-a d)}+\frac {2 b (2 b c-a d)}{a (a+b x) (c+d x)^{3/2} (b c-a d)}}{2 a c}-\frac {1}{a c x (a+b x) (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {\frac {\frac {\frac {\frac {2 (b c-a d)^3 (5 a d+4 b c) \int \frac {1}{\frac {c+d x}{d}-\frac {c}{d}}d\sqrt {c+d x}}{a d}-\frac {2 b^4 c^3 (4 b c-9 a d) \int \frac {1}{a+\frac {b (c+d x)}{d}-\frac {b c}{d}}d\sqrt {c+d x}}{a d}}{c (b c-a d)}+\frac {2 d (2 b c-a d) \left (5 a^2 d^2-a b c d+b^2 c^2\right )}{c \sqrt {c+d x} (b c-a d)}}{c (b c-a d)}+\frac {2 d \left (5 a^2 d^2-6 a b c d+6 b^2 c^2\right )}{3 c (c+d x)^{3/2} (b c-a d)}}{a (b c-a d)}+\frac {2 b (2 b c-a d)}{a (a+b x) (c+d x)^{3/2} (b c-a d)}}{2 a c}-\frac {1}{a c x (a+b x) (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {\frac {\frac {\frac {2 d (2 b c-a d) \left (5 a^2 d^2-a b c d+b^2 c^2\right )}{c \sqrt {c+d x} (b c-a d)}+\frac {\frac {2 b^{7/2} c^3 (4 b c-9 a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{a \sqrt {b c-a d}}-\frac {2 (b c-a d)^3 (5 a d+4 b c) \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a \sqrt {c}}}{c (b c-a d)}}{c (b c-a d)}+\frac {2 d \left (5 a^2 d^2-6 a b c d+6 b^2 c^2\right )}{3 c (c+d x)^{3/2} (b c-a d)}}{a (b c-a d)}+\frac {2 b (2 b c-a d)}{a (a+b x) (c+d x)^{3/2} (b c-a d)}}{2 a c}-\frac {1}{a c x (a+b x) (c+d x)^{3/2}}\) |
-(1/(a*c*x*(a + b*x)*(c + d*x)^(3/2))) - ((2*b*(2*b*c - a*d))/(a*(b*c - a* d)*(a + b*x)*(c + d*x)^(3/2)) + ((2*d*(6*b^2*c^2 - 6*a*b*c*d + 5*a^2*d^2)) /(3*c*(b*c - a*d)*(c + d*x)^(3/2)) + ((2*d*(2*b*c - a*d)*(b^2*c^2 - a*b*c* d + 5*a^2*d^2))/(c*(b*c - a*d)*Sqrt[c + d*x]) + ((-2*(b*c - a*d)^3*(4*b*c + 5*a*d)*ArcTanh[Sqrt[c + d*x]/Sqrt[c]])/(a*Sqrt[c]) + (2*b^(7/2)*c^3*(4*b *c - 9*a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(a*Sqrt[b*c - a*d]))/(c*(b*c - a*d)))/(c*(b*c - a*d)))/(a*(b*c - a*d)))/(2*a*c)
3.5.78.3.1 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_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n *(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n *(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[ 2*m, 2*n, 2*p]
Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* ((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d) Int[(e + f*x)^ p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d) Int[(e + f*x)^p/(c + d *x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, 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]
Time = 0.77 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.74
| method | result | size |
| derivativedivides | \(2 d^{3} \left (-\frac {1}{3 c^{2} \left (a d -b c \right )^{2} \left (d x +c \right )^{\frac {3}{2}}}-\frac {2 a d -4 b c}{c^{3} \left (a d -b c \right )^{3} \sqrt {d x +c}}+\frac {-\frac {a \sqrt {d x +c}}{2 x}+\frac {\left (5 a d +4 b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{2 \sqrt {c}}}{a^{3} c^{3} d^{3}}+\frac {b^{4} \left (\frac {\sqrt {d x +c}\, a d}{2 \left (d x +c \right ) b +2 a d -2 b c}+\frac {\left (9 a d -4 b c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \sqrt {\left (a d -b c \right ) b}}\right )}{d^{3} a^{3} \left (a d -b c \right )^{3}}\right )\) | \(204\) |
| default | \(2 d^{3} \left (-\frac {1}{3 c^{2} \left (a d -b c \right )^{2} \left (d x +c \right )^{\frac {3}{2}}}-\frac {2 a d -4 b c}{c^{3} \left (a d -b c \right )^{3} \sqrt {d x +c}}+\frac {-\frac {a \sqrt {d x +c}}{2 x}+\frac {\left (5 a d +4 b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{2 \sqrt {c}}}{a^{3} c^{3} d^{3}}+\frac {b^{4} \left (\frac {\sqrt {d x +c}\, a d}{2 \left (d x +c \right ) b +2 a d -2 b c}+\frac {\left (9 a d -4 b c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \sqrt {\left (a d -b c \right ) b}}\right )}{d^{3} a^{3} \left (a d -b c \right )^{3}}\right )\) | \(204\) |
| risch | \(-\frac {\sqrt {d x +c}}{c^{3} a^{2} x}-\frac {d \left (-\frac {2 b^{4} c^{3} \left (\frac {\sqrt {d x +c}\, a d}{2 \left (d x +c \right ) b +2 a d -2 b c}+\frac {\left (9 a d -4 b c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \sqrt {\left (a d -b c \right ) b}}\right )}{\left (a d -b c \right )^{3} a d}-\frac {\left (5 a d +4 b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{a d \sqrt {c}}+\frac {4 a^{2} d^{2} \left (a d -2 b c \right )}{\left (a d -b c \right )^{3} \sqrt {d x +c}}+\frac {2 c \,a^{2} d^{2}}{3 \left (a d -b c \right )^{2} \left (d x +c \right )^{\frac {3}{2}}}\right )}{c^{3} a^{2}}\) | \(219\) |
| pseudoelliptic | \(d^{3} \left (\frac {\sqrt {d x +c}\, b^{4}}{a^{2} d^{3} \left (b x +a \right ) \left (a d -b c \right )^{3}}+\frac {9 \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right ) b^{4}}{\sqrt {\left (a d -b c \right ) b}\, a^{2} d^{2} \left (a d -b c \right )^{3}}-\frac {4 \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right ) b^{5} c}{\sqrt {\left (a d -b c \right ) b}\, a^{3} d^{3} \left (a d -b c \right )^{3}}+\frac {5 \,\operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right ) a d x +4 \,\operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right ) b c x -a \sqrt {d x +c}\, \sqrt {c}}{x \,c^{\frac {7}{2}} a^{3} d^{3}}-\frac {2}{3 c^{2} \left (a d -b c \right )^{2} \left (d x +c \right )^{\frac {3}{2}}}-\frac {4 \left (a d -2 b c \right )}{c^{3} \left (a d -b c \right )^{3} \sqrt {d x +c}}\right )\) | \(264\) |
2*d^3*(-1/3/c^2/(a*d-b*c)^2/(d*x+c)^(3/2)-(2*a*d-4*b*c)/c^3/(a*d-b*c)^3/(d *x+c)^(1/2)+1/a^3/c^3/d^3*(-1/2*a*(d*x+c)^(1/2)/x+1/2*(5*a*d+4*b*c)/c^(1/2 )*arctanh((d*x+c)^(1/2)/c^(1/2)))+b^4/d^3/a^3/(a*d-b*c)^3*(1/2*a*d*(d*x+c) ^(1/2)/((d*x+c)*b+a*d-b*c)+1/2*(9*a*d-4*b*c)/((a*d-b*c)*b)^(1/2)*arctan(b* (d*x+c)^(1/2)/((a*d-b*c)*b)^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 960 vs. \(2 (251) = 502\).
Time = 1.50 (sec) , antiderivative size = 3872, normalized size of antiderivative = 13.98 \[ \int \frac {1}{x^2 (a+b x)^2 (c+d x)^{5/2}} \, dx=\text {Too large to display} \]
[1/6*(3*((4*b^5*c^5*d^2 - 9*a*b^4*c^4*d^3)*x^4 + (8*b^5*c^6*d - 14*a*b^4*c ^5*d^2 - 9*a^2*b^3*c^4*d^3)*x^3 + (4*b^5*c^7 - a*b^4*c^6*d - 18*a^2*b^3*c^ 5*d^2)*x^2 + (4*a*b^4*c^7 - 9*a^2*b^3*c^6*d)*x)*sqrt(b/(b*c - a*d))*log((b *d*x + 2*b*c - a*d - 2*(b*c - a*d)*sqrt(d*x + c)*sqrt(b/(b*c - a*d)))/(b*x + a)) + 3*((4*b^5*c^4*d^2 - 7*a*b^4*c^3*d^3 - 3*a^2*b^3*c^2*d^4 + 11*a^3* b^2*c*d^5 - 5*a^4*b*d^6)*x^4 + (8*b^5*c^5*d - 10*a*b^4*c^4*d^2 - 13*a^2*b^ 3*c^3*d^3 + 19*a^3*b^2*c^2*d^4 + a^4*b*c*d^5 - 5*a^5*d^6)*x^3 + (4*b^5*c^6 + a*b^4*c^5*d - 17*a^2*b^3*c^4*d^2 + 5*a^3*b^2*c^3*d^3 + 17*a^4*b*c^2*d^4 - 10*a^5*c*d^5)*x^2 + (4*a*b^4*c^6 - 7*a^2*b^3*c^5*d - 3*a^3*b^2*c^4*d^2 + 11*a^4*b*c^3*d^3 - 5*a^5*c^2*d^4)*x)*sqrt(c)*log((d*x + 2*sqrt(d*x + c)* sqrt(c) + 2*c)/x) - 2*(3*a^2*b^3*c^6 - 9*a^3*b^2*c^5*d + 9*a^4*b*c^4*d^2 - 3*a^5*c^3*d^3 + 3*(2*a*b^4*c^4*d^2 - 3*a^2*b^3*c^3*d^3 + 11*a^3*b^2*c^2*d ^4 - 5*a^4*b*c*d^5)*x^3 + (12*a*b^4*c^5*d - 15*a^2*b^3*c^4*d^2 + 35*a^3*b^ 2*c^3*d^3 + 13*a^4*b*c^2*d^4 - 15*a^5*c*d^5)*x^2 + (6*a*b^4*c^6 - 3*a^2*b^ 3*c^5*d - 9*a^3*b^2*c^4*d^2 + 41*a^4*b*c^3*d^3 - 20*a^5*c^2*d^4)*x)*sqrt(d *x + c))/((a^3*b^4*c^7*d^2 - 3*a^4*b^3*c^6*d^3 + 3*a^5*b^2*c^5*d^4 - a^6*b *c^4*d^5)*x^4 + (2*a^3*b^4*c^8*d - 5*a^4*b^3*c^7*d^2 + 3*a^5*b^2*c^6*d^3 + a^6*b*c^5*d^4 - a^7*c^4*d^5)*x^3 + (a^3*b^4*c^9 - a^4*b^3*c^8*d - 3*a^5*b ^2*c^7*d^2 + 5*a^6*b*c^6*d^3 - 2*a^7*c^5*d^4)*x^2 + (a^4*b^3*c^9 - 3*a^5*b ^2*c^8*d + 3*a^6*b*c^7*d^2 - a^7*c^6*d^3)*x), -1/6*(6*((4*b^5*c^5*d^2 -...
\[ \int \frac {1}{x^2 (a+b x)^2 (c+d x)^{5/2}} \, dx=\int \frac {1}{x^{2} \left (a + b x\right )^{2} \left (c + d x\right )^{\frac {5}{2}}}\, dx \]
Exception generated. \[ \int \frac {1}{x^2 (a+b x)^2 (c+d x)^{5/2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for m ore detail
Time = 0.30 (sec) , antiderivative size = 470, normalized size of antiderivative = 1.70 \[ \int \frac {1}{x^2 (a+b x)^2 (c+d x)^{5/2}} \, dx=\frac {{\left (4 \, b^{5} c - 9 \, a b^{4} d\right )} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{{\left (a^{3} b^{3} c^{3} - 3 \, a^{4} b^{2} c^{2} d + 3 \, a^{5} b c d^{2} - a^{6} d^{3}\right )} \sqrt {-b^{2} c + a b d}} - \frac {2 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{4} c^{3} d - 2 \, \sqrt {d x + c} b^{4} c^{4} d - 3 \, {\left (d x + c\right )}^{\frac {3}{2}} a b^{3} c^{2} d^{2} + 4 \, \sqrt {d x + c} a b^{3} c^{3} d^{2} + 3 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{2} b^{2} c d^{3} - 6 \, \sqrt {d x + c} a^{2} b^{2} c^{2} d^{3} - {\left (d x + c\right )}^{\frac {3}{2}} a^{3} b d^{4} + 4 \, \sqrt {d x + c} a^{3} b c d^{4} - \sqrt {d x + c} a^{4} d^{5}}{{\left (a^{2} b^{3} c^{6} - 3 \, a^{3} b^{2} c^{5} d + 3 \, a^{4} b c^{4} d^{2} - a^{5} c^{3} d^{3}\right )} {\left ({\left (d x + c\right )}^{2} b - 2 \, {\left (d x + c\right )} b c + b c^{2} + {\left (d x + c\right )} a d - a c d\right )}} - \frac {2 \, {\left (12 \, {\left (d x + c\right )} b c d^{3} + b c^{2} d^{3} - 6 \, {\left (d x + c\right )} a d^{4} - a c d^{4}\right )}}{3 \, {\left (b^{3} c^{6} - 3 \, a b^{2} c^{5} d + 3 \, a^{2} b c^{4} d^{2} - a^{3} c^{3} d^{3}\right )} {\left (d x + c\right )}^{\frac {3}{2}}} - \frac {{\left (4 \, b c + 5 \, a d\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-c}}\right )}{a^{3} \sqrt {-c} c^{3}} \]
(4*b^5*c - 9*a*b^4*d)*arctan(sqrt(d*x + c)*b/sqrt(-b^2*c + a*b*d))/((a^3*b ^3*c^3 - 3*a^4*b^2*c^2*d + 3*a^5*b*c*d^2 - a^6*d^3)*sqrt(-b^2*c + a*b*d)) - (2*(d*x + c)^(3/2)*b^4*c^3*d - 2*sqrt(d*x + c)*b^4*c^4*d - 3*(d*x + c)^( 3/2)*a*b^3*c^2*d^2 + 4*sqrt(d*x + c)*a*b^3*c^3*d^2 + 3*(d*x + c)^(3/2)*a^2 *b^2*c*d^3 - 6*sqrt(d*x + c)*a^2*b^2*c^2*d^3 - (d*x + c)^(3/2)*a^3*b*d^4 + 4*sqrt(d*x + c)*a^3*b*c*d^4 - sqrt(d*x + c)*a^4*d^5)/((a^2*b^3*c^6 - 3*a^ 3*b^2*c^5*d + 3*a^4*b*c^4*d^2 - a^5*c^3*d^3)*((d*x + c)^2*b - 2*(d*x + c)* b*c + b*c^2 + (d*x + c)*a*d - a*c*d)) - 2/3*(12*(d*x + c)*b*c*d^3 + b*c^2* d^3 - 6*(d*x + c)*a*d^4 - a*c*d^4)/((b^3*c^6 - 3*a*b^2*c^5*d + 3*a^2*b*c^4 *d^2 - a^3*c^3*d^3)*(d*x + c)^(3/2)) - (4*b*c + 5*a*d)*arctan(sqrt(d*x + c )/sqrt(-c))/(a^3*sqrt(-c)*c^3)
Time = 3.50 (sec) , antiderivative size = 5736, normalized size of antiderivative = 20.71 \[ \int \frac {1}{x^2 (a+b x)^2 (c+d x)^{5/2}} \, dx=\text {Too large to display} \]
((10*d^3*(a*d - 2*b*c)*(c + d*x))/(3*(b*c^2 - a*c*d)^2) - (2*d^3)/(3*(b*c^ 2 - a*c*d)) + (d*(c + d*x)^2*(15*a^4*d^4 + 6*b^4*c^4 + 64*a^2*b^2*c^2*d^2 - 12*a*b^3*c^3*d - 58*a^3*b*c*d^3))/(3*a^2*(b*c^2 - a*c*d)^3) + (d*(a*d - 2*b*c)*(c + d*x)^3*(b^3*c^2 + 5*a^2*b*d^2 - a*b^2*c*d))/(a^2*(b*c^2 - a*c* d)^3))/(b*(c + d*x)^(7/2) + (b*c^2 - a*c*d)*(c + d*x)^(3/2) + (a*d - 2*b*c )*(c + d*x)^(5/2)) - (atan((a^19*c^15*d^19*(c + d*x)^(1/2)*125i - a^18*b*c ^16*d^18*(c + d*x)^(1/2)*1700i + a^3*b^16*c^31*d^3*(c + d*x)^(1/2)*420i - a^4*b^15*c^30*d^4*(c + d*x)^(1/2)*4515i + a^5*b^14*c^29*d^5*(c + d*x)^(1/2 )*20916i - a^6*b^13*c^28*d^6*(c + d*x)^(1/2)*52836i + a^7*b^12*c^27*d^7*(c + d*x)^(1/2)*71070i - a^8*b^11*c^26*d^8*(c + d*x)^(1/2)*19530i - a^9*b^10 *c^25*d^9*(c + d*x)^(1/2)*107740i + a^10*b^9*c^24*d^10*(c + d*x)^(1/2)*212 608i - a^11*b^8*c^23*d^11*(c + d*x)^(1/2)*184563i + a^12*b^7*c^22*d^12*(c + d*x)^(1/2)*40965i + a^13*b^6*c^21*d^13*(c + d*x)^(1/2)*91560i - a^14*b^5 *c^20*d^14*(c + d*x)^(1/2)*126720i + a^15*b^4*c^19*d^15*(c + d*x)^(1/2)*87 276i - a^16*b^3*c^18*d^16*(c + d*x)^(1/2)*37776i + a^17*b^2*c^17*d^17*(c + d*x)^(1/2)*10440i)/(c^7*(c^7)^(1/2)*(c^7*(c^7*(212608*a^10*b^9*d^10 - 107 740*a^9*b^10*c*d^9 + 420*a^3*b^16*c^7*d^3 - 4515*a^4*b^15*c^6*d^4 + 20916* a^5*b^14*c^5*d^5 - 52836*a^6*b^13*c^4*d^6 + 71070*a^7*b^12*c^3*d^7 - 19530 *a^8*b^11*c^2*d^8) + 10440*a^17*b^2*d^17 - 37776*a^16*b^3*c*d^16 - 184563* a^11*b^8*c^6*d^11 + 40965*a^12*b^7*c^5*d^12 + 91560*a^13*b^6*c^4*d^13 -...