Integrand size = 20, antiderivative size = 216 \[ \int \frac {1}{x^2 (a+b x)^2 (c+d x)^{3/2}} \, dx=-\frac {d \left (2 b^2 c^2-2 a b c d+3 a^2 d^2\right )}{a^2 c^2 (b c-a d)^2 \sqrt {c+d x}}-\frac {b (2 b c-a d)}{a^2 c (b c-a d) (a+b x) \sqrt {c+d x}}-\frac {1}{a c x (a+b x) \sqrt {c+d x}}+\frac {(4 b c+3 a d) \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^3 c^{5/2}}-\frac {b^{5/2} (4 b c-7 a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{a^3 (b c-a d)^{5/2}} \]
(3*a*d+4*b*c)*arctanh((d*x+c)^(1/2)/c^(1/2))/a^3/c^(5/2)-b^(5/2)*(-7*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)^(5/2) -d*(3*a^2*d^2-2*a*b*c*d+2*b^2*c^2)/a^2/c^2/(-a*d+b*c)^2/(d*x+c)^(1/2)-b*(- a*d+2*b*c)/a^2/c/(-a*d+b*c)/(b*x+a)/(d*x+c)^(1/2)-1/a/c/x/(b*x+a)/(d*x+c)^ (1/2)
Time = 0.95 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.94 \[ \int \frac {1}{x^2 (a+b x)^2 (c+d x)^{3/2}} \, dx=\frac {-\frac {a \left (2 b^3 c^2 x (c+d x)+a^3 d^2 (c+3 d x)+a b^2 c \left (c^2-c d x-2 d^2 x^2\right )+a^2 b d \left (-2 c^2-c d x+3 d^2 x^2\right )\right )}{c^2 (b c-a d)^2 x (a+b x) \sqrt {c+d x}}+\frac {b^{5/2} (4 b c-7 a d) \arctan \left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{(-b c+a d)^{5/2}}+\frac {(4 b c+3 a d) \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{c^{5/2}}}{a^3} \]
(-((a*(2*b^3*c^2*x*(c + d*x) + a^3*d^2*(c + 3*d*x) + a*b^2*c*(c^2 - c*d*x - 2*d^2*x^2) + a^2*b*d*(-2*c^2 - c*d*x + 3*d^2*x^2)))/(c^2*(b*c - a*d)^2*x *(a + b*x)*Sqrt[c + d*x])) + (b^(5/2)*(4*b*c - 7*a*d)*ArcTan[(Sqrt[b]*Sqrt [c + d*x])/Sqrt[-(b*c) + a*d]])/(-(b*c) + a*d)^(5/2) + ((4*b*c + 3*a*d)*Ar cTanh[Sqrt[c + d*x]/Sqrt[c]])/c^(5/2))/a^3
Time = 0.42 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.23, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {114, 27, 168, 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.
| \(\displaystyle \int \frac {1}{x^2 (a+b x)^2 (c+d x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle -\frac {\int \frac {4 b c+3 a d+5 b d x}{2 x (a+b x)^2 (c+d x)^{3/2}}dx}{a c}-\frac {1}{a c x (a+b x) \sqrt {c+d x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {4 b c+3 a d+5 b d x}{x (a+b x)^2 (c+d x)^{3/2}}dx}{2 a c}-\frac {1}{a c x (a+b x) \sqrt {c+d x}}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle -\frac {\frac {\int \frac {(b c-a d) (4 b c+3 a d)+3 b d (2 b c-a d) x}{x (a+b x) (c+d x)^{3/2}}dx}{a (b c-a d)}+\frac {2 b (2 b c-a d)}{a (a+b x) \sqrt {c+d x} (b c-a d)}}{2 a c}-\frac {1}{a c x (a+b x) \sqrt {c+d x}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {\frac {\frac {2 d \left (3 a^2 d^2-2 a b c d+2 b^2 c^2\right )}{c \sqrt {c+d x} (b c-a d)}-\frac {2 \int -\frac {(4 b c+3 a d) (b c-a d)^2+b d \left (2 b^2 c^2-2 a b d c+3 a^2 d^2\right ) x}{2 x (a+b x) \sqrt {c+d x}}dx}{c (b c-a d)}}{a (b c-a d)}+\frac {2 b (2 b c-a d)}{a (a+b x) \sqrt {c+d x} (b c-a d)}}{2 a c}-\frac {1}{a c x (a+b x) \sqrt {c+d x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\frac {\int \frac {(4 b c+3 a d) (b c-a d)^2+b d \left (2 b^2 c^2-2 a b d c+3 a^2 d^2\right ) x}{x (a+b x) \sqrt {c+d x}}dx}{c (b c-a d)}+\frac {2 d \left (3 a^2 d^2-2 a b c d+2 b^2 c^2\right )}{c \sqrt {c+d x} (b c-a d)}}{a (b c-a d)}+\frac {2 b (2 b c-a d)}{a (a+b x) \sqrt {c+d x} (b c-a d)}}{2 a c}-\frac {1}{a c x (a+b x) \sqrt {c+d x}}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle -\frac {\frac {\frac {\frac {(b c-a d)^2 (3 a d+4 b c) \int \frac {1}{x \sqrt {c+d x}}dx}{a}-\frac {b^3 c^2 (4 b c-7 a d) \int \frac {1}{(a+b x) \sqrt {c+d x}}dx}{a}}{c (b c-a d)}+\frac {2 d \left (3 a^2 d^2-2 a b c d+2 b^2 c^2\right )}{c \sqrt {c+d x} (b c-a d)}}{a (b c-a d)}+\frac {2 b (2 b c-a d)}{a (a+b x) \sqrt {c+d x} (b c-a d)}}{2 a c}-\frac {1}{a c x (a+b x) \sqrt {c+d x}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {\frac {\frac {\frac {2 (b c-a d)^2 (3 a d+4 b c) \int \frac {1}{\frac {c+d x}{d}-\frac {c}{d}}d\sqrt {c+d x}}{a d}-\frac {2 b^3 c^2 (4 b c-7 a d) \int \frac {1}{a+\frac {b (c+d x)}{d}-\frac {b c}{d}}d\sqrt {c+d x}}{a d}}{c (b c-a d)}+\frac {2 d \left (3 a^2 d^2-2 a b c d+2 b^2 c^2\right )}{c \sqrt {c+d x} (b c-a d)}}{a (b c-a d)}+\frac {2 b (2 b c-a d)}{a (a+b x) \sqrt {c+d x} (b c-a d)}}{2 a c}-\frac {1}{a c x (a+b x) \sqrt {c+d x}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {\frac {\frac {2 d \left (3 a^2 d^2-2 a b c d+2 b^2 c^2\right )}{c \sqrt {c+d x} (b c-a d)}+\frac {\frac {2 b^{5/2} c^2 (4 b c-7 a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{a \sqrt {b c-a d}}-\frac {2 (b c-a d)^2 (3 a d+4 b c) \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a \sqrt {c}}}{c (b c-a d)}}{a (b c-a d)}+\frac {2 b (2 b c-a d)}{a (a+b x) \sqrt {c+d x} (b c-a d)}}{2 a c}-\frac {1}{a c x (a+b x) \sqrt {c+d x}}\) |
-(1/(a*c*x*(a + b*x)*Sqrt[c + d*x])) - ((2*b*(2*b*c - a*d))/(a*(b*c - a*d) *(a + b*x)*Sqrt[c + d*x]) + ((2*d*(2*b^2*c^2 - 2*a*b*c*d + 3*a^2*d^2))/(c* (b*c - a*d)*Sqrt[c + d*x]) + ((-2*(b*c - a*d)^2*(4*b*c + 3*a*d)*ArcTanh[Sq rt[c + d*x]/Sqrt[c]])/(a*Sqrt[c]) + (2*b^(5/2)*c^2*(4*b*c - 7*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)))/(a*(b*c - a*d)))/(2*a*c)
3.5.77.3.1 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_))^(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 = 0.69 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.81
| method | result | size |
| derivativedivides | \(2 d^{3} \left (-\frac {1}{c^{2} \left (a d -b c \right )^{2} \sqrt {d x +c}}+\frac {-\frac {a \sqrt {d x +c}}{2 x}+\frac {\left (3 a d +4 b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{2 \sqrt {c}}}{c^{2} a^{3} d^{3}}-\frac {b^{3} \left (\frac {\sqrt {d x +c}\, a d}{2 \left (d x +c \right ) b +2 a d -2 b c}+\frac {\left (7 a d -4 b c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \sqrt {\left (a d -b c \right ) b}}\right )}{a^{3} d^{3} \left (a d -b c \right )^{2}}\right )\) | \(174\) |
| default | \(2 d^{3} \left (-\frac {1}{c^{2} \left (a d -b c \right )^{2} \sqrt {d x +c}}+\frac {-\frac {a \sqrt {d x +c}}{2 x}+\frac {\left (3 a d +4 b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{2 \sqrt {c}}}{c^{2} a^{3} d^{3}}-\frac {b^{3} \left (\frac {\sqrt {d x +c}\, a d}{2 \left (d x +c \right ) b +2 a d -2 b c}+\frac {\left (7 a d -4 b c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \sqrt {\left (a d -b c \right ) b}}\right )}{a^{3} d^{3} \left (a d -b c \right )^{2}}\right )\) | \(174\) |
| risch | \(-\frac {\sqrt {d x +c}}{c^{2} a^{2} x}-\frac {d \left (\frac {2 b^{3} c^{2} \left (\frac {\sqrt {d x +c}\, a d}{2 \left (d x +c \right ) b +2 a d -2 b c}+\frac {\left (7 a d -4 b c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \sqrt {\left (a d -b c \right ) b}}\right )}{\left (a d -b c \right )^{2} a d}-\frac {\left (3 a d +4 b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{a d \sqrt {c}}+\frac {2 a^{2} d^{2}}{\left (a d -b c \right )^{2} \sqrt {d x +c}}\right )}{a^{2} c^{2}}\) | \(185\) |
| pseudoelliptic | \(-\frac {-4 x \left (b c -\frac {7 a d}{4}\right ) b^{3} \sqrt {d x +c}\, \left (b x +a \right ) c^{\frac {9}{2}} \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )+\left (-3 x \left (a d -b c \right )^{2} \sqrt {d x +c}\, \left (b x +a \right ) c^{2} \left (a d +\frac {4 b c}{3}\right ) \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )+\left (\left (2 c^{2} d \,x^{2}+2 c^{3} x \right ) b^{3}+a c \left (d x +c \right ) \left (-2 d x +c \right ) b^{2}-2 d \left (-d x +c \right ) a^{2} \left (\frac {3 d x}{2}+c \right ) b +a^{3} d^{2} \left (3 d x +c \right )\right ) a \,c^{\frac {5}{2}}\right ) \sqrt {\left (a d -b c \right ) b}}{\sqrt {d x +c}\, \sqrt {\left (a d -b c \right ) b}\, a^{3} \left (b x +a \right ) \left (a d -b c \right )^{2} x \,c^{\frac {9}{2}}}\) | \(237\) |
2*d^3*(-1/c^2/(a*d-b*c)^2/(d*x+c)^(1/2)+1/c^2/a^3/d^3*(-1/2*a*(d*x+c)^(1/2 )/x+1/2*(3*a*d+4*b*c)/c^(1/2)*arctanh((d*x+c)^(1/2)/c^(1/2)))-b^3/a^3/d^3/ (a*d-b*c)^2*(1/2*a*d*(d*x+c)^(1/2)/((d*x+c)*b+a*d-b*c)+1/2*(7*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. 569 vs. \(2 (194) = 388\).
Time = 0.57 (sec) , antiderivative size = 2312, normalized size of antiderivative = 10.70 \[ \int \frac {1}{x^2 (a+b x)^2 (c+d x)^{3/2}} \, dx=\text {Too large to display} \]
[-1/2*(((4*b^4*c^4*d - 7*a*b^3*c^3*d^2)*x^3 + (4*b^4*c^5 - 3*a*b^3*c^4*d - 7*a^2*b^2*c^3*d^2)*x^2 + (4*a*b^3*c^5 - 7*a^2*b^2*c^4*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)) - ((4*b^4*c^3*d - 5*a*b^3*c^2*d^2 - 2*a^2*b^2*c*d^3 + 3*a^3*b*d^4)*x^3 + (4*b^4*c^4 - a*b^3*c^3*d - 7*a^2*b^2*c^2*d^2 + a^3*b*c* d^3 + 3*a^4*d^4)*x^2 + (4*a*b^3*c^4 - 5*a^2*b^2*c^3*d - 2*a^3*b*c^2*d^2 + 3*a^4*c*d^3)*x)*sqrt(c)*log((d*x + 2*sqrt(d*x + c)*sqrt(c) + 2*c)/x) + 2*( a^2*b^2*c^4 - 2*a^3*b*c^3*d + a^4*c^2*d^2 + (2*a*b^3*c^3*d - 2*a^2*b^2*c^2 *d^2 + 3*a^3*b*c*d^3)*x^2 + (2*a*b^3*c^4 - a^2*b^2*c^3*d - a^3*b*c^2*d^2 + 3*a^4*c*d^3)*x)*sqrt(d*x + c))/((a^3*b^3*c^5*d - 2*a^4*b^2*c^4*d^2 + a^5* b*c^3*d^3)*x^3 + (a^3*b^3*c^6 - a^4*b^2*c^5*d - a^5*b*c^4*d^2 + a^6*c^3*d^ 3)*x^2 + (a^4*b^2*c^6 - 2*a^5*b*c^5*d + a^6*c^4*d^2)*x), -1/2*(2*((4*b^4*c ^4*d - 7*a*b^3*c^3*d^2)*x^3 + (4*b^4*c^5 - 3*a*b^3*c^4*d - 7*a^2*b^2*c^3*d ^2)*x^2 + (4*a*b^3*c^5 - 7*a^2*b^2*c^4*d)*x)*sqrt(-b/(b*c - a*d))*arctan(- (b*c - a*d)*sqrt(d*x + c)*sqrt(-b/(b*c - a*d))/(b*d*x + b*c)) - ((4*b^4*c^ 3*d - 5*a*b^3*c^2*d^2 - 2*a^2*b^2*c*d^3 + 3*a^3*b*d^4)*x^3 + (4*b^4*c^4 - a*b^3*c^3*d - 7*a^2*b^2*c^2*d^2 + a^3*b*c*d^3 + 3*a^4*d^4)*x^2 + (4*a*b^3* c^4 - 5*a^2*b^2*c^3*d - 2*a^3*b*c^2*d^2 + 3*a^4*c*d^3)*x)*sqrt(c)*log((d*x + 2*sqrt(d*x + c)*sqrt(c) + 2*c)/x) + 2*(a^2*b^2*c^4 - 2*a^3*b*c^3*d + a^ 4*c^2*d^2 + (2*a*b^3*c^3*d - 2*a^2*b^2*c^2*d^2 + 3*a^3*b*c*d^3)*x^2 + (...
\[ \int \frac {1}{x^2 (a+b x)^2 (c+d x)^{3/2}} \, dx=\int \frac {1}{x^{2} \left (a + b x\right )^{2} \left (c + d x\right )^{\frac {3}{2}}}\, dx \]
Exception generated. \[ \int \frac {1}{x^2 (a+b x)^2 (c+d x)^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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 = 0.29 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.56 \[ \int \frac {1}{x^2 (a+b x)^2 (c+d x)^{3/2}} \, dx=\frac {{\left (4 \, b^{4} c - 7 \, a b^{3} d\right )} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{{\left (a^{3} b^{2} c^{2} - 2 \, a^{4} b c d + a^{5} d^{2}\right )} \sqrt {-b^{2} c + a b d}} - \frac {2 \, {\left (d x + c\right )}^{2} b^{3} c^{2} d - 2 \, {\left (d x + c\right )} b^{3} c^{3} d - 2 \, {\left (d x + c\right )}^{2} a b^{2} c d^{2} + 3 \, {\left (d x + c\right )} a b^{2} c^{2} d^{2} + 3 \, {\left (d x + c\right )}^{2} a^{2} b d^{3} - 7 \, {\left (d x + c\right )} a^{2} b c d^{3} + 2 \, a^{2} b c^{2} d^{3} + 3 \, {\left (d x + c\right )} a^{3} d^{4} - 2 \, a^{3} c d^{4}}{{\left (a^{2} b^{2} c^{4} - 2 \, a^{3} b c^{3} d + a^{4} c^{2} d^{2}\right )} {\left ({\left (d x + c\right )}^{\frac {5}{2}} b - 2 \, {\left (d x + c\right )}^{\frac {3}{2}} b c + \sqrt {d x + c} b c^{2} + {\left (d x + c\right )}^{\frac {3}{2}} a d - \sqrt {d x + c} a c d\right )}} - \frac {{\left (4 \, b c + 3 \, a d\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-c}}\right )}{a^{3} \sqrt {-c} c^{2}} \]
(4*b^4*c - 7*a*b^3*d)*arctan(sqrt(d*x + c)*b/sqrt(-b^2*c + a*b*d))/((a^3*b ^2*c^2 - 2*a^4*b*c*d + a^5*d^2)*sqrt(-b^2*c + a*b*d)) - (2*(d*x + c)^2*b^3 *c^2*d - 2*(d*x + c)*b^3*c^3*d - 2*(d*x + c)^2*a*b^2*c*d^2 + 3*(d*x + c)*a *b^2*c^2*d^2 + 3*(d*x + c)^2*a^2*b*d^3 - 7*(d*x + c)*a^2*b*c*d^3 + 2*a^2*b *c^2*d^3 + 3*(d*x + c)*a^3*d^4 - 2*a^3*c*d^4)/((a^2*b^2*c^4 - 2*a^3*b*c^3* d + a^4*c^2*d^2)*((d*x + c)^(5/2)*b - 2*(d*x + c)^(3/2)*b*c + sqrt(d*x + c )*b*c^2 + (d*x + c)^(3/2)*a*d - sqrt(d*x + c)*a*c*d)) - (4*b*c + 3*a*d)*ar ctan(sqrt(d*x + c)/sqrt(-c))/(a^3*sqrt(-c)*c^2)
Time = 2.64 (sec) , antiderivative size = 4234, normalized size of antiderivative = 19.60 \[ \int \frac {1}{x^2 (a+b x)^2 (c+d x)^{3/2}} \, dx=\text {Too large to display} \]
(atan((((-b^5*(a*d - b*c)^5)^(1/2)*((c + d*x)^(1/2)*(64*a^6*b^15*c^18*d^2 - 576*a^7*b^14*c^17*d^3 + 2228*a^8*b^13*c^16*d^4 - 4768*a^9*b^12*c^15*d^5 + 5960*a^10*b^11*c^14*d^6 - 3976*a^11*b^10*c^13*d^7 + 578*a^12*b^9*c^12*d^ 8 + 1004*a^13*b^8*c^11*d^9 - 442*a^14*b^7*c^10*d^10 - 320*a^15*b^6*c^9*d^1 1 + 362*a^16*b^5*c^8*d^12 - 132*a^17*b^4*c^7*d^13 + 18*a^18*b^3*c^6*d^14) + ((-b^5*(a*d - b*c)^5)^(1/2)*(7*a*d - 4*b*c)*(8*a^10*b^13*c^19*d^3 - 76*a ^11*b^12*c^18*d^4 + 300*a^12*b^11*c^17*d^5 - 612*a^13*b^10*c^16*d^6 + 576* a^14*b^9*c^15*d^7 + 168*a^15*b^8*c^14*d^8 - 1176*a^16*b^7*c^13*d^9 + 1560* a^17*b^6*c^12*d^10 - 1128*a^18*b^5*c^11*d^11 + 484*a^19*b^4*c^10*d^12 - 11 6*a^20*b^3*c^9*d^13 + 12*a^21*b^2*c^8*d^14 - ((-b^5*(a*d - b*c)^5)^(1/2)*( 7*a*d - 4*b*c)*(c + d*x)^(1/2)*(16*a^12*b^13*c^21*d^2 - 168*a^13*b^12*c^20 *d^3 + 800*a^14*b^11*c^19*d^4 - 2280*a^15*b^10*c^18*d^5 + 4320*a^16*b^9*c^ 17*d^6 - 5712*a^17*b^8*c^16*d^7 + 5376*a^18*b^7*c^15*d^8 - 3600*a^19*b^6*c ^14*d^9 + 1680*a^20*b^5*c^13*d^10 - 520*a^21*b^4*c^12*d^11 + 96*a^22*b^3*c ^11*d^12 - 8*a^23*b^2*c^10*d^13))/(2*(a^8*d^5 - a^3*b^5*c^5 + 5*a^4*b^4*c^ 4*d - 10*a^5*b^3*c^3*d^2 + 10*a^6*b^2*c^2*d^3 - 5*a^7*b*c*d^4))))/(2*(a^8* d^5 - a^3*b^5*c^5 + 5*a^4*b^4*c^4*d - 10*a^5*b^3*c^3*d^2 + 10*a^6*b^2*c^2* d^3 - 5*a^7*b*c*d^4)))*(7*a*d - 4*b*c)*1i)/(2*(a^8*d^5 - a^3*b^5*c^5 + 5*a ^4*b^4*c^4*d - 10*a^5*b^3*c^3*d^2 + 10*a^6*b^2*c^2*d^3 - 5*a^7*b*c*d^4)) + ((-b^5*(a*d - b*c)^5)^(1/2)*((c + d*x)^(1/2)*(64*a^6*b^15*c^18*d^2 - 5...