Integrand size = 20, antiderivative size = 164 \[ \int \frac {1}{x^2 (a+b x)^2 \sqrt {c+d x}} \, dx=-\frac {b (2 b c-a d) \sqrt {c+d x}}{a^2 c (b c-a d) (a+b x)}-\frac {\sqrt {c+d x}}{a c x (a+b x)}+\frac {(4 b c+a d) \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^3 c^{3/2}}-\frac {b^{3/2} (4 b c-5 a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{a^3 (b c-a d)^{3/2}} \]
(a*d+4*b*c)*arctanh((d*x+c)^(1/2)/c^(1/2))/a^3/c^(3/2)-b^(3/2)*(-5*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)^(3/2)-b *(-a*d+2*b*c)*(d*x+c)^(1/2)/a^2/c/(-a*d+b*c)/(b*x+a)-(d*x+c)^(1/2)/a/c/x/( b*x+a)
Time = 0.64 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.91 \[ \int \frac {1}{x^2 (a+b x)^2 \sqrt {c+d x}} \, dx=\frac {-\frac {a \sqrt {c+d x} \left (-a b c+a^2 d-2 b^2 c x+a b d x\right )}{c (-b c+a d) x (a+b x)}-\frac {b^{3/2} (4 b c-5 a d) \arctan \left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{(-b c+a d)^{3/2}}+\frac {(4 b c+a d) \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{c^{3/2}}}{a^3} \]
(-((a*Sqrt[c + d*x]*(-(a*b*c) + a^2*d - 2*b^2*c*x + a*b*d*x))/(c*(-(b*c) + a*d)*x*(a + b*x))) - (b^(3/2)*(4*b*c - 5*a*d)*ArcTan[(Sqrt[b]*Sqrt[c + d* x])/Sqrt[-(b*c) + a*d]])/(-(b*c) + a*d)^(3/2) + ((4*b*c + a*d)*ArcTanh[Sqr t[c + d*x]/Sqrt[c]])/c^(3/2))/a^3
Time = 0.33 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.20, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {114, 27, 168, 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.
| \(\displaystyle \int \frac {1}{x^2 (a+b x)^2 \sqrt {c+d x}} \, dx\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle -\frac {\int \frac {4 b c+a d+3 b d x}{2 x (a+b x)^2 \sqrt {c+d x}}dx}{a c}-\frac {\sqrt {c+d x}}{a c x (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {4 b c+a d+3 b d x}{x (a+b x)^2 \sqrt {c+d x}}dx}{2 a c}-\frac {\sqrt {c+d x}}{a c x (a+b x)}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle -\frac {\frac {\int \frac {(b c-a d) (4 b c+a d)+b d (2 b c-a d) x}{x (a+b x) \sqrt {c+d x}}dx}{a (b c-a d)}+\frac {2 b \sqrt {c+d x} (2 b c-a d)}{a (a+b x) (b c-a d)}}{2 a c}-\frac {\sqrt {c+d x}}{a c x (a+b x)}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle -\frac {\frac {\frac {(b c-a d) (a d+4 b c) \int \frac {1}{x \sqrt {c+d x}}dx}{a}-\frac {b^2 c (4 b c-5 a d) \int \frac {1}{(a+b x) \sqrt {c+d x}}dx}{a}}{a (b c-a d)}+\frac {2 b \sqrt {c+d x} (2 b c-a d)}{a (a+b x) (b c-a d)}}{2 a c}-\frac {\sqrt {c+d x}}{a c x (a+b x)}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {\frac {\frac {2 (b c-a d) (a d+4 b c) \int \frac {1}{\frac {c+d x}{d}-\frac {c}{d}}d\sqrt {c+d x}}{a d}-\frac {2 b^2 c (4 b c-5 a d) \int \frac {1}{a+\frac {b (c+d x)}{d}-\frac {b c}{d}}d\sqrt {c+d x}}{a d}}{a (b c-a d)}+\frac {2 b \sqrt {c+d x} (2 b c-a d)}{a (a+b x) (b c-a d)}}{2 a c}-\frac {\sqrt {c+d x}}{a c x (a+b x)}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {\frac {\frac {2 b^{3/2} c (4 b c-5 a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{a \sqrt {b c-a d}}-\frac {2 (b c-a d) (a d+4 b c) \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a \sqrt {c}}}{a (b c-a d)}+\frac {2 b \sqrt {c+d x} (2 b c-a d)}{a (a+b x) (b c-a d)}}{2 a c}-\frac {\sqrt {c+d x}}{a c x (a+b x)}\) |
-(Sqrt[c + d*x]/(a*c*x*(a + b*x))) - ((2*b*(2*b*c - a*d)*Sqrt[c + d*x])/(a *(b*c - a*d)*(a + b*x)) + ((-2*(b*c - a*d)*(4*b*c + a*d)*ArcTanh[Sqrt[c + d*x]/Sqrt[c]])/(a*Sqrt[c]) + (2*b^(3/2)*c*(4*b*c - 5*a*d)*ArcTanh[(Sqrt[b] *Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(a*Sqrt[b*c - a*d]))/(a*(b*c - a*d)))/(2 *a*c)
3.5.76.3.1 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[(((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 = 0.64 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.98
| method | result | size |
| derivativedivides | \(2 d^{3} \left (\frac {-\frac {a \sqrt {d x +c}}{2 c x}+\frac {\left (a d +4 b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{2 c^{\frac {3}{2}}}}{a^{3} d^{3}}+\frac {b^{2} \left (\frac {a d \sqrt {d x +c}}{2 \left (a d -b c \right ) \left (\left (d x +c \right ) b +a d -b c \right )}+\frac {\left (5 a d -4 b c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \left (a d -b c \right ) \sqrt {\left (a d -b c \right ) b}}\right )}{a^{3} d^{3}}\right )\) | \(160\) |
| default | \(2 d^{3} \left (\frac {-\frac {a \sqrt {d x +c}}{2 c x}+\frac {\left (a d +4 b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{2 c^{\frac {3}{2}}}}{a^{3} d^{3}}+\frac {b^{2} \left (\frac {a d \sqrt {d x +c}}{2 \left (a d -b c \right ) \left (\left (d x +c \right ) b +a d -b c \right )}+\frac {\left (5 a d -4 b c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \left (a d -b c \right ) \sqrt {\left (a d -b c \right ) b}}\right )}{a^{3} d^{3}}\right )\) | \(160\) |
| risch | \(-\frac {\sqrt {d x +c}}{c \,a^{2} x}-\frac {d \left (-\frac {2 b^{2} c \left (\frac {a d \sqrt {d x +c}}{2 \left (a d -b c \right ) \left (\left (d x +c \right ) b +a d -b c \right )}+\frac {\left (5 a d -4 b c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \left (a d -b c \right ) \sqrt {\left (a d -b c \right ) b}}\right )}{a d}-\frac {\left (a d +4 b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{a d \sqrt {c}}\right )}{c \,a^{2}}\) | \(167\) |
| pseudoelliptic | \(-\frac {4 \left (x \,b^{2} c^{\frac {5}{2}} \left (b x +a \right ) \left (b c -\frac {5 a d}{4}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )-\frac {\left (c x \left (a d +4 b c \right ) \left (a d -b c \right ) \left (b x +a \right ) \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )+c^{\frac {3}{2}} \left (2 b^{2} c x +a \left (-d x +c \right ) b -a^{2} d \right ) a \sqrt {d x +c}\right ) \sqrt {\left (a d -b c \right ) b}}{4}\right )}{\sqrt {\left (a d -b c \right ) b}\, a^{3} \left (a d -b c \right ) \left (b x +a \right ) c^{\frac {5}{2}} x}\) | \(171\) |
2*d^3*(1/a^3/d^3*(-1/2*a/c*(d*x+c)^(1/2)/x+1/2*(a*d+4*b*c)/c^(3/2)*arctanh ((d*x+c)^(1/2)/c^(1/2)))+b^2/a^3/d^3*(1/2*a*d/(a*d-b*c)*(d*x+c)^(1/2)/((d* x+c)*b+a*d-b*c)+1/2*(5*a*d-4*b*c)/(a*d-b*c)/((a*d-b*c)*b)^(1/2)*arctan(b*( d*x+c)^(1/2)/((a*d-b*c)*b)^(1/2))))
Time = 0.32 (sec) , antiderivative size = 1164, normalized size of antiderivative = 7.10 \[ \int \frac {1}{x^2 (a+b x)^2 \sqrt {c+d x}} \, dx=\text {Too large to display} \]
[1/2*(((4*b^3*c^3 - 5*a*b^2*c^2*d)*x^2 + (4*a*b^2*c^3 - 5*a^2*b*c^2*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^3*c^2 - 3*a*b^2*c*d - a^2*b*d^2)* x^2 + (4*a*b^2*c^2 - 3*a^2*b*c*d - a^3*d^2)*x)*sqrt(c)*log((d*x + 2*sqrt(d *x + c)*sqrt(c) + 2*c)/x) - 2*(a^2*b*c^2 - a^3*c*d + (2*a*b^2*c^2 - a^2*b* c*d)*x)*sqrt(d*x + c))/((a^3*b^2*c^3 - a^4*b*c^2*d)*x^2 + (a^4*b*c^3 - a^5 *c^2*d)*x), -1/2*(2*((4*b^3*c^3 - 5*a*b^2*c^2*d)*x^2 + (4*a*b^2*c^3 - 5*a^ 2*b*c^2*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^3*c^2 - 3*a*b^2*c*d - a^2*b*d^2)*x^ 2 + (4*a*b^2*c^2 - 3*a^2*b*c*d - a^3*d^2)*x)*sqrt(c)*log((d*x + 2*sqrt(d*x + c)*sqrt(c) + 2*c)/x) + 2*(a^2*b*c^2 - a^3*c*d + (2*a*b^2*c^2 - a^2*b*c* d)*x)*sqrt(d*x + c))/((a^3*b^2*c^3 - a^4*b*c^2*d)*x^2 + (a^4*b*c^3 - a^5*c ^2*d)*x), -1/2*(2*((4*b^3*c^2 - 3*a*b^2*c*d - a^2*b*d^2)*x^2 + (4*a*b^2*c^ 2 - 3*a^2*b*c*d - a^3*d^2)*x)*sqrt(-c)*arctan(sqrt(d*x + c)*sqrt(-c)/c) - ((4*b^3*c^3 - 5*a*b^2*c^2*d)*x^2 + (4*a*b^2*c^3 - 5*a^2*b*c^2*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)) + 2*(a^2*b*c^2 - a^3*c*d + (2*a*b^2*c^2 - a^2*b *c*d)*x)*sqrt(d*x + c))/((a^3*b^2*c^3 - a^4*b*c^2*d)*x^2 + (a^4*b*c^3 - a^ 5*c^2*d)*x), -(((4*b^3*c^3 - 5*a*b^2*c^2*d)*x^2 + (4*a*b^2*c^3 - 5*a^2*b*c ^2*d)*x)*sqrt(-b/(b*c - a*d))*arctan(-(b*c - a*d)*sqrt(d*x + c)*sqrt(-b...
\[ \int \frac {1}{x^2 (a+b x)^2 \sqrt {c+d x}} \, dx=\int \frac {1}{x^{2} \left (a + b x\right )^{2} \sqrt {c + d x}}\, dx \]
Exception generated. \[ \int \frac {1}{x^2 (a+b x)^2 \sqrt {c+d x}} \, dx=\text {Exception raised: ValueError} \]
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 = 0.28 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.44 \[ \int \frac {1}{x^2 (a+b x)^2 \sqrt {c+d x}} \, dx=\frac {{\left (4 \, b^{3} c - 5 \, a b^{2} d\right )} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{{\left (a^{3} b c - a^{4} d\right )} \sqrt {-b^{2} c + a b d}} - \frac {2 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{2} c d - 2 \, \sqrt {d x + c} b^{2} c^{2} d - {\left (d x + c\right )}^{\frac {3}{2}} a b d^{2} + 2 \, \sqrt {d x + c} a b c d^{2} - \sqrt {d x + c} a^{2} d^{3}}{{\left (a^{2} b c^{2} - a^{3} c d\right )} {\left ({\left (d x + c\right )}^{2} b - 2 \, {\left (d x + c\right )} b c + b c^{2} + {\left (d x + c\right )} a d - a c d\right )}} - \frac {{\left (4 \, b c + a d\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-c}}\right )}{a^{3} \sqrt {-c} c} \]
(4*b^3*c - 5*a*b^2*d)*arctan(sqrt(d*x + c)*b/sqrt(-b^2*c + a*b*d))/((a^3*b *c - a^4*d)*sqrt(-b^2*c + a*b*d)) - (2*(d*x + c)^(3/2)*b^2*c*d - 2*sqrt(d* x + c)*b^2*c^2*d - (d*x + c)^(3/2)*a*b*d^2 + 2*sqrt(d*x + c)*a*b*c*d^2 - s qrt(d*x + c)*a^2*d^3)/((a^2*b*c^2 - a^3*c*d)*((d*x + c)^2*b - 2*(d*x + c)* b*c + b*c^2 + (d*x + c)*a*d - a*c*d)) - (4*b*c + a*d)*arctan(sqrt(d*x + c) /sqrt(-c))/(a^3*sqrt(-c)*c)
Time = 1.45 (sec) , antiderivative size = 3784, normalized size of antiderivative = 23.07 \[ \int \frac {1}{x^2 (a+b x)^2 \sqrt {c+d x}} \, dx=\text {Too large to display} \]
(((c + d*x)^(1/2)*(a^2*d^3 + 2*b^2*c^2*d - 2*a*b*c*d^2))/(a^2*(b*c^2 - a*c *d)) + (b*(a*d^2 - 2*b*c*d)*(c + d*x)^(3/2))/(a^2*(b*c^2 - a*c*d)))/((a*d - 2*b*c)*(c + d*x) + b*(c + d*x)^2 + b*c^2 - a*c*d) + (atan(((((2*(c + d*x )^(1/2)*(a^4*b^3*d^6 + 32*b^7*c^4*d^2 - 64*a*b^6*c^3*d^3 + 6*a^3*b^4*c*d^5 + 26*a^2*b^5*c^2*d^4))/(a^4*b^2*c^4 + a^6*c^2*d^2 - 2*a^5*b*c^3*d) - ((-b ^3*(a*d - b*c)^3)^(1/2)*((4*a^9*b^2*c*d^6 + 8*a^6*b^5*c^4*d^3 - 16*a^7*b^4 *c^3*d^4 + 4*a^8*b^3*c^2*d^5)/(a^6*b^2*c^4 + a^8*c^2*d^2 - 2*a^7*b*c^3*d) + ((-b^3*(a*d - b*c)^3)^(1/2)*(5*a*d - 4*b*c)*(c + d*x)^(1/2)*(8*a^6*b^5*c ^5*d^2 - 20*a^7*b^4*c^4*d^3 + 16*a^8*b^3*c^3*d^4 - 4*a^9*b^2*c^2*d^5))/((a ^4*b^2*c^4 + a^6*c^2*d^2 - 2*a^5*b*c^3*d)*(a^6*d^3 - a^3*b^3*c^3 + 3*a^4*b ^2*c^2*d - 3*a^5*b*c*d^2)))*(5*a*d - 4*b*c))/(2*(a^6*d^3 - a^3*b^3*c^3 + 3 *a^4*b^2*c^2*d - 3*a^5*b*c*d^2)))*(-b^3*(a*d - b*c)^3)^(1/2)*(5*a*d - 4*b* c)*1i)/(2*(a^6*d^3 - a^3*b^3*c^3 + 3*a^4*b^2*c^2*d - 3*a^5*b*c*d^2)) + ((( 2*(c + d*x)^(1/2)*(a^4*b^3*d^6 + 32*b^7*c^4*d^2 - 64*a*b^6*c^3*d^3 + 6*a^3 *b^4*c*d^5 + 26*a^2*b^5*c^2*d^4))/(a^4*b^2*c^4 + a^6*c^2*d^2 - 2*a^5*b*c^3 *d) + ((-b^3*(a*d - b*c)^3)^(1/2)*((4*a^9*b^2*c*d^6 + 8*a^6*b^5*c^4*d^3 - 16*a^7*b^4*c^3*d^4 + 4*a^8*b^3*c^2*d^5)/(a^6*b^2*c^4 + a^8*c^2*d^2 - 2*a^7 *b*c^3*d) - ((-b^3*(a*d - b*c)^3)^(1/2)*(5*a*d - 4*b*c)*(c + d*x)^(1/2)*(8 *a^6*b^5*c^5*d^2 - 20*a^7*b^4*c^4*d^3 + 16*a^8*b^3*c^3*d^4 - 4*a^9*b^2*c^2 *d^5))/((a^4*b^2*c^4 + a^6*c^2*d^2 - 2*a^5*b*c^3*d)*(a^6*d^3 - a^3*b^3*...