Integrand size = 21, antiderivative size = 146 \[ \int \frac {1}{x^3 \left (a+\frac {b}{c+d x^2}\right )^{3/2}} \, dx=\frac {3 b d}{2 (b+a c)^2 \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}-\frac {c+d x^2}{2 (b+a c) x^2 \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}-\frac {3 b \sqrt {c} d \text {arctanh}\left (\frac {\sqrt {c} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{\sqrt {b+a c}}\right )}{2 (b+a c)^{5/2}} \]
-3/2*b*d*arctanh(c^(1/2)*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)/(a*c+b)^(1/2))* c^(1/2)/(a*c+b)^(5/2)+3/2*b*d/(a*c+b)^2/((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)+ 1/2*(-d*x^2-c)/(a*c+b)/x^2/((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)
Time = 0.24 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.01 \[ \int \frac {1}{x^3 \left (a+\frac {b}{c+d x^2}\right )^{3/2}} \, dx=-\frac {\left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \left (b \left (c-2 d x^2\right )+a c \left (c+d x^2\right )\right )}{2 (b+a c)^2 x^2 \left (b+a \left (c+d x^2\right )\right )}+\frac {3 b \sqrt {c} d \arctan \left (\frac {\sqrt {c} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{\sqrt {-b-a c}}\right )}{2 (-b-a c)^{5/2}} \]
-1/2*((c + d*x^2)*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(b*(c - 2*d*x^2) + a*c*(c + d*x^2)))/((b + a*c)^2*x^2*(b + a*(c + d*x^2))) + (3*b*Sqrt[c]*d* ArcTan[(Sqrt[c]*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)])/Sqrt[-b - a*c]])/(2 *(-b - a*c)^(5/2))
Time = 0.36 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.77, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2057, 2053, 2052, 253, 264, 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.
| \(\displaystyle \int \frac {1}{x^3 \left (a+\frac {b}{c+d x^2}\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 2057 |
| \(\displaystyle \int \frac {1}{x^3 \left (\frac {a c+a d x^2+b}{c+d x^2}\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 2053 |
| \(\displaystyle \frac {1}{2} \int \frac {1}{x^4 \left (\frac {a d x^2+b+a c}{d x^2+c}\right )^{3/2}}dx^2\) |
| \(\Big \downarrow \) 2052 |
| \(\displaystyle -b d \int \frac {1}{x^4 \left (-c x^4+b+a c\right )^2}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle -b d \left (\frac {3 \int \frac {1}{x^4 \left (-c x^4+b+a c\right )}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}}{2 (a c+b)}+\frac {1}{2 x^2 (a c+b) \left (a c+b-c x^4\right )}\right )\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle -b d \left (\frac {3 \left (\frac {c \int \frac {1}{-c x^4+b+a c}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}}{a c+b}-\frac {1}{x^2 (a c+b)}\right )}{2 (a c+b)}+\frac {1}{2 x^2 (a c+b) \left (a c+b-c x^4\right )}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -b d \left (\frac {3 \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{\sqrt {a c+b}}\right )}{(a c+b)^{3/2}}-\frac {1}{x^2 (a c+b)}\right )}{2 (a c+b)}+\frac {1}{2 x^2 (a c+b) \left (a c+b-c x^4\right )}\right )\) |
-(b*d*(1/(2*(b + a*c)*x^2*(b + a*c - c*x^4)) + (3*(-(1/((b + a*c)*x^2)) + (Sqrt[c]*ArcTanh[(Sqrt[c]*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)])/Sqrt[b + a*c]])/(b + a*c)^(3/2)))/(2*(b + a*c))))
3.4.59.3.1 Defintions of rubi rules used
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[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(c*x )^(m + 1))*((a + b*x^2)^(p + 1)/(2*a*c*(p + 1))), x] + Simp[(m + 2*p + 3)/( 2*a*(p + 1)) Int[(c*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, m }, x] && LtQ[p, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c ^2*(m + 1))) Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p }, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[(x_)^(m_.)*(((e_.)*((a_.) + (b_.)*(x_)))/((c_) + (d_.)*(x_)))^(p_), x_S ymbol] :> With[{q = Denominator[p]}, Simp[q*e*(b*c - a*d) Subst[Int[x^(q* (p + 1) - 1)*(((-a)*e + c*x^q)^m/(b*e - d*x^q)^(m + 2)), x], x, (e*((a + b* x)/(c + d*x)))^(1/q)], x]] /; FreeQ[{a, b, c, d, e, m}, x] && FractionQ[p] && IntegerQ[m]
Int[(x_)^(m_.)*(((e_.)*((a_.) + (b_.)*(x_)^(n_.)))/((c_) + (d_.)*(x_)^(n_.) ))^(p_), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(e*( (a + b*x)/(c + d*x)))^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Int[(u_.)*((a_) + (b_.)/((c_) + (d_.)*(x_)^(n_)))^(p_), x_Symbol] :> Int[u* ((b + a*c + a*d*x^n)/(c + d*x^n))^p, x] /; FreeQ[{a, b, c, d, n, p}, x]
Time = 0.21 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.67
| method | result | size |
| risch | \(-\frac {c \left (a d \,x^{2}+a c +b \right )}{2 \left (a c +b \right )^{2} x^{2} \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}+\frac {d b \left (-\frac {3 c \ln \left (\frac {2 a \,c^{2}+2 b c +\left (2 a c d +b d \right ) x^{2}+2 \sqrt {a \,c^{2}+b c}\, \sqrt {a \,c^{2}+b c +\left (2 a c d +b d \right ) x^{2}+a \,d^{2} x^{4}}}{x^{2}}\right )}{2 \sqrt {a \,c^{2}+b c}}+\frac {2 d \,x^{2}+2 c}{\sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}}\right ) \sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}}{2 \left (a c +b \right )^{2} \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right )}\) | \(244\) |
| default | \(\text {Expression too large to display}\) | \(1088\) |
-1/2/(a*c+b)^2*c*(a*d*x^2+a*c+b)/x^2/((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)+1/2 *d*b/(a*c+b)^2*(-3/2*c/(a*c^2+b*c)^(1/2)*ln((2*a*c^2+2*b*c+(2*a*c*d+b*d)*x ^2+2*(a*c^2+b*c)^(1/2)*(a*c^2+b*c+(2*a*c*d+b*d)*x^2+a*d^2*x^4)^(1/2))/x^2) +2*(d*x^2+c)/(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2))/((a*d*x^2+a* c+b)/(d*x^2+c))^(1/2)*((a*d*x^2+a*c+b)*(d*x^2+c))^(1/2)/(d*x^2+c)
Time = 0.39 (sec) , antiderivative size = 599, normalized size of antiderivative = 4.10 \[ \int \frac {1}{x^3 \left (a+\frac {b}{c+d x^2}\right )^{3/2}} \, dx=\left [\frac {3 \, {\left (a b d^{2} x^{4} + {\left (a b c + b^{2}\right )} d x^{2}\right )} \sqrt {\frac {c}{a c + b}} \log \left (\frac {{\left (8 \, a^{2} c^{2} + 8 \, a b c + b^{2}\right )} d^{2} x^{4} + 8 \, a^{2} c^{4} + 16 \, a b c^{3} + 8 \, b^{2} c^{2} + 8 \, {\left (2 \, a^{2} c^{3} + 3 \, a b c^{2} + b^{2} c\right )} d x^{2} - 4 \, {\left ({\left (2 \, a^{2} c^{2} + 3 \, a b c + b^{2}\right )} d^{2} x^{4} + 2 \, a^{2} c^{4} + 4 \, a b c^{3} + 2 \, b^{2} c^{2} + {\left (4 \, a^{2} c^{3} + 7 \, a b c^{2} + 3 \, b^{2} c\right )} d x^{2}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}} \sqrt {\frac {c}{a c + b}}}{x^{4}}\right ) - 4 \, {\left ({\left (a c - 2 \, b\right )} d^{2} x^{4} + a c^{3} + {\left (2 \, a c^{2} - b c\right )} d x^{2} + b c^{2}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{8 \, {\left ({\left (a^{3} c^{2} + 2 \, a^{2} b c + a b^{2}\right )} d x^{4} + {\left (a^{3} c^{3} + 3 \, a^{2} b c^{2} + 3 \, a b^{2} c + b^{3}\right )} x^{2}\right )}}, \frac {3 \, {\left (a b d^{2} x^{4} + {\left (a b c + b^{2}\right )} d x^{2}\right )} \sqrt {-\frac {c}{a c + b}} \arctan \left (\frac {{\left ({\left (2 \, a c + b\right )} d x^{2} + 2 \, a c^{2} + 2 \, b c\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}} \sqrt {-\frac {c}{a c + b}}}{2 \, {\left (a c d x^{2} + a c^{2} + b c\right )}}\right ) - 2 \, {\left ({\left (a c - 2 \, b\right )} d^{2} x^{4} + a c^{3} + {\left (2 \, a c^{2} - b c\right )} d x^{2} + b c^{2}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{4 \, {\left ({\left (a^{3} c^{2} + 2 \, a^{2} b c + a b^{2}\right )} d x^{4} + {\left (a^{3} c^{3} + 3 \, a^{2} b c^{2} + 3 \, a b^{2} c + b^{3}\right )} x^{2}\right )}}\right ] \]
[1/8*(3*(a*b*d^2*x^4 + (a*b*c + b^2)*d*x^2)*sqrt(c/(a*c + b))*log(((8*a^2* c^2 + 8*a*b*c + b^2)*d^2*x^4 + 8*a^2*c^4 + 16*a*b*c^3 + 8*b^2*c^2 + 8*(2*a ^2*c^3 + 3*a*b*c^2 + b^2*c)*d*x^2 - 4*((2*a^2*c^2 + 3*a*b*c + b^2)*d^2*x^4 + 2*a^2*c^4 + 4*a*b*c^3 + 2*b^2*c^2 + (4*a^2*c^3 + 7*a*b*c^2 + 3*b^2*c)*d *x^2)*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c))*sqrt(c/(a*c + b)))/x^4) - 4*(( a*c - 2*b)*d^2*x^4 + a*c^3 + (2*a*c^2 - b*c)*d*x^2 + b*c^2)*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c)))/((a^3*c^2 + 2*a^2*b*c + a*b^2)*d*x^4 + (a^3*c^3 + 3*a^2*b*c^2 + 3*a*b^2*c + b^3)*x^2), 1/4*(3*(a*b*d^2*x^4 + (a*b*c + b^2)* d*x^2)*sqrt(-c/(a*c + b))*arctan(1/2*((2*a*c + b)*d*x^2 + 2*a*c^2 + 2*b*c) *sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c))*sqrt(-c/(a*c + b))/(a*c*d*x^2 + a*c ^2 + b*c)) - 2*((a*c - 2*b)*d^2*x^4 + a*c^3 + (2*a*c^2 - b*c)*d*x^2 + b*c^ 2)*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c)))/((a^3*c^2 + 2*a^2*b*c + a*b^2)*d *x^4 + (a^3*c^3 + 3*a^2*b*c^2 + 3*a*b^2*c + b^3)*x^2)]
\[ \int \frac {1}{x^3 \left (a+\frac {b}{c+d x^2}\right )^{3/2}} \, dx=\int \frac {1}{x^{3} \left (\frac {a c + a d x^{2} + b}{c + d x^{2}}\right )^{\frac {3}{2}}}\, dx \]
Time = 0.30 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.69 \[ \int \frac {1}{x^3 \left (a+\frac {b}{c+d x^2}\right )^{3/2}} \, dx=\frac {3 \, b c d \log \left (\frac {c \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}} - \sqrt {{\left (a c + b\right )} c}}{c \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}} + \sqrt {{\left (a c + b\right )} c}}\right )}{4 \, {\left (a^{2} c^{2} + 2 \, a b c + b^{2}\right )} \sqrt {{\left (a c + b\right )} c}} + \frac {\frac {3 \, {\left (a d x^{2} + a c + b\right )} b c d}{d x^{2} + c} - 2 \, {\left (a b c + b^{2}\right )} d}{2 \, {\left ({\left (a^{2} c^{3} + 2 \, a b c^{2} + b^{2} c\right )} \left (\frac {a d x^{2} + a c + b}{d x^{2} + c}\right )^{\frac {3}{2}} - {\left (a^{3} c^{3} + 3 \, a^{2} b c^{2} + 3 \, a b^{2} c + b^{3}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}\right )}} \]
3/4*b*c*d*log((c*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c)) - sqrt((a*c + b)*c) )/(c*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c)) + sqrt((a*c + b)*c)))/((a^2*c^2 + 2*a*b*c + b^2)*sqrt((a*c + b)*c)) + 1/2*(3*(a*d*x^2 + a*c + b)*b*c*d/(d *x^2 + c) - 2*(a*b*c + b^2)*d)/((a^2*c^3 + 2*a*b*c^2 + b^2*c)*((a*d*x^2 + a*c + b)/(d*x^2 + c))^(3/2) - (a^3*c^3 + 3*a^2*b*c^2 + 3*a*b^2*c + b^3)*sq rt((a*d*x^2 + a*c + b)/(d*x^2 + c)))
\[ \int \frac {1}{x^3 \left (a+\frac {b}{c+d x^2}\right )^{3/2}} \, dx=\int { \frac {1}{{\left (a + \frac {b}{d x^{2} + c}\right )}^{\frac {3}{2}} x^{3}} \,d x } \]
Timed out. \[ \int \frac {1}{x^3 \left (a+\frac {b}{c+d x^2}\right )^{3/2}} \, dx=\int \frac {1}{x^3\,{\left (a+\frac {b}{d\,x^2+c}\right )}^{3/2}} \,d x \]