Integrand size = 24, antiderivative size = 101 \[ \int \frac {A+B x^2}{x^6 \left (b x^2+c x^4\right )} \, dx=-\frac {A}{7 b x^7}-\frac {b B-A c}{5 b^2 x^5}+\frac {c (b B-A c)}{3 b^3 x^3}-\frac {c^2 (b B-A c)}{b^4 x}-\frac {c^{5/2} (b B-A c) \arctan \left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{b^{9/2}} \] Output:
-1/7*A/b/x^7-1/5*(-A*c+B*b)/b^2/x^5+1/3*c*(-A*c+B*b)/b^3/x^3-c^2*(-A*c+B*b )/b^4/x-c^(5/2)*(-A*c+B*b)*arctan(c^(1/2)*x/b^(1/2))/b^(9/2)
Time = 0.05 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00 \[ \int \frac {A+B x^2}{x^6 \left (b x^2+c x^4\right )} \, dx=-\frac {A}{7 b x^7}+\frac {-b B+A c}{5 b^2 x^5}+\frac {c (b B-A c)}{3 b^3 x^3}-\frac {c^2 (b B-A c)}{b^4 x}-\frac {c^{5/2} (b B-A c) \arctan \left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{b^{9/2}} \] Input:
Integrate[(A + B*x^2)/(x^6*(b*x^2 + c*x^4)),x]
Output:
-1/7*A/(b*x^7) + (-(b*B) + A*c)/(5*b^2*x^5) + (c*(b*B - A*c))/(3*b^3*x^3) - (c^2*(b*B - A*c))/(b^4*x) - (c^(5/2)*(b*B - A*c)*ArcTan[(Sqrt[c]*x)/Sqrt [b]])/b^(9/2)
Time = 0.33 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.91, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {9, 359, 264, 264, 264, 218}
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.
\(\displaystyle \int \frac {A+B x^2}{x^6 \left (b x^2+c x^4\right )} \, dx\) |
\(\Big \downarrow \) 9 |
\(\displaystyle \int \frac {A+B x^2}{x^8 \left (b+c x^2\right )}dx\) |
\(\Big \downarrow \) 359 |
\(\displaystyle \frac {(b B-A c) \int \frac {1}{x^6 \left (c x^2+b\right )}dx}{b}-\frac {A}{7 b x^7}\) |
\(\Big \downarrow \) 264 |
\(\displaystyle \frac {(b B-A c) \left (-\frac {c \int \frac {1}{x^4 \left (c x^2+b\right )}dx}{b}-\frac {1}{5 b x^5}\right )}{b}-\frac {A}{7 b x^7}\) |
\(\Big \downarrow \) 264 |
\(\displaystyle \frac {(b B-A c) \left (-\frac {c \left (-\frac {c \int \frac {1}{x^2 \left (c x^2+b\right )}dx}{b}-\frac {1}{3 b x^3}\right )}{b}-\frac {1}{5 b x^5}\right )}{b}-\frac {A}{7 b x^7}\) |
\(\Big \downarrow \) 264 |
\(\displaystyle \frac {(b B-A c) \left (-\frac {c \left (-\frac {c \left (-\frac {c \int \frac {1}{c x^2+b}dx}{b}-\frac {1}{b x}\right )}{b}-\frac {1}{3 b x^3}\right )}{b}-\frac {1}{5 b x^5}\right )}{b}-\frac {A}{7 b x^7}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {(b B-A c) \left (-\frac {c \left (-\frac {c \left (-\frac {\sqrt {c} \arctan \left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{b^{3/2}}-\frac {1}{b x}\right )}{b}-\frac {1}{3 b x^3}\right )}{b}-\frac {1}{5 b x^5}\right )}{b}-\frac {A}{7 b x^7}\) |
Input:
Int[(A + B*x^2)/(x^6*(b*x^2 + c*x^4)),x]
Output:
-1/7*A/(b*x^7) + ((b*B - A*c)*(-1/5*1/(b*x^5) - (c*(-1/3*1/(b*x^3) - (c*(- (1/(b*x)) - (Sqrt[c]*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/b^(3/2)))/b))/b))/b
Int[(u_.)*(Px_)^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[1/e^(p*r) Int[u*(e*x)^(m + p*r)*ExpandToSum[Px/x^r, x]^p, x], x] /; IGtQ[r, 0]] /; FreeQ[{e, m}, x] && PolyQ[Px, x] && IntegerQ[p] && !MonomialQ[Px, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
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[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x _Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] + Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(a*e^2*(m + 1)) Int[(e*x)^(m + 2)* (a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && !ILtQ[p, -1]
Time = 0.44 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.90
method | result | size |
default | \(-\frac {A}{7 b \,x^{7}}-\frac {-A c +B b}{5 b^{2} x^{5}}-\frac {\left (A c -B b \right ) c}{3 b^{3} x^{3}}+\frac {\left (A c -B b \right ) c^{2}}{b^{4} x}+\frac {c^{3} \left (A c -B b \right ) \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{b^{4} \sqrt {b c}}\) | \(91\) |
risch | \(\frac {\frac {\left (A c -B b \right ) c^{2} x^{6}}{b^{4}}-\frac {\left (A c -B b \right ) c \,x^{4}}{3 b^{3}}+\frac {\left (A c -B b \right ) x^{2}}{5 b^{2}}-\frac {A}{7 b}}{x^{7}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b^{9} \textit {\_Z}^{2}+A^{2} c^{7}-2 A B b \,c^{6}+B^{2} b^{2} c^{5}\right )}{\sum }\textit {\_R} \ln \left (\left (3 \textit {\_R}^{2} b^{9}+2 A^{2} c^{7}-4 A B b \,c^{6}+2 B^{2} b^{2} c^{5}\right ) x +\left (-A \,b^{5} c^{3}+B \,b^{6} c^{2}\right ) \textit {\_R} \right )\right )}{2}\) | \(165\) |
Input:
int((B*x^2+A)/x^6/(c*x^4+b*x^2),x,method=_RETURNVERBOSE)
Output:
-1/7*A/b/x^7-1/5*(-A*c+B*b)/b^2/x^5-1/3*(A*c-B*b)/b^3*c/x^3+(A*c-B*b)/b^4* c^2/x+c^3*(A*c-B*b)/b^4/(b*c)^(1/2)*arctan(c*x/(b*c)^(1/2))
Time = 0.08 (sec) , antiderivative size = 234, normalized size of antiderivative = 2.32 \[ \int \frac {A+B x^2}{x^6 \left (b x^2+c x^4\right )} \, dx=\left [-\frac {105 \, {\left (B b c^{2} - A c^{3}\right )} x^{7} \sqrt {-\frac {c}{b}} \log \left (\frac {c x^{2} + 2 \, b x \sqrt {-\frac {c}{b}} - b}{c x^{2} + b}\right ) + 210 \, {\left (B b c^{2} - A c^{3}\right )} x^{6} - 70 \, {\left (B b^{2} c - A b c^{2}\right )} x^{4} + 30 \, A b^{3} + 42 \, {\left (B b^{3} - A b^{2} c\right )} x^{2}}{210 \, b^{4} x^{7}}, -\frac {105 \, {\left (B b c^{2} - A c^{3}\right )} x^{7} \sqrt {\frac {c}{b}} \arctan \left (x \sqrt {\frac {c}{b}}\right ) + 105 \, {\left (B b c^{2} - A c^{3}\right )} x^{6} - 35 \, {\left (B b^{2} c - A b c^{2}\right )} x^{4} + 15 \, A b^{3} + 21 \, {\left (B b^{3} - A b^{2} c\right )} x^{2}}{105 \, b^{4} x^{7}}\right ] \] Input:
integrate((B*x^2+A)/x^6/(c*x^4+b*x^2),x, algorithm="fricas")
Output:
[-1/210*(105*(B*b*c^2 - A*c^3)*x^7*sqrt(-c/b)*log((c*x^2 + 2*b*x*sqrt(-c/b ) - b)/(c*x^2 + b)) + 210*(B*b*c^2 - A*c^3)*x^6 - 70*(B*b^2*c - A*b*c^2)*x ^4 + 30*A*b^3 + 42*(B*b^3 - A*b^2*c)*x^2)/(b^4*x^7), -1/105*(105*(B*b*c^2 - A*c^3)*x^7*sqrt(c/b)*arctan(x*sqrt(c/b)) + 105*(B*b*c^2 - A*c^3)*x^6 - 3 5*(B*b^2*c - A*b*c^2)*x^4 + 15*A*b^3 + 21*(B*b^3 - A*b^2*c)*x^2)/(b^4*x^7) ]
Leaf count of result is larger than twice the leaf count of optimal. 187 vs. \(2 (87) = 174\).
Time = 0.28 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.85 \[ \int \frac {A+B x^2}{x^6 \left (b x^2+c x^4\right )} \, dx=\frac {\sqrt {- \frac {c^{5}}{b^{9}}} \left (- A c + B b\right ) \log {\left (- \frac {b^{5} \sqrt {- \frac {c^{5}}{b^{9}}} \left (- A c + B b\right )}{- A c^{4} + B b c^{3}} + x \right )}}{2} - \frac {\sqrt {- \frac {c^{5}}{b^{9}}} \left (- A c + B b\right ) \log {\left (\frac {b^{5} \sqrt {- \frac {c^{5}}{b^{9}}} \left (- A c + B b\right )}{- A c^{4} + B b c^{3}} + x \right )}}{2} + \frac {- 15 A b^{3} + x^{6} \cdot \left (105 A c^{3} - 105 B b c^{2}\right ) + x^{4} \left (- 35 A b c^{2} + 35 B b^{2} c\right ) + x^{2} \cdot \left (21 A b^{2} c - 21 B b^{3}\right )}{105 b^{4} x^{7}} \] Input:
integrate((B*x**2+A)/x**6/(c*x**4+b*x**2),x)
Output:
sqrt(-c**5/b**9)*(-A*c + B*b)*log(-b**5*sqrt(-c**5/b**9)*(-A*c + B*b)/(-A* c**4 + B*b*c**3) + x)/2 - sqrt(-c**5/b**9)*(-A*c + B*b)*log(b**5*sqrt(-c** 5/b**9)*(-A*c + B*b)/(-A*c**4 + B*b*c**3) + x)/2 + (-15*A*b**3 + x**6*(105 *A*c**3 - 105*B*b*c**2) + x**4*(-35*A*b*c**2 + 35*B*b**2*c) + x**2*(21*A*b **2*c - 21*B*b**3))/(105*b**4*x**7)
Time = 0.11 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.02 \[ \int \frac {A+B x^2}{x^6 \left (b x^2+c x^4\right )} \, dx=-\frac {{\left (B b c^{3} - A c^{4}\right )} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{\sqrt {b c} b^{4}} - \frac {105 \, {\left (B b c^{2} - A c^{3}\right )} x^{6} - 35 \, {\left (B b^{2} c - A b c^{2}\right )} x^{4} + 15 \, A b^{3} + 21 \, {\left (B b^{3} - A b^{2} c\right )} x^{2}}{105 \, b^{4} x^{7}} \] Input:
integrate((B*x^2+A)/x^6/(c*x^4+b*x^2),x, algorithm="maxima")
Output:
-(B*b*c^3 - A*c^4)*arctan(c*x/sqrt(b*c))/(sqrt(b*c)*b^4) - 1/105*(105*(B*b *c^2 - A*c^3)*x^6 - 35*(B*b^2*c - A*b*c^2)*x^4 + 15*A*b^3 + 21*(B*b^3 - A* b^2*c)*x^2)/(b^4*x^7)
Time = 0.20 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.05 \[ \int \frac {A+B x^2}{x^6 \left (b x^2+c x^4\right )} \, dx=-\frac {{\left (B b c^{3} - A c^{4}\right )} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{\sqrt {b c} b^{4}} - \frac {105 \, B b c^{2} x^{6} - 105 \, A c^{3} x^{6} - 35 \, B b^{2} c x^{4} + 35 \, A b c^{2} x^{4} + 21 \, B b^{3} x^{2} - 21 \, A b^{2} c x^{2} + 15 \, A b^{3}}{105 \, b^{4} x^{7}} \] Input:
integrate((B*x^2+A)/x^6/(c*x^4+b*x^2),x, algorithm="giac")
Output:
-(B*b*c^3 - A*c^4)*arctan(c*x/sqrt(b*c))/(sqrt(b*c)*b^4) - 1/105*(105*B*b* c^2*x^6 - 105*A*c^3*x^6 - 35*B*b^2*c*x^4 + 35*A*b*c^2*x^4 + 21*B*b^3*x^2 - 21*A*b^2*c*x^2 + 15*A*b^3)/(b^4*x^7)
Time = 9.16 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.88 \[ \int \frac {A+B x^2}{x^6 \left (b x^2+c x^4\right )} \, dx=\frac {c^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {b}}\right )\,\left (A\,c-B\,b\right )}{b^{9/2}}-\frac {\frac {A}{7\,b}-\frac {x^2\,\left (A\,c-B\,b\right )}{5\,b^2}-\frac {c^2\,x^6\,\left (A\,c-B\,b\right )}{b^4}+\frac {c\,x^4\,\left (A\,c-B\,b\right )}{3\,b^3}}{x^7} \] Input:
int((A + B*x^2)/(x^6*(b*x^2 + c*x^4)),x)
Output:
(c^(5/2)*atan((c^(1/2)*x)/b^(1/2))*(A*c - B*b))/b^(9/2) - (A/(7*b) - (x^2* (A*c - B*b))/(5*b^2) - (c^2*x^6*(A*c - B*b))/b^4 + (c*x^4*(A*c - B*b))/(3* b^3))/x^7
Time = 0.19 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.26 \[ \int \frac {A+B x^2}{x^6 \left (b x^2+c x^4\right )} \, dx=\frac {105 \sqrt {c}\, \sqrt {b}\, \mathit {atan} \left (\frac {c x}{\sqrt {c}\, \sqrt {b}}\right ) a \,c^{3} x^{7}-105 \sqrt {c}\, \sqrt {b}\, \mathit {atan} \left (\frac {c x}{\sqrt {c}\, \sqrt {b}}\right ) b^{2} c^{2} x^{7}-15 a \,b^{4}+21 a \,b^{3} c \,x^{2}-35 a \,b^{2} c^{2} x^{4}+105 a b \,c^{3} x^{6}-21 b^{5} x^{2}+35 b^{4} c \,x^{4}-105 b^{3} c^{2} x^{6}}{105 b^{5} x^{7}} \] Input:
int((B*x^2+A)/x^6/(c*x^4+b*x^2),x)
Output:
(105*sqrt(c)*sqrt(b)*atan((c*x)/(sqrt(c)*sqrt(b)))*a*c**3*x**7 - 105*sqrt( c)*sqrt(b)*atan((c*x)/(sqrt(c)*sqrt(b)))*b**2*c**2*x**7 - 15*a*b**4 + 21*a *b**3*c*x**2 - 35*a*b**2*c**2*x**4 + 105*a*b*c**3*x**6 - 21*b**5*x**2 + 35 *b**4*c*x**4 - 105*b**3*c**2*x**6)/(105*b**5*x**7)