Integrand size = 22, antiderivative size = 98 \[ \int \frac {\left (a+b x^2\right )^2}{x^7 \left (c+d x^2\right )} \, dx=-\frac {a^2}{6 c x^6}-\frac {a (2 b c-a d)}{4 c^2 x^4}-\frac {(b c-a d)^2}{2 c^3 x^2}-\frac {d (b c-a d)^2 \log (x)}{c^4}+\frac {d (b c-a d)^2 \log \left (c+d x^2\right )}{2 c^4} \]
-1/6*a^2/c/x^6-1/4*a*(-a*d+2*b*c)/c^2/x^4-1/2*(-a*d+b*c)^2/c^3/x^2-d*(-a*d +b*c)^2*ln(x)/c^4+1/2*d*(-a*d+b*c)^2*ln(d*x^2+c)/c^4
Time = 0.04 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.10 \[ \int \frac {\left (a+b x^2\right )^2}{x^7 \left (c+d x^2\right )} \, dx=-\frac {c \left (6 b^2 c^2 x^4+6 a b c x^2 \left (c-2 d x^2\right )+a^2 \left (2 c^2-3 c d x^2+6 d^2 x^4\right )\right )+12 d (b c-a d)^2 x^6 \log (x)-6 d (b c-a d)^2 x^6 \log \left (c+d x^2\right )}{12 c^4 x^6} \]
-1/12*(c*(6*b^2*c^2*x^4 + 6*a*b*c*x^2*(c - 2*d*x^2) + a^2*(2*c^2 - 3*c*d*x ^2 + 6*d^2*x^4)) + 12*d*(b*c - a*d)^2*x^6*Log[x] - 6*d*(b*c - a*d)^2*x^6*L og[c + d*x^2])/(c^4*x^6)
Time = 0.26 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.01, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {354, 99, 2009}
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.
\(\displaystyle \int \frac {\left (a+b x^2\right )^2}{x^7 \left (c+d x^2\right )} \, dx\) |
\(\Big \downarrow \) 354 |
\(\displaystyle \frac {1}{2} \int \frac {\left (b x^2+a\right )^2}{x^8 \left (d x^2+c\right )}dx^2\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \frac {1}{2} \int \left (\frac {a^2}{c x^8}-\frac {(a d-2 b c) a}{c^2 x^6}+\frac {d^2 (b c-a d)^2}{c^4 \left (d x^2+c\right )}-\frac {d (b c-a d)^2}{c^4 x^2}+\frac {(b c-a d)^2}{c^3 x^4}\right )dx^2\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (-\frac {a^2}{3 c x^6}-\frac {d \log \left (x^2\right ) (b c-a d)^2}{c^4}+\frac {d (b c-a d)^2 \log \left (c+d x^2\right )}{c^4}-\frac {(b c-a d)^2}{c^3 x^2}-\frac {a (2 b c-a d)}{2 c^2 x^4}\right )\) |
(-1/3*a^2/(c*x^6) - (a*(2*b*c - a*d))/(2*c^2*x^4) - (b*c - a*d)^2/(c^3*x^2 ) - (d*(b*c - a*d)^2*Log[x^2])/c^4 + (d*(b*c - a*d)^2*Log[c + d*x^2])/c^4) /2
3.2.79.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S ymbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x , x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ [(m - 1)/2]
Time = 2.75 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.26
method | result | size |
default | \(-\frac {a^{2}}{6 c \,x^{6}}-\frac {a^{2} d^{2}-2 a b c d +b^{2} c^{2}}{2 c^{3} x^{2}}+\frac {a \left (a d -2 b c \right )}{4 c^{2} x^{4}}-\frac {d \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (x \right )}{c^{4}}+\frac {d \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (d \,x^{2}+c \right )}{2 c^{4}}\) | \(123\) |
norman | \(\frac {-\frac {a^{2}}{6 c}-\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) x^{4}}{2 c^{3}}+\frac {a \left (a d -2 b c \right ) x^{2}}{4 c^{2}}}{x^{6}}-\frac {d \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (x \right )}{c^{4}}+\frac {d \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (d \,x^{2}+c \right )}{2 c^{4}}\) | \(125\) |
risch | \(\frac {-\frac {a^{2}}{6 c}-\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) x^{4}}{2 c^{3}}+\frac {a \left (a d -2 b c \right ) x^{2}}{4 c^{2}}}{x^{6}}-\frac {d^{3} \ln \left (x \right ) a^{2}}{c^{4}}+\frac {2 d^{2} \ln \left (x \right ) a b}{c^{3}}-\frac {d \ln \left (x \right ) b^{2}}{c^{2}}+\frac {d^{3} \ln \left (-d \,x^{2}-c \right ) a^{2}}{2 c^{4}}-\frac {d^{2} \ln \left (-d \,x^{2}-c \right ) a b}{c^{3}}+\frac {d \ln \left (-d \,x^{2}-c \right ) b^{2}}{2 c^{2}}\) | \(160\) |
parallelrisch | \(-\frac {12 \ln \left (x \right ) x^{6} a^{2} d^{3}-24 \ln \left (x \right ) x^{6} a b c \,d^{2}+12 \ln \left (x \right ) x^{6} b^{2} c^{2} d -6 \ln \left (d \,x^{2}+c \right ) x^{6} a^{2} d^{3}+12 \ln \left (d \,x^{2}+c \right ) x^{6} a b c \,d^{2}-6 \ln \left (d \,x^{2}+c \right ) x^{6} b^{2} c^{2} d +6 a^{2} c \,d^{2} x^{4}-12 a b \,c^{2} d \,x^{4}+6 b^{2} c^{3} x^{4}-3 a^{2} c^{2} d \,x^{2}+6 a b \,c^{3} x^{2}+2 a^{2} c^{3}}{12 c^{4} x^{6}}\) | \(172\) |
-1/6*a^2/c/x^6-1/2*(a^2*d^2-2*a*b*c*d+b^2*c^2)/c^3/x^2+1/4*a*(a*d-2*b*c)/c ^2/x^4-d*(a^2*d^2-2*a*b*c*d+b^2*c^2)/c^4*ln(x)+1/2*d*(a^2*d^2-2*a*b*c*d+b^ 2*c^2)/c^4*ln(d*x^2+c)
Time = 0.26 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.39 \[ \int \frac {\left (a+b x^2\right )^2}{x^7 \left (c+d x^2\right )} \, dx=\frac {6 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x^{6} \log \left (d x^{2} + c\right ) - 12 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x^{6} \log \left (x\right ) - 2 \, a^{2} c^{3} - 6 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} x^{4} - 3 \, {\left (2 \, a b c^{3} - a^{2} c^{2} d\right )} x^{2}}{12 \, c^{4} x^{6}} \]
1/12*(6*(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*x^6*log(d*x^2 + c) - 12*(b^2*c ^2*d - 2*a*b*c*d^2 + a^2*d^3)*x^6*log(x) - 2*a^2*c^3 - 6*(b^2*c^3 - 2*a*b* c^2*d + a^2*c*d^2)*x^4 - 3*(2*a*b*c^3 - a^2*c^2*d)*x^2)/(c^4*x^6)
Time = 0.88 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.07 \[ \int \frac {\left (a+b x^2\right )^2}{x^7 \left (c+d x^2\right )} \, dx=\frac {- 2 a^{2} c^{2} + x^{4} \left (- 6 a^{2} d^{2} + 12 a b c d - 6 b^{2} c^{2}\right ) + x^{2} \cdot \left (3 a^{2} c d - 6 a b c^{2}\right )}{12 c^{3} x^{6}} - \frac {d \left (a d - b c\right )^{2} \log {\left (x \right )}}{c^{4}} + \frac {d \left (a d - b c\right )^{2} \log {\left (\frac {c}{d} + x^{2} \right )}}{2 c^{4}} \]
(-2*a**2*c**2 + x**4*(-6*a**2*d**2 + 12*a*b*c*d - 6*b**2*c**2) + x**2*(3*a **2*c*d - 6*a*b*c**2))/(12*c**3*x**6) - d*(a*d - b*c)**2*log(x)/c**4 + d*( a*d - b*c)**2*log(c/d + x**2)/(2*c**4)
Time = 0.20 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.37 \[ \int \frac {\left (a+b x^2\right )^2}{x^7 \left (c+d x^2\right )} \, dx=\frac {{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} \log \left (d x^{2} + c\right )}{2 \, c^{4}} - \frac {{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} \log \left (x^{2}\right )}{2 \, c^{4}} - \frac {6 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{4} + 2 \, a^{2} c^{2} + 3 \, {\left (2 \, a b c^{2} - a^{2} c d\right )} x^{2}}{12 \, c^{3} x^{6}} \]
1/2*(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*log(d*x^2 + c)/c^4 - 1/2*(b^2*c^2* d - 2*a*b*c*d^2 + a^2*d^3)*log(x^2)/c^4 - 1/12*(6*(b^2*c^2 - 2*a*b*c*d + a ^2*d^2)*x^4 + 2*a^2*c^2 + 3*(2*a*b*c^2 - a^2*c*d)*x^2)/(c^3*x^6)
Leaf count of result is larger than twice the leaf count of optimal. 184 vs. \(2 (90) = 180\).
Time = 0.28 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.88 \[ \int \frac {\left (a+b x^2\right )^2}{x^7 \left (c+d x^2\right )} \, dx=-\frac {{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} \log \left (x^{2}\right )}{2 \, c^{4}} + \frac {{\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} \log \left ({\left | d x^{2} + c \right |}\right )}{2 \, c^{4} d} + \frac {11 \, b^{2} c^{2} d x^{6} - 22 \, a b c d^{2} x^{6} + 11 \, a^{2} d^{3} x^{6} - 6 \, b^{2} c^{3} x^{4} + 12 \, a b c^{2} d x^{4} - 6 \, a^{2} c d^{2} x^{4} - 6 \, a b c^{3} x^{2} + 3 \, a^{2} c^{2} d x^{2} - 2 \, a^{2} c^{3}}{12 \, c^{4} x^{6}} \]
-1/2*(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*log(x^2)/c^4 + 1/2*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*log(abs(d*x^2 + c))/(c^4*d) + 1/12*(11*b^2*c^2*d*x ^6 - 22*a*b*c*d^2*x^6 + 11*a^2*d^3*x^6 - 6*b^2*c^3*x^4 + 12*a*b*c^2*d*x^4 - 6*a^2*c*d^2*x^4 - 6*a*b*c^3*x^2 + 3*a^2*c^2*d*x^2 - 2*a^2*c^3)/(c^4*x^6)
Time = 0.14 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.32 \[ \int \frac {\left (a+b x^2\right )^2}{x^7 \left (c+d x^2\right )} \, dx=\frac {\ln \left (d\,x^2+c\right )\,\left (a^2\,d^3-2\,a\,b\,c\,d^2+b^2\,c^2\,d\right )}{2\,c^4}-\frac {\frac {a^2}{6\,c}+\frac {x^4\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{2\,c^3}-\frac {a\,x^2\,\left (a\,d-2\,b\,c\right )}{4\,c^2}}{x^6}-\frac {\ln \left (x\right )\,\left (a^2\,d^3-2\,a\,b\,c\,d^2+b^2\,c^2\,d\right )}{c^4} \]