Integrand size = 20, antiderivative size = 115 \[ \int \frac {\left (a+b x^2\right )^2}{x^5 (c+d x)} \, dx=-\frac {a^2}{4 c x^4}+\frac {a^2 d}{3 c^2 x^3}-\frac {a \left (2 b c^2+a d^2\right )}{2 c^3 x^2}+\frac {a d \left (2 b c^2+a d^2\right )}{c^4 x}+\frac {\left (b c^2+a d^2\right )^2 \log (x)}{c^5}-\frac {\left (b c^2+a d^2\right )^2 \log (c+d x)}{c^5} \] Output:
-1/4*a^2/c/x^4+1/3*a^2*d/c^2/x^3-1/2*a*(a*d^2+2*b*c^2)/c^3/x^2+a*d*(a*d^2+ 2*b*c^2)/c^4/x+(a*d^2+b*c^2)^2*ln(x)/c^5-(a*d^2+b*c^2)^2*ln(d*x+c)/c^5
Time = 0.06 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+b x^2\right )^2}{x^5 (c+d x)} \, dx=\frac {\frac {a c \left (-12 b c^2 x^2 (c-2 d x)+a \left (-3 c^3+4 c^2 d x-6 c d^2 x^2+12 d^3 x^3\right )\right )}{x^4}+12 \left (b c^2+a d^2\right )^2 \log (x)-12 \left (b c^2+a d^2\right )^2 \log (c+d x)}{12 c^5} \] Input:
Integrate[(a + b*x^2)^2/(x^5*(c + d*x)),x]
Output:
((a*c*(-12*b*c^2*x^2*(c - 2*d*x) + a*(-3*c^3 + 4*c^2*d*x - 6*c*d^2*x^2 + 1 2*d^3*x^3)))/x^4 + 12*(b*c^2 + a*d^2)^2*Log[x] - 12*(b*c^2 + a*d^2)^2*Log[ c + d*x])/(12*c^5)
Time = 0.46 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {522, 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.
\(\displaystyle \int \frac {\left (a+b x^2\right )^2}{x^5 (c+d x)} \, dx\) |
\(\Big \downarrow \) 522 |
\(\displaystyle \int \left (-\frac {a^2 d}{c^2 x^4}+\frac {a^2}{c x^5}+\frac {\left (a d^2+b c^2\right )^2}{c^5 x}-\frac {d \left (a d^2+b c^2\right )^2}{c^5 (c+d x)}-\frac {a d \left (a d^2+2 b c^2\right )}{c^4 x^2}+\frac {a \left (a d^2+2 b c^2\right )}{c^3 x^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a^2 d}{3 c^2 x^3}-\frac {a^2}{4 c x^4}+\frac {\log (x) \left (a d^2+b c^2\right )^2}{c^5}-\frac {\left (a d^2+b c^2\right )^2 \log (c+d x)}{c^5}+\frac {a d \left (a d^2+2 b c^2\right )}{c^4 x}-\frac {a \left (a d^2+2 b c^2\right )}{2 c^3 x^2}\) |
Input:
Int[(a + b*x^2)^2/(x^5*(c + d*x)),x]
Output:
-1/4*a^2/(c*x^4) + (a^2*d)/(3*c^2*x^3) - (a*(2*b*c^2 + a*d^2))/(2*c^3*x^2) + (a*d*(2*b*c^2 + a*d^2))/(c^4*x) + ((b*c^2 + a*d^2)^2*Log[x])/c^5 - ((b* c^2 + a*d^2)^2*Log[c + d*x])/c^5
Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_. ), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
Time = 0.26 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.17
method | result | size |
default | \(-\frac {a^{2}}{4 c \,x^{4}}+\frac {\left (a^{2} d^{4}+2 b \,c^{2} d^{2} a +b^{2} c^{4}\right ) \ln \left (x \right )}{c^{5}}-\frac {a \left (a \,d^{2}+2 b \,c^{2}\right )}{2 c^{3} x^{2}}+\frac {a^{2} d}{3 c^{2} x^{3}}+\frac {a d \left (a \,d^{2}+2 b \,c^{2}\right )}{c^{4} x}-\frac {\left (a^{2} d^{4}+2 b \,c^{2} d^{2} a +b^{2} c^{4}\right ) \ln \left (d x +c \right )}{c^{5}}\) | \(134\) |
norman | \(\frac {\frac {a d \left (a \,d^{2}+2 b \,c^{2}\right ) x^{3}}{c^{4}}-\frac {a^{2}}{4 c}-\frac {a \left (a \,d^{2}+2 b \,c^{2}\right ) x^{2}}{2 c^{3}}+\frac {d \,a^{2} x}{3 c^{2}}}{x^{4}}+\frac {\left (a^{2} d^{4}+2 b \,c^{2} d^{2} a +b^{2} c^{4}\right ) \ln \left (x \right )}{c^{5}}-\frac {\left (a^{2} d^{4}+2 b \,c^{2} d^{2} a +b^{2} c^{4}\right ) \ln \left (d x +c \right )}{c^{5}}\) | \(134\) |
risch | \(\frac {\frac {a d \left (a \,d^{2}+2 b \,c^{2}\right ) x^{3}}{c^{4}}-\frac {a^{2}}{4 c}-\frac {a \left (a \,d^{2}+2 b \,c^{2}\right ) x^{2}}{2 c^{3}}+\frac {d \,a^{2} x}{3 c^{2}}}{x^{4}}-\frac {\ln \left (d x +c \right ) a^{2} d^{4}}{c^{5}}-\frac {2 \ln \left (d x +c \right ) b \,d^{2} a}{c^{3}}-\frac {\ln \left (d x +c \right ) b^{2}}{c}+\frac {\ln \left (-x \right ) a^{2} d^{4}}{c^{5}}+\frac {2 \ln \left (-x \right ) b \,d^{2} a}{c^{3}}+\frac {\ln \left (-x \right ) b^{2}}{c}\) | \(153\) |
parallelrisch | \(\frac {12 \ln \left (x \right ) x^{4} a^{2} d^{4}+24 \ln \left (x \right ) x^{4} a b \,c^{2} d^{2}+12 \ln \left (x \right ) x^{4} b^{2} c^{4}-12 \ln \left (d x +c \right ) x^{4} a^{2} d^{4}-24 \ln \left (d x +c \right ) x^{4} a b \,c^{2} d^{2}-12 \ln \left (d x +c \right ) x^{4} b^{2} c^{4}+12 a^{2} c \,d^{3} x^{3}+24 a b \,c^{3} d \,x^{3}-6 a^{2} c^{2} d^{2} x^{2}-12 a b \,c^{4} x^{2}+4 a^{2} d \,c^{3} x -3 a^{2} c^{4}}{12 c^{5} x^{4}}\) | \(169\) |
Input:
int((b*x^2+a)^2/x^5/(d*x+c),x,method=_RETURNVERBOSE)
Output:
-1/4*a^2/c/x^4+(a^2*d^4+2*a*b*c^2*d^2+b^2*c^4)/c^5*ln(x)-1/2*a*(a*d^2+2*b* c^2)/c^3/x^2+1/3*a^2*d/c^2/x^3+a*d*(a*d^2+2*b*c^2)/c^4/x-(a^2*d^4+2*a*b*c^ 2*d^2+b^2*c^4)/c^5*ln(d*x+c)
Time = 0.12 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.22 \[ \int \frac {\left (a+b x^2\right )^2}{x^5 (c+d x)} \, dx=\frac {4 \, a^{2} c^{3} d x - 3 \, a^{2} c^{4} - 12 \, {\left (b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}\right )} x^{4} \log \left (d x + c\right ) + 12 \, {\left (b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}\right )} x^{4} \log \left (x\right ) + 12 \, {\left (2 \, a b c^{3} d + a^{2} c d^{3}\right )} x^{3} - 6 \, {\left (2 \, a b c^{4} + a^{2} c^{2} d^{2}\right )} x^{2}}{12 \, c^{5} x^{4}} \] Input:
integrate((b*x^2+a)^2/x^5/(d*x+c),x, algorithm="fricas")
Output:
1/12*(4*a^2*c^3*d*x - 3*a^2*c^4 - 12*(b^2*c^4 + 2*a*b*c^2*d^2 + a^2*d^4)*x ^4*log(d*x + c) + 12*(b^2*c^4 + 2*a*b*c^2*d^2 + a^2*d^4)*x^4*log(x) + 12*( 2*a*b*c^3*d + a^2*c*d^3)*x^3 - 6*(2*a*b*c^4 + a^2*c^2*d^2)*x^2)/(c^5*x^4)
Leaf count of result is larger than twice the leaf count of optimal. 253 vs. \(2 (105) = 210\).
Time = 0.53 (sec) , antiderivative size = 253, normalized size of antiderivative = 2.20 \[ \int \frac {\left (a+b x^2\right )^2}{x^5 (c+d x)} \, dx=\frac {- 3 a^{2} c^{3} + 4 a^{2} c^{2} d x + x^{3} \cdot \left (12 a^{2} d^{3} + 24 a b c^{2} d\right ) + x^{2} \left (- 6 a^{2} c d^{2} - 12 a b c^{3}\right )}{12 c^{4} x^{4}} + \frac {\left (a d^{2} + b c^{2}\right )^{2} \log {\left (x + \frac {a^{2} c d^{4} + 2 a b c^{3} d^{2} + b^{2} c^{5} - c \left (a d^{2} + b c^{2}\right )^{2}}{2 a^{2} d^{5} + 4 a b c^{2} d^{3} + 2 b^{2} c^{4} d} \right )}}{c^{5}} - \frac {\left (a d^{2} + b c^{2}\right )^{2} \log {\left (x + \frac {a^{2} c d^{4} + 2 a b c^{3} d^{2} + b^{2} c^{5} + c \left (a d^{2} + b c^{2}\right )^{2}}{2 a^{2} d^{5} + 4 a b c^{2} d^{3} + 2 b^{2} c^{4} d} \right )}}{c^{5}} \] Input:
integrate((b*x**2+a)**2/x**5/(d*x+c),x)
Output:
(-3*a**2*c**3 + 4*a**2*c**2*d*x + x**3*(12*a**2*d**3 + 24*a*b*c**2*d) + x* *2*(-6*a**2*c*d**2 - 12*a*b*c**3))/(12*c**4*x**4) + (a*d**2 + b*c**2)**2*l og(x + (a**2*c*d**4 + 2*a*b*c**3*d**2 + b**2*c**5 - c*(a*d**2 + b*c**2)**2 )/(2*a**2*d**5 + 4*a*b*c**2*d**3 + 2*b**2*c**4*d))/c**5 - (a*d**2 + b*c**2 )**2*log(x + (a**2*c*d**4 + 2*a*b*c**3*d**2 + b**2*c**5 + c*(a*d**2 + b*c* *2)**2)/(2*a**2*d**5 + 4*a*b*c**2*d**3 + 2*b**2*c**4*d))/c**5
Time = 0.03 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.19 \[ \int \frac {\left (a+b x^2\right )^2}{x^5 (c+d x)} \, dx=-\frac {{\left (b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}\right )} \log \left (d x + c\right )}{c^{5}} + \frac {{\left (b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}\right )} \log \left (x\right )}{c^{5}} + \frac {4 \, a^{2} c^{2} d x - 3 \, a^{2} c^{3} + 12 \, {\left (2 \, a b c^{2} d + a^{2} d^{3}\right )} x^{3} - 6 \, {\left (2 \, a b c^{3} + a^{2} c d^{2}\right )} x^{2}}{12 \, c^{4} x^{4}} \] Input:
integrate((b*x^2+a)^2/x^5/(d*x+c),x, algorithm="maxima")
Output:
-(b^2*c^4 + 2*a*b*c^2*d^2 + a^2*d^4)*log(d*x + c)/c^5 + (b^2*c^4 + 2*a*b*c ^2*d^2 + a^2*d^4)*log(x)/c^5 + 1/12*(4*a^2*c^2*d*x - 3*a^2*c^3 + 12*(2*a*b *c^2*d + a^2*d^3)*x^3 - 6*(2*a*b*c^3 + a^2*c*d^2)*x^2)/(c^4*x^4)
Time = 0.12 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.27 \[ \int \frac {\left (a+b x^2\right )^2}{x^5 (c+d x)} \, dx=\frac {{\left (b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}\right )} \log \left ({\left | x \right |}\right )}{c^{5}} - \frac {{\left (b^{2} c^{4} d + 2 \, a b c^{2} d^{3} + a^{2} d^{5}\right )} \log \left ({\left | d x + c \right |}\right )}{c^{5} d} + \frac {4 \, a^{2} c^{3} d x - 3 \, a^{2} c^{4} + 12 \, {\left (2 \, a b c^{3} d + a^{2} c d^{3}\right )} x^{3} - 6 \, {\left (2 \, a b c^{4} + a^{2} c^{2} d^{2}\right )} x^{2}}{12 \, c^{5} x^{4}} \] Input:
integrate((b*x^2+a)^2/x^5/(d*x+c),x, algorithm="giac")
Output:
(b^2*c^4 + 2*a*b*c^2*d^2 + a^2*d^4)*log(abs(x))/c^5 - (b^2*c^4*d + 2*a*b*c ^2*d^3 + a^2*d^5)*log(abs(d*x + c))/(c^5*d) + 1/12*(4*a^2*c^3*d*x - 3*a^2* c^4 + 12*(2*a*b*c^3*d + a^2*c*d^3)*x^3 - 6*(2*a*b*c^4 + a^2*c^2*d^2)*x^2)/ (c^5*x^4)
Time = 7.07 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.19 \[ \int \frac {\left (a+b x^2\right )^2}{x^5 (c+d x)} \, dx=-\frac {\frac {a^2}{4\,c}-\frac {a^2\,d\,x}{3\,c^2}+\frac {a\,x^2\,\left (2\,b\,c^2+a\,d^2\right )}{2\,c^3}-\frac {a\,d\,x^3\,\left (2\,b\,c^2+a\,d^2\right )}{c^4}}{x^4}-\frac {2\,\mathrm {atanh}\left (\frac {{\left (b\,c^2+a\,d^2\right )}^2\,\left (c+2\,d\,x\right )}{c\,\left (a^2\,d^4+2\,a\,b\,c^2\,d^2+b^2\,c^4\right )}\right )\,{\left (b\,c^2+a\,d^2\right )}^2}{c^5} \] Input:
int((a + b*x^2)^2/(x^5*(c + d*x)),x)
Output:
- (a^2/(4*c) - (a^2*d*x)/(3*c^2) + (a*x^2*(a*d^2 + 2*b*c^2))/(2*c^3) - (a* d*x^3*(a*d^2 + 2*b*c^2))/c^4)/x^4 - (2*atanh(((a*d^2 + b*c^2)^2*(c + 2*d*x ))/(c*(a^2*d^4 + b^2*c^4 + 2*a*b*c^2*d^2)))*(a*d^2 + b*c^2)^2)/c^5
Time = 0.18 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.46 \[ \int \frac {\left (a+b x^2\right )^2}{x^5 (c+d x)} \, dx=\frac {-12 \,\mathrm {log}\left (d x +c \right ) a^{2} d^{4} x^{4}-24 \,\mathrm {log}\left (d x +c \right ) a b \,c^{2} d^{2} x^{4}-12 \,\mathrm {log}\left (d x +c \right ) b^{2} c^{4} x^{4}+12 \,\mathrm {log}\left (x \right ) a^{2} d^{4} x^{4}+24 \,\mathrm {log}\left (x \right ) a b \,c^{2} d^{2} x^{4}+12 \,\mathrm {log}\left (x \right ) b^{2} c^{4} x^{4}-3 a^{2} c^{4}+4 a^{2} c^{3} d x -6 a^{2} c^{2} d^{2} x^{2}+12 a^{2} c \,d^{3} x^{3}-12 a b \,c^{4} x^{2}+24 a b \,c^{3} d \,x^{3}}{12 c^{5} x^{4}} \] Input:
int((b*x^2+a)^2/x^5/(d*x+c),x)
Output:
( - 12*log(c + d*x)*a**2*d**4*x**4 - 24*log(c + d*x)*a*b*c**2*d**2*x**4 - 12*log(c + d*x)*b**2*c**4*x**4 + 12*log(x)*a**2*d**4*x**4 + 24*log(x)*a*b* c**2*d**2*x**4 + 12*log(x)*b**2*c**4*x**4 - 3*a**2*c**4 + 4*a**2*c**3*d*x - 6*a**2*c**2*d**2*x**2 + 12*a**2*c*d**3*x**3 - 12*a*b*c**4*x**2 + 24*a*b* c**3*d*x**3)/(12*c**5*x**4)