Integrand size = 23, antiderivative size = 95 \[ \int \frac {x^4 (c+d x)}{a^2-b^2 x^2} \, dx=-\frac {a^2 c x}{b^4}-\frac {a^2 d x^2}{2 b^4}-\frac {c x^3}{3 b^2}-\frac {d x^4}{4 b^2}-\frac {a^3 (b c+a d) \log (a-b x)}{2 b^6}+\frac {a^3 (b c-a d) \log (a+b x)}{2 b^6} \] Output:
-a^2*c*x/b^4-1/2*a^2*d*x^2/b^4-1/3*c*x^3/b^2-1/4*d*x^4/b^2-1/2*a^3*(a*d+b* c)*ln(-b*x+a)/b^6+1/2*a^3*(-a*d+b*c)*ln(b*x+a)/b^6
Time = 0.02 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.91 \[ \int \frac {x^4 (c+d x)}{a^2-b^2 x^2} \, dx=-\frac {a^2 c x}{b^4}-\frac {a^2 d x^2}{2 b^4}-\frac {c x^3}{3 b^2}-\frac {d x^4}{4 b^2}+\frac {a^3 c \text {arctanh}\left (\frac {b x}{a}\right )}{b^5}-\frac {a^4 d \log \left (a^2-b^2 x^2\right )}{2 b^6} \] Input:
Integrate[(x^4*(c + d*x))/(a^2 - b^2*x^2),x]
Output:
-((a^2*c*x)/b^4) - (a^2*d*x^2)/(2*b^4) - (c*x^3)/(3*b^2) - (d*x^4)/(4*b^2) + (a^3*c*ArcTanh[(b*x)/a])/b^5 - (a^4*d*Log[a^2 - b^2*x^2])/(2*b^6)
Time = 0.39 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.91, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {523, 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.
\(\displaystyle \int \frac {x^4 (c+d x)}{a^2-b^2 x^2} \, dx\) |
\(\Big \downarrow \) 523 |
\(\displaystyle \int \left (-\frac {a^2 c}{b^4}-\frac {a^2 d x}{b^4}+\frac {a^4 c+a^4 d x}{b^4 \left (a^2-b^2 x^2\right )}-\frac {c x^2}{b^2}-\frac {d x^3}{b^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a^3 c \text {arctanh}\left (\frac {b x}{a}\right )}{b^5}-\frac {a^2 c x}{b^4}-\frac {a^2 d x^2}{2 b^4}-\frac {a^4 d \log \left (a^2-b^2 x^2\right )}{2 b^6}-\frac {c x^3}{3 b^2}-\frac {d x^4}{4 b^2}\) |
Input:
Int[(x^4*(c + d*x))/(a^2 - b^2*x^2),x]
Output:
-((a^2*c*x)/b^4) - (a^2*d*x^2)/(2*b^4) - (c*x^3)/(3*b^2) - (d*x^4)/(4*b^2) + (a^3*c*ArcTanh[(b*x)/a])/b^5 - (a^4*d*Log[a^2 - b^2*x^2])/(2*b^6)
Int[((x_)^(m_.)*((c_) + (d_.)*(x_)))/((a_) + (b_.)*(x_)^2), x_Symbol] :> In t[ExpandIntegrand[x^m*((c + d*x)/(a + b*x^2)), x], x] /; FreeQ[{a, b, c, d} , x] && IntegerQ[m]
Time = 0.28 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.89
method | result | size |
default | \(-\frac {\frac {1}{4} b^{2} d \,x^{4}+\frac {1}{3} b^{2} c \,x^{3}+\frac {1}{2} a^{2} d \,x^{2}+a^{2} c x}{b^{4}}-\frac {a^{3} \left (a d -b c \right ) \ln \left (b x +a \right )}{2 b^{6}}-\frac {a^{3} \left (a d +b c \right ) \ln \left (-b x +a \right )}{2 b^{6}}\) | \(85\) |
norman | \(-\frac {c \,x^{3}}{3 b^{2}}-\frac {d \,x^{4}}{4 b^{2}}-\frac {a^{2} c x}{b^{4}}-\frac {a^{2} d \,x^{2}}{2 b^{4}}-\frac {a^{3} \left (a d -b c \right ) \ln \left (b x +a \right )}{2 b^{6}}-\frac {a^{3} \left (a d +b c \right ) \ln \left (-b x +a \right )}{2 b^{6}}\) | \(86\) |
parallelrisch | \(-\frac {3 d \,x^{4} b^{4}+4 c \,x^{3} b^{4}+6 a^{2} d \,x^{2} b^{2}+6 \ln \left (b x -a \right ) a^{4} d +6 \ln \left (b x -a \right ) a^{3} b c +6 \ln \left (b x +a \right ) a^{4} d -6 \ln \left (b x +a \right ) a^{3} b c +12 a^{2} c x \,b^{2}}{12 b^{6}}\) | \(101\) |
risch | \(-\frac {c \,x^{3}}{3 b^{2}}-\frac {d \,x^{4}}{4 b^{2}}-\frac {a^{2} c x}{b^{4}}-\frac {a^{2} d \,x^{2}}{2 b^{4}}-\frac {a^{4} \ln \left (b x +a \right ) d}{2 b^{6}}+\frac {a^{3} \ln \left (b x +a \right ) c}{2 b^{5}}-\frac {a^{4} \ln \left (-b x +a \right ) d}{2 b^{6}}-\frac {a^{3} \ln \left (-b x +a \right ) c}{2 b^{5}}\) | \(104\) |
Input:
int(x^4*(d*x+c)/(-b^2*x^2+a^2),x,method=_RETURNVERBOSE)
Output:
-1/b^4*(1/4*b^2*d*x^4+1/3*b^2*c*x^3+1/2*a^2*d*x^2+a^2*c*x)-1/2/b^6*a^3*(a* d-b*c)*ln(b*x+a)-1/2*a^3*(a*d+b*c)*ln(-b*x+a)/b^6
Time = 0.12 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.94 \[ \int \frac {x^4 (c+d x)}{a^2-b^2 x^2} \, dx=-\frac {3 \, b^{4} d x^{4} + 4 \, b^{4} c x^{3} + 6 \, a^{2} b^{2} d x^{2} + 12 \, a^{2} b^{2} c x - 6 \, {\left (a^{3} b c - a^{4} d\right )} \log \left (b x + a\right ) + 6 \, {\left (a^{3} b c + a^{4} d\right )} \log \left (b x - a\right )}{12 \, b^{6}} \] Input:
integrate(x^4*(d*x+c)/(-b^2*x^2+a^2),x, algorithm="fricas")
Output:
-1/12*(3*b^4*d*x^4 + 4*b^4*c*x^3 + 6*a^2*b^2*d*x^2 + 12*a^2*b^2*c*x - 6*(a ^3*b*c - a^4*d)*log(b*x + a) + 6*(a^3*b*c + a^4*d)*log(b*x - a))/b^6
Time = 0.22 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.36 \[ \int \frac {x^4 (c+d x)}{a^2-b^2 x^2} \, dx=- \frac {a^{3} \left (a d - b c\right ) \log {\left (x + \frac {a^{4} d - a^{3} \left (a d - b c\right )}{a^{2} b^{2} c} \right )}}{2 b^{6}} - \frac {a^{3} \left (a d + b c\right ) \log {\left (x + \frac {a^{4} d - a^{3} \left (a d + b c\right )}{a^{2} b^{2} c} \right )}}{2 b^{6}} - \frac {a^{2} c x}{b^{4}} - \frac {a^{2} d x^{2}}{2 b^{4}} - \frac {c x^{3}}{3 b^{2}} - \frac {d x^{4}}{4 b^{2}} \] Input:
integrate(x**4*(d*x+c)/(-b**2*x**2+a**2),x)
Output:
-a**3*(a*d - b*c)*log(x + (a**4*d - a**3*(a*d - b*c))/(a**2*b**2*c))/(2*b* *6) - a**3*(a*d + b*c)*log(x + (a**4*d - a**3*(a*d + b*c))/(a**2*b**2*c))/ (2*b**6) - a**2*c*x/b**4 - a**2*d*x**2/(2*b**4) - c*x**3/(3*b**2) - d*x**4 /(4*b**2)
Time = 0.03 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.95 \[ \int \frac {x^4 (c+d x)}{a^2-b^2 x^2} \, dx=-\frac {3 \, b^{2} d x^{4} + 4 \, b^{2} c x^{3} + 6 \, a^{2} d x^{2} + 12 \, a^{2} c x}{12 \, b^{4}} + \frac {{\left (a^{3} b c - a^{4} d\right )} \log \left (b x + a\right )}{2 \, b^{6}} - \frac {{\left (a^{3} b c + a^{4} d\right )} \log \left (b x - a\right )}{2 \, b^{6}} \] Input:
integrate(x^4*(d*x+c)/(-b^2*x^2+a^2),x, algorithm="maxima")
Output:
-1/12*(3*b^2*d*x^4 + 4*b^2*c*x^3 + 6*a^2*d*x^2 + 12*a^2*c*x)/b^4 + 1/2*(a^ 3*b*c - a^4*d)*log(b*x + a)/b^6 - 1/2*(a^3*b*c + a^4*d)*log(b*x - a)/b^6
Time = 0.12 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.03 \[ \int \frac {x^4 (c+d x)}{a^2-b^2 x^2} \, dx=\frac {{\left (a^{3} b c - a^{4} d\right )} \log \left ({\left | b x + a \right |}\right )}{2 \, b^{6}} - \frac {{\left (a^{3} b c + a^{4} d\right )} \log \left ({\left | b x - a \right |}\right )}{2 \, b^{6}} - \frac {3 \, b^{6} d x^{4} + 4 \, b^{6} c x^{3} + 6 \, a^{2} b^{4} d x^{2} + 12 \, a^{2} b^{4} c x}{12 \, b^{8}} \] Input:
integrate(x^4*(d*x+c)/(-b^2*x^2+a^2),x, algorithm="giac")
Output:
1/2*(a^3*b*c - a^4*d)*log(abs(b*x + a))/b^6 - 1/2*(a^3*b*c + a^4*d)*log(ab s(b*x - a))/b^6 - 1/12*(3*b^6*d*x^4 + 4*b^6*c*x^3 + 6*a^2*b^4*d*x^2 + 12*a ^2*b^4*c*x)/b^8
Time = 0.13 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.94 \[ \int \frac {x^4 (c+d x)}{a^2-b^2 x^2} \, dx=-\frac {\ln \left (a+b\,x\right )\,\left (a^4\,d-a^3\,b\,c\right )}{2\,b^6}-\frac {\ln \left (a-b\,x\right )\,\left (d\,a^4+b\,c\,a^3\right )}{2\,b^6}-\frac {c\,x^3}{3\,b^2}-\frac {d\,x^4}{4\,b^2}-\frac {a^2\,d\,x^2}{2\,b^4}-\frac {a^2\,c\,x}{b^4} \] Input:
int((x^4*(c + d*x))/(a^2 - b^2*x^2),x)
Output:
- (log(a + b*x)*(a^4*d - a^3*b*c))/(2*b^6) - (log(a - b*x)*(a^4*d + a^3*b* c))/(2*b^6) - (c*x^3)/(3*b^2) - (d*x^4)/(4*b^2) - (a^2*d*x^2)/(2*b^4) - (a ^2*c*x)/b^4
Time = 0.19 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.09 \[ \int \frac {x^4 (c+d x)}{a^2-b^2 x^2} \, dx=\frac {-6 \,\mathrm {log}\left (-b x -a \right ) a^{4} d +6 \,\mathrm {log}\left (-b x -a \right ) a^{3} b c -6 \,\mathrm {log}\left (-b x +a \right ) a^{4} d -6 \,\mathrm {log}\left (-b x +a \right ) a^{3} b c -12 a^{2} b^{2} c x -6 a^{2} b^{2} d \,x^{2}-4 b^{4} c \,x^{3}-3 b^{4} d \,x^{4}}{12 b^{6}} \] Input:
int(x^4*(d*x+c)/(-b^2*x^2+a^2),x)
Output:
( - 6*log( - a - b*x)*a**4*d + 6*log( - a - b*x)*a**3*b*c - 6*log(a - b*x) *a**4*d - 6*log(a - b*x)*a**3*b*c - 12*a**2*b**2*c*x - 6*a**2*b**2*d*x**2 - 4*b**4*c*x**3 - 3*b**4*d*x**4)/(12*b**6)