Integrand size = 23, antiderivative size = 108 \[ \int \frac {c+d x}{x^3 \left (a^2-b^2 x^2\right )^2} \, dx=-\frac {c}{2 a^4 x^2}-\frac {d}{a^4 x}+\frac {b^2 (c+d x)}{2 a^4 \left (a^2-b^2 x^2\right )}+\frac {2 b^2 c \log (x)}{a^6}-\frac {b (4 b c+3 a d) \log (a-b x)}{4 a^6}-\frac {b (4 b c-3 a d) \log (a+b x)}{4 a^6} \] Output:
-1/2*c/a^4/x^2-d/a^4/x+1/2*b^2*(d*x+c)/a^4/(-b^2*x^2+a^2)+2*b^2*c*ln(x)/a^ 6-1/4*b*(3*a*d+4*b*c)*ln(-b*x+a)/a^6-1/4*b*(-3*a*d+4*b*c)*ln(b*x+a)/a^6
Time = 0.06 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.84 \[ \int \frac {c+d x}{x^3 \left (a^2-b^2 x^2\right )^2} \, dx=\frac {-\frac {a^2 c}{x^2}-\frac {2 a^2 d}{x}+\frac {a^2 b^2 (c+d x)}{a^2-b^2 x^2}+3 a b d \text {arctanh}\left (\frac {b x}{a}\right )+4 b^2 c \log (x)-2 b^2 c \log \left (a^2-b^2 x^2\right )}{2 a^6} \] Input:
Integrate[(c + d*x)/(x^3*(a^2 - b^2*x^2)^2),x]
Output:
(-((a^2*c)/x^2) - (2*a^2*d)/x + (a^2*b^2*(c + d*x))/(a^2 - b^2*x^2) + 3*a* b*d*ArcTanh[(b*x)/a] + 4*b^2*c*Log[x] - 2*b^2*c*Log[a^2 - b^2*x^2])/(2*a^6 )
Time = 0.63 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.06, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {532, 25, 2333, 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.
\(\displaystyle \int \frac {c+d x}{x^3 \left (a^2-b^2 x^2\right )^2} \, dx\) |
\(\Big \downarrow \) 532 |
\(\displaystyle \frac {b^2 (c+d x)}{2 a^4 \left (a^2-b^2 x^2\right )}-\frac {\int -\frac {\frac {b^2 d x^3}{a^2}+\frac {2 b^2 c x^2}{a^2}+2 d x+2 c}{x^3 \left (a^2-b^2 x^2\right )}dx}{2 a^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \frac {\frac {b^2 d x^3}{a^2}+\frac {2 b^2 c x^2}{a^2}+2 d x+2 c}{x^3 \left (a^2-b^2 x^2\right )}dx}{2 a^2}+\frac {b^2 (c+d x)}{2 a^4 \left (a^2-b^2 x^2\right )}\) |
\(\Big \downarrow \) 2333 |
\(\displaystyle \frac {\int \left (\frac {4 c b^2}{a^4 x}+\frac {(4 b c+3 a d) b^2}{2 a^4 (a-b x)}+\frac {(3 a d-4 b c) b^2}{2 a^4 (a+b x)}+\frac {2 d}{a^2 x^2}+\frac {2 c}{a^2 x^3}\right )dx}{2 a^2}+\frac {b^2 (c+d x)}{2 a^4 \left (a^2-b^2 x^2\right )}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {b^2 (c+d x)}{2 a^4 \left (a^2-b^2 x^2\right )}+\frac {\frac {4 b^2 c \log (x)}{a^4}-\frac {b (3 a d+4 b c) \log (a-b x)}{2 a^4}-\frac {b (4 b c-3 a d) \log (a+b x)}{2 a^4}-\frac {c}{a^2 x^2}-\frac {2 d}{a^2 x}}{2 a^2}\) |
Input:
Int[(c + d*x)/(x^3*(a^2 - b^2*x^2)^2),x]
Output:
(b^2*(c + d*x))/(2*a^4*(a^2 - b^2*x^2)) + (-(c/(a^2*x^2)) - (2*d)/(a^2*x) + (4*b^2*c*Log[x])/a^4 - (b*(4*b*c + 3*a*d)*Log[a - b*x])/(2*a^4) - (b*(4* b*c - 3*a*d)*Log[a + b*x])/(2*a^4))/(2*a^2)
Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> With[{Qx = PolynomialQuotient[x^m*(c + d*x)^n, a + b*x^2, x], e = Coe ff[PolynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 0], f = Coeff[Pol ynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 1]}, Simp[(a*f - b*e*x) *((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1)) Int[x^m *(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*(Qx/x^m) + e*((2*p + 3)/x^m), x], x], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && ILtQ[m, 0] && LtQ[p, -1] && IntegerQ[2*p]
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ ExpandIntegrand[(c*x)^m*Pq*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
Time = 0.27 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.03
method | result | size |
norman | \(\frac {-\frac {c}{2 a^{2}}-\frac {d x}{a^{2}}+\frac {3 b^{2} d \,x^{3}}{2 a^{4}}+\frac {b^{4} c \,x^{4}}{a^{6}}}{x^{2} \left (-b^{2} x^{2}+a^{2}\right )}+\frac {b \left (3 a d -4 b c \right ) \ln \left (b x +a \right )}{4 a^{6}}-\frac {b \left (3 a d +4 b c \right ) \ln \left (-b x +a \right )}{4 a^{6}}+\frac {2 b^{2} c \ln \left (x \right )}{a^{6}}\) | \(111\) |
default | \(\frac {b \left (3 a d -4 b c \right ) \ln \left (b x +a \right )}{4 a^{6}}-\frac {\left (a d -b c \right ) b}{4 a^{5} \left (b x +a \right )}-\frac {c}{2 a^{4} x^{2}}-\frac {d}{a^{4} x}+\frac {2 b^{2} c \ln \left (x \right )}{a^{6}}-\frac {b \left (3 a d +4 b c \right ) \ln \left (-b x +a \right )}{4 a^{6}}+\frac {\left (a d +b c \right ) b}{4 a^{5} \left (-b x +a \right )}\) | \(116\) |
risch | \(\frac {\frac {3 b^{2} d \,x^{3}}{2 a^{4}}+\frac {b^{2} x^{2} c}{a^{4}}-\frac {d x}{a^{2}}-\frac {c}{2 a^{2}}}{x^{2} \left (-b^{2} x^{2}+a^{2}\right )}-\frac {3 b \ln \left (b x -a \right ) d}{4 a^{5}}-\frac {b^{2} \ln \left (b x -a \right ) c}{a^{6}}+\frac {3 b \ln \left (b x +a \right ) d}{4 a^{5}}-\frac {b^{2} \ln \left (b x +a \right ) c}{a^{6}}+\frac {2 c \,b^{2} \ln \left (-x \right )}{a^{6}}\) | \(130\) |
parallelrisch | \(\frac {8 \ln \left (x \right ) x^{4} b^{4} c -3 \ln \left (b x -a \right ) x^{4} a \,b^{3} d -4 \ln \left (b x -a \right ) x^{4} b^{4} c +3 \ln \left (b x +a \right ) x^{4} a \,b^{3} d -4 \ln \left (b x +a \right ) x^{4} b^{4} c -4 x^{4} b^{4} c -8 \ln \left (x \right ) x^{2} a^{2} b^{2} c +3 \ln \left (b x -a \right ) x^{2} a^{3} b d +4 \ln \left (b x -a \right ) x^{2} a^{2} b^{2} c -3 \ln \left (b x +a \right ) x^{2} a^{3} b d +4 \ln \left (b x +a \right ) x^{2} a^{2} b^{2} c -6 x^{3} a^{2} b^{2} d +4 a^{4} d x +2 c \,a^{4}}{4 a^{6} x^{2} \left (b^{2} x^{2}-a^{2}\right )}\) | \(222\) |
Input:
int((d*x+c)/x^3/(-b^2*x^2+a^2)^2,x,method=_RETURNVERBOSE)
Output:
(-1/2*c/a^2-d/a^2*x+3/2*b^2*d/a^4*x^3+b^4*c/a^6*x^4)/x^2/(-b^2*x^2+a^2)+1/ 4*b*(3*a*d-4*b*c)/a^6*ln(b*x+a)-1/4*b*(3*a*d+4*b*c)*ln(-b*x+a)/a^6+2*b^2*c *ln(x)/a^6
Time = 0.14 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.70 \[ \int \frac {c+d x}{x^3 \left (a^2-b^2 x^2\right )^2} \, dx=-\frac {6 \, a^{2} b^{2} d x^{3} + 4 \, a^{2} b^{2} c x^{2} - 4 \, a^{4} d x - 2 \, a^{4} c + {\left ({\left (4 \, b^{4} c - 3 \, a b^{3} d\right )} x^{4} - {\left (4 \, a^{2} b^{2} c - 3 \, a^{3} b d\right )} x^{2}\right )} \log \left (b x + a\right ) + {\left ({\left (4 \, b^{4} c + 3 \, a b^{3} d\right )} x^{4} - {\left (4 \, a^{2} b^{2} c + 3 \, a^{3} b d\right )} x^{2}\right )} \log \left (b x - a\right ) - 8 \, {\left (b^{4} c x^{4} - a^{2} b^{2} c x^{2}\right )} \log \left (x\right )}{4 \, {\left (a^{6} b^{2} x^{4} - a^{8} x^{2}\right )}} \] Input:
integrate((d*x+c)/x^3/(-b^2*x^2+a^2)^2,x, algorithm="fricas")
Output:
-1/4*(6*a^2*b^2*d*x^3 + 4*a^2*b^2*c*x^2 - 4*a^4*d*x - 2*a^4*c + ((4*b^4*c - 3*a*b^3*d)*x^4 - (4*a^2*b^2*c - 3*a^3*b*d)*x^2)*log(b*x + a) + ((4*b^4*c + 3*a*b^3*d)*x^4 - (4*a^2*b^2*c + 3*a^3*b*d)*x^2)*log(b*x - a) - 8*(b^4*c *x^4 - a^2*b^2*c*x^2)*log(x))/(a^6*b^2*x^4 - a^8*x^2)
Leaf count of result is larger than twice the leaf count of optimal. 311 vs. \(2 (100) = 200\).
Time = 1.29 (sec) , antiderivative size = 311, normalized size of antiderivative = 2.88 \[ \int \frac {c+d x}{x^3 \left (a^2-b^2 x^2\right )^2} \, dx=\frac {a^{2} c + 2 a^{2} d x - 2 b^{2} c x^{2} - 3 b^{2} d x^{3}}{- 2 a^{6} x^{2} + 2 a^{4} b^{2} x^{4}} + \frac {2 b^{2} c \log {\left (x \right )}}{a^{6}} + \frac {b \left (3 a d - 4 b c\right ) \log {\left (x + \frac {- 24 a^{2} b^{2} c d^{2} + 3 a^{2} b d^{2} \cdot \left (3 a d - 4 b c\right ) - 128 b^{4} c^{3} - 16 b^{3} c^{2} \cdot \left (3 a d - 4 b c\right ) + 4 b^{2} c \left (3 a d - 4 b c\right )^{2}}{9 a^{2} b^{2} d^{3} - 144 b^{4} c^{2} d} \right )}}{4 a^{6}} - \frac {b \left (3 a d + 4 b c\right ) \log {\left (x + \frac {- 24 a^{2} b^{2} c d^{2} - 3 a^{2} b d^{2} \cdot \left (3 a d + 4 b c\right ) - 128 b^{4} c^{3} + 16 b^{3} c^{2} \cdot \left (3 a d + 4 b c\right ) + 4 b^{2} c \left (3 a d + 4 b c\right )^{2}}{9 a^{2} b^{2} d^{3} - 144 b^{4} c^{2} d} \right )}}{4 a^{6}} \] Input:
integrate((d*x+c)/x**3/(-b**2*x**2+a**2)**2,x)
Output:
(a**2*c + 2*a**2*d*x - 2*b**2*c*x**2 - 3*b**2*d*x**3)/(-2*a**6*x**2 + 2*a* *4*b**2*x**4) + 2*b**2*c*log(x)/a**6 + b*(3*a*d - 4*b*c)*log(x + (-24*a**2 *b**2*c*d**2 + 3*a**2*b*d**2*(3*a*d - 4*b*c) - 128*b**4*c**3 - 16*b**3*c** 2*(3*a*d - 4*b*c) + 4*b**2*c*(3*a*d - 4*b*c)**2)/(9*a**2*b**2*d**3 - 144*b **4*c**2*d))/(4*a**6) - b*(3*a*d + 4*b*c)*log(x + (-24*a**2*b**2*c*d**2 - 3*a**2*b*d**2*(3*a*d + 4*b*c) - 128*b**4*c**3 + 16*b**3*c**2*(3*a*d + 4*b* c) + 4*b**2*c*(3*a*d + 4*b*c)**2)/(9*a**2*b**2*d**3 - 144*b**4*c**2*d))/(4 *a**6)
Time = 0.03 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.06 \[ \int \frac {c+d x}{x^3 \left (a^2-b^2 x^2\right )^2} \, dx=-\frac {3 \, b^{2} d x^{3} + 2 \, b^{2} c x^{2} - 2 \, a^{2} d x - a^{2} c}{2 \, {\left (a^{4} b^{2} x^{4} - a^{6} x^{2}\right )}} + \frac {2 \, b^{2} c \log \left (x\right )}{a^{6}} - \frac {{\left (4 \, b^{2} c - 3 \, a b d\right )} \log \left (b x + a\right )}{4 \, a^{6}} - \frac {{\left (4 \, b^{2} c + 3 \, a b d\right )} \log \left (b x - a\right )}{4 \, a^{6}} \] Input:
integrate((d*x+c)/x^3/(-b^2*x^2+a^2)^2,x, algorithm="maxima")
Output:
-1/2*(3*b^2*d*x^3 + 2*b^2*c*x^2 - 2*a^2*d*x - a^2*c)/(a^4*b^2*x^4 - a^6*x^ 2) + 2*b^2*c*log(x)/a^6 - 1/4*(4*b^2*c - 3*a*b*d)*log(b*x + a)/a^6 - 1/4*( 4*b^2*c + 3*a*b*d)*log(b*x - a)/a^6
Time = 0.12 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.25 \[ \int \frac {c+d x}{x^3 \left (a^2-b^2 x^2\right )^2} \, dx=\frac {2 \, b^{2} c \log \left ({\left | x \right |}\right )}{a^{6}} - \frac {{\left (4 \, b^{3} c - 3 \, a b^{2} d\right )} \log \left ({\left | b x + a \right |}\right )}{4 \, a^{6} b} - \frac {{\left (4 \, b^{3} c + 3 \, a b^{2} d\right )} \log \left ({\left | b x - a \right |}\right )}{4 \, a^{6} b} - \frac {3 \, a^{2} b^{2} d x^{3} + 2 \, a^{2} b^{2} c x^{2} - 2 \, a^{4} d x - a^{4} c}{2 \, {\left (b x + a\right )} {\left (b x - a\right )} a^{6} x^{2}} \] Input:
integrate((d*x+c)/x^3/(-b^2*x^2+a^2)^2,x, algorithm="giac")
Output:
2*b^2*c*log(abs(x))/a^6 - 1/4*(4*b^3*c - 3*a*b^2*d)*log(abs(b*x + a))/(a^6 *b) - 1/4*(4*b^3*c + 3*a*b^2*d)*log(abs(b*x - a))/(a^6*b) - 1/2*(3*a^2*b^2 *d*x^3 + 2*a^2*b^2*c*x^2 - 2*a^4*d*x - a^4*c)/((b*x + a)*(b*x - a)*a^6*x^2 )
Time = 7.84 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.07 \[ \int \frac {c+d x}{x^3 \left (a^2-b^2 x^2\right )^2} \, dx=\frac {2\,b^2\,c\,\ln \left (x\right )}{a^6}-\frac {\ln \left (a+b\,x\right )\,\left (4\,b^2\,c-3\,a\,b\,d\right )}{4\,a^6}-\frac {\ln \left (a-b\,x\right )\,\left (4\,c\,b^2+3\,a\,d\,b\right )}{4\,a^6}-\frac {\frac {c}{2\,a^2}+\frac {d\,x}{a^2}-\frac {b^2\,c\,x^2}{a^4}-\frac {3\,b^2\,d\,x^3}{2\,a^4}}{a^2\,x^2-b^2\,x^4} \] Input:
int((c + d*x)/(x^3*(a^2 - b^2*x^2)^2),x)
Output:
(2*b^2*c*log(x))/a^6 - (log(a + b*x)*(4*b^2*c - 3*a*b*d))/(4*a^6) - (log(a - b*x)*(4*b^2*c + 3*a*b*d))/(4*a^6) - (c/(2*a^2) + (d*x)/a^2 - (b^2*c*x^2 )/a^4 - (3*b^2*d*x^3)/(2*a^4))/(a^2*x^2 - b^2*x^4)
Time = 0.20 (sec) , antiderivative size = 228, normalized size of antiderivative = 2.11 \[ \int \frac {c+d x}{x^3 \left (a^2-b^2 x^2\right )^2} \, dx=\frac {3 \,\mathrm {log}\left (-b x -a \right ) a^{3} b d \,x^{2}-4 \,\mathrm {log}\left (-b x -a \right ) a^{2} b^{2} c \,x^{2}-3 \,\mathrm {log}\left (-b x -a \right ) a \,b^{3} d \,x^{4}+4 \,\mathrm {log}\left (-b x -a \right ) b^{4} c \,x^{4}-3 \,\mathrm {log}\left (-b x +a \right ) a^{3} b d \,x^{2}-4 \,\mathrm {log}\left (-b x +a \right ) a^{2} b^{2} c \,x^{2}+3 \,\mathrm {log}\left (-b x +a \right ) a \,b^{3} d \,x^{4}+4 \,\mathrm {log}\left (-b x +a \right ) b^{4} c \,x^{4}+8 \,\mathrm {log}\left (x \right ) a^{2} b^{2} c \,x^{2}-8 \,\mathrm {log}\left (x \right ) b^{4} c \,x^{4}-2 a^{4} c -4 a^{4} d x +6 a^{2} b^{2} d \,x^{3}+4 b^{4} c \,x^{4}}{4 a^{6} x^{2} \left (-b^{2} x^{2}+a^{2}\right )} \] Input:
int((d*x+c)/x^3/(-b^2*x^2+a^2)^2,x)
Output:
(3*log( - a - b*x)*a**3*b*d*x**2 - 4*log( - a - b*x)*a**2*b**2*c*x**2 - 3* log( - a - b*x)*a*b**3*d*x**4 + 4*log( - a - b*x)*b**4*c*x**4 - 3*log(a - b*x)*a**3*b*d*x**2 - 4*log(a - b*x)*a**2*b**2*c*x**2 + 3*log(a - b*x)*a*b* *3*d*x**4 + 4*log(a - b*x)*b**4*c*x**4 + 8*log(x)*a**2*b**2*c*x**2 - 8*log (x)*b**4*c*x**4 - 2*a**4*c - 4*a**4*d*x + 6*a**2*b**2*d*x**3 + 4*b**4*c*x* *4)/(4*a**6*x**2*(a**2 - b**2*x**2))