Integrand size = 23, antiderivative size = 108 \[ \int \frac {d+e x}{x^3 \left (a^2-c^2 x^2\right )^2} \, dx=-\frac {d}{a^4 x^2}-\frac {3 e}{2 a^4 x}+\frac {d+e x}{2 a^2 x^2 \left (a^2-c^2 x^2\right )}+\frac {2 c^2 d \log (x)}{a^6}-\frac {c (4 c d+3 a e) \log (a-c x)}{4 a^6}-\frac {c (4 c d-3 a e) \log (a+c x)}{4 a^6} \]
-d/a^4/x^2-3/2*e/a^4/x+1/2*(e*x+d)/a^2/x^2/(-c^2*x^2+a^2)+2*c^2*d*ln(x)/a^ 6-1/4*c*(3*a*e+4*c*d)*ln(-c*x+a)/a^6-1/4*c*(-3*a*e+4*c*d)*ln(c*x+a)/a^6
Time = 0.05 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.84 \[ \int \frac {d+e x}{x^3 \left (a^2-c^2 x^2\right )^2} \, dx=\frac {-\frac {a^2 d}{x^2}-\frac {2 a^2 e}{x}+\frac {a^2 c^2 (d+e x)}{a^2-c^2 x^2}+3 a c e \text {arctanh}\left (\frac {c x}{a}\right )+4 c^2 d \log (x)-2 c^2 d \log \left (a^2-c^2 x^2\right )}{2 a^6} \]
(-((a^2*d)/x^2) - (2*a^2*e)/x + (a^2*c^2*(d + e*x))/(a^2 - c^2*x^2) + 3*a* c*e*ArcTanh[(c*x)/a] + 4*c^2*d*Log[x] - 2*c^2*d*Log[a^2 - c^2*x^2])/(2*a^6 )
Time = 0.38 (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.
\(\displaystyle \int \frac {d+e x}{x^3 \left (a^2-c^2 x^2\right )^2} \, dx\) |
\(\Big \downarrow \) 532 |
\(\displaystyle \frac {c^2 (d+e x)}{2 a^4 \left (a^2-c^2 x^2\right )}-\frac {\int -\frac {\frac {c^2 e x^3}{a^2}+\frac {2 c^2 d x^2}{a^2}+2 e x+2 d}{x^3 \left (a^2-c^2 x^2\right )}dx}{2 a^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \frac {\frac {c^2 e x^3}{a^2}+\frac {2 c^2 d x^2}{a^2}+2 e x+2 d}{x^3 \left (a^2-c^2 x^2\right )}dx}{2 a^2}+\frac {c^2 (d+e x)}{2 a^4 \left (a^2-c^2 x^2\right )}\) |
\(\Big \downarrow \) 2333 |
\(\displaystyle \frac {\int \left (\frac {4 d c^2}{a^4 x}+\frac {(4 c d+3 a e) c^2}{2 a^4 (a-c x)}+\frac {(3 a e-4 c d) c^2}{2 a^4 (a+c x)}+\frac {2 e}{a^2 x^2}+\frac {2 d}{a^2 x^3}\right )dx}{2 a^2}+\frac {c^2 (d+e x)}{2 a^4 \left (a^2-c^2 x^2\right )}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {c^2 (d+e x)}{2 a^4 \left (a^2-c^2 x^2\right )}+\frac {\frac {4 c^2 d \log (x)}{a^4}-\frac {c (3 a e+4 c d) \log (a-c x)}{2 a^4}-\frac {c (4 c d-3 a e) \log (a+c x)}{2 a^4}-\frac {d}{a^2 x^2}-\frac {2 e}{a^2 x}}{2 a^2}\) |
(c^2*(d + e*x))/(2*a^4*(a^2 - c^2*x^2)) + (-(d/(a^2*x^2)) - (2*e)/(a^2*x) + (4*c^2*d*Log[x])/a^4 - (c*(4*c*d + 3*a*e)*Log[a - c*x])/(2*a^4) - (c*(4* c*d - 3*a*e)*Log[a + c*x])/(2*a^4))/(2*a^2)
3.4.13.3.1 Defintions of rubi rules used
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 = 0.10 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.03
method | result | size |
norman | \(\frac {-\frac {d}{2 a^{2}}-\frac {e x}{a^{2}}+\frac {3 c^{2} e \,x^{3}}{2 a^{4}}+\frac {c^{4} d \,x^{4}}{a^{6}}}{x^{2} \left (-c^{2} x^{2}+a^{2}\right )}+\frac {c \left (3 a e -4 c d \right ) \ln \left (c x +a \right )}{4 a^{6}}-\frac {c \left (3 a e +4 c d \right ) \ln \left (-c x +a \right )}{4 a^{6}}+\frac {2 c^{2} d \ln \left (x \right )}{a^{6}}\) | \(111\) |
default | \(\frac {c \left (3 a e -4 c d \right ) \ln \left (c x +a \right )}{4 a^{6}}-\frac {\left (a e -c d \right ) c}{4 a^{5} \left (c x +a \right )}-\frac {e}{a^{4} x}-\frac {d}{2 a^{4} x^{2}}+\frac {2 c^{2} d \ln \left (x \right )}{a^{6}}-\frac {c \left (3 a e +4 c d \right ) \ln \left (-c x +a \right )}{4 a^{6}}+\frac {\left (a e +c d \right ) c}{4 a^{5} \left (-c x +a \right )}\) | \(116\) |
risch | \(\frac {\frac {3 c^{2} e \,x^{3}}{2 a^{4}}+\frac {c^{2} d \,x^{2}}{a^{4}}-\frac {e x}{a^{2}}-\frac {d}{2 a^{2}}}{x^{2} \left (-c^{2} x^{2}+a^{2}\right )}+\frac {3 c \ln \left (c x +a \right ) e}{4 a^{5}}-\frac {c^{2} \ln \left (c x +a \right ) d}{a^{6}}-\frac {3 c \ln \left (c x -a \right ) e}{4 a^{5}}-\frac {c^{2} \ln \left (c x -a \right ) d}{a^{6}}+\frac {2 d \,c^{2} \ln \left (-x \right )}{a^{6}}\) | \(130\) |
parallelrisch | \(\frac {8 \ln \left (x \right ) x^{4} c^{4} d -3 \ln \left (c x -a \right ) x^{4} a \,c^{3} e -4 \ln \left (c x -a \right ) x^{4} c^{4} d +3 \ln \left (c x +a \right ) x^{4} a \,c^{3} e -4 \ln \left (c x +a \right ) x^{4} c^{4} d -4 x^{4} c^{4} d -8 \ln \left (x \right ) x^{2} a^{2} c^{2} d +3 \ln \left (c x -a \right ) x^{2} a^{3} c e +4 \ln \left (c x -a \right ) x^{2} a^{2} c^{2} d -3 \ln \left (c x +a \right ) x^{2} a^{3} c e +4 \ln \left (c x +a \right ) x^{2} a^{2} c^{2} d -6 x^{3} a^{2} c^{2} e +4 x \,a^{4} e +2 a^{4} d}{4 a^{6} x^{2} \left (c^{2} x^{2}-a^{2}\right )}\) | \(222\) |
(-1/2*d/a^2-e/a^2*x+3/2*c^2*e/a^4*x^3+c^4*d/a^6*x^4)/x^2/(-c^2*x^2+a^2)+1/ 4*c*(3*a*e-4*c*d)/a^6*ln(c*x+a)-1/4*c*(3*a*e+4*c*d)*ln(-c*x+a)/a^6+2*c^2*d *ln(x)/a^6
Time = 0.27 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.70 \[ \int \frac {d+e x}{x^3 \left (a^2-c^2 x^2\right )^2} \, dx=-\frac {6 \, a^{2} c^{2} e x^{3} + 4 \, a^{2} c^{2} d x^{2} - 4 \, a^{4} e x - 2 \, a^{4} d + {\left ({\left (4 \, c^{4} d - 3 \, a c^{3} e\right )} x^{4} - {\left (4 \, a^{2} c^{2} d - 3 \, a^{3} c e\right )} x^{2}\right )} \log \left (c x + a\right ) + {\left ({\left (4 \, c^{4} d + 3 \, a c^{3} e\right )} x^{4} - {\left (4 \, a^{2} c^{2} d + 3 \, a^{3} c e\right )} x^{2}\right )} \log \left (c x - a\right ) - 8 \, {\left (c^{4} d x^{4} - a^{2} c^{2} d x^{2}\right )} \log \left (x\right )}{4 \, {\left (a^{6} c^{2} x^{4} - a^{8} x^{2}\right )}} \]
-1/4*(6*a^2*c^2*e*x^3 + 4*a^2*c^2*d*x^2 - 4*a^4*e*x - 2*a^4*d + ((4*c^4*d - 3*a*c^3*e)*x^4 - (4*a^2*c^2*d - 3*a^3*c*e)*x^2)*log(c*x + a) + ((4*c^4*d + 3*a*c^3*e)*x^4 - (4*a^2*c^2*d + 3*a^3*c*e)*x^2)*log(c*x - a) - 8*(c^4*d *x^4 - a^2*c^2*d*x^2)*log(x))/(a^6*c^2*x^4 - a^8*x^2)
Leaf count of result is larger than twice the leaf count of optimal. 311 vs. \(2 (104) = 208\).
Time = 1.63 (sec) , antiderivative size = 311, normalized size of antiderivative = 2.88 \[ \int \frac {d+e x}{x^3 \left (a^2-c^2 x^2\right )^2} \, dx=\frac {a^{2} d + 2 a^{2} e x - 2 c^{2} d x^{2} - 3 c^{2} e x^{3}}{- 2 a^{6} x^{2} + 2 a^{4} c^{2} x^{4}} + \frac {2 c^{2} d \log {\left (x \right )}}{a^{6}} + \frac {c \left (3 a e - 4 c d\right ) \log {\left (x + \frac {- 24 a^{2} c^{2} d e^{2} + 3 a^{2} c e^{2} \cdot \left (3 a e - 4 c d\right ) - 128 c^{4} d^{3} - 16 c^{3} d^{2} \cdot \left (3 a e - 4 c d\right ) + 4 c^{2} d \left (3 a e - 4 c d\right )^{2}}{9 a^{2} c^{2} e^{3} - 144 c^{4} d^{2} e} \right )}}{4 a^{6}} - \frac {c \left (3 a e + 4 c d\right ) \log {\left (x + \frac {- 24 a^{2} c^{2} d e^{2} - 3 a^{2} c e^{2} \cdot \left (3 a e + 4 c d\right ) - 128 c^{4} d^{3} + 16 c^{3} d^{2} \cdot \left (3 a e + 4 c d\right ) + 4 c^{2} d \left (3 a e + 4 c d\right )^{2}}{9 a^{2} c^{2} e^{3} - 144 c^{4} d^{2} e} \right )}}{4 a^{6}} \]
(a**2*d + 2*a**2*e*x - 2*c**2*d*x**2 - 3*c**2*e*x**3)/(-2*a**6*x**2 + 2*a* *4*c**2*x**4) + 2*c**2*d*log(x)/a**6 + c*(3*a*e - 4*c*d)*log(x + (-24*a**2 *c**2*d*e**2 + 3*a**2*c*e**2*(3*a*e - 4*c*d) - 128*c**4*d**3 - 16*c**3*d** 2*(3*a*e - 4*c*d) + 4*c**2*d*(3*a*e - 4*c*d)**2)/(9*a**2*c**2*e**3 - 144*c **4*d**2*e))/(4*a**6) - c*(3*a*e + 4*c*d)*log(x + (-24*a**2*c**2*d*e**2 - 3*a**2*c*e**2*(3*a*e + 4*c*d) - 128*c**4*d**3 + 16*c**3*d**2*(3*a*e + 4*c* d) + 4*c**2*d*(3*a*e + 4*c*d)**2)/(9*a**2*c**2*e**3 - 144*c**4*d**2*e))/(4 *a**6)
Time = 0.20 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.06 \[ \int \frac {d+e x}{x^3 \left (a^2-c^2 x^2\right )^2} \, dx=-\frac {3 \, c^{2} e x^{3} + 2 \, c^{2} d x^{2} - 2 \, a^{2} e x - a^{2} d}{2 \, {\left (a^{4} c^{2} x^{4} - a^{6} x^{2}\right )}} + \frac {2 \, c^{2} d \log \left (x\right )}{a^{6}} - \frac {{\left (4 \, c^{2} d - 3 \, a c e\right )} \log \left (c x + a\right )}{4 \, a^{6}} - \frac {{\left (4 \, c^{2} d + 3 \, a c e\right )} \log \left (c x - a\right )}{4 \, a^{6}} \]
-1/2*(3*c^2*e*x^3 + 2*c^2*d*x^2 - 2*a^2*e*x - a^2*d)/(a^4*c^2*x^4 - a^6*x^ 2) + 2*c^2*d*log(x)/a^6 - 1/4*(4*c^2*d - 3*a*c*e)*log(c*x + a)/a^6 - 1/4*( 4*c^2*d + 3*a*c*e)*log(c*x - a)/a^6
Time = 0.28 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.25 \[ \int \frac {d+e x}{x^3 \left (a^2-c^2 x^2\right )^2} \, dx=\frac {2 \, c^{2} d \log \left ({\left | x \right |}\right )}{a^{6}} - \frac {{\left (4 \, c^{3} d - 3 \, a c^{2} e\right )} \log \left ({\left | c x + a \right |}\right )}{4 \, a^{6} c} - \frac {{\left (4 \, c^{3} d + 3 \, a c^{2} e\right )} \log \left ({\left | c x - a \right |}\right )}{4 \, a^{6} c} - \frac {3 \, a^{2} c^{2} e x^{3} + 2 \, a^{2} c^{2} d x^{2} - 2 \, a^{4} e x - a^{4} d}{2 \, {\left (c x + a\right )} {\left (c x - a\right )} a^{6} x^{2}} \]
2*c^2*d*log(abs(x))/a^6 - 1/4*(4*c^3*d - 3*a*c^2*e)*log(abs(c*x + a))/(a^6 *c) - 1/4*(4*c^3*d + 3*a*c^2*e)*log(abs(c*x - a))/(a^6*c) - 1/2*(3*a^2*c^2 *e*x^3 + 2*a^2*c^2*d*x^2 - 2*a^4*e*x - a^4*d)/((c*x + a)*(c*x - a)*a^6*x^2 )
Time = 10.25 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.07 \[ \int \frac {d+e x}{x^3 \left (a^2-c^2 x^2\right )^2} \, dx=\frac {2\,c^2\,d\,\ln \left (x\right )}{a^6}-\frac {\ln \left (a+c\,x\right )\,\left (4\,c^2\,d-3\,a\,c\,e\right )}{4\,a^6}-\frac {\ln \left (a-c\,x\right )\,\left (4\,d\,c^2+3\,a\,e\,c\right )}{4\,a^6}-\frac {\frac {d}{2\,a^2}+\frac {e\,x}{a^2}-\frac {c^2\,d\,x^2}{a^4}-\frac {3\,c^2\,e\,x^3}{2\,a^4}}{a^2\,x^2-c^2\,x^4} \]