Integrand size = 26, antiderivative size = 103 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^4} \, dx=\frac {b^4 x}{e^4}-\frac {(b d-a e)^4}{3 e^5 (d+e x)^3}+\frac {2 b (b d-a e)^3}{e^5 (d+e x)^2}-\frac {6 b^2 (b d-a e)^2}{e^5 (d+e x)}-\frac {4 b^3 (b d-a e) \log (d+e x)}{e^5} \] Output:
b^4*x/e^4-1/3*(-a*e+b*d)^4/e^5/(e*x+d)^3+2*b*(-a*e+b*d)^3/e^5/(e*x+d)^2-6* b^2*(-a*e+b*d)^2/e^5/(e*x+d)-4*b^3*(-a*e+b*d)*ln(e*x+d)/e^5
Time = 0.07 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.58 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^4} \, dx=-\frac {a^4 e^4+2 a^3 b e^3 (d+3 e x)+6 a^2 b^2 e^2 \left (d^2+3 d e x+3 e^2 x^2\right )-2 a b^3 d e \left (11 d^2+27 d e x+18 e^2 x^2\right )+b^4 \left (13 d^4+27 d^3 e x+9 d^2 e^2 x^2-9 d e^3 x^3-3 e^4 x^4\right )+12 b^3 (b d-a e) (d+e x)^3 \log (d+e x)}{3 e^5 (d+e x)^3} \] Input:
Integrate[(a^2 + 2*a*b*x + b^2*x^2)^2/(d + e*x)^4,x]
Output:
-1/3*(a^4*e^4 + 2*a^3*b*e^3*(d + 3*e*x) + 6*a^2*b^2*e^2*(d^2 + 3*d*e*x + 3 *e^2*x^2) - 2*a*b^3*d*e*(11*d^2 + 27*d*e*x + 18*e^2*x^2) + b^4*(13*d^4 + 2 7*d^3*e*x + 9*d^2*e^2*x^2 - 9*d*e^3*x^3 - 3*e^4*x^4) + 12*b^3*(b*d - a*e)* (d + e*x)^3*Log[d + e*x])/(e^5*(d + e*x)^3)
Time = 0.48 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1098, 27, 49, 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.
\(\displaystyle \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^4} \, dx\) |
\(\Big \downarrow \) 1098 |
\(\displaystyle \frac {\int \frac {b^4 (a+b x)^4}{(d+e x)^4}dx}{b^4}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int \frac {(a+b x)^4}{(d+e x)^4}dx\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \int \left (-\frac {4 b^3 (b d-a e)}{e^4 (d+e x)}+\frac {6 b^2 (b d-a e)^2}{e^4 (d+e x)^2}-\frac {4 b (b d-a e)^3}{e^4 (d+e x)^3}+\frac {(a e-b d)^4}{e^4 (d+e x)^4}+\frac {b^4}{e^4}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {4 b^3 (b d-a e) \log (d+e x)}{e^5}-\frac {6 b^2 (b d-a e)^2}{e^5 (d+e x)}+\frac {2 b (b d-a e)^3}{e^5 (d+e x)^2}-\frac {(b d-a e)^4}{3 e^5 (d+e x)^3}+\frac {b^4 x}{e^4}\) |
Input:
Int[(a^2 + 2*a*b*x + b^2*x^2)^2/(d + e*x)^4,x]
Output:
(b^4*x)/e^4 - (b*d - a*e)^4/(3*e^5*(d + e*x)^3) + (2*b*(b*d - a*e)^3)/(e^5 *(d + e*x)^2) - (6*b^2*(b*d - a*e)^2)/(e^5*(d + e*x)) - (4*b^3*(b*d - a*e) *Log[d + e*x])/e^5
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[((d_.) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_ Symbol] :> Simp[1/c^p Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[ {a, b, c, d, e, m}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Time = 1.14 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.73
method | result | size |
default | \(\frac {b^{4} x}{e^{4}}-\frac {a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}}{3 e^{5} \left (e x +d \right )^{3}}+\frac {4 b^{3} \left (a e -b d \right ) \ln \left (e x +d \right )}{e^{5}}-\frac {6 b^{2} \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )}{e^{5} \left (e x +d \right )}-\frac {2 b \left (e^{3} a^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}{e^{5} \left (e x +d \right )^{2}}\) | \(178\) |
norman | \(\frac {\frac {b^{4} x^{4}}{e}-\frac {a^{4} e^{4}+2 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-22 a \,b^{3} d^{3} e +22 b^{4} d^{4}}{3 e^{5}}-\frac {3 \left (2 a^{2} b^{2} e^{2}-4 a \,b^{3} d e +4 d^{2} b^{4}\right ) x^{2}}{e^{3}}-\frac {\left (2 a^{3} b \,e^{3}+6 a^{2} b^{2} d \,e^{2}-18 d^{2} e a \,b^{3}+18 b^{4} d^{3}\right ) x}{e^{4}}}{\left (e x +d \right )^{3}}+\frac {4 b^{3} \left (a e -b d \right ) \ln \left (e x +d \right )}{e^{5}}\) | \(180\) |
risch | \(\frac {b^{4} x}{e^{4}}+\frac {\left (-6 a^{2} b^{2} e^{3}+12 d \,e^{2} a \,b^{3}-6 d^{2} e \,b^{4}\right ) x^{2}-2 b \left (e^{3} a^{3}+3 a^{2} b d \,e^{2}-9 a \,b^{2} d^{2} e +5 b^{3} d^{3}\right ) x -\frac {a^{4} e^{4}+2 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-22 a \,b^{3} d^{3} e +13 b^{4} d^{4}}{3 e}}{e^{4} \left (e x +d \right )^{3}}+\frac {4 b^{3} \ln \left (e x +d \right ) a}{e^{4}}-\frac {4 b^{4} \ln \left (e x +d \right ) d}{e^{5}}\) | \(182\) |
parallelrisch | \(\frac {-a^{4} e^{4}+36 \ln \left (e x +d \right ) x^{2} a \,b^{3} d \,e^{3}+36 \ln \left (e x +d \right ) x a \,b^{3} d^{2} e^{2}-2 a^{3} b d \,e^{3}+22 a \,b^{3} d^{3} e -6 a^{2} b^{2} d^{2} e^{2}-22 b^{4} d^{4}-36 \ln \left (e x +d \right ) x \,b^{4} d^{3} e -36 \ln \left (e x +d \right ) x^{2} b^{4} d^{2} e^{2}-36 x^{2} b^{4} d^{2} e^{2}-6 x \,a^{3} b \,e^{4}-54 x \,b^{4} d^{3} e +36 x^{2} a \,b^{3} d \,e^{3}+12 \ln \left (e x +d \right ) a \,b^{3} d^{3} e -18 x \,a^{2} b^{2} d \,e^{3}+54 x a \,b^{3} d^{2} e^{2}+12 \ln \left (e x +d \right ) x^{3} a \,b^{3} e^{4}-12 \ln \left (e x +d \right ) x^{3} b^{4} d \,e^{3}-18 x^{2} a^{2} b^{2} e^{4}+3 b^{4} x^{4} e^{4}-12 \ln \left (e x +d \right ) b^{4} d^{4}}{3 e^{5} \left (e x +d \right )^{3}}\) | \(302\) |
Input:
int((b^2*x^2+2*a*b*x+a^2)^2/(e*x+d)^4,x,method=_RETURNVERBOSE)
Output:
b^4*x/e^4-1/3*(a^4*e^4-4*a^3*b*d*e^3+6*a^2*b^2*d^2*e^2-4*a*b^3*d^3*e+b^4*d ^4)/e^5/(e*x+d)^3+4*b^3/e^5*(a*e-b*d)*ln(e*x+d)-6*b^2/e^5*(a^2*e^2-2*a*b*d *e+b^2*d^2)/(e*x+d)-2*b/e^5*(a^3*e^3-3*a^2*b*d*e^2+3*a*b^2*d^2*e-b^3*d^3)/ (e*x+d)^2
Leaf count of result is larger than twice the leaf count of optimal. 292 vs. \(2 (101) = 202\).
Time = 0.09 (sec) , antiderivative size = 292, normalized size of antiderivative = 2.83 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^4} \, dx=\frac {3 \, b^{4} e^{4} x^{4} + 9 \, b^{4} d e^{3} x^{3} - 13 \, b^{4} d^{4} + 22 \, a b^{3} d^{3} e - 6 \, a^{2} b^{2} d^{2} e^{2} - 2 \, a^{3} b d e^{3} - a^{4} e^{4} - 9 \, {\left (b^{4} d^{2} e^{2} - 4 \, a b^{3} d e^{3} + 2 \, a^{2} b^{2} e^{4}\right )} x^{2} - 3 \, {\left (9 \, b^{4} d^{3} e - 18 \, a b^{3} d^{2} e^{2} + 6 \, a^{2} b^{2} d e^{3} + 2 \, a^{3} b e^{4}\right )} x - 12 \, {\left (b^{4} d^{4} - a b^{3} d^{3} e + {\left (b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} + 3 \, {\left (b^{4} d^{2} e^{2} - a b^{3} d e^{3}\right )} x^{2} + 3 \, {\left (b^{4} d^{3} e - a b^{3} d^{2} e^{2}\right )} x\right )} \log \left (e x + d\right )}{3 \, {\left (e^{8} x^{3} + 3 \, d e^{7} x^{2} + 3 \, d^{2} e^{6} x + d^{3} e^{5}\right )}} \] Input:
integrate((b^2*x^2+2*a*b*x+a^2)^2/(e*x+d)^4,x, algorithm="fricas")
Output:
1/3*(3*b^4*e^4*x^4 + 9*b^4*d*e^3*x^3 - 13*b^4*d^4 + 22*a*b^3*d^3*e - 6*a^2 *b^2*d^2*e^2 - 2*a^3*b*d*e^3 - a^4*e^4 - 9*(b^4*d^2*e^2 - 4*a*b^3*d*e^3 + 2*a^2*b^2*e^4)*x^2 - 3*(9*b^4*d^3*e - 18*a*b^3*d^2*e^2 + 6*a^2*b^2*d*e^3 + 2*a^3*b*e^4)*x - 12*(b^4*d^4 - a*b^3*d^3*e + (b^4*d*e^3 - a*b^3*e^4)*x^3 + 3*(b^4*d^2*e^2 - a*b^3*d*e^3)*x^2 + 3*(b^4*d^3*e - a*b^3*d^2*e^2)*x)*log (e*x + d))/(e^8*x^3 + 3*d*e^7*x^2 + 3*d^2*e^6*x + d^3*e^5)
Leaf count of result is larger than twice the leaf count of optimal. 209 vs. \(2 (94) = 188\).
Time = 0.95 (sec) , antiderivative size = 209, normalized size of antiderivative = 2.03 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^4} \, dx=\frac {b^{4} x}{e^{4}} + \frac {4 b^{3} \left (a e - b d\right ) \log {\left (d + e x \right )}}{e^{5}} + \frac {- a^{4} e^{4} - 2 a^{3} b d e^{3} - 6 a^{2} b^{2} d^{2} e^{2} + 22 a b^{3} d^{3} e - 13 b^{4} d^{4} + x^{2} \left (- 18 a^{2} b^{2} e^{4} + 36 a b^{3} d e^{3} - 18 b^{4} d^{2} e^{2}\right ) + x \left (- 6 a^{3} b e^{4} - 18 a^{2} b^{2} d e^{3} + 54 a b^{3} d^{2} e^{2} - 30 b^{4} d^{3} e\right )}{3 d^{3} e^{5} + 9 d^{2} e^{6} x + 9 d e^{7} x^{2} + 3 e^{8} x^{3}} \] Input:
integrate((b**2*x**2+2*a*b*x+a**2)**2/(e*x+d)**4,x)
Output:
b**4*x/e**4 + 4*b**3*(a*e - b*d)*log(d + e*x)/e**5 + (-a**4*e**4 - 2*a**3* b*d*e**3 - 6*a**2*b**2*d**2*e**2 + 22*a*b**3*d**3*e - 13*b**4*d**4 + x**2* (-18*a**2*b**2*e**4 + 36*a*b**3*d*e**3 - 18*b**4*d**2*e**2) + x*(-6*a**3*b *e**4 - 18*a**2*b**2*d*e**3 + 54*a*b**3*d**2*e**2 - 30*b**4*d**3*e))/(3*d* *3*e**5 + 9*d**2*e**6*x + 9*d*e**7*x**2 + 3*e**8*x**3)
Time = 0.04 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.95 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^4} \, dx=\frac {b^{4} x}{e^{4}} - \frac {13 \, b^{4} d^{4} - 22 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} + 2 \, a^{3} b d e^{3} + a^{4} e^{4} + 18 \, {\left (b^{4} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 6 \, {\left (5 \, b^{4} d^{3} e - 9 \, a b^{3} d^{2} e^{2} + 3 \, a^{2} b^{2} d e^{3} + a^{3} b e^{4}\right )} x}{3 \, {\left (e^{8} x^{3} + 3 \, d e^{7} x^{2} + 3 \, d^{2} e^{6} x + d^{3} e^{5}\right )}} - \frac {4 \, {\left (b^{4} d - a b^{3} e\right )} \log \left (e x + d\right )}{e^{5}} \] Input:
integrate((b^2*x^2+2*a*b*x+a^2)^2/(e*x+d)^4,x, algorithm="maxima")
Output:
b^4*x/e^4 - 1/3*(13*b^4*d^4 - 22*a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 + 2*a^3*b *d*e^3 + a^4*e^4 + 18*(b^4*d^2*e^2 - 2*a*b^3*d*e^3 + a^2*b^2*e^4)*x^2 + 6* (5*b^4*d^3*e - 9*a*b^3*d^2*e^2 + 3*a^2*b^2*d*e^3 + a^3*b*e^4)*x)/(e^8*x^3 + 3*d*e^7*x^2 + 3*d^2*e^6*x + d^3*e^5) - 4*(b^4*d - a*b^3*e)*log(e*x + d)/ e^5
Time = 0.17 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.72 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^4} \, dx=\frac {b^{4} x}{e^{4}} - \frac {4 \, {\left (b^{4} d - a b^{3} e\right )} \log \left ({\left | e x + d \right |}\right )}{e^{5}} - \frac {13 \, b^{4} d^{4} - 22 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} + 2 \, a^{3} b d e^{3} + a^{4} e^{4} + 18 \, {\left (b^{4} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 6 \, {\left (5 \, b^{4} d^{3} e - 9 \, a b^{3} d^{2} e^{2} + 3 \, a^{2} b^{2} d e^{3} + a^{3} b e^{4}\right )} x}{3 \, {\left (e x + d\right )}^{3} e^{5}} \] Input:
integrate((b^2*x^2+2*a*b*x+a^2)^2/(e*x+d)^4,x, algorithm="giac")
Output:
b^4*x/e^4 - 4*(b^4*d - a*b^3*e)*log(abs(e*x + d))/e^5 - 1/3*(13*b^4*d^4 - 22*a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 + 2*a^3*b*d*e^3 + a^4*e^4 + 18*(b^4*d^2 *e^2 - 2*a*b^3*d*e^3 + a^2*b^2*e^4)*x^2 + 6*(5*b^4*d^3*e - 9*a*b^3*d^2*e^2 + 3*a^2*b^2*d*e^3 + a^3*b*e^4)*x)/((e*x + d)^3*e^5)
Time = 9.04 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.98 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^4} \, dx=\frac {b^4\,x}{e^4}-\frac {\ln \left (d+e\,x\right )\,\left (4\,b^4\,d-4\,a\,b^3\,e\right )}{e^5}-\frac {\frac {a^4\,e^4+2\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-22\,a\,b^3\,d^3\,e+13\,b^4\,d^4}{3\,e}+x\,\left (2\,a^3\,b\,e^3+6\,a^2\,b^2\,d\,e^2-18\,a\,b^3\,d^2\,e+10\,b^4\,d^3\right )+x^2\,\left (6\,a^2\,b^2\,e^3-12\,a\,b^3\,d\,e^2+6\,b^4\,d^2\,e\right )}{d^3\,e^4+3\,d^2\,e^5\,x+3\,d\,e^6\,x^2+e^7\,x^3} \] Input:
int((a^2 + b^2*x^2 + 2*a*b*x)^2/(d + e*x)^4,x)
Output:
(b^4*x)/e^4 - (log(d + e*x)*(4*b^4*d - 4*a*b^3*e))/e^5 - ((a^4*e^4 + 13*b^ 4*d^4 + 6*a^2*b^2*d^2*e^2 - 22*a*b^3*d^3*e + 2*a^3*b*d*e^3)/(3*e) + x*(10* b^4*d^3 + 2*a^3*b*e^3 + 6*a^2*b^2*d*e^2 - 18*a*b^3*d^2*e) + x^2*(6*b^4*d^2 *e + 6*a^2*b^2*e^3 - 12*a*b^3*d*e^2))/(d^3*e^4 + e^7*x^3 + 3*d^2*e^5*x + 3 *d*e^6*x^2)
Time = 0.20 (sec) , antiderivative size = 309, normalized size of antiderivative = 3.00 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^4} \, dx=\frac {12 \,\mathrm {log}\left (e x +d \right ) a \,b^{3} d^{4} e +36 \,\mathrm {log}\left (e x +d \right ) a \,b^{3} d^{3} e^{2} x +36 \,\mathrm {log}\left (e x +d \right ) a \,b^{3} d^{2} e^{3} x^{2}+12 \,\mathrm {log}\left (e x +d \right ) a \,b^{3} d \,e^{4} x^{3}-12 \,\mathrm {log}\left (e x +d \right ) b^{4} d^{5}-36 \,\mathrm {log}\left (e x +d \right ) b^{4} d^{4} e x -36 \,\mathrm {log}\left (e x +d \right ) b^{4} d^{3} e^{2} x^{2}-12 \,\mathrm {log}\left (e x +d \right ) b^{4} d^{2} e^{3} x^{3}-a^{4} d \,e^{4}-2 a^{3} b \,d^{2} e^{3}-6 a^{3} b d \,e^{4} x +6 a^{2} b^{2} e^{5} x^{3}+10 a \,b^{3} d^{4} e +18 a \,b^{3} d^{3} e^{2} x -12 a \,b^{3} d \,e^{4} x^{3}-10 b^{4} d^{5}-18 b^{4} d^{4} e x +12 b^{4} d^{2} e^{3} x^{3}+3 b^{4} d \,e^{4} x^{4}}{3 d \,e^{5} \left (e^{3} x^{3}+3 d \,e^{2} x^{2}+3 d^{2} e x +d^{3}\right )} \] Input:
int((b^2*x^2+2*a*b*x+a^2)^2/(e*x+d)^4,x)
Output:
(12*log(d + e*x)*a*b**3*d**4*e + 36*log(d + e*x)*a*b**3*d**3*e**2*x + 36*l og(d + e*x)*a*b**3*d**2*e**3*x**2 + 12*log(d + e*x)*a*b**3*d*e**4*x**3 - 1 2*log(d + e*x)*b**4*d**5 - 36*log(d + e*x)*b**4*d**4*e*x - 36*log(d + e*x) *b**4*d**3*e**2*x**2 - 12*log(d + e*x)*b**4*d**2*e**3*x**3 - a**4*d*e**4 - 2*a**3*b*d**2*e**3 - 6*a**3*b*d*e**4*x + 6*a**2*b**2*e**5*x**3 + 10*a*b** 3*d**4*e + 18*a*b**3*d**3*e**2*x - 12*a*b**3*d*e**4*x**3 - 10*b**4*d**5 - 18*b**4*d**4*e*x + 12*b**4*d**2*e**3*x**3 + 3*b**4*d*e**4*x**4)/(3*d*e**5* (d**3 + 3*d**2*e*x + 3*d*e**2*x**2 + e**3*x**3))