\(\int \frac {(a^2+2 a b x+b^2 x^2)^2}{(d+e x)^4} \, dx\) [1473]

Optimal result

Integrand size = 26, antiderivative size = 103 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^4} \, dx=\frac {b^4 x}{e^4}-\frac {(b d-a e)^4}{3 e^5 (d+e x)^3}+\frac {2 b (b d-a e)^3}{e^5 (d+e x)^2}-\frac {6 b^2 (b d-a e)^2}{e^5 (d+e x)}-\frac {4 b^3 (b d-a e) \log (d+e x)}{e^5} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {27, 45} \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^4} \, dx=-\frac {4 b^3 (b d-a e) \log (d+e x)}{e^5}-\frac {6 b^2 (b d-a e)^2}{e^5 (d+e x)}+\frac {2 b (b d-a e)^3}{e^5 (d+e x)^2}-\frac {(b d-a e)^4}{3 e^5 (d+e x)^3}+\frac {b^4 x}{e^4} \]

[In]

Int[(a^2 + 2*a*b*x + b^2*x^2)^2/(d + e*x)^4,x]

[Out]

(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

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 45

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps \begin{align*} \text {integral}& = \int \frac {(a+b x)^4}{(d+e x)^4} \, dx \\ & = \int \left (\frac {b^4}{e^4}+\frac {(-b d+a e)^4}{e^4 (d+e x)^4}-\frac {4 b (b d-a e)^3}{e^4 (d+e x)^3}+\frac {6 b^2 (b d-a e)^2}{e^4 (d+e x)^2}-\frac {4 b^3 (b d-a e)}{e^4 (d+e x)}\right ) \, 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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.58 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^4} \, dx=-\frac {a^4 e^4+2 a^3 b e^3 (d+3 e x)+6 a^2 b^2 e^2 \left (d^2+3 d e x+3 e^2 x^2\right )-2 a b^3 d e \left (11 d^2+27 d e x+18 e^2 x^2\right )+b^4 \left (13 d^4+27 d^3 e x+9 d^2 e^2 x^2-9 d e^3 x^3-3 e^4 x^4\right )+12 b^3 (b d-a e) (d+e x)^3 \log (d+e x)}{3 e^5 (d+e x)^3} \]

[In]

Integrate[(a^2 + 2*a*b*x + b^2*x^2)^2/(d + e*x)^4,x]

[Out]

-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 + 27*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)

Maple [A] (verified)

Time = 2.23 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.73

method	result	size
default	\(\frac {b^{4} x}{e^{4}}-\frac {6 b^{2} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}{e^{5} \left (e x +d \right )}-\frac {e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}}{3 e^{5} \left (e x +d \right )^{3}}-\frac {2 b \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}{e^{5} \left (e x +d \right )^{2}}+\frac {4 b^{3} \left (a e -b d \right ) \ln \left (e x +d \right )}{e^{5}}\)	\(178\)
norman	\(\frac {\frac {b^{4} x^{4}}{e}-\frac {e^{4} a^{4}+2 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-22 a \,b^{3} d^{3} e +22 b^{4} d^{4}}{3 e^{5}}-\frac {3 \left (2 a^{2} b^{2} e^{2}-4 a \,b^{3} d e +4 b^{4} d^{2}\right ) x^{2}}{e^{3}}-\frac {\left (2 a^{3} b \,e^{3}+6 a^{2} b^{2} d \,e^{2}-18 d^{2} e a \,b^{3}+18 b^{4} d^{3}\right ) x}{e^{4}}}{\left (e x +d \right )^{3}}+\frac {4 b^{3} \left (a e -b d \right ) \ln \left (e x +d \right )}{e^{5}}\)	\(180\)
risch	\(\frac {b^{4} x}{e^{4}}+\frac {\left (-6 a^{2} b^{2} e^{3}+12 d \,e^{2} a \,b^{3}-6 d^{2} e \,b^{4}\right ) x^{2}-2 b \left (a^{3} e^{3}+3 a^{2} b d \,e^{2}-9 a \,b^{2} d^{2} e +5 b^{3} d^{3}\right ) x -\frac {e^{4} a^{4}+2 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-22 a \,b^{3} d^{3} e +13 b^{4} d^{4}}{3 e}}{e^{4} \left (e x +d \right )^{3}}+\frac {4 b^{3} \ln \left (e x +d \right ) a}{e^{4}}-\frac {4 b^{4} \ln \left (e x +d \right ) d}{e^{5}}\)	\(182\)
parallelrisch	\(\frac {-e^{4} a^{4}-22 b^{4} d^{4}+22 a \,b^{3} d^{3} e -6 b^{2} e^{2} d^{2} a^{2}-2 b \,e^{3} d \,a^{3}+36 \ln \left (e x +d \right ) x^{2} a \,b^{3} d \,e^{3}-54 x \,b^{4} d^{3} e +36 \ln \left (e x +d \right ) x a \,b^{3} d^{2} e^{2}-12 \ln \left (e x +d \right ) b^{4} d^{4}-36 \ln \left (e x +d \right ) x^{2} b^{4} d^{2} e^{2}+12 \ln \left (e x +d \right ) x^{3} a \,b^{3} e^{4}-12 \ln \left (e x +d \right ) x^{3} b^{4} d \,e^{3}+3 b^{4} x^{4} e^{4}-18 x^{2} a^{2} b^{2} e^{4}-36 x^{2} b^{4} d^{2} e^{2}-6 x \,a^{3} b \,e^{4}+36 x^{2} a \,b^{3} d \,e^{3}-36 \ln \left (e x +d \right ) x \,b^{4} d^{3} e +12 \ln \left (e x +d \right ) a \,b^{3} d^{3} e -18 x \,a^{2} b^{2} d \,e^{3}+54 x a \,b^{3} d^{2} e^{2}}{3 e^{5} \left (e x +d \right )^{3}}\)	\(302\)

[In]

int((b^2*x^2+2*a*b*x+a^2)^2/(e*x+d)^4,x,method=_RETURNVERBOSE)

[Out]

b^4*x/e^4-6*b^2/e^5*(a^2*e^2-2*a*b*d*e+b^2*d^2)/(e*x+d)-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)^3-2*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+4*b^3/e^5*(a*e-b*d
)*ln(e*x+d)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 292 vs. \(2 (101) = 202\).

Time = 0.29 (sec) , antiderivative size = 292, normalized size of antiderivative = 2.83 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^4} \, dx=\frac {3 \, b^{4} e^{4} x^{4} + 9 \, b^{4} d e^{3} x^{3} - 13 \, b^{4} d^{4} + 22 \, a b^{3} d^{3} e - 6 \, a^{2} b^{2} d^{2} e^{2} - 2 \, a^{3} b d e^{3} - a^{4} e^{4} - 9 \, {\left (b^{4} d^{2} e^{2} - 4 \, a b^{3} d e^{3} + 2 \, a^{2} b^{2} e^{4}\right )} x^{2} - 3 \, {\left (9 \, b^{4} d^{3} e - 18 \, a b^{3} d^{2} e^{2} + 6 \, a^{2} b^{2} d e^{3} + 2 \, a^{3} b e^{4}\right )} x - 12 \, {\left (b^{4} d^{4} - a b^{3} d^{3} e + {\left (b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} + 3 \, {\left (b^{4} d^{2} e^{2} - a b^{3} d e^{3}\right )} x^{2} + 3 \, {\left (b^{4} d^{3} e - a b^{3} d^{2} e^{2}\right )} x\right )} \log \left (e x + d\right )}{3 \, {\left (e^{8} x^{3} + 3 \, d e^{7} x^{2} + 3 \, d^{2} e^{6} x + d^{3} e^{5}\right )}} \]

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^2/(e*x+d)^4,x, algorithm="fricas")

[Out]

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)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 209 vs. \(2 (94) = 188\).

Time = 1.03 (sec) , antiderivative size = 209, normalized size of antiderivative = 2.03 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^4} \, dx=\frac {b^{4} x}{e^{4}} + \frac {4 b^{3} \left (a e - b d\right ) \log {\left (d + e x \right )}}{e^{5}} + \frac {- a^{4} e^{4} - 2 a^{3} b d e^{3} - 6 a^{2} b^{2} d^{2} e^{2} + 22 a b^{3} d^{3} e - 13 b^{4} d^{4} + x^{2} \left (- 18 a^{2} b^{2} e^{4} + 36 a b^{3} d e^{3} - 18 b^{4} d^{2} e^{2}\right ) + x \left (- 6 a^{3} b e^{4} - 18 a^{2} b^{2} d e^{3} + 54 a b^{3} d^{2} e^{2} - 30 b^{4} d^{3} e\right )}{3 d^{3} e^{5} + 9 d^{2} e^{6} x + 9 d e^{7} x^{2} + 3 e^{8} x^{3}} \]

[In]

integrate((b**2*x**2+2*a*b*x+a**2)**2/(e*x+d)**4,x)

[Out]

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 + 2
2*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)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.95 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^4} \, dx=\frac {b^{4} x}{e^{4}} - \frac {13 \, b^{4} d^{4} - 22 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} + 2 \, a^{3} b d e^{3} + a^{4} e^{4} + 18 \, {\left (b^{4} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 6 \, {\left (5 \, b^{4} d^{3} e - 9 \, a b^{3} d^{2} e^{2} + 3 \, a^{2} b^{2} d e^{3} + a^{3} b e^{4}\right )} x}{3 \, {\left (e^{8} x^{3} + 3 \, d e^{7} x^{2} + 3 \, d^{2} e^{6} x + d^{3} e^{5}\right )}} - \frac {4 \, {\left (b^{4} d - a b^{3} e\right )} \log \left (e x + d\right )}{e^{5}} \]

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^2/(e*x+d)^4,x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.72 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^4} \, dx=\frac {b^{4} x}{e^{4}} - \frac {4 \, {\left (b^{4} d - a b^{3} e\right )} \log \left ({\left | e x + d \right |}\right )}{e^{5}} - \frac {13 \, b^{4} d^{4} - 22 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} + 2 \, a^{3} b d e^{3} + a^{4} e^{4} + 18 \, {\left (b^{4} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 6 \, {\left (5 \, b^{4} d^{3} e - 9 \, a b^{3} d^{2} e^{2} + 3 \, a^{2} b^{2} d e^{3} + a^{3} b e^{4}\right )} x}{3 \, {\left (e x + d\right )}^{3} e^{5}} \]

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^2/(e*x+d)^4,x, algorithm="giac")

[Out]

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)

Mupad [B] (verification not implemented)

Time = 9.68 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.98 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^4} \, dx=\frac {b^4\,x}{e^4}-\frac {\ln \left (d+e\,x\right )\,\left (4\,b^4\,d-4\,a\,b^3\,e\right )}{e^5}-\frac {\frac {a^4\,e^4+2\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-22\,a\,b^3\,d^3\,e+13\,b^4\,d^4}{3\,e}+x\,\left (2\,a^3\,b\,e^3+6\,a^2\,b^2\,d\,e^2-18\,a\,b^3\,d^2\,e+10\,b^4\,d^3\right )+x^2\,\left (6\,a^2\,b^2\,e^3-12\,a\,b^3\,d\,e^2+6\,b^4\,d^2\,e\right )}{d^3\,e^4+3\,d^2\,e^5\,x+3\,d\,e^6\,x^2+e^7\,x^3} \]

[In]

int((a^2 + b^2*x^2 + 2*a*b*x)^2/(d + e*x)^4,x)

[Out]

(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)