\(\int \frac {(a^2+2 a b x+b^2 x^2)^2}{(d+e x)^6} \, dx\) [1475]

Optimal result

Integrand size = 26, antiderivative size = 28 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^6} \, dx=\frac {(a+b x)^5}{5 (b d-a e) (d+e x)^5} \]

[Out]

1/5*(b*x+a)^5/(-a*e+b*d)/(e*x+d)^5

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {27, 37} \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^6} \, dx=\frac {(a+b x)^5}{5 (d+e x)^5 (b d-a e)} \]

[In]

Int[(a^2 + 2*a*b*x + b^2*x^2)^2/(d + e*x)^6,x]

[Out]

(a + b*x)^5/(5*(b*d - a*e)*(d + e*x)^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 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps \begin{align*} \text {integral}& = \int \frac {(a+b x)^4}{(d+e x)^6} \, dx \\ & = \frac {(a+b x)^5}{5 (b d-a e) (d+e x)^5} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(140\) vs. \(2(28)=56\).

Time = 0.03 (sec) , antiderivative size = 140, normalized size of antiderivative = 5.00 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^6} \, dx=-\frac {a^4 e^4+a^3 b e^3 (d+5 e x)+a^2 b^2 e^2 \left (d^2+5 d e x+10 e^2 x^2\right )+a b^3 e \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )+b^4 \left (d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right )}{5 e^5 (d+e x)^5} \]

[In]

Integrate[(a^2 + 2*a*b*x + b^2*x^2)^2/(d + e*x)^6,x]

[Out]

-1/5*(a^4*e^4 + a^3*b*e^3*(d + 5*e*x) + a^2*b^2*e^2*(d^2 + 5*d*e*x + 10*e^2*x^2) + a*b^3*e*(d^3 + 5*d^2*e*x +
10*d*e^2*x^2 + 10*e^3*x^3) + b^4*(d^4 + 5*d^3*e*x + 10*d^2*e^2*x^2 + 10*d*e^3*x^3 + 5*e^4*x^4))/(e^5*(d + e*x)
^5)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(160\) vs. \(2(26)=52\).

Time = 2.23 (sec) , antiderivative size = 161, normalized size of antiderivative = 5.75

method	result	size
risch	\(\frac {-\frac {b^{4} x^{4}}{e}-\frac {2 b^{3} \left (a e +b d \right ) x^{3}}{e^{2}}-\frac {2 b^{2} \left (a^{2} e^{2}+a b d e +b^{2} d^{2}\right ) x^{2}}{e^{3}}-\frac {b \left (a^{3} e^{3}+a^{2} b d \,e^{2}+a \,b^{2} d^{2} e +b^{3} d^{3}\right ) x}{e^{4}}-\frac {e^{4} a^{4}+b \,e^{3} d \,a^{3}+b^{2} e^{2} d^{2} a^{2}+a \,b^{3} d^{3} e +b^{4} d^{4}}{5 e^{5}}}{\left (e x +d \right )^{5}}\)	\(161\)
norman	\(\frac {-\frac {b^{4} x^{4}}{e}-\frac {2 \left (e a \,b^{3}+d \,b^{4}\right ) x^{3}}{e^{2}}-\frac {2 \left (a^{2} b^{2} e^{2}+a \,b^{3} d e +b^{4} d^{2}\right ) x^{2}}{e^{3}}-\frac {\left (a^{3} b \,e^{3}+a^{2} b^{2} d \,e^{2}+d^{2} e a \,b^{3}+b^{4} d^{3}\right ) x}{e^{4}}-\frac {e^{4} a^{4}+b \,e^{3} d \,a^{3}+b^{2} e^{2} d^{2} a^{2}+a \,b^{3} d^{3} e +b^{4} d^{4}}{5 e^{5}}}{\left (e x +d \right )^{5}}\)	\(167\)
gosper	\(-\frac {5 b^{4} x^{4} e^{4}+10 x^{3} a \,b^{3} e^{4}+10 x^{3} b^{4} d \,e^{3}+10 x^{2} a^{2} b^{2} e^{4}+10 x^{2} a \,b^{3} d \,e^{3}+10 x^{2} b^{4} d^{2} e^{2}+5 x \,a^{3} b \,e^{4}+5 x \,a^{2} b^{2} d \,e^{3}+5 x a \,b^{3} d^{2} e^{2}+5 x \,b^{4} d^{3} e +e^{4} a^{4}+b \,e^{3} d \,a^{3}+b^{2} e^{2} d^{2} a^{2}+a \,b^{3} d^{3} e +b^{4} d^{4}}{5 e^{5} \left (e x +d \right )^{5}}\)	\(181\)
default	\(-\frac {e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}}{5 e^{5} \left (e x +d \right )^{5}}-\frac {b^{4}}{e^{5} \left (e x +d \right )}-\frac {2 b^{2} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}{e^{5} \left (e x +d \right )^{3}}-\frac {b \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}{e^{5} \left (e x +d \right )^{4}}-\frac {2 b^{3} \left (a e -b d \right )}{e^{5} \left (e x +d \right )^{2}}\)	\(186\)
parallelrisch	\(\frac {-5 b^{4} x^{4} e^{4}-10 x^{3} a \,b^{3} e^{4}-10 x^{3} b^{4} d \,e^{3}-10 x^{2} a^{2} b^{2} e^{4}-10 x^{2} a \,b^{3} d \,e^{3}-10 x^{2} b^{4} d^{2} e^{2}-5 x \,a^{3} b \,e^{4}-5 x \,a^{2} b^{2} d \,e^{3}-5 x a \,b^{3} d^{2} e^{2}-5 x \,b^{4} d^{3} e -e^{4} a^{4}-b \,e^{3} d \,a^{3}-b^{2} e^{2} d^{2} a^{2}-a \,b^{3} d^{3} e -b^{4} d^{4}}{5 e^{5} \left (e x +d \right )^{5}}\)	\(186\)

[In]

int((b^2*x^2+2*a*b*x+a^2)^2/(e*x+d)^6,x,method=_RETURNVERBOSE)

[Out]

(-1/e*b^4*x^4-2*b^3*(a*e+b*d)/e^2*x^3-2*b^2*(a^2*e^2+a*b*d*e+b^2*d^2)/e^3*x^2-b*(a^3*e^3+a^2*b*d*e^2+a*b^2*d^2
*e+b^3*d^3)/e^4*x-1/5*(a^4*e^4+a^3*b*d*e^3+a^2*b^2*d^2*e^2+a*b^3*d^3*e+b^4*d^4)/e^5)/(e*x+d)^5

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 215 vs. \(2 (26) = 52\).

Time = 0.29 (sec) , antiderivative size = 215, normalized size of antiderivative = 7.68 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^6} \, dx=-\frac {5 \, b^{4} e^{4} x^{4} + b^{4} d^{4} + a b^{3} d^{3} e + a^{2} b^{2} d^{2} e^{2} + a^{3} b d e^{3} + a^{4} e^{4} + 10 \, {\left (b^{4} d e^{3} + a b^{3} e^{4}\right )} x^{3} + 10 \, {\left (b^{4} d^{2} e^{2} + a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 5 \, {\left (b^{4} d^{3} e + a b^{3} d^{2} e^{2} + a^{2} b^{2} d e^{3} + a^{3} b e^{4}\right )} x}{5 \, {\left (e^{10} x^{5} + 5 \, d e^{9} x^{4} + 10 \, d^{2} e^{8} x^{3} + 10 \, d^{3} e^{7} x^{2} + 5 \, d^{4} e^{6} x + d^{5} e^{5}\right )}} \]

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^2/(e*x+d)^6,x, algorithm="fricas")

[Out]

-1/5*(5*b^4*e^4*x^4 + b^4*d^4 + a*b^3*d^3*e + a^2*b^2*d^2*e^2 + a^3*b*d*e^3 + a^4*e^4 + 10*(b^4*d*e^3 + a*b^3*
e^4)*x^3 + 10*(b^4*d^2*e^2 + a*b^3*d*e^3 + a^2*b^2*e^4)*x^2 + 5*(b^4*d^3*e + a*b^3*d^2*e^2 + a^2*b^2*d*e^3 + a
^3*b*e^4)*x)/(e^10*x^5 + 5*d*e^9*x^4 + 10*d^2*e^8*x^3 + 10*d^3*e^7*x^2 + 5*d^4*e^6*x + d^5*e^5)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 236 vs. \(2 (20) = 40\).

Time = 4.61 (sec) , antiderivative size = 236, normalized size of antiderivative = 8.43 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^6} \, dx=\frac {- a^{4} e^{4} - a^{3} b d e^{3} - a^{2} b^{2} d^{2} e^{2} - a b^{3} d^{3} e - b^{4} d^{4} - 5 b^{4} e^{4} x^{4} + x^{3} \left (- 10 a b^{3} e^{4} - 10 b^{4} d e^{3}\right ) + x^{2} \left (- 10 a^{2} b^{2} e^{4} - 10 a b^{3} d e^{3} - 10 b^{4} d^{2} e^{2}\right ) + x \left (- 5 a^{3} b e^{4} - 5 a^{2} b^{2} d e^{3} - 5 a b^{3} d^{2} e^{2} - 5 b^{4} d^{3} e\right )}{5 d^{5} e^{5} + 25 d^{4} e^{6} x + 50 d^{3} e^{7} x^{2} + 50 d^{2} e^{8} x^{3} + 25 d e^{9} x^{4} + 5 e^{10} x^{5}} \]

[In]

integrate((b**2*x**2+2*a*b*x+a**2)**2/(e*x+d)**6,x)

[Out]

(-a**4*e**4 - a**3*b*d*e**3 - a**2*b**2*d**2*e**2 - a*b**3*d**3*e - b**4*d**4 - 5*b**4*e**4*x**4 + x**3*(-10*a
*b**3*e**4 - 10*b**4*d*e**3) + x**2*(-10*a**2*b**2*e**4 - 10*a*b**3*d*e**3 - 10*b**4*d**2*e**2) + x*(-5*a**3*b
*e**4 - 5*a**2*b**2*d*e**3 - 5*a*b**3*d**2*e**2 - 5*b**4*d**3*e))/(5*d**5*e**5 + 25*d**4*e**6*x + 50*d**3*e**7
*x**2 + 50*d**2*e**8*x**3 + 25*d*e**9*x**4 + 5*e**10*x**5)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 215 vs. \(2 (26) = 52\).

Time = 0.21 (sec) , antiderivative size = 215, normalized size of antiderivative = 7.68 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^6} \, dx=-\frac {5 \, b^{4} e^{4} x^{4} + b^{4} d^{4} + a b^{3} d^{3} e + a^{2} b^{2} d^{2} e^{2} + a^{3} b d e^{3} + a^{4} e^{4} + 10 \, {\left (b^{4} d e^{3} + a b^{3} e^{4}\right )} x^{3} + 10 \, {\left (b^{4} d^{2} e^{2} + a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 5 \, {\left (b^{4} d^{3} e + a b^{3} d^{2} e^{2} + a^{2} b^{2} d e^{3} + a^{3} b e^{4}\right )} x}{5 \, {\left (e^{10} x^{5} + 5 \, d e^{9} x^{4} + 10 \, d^{2} e^{8} x^{3} + 10 \, d^{3} e^{7} x^{2} + 5 \, d^{4} e^{6} x + d^{5} e^{5}\right )}} \]

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^2/(e*x+d)^6,x, algorithm="maxima")

[Out]

-1/5*(5*b^4*e^4*x^4 + b^4*d^4 + a*b^3*d^3*e + a^2*b^2*d^2*e^2 + a^3*b*d*e^3 + a^4*e^4 + 10*(b^4*d*e^3 + a*b^3*
e^4)*x^3 + 10*(b^4*d^2*e^2 + a*b^3*d*e^3 + a^2*b^2*e^4)*x^2 + 5*(b^4*d^3*e + a*b^3*d^2*e^2 + a^2*b^2*d*e^3 + a
^3*b*e^4)*x)/(e^10*x^5 + 5*d*e^9*x^4 + 10*d^2*e^8*x^3 + 10*d^3*e^7*x^2 + 5*d^4*e^6*x + d^5*e^5)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 180 vs. \(2 (26) = 52\).

Time = 0.26 (sec) , antiderivative size = 180, normalized size of antiderivative = 6.43 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^6} \, dx=-\frac {5 \, b^{4} e^{4} x^{4} + 10 \, b^{4} d e^{3} x^{3} + 10 \, a b^{3} e^{4} x^{3} + 10 \, b^{4} d^{2} e^{2} x^{2} + 10 \, a b^{3} d e^{3} x^{2} + 10 \, a^{2} b^{2} e^{4} x^{2} + 5 \, b^{4} d^{3} e x + 5 \, a b^{3} d^{2} e^{2} x + 5 \, a^{2} b^{2} d e^{3} x + 5 \, a^{3} b e^{4} x + b^{4} d^{4} + a b^{3} d^{3} e + a^{2} b^{2} d^{2} e^{2} + a^{3} b d e^{3} + a^{4} e^{4}}{5 \, {\left (e x + d\right )}^{5} e^{5}} \]

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^2/(e*x+d)^6,x, algorithm="giac")

[Out]

-1/5*(5*b^4*e^4*x^4 + 10*b^4*d*e^3*x^3 + 10*a*b^3*e^4*x^3 + 10*b^4*d^2*e^2*x^2 + 10*a*b^3*d*e^3*x^2 + 10*a^2*b
^2*e^4*x^2 + 5*b^4*d^3*e*x + 5*a*b^3*d^2*e^2*x + 5*a^2*b^2*d*e^3*x + 5*a^3*b*e^4*x + b^4*d^4 + a*b^3*d^3*e + a
^2*b^2*d^2*e^2 + a^3*b*d*e^3 + a^4*e^4)/((e*x + d)^5*e^5)

Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 203, normalized size of antiderivative = 7.25 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^6} \, dx=-\frac {\frac {a^4\,e^4+a^3\,b\,d\,e^3+a^2\,b^2\,d^2\,e^2+a\,b^3\,d^3\,e+b^4\,d^4}{5\,e^5}+\frac {b^4\,x^4}{e}+\frac {2\,b^3\,x^3\,\left (a\,e+b\,d\right )}{e^2}+\frac {b\,x\,\left (a^3\,e^3+a^2\,b\,d\,e^2+a\,b^2\,d^2\,e+b^3\,d^3\right )}{e^4}+\frac {2\,b^2\,x^2\,\left (a^2\,e^2+a\,b\,d\,e+b^2\,d^2\right )}{e^3}}{d^5+5\,d^4\,e\,x+10\,d^3\,e^2\,x^2+10\,d^2\,e^3\,x^3+5\,d\,e^4\,x^4+e^5\,x^5} \]

[In]

int((a^2 + b^2*x^2 + 2*a*b*x)^2/(d + e*x)^6,x)

[Out]

-((a^4*e^4 + b^4*d^4 + a^2*b^2*d^2*e^2 + a*b^3*d^3*e + a^3*b*d*e^3)/(5*e^5) + (b^4*x^4)/e + (2*b^3*x^3*(a*e +
b*d))/e^2 + (b*x*(a^3*e^3 + b^3*d^3 + a*b^2*d^2*e + a^2*b*d*e^2))/e^4 + (2*b^2*x^2*(a^2*e^2 + b^2*d^2 + a*b*d*
e))/e^3)/(d^5 + e^5*x^5 + 5*d*e^4*x^4 + 10*d^3*e^2*x^2 + 10*d^2*e^3*x^3 + 5*d^4*e*x)