\(\int \frac {a^2+2 a b x+b^2 x^2}{(d+e x)^6} \, dx\) [1461]

Optimal result

Integrand size = 24, antiderivative size = 65 \[ \int \frac {a^2+2 a b x+b^2 x^2}{(d+e x)^6} \, dx=-\frac {(b d-a e)^2}{5 e^3 (d+e x)^5}+\frac {b (b d-a e)}{2 e^3 (d+e x)^4}-\frac {b^2}{3 e^3 (d+e x)^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {27, 45} \[ \int \frac {a^2+2 a b x+b^2 x^2}{(d+e x)^6} \, dx=\frac {b (b d-a e)}{2 e^3 (d+e x)^4}-\frac {(b d-a e)^2}{5 e^3 (d+e x)^5}-\frac {b^2}{3 e^3 (d+e x)^3} \]

[In]

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

[Out]

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

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)^2}{(d+e x)^6} \, dx \\ & = \int \left (\frac {(-b d+a e)^2}{e^2 (d+e x)^6}-\frac {2 b (b d-a e)}{e^2 (d+e x)^5}+\frac {b^2}{e^2 (d+e x)^4}\right ) \, dx \\ & = -\frac {(b d-a e)^2}{5 e^3 (d+e x)^5}+\frac {b (b d-a e)}{2 e^3 (d+e x)^4}-\frac {b^2}{3 e^3 (d+e x)^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.85 \[ \int \frac {a^2+2 a b x+b^2 x^2}{(d+e x)^6} \, dx=-\frac {6 a^2 e^2+3 a b e (d+5 e x)+b^2 \left (d^2+5 d e x+10 e^2 x^2\right )}{30 e^3 (d+e x)^5} \]

[In]

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

[Out]

-1/30*(6*a^2*e^2 + 3*a*b*e*(d + 5*e*x) + b^2*(d^2 + 5*d*e*x + 10*e^2*x^2))/(e^3*(d + e*x)^5)

Maple [A] (verified)

Time = 2.22 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.95

method	result	size
gosper	\(-\frac {10 x^{2} b^{2} e^{2}+15 x a b \,e^{2}+5 b^{2} d e x +6 a^{2} e^{2}+3 a b d e +b^{2} d^{2}}{30 e^{3} \left (e x +d \right )^{5}}\)	\(62\)
risch	\(\frac {-\frac {b^{2} x^{2}}{3 e}-\frac {b \left (3 a e +b d \right ) x}{6 e^{2}}-\frac {6 a^{2} e^{2}+3 a b d e +b^{2} d^{2}}{30 e^{3}}}{\left (e x +d \right )^{5}}\)	\(63\)
parallelrisch	\(\frac {-10 b^{2} x^{2} e^{4}-15 a b \,e^{4} x -5 b^{2} d \,e^{3} x -6 a^{2} e^{4}-3 a b d \,e^{3}-b^{2} d^{2} e^{2}}{30 e^{5} \left (e x +d \right )^{5}}\)	\(70\)
default	\(-\frac {a^{2} e^{2}-2 a b d e +b^{2} d^{2}}{5 e^{3} \left (e x +d \right )^{5}}-\frac {b^{2}}{3 e^{3} \left (e x +d \right )^{3}}-\frac {b \left (a e -b d \right )}{2 e^{3} \left (e x +d \right )^{4}}\)	\(71\)
norman	\(\frac {-\frac {b^{2} x^{2}}{3 e}-\frac {\left (3 a b \,e^{3}+b^{2} d \,e^{2}\right ) x}{6 e^{4}}-\frac {6 a^{2} e^{4}+3 a b d \,e^{3}+b^{2} d^{2} e^{2}}{30 e^{5}}}{\left (e x +d \right )^{5}}\)	\(75\)

[In]

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

[Out]

-1/30/e^3*(10*b^2*e^2*x^2+15*a*b*e^2*x+5*b^2*d*e*x+6*a^2*e^2+3*a*b*d*e+b^2*d^2)/(e*x+d)^5

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.68 \[ \int \frac {a^2+2 a b x+b^2 x^2}{(d+e x)^6} \, dx=-\frac {10 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 3 \, a b d e + 6 \, a^{2} e^{2} + 5 \, {\left (b^{2} d e + 3 \, a b e^{2}\right )} x}{30 \, {\left (e^{8} x^{5} + 5 \, d e^{7} x^{4} + 10 \, d^{2} e^{6} x^{3} + 10 \, d^{3} e^{5} x^{2} + 5 \, d^{4} e^{4} x + d^{5} e^{3}\right )}} \]

[In]

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

[Out]

-1/30*(10*b^2*e^2*x^2 + b^2*d^2 + 3*a*b*d*e + 6*a^2*e^2 + 5*(b^2*d*e + 3*a*b*e^2)*x)/(e^8*x^5 + 5*d*e^7*x^4 +
10*d^2*e^6*x^3 + 10*d^3*e^5*x^2 + 5*d^4*e^4*x + d^5*e^3)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 116 vs. \(2 (54) = 108\).

Time = 0.52 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.78 \[ \int \frac {a^2+2 a b x+b^2 x^2}{(d+e x)^6} \, dx=\frac {- 6 a^{2} e^{2} - 3 a b d e - b^{2} d^{2} - 10 b^{2} e^{2} x^{2} + x \left (- 15 a b e^{2} - 5 b^{2} d e\right )}{30 d^{5} e^{3} + 150 d^{4} e^{4} x + 300 d^{3} e^{5} x^{2} + 300 d^{2} e^{6} x^{3} + 150 d e^{7} x^{4} + 30 e^{8} x^{5}} \]

[In]

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

[Out]

(-6*a**2*e**2 - 3*a*b*d*e - b**2*d**2 - 10*b**2*e**2*x**2 + x*(-15*a*b*e**2 - 5*b**2*d*e))/(30*d**5*e**3 + 150
*d**4*e**4*x + 300*d**3*e**5*x**2 + 300*d**2*e**6*x**3 + 150*d*e**7*x**4 + 30*e**8*x**5)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.68 \[ \int \frac {a^2+2 a b x+b^2 x^2}{(d+e x)^6} \, dx=-\frac {10 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 3 \, a b d e + 6 \, a^{2} e^{2} + 5 \, {\left (b^{2} d e + 3 \, a b e^{2}\right )} x}{30 \, {\left (e^{8} x^{5} + 5 \, d e^{7} x^{4} + 10 \, d^{2} e^{6} x^{3} + 10 \, d^{3} e^{5} x^{2} + 5 \, d^{4} e^{4} x + d^{5} e^{3}\right )}} \]

[In]

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

[Out]

-1/30*(10*b^2*e^2*x^2 + b^2*d^2 + 3*a*b*d*e + 6*a^2*e^2 + 5*(b^2*d*e + 3*a*b*e^2)*x)/(e^8*x^5 + 5*d*e^7*x^4 +
10*d^2*e^6*x^3 + 10*d^3*e^5*x^2 + 5*d^4*e^4*x + d^5*e^3)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.94 \[ \int \frac {a^2+2 a b x+b^2 x^2}{(d+e x)^6} \, dx=-\frac {10 \, b^{2} e^{2} x^{2} + 5 \, b^{2} d e x + 15 \, a b e^{2} x + b^{2} d^{2} + 3 \, a b d e + 6 \, a^{2} e^{2}}{30 \, {\left (e x + d\right )}^{5} e^{3}} \]

[In]

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

[Out]

-1/30*(10*b^2*e^2*x^2 + 5*b^2*d*e*x + 15*a*b*e^2*x + b^2*d^2 + 3*a*b*d*e + 6*a^2*e^2)/((e*x + d)^5*e^3)

Mupad [B] (verification not implemented)

Time = 9.96 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.65 \[ \int \frac {a^2+2 a b x+b^2 x^2}{(d+e x)^6} \, dx=-\frac {\frac {6\,a^2\,e^2+3\,a\,b\,d\,e+b^2\,d^2}{30\,e^3}+\frac {b^2\,x^2}{3\,e}+\frac {b\,x\,\left (3\,a\,e+b\,d\right )}{6\,e^2}}{d^5+5\,d^4\,e\,x+10\,d^3\,e^2\,x^2+10\,d^2\,e^3\,x^3+5\,d\,e^4\,x^4+e^5\,x^5} \]

[In]

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

[Out]

-((6*a^2*e^2 + b^2*d^2 + 3*a*b*d*e)/(30*e^3) + (b^2*x^2)/(3*e) + (b*x*(3*a*e + b*d))/(6*e^2))/(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)