\(\int \frac {a^2+2 a b x+b^2 x^2}{(d+e x)^4} \, dx\) [1459]

Optimal result

Integrand size = 24, antiderivative size = 28 \[ \int \frac {a^2+2 a b x+b^2 x^2}{(d+e x)^4} \, dx=\frac {(a+b x)^3}{3 (b d-a e) (d+e x)^3} \]

[Out]

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

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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {27, 37} \[ \int \frac {a^2+2 a b x+b^2 x^2}{(d+e x)^4} \, dx=\frac {(a+b x)^3}{3 (d+e x)^3 (b d-a e)} \]

[In]

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

[Out]

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.89 \[ \int \frac {a^2+2 a b x+b^2 x^2}{(d+e x)^4} \, dx=-\frac {a^2 e^2+a b e (d+3 e x)+b^2 \left (d^2+3 d e x+3 e^2 x^2\right )}{3 e^3 (d+e x)^3} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(59\) vs. \(2(26)=52\).

Time = 2.05 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.14

method	result	size
gosper	\(-\frac {3 x^{2} b^{2} e^{2}+3 x a b \,e^{2}+3 b^{2} d e x +a^{2} e^{2}+a b d e +b^{2} d^{2}}{3 e^{3} \left (e x +d \right )^{3}}\)	\(60\)
risch	\(\frac {-\frac {b^{2} x^{2}}{e}-\frac {b \left (a e +b d \right ) x}{e^{2}}-\frac {a^{2} e^{2}+a b d e +b^{2} d^{2}}{3 e^{3}}}{\left (e x +d \right )^{3}}\)	\(60\)
norman	\(\frac {-\frac {b^{2} x^{2}}{e}-\frac {\left (a e b +b^{2} d \right ) x}{e^{2}}-\frac {a^{2} e^{2}+a b d e +b^{2} d^{2}}{3 e^{3}}}{\left (e x +d \right )^{3}}\)	\(62\)
parallelrisch	\(\frac {-3 x^{2} b^{2} e^{2}-3 x a b \,e^{2}-3 b^{2} d e x -a^{2} e^{2}-a b d e -b^{2} d^{2}}{3 e^{3} \left (e x +d \right )^{3}}\)	\(63\)
default	\(-\frac {b^{2}}{e^{3} \left (e x +d \right )}-\frac {a^{2} e^{2}-2 a b d e +b^{2} d^{2}}{3 e^{3} \left (e x +d \right )^{3}}-\frac {b \left (a e -b d \right )}{e^{3} \left (e x +d \right )^{2}}\)	\(71\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 84 vs. \(2 (26) = 52\).

Time = 0.29 (sec) , antiderivative size = 84, normalized size of antiderivative = 3.00 \[ \int \frac {a^2+2 a b x+b^2 x^2}{(d+e x)^4} \, dx=-\frac {3 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + a b d e + a^{2} e^{2} + 3 \, {\left (b^{2} d e + a b e^{2}\right )} x}{3 \, {\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (20) = 40\).

Time = 0.32 (sec) , antiderivative size = 88, normalized size of antiderivative = 3.14 \[ \int \frac {a^2+2 a b x+b^2 x^2}{(d+e x)^4} \, dx=\frac {- a^{2} e^{2} - a b d e - b^{2} d^{2} - 3 b^{2} e^{2} x^{2} + x \left (- 3 a b e^{2} - 3 b^{2} d e\right )}{3 d^{3} e^{3} + 9 d^{2} e^{4} x + 9 d e^{5} x^{2} + 3 e^{6} x^{3}} \]

[In]

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

[Out]

(-a**2*e**2 - a*b*d*e - b**2*d**2 - 3*b**2*e**2*x**2 + x*(-3*a*b*e**2 - 3*b**2*d*e))/(3*d**3*e**3 + 9*d**2*e**
4*x + 9*d*e**5*x**2 + 3*e**6*x**3)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 84 vs. \(2 (26) = 52\).

Time = 0.19 (sec) , antiderivative size = 84, normalized size of antiderivative = 3.00 \[ \int \frac {a^2+2 a b x+b^2 x^2}{(d+e x)^4} \, dx=-\frac {3 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + a b d e + a^{2} e^{2} + 3 \, {\left (b^{2} d e + a b e^{2}\right )} x}{3 \, {\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (26) = 52\).

Time = 0.26 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.11 \[ \int \frac {a^2+2 a b x+b^2 x^2}{(d+e x)^4} \, dx=-\frac {3 \, b^{2} e^{2} x^{2} + 3 \, b^{2} d e x + 3 \, a b e^{2} x + b^{2} d^{2} + a b d e + a^{2} e^{2}}{3 \, {\left (e x + d\right )}^{3} e^{3}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.86 \[ \int \frac {a^2+2 a b x+b^2 x^2}{(d+e x)^4} \, dx=-\frac {\frac {a^2\,e^2+a\,b\,d\,e+b^2\,d^2}{3\,e^3}+\frac {b^2\,x^2}{e}+\frac {b\,x\,\left (a\,e+b\,d\right )}{e^2}}{d^3+3\,d^2\,e\,x+3\,d\,e^2\,x^2+e^3\,x^3} \]

[In]

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

[Out]

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