\(\int \frac {a+b x+c x^2}{(d+e x)^4} \, dx\) [2115]

Optimal result

Integrand size = 18, antiderivative size = 67 \[ \int \frac {a+b x+c x^2}{(d+e x)^4} \, dx=-\frac {c d^2-b d e+a e^2}{3 e^3 (d+e x)^3}+\frac {2 c d-b e}{2 e^3 (d+e x)^2}-\frac {c}{e^3 (d+e x)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {712} \[ \int \frac {a+b x+c x^2}{(d+e x)^4} \, dx=-\frac {a e^2-b d e+c d^2}{3 e^3 (d+e x)^3}+\frac {2 c d-b e}{2 e^3 (d+e x)^2}-\frac {c}{e^3 (d+e x)} \]

[In]

Int[(a + b*x + c*x^2)/(d + e*x)^4,x]

[Out]

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

Rule 712

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.75 \[ \int \frac {a+b x+c x^2}{(d+e x)^4} \, dx=-\frac {2 c \left (d^2+3 d e x+3 e^2 x^2\right )+e (2 a e+b (d+3 e x))}{6 e^3 (d+e x)^3} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 2.59 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.78

method	result	size
gosper	\(-\frac {6 c \,x^{2} e^{2}+3 x b \,e^{2}+6 x c d e +2 e^{2} a +b d e +2 c \,d^{2}}{6 \left (e x +d \right )^{3} e^{3}}\)	\(52\)
parallelrisch	\(\frac {-6 c \,x^{2} e^{2}-3 x b \,e^{2}-6 x c d e -2 e^{2} a -b d e -2 c \,d^{2}}{6 e^{3} \left (e x +d \right )^{3}}\)	\(53\)
norman	\(\frac {-\frac {c \,x^{2}}{e}-\frac {\left (b e +2 c d \right ) x}{2 e^{2}}-\frac {2 e^{2} a +b d e +2 c \,d^{2}}{6 e^{3}}}{\left (e x +d \right )^{3}}\)	\(55\)
risch	\(\frac {-\frac {c \,x^{2}}{e}-\frac {\left (b e +2 c d \right ) x}{2 e^{2}}-\frac {2 e^{2} a +b d e +2 c \,d^{2}}{6 e^{3}}}{\left (e x +d \right )^{3}}\)	\(55\)
default	\(-\frac {c}{e^{3} \left (e x +d \right )}-\frac {e^{2} a -b d e +c \,d^{2}}{3 e^{3} \left (e x +d \right )^{3}}-\frac {b e -2 c d}{2 e^{3} \left (e x +d \right )^{2}}\)	\(63\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.15 \[ \int \frac {a+b x+c x^2}{(d+e x)^4} \, dx=-\frac {6 \, c e^{2} x^{2} + 2 \, c d^{2} + b d e + 2 \, a e^{2} + 3 \, {\left (2 \, c d e + b e^{2}\right )} x}{6 \, {\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} \]

[In]

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

[Out]

-1/6*(6*c*e^2*x^2 + 2*c*d^2 + b*d*e + 2*a*e^2 + 3*(2*c*d*e + 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 [A] (verification not implemented)

Time = 0.41 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.22 \[ \int \frac {a+b x+c x^2}{(d+e x)^4} \, dx=\frac {- 2 a e^{2} - b d e - 2 c d^{2} - 6 c e^{2} x^{2} + x \left (- 3 b e^{2} - 6 c d e\right )}{6 d^{3} e^{3} + 18 d^{2} e^{4} x + 18 d e^{5} x^{2} + 6 e^{6} x^{3}} \]

[In]

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

[Out]

(-2*a*e**2 - b*d*e - 2*c*d**2 - 6*c*e**2*x**2 + x*(-3*b*e**2 - 6*c*d*e))/(6*d**3*e**3 + 18*d**2*e**4*x + 18*d*
e**5*x**2 + 6*e**6*x**3)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.15 \[ \int \frac {a+b x+c x^2}{(d+e x)^4} \, dx=-\frac {6 \, c e^{2} x^{2} + 2 \, c d^{2} + b d e + 2 \, a e^{2} + 3 \, {\left (2 \, c d e + b e^{2}\right )} x}{6 \, {\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} \]

[In]

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

[Out]

-1/6*(6*c*e^2*x^2 + 2*c*d^2 + b*d*e + 2*a*e^2 + 3*(2*c*d*e + 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 [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.76 \[ \int \frac {a+b x+c x^2}{(d+e x)^4} \, dx=-\frac {6 \, c e^{2} x^{2} + 6 \, c d e x + 3 \, b e^{2} x + 2 \, c d^{2} + b d e + 2 \, a e^{2}}{6 \, {\left (e x + d\right )}^{3} e^{3}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.13 \[ \int \frac {a+b x+c x^2}{(d+e x)^4} \, dx=-\frac {\frac {2\,c\,d^2+b\,d\,e+2\,a\,e^2}{6\,e^3}+\frac {x\,\left (b\,e+2\,c\,d\right )}{2\,e^2}+\frac {c\,x^2}{e}}{d^3+3\,d^2\,e\,x+3\,d\,e^2\,x^2+e^3\,x^3} \]

[In]

int((a + b*x + c*x^2)/(d + e*x)^4,x)

[Out]

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