\(\int \frac {a+c x^2}{(d+e x)^4} \, dx\) [458]

Optimal result

Integrand size = 15, antiderivative size = 54 \[ \int \frac {a+c x^2}{(d+e x)^4} \, dx=\frac {-c d^2-a e^2}{3 e^3 (d+e x)^3}+\frac {c d}{e^3 (d+e x)^2}-\frac {c}{e^3 (d+e x)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.96, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {711} \[ \int \frac {a+c x^2}{(d+e x)^4} \, dx=-\frac {a e^2+c d^2}{3 e^3 (d+e x)^3}-\frac {c}{e^3 (d+e x)}+\frac {c d}{e^3 (d+e x)^2} \]

[In]

Int[(a + c*x^2)/(d + e*x)^4,x]

[Out]

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

Rule 711

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

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.72 \[ \int \frac {a+c x^2}{(d+e x)^4} \, dx=-\frac {a e^2+c \left (d^2+3 d e x+3 e^2 x^2\right )}{3 e^3 (d+e x)^3} \]

[In]

Integrate[(a + c*x^2)/(d + e*x)^4,x]

[Out]

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

Maple [A] (verified)

Time = 2.06 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.72

method	result	size
gosper	\(-\frac {3 c \,x^{2} e^{2}+3 x c d e +e^{2} a +c \,d^{2}}{3 \left (e x +d \right )^{3} e^{3}}\)	\(39\)
parallelrisch	\(\frac {-3 c \,x^{2} e^{2}-3 x c d e -e^{2} a -c \,d^{2}}{3 e^{3} \left (e x +d \right )^{3}}\)	\(41\)
norman	\(\frac {-\frac {c \,x^{2}}{e}-\frac {c d x}{e^{2}}-\frac {e^{2} a +c \,d^{2}}{3 e^{3}}}{\left (e x +d \right )^{3}}\)	\(43\)
risch	\(\frac {-\frac {c \,x^{2}}{e}-\frac {c d x}{e^{2}}-\frac {e^{2} a +c \,d^{2}}{3 e^{3}}}{\left (e x +d \right )^{3}}\)	\(43\)
default	\(-\frac {c}{e^{3} \left (e x +d \right )}-\frac {e^{2} a +c \,d^{2}}{3 e^{3} \left (e x +d \right )^{3}}+\frac {c d}{e^{3} \left (e x +d \right )^{2}}\)	\(51\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.17 \[ \int \frac {a+c x^2}{(d+e x)^4} \, dx=-\frac {3 \, c e^{2} x^{2} + 3 \, c d e x + c d^{2} + a e^{2}}{3 \, {\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} \]

[In]

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

[Out]

-1/3*(3*c*e^2*x^2 + 3*c*d*e*x + c*d^2 + a*e^2)/(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 = 0.23 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.22 \[ \int \frac {a+c x^2}{(d+e x)^4} \, dx=\frac {- a e^{2} - c d^{2} - 3 c d e x - 3 c e^{2} x^{2}}{3 d^{3} e^{3} + 9 d^{2} e^{4} x + 9 d e^{5} x^{2} + 3 e^{6} x^{3}} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.17 \[ \int \frac {a+c x^2}{(d+e x)^4} \, dx=-\frac {3 \, c e^{2} x^{2} + 3 \, c d e x + c d^{2} + a e^{2}}{3 \, {\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} \]

[In]

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

[Out]

-1/3*(3*c*e^2*x^2 + 3*c*d*e*x + c*d^2 + a*e^2)/(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 = 0.27 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.70 \[ \int \frac {a+c x^2}{(d+e x)^4} \, dx=-\frac {3 \, c e^{2} x^{2} + 3 \, c d e x + c d^{2} + a e^{2}}{3 \, {\left (e x + d\right )}^{3} e^{3}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.17 \[ \int \frac {a+c x^2}{(d+e x)^4} \, dx=-\frac {\frac {c\,d^2+a\,e^2}{3\,e^3}+\frac {c\,x^2}{e}+\frac {c\,d\,x}{e^2}}{d^3+3\,d^2\,e\,x+3\,d\,e^2\,x^2+e^3\,x^3} \]

[In]

int((a + c*x^2)/(d + e*x)^4,x)

[Out]

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