\(\int \frac {a+c x^2}{(d+e x)^5} \, dx\) [459]

Optimal result

Integrand size = 15, antiderivative size = 59 \[ \int \frac {a+c x^2}{(d+e x)^5} \, dx=\frac {-c d^2-a e^2}{4 e^3 (d+e x)^4}+\frac {2 c d}{3 e^3 (d+e x)^3}-\frac {c}{2 e^3 (d+e x)^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.97, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {711} \[ \int \frac {a+c x^2}{(d+e x)^5} \, dx=-\frac {a e^2+c d^2}{4 e^3 (d+e x)^4}-\frac {c}{2 e^3 (d+e x)^2}+\frac {2 c d}{3 e^3 (d+e x)^3} \]

[In]

Int[(a + c*x^2)/(d + e*x)^5,x]

[Out]

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

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.68 \[ \int \frac {a+c x^2}{(d+e x)^5} \, dx=-\frac {3 a e^2+c \left (d^2+4 d e x+6 e^2 x^2\right )}{12 e^3 (d+e x)^4} \]

[In]

Integrate[(a + c*x^2)/(d + e*x)^5,x]

[Out]

-1/12*(3*a*e^2 + c*(d^2 + 4*d*e*x + 6*e^2*x^2))/(e^3*(d + e*x)^4)

Maple [A] (verified)

Time = 2.07 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.68

method	result	size
gosper	\(-\frac {6 c \,x^{2} e^{2}+4 x c d e +3 e^{2} a +c \,d^{2}}{12 e^{3} \left (e x +d \right )^{4}}\)	\(40\)
risch	\(\frac {-\frac {c \,x^{2}}{2 e}-\frac {c d x}{3 e^{2}}-\frac {3 e^{2} a +c \,d^{2}}{12 e^{3}}}{\left (e x +d \right )^{4}}\)	\(44\)
parallelrisch	\(\frac {-6 c \,x^{2} e^{3}-4 c d x \,e^{2}-3 a \,e^{3}-d^{2} e c}{12 e^{4} \left (e x +d \right )^{4}}\)	\(44\)
norman	\(\frac {-\frac {c \,x^{2}}{2 e}-\frac {c d x}{3 e^{2}}-\frac {3 a \,e^{3}+d^{2} e c}{12 e^{4}}}{\left (e x +d \right )^{4}}\)	\(45\)
default	\(\frac {2 c d}{3 e^{3} \left (e x +d \right )^{3}}-\frac {e^{2} a +c \,d^{2}}{4 e^{3} \left (e x +d \right )^{4}}-\frac {c}{2 e^{3} \left (e x +d \right )^{2}}\)	\(52\)

[In]

int((c*x^2+a)/(e*x+d)^5,x,method=_RETURNVERBOSE)

[Out]

-1/12/e^3*(6*c*e^2*x^2+4*c*d*e*x+3*a*e^2+c*d^2)/(e*x+d)^4

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.27 \[ \int \frac {a+c x^2}{(d+e x)^5} \, dx=-\frac {6 \, c e^{2} x^{2} + 4 \, c d e x + c d^{2} + 3 \, a e^{2}}{12 \, {\left (e^{7} x^{4} + 4 \, d e^{6} x^{3} + 6 \, d^{2} e^{5} x^{2} + 4 \, d^{3} e^{4} x + d^{4} e^{3}\right )}} \]

[In]

integrate((c*x^2+a)/(e*x+d)^5,x, algorithm="fricas")

[Out]

-1/12*(6*c*e^2*x^2 + 4*c*d*e*x + c*d^2 + 3*a*e^2)/(e^7*x^4 + 4*d*e^6*x^3 + 6*d^2*e^5*x^2 + 4*d^3*e^4*x + d^4*e
^3)

Sympy [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.36 \[ \int \frac {a+c x^2}{(d+e x)^5} \, dx=\frac {- 3 a e^{2} - c d^{2} - 4 c d e x - 6 c e^{2} x^{2}}{12 d^{4} e^{3} + 48 d^{3} e^{4} x + 72 d^{2} e^{5} x^{2} + 48 d e^{6} x^{3} + 12 e^{7} x^{4}} \]

[In]

integrate((c*x**2+a)/(e*x+d)**5,x)

[Out]

(-3*a*e**2 - c*d**2 - 4*c*d*e*x - 6*c*e**2*x**2)/(12*d**4*e**3 + 48*d**3*e**4*x + 72*d**2*e**5*x**2 + 48*d*e**
6*x**3 + 12*e**7*x**4)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.27 \[ \int \frac {a+c x^2}{(d+e x)^5} \, dx=-\frac {6 \, c e^{2} x^{2} + 4 \, c d e x + c d^{2} + 3 \, a e^{2}}{12 \, {\left (e^{7} x^{4} + 4 \, d e^{6} x^{3} + 6 \, d^{2} e^{5} x^{2} + 4 \, d^{3} e^{4} x + d^{4} e^{3}\right )}} \]

[In]

integrate((c*x^2+a)/(e*x+d)^5,x, algorithm="maxima")

[Out]

-1/12*(6*c*e^2*x^2 + 4*c*d*e*x + c*d^2 + 3*a*e^2)/(e^7*x^4 + 4*d*e^6*x^3 + 6*d^2*e^5*x^2 + 4*d^3*e^4*x + d^4*e
^3)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00 \[ \int \frac {a+c x^2}{(d+e x)^5} \, dx=-\frac {\frac {3 \, a}{{\left (e x + d\right )}^{4}} + \frac {6 \, c}{{\left (e x + d\right )}^{2} e^{2}} - \frac {8 \, c d}{{\left (e x + d\right )}^{3} e^{2}} + \frac {3 \, c d^{2}}{{\left (e x + d\right )}^{4} e^{2}}}{12 \, e} \]

[In]

integrate((c*x^2+a)/(e*x+d)^5,x, algorithm="giac")

[Out]

-1/12*(3*a/(e*x + d)^4 + 6*c/((e*x + d)^2*e^2) - 8*c*d/((e*x + d)^3*e^2) + 3*c*d^2/((e*x + d)^4*e^2))/e

Mupad [B] (verification not implemented)

Time = 9.37 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.31 \[ \int \frac {a+c x^2}{(d+e x)^5} \, dx=-\frac {\frac {c\,d^2+3\,a\,e^2}{12\,e^3}+\frac {c\,x^2}{2\,e}+\frac {c\,d\,x}{3\,e^2}}{d^4+4\,d^3\,e\,x+6\,d^2\,e^2\,x^2+4\,d\,e^3\,x^3+e^4\,x^4} \]

[In]

int((a + c*x^2)/(d + e*x)^5,x)

[Out]

-((3*a*e^2 + c*d^2)/(12*e^3) + (c*x^2)/(2*e) + (c*d*x)/(3*e^2))/(d^4 + e^4*x^4 + 4*d*e^3*x^3 + 6*d^2*e^2*x^2 +
4*d^3*e*x)