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

Optimal. Leaf size=67 \[ -\frac {c d^2-b d e+a e^2}{3 e^3 (d+e x)^3}+\frac {2 c d-b e}{2 e^3 (d+e x)^2}-\frac {c}{e^3 (d+e x)} \]

[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]
time = 0.03, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {712} \begin {gather*} -\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)} \end {gather*}

Antiderivative was successfully verified.

[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*} \int \frac {a+b x+c x^2}{(d+e x)^4} \, dx &=\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]
time = 0.01, size = 50, normalized size = 0.75 \begin {gather*} -\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} \end {gather*}

Antiderivative was successfully verified.

[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]
time = 0.47, size = 63, normalized size = 0.94

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.34, size = 72, normalized size = 1.07 \begin {gather*} -\frac {6 \, c x^{2} e^{2} + 2 \, c d^{2} + b d e + 3 \, {\left (2 \, c d e + b e^{2}\right )} x + 2 \, a e^{2}}{6 \, {\left (x^{3} e^{6} + 3 \, d x^{2} e^{5} + 3 \, d^{2} x e^{4} + d^{3} e^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 2.81, size = 69, normalized size = 1.03 \begin {gather*} -\frac {2 \, c d^{2} + {\left (6 \, c x^{2} + 3 \, b x + 2 \, a\right )} e^{2} + {\left (6 \, c d x + b d\right )} e}{6 \, {\left (x^{3} e^{6} + 3 \, d x^{2} e^{5} + 3 \, d^{2} x e^{4} + d^{3} e^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]
time = 0.47, size = 82, normalized size = 1.22 \begin {gather*} \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Giac [A]
time = 1.78, size = 50, normalized size = 0.75 \begin {gather*} -\frac {{\left (6 \, c x^{2} e^{2} + 6 \, c d x e + 2 \, c d^{2} + 3 \, b x e^{2} + b d e + 2 \, a e^{2}\right )} e^{\left (-3\right )}}{6 \, {\left (x e + d\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [B]
time = 0.04, size = 76, normalized size = 1.13 \begin {gather*} -\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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________