3.3.29 \(\int \frac {b x+c x^2}{(d+e x)^4} \, dx\) [229]

Optimal. Leaf size=60 \[ -\frac {d (c d-b e)}{3 e^3 (d+e x)^3}+\frac {2 c d-b e}{2 e^3 (d+e x)^2}-\frac {c}{e^3 (d+e x)} \]

[Out]

-1/3*d*(-b*e+c*d)/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)

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Rubi [A]
time = 0.03, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {712} \begin {gather*} \frac {2 c d-b e}{2 e^3 (d+e x)^2}-\frac {d (c d-b e)}{3 e^3 (d+e x)^3}-\frac {c}{e^3 (d+e x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b*x + c*x^2)/(d + e*x)^4,x]

[Out]

-1/3*(d*(c*d - b*e))/(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 {b x+c x^2}{(d+e x)^4} \, dx &=\int \left (\frac {d (c d-b e)}{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 {d (c d-b e)}{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*}

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Mathematica [A]
time = 0.01, size = 44, normalized size = 0.73 \begin {gather*} -\frac {b e (d+3 e x)+2 c \left (d^2+3 d e x+3 e^2 x^2\right )}{6 e^3 (d+e x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.45, size = 56, normalized size = 0.93

method	result	size
gosper	\(-\frac {6 c \,x^{2} e^{2}+3 b \,e^{2} x +6 c d e x +b d e +2 c \,d^{2}}{6 e^{3} \left (e x +d \right )^{3}}\)	\(46\)
norman	\(\frac {-\frac {c \,x^{2}}{e}-\frac {\left (b e +2 c d \right ) x}{2 e^{2}}-\frac {d \left (b e +2 c d \right )}{6 e^{3}}}{\left (e x +d \right )^{3}}\)	\(47\)
risch	\(\frac {-\frac {c \,x^{2}}{e}-\frac {\left (b e +2 c d \right ) x}{2 e^{2}}-\frac {d \left (b e +2 c d \right )}{6 e^{3}}}{\left (e x +d \right )^{3}}\)	\(47\)
default	\(-\frac {b e -2 c d}{2 e^{3} \left (e x +d \right )^{2}}+\frac {d \left (b e -c d \right )}{3 e^{3} \left (e x +d \right )^{3}}-\frac {c}{e^{3} \left (e x +d \right )}\)	\(56\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 67, normalized size = 1.12 \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}{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)/(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)/(x^3*e^6 + 3*d*x^2*e^5 + 3*d^2*x*e^4 + d^3*e^3)

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Fricas [A]
time = 1.69, size = 66, normalized size = 1.10 \begin {gather*} -\frac {2 \, c d^{2} + 3 \, {\left (2 \, c x^{2} + b x\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)/(e*x+d)^4,x, algorithm="fricas")

[Out]

-1/6*(2*c*d^2 + 3*(2*c*x^2 + b*x)*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)

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Sympy [A]
time = 0.22, size = 75, normalized size = 1.25 \begin {gather*} \frac {- 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)/(e*x+d)**4,x)

[Out]

(-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)

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Giac [A]
time = 0.73, size = 45, 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\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)/(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)*e^(-3)/(x*e + d)^3

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Mupad [B]
time = 0.05, size = 68, normalized size = 1.13 \begin {gather*} -\frac {\frac {d\,\left (b\,e+2\,c\,d\right )}{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((b*x + c*x^2)/(d + e*x)^4,x)

[Out]

-((d*(b*e + 2*c*d))/(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)

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