3.1024 \(\int \frac {(d+e x)^2}{(c d^2+2 c d e x+c e^2 x^2)^3} \, dx\)

Optimal. Leaf size=17 \[ -\frac {1}{3 c^3 e (d+e x)^3} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {27, 12, 32} \[ -\frac {1}{3 c^3 e (d+e x)^3} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^2/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^3,x]

[Out]

-1/(3*c^3*e*(d + e*x)^3)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 17, normalized size = 1.00 \[ -\frac {1}{3 c^3 e (d+e x)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^2/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^3,x]

[Out]

-1/3*1/(c^3*e*(d + e*x)^3)

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fricas [B]  time = 1.05, size = 47, normalized size = 2.76 \[ -\frac {1}{3 \, {\left (c^{3} e^{4} x^{3} + 3 \, c^{3} d e^{3} x^{2} + 3 \, c^{3} d^{2} e^{2} x + c^{3} d^{3} e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2/(c*e^2*x^2+2*c*d*e*x+c*d^2)^3,x, algorithm="fricas")

[Out]

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

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2/(c*e^2*x^2+2*c*d*e*x+c*d^2)^3,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: (-3*x^3*exp(2)^2+2*x^3*exp(2)*exp(1)^2-9
*x^2*exp(2)*d*exp(1)+6*x^2*d*exp(1)^3-5*x*exp(2)*d^2+2*x*d^2*exp(1)^2-d^3*exp(1))/(-8*c^3*exp(2)*d^2+8*c^3*d^2
*exp(1)^2)/(-x^2*exp(2)-2*x*d*exp(1)-d^2)^2+(3*exp(2)-2*exp(1)^2)*1/2/(4*c^3*exp(2)*d^2-4*c^3*d^2*exp(1)^2)/d/
sqrt(-exp(1)^2+exp(2))*atan((d*exp(1)+x*exp(2))/d/sqrt(-exp(1)^2+exp(2)))

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maple [A]  time = 0.05, size = 16, normalized size = 0.94 \[ -\frac {1}{3 \left (e x +d \right )^{3} c^{3} e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^2/(c*e^2*x^2+2*c*d*e*x+c*d^2)^3,x)

[Out]

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

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maxima [B]  time = 1.38, size = 47, normalized size = 2.76 \[ -\frac {1}{3 \, {\left (c^{3} e^{4} x^{3} + 3 \, c^{3} d e^{3} x^{2} + 3 \, c^{3} d^{2} e^{2} x + c^{3} d^{3} e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2/(c*e^2*x^2+2*c*d*e*x+c*d^2)^3,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.41, size = 49, normalized size = 2.88 \[ -\frac {1}{3\,c^3\,d^3\,e+9\,c^3\,d^2\,e^2\,x+9\,c^3\,d\,e^3\,x^2+3\,c^3\,e^4\,x^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^2/(c*d^2 + c*e^2*x^2 + 2*c*d*e*x)^3,x)

[Out]

-1/(3*c^3*d^3*e + 3*c^3*e^4*x^3 + 9*c^3*d^2*e^2*x + 9*c^3*d*e^3*x^2)

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sympy [B]  time = 0.31, size = 51, normalized size = 3.00 \[ - \frac {1}{3 c^{3} d^{3} e + 9 c^{3} d^{2} e^{2} x + 9 c^{3} d e^{3} x^{2} + 3 c^{3} e^{4} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**2/(c*e**2*x**2+2*c*d*e*x+c*d**2)**3,x)

[Out]

-1/(3*c**3*d**3*e + 9*c**3*d**2*e**2*x + 9*c**3*d*e**3*x**2 + 3*c**3*e**4*x**3)

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