3.9.22 \(\int \frac {(d+e x)^4}{(c d^2+2 c d e x+c e^2 x^2)^2} \, dx\)

Optimal. Leaf size=5 \[ \frac {x}{c^2} \]

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Rubi [A]  time = 0.00, antiderivative size = 5, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {27, 8} \begin {gather*} \frac {x}{c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x/c^2

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, 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]

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 5, normalized size = 1.00 \begin {gather*} \frac {x}{c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x/c^2

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(d+e x)^4}{\left (c d^2+2 c d e x+c e^2 x^2\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.38, size = 5, normalized size = 1.00 \begin {gather*} \frac {x}{c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/c^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: x*exp(1)^4/c^2/exp(2)^2+((exp(2)^3*d^2-9
*exp(2)^2*d^2*exp(1)^2+16*exp(2)*d^2*exp(1)^4-8*d^2*exp(1)^6)/exp(2)*x+(-3*exp(2)^2*d^3*exp(1)+7*exp(2)*d^3*ex
p(1)^3-4*d^3*exp(1)^5)/exp(2))/2/c^2/exp(2)^2/(2*exp(1)*d*x+exp(2)*x^2+d^2)+(2*exp(2)*d*exp(1)^3-2*d*exp(1)^5)
/c^2/exp(2)^3*ln(x^2*exp(2)+2*x*d*exp(1)+d^2)+(exp(2)^3*d^2+3*exp(2)^2*d^2*exp(1)^2-12*exp(2)*d^2*exp(1)^4+8*d
^2*exp(1)^6)/c^2/exp(2)^3*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 = 6, normalized size = 1.20 \begin {gather*} \frac {x}{c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^4/(c*e^2*x^2+2*c*d*e*x+c*d^2)^2,x)

[Out]

x/c^2

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maxima [A]  time = 1.36, size = 5, normalized size = 1.00 \begin {gather*} \frac {x}{c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/c^2

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mupad [B]  time = 0.01, size = 5, normalized size = 1.00 \begin {gather*} \frac {x}{c^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^4/(c*d^2 + c*e^2*x^2 + 2*c*d*e*x)^2,x)

[Out]

x/c^2

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sympy [A]  time = 0.10, size = 3, normalized size = 0.60 \begin {gather*} \frac {x}{c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/c**2

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