3.11.1 \(\int \frac {(d+e x)^2}{c d^2+2 c d e x+c e^2 x^2} \, dx\) [1001]

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

[Out]

x/c

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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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x/c

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

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

Antiderivative was successfully verified.

[In]

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

[Out]

x/c

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Maple [A]
time = 0.57, size = 6, normalized size = 1.20

method	result	size
default	\(\frac {x}{c}\)	\(6\)
risch	\(\frac {x}{c}\)	\(6\)
norman	\(\frac {\frac {e \,x^{2}}{c}+\frac {x d}{c}}{e x +d}\)	\(24\)

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),x,method=_RETURNVERBOSE)

[Out]

x/c

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Maxima [A]
time = 0.29, size = 5, normalized size = 1.00 \begin {gather*} \frac {x}{c} \end {gather*}

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),x, algorithm="maxima")

[Out]

x/c

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Fricas [A]
time = 3.41, size = 5, normalized size = 1.00 \begin {gather*} \frac {x}{c} \end {gather*}

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),x, algorithm="fricas")

[Out]

x/c

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Sympy [A]
time = 0.01, size = 2, normalized size = 0.40 \begin {gather*} \frac {x}{c} \end {gather*}

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),x)

[Out]

x/c

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Giac [A]
time = 3.72, size = 5, normalized size = 1.00 \begin {gather*} \frac {x}{c} \end {gather*}

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),x, algorithm="giac")

[Out]

x/c

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

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),x)

[Out]

x/c

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