3.1020 \(\int \frac {(d+e x)^6}{(c d^2+2 c d e x+c e^2 x^2)^3} \, dx\)

Optimal. Leaf size=5 \[ \frac {x}{c^3} \]

[Out]

x/c^3

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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} \[ \frac {x}{c^3} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^6/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^3,x]

[Out]

x/c^3

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)^6}{\left (c d^2+2 c d e x+c e^2 x^2\right )^3} \, dx &=\int \frac {1}{c^3} \, dx\\ &=\frac {x}{c^3}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 5, normalized size = 1.00 \[ \frac {x}{c^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^6/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^3,x]

[Out]

x/c^3

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fricas [A]  time = 0.99, size = 5, normalized size = 1.00 \[ \frac {x}{c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^6/(c*e^2*x^2+2*c*d*e*x+c*d^2)^3,x, algorithm="fricas")

[Out]

x/c^3

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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)^6/(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: x*exp(1)^6/c^3/exp(2)^3+((3*exp(2)^5*d^2
+3*exp(2)^4*d^2*exp(1)^2-102*exp(2)^3*d^2*exp(1)^4+192*exp(2)^2*d^2*exp(1)^6-96*exp(2)*d^2*exp(1)^8)*x^3+(9*ex
p(2)^4*d^3*exp(1)-71*exp(2)^3*d^3*exp(1)^3-18*exp(2)^2*d^3*exp(1)^5+240*exp(2)*d^3*exp(1)^7-160*d^3*exp(1)^9)*
x^2+(5*exp(2)^4*d^4-31*exp(2)^3*d^4*exp(1)^2-86*exp(2)^2*d^4*exp(1)^4+272*exp(2)*d^4*exp(1)^6-160*d^4*exp(1)^8
)*x-7*exp(2)^3*d^5*exp(1)-17*exp(2)^2*d^5*exp(1)^3+64*exp(2)*d^5*exp(1)^5-40*d^5*exp(1)^7)/8/exp(2)^4/c^3/(2*e
xp(1)*d*x+exp(2)*x^2+d^2)^2-(-3*exp(2)*d*exp(1)^5+3*d*exp(1)^7)/c^3/exp(2)^4*ln(x^2*exp(2)+2*x*d*exp(1)+d^2)-(
-3*exp(2)^4*d^2-3*exp(2)^3*d^2*exp(1)^2-18*exp(2)^2*d^2*exp(1)^4+72*exp(2)*d^2*exp(1)^6-48*d^2*exp(1)^8)*1/4/c
^3/exp(2)^4*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 = 0.04, size = 6, normalized size = 1.20 \[ \frac {x}{c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^6/(c*e^2*x^2+2*c*d*e*x+c*d^2)^3,x)

[Out]

x/c^3

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maxima [A]  time = 1.31, size = 5, normalized size = 1.00 \[ \frac {x}{c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^6/(c*e^2*x^2+2*c*d*e*x+c*d^2)^3,x, algorithm="maxima")

[Out]

x/c^3

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mupad [B]  time = 0.01, size = 5, normalized size = 1.00 \[ \frac {x}{c^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^6/(c*d^2 + c*e^2*x^2 + 2*c*d*e*x)^3,x)

[Out]

x/c^3

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sympy [A]  time = 0.15, size = 3, normalized size = 0.60 \[ \frac {x}{c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**6/(c*e**2*x**2+2*c*d*e*x+c*d**2)**3,x)

[Out]

x/c**3

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