3.1008 \(\int \frac {(d+e x)^6}{(c d^2+2 c d e x+c e^2 x^2)^2} \, dx\)

Optimal. Leaf size=17 \[ \frac {(d+e x)^3}{3 c^2 e} \]

[Out]

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

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Rubi [A]  time = 0.00, 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} \[ \frac {(d+e x)^3}{3 c^2 e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d + e*x)^3/(3*c^2*e)

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

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Mathematica [A]  time = 0.00, size = 17, normalized size = 1.00 \[ \frac {(d+e x)^3}{3 c^2 e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d + e*x)^3/(3*c^2*e)

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fricas [A]  time = 0.99, size = 26, normalized size = 1.53 \[ \frac {e^{2} x^{3} + 3 \, d e x^{2} + 3 \, d^{2} x}{3 \, c^{2}} \]

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)^2,x, algorithm="fricas")

[Out]

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

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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)^2,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: (1/3*x^3*c^4*exp(2)^4*exp(1)^6+3*x^2*c^4
*exp(2)^4*d*exp(1)^5-2*x^2*c^4*exp(2)^3*d*exp(1)^7+15*x*c^4*exp(2)^4*d^2*exp(1)^4-26*x*c^4*exp(2)^3*d^2*exp(1)
^6+12*x*c^4*exp(2)^2*d^2*exp(1)^8)/c^6/exp(2)^6+(-5*exp(2)^4*d^5*exp(1)+30*exp(2)^3*d^5*exp(1)^3-61*exp(2)^2*d
^5*exp(1)^5+52*exp(2)*d^5*exp(1)^7-16*d^5*exp(1)^9+(exp(2)^5*d^4-20*exp(2)^4*d^4*exp(1)^2+85*exp(2)^3*d^4*exp(
1)^4-146*exp(2)^2*d^4*exp(1)^6+112*exp(2)*d^4*exp(1)^8-32*d^4*exp(1)^10)*x)/2/exp(2)^5/c^2/(2*exp(1)*d*x+exp(2
)*x^2+d^2)+(10*exp(2)^3*d^3*exp(1)^3-36*exp(2)^2*d^3*exp(1)^5+42*exp(2)*d^3*exp(1)^7-16*d^3*exp(1)^9)/c^2/exp(
2)^5*ln(x^2*exp(2)+2*x*d*exp(1)+d^2)+(exp(2)^5*d^4+10*exp(2)^4*d^4*exp(1)^2-95*exp(2)^3*d^4*exp(1)^4+220*exp(2
)^2*d^4*exp(1)^6-200*exp(2)*d^4*exp(1)^8+64*d^4*exp(1)^10)/c^2/exp(2)^5*1/2/d/sqrt(-exp(1)^2+exp(2))*atan((d*e
xp(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 \[ \frac {\left (e x +d \right )^{3}}{3 c^{2} e} \]

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)^2,x)

[Out]

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

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maxima [A]  time = 1.32, size = 26, normalized size = 1.53 \[ \frac {e^{2} x^{3} + 3 \, d e x^{2} + 3 \, d^{2} x}{3 \, c^{2}} \]

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)^2,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.03, size = 24, normalized size = 1.41 \[ \frac {x\,\left (3\,d^2+3\,d\,e\,x+e^2\,x^2\right )}{3\,c^2} \]

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)^2,x)

[Out]

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

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sympy [B]  time = 0.12, size = 29, normalized size = 1.71 \[ \frac {d^{2} x}{c^{2}} + \frac {d e x^{2}}{c^{2}} + \frac {e^{2} x^{3}}{3 c^{2}} \]

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)**2,x)

[Out]

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

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