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

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

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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} \begin {gather*} \frac {(d+e x)^3}{3 c e} \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),x]

[Out]

(d + e*x)^3/(3*c*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)^4}{c d^2+2 c d e x+c e^2 x^2} \, dx &=\int \frac {(d+e x)^2}{c} \, dx\\ &=\frac {\int (d+e x)^2 \, dx}{c}\\ &=\frac {(d+e x)^3}{3 c e}\\ \end {align*}

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

[Out]

(d + e*x)^3/(3*c*e)

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

[Out]

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

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fricas [A]  time = 0.37, size = 26, normalized size = 1.53 \begin {gather*} \frac {e^{2} x^{3} + 3 \, d e x^{2} + 3 \, d^{2} x}{3 \, c} \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),x, algorithm="fricas")

[Out]

1/3*(e^2*x^3 + 3*d*e*x^2 + 3*d^2*x)/c

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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),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: (2*exp(2)^3*d^3*exp(1)-8*exp(2)^2*d^3*ex
p(1)^3+10*exp(2)*d^3*exp(1)^5-4*d^3*exp(1)^7)/c/exp(2)^4*ln(x^2*exp(2)+2*x*d*exp(1)+d^2)+(2*exp(2)^4*d^4-20*ex
p(2)^3*d^4*exp(1)^2+50*exp(2)^2*d^4*exp(1)^4-48*exp(2)*d^4*exp(1)^6+16*d^4*exp(1)^8)/c/exp(2)^4*1/2/d/sqrt(-ex
p(1)^2+exp(2))*atan((d*exp(1)+x*exp(2))/d/sqrt(-exp(1)^2+exp(2)))+(1/3*x^3*c^2*exp(2)^2*exp(1)^4+2*x^2*c^2*exp
(2)^2*d*exp(1)^3-x^2*c^2*exp(2)*d*exp(1)^5+6*x*c^2*exp(2)^2*d^2*exp(1)^2-9*x*c^2*exp(2)*d^2*exp(1)^4+4*x*c^2*d
^2*exp(1)^6)/c^3/exp(2)^3

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

[Out]

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

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maxima [A]  time = 1.26, size = 26, normalized size = 1.53 \begin {gather*} \frac {e^{2} x^{3} + 3 \, d e x^{2} + 3 \, d^{2} x}{3 \, c} \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),x, algorithm="maxima")

[Out]

1/3*(e^2*x^3 + 3*d*e*x^2 + 3*d^2*x)/c

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

[Out]

(x*(3*d^2 + e^2*x^2 + 3*d*e*x))/(3*c)

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sympy [B]  time = 0.10, size = 24, normalized size = 1.41 \begin {gather*} \frac {d^{2} x}{c} + \frac {d e x^{2}}{c} + \frac {e^{2} x^{3}}{3 c} \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),x)

[Out]

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

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