3.9.10 \(\int \frac {(d+e x)^5}{c d^2+2 c d e x+c e^2 x^2} \, dx\)

Optimal. Leaf size=17 \[ \frac {(d+e x)^4}{4 c e} \]

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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)^4}{4 c e} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^5/(c*d^2 + 2*c*d*e*x + c*e^2*x^2),x]

[Out]

(d + e*x)^4/(4*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)^5}{c d^2+2 c d e x+c e^2 x^2} \, dx &=\int \frac {(d+e x)^3}{c} \, dx\\ &=\frac {\int (d+e x)^3 \, dx}{c}\\ &=\frac {(d+e x)^4}{4 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)^4}{4 c e} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^5/(c*d^2 + 2*c*d*e*x + c*e^2*x^2),x]

[Out]

(d + e*x)^4/(4*c*e)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(d+e x)^5}{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)^5/(c*d^2 + 2*c*d*e*x + c*e^2*x^2),x]

[Out]

IntegrateAlgebraic[(d + e*x)^5/(c*d^2 + 2*c*d*e*x + c*e^2*x^2), x]

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fricas [B]  time = 0.39, size = 37, normalized size = 2.18 \begin {gather*} \frac {e^{3} x^{4} + 4 \, d e^{2} x^{3} + 6 \, d^{2} e x^{2} + 4 \, d^{3} x}{4 \, c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^5/(c*e^2*x^2+2*c*d*e*x+c*d^2),x, algorithm="fricas")

[Out]

1/4*(e^3*x^4 + 4*d*e^2*x^3 + 6*d^2*e*x^2 + 4*d^3*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)^5/(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: (5*exp(2)^4*d^4*exp(1)-30*exp(2)^3*d^4*e
xp(1)^3+61*exp(2)^2*d^4*exp(1)^5-52*exp(2)*d^4*exp(1)^7+16*d^4*exp(1)^9)*1/2/c/exp(2)^5*ln(x^2*exp(2)+2*x*d*ex
p(1)+d^2)+(2*exp(2)^5*d^5-30*exp(2)^4*d^5*exp(1)^2+110*exp(2)^3*d^5*exp(1)^4-170*exp(2)^2*d^5*exp(1)^6+120*exp
(2)*d^5*exp(1)^8-32*d^5*exp(1)^10)/c/exp(2)^5*1/2/d/sqrt(-exp(1)^2+exp(2))*atan((d*exp(1)+x*exp(2))/d/sqrt(-ex
p(1)^2+exp(2)))+(1/4*x^4*c^3*exp(2)^3*exp(1)^5+5/3*x^3*c^3*exp(2)^3*d*exp(1)^4-2/3*x^3*c^3*exp(2)^2*d*exp(1)^6
+5*x^2*c^3*exp(2)^3*d^2*exp(1)^3-11/2*x^2*c^3*exp(2)^2*d^2*exp(1)^5+2*x^2*c^3*exp(2)*d^2*exp(1)^7+10*x*c^3*exp
(2)^3*d^3*exp(1)^2-25*x*c^3*exp(2)^2*d^3*exp(1)^4+24*x*c^3*exp(2)*d^3*exp(1)^6-8*x*c^3*d^3*exp(1)^8)/c^4/exp(2
)^4

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maple [A]  time = 0.06, size = 16, normalized size = 0.94 \begin {gather*} \frac {\left (e x +d \right )^{4}}{4 c e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^5/(c*e^2*x^2+2*c*d*e*x+c*d^2),x)

[Out]

1/4*(e*x+d)^4/c/e

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maxima [B]  time = 1.31, size = 37, normalized size = 2.18 \begin {gather*} \frac {e^{3} x^{4} + 4 \, d e^{2} x^{3} + 6 \, d^{2} e x^{2} + 4 \, d^{3} x}{4 \, c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^5/(c*e^2*x^2+2*c*d*e*x+c*d^2),x, algorithm="maxima")

[Out]

1/4*(e^3*x^4 + 4*d*e^2*x^3 + 6*d^2*e*x^2 + 4*d^3*x)/c

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mupad [B]  time = 0.05, size = 43, normalized size = 2.53 \begin {gather*} \frac {d^3\,x}{c}+\frac {e^3\,x^4}{4\,c}+\frac {3\,d^2\,e\,x^2}{2\,c}+\frac {d\,e^2\,x^3}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^5/(c*d^2 + c*e^2*x^2 + 2*c*d*e*x),x)

[Out]

(d^3*x)/c + (e^3*x^4)/(4*c) + (3*d^2*e*x^2)/(2*c) + (d*e^2*x^3)/c

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sympy [B]  time = 0.10, size = 39, normalized size = 2.29 \begin {gather*} \frac {d^{3} x}{c} + \frac {3 d^{2} e x^{2}}{2 c} + \frac {d e^{2} x^{3}}{c} + \frac {e^{3} x^{4}}{4 c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**5/(c*e**2*x**2+2*c*d*e*x+c*d**2),x)

[Out]

d**3*x/c + 3*d**2*e*x**2/(2*c) + d*e**2*x**3/c + e**3*x**4/(4*c)

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