3.1021 \(\int \frac {(d+e x)^5}{(c d^2+2 c d e x+c e^2 x^2)^3} \, dx\)

Optimal. Leaf size=13 \[ \frac {\log (d+e x)}{c^3 e} \]

[Out]

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

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Rubi [A]  time = 0.00, antiderivative size = 13, 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, 31} \[ \frac {\log (d+e x)}{c^3 e} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^5/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^3,x]

[Out]

Log[d + e*x]/(c^3*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 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin {align*} \int \frac {(d+e x)^5}{\left (c d^2+2 c d e x+c e^2 x^2\right )^3} \, dx &=\int \frac {1}{c^3 (d+e x)} \, dx\\ &=\frac {\int \frac {1}{d+e x} \, dx}{c^3}\\ &=\frac {\log (d+e x)}{c^3 e}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 13, normalized size = 1.00 \[ \frac {\log (d+e x)}{c^3 e} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^5/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^3,x]

[Out]

Log[d + e*x]/(c^3*e)

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fricas [A]  time = 1.34, size = 13, normalized size = 1.00 \[ \frac {\log \left (e x + d\right )}{c^{3} e} \]

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)^3,x, algorithm="fricas")

[Out]

log(e*x + d)/(c^3*e)

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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)^5/(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: exp(1)^5*1/2/c^3/exp(2)^3*ln(x^2*exp(2)+
2*x*d*exp(1)+d^2)-(-3*exp(2)^3*d-exp(2)^2*d*exp(1)^2-4*exp(2)*d*exp(1)^4+8*d*exp(1)^6)*1/4/c^3/exp(2)^3*1/2/d/
sqrt(-exp(1)^2+exp(2))*atan((d*exp(1)+x*exp(2))/d/sqrt(-exp(1)^2+exp(2)))+((3*exp(2)^4*d+exp(2)^3*exp(1)^2*d-3
6*exp(2)^2*exp(1)^4*d+32*exp(2)*exp(1)^6*d)*x^3+(9*exp(2)^3*exp(1)*d^2-37*exp(2)^2*exp(1)^3*d^2-20*exp(2)*exp(
1)^5*d^2+48*exp(1)^7*d^2)*x^2+(5*exp(2)^3*d^3-21*exp(2)^2*exp(1)^2*d^3-32*exp(2)*exp(1)^4*d^3+48*exp(1)^6*d^3)
*x-5*exp(2)^2*exp(1)*d^4-7*exp(2)*exp(1)^3*d^4+12*exp(1)^5*d^4)/8/c^3/exp(2)^3/(2*exp(1)*x*d+exp(2)*x^2+d^2)^2

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maple [A]  time = 0.05, size = 14, normalized size = 1.08 \[ \frac {\ln \left (e x +d \right )}{c^{3} e} \]

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)^3,x)

[Out]

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

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maxima [A]  time = 1.36, size = 13, normalized size = 1.00 \[ \frac {\log \left (e x + d\right )}{c^{3} e} \]

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)^3,x, algorithm="maxima")

[Out]

log(e*x + d)/(c^3*e)

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mupad [B]  time = 0.41, size = 13, normalized size = 1.00 \[ \frac {\ln \left (d+e\,x\right )}{c^3\,e} \]

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)^3,x)

[Out]

log(d + e*x)/(c^3*e)

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sympy [A]  time = 0.12, size = 17, normalized size = 1.31 \[ \frac {\log {\left (c^{3} d + c^{3} e x \right )}}{c^{3} e} \]

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)**3,x)

[Out]

log(c**3*d + c**3*e*x)/(c**3*e)

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